[    {        "title": "1207.1525v1.Double_Exchange_Ferromagnetism_in_the_Peierls_Insulator_State.pdf",        "content": "arXiv:1207.1525v1  [cond-mat.str-el]  6 Jul 2012Double Exchange Ferromagnetism in the Peierls Insulator St ate\nS. Nishimoto1and Y. Ohta2\n1Institut f¨ ur Theoretische Festk¨ orperphysik, IFW Dresde n, 01171 Dresden, Germany and\n2Department of Physics, Chiba University, Chiba 263-8522, J apan\n(Dated: November 8, 2018)\nWe studythe effects of opening of theband gap on the double exc hange ferromagnetism. Applying\nthe density-matrix renormalization group method and an ana lytical expansion from the dimer limit\nto the one-dimensional double exchange model, we demonstra te for a relevant region of the exchange\ncoupling that, in the weak dimerization regime, the Peierls gap opens in the fully spin-polarized con-\nduction band without affecting its ferromagnetism, whereas in the strong dimerization regime, the\nferromagnetism is destroyed and the Mott gap opens instead, leading the system to the antiferro-\nmagnetic quasi-long-range order. An insulator version of t he double exchange ferromagnetism is\nthus established.\nPACS numbers: 71.10.Fd, 75.10.-b, 71.30.+h\nMagnetism and electronic transport properties of ma-\nterials are closely related to each other, e.g., insulat-\ning transition-metal oxides are typically antiferromag-\nnetic and ferromagnetismusually goeshand in hand with\nmetallicity [1], and elucidation of the mechanisms of this\nrelationship is one of the major issues in the field of\nphysics of strong electron correlations. A well-known ex-\nampleis the doubleexchangeferromagnetismthat occurs\nin mixed systems of localized spins and itinerant elec-\ntrons interacting via the Hund’s rule coupling, where the\ncoherent motion of the itinerant electrons aligns the lo-\ncalized spins ferromagnetically to gain in kinetic energy\nof the itinerant electrons [2–4].\nA question then arises as to what happens in this fer-\nromagnetism if the coherent motion of electrons ceases\ndue, e.g., to the opening of the band gap. This issue\ncomes into a real question in ferromagnetic hollandite\nK2Cr8O16[5], where the double exchange mechanism in-\nduces the three-dimensional (3D) full spin polarization\nin the system [6], and then the metal-insulator transition\nfollows in its fully spin-polarized quasi-one-dimensional\n(1D) conduction band by the Peierls mechanism [7, 8],\nwithout affecting its 3D ferromagnetism. Thus, the un-\ncommon ferromagnetic insulating (FI) state is realized in\nthis material [7].\nA naive answer to the above frequently asked question\nmay then be that the ferromagnetism can survive if the\nbandgapissmallenoughincomparisonwiththewidthof\nthe conduction band and therefore the motion of conduc-\ntion electrons, though not coherent, is not significantly\nsuppressed. However, to the best of our knowledge, no\nquantitative theoretical studies have been made on this\nissue whether this is actually the case.\nIn this paper, we address this issue from the theoret-\nical standpoint. We apply the numerical density-matrix\nrenormalization group (DMRG) technique [9] and ana-\nlytical expansion from the dimer limit to the 1D dou-\nble exchange model that simulates the quasi-1D chain of\nK2Cr8O16and study the effects of opening of the bandgap on the double exchange ferromagnetism. We calcu-\nlate the total-spin quantum number and charge gap of\nthe system and extract the ground-state phase diagram\nof the model.\nWe will thereby demonstrate for a relevant region of\nthe exchange coupling that, in the weak dimerization\nregime, the Peierls gap opens in the fully spin-polarized\nconduction band without affecting its ferromagnetism,\nwhereas in the strong dimerization regime, the ferromag-\nnetism is destroyed and the Mott gap opens instead, due\nto the effective “on-dimer” Coulomb interaction, which\nleads the system to the antiferromagnetic quasi-long-\nrange order. The metallicity itself is not therefore a\nnecessary condition for the realization of the double ex-\nchangeferromagnetismandthusan“insulatorversion”of\nthe double exchange ferromagnetism is established. This\nis a route to realization of insulating ferromagnets, dif-\nferent from those discussed in doped LaMnO 3where the\norbital ordering plays an essential role [10–12].\nOur model (see Fig. 1) contains the terms of Peierls\ndimerization and Hund’s rule coupling and is defined by\nthe Hamiltonian\nH=−L/summationdisplay\ni=1,σti,i+1(c†\ni,σci+1,σ+H.c.)−JHL/summationdisplay\ni=1si·Si\nsi=1\n2/summationdisplay\nσ,σ′c†\ni,στσ,σ′ci,σ′, (1)\nwherec†\ni,σis the creation operator of an electron with\nspinσ(=↑,↓) at site i,siis the spin operator of a\nconduction electron at site i,τis the vector of Pauli\nmatrices, and Siis the quantum spin operator (of spin\n1/2) of a localized electron at site i. The hopping pa-\nrameter between the nearest-neighbor sites is defined as\nti,i+1=/bracketleftbig\n1−(−1)iδ/2/bracketrightbig\ntwith the dimerization strength\nδof 0≤δ <2; we in particular define t±= (1±δ/2)t.\nJHis the strength of the Hund’s rule coupling. Lis the\nnumber of sites in the system, where the site contains a\nconduction orbital and a localized spin. We confine our-\nselvestothecaseatquarterfillingofconductionelectrons2\n-1 0 1-202(a)\n(b)\n(c)\n/c50/c100strong-dimerization limit\nweak-dimerization limit\nk//c112/c101( )/k tt-t+\nJH\nFIG. 1: (Color online) Schematic representations of the dou -\nble exchange model in (a) the weak dimerization regime and\n(b) strong dimerization regime. In (c), we illustrate the no n-\ninteracting band structure ε(k) of our model with the lattice\ndimerization δin the unfolded Brillouin zone.\nn=N/L= 1/2, where Nis the number of conduction\nelectrons in the system. We introduce δas a controllable\ninput parameter rather than the order parameter of the\nspontaneous lattice dimerization since our purpose is to\nstudytheeffectsof δontheelectronicstatesofthemodel.\nWe use the DMRG method to investigate the ground-\nstate properties of the system Eq. (1), where the open-\nend boundary conditions are applied. We study the\nmodel with several lengths of L= 8 to 20 with keeping\nm= 1200to3200density-matrixeigenstatesintherenor-\nmalization procedure; in this way, the largest truncation\nerror, or the discarded weight, is of the order of 10−11.\nNote that we have to keep relatively-numerous density-\nmatrix eigenstates to extract the true ground state from\na number of nearly degenerate magnetic states, in par-\nticular, in the vicinity of the phase boundaries. The ex-\ntrapolation to the thermodynamic limit L→ ∞is made\nin the results presented unless otherwise indicated.\nFirst, let us present the total-spin quantum number\nSof the system, which is calculated directly from the\nground-state expectation value of S2defined as ∝angbracketleftS2∝angbracketright=\nS(S+1) =/summationtext\nij/parenleftbig\n∝angbracketleftsi·sj∝angbracketright+2∝angbracketleftsi·Sj∝angbracketright+∝angbracketleftSi·Sj∝angbracketright/parenrightbig\n. Theresults\nare given in Fig. 2 as a ground-state phase diagram. We\nfind that there are three phases, S= 0, 0< S < S max,\nandS=Smax, depending on the values of JHandδ,\nwhereSmaxdenotes the full spin polarization. As ex-\npected, and in agreement with the previous calculations\natδ= 0 [13–16], the S=Smaxphase appears when\nJHis large and S= 0 phase appears when JHis small.\nAnd in between, the phase with the intermediate spinFI ( = )S Smax\nPI ( =0)SFM\nPMFI (0< < ) S Smax\n/c100JH/t\n0 1 20102030\nFIG. 2: (Color online) Calculated ground-state phase dia-\ngram of the 1D double exchange model at quarter filling with\nthe lattice dimerization δ. We find the FI (ferromagnetic in-\nsulating), PI (paramagnetic insulating), FM (ferromagnet ic\nmetallic), and PM (paramagnetic metallic) phases. The FI-\nPI phase boundary at δ→2 is determined by the analytical\nexpansion from the dimer limit and is shown by the open di-\namond and dashed line. The FM and PM phases at δ= 0 or\nJH= 0 are also indicated by the thick lines.\npolarization also appears as in Ref. 15 at δ= 0 (see also\nFig. 3(a) below). These phases are retained even when\nthe dimerization δis introduced. We note that the two\ncritical values of JHthat separate between the S=Smax\nandS= 0 phases increase with increasing δand that the\nregion with the intermediate spin polarization becomes\nnarrower and vanishes at δ→2 (see below).\nNext, let us calculate the charge gap ∆ defined as ∆ =\nlimL→∞∆(L) with ∆( L) =1\n2/bracketleftbig\nEN+2\n0(L) +EN−2\n0(L)−\n2EN\n0(L)/bracketrightbig\n, whereEN\n0(L) is the ground-state energy of the\nsystem of size LwithNelectrons. The gap ∆ is defined\nwith the prefactor 1 /2, so that the single-particle band\ngap is identical to ∆ in the present case where pairing\ninteractions are absent. The results are shown in Figs. 3\n(b)-(d). We find that the charge gap opens in the entire\nparameter space except at the lines δ= 0 and JH= 0.\nThe model at JH= 0 is trivial, where the conduction\nelectrons, decoupled completely from the localized spins,\nbehave just as the noninteracting electrons, resulting in\nthe PM phase.\nThe model Eq. (1) at δ= 0 on the other hand is highly\nnontrivial and much work has been done in recent years\n[13–16]: basically,thereappearstheferromagneticmetal-\nlic(FM) phaseforthelarge JHregion,whichchangesinto\nthe paramagnetic metallic (PM) phase when JHbecomes\nsmall. In addition, it has been claimed [15, 17, 18] that\nthere appears the region of phase separation in particu-3\nlar near half filling and the “spiral” phase with a long-\nwavelengthantiferromagneticcorrelationsat the FM-PM\nphase boundary. Although we have not detected any in-\ndications of the phase separation at least at quarter fill-\ning in our accurate DMRG calculations with very large\nmvalues, our results obtained are consistent with the\nresults of the previous work [13–16]: we find that either\nthe FM phase (when JHis large) or PM phase (when JH\nis small) is realized, and in between there is the partially\nspin-polarized metallic phase (see Figs. 2 and 3(a)) that\nmay correspond to the spiral phase predicted in Ref. 15.\n0 0.5 1012\n0 1 200.10.20.30.40.5\n0 1 200.20.40.6\nL=16\nL=12\nL=80 1 200.20.40.610 15 2000.20.40.60.81\nL=16\nL=12\nL=8\n0 1 200.20.40.6\nL=16\nL=12\nL=800.05 0.100.5\n/c100=0.1/c100=1.900.05 0.100.511.5\n/c100=0.1\n/c100/c100\nJH/t/c68/t\n/c68/t/c68/t\nS S/max\nS q( )S q( )(a)\n(b)\n(c)\n(d)\n(e) (f)JH=20/t\nJH=1 /t\n/c100/c61/c49/c46/c53JH/t/c100/c61/c48/c46/c49\nJ tH/ =2, /c100/c61/c48 J tH/ =2, /c100/c61/c48/c46/c53\nq//c112 q//c112/c68 /c61/c50/c100/t\ndimer limit1/L1/L\nFIG. 3: (Color online) (a) Calculated normalized total-spi n\nquantum number S/Smaxas a function of JH/t. Also shown\nare the calculated results for the charge gap ∆: (b) δdepen-\ndenceintheFIphase, (c) δdependenceinthePIphase(where\nthe dotted line indicates the result of the strong-dimeriza tion\nexpansion), and (d) JH/tdependence in the PI phase. Inset\nof (b) and (c) shows examples of the finite-size scaling of the\ngap. In (e) and (f), the calculated spin structure factors S(q)\nin the PI phase without and with the lattice dimerization,\nrespectively, are shown.\nLet us then introduce the lattice dimerization δ >0.\nThe results are the following: In the S=Smaxregion,\nwe find that the charge gap of ∆ /t= 2δopens as shownin Fig. 3(b). This can simply be understood because in\nthis region we have the noninteracting band of spinless\nfermions at half filling, and therefore the lattice dimer-\nizationδopens the band gap of the size ∆ /t= 2δ. The\nFI phase is thus realized. Since the FM phase due to\nthe double exchange mechanism is continuous to this FI\nphase, we may naturally refer to it as the insulator ver-\nsion of the double exchange ferromagnetism.\nIn theS= 0 region, we find that the gap actually\nopens as ∆ ∝δin the weak dimerization limit as shown\nin Fig. 3(c). The size of the gap increases as the value\nofJHincreases as shown in Fig. 3(d). This phase with\n∆>0 may then be denoted as the paramagnetic insu-\nlating (PI) phase. The Fourier transform of the spin-\nspin correlation function for the conduction electrons\nS(q) =1\nL/summationtext\ni,jeiq(Rj−Ri)∝angbracketleftsi·sj∝angbracketright(as well as that for the\nlocalizedelectrons, seeFig.2ofRef. 15) maycharacterize\nthis phase. The calculated results are shown in Figs. 3\n(e) and (f), wherewefind that the antiferromagneticspin\ncorrelation of the wavevector of q=π/2 is enhanced and\nthat the lattice dimerization inducing the localization of\nconduction electrons further enhances this correlation.\nThis results may therefore be interpreted as an enhance-\nment by the lattice dimerization of the “island” state\npredicted in Ref. 15, where the high-spin ( S= 3/2) clus-\nters, formed by a conduction electron coupled ferromag-\nnetically with the two neighboring localized spins, are\narrangedantiferromagneticallywith the quasi-long-range\norder [15, 19]. In higher spatial dimensions, this phase\nmay well fall into the true long-range antiferromagnetic\norder, resultingin the antiferromagneticinsulating(AFI)\nphase. Thissituationresemblesthatofthe“dimer-Mott”\nphase[20, 21] in the dimerized Hubbard model atquarter\nfilling although in the latter the spins are of S= 1/2.\nIn the 0< S < S maxregion, the charge gap also opens\natδ >0, where its size increases rapidly with increasing\nJH/t. We thus have the FI phase here as well, which may\nbe the insulating spiral phase continuous to the metallic\none predicted in Ref. 15.\nNow, let us discuss the strong dimerization limit,\nwhere we start with the highly correlated clusters Cl\n(l= 1,···,L/2) coupled weakly to each other through\nthe hopping parameter t−(see Fig. 1(b)). Each of the\nclusters consists of the two conduction orbitals and two\nlocalized spins. In the ground state, the cluster contains\noneconduction electron(andtwolocalizedspins) and the\ninternal three spins are fully polarized. The lowest en-\nergy of the single cluster is e(1) =−JH/4−t+, where\nthe conduction electron is in the bonding state of the\ntwo conduction orbitals. The eigenstates of the cluster\nwith the 0 and 2 conduction electrons are also derived\nexactly and the lowest energies are found to be e(0) = 0\nande(2) =−/radicalBig\nJ2\nH+16t2\n+/slashbig\n2, respectively. In the strong-\ndimerization limit of the PI phase, we can therefore map\nour system Eq. (1) onto an effective single-band Hub-4\nbard model defined in terms of the bonding orbital on\neach dimer. The Hamiltonian may be written as\nHeff=teffL/2/summationdisplay\ni=1,σ(b†\ni,σbi+1,σ+H.c.)+UeffL/2/summationdisplay\ni=1nb\ni,↑nb\ni,↓,(2)\nwith the creation operator of an electron on the bond-\ning orbital b†\ni= (c†\n2i−1,σ+c†\n2i,σ)/√\n2 andnb\ni,σ=b†\ni,σbi,σ.\nWe obtain the effective hopping integral teff=t−/2\nand effective “on-dimer”” Coulomb interaction Ueff=\ne(2)+e(0)−2e(1). In this mapping, the localized spins\ncontribute only to Ueffand their degrees of freedom are\nnot explicitly involved in the operator b†\ni,σ. An analytical\nexpression for the charge gap may thus be derived as\n∆ =Ueff−2t−\n=−/radicalBig\nJ2\nH+4t2(2+δ)2/2+2tδ+JH/2,(3)\nwhichisvaliduptothefirstorderof2 −δ. Theresultthus\nobtained is shown in Fig. 3(c) as a dotted line, where we\nfind that the agreement with the DMRG result is very\ngood. In the FI phase, on the other hand, the charge\ngap is always ∆ /t= 2δin its entire region, independent\nofJH. The two gaps are therefore discontinuous at the\nphase boundary.\nNext, we derive the effective exchange interaction Jeff\nbetween conduction electrons on the neighboring clus-\nters. We first take a direct product of the isolated clus-\nters (/producttextL/2\nl=1Cl) to be the unperturbed ground state of the\nsystem. Then, taking into account all the processes up\nto the second order of the hopping t−, we obtain the\nexpression for Jeffas\nJeff=t2\n−/bracketleftBig(J2\nH+8t2\n+)\n(JH+4t+)(J2\nH+16t2\n+)+1\n8(JH+2t+)\n+(/radicalBig\nJ2\nH+16t2\n++4t+)2(JH+4t+)\n16JHt+(J2\nH+16t2\n+)−3\n16t+/bracketrightBig\n,(4)\nwhere we should note that, if only the lowest intermedi-\nate state is taken into account in the second-order pro-\ncess, we obtain Jeff= 4t2\n−/Ueff, which is always positive\n(or antiferromagnetic). We thus find that, depending on\nthe value of Jeff(either positive or negative), the ground\nstate realized is either the PI phase or the FI phase. The\ncritical value of JHatδ→2 is found to be 29 .004t,which\ndetermines the FI-PI phase boundary at δ→2. The re-\ngion of the intermediate spin state does not appear here.\nThe result for the phase boundary obtained from Eq. (4)\nis shown as a dashed line in Fig. 2, where we find again\nthat the agreement with our DMRG result is very good,\nreinforcing the validity of our phase diagram of Fig. 2.\nIn summary, we studied the effects of opening of the\nband gap on the double exchange ferromagnetism. We\napplied the DMRG technique and analytical expansion\nfrom the dimer limit to the 1D double exchange modelat quarter filling with lattice dimerization and obtained\nthe ground-state phase diagram. We found three phases:\nthe FI phase with the Peierls gap and full spin polariza-\ntion, the PI phase with the Mott gap and dominant an-\ntiferromagnetic spin correlations, and the FI phase with\nthe partial spin polarization. The results for JH/t/greaterorsimilar15\ndemonstrated that, in the weak dimerization regime, the\nPeierls gap opens in the fully spin-polarized conduction\nband without affecting its ferromagnetism. Therefore,\nthe metallicity itself is not a necessary condition for\nthe realization of the double exchange ferromagnetism.\nThe concept of the insulator version of the double ex-\nchange ferromagnetism was thus established. In the\nstrong dimerization regime, on the other hand, the ferro-\nmagnetism is destroyed at JH/t/lessorsimilar29 and the Mott gap\ndue to the effective on-dimer Coulomb interaction opens\nthere with the antiferromagnetic quasi-long-range order\nin the system.\nThe uncommon FI state realized in K 2Cr8O16[7] then\nmeans that this material is in the weak dimerization\nregime with the Peierls gap and full spin polarization.\nA recent experiment suggests [22] that, by applying high\npressures of /greaterorsimilar2 GPa, the FM phase is suppressed very\nrapidly, while the metal-insulator transition remains al-\nmost unchanged, leading to the transition from the PM\nphase to the PI or AFI phase by lowering temperature.\nIt may then be quite interesting to point out that, if the\napplied pressure decreases the value of JH/t, this might\ncorrespond to the phase change in the quasi-1D chains\nfrom the FI to PI phase as in our phase diagram given\nin Fig. 2, where the intermediate spin state, or the spiral\nstate of Ref. 15, may also be predicted to appear under\nhigh pressure. We hope that ourworkpresentedhere will\nstimulate further searches for new phenomena and ma-\nterials with intriguing magnetic and transport properties\nderived from the interplay between the double exchange\nand Peierls/Mott mechanisms.\nEnlightening discussions with R. Eder, D. I. Khomskii,\nT. Konishi, K. Nakano, T. Toriyama, Y. Ueda, and T.\nYamauchi are gratefully acknowledged. This work was\nsupported in part by a Kakenhi Grant No. 22540363 of\nJapan. A part of computations was carried out at the\nResearchCenter for Computational Science, OkazakiRe-\nsearch Facilities, Japan.\n[1] D.I.KhomskiiandG.A.Sawatzky, SolidStateCommun.\n102, 87 (1997).\n[2] C. Zener, Phys. Rev. 81, 440 (1951); 82, 403 (1951).\n[3] P. W. Anderson and H. Hasegawa, Phys. Rev. 100, 675\n(1955).\n[4] P.-G. de Gennes, Phys. Rev. 118, 141 (1960).\n[5] K. Hasegawa, M. Isobe, T. Yamauchi, H. Ueda, J. Ya-\nmaura, H. Gotou, T. Yagi, H. Sato, and Y. Ueda, Phys.\nRev. Lett. 103, 146403 (2009).5\n[6] M. Sakamaki, T. Konishi, and Y. Ohta, Phys. Rev. B 80,\n024416 (2009); 82, 099903(E) (2010).\n[7] T. Toriyama, A. Nakao, Y. Yamaki, H. Nakao, Y. Mu-\nrakami, K. Hasegawa, M. Isobe, Y. Ueda, A. V. Ushakov,\nD. I. Khomskii, S. V. Streltsov, T. Konishi, and Y. Ohta,\nPhys. Rev. Lett. 107, 266402 (2011).\n[8] A. Nakao, Y. Yamaki, H. Nakao, Y. Murakami, K.\nHasegawa, M. Isobe, and Y. Ueda, J. Phys. Soc. Jpn.\n81, 054710 (2012).\n[9] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys.\nRev. B48, 10345 (1993).\n[10] H. Y. Hwang, S.-W. Cheong, P. G. Radaelli, M. Marezio,\nand B. Batlogg, Phys. Rev. Lett. 75, 914 (1995).\n[11] T. Mizokawa, D. I. Khomskii, and G. A. Sawatzky, Phys.\nRev. B61, R3776 (2000).\n[12] J. Geck, P. Wochner, S. Kiele, R. Klingeler, A.\nRevcolevschi, M. v. Zimmermann, B. B¨ uchner, and P\nReutler, New J. Phys. 6, 152 (2004).\n[13] J. Kienert and W. Nolting, Phys. Rev. B 73, 224405\n(2006).[14] S. Henning and W. Nolting, Phys. Rev. B 79, 064411\n(2009).\n[15] D. J. Garcia, K. Hallberg, B. Alascio, and M. Avignon,\nPhys. Rev. Lett. 93, 177204 (2004).\n[16] M. Gul´ acsi, Adv. Phys. 53, 769 (2004).\n[17] S. Yunoki,J. Hu, A.L.Malvezzi, A.Moreo, N.Furukawa,\nand E. Dagotto, Phys. Rev. Lett. 80, 845 (1998).\n[18] E. Dagotto, S. Yunoki, A. L. Malvezzi, A. Moreo, J. Hu,\nS. Capponi, D. Poilblanc, and N. Furukawa, Phys. Rev.\nB58, 6414 (1998)\n[19] See, e.g., P. Fazekas, Lecture Notes on Electron Correla-\ntion and Magnetism (World Scientific, Singapore, 1999),\npp. 383-384.\n[20] K. Penc and F. Mila, Phys. Rev. B 50, 11429 (1994).\n[21] H. Seo, J. Merino, H. Yoshioka, and M. Ogata, J. Phys.\nSoc. Jpn. 75, 051009 (2006).\n[22] T. Yamauchi, K. Hasegawa, H. Ueda, M. Isobe, and Y.\nUeda, unpublished."    },    {        "title": "2008.06757v1.The_inverse_proximity_effect_in_strong_ferromagnet_superconductor_structures.pdf",        "content": " 1  \n The inverse proximity effect in strong \nferromagnet -superconductor structures  \nV. O. Yagovtsev1, N. G. Pugach1,2 and M. Eschrig3 \n1 National Research University Higher School of Economics,  Moscow, 101000 , Russia  \n2 Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Leninskie Gory, 1 (2), Moscow GSP -2, 119991 , Russia  \n3 Institut für Physik, Universität Greifswald, D-17489 Greifswald , Germany  \n \nE-mail: vyagovtsev @hse.ru  \n \nAbstract  \nThe magnetization in a superconductor induced due to the inverse proximity effect is investigated in hybrid \nbilayers containing a superconductor and a ferromagnetic insulator or a strongly spin -polarized ferromagnetic \nmetal. The study is performe d within a quasiclassical Green function framework, wherein Usadel equations are \nsolved with boundary conditions appropriate for strongly spin -polarized ferromagnetic materials. A comparison \nwith recent experimental data is presented . The singlet to triple t conversion of the superconducting correlations as \na result of the proximity effect with a ferromagnet is studied.  \nKeywords: superconductivity, superconductor -ferromagnet, superconductor -ferromagnetic insulator, inverse \nproximity effect, induced magnetiza tion \n \n1. Introduction  \nIn classical electronics, charge currents are used to transfer \ninformation, while in spintronics, spin -polarized currents are \nused for this purpose [1]. The heat, generated in the process of \nusing spin -polarized  currents , can be an undesirable spurious  \neffect. The use of superconductors in magnetic nanostructures \ncan greatly reduce this heating, increasing the energy \nefficiency of spintronic devices. This idea underlies \nsuperconducting spintronics.  \nIn the last 30 years, superconducting spintronics has b een \nactively studied by many experimental and theoretical groups \n[2–6]. The main focus of the research was the theoretical  and \nexperimental  description of hybrid nanostructures. In \nparticular, the proximity effect in superconductor -ferromagnet \n(SF) structures was inves tigated. Its main features are spatial \noscillations of the amplitude of the superconducting \ncorrelation  penetrating into the ferromagnet  [4,5,7 –9] and the \nappearance of triplet superconducting correlations with \ninhomogeneous magnetization of the layers [6,10] . It leads to \na long -range proximity effect [5,6] . This was discovered in \nexperimental studies and it is now known that the long -range proximity effect leads to change  in the critical temperature of \nsuperconductors in bilayer and multilayer SF structures, \noscillations of the Josephson current in the presence of a \nferromagnet, and the long -range Josephson effect [2–5]. \nDevices of superconducting spintron ics are considered as \npromising  to create sensitive sensors and element base for a \nquantum computer [5,11] . \nThe inverse proximity effect in a superconductor in contact \nwith a ferromagnet was  first described in the work   [12]. \nTriplet superconducting correlations are the origin of the non -\nzero induced magnetization in the superconductor due to \ninverse proximity effect. Triplet Cooper pairs are symmetric \nin spin space   [13–16]. In theoretical work [17] the induced \nmagnetization in such structures was estimated in the presence \nof long -range triplet supe rconducting correlations . In \nexperimental studies  [18,19]  evidence of the presence of \ninduced magnetization in the proximity of a superconductor to \na ferromagnet ic metal  was demonstrated.  \nThe origin of the magnetization is a spin -splitting of the \nlocal density of states in the superconductor near the SF \ninterface. This spin -splitting is directly related to the spin -\npolarization of the Cooper pair . As a result, they  will create a    \n 2  \n noticeable magnetic field, which has the feature that it depends \non the concentration of Cooper pairs. In the absence of \nsuperconductivity, this magnetic field will be restricted to a \nmuch shorter length scale, related to the Thomas -Fermi length  \n[20]. \nAnother way to explain the appearance of magnetization is \nthe spin mixing  angle  [21]. It quantifies  how large the  relative  \nscattering phase shift between the electrons of the Cooper pair \nis after they are reflected from an S–F interface. This \ndifference between phase of the spin -up and spin -down \nelectrons of the Cooper pair results in an imbalance in the \nDOS for electrons with different spin , and it leads to the \nappearance of an induced magnetization.  \nHowever, most of these t heoretic al and experimental \nstudies were focused on structures with ferromagnetic metals \n(FM). Later phenomen a associated with the inverse proximity \neffect in structures with ferromagnetic insulators (FI) were \nexperimentally demonstrated [21,22] . In addition, in \nexperiments [23–25] phenomena related to the proximity \neffect of such structures were demon strated. They include \nsplitting in the density of states of the superconductor due to \nthe effective exchange field created by the proximity with a \nferromagnetic insulator [23] and the possibility of \nmanipulating spin transfer by adding ferromagnetic insulators \nto the s uperconducting layer [25].  \nThe present  work is devoted to the theoretical descr iption \nof induced magnetization in these structures. In Section 3, a \ncomparison is made with experimental studies [25–27], where \nthe value of the induced magnetization is obtained. In the \ntheoretical model of induced magnetization in S -FI and S -FI-\nS structures [28,29]  the Hamiltonian consists of a general \nHamiltonians of the S layers and a tunnelling  term with spin -\ndependent matrix elements. The  latter  represent the band \nsplitting in the ferromagnetic barrier which leads to different \ntunnel barrier heights for spin -up and spin -down electrons, \nand consequently, to differences between tunnelling  \namplitudes for electrons with different spin  projections . \nElectrons with spin -up and spin -down will penetrate the \ninsulating barrier  of the S-FI bilayer to different depths. In \nboth cases this will lead to a spin imbalance in the S layer near \nthe S-FI interface  and produce spin magnetization  in the \nsuperconductor. And again, absence of Cooper pairs will \nresult in absence of this mechanism for  induced \nmagnetization.  \nOur work also describes induced magnetization in \nstructures with strongly ferromagnetic metals (Section 2.3). \nThe work is based on the quasiclassical app roximation using \nUsadel equations  with boundary conditions appropriate for the \ncase of strong spin polarization  of the ferromagnet [21,30] . \nThese equations are applicable in  the dirty limit of a \nsuperconductor, in which the mean free path of an electron is \nmuch l ess than the coherence length of a Cooper pair .  \nFigure  1. The schematic view of the simulated \nsuperconducting -ferromagnet structure.  \nThe structure we describe is shown schematically in Fig. 1. \nLayer S is the superconductor layer, and F is a ferromagnetic \ninsulator or ferromagnetic metal. Here L is the  thickness of the  \nsuperconduct ing layer . \nSection 2.1 provides a model describing the S -F bilayer \nusing linearized U sadel equations. The solution to these \nequations serves as the basis for calculating the magnetization \ninduced in the superconductor. Section 2.2 shows boundary \nconditions and results for the induced magnetization for the S-\nFI case, Section 2.3 .1 shows them for the S -FM case  at the \ntemper ature close to its critical value and Section 2.3 .2 \npresents the equations for S-FM case  in the limit of weak \nproximity effect . \nSection 3.1 presents the determination of the angle of spin \nmixing from comparison with  the theoretical model [31], and \nSection 3.2 presents the determination of the angle using \nrelated  experimental data [25–27]. Section 3.3 is devoted to \nthe results of the numerical calculations and  shows how the \nmagnetization depends on various parameters of the model. \nSection 4 discusses the work’s main results.  \n   \n 3  \n 2. Model  \n2.1. Equations for superconductor in contact with \nferromagnet  \nTo estimate the value of the induced magnetization in the \nsupercon ductor we use the quasiclassical approximation in the \nform of  Usadel equations, assuming that the mean free path is \nthe smallest characteristic length in all layers of our structure. \nIn the ferromagnet, uniform magnetization is assumed over \nthe entire volume of  the material . It is assumed that the \ndiffus ive limit is realized in the structure, as real \nsuperconducting nanostructures created by sputtering usually \nsatisfy the conditions of the diffusive, or “dirty” limit . The \ncoherence length in the superconductor is 𝜉=\n√ℏ𝐷𝑠2𝜋𝑘B𝑇cb ⁄ , where ℏ is the reduced Planck constant,  𝑘B \nis the Boltzmann constant  (we use  units  where  ℏ=𝑘B=\n1), 𝐷𝑠 is the  electron diffusion coefficient of the \nsuperconductor , and Tcb is the criti cal temperature of the bulk \nsuperconductor.  \nThe linearized U sadel transport equations within the \nsuperconductor, applicable in the dirty limit [32] with \ntemperature in the vicinity of  the S film’s critical \nsuperconducting temperature Tс, are:  \n(𝐷𝑠𝛁2−2|𝜔𝑛|)𝑓𝑠=−2𝜋Δ,\n(𝐷𝑠𝛁2−2|𝜔𝑛|)𝐟𝑡=0.                     (1) \nHere the Matsubara frequencies are  𝜔𝑛=𝜋𝑇(2𝑛+1),𝑛 is \nan integer, T is the temperature, 𝑓𝑠 and 𝐟𝑡 are the singlet and \ntriplet components of the anomalous Green function, \nrespectively, and 𝐟𝑡 is the vector (𝑓𝑡𝑥,𝑓𝑡𝑦,𝑓𝑡𝑧), Δ is the scalar  \nsuperconducting order parameter which we assu me to be real -\nvalued, and in the bulk equal to the superconducting energy  \ngap. As a result of the influence of the ferromagnetic layer, the  \nsinglet superconducting correlations near the interface with \nthe insulator is suppressed  and converted to the triplet ones,  its \nvalue depends on the distance to the S -F interface . The F layer \nmagnetization is considered to be aligned  along the z axis.  \nThe boundary condition at the boundary of the \nsuperconductor with the (outer) environment  has the form  \n𝜕𝑓𝑠\n∂𝑥=𝜕𝑓𝑡𝑧\n∂𝑥=0, \nreflecting zero current flow through this interface.  \nIn both cases  of FI and FM, after finding the Green's \nfunctions by solving the Usadel equation with boundary \nconditions, it is necessary  to take into account the suppression \nof the superconducting correlations. The suppression of the \nsinglet component at the interface in the  limit of small \ntransparency of the S -F interface goes like  the square of the \nspin-mixing angle, and can thus be neglected because we are \ninterested only in effects of linear order in the spin -mixing \nangle [33]. To show this, we calculate the singlet and triplet \npair amplitudes and make sure that the order parameter  has almost constant value. In order to do this, the self -consistent \nequation [15]  \n∆(x)ln𝑇c\n𝑇=𝜋𝑇∑(∆(x)\n𝜔𝑛−𝑓𝑠(𝑥)\n𝜋).\n𝑛>0              (2) \nIs also solved approximately in the appropriate case . The \ninduced magnetization i s calculated by the formula [15]: \n𝛿𝑀(𝑥)=2𝜇𝐵𝑁0π𝑘B𝑇∑𝐠(𝑥,𝜔𝑛)\n𝑛>0,                (3) \nwhere 𝜇𝐵 is the Bohr magneton, 𝑁0 is the density of states  \nat the Fermi level  in the normal state . \nEquations (1) are supplemented  by the boundary condition \nfor the boundaries parallel to the YOZ plane. In our work, we \nconsider boundary conditions  appropriate  for a ferromagnetic \ninsulator and for a ferromagnetic metal.  \n2.2. Bilayer with f erromagnetic insulator  \nIn this case  the bound ary condition [21,30]  at the boundary \nof the S and FI layers is:  \n𝐴𝜎𝑆g0𝜕(𝑓𝑡𝑧+𝑓𝑠)\n∂𝑥=−2𝑁𝐺𝑄𝑃(φ),                 (4) \nwhere A is the contact area,  𝜎𝑆 is the conductivity of the \nsuperconductor in the non -superconducting state, g0 is the \nsinglet component of the Green function g, N is the number of \nconduction channels at the boundary (in the general case, it is \ndetermined by the polarization of the ferromagnet, the \ncoefficient of particle transmission through the boundary, and \nthe conductivity of the boundary ), 𝐺𝑄=𝑒2\n𝜋⁄ is the quantum \nof conductivity. P(φ) is the function defined in [21], which \ndepends on the value of φ, the angle of spin mixing.  \nThe spin mixing angle describes how s trongly  the magnetic  \nexchange  field affects the phase diff erence for electrons with \nspin-up and spin -down  at the superconductor -ferromagnet  \ninterface. Since the ferromagnetic  insulator may be considered \n[28,29]  as a spin-dependent  potential barrier , spin-up and spin -\ndown electrons  penet rate into FI at different depths. Th e \nsimplest approach leads to the assumption that the stronger the \nmagnetic  exchange  field in the ferromagnetic material, the \ngreater the difference in penetration depth , and t herefore, the \nphase difference created by the  boundary will be larger. \nHowever , it was shown [34] that the dependence of the created \nphase difference on the exchange splitting  is nonlinear  at large \nsplitting . The angle of spin mixing φ can be estimated using \nthe model given  in the work [31]. \nThe vector component of the normal Green function g is \nexpressed through the combination of the components of the \nanomalous Green function,  \n𝑓𝑓̃+g2=σ0. \nHere σ0 is a unit matrix, g=g0+(𝐠∙𝛔),𝑓=[𝑓𝑠+\n(𝒇𝑡∙𝛔)]𝑖σy, where g is the  spin vector  part of the normal \nGreen function g, 𝛔 is the vector of spin Pauli matrices.   \nA linear approximation was used, which is justifie d by a \nsmall change in the normal (diagonal part of the Nambu -   \n 4  \n Gor’kov matrix) Green function and the smallness of the \nanomalous (off-diagonal) Green function . To obtain an \nanalytic solution of the Usadel equation with two boundary \nconditions, the limit of temperature close to the critical \ntemperature was taken.  \nAfter expanding the normal Green function g into a Taylor \nseries  over the anomalous Green function components  and \ndiscarding the terms of third and higher order using equation \n(3), one obtain s the following expression for the induced \nmagnetization [17]: \n𝛿𝑴(𝑥)=−4𝜇𝐵𝑁0π𝑘𝐵𝑇∑Im(𝑓𝑠(𝑥,𝜔𝑛)𝑓̃𝑡𝑧(𝑥,𝜔𝑛)).\n𝑛>0(5) \nNote that neither a pure singlet nor a pure triplet pair can \ncreate such a spin -splitting, only the simultaneous presence of \nboth, see discussion around Eq. (87) in review [5]. \nWe assume that i t is possible to neglect  the Meissner effect \nfor this structure, which acts only at lengths of the order of the \nLondon penetration depth λ, [35] i.e. it is assumed that the \nthickness of the S layer is 𝐿~𝜉𝑆≪λ. \n \n2.3. Ferromagn etic metal case  \n2.3.1 Limit of temperature close to the critical \ntemperature  \nNow we turn to the case of a ferromagnetic metal (FM)  as \nmagnetic element in the bilayer . We consider the case of a \nstrongly spin -polarized ferromagnet where the period of space \noscillations of the superconducting correlations and its decay \nlength are much smaller than the mean free path lF and the \nlayer thi ckness L. It means that the exchange magnetic energy \nH is large enough to satisfy the inequality ξH <<lF,L, where ξ H \n= ħvF/H for the magnetic length,  where  vF is Fermi velocity in \nthe ferromagnet. The anomalous Green function describes the \nsuperconducting correlation s, which penetrate in the F layer \ndue to the proximity effect. The mean value of the short -range \ncomponents of the anomalous Green function is negligible  due \nto its fast oscillations in the strong ferromagnet.  \nThus, on a length -scale considerably larger than the mean \nfree path, we may assume that the short -range components of \nthe anomalous Green function are zero and normal Green \nfunction is a consta nt in the ferromagnet. In this case we can \nconsider the ferromagnet as semi -infinite and focus our \nattention on the S layer with appropriate boundary conditions \nat the SF interface. We assume that the ferromagnet occupies \nthe semi -space 𝑥>𝐿. For the same  reason as for S -FI case, \nwe assume that 𝐿 is much less than the London penetration \ndepth λ. We also consider limit of temperature close to the \ncritical temperature.  \nThe Usadel equation in the superconductor has the s imilar  \nform as (1),  \n(𝐷𝑠\n𝜋g𝑑2\n𝑑𝑥2+2𝑖𝜔)(𝑓𝑠±𝑓𝑡𝑧)=−2Δg, but the boundary condition  at the boundary of the S and F M \nlayers is  different from the S -FI case. F or the ferromagnetic \nmetal the boundary condition [36] at the interface ( 𝑥=𝐿) may \nbe written in the simple closed form  \nℏ𝐴𝜎𝑆\n𝑒2𝜋𝜔𝑛\n√𝜔𝑛2+∆2𝜕(𝑓𝑠±𝑓𝑡𝑧)\n∂𝑥=(𝑡∓𝑖φ)(𝑓𝑠±𝑓𝑡𝑧),    (6) \nwhere 𝑡=𝑡↑+𝑡↓ is the probability for the particles  to be \ntransmit ted through the S-FM interface . Here, 𝑡↑=\n∑|𝑡↑↑𝑛|2\n𝑛↑ ,𝑡↓=∑|𝑡↓↓𝑛|2\n𝑛↓  are the transmission probability \nmatrix elements for each of the 𝑛↑ or 𝑛↓ transmission channels \nwith spin up ( ↑) or down ( ↓), respectivel y [36]. We assume no \nspin-flip scattering on the SF interface, therefore the \ntransmission matrix has a diagonal form. There are different \nnumbers of transmission channels  for spin -up and spin -down \nin the ferromagnet, therefore 𝑛↑≠𝑛↓. \nThe imaginary part 𝑖φ appears due to the spin splitting in \nthe ferromagnet. For a structure without Josephson current, \nwhen ∆ may be assumed to be real, (𝑓𝑠−𝑓𝑡𝑧)=(𝑓𝑠+𝑓𝑡𝑧)∗. \nEarlier calculation s of the proximity effect between a \nsuperconductor an d a strong ferromagnet  [37,38]  were \nperformed  for only a very thin F layer L ~ξH. In this case the \nboundary conditions at the SF interface had approximately the \nsame form  [38] with a complex coefficient between the \nfunctions (𝑓𝑠±𝑓𝑡𝑧) and their derivati ves, but these \ncoefficients oscillated with thickness dF. \nThe inverse proximity effect leads to the singl et to triplet \ntransform of the superconducting correlations near the SF \ninterface. If L~ξ, the order parameter and the anomalous Green \nfunction 𝑓𝑠+𝑓𝑡𝑧=𝜋∆(𝑥)\n|𝜔𝑛|+𝐷𝑠𝑘2/2 may be determined by the \nansatz [38] \n∆(𝑥)=∆0cos[𝑘(𝑥−𝐿)].                       (7) \nThis ansatz satisfies the Usadel equation (1) with the \nboundary condition at the free interface. For this case, for the \ndependence of all the quantities of interest, we will \nconcentrate  only on thicknesses of the order of the coherence \nlength, since the ansatz does not work well  at larger  \nthicknesses .  \nThe value of k is found from the boundary condition (6) that \nleads t o the equation  \n𝑘∙tan(𝑘𝐿)=𝑒2\n2𝜋2ℏ𝐴𝜎𝑆(𝑡−𝑖φ).                  (8) \nNote that here k depends on the values 𝐿, 𝑡 and φ, and does \nnot depend on 𝜔𝑛. Using the obtained value of k we may \ncalculate properties of the proximity structure (consisting of a \nsuperconductor – strongly spin -polarized ferromagnet bilayer) \nas functions of the parameters of the SF boundary , 𝑡 and φ. \nAs it was mentioned above, the inverse proximity effect \nbetween a ferromagnet and a superconductor leads to an \ninduced magnetization at the S side  [14]. Such induced \nmagnetization was detected in experiments using the nuclear \nmagnetic resonance  [19], and the polar Kerr effect  [39]. The \nvector of this induced magnetization ma y be found by the    \n 5  \n formula  (3). Using  this formula, we calculated the \nmagnetization at the S -FM interface depending on the \nthickness of the superconductor L. \n \n2.3.2 The weak proximity effect limit  \nHere w e assume that the transparency of the SF interface is \nsmall, and therefore the Green functions weakly differ from \ntheir bulk values  g=−𝑖𝜔𝑛\n√𝜔𝑛2+∆2, 𝑓=∆\n√𝜔𝑛2+∆2  and 𝑓𝑡𝑧,𝑓𝑠 are \ndefined b y the boundary condition  (6). \nThe corrections to the bulk values of functions g and 𝑓 \nappear only in the second order on 𝑓𝑡𝑧 and may be neglected. \nThe solution of the Usadel equation ( 1) satisfying boundary \ncondition at the free interface has the form  𝑓𝑡𝑧=\n𝐶cosh 𝑘𝑆(𝑥−𝐿), where 𝑘𝑆=2√𝜔𝑛2+∆2ℏ𝐷𝑠 ⁄ . The \nconstant C may be found from th e boundary condition at the \nSF interface ( 6), \n𝐶=𝑖𝑓φ𝑠𝑖𝑔𝑛(𝜔𝑛)(𝜋𝐴𝜎𝑆\n𝑒22|𝜔𝑛|\n𝐷𝑠𝑘𝑆sinh 𝑘𝑆𝐿−𝑡cosh 𝑘𝑆𝐿)−1\n. \nAs it is expected, the amplitude 𝐶 of the triplet component \nis proportional to the phase shift φ, which the anomalous \nfunction  experiences  at the interface with the ferromagnet. \nThis value depends on the exchange field 𝐻 and is equal to \nzero in the absence of 𝐻. The expression for the induced \nmagnetization ( 5) may be written as  \n𝛿𝑀(𝑥)=−4𝜇𝐵𝑁0π𝑘𝐵𝑇∙ \n∙∑∆2\n𝜔𝑛√𝜔𝑛2+∆2\n𝜔𝑛>0φcosh [𝑘𝑆(𝑥−𝐿)]\n𝜋𝐴𝜎𝑆\n𝑒22𝜔𝑛\n𝐷𝑠𝑘𝑆sinh 𝑘𝑆𝐿−𝑡cosh 𝑘𝑆𝐿. \nFor a thick superconductor for 𝐿»𝜉 we may write this \nequation in the form:  \n𝛿𝑀(𝑥)=−4𝜇𝐵𝑁0π𝑘𝐵𝑇∑∆2\n𝜔𝑛√𝜔𝑛2+∆2\n𝜔𝑛>0φexp(−𝑘𝑆𝑥)\n𝜋𝐴𝜎𝑆\n𝑒22𝜔𝑛\n𝐷𝑠𝑘𝑆−𝑡. \nThe induced magnetization decreases exponentially from  \nthe SF boundary at a distance of the order of the \nsuperconducting coherence length  𝜉 at low temperature.  \n3. Results  \n3.1. Comparison with experiment  \nThe spin mixing angle can be estimated by comparing the \nmagnetization obtained for different values of the angle with \nthe magnetization which was taken from experimental studies \n[25–27]. In these works, the conductivity of the structures \nunder study was measured, and the densities of states  (DOS)  \nwere obtained. The values of the exchange field in the \nsuperconductor were obtained from DOS.  \nTable 1 shows values of the spin -mixing angle that were \nobtained from experimental studies as a result of fitting the \ndata. In these works, Al was taken as a s uperconductor and EuS was taken as a ferromagnetic insulator. 𝜎𝑆 was taken as \n3.8∙107 𝑂ℎ𝑚−1∙𝑚−1. If it were not mentioned in the article, \n𝐷𝑠 was taken as 8.68∙10−3𝑠−1∙𝑚2, because we can assume \nthat its value in all articles is approximately the sa me due to \nalmost identical production conditions.  \n \n№ L, nm  𝐷𝑠, s−1∙m2 𝐵, T φ \n22 10 2∙10−3 1.75 0.033  \n23 10-15 8.68∙10−3 2.25-4.5 0.14-0.27 \n24 20 8.68∙10−3 0.25 0.8 \nTable 1.  Comparison with the experimental works.  \nOne can see that  the spin mixing  angle does not reach its \nmaximum value π/2. This is consistent with the theoretical \ndependence of the magnetization on the spin mixing  angle . \nThe model calculation fit with the experimental data yields \nsome variation  of the spin mixing angle for different  \nsuperconductor thicknesses , although the spin-mixing angle \ndepends only on the properties of the interface between \nmaterials. Probably this is a consequence of the constant order \nparameter approximation, which nevertheless allows \nestimation of φ. \n3.2. Estimation of the angle of spin mixing  \nTo determine the spin mixing  angle , we used a comparison \nwith the boundary  condition [31] written in a linearized form.  \nAfter carrying out the corresponding transformations, we \nobtain the following relationship between the \nphenomenological parameters:  \nsin(φ)=−𝐴g0\n2𝑁GQ𝑅𝐵𝐿GΦ\nGT, \nHere, 𝑅𝐵 is the barrier resistance, which is  a \nphenomenological parameter,  and GΦ and GT are the \nphenomenological parameters given in the work  [31]. \nThe following estimat es were obtained for the spin mixing  \nangle  in radians: φ=0.82 at 𝑁𝐴⁄=2.48∙1019 μm−2, 𝑇=\n0.8∙𝑇𝑐, 𝑇𝑐=1.2 К, 𝐿=5𝜉, 𝜉= 93 nm. \nParameters appropriate for aluminium  are taken for the \nsuperconductor. Aluminium  has 𝑇𝑐=1.2 К, and in the \ncurrently used preparation technology  𝜎=3.8∙107 Ohm−1∙\nm−1, 𝐷=8.68∙10−3 s−1∙m2. Parameters appropriate for \ncobalt  are taken for the ferromagnet metal in our work.  \n3.3. Numerical results and discussion  \nThe magnetization  profile  was obtained over the thickness \nof the superconducting layer as a function o f the spin mixing  \nangle . The structure was modelled  with the following \nparameters: 𝑁𝐴⁄=2.48∙1019 μm−2, 𝑇=0.8∙𝑇c, 𝑇c is 1.2 \nK, 𝜎=3.8∙107 Ohm−1∙m−1, 𝐷=8.68∙10−3 s−1∙m2,    \n 6  \n 𝐿=4𝜉, 𝜉= 93 nm, 𝑡=0.7. All dependencies are presented  \nfor given parameter values, unless otherwise indicated.  \nThe calculation of pairing amplitudes was carried out: the \nvalues of Green functions with a constant order parameter \nwere substituted into the equation (2). The result is presented \nin Fig. 2. It allo ws estimation of the suppression of the \nsuperconducting correlations at the boundary. It is seen that \nthe superconducting correlations, and consequently, the order \nparameter of the superconductor is weakly suppressed by the \nmagnetization of the neighbourin g insulator layer at small spin \nmixing angles.  \n \n \nFigure 2. Pairing amplitudes  of the superconductor  for S -FI \ncase as a function of  the coordinate for various values of the \nangle of spin mixing, φ.  \nThe obtained dependences of the components of the \nanomalou s Green function are presented in  \nFig. 3.  \nThe relations between the various components of the \nanomalous Green function are: \n𝑓𝑆=𝑓↑↓−𝑓↓↑\n2,𝑓𝑡𝑧=𝑓↑↓+𝑓↓↑\n2.  \nThe dependences of the components 𝑓𝑆 and 𝑓𝑡𝑧 of the \nanomalous Green function on the x coordinate, shown in Fig. \n3, demonstrate  that the singlet component of the Green \nfunction is suppressed when approaching the F boundary . The \ntriplet component, on the contrary, decreases when moving \naway from the S -FI boundary. This corresponds to physical \nexpectations, according to which, singlet Cooper pairs are \ntransformed into triplet pairs  at the S -FI border.  \nThe functions shown in fig. 3 were substituted into formula \n(5). As a result, the induced magnetization  values were  \nobtained.  \n  \nFigure  3. Real and imaginary components fs and ftz of the \nanomalous Green functions  in the superconductor  as \nfunctions  of the coordinates , of (a) S -FI interface, (b) S-FM \ninterface for the temperature close to the critical one, (c ) S-FM \ninterface for the weak proximity effect.  \n   \n 7  \n  \nFigure  4. Magnetization versus the x coordinate for various \nvalues of the angle of spin mixing, φ  in the S layer of (a) S-FI \nbilayer , (b) S-FM bilayer  for the temperature close to the \ncritical limit, (c) S-FM bilayer  for the weak proximity effect \nlimit.  As it can be seen from Fig. 4(a), a local maximum of \nmagnetization appears on the curve at the extremal value of \nthe coordinate x.  It becomes clear that such a behaviour  of \nmagnetization should have been e xpected. The reason for this \nbehaviour  is the decrease of the singlet component and the \ngrowth of the triplet component when approaching the \nsuperconductor -ferromagnetic interface.  \nWe can see from Fig. 4 (b, c) the absence of a local extreme \npoint of the i nduced magnetization profile at a contact with a \nferromagnetic metal. We may assume that this is a \nconsequence of the difference in boundary conditions, \ndescribing a nonzero transparency  of the interface for \nelectrons in the metallic case. The S -FM interfa ce provides not \nonly an inverse, but also a direct proximity effect with \ndrainage of Cooper pairs from S into the ferromagnet. The S -\nFI interface produces only singlet -to-triplet conversion of \nCooper pairs. This is the reason for different behaviour  of the  \ninduced magnetization in the superconductor in the vicinity of \nthese interfaces.  \n \nFigure  5. Magnetization in the S layer near the S -FM interface \nversus the superconductor thickness  L for various values of \nthe angle, 𝜑, (a) for the temperature close t o the critical limit, \n(b) for the weak proximity effect limit, here 𝑇=0.8∙𝑇𝑐, 𝑡=\n0.7 for (a), 𝑡=0.1 for (b).  \n   \n 8  \n Fig. 5 (a) shows that, as one would expect, the \nmagnetization at the S -FM boundary for small L is minuscule, \nbecause superconductivity is suppressed by the magnetization \nof a ferromagnet. The higher the angle, the less magnetization \noccurs. It is clear that growth of the ferromagnetic insulator’s \nmagnetization increases the suppression of superconductivity \nin the S -layer.  \nFor the S -FM case w ithin weak proximity effect, see Fig.5 \n(b). In this case, for small thickness of superconductor the \nmagnetization is bigger than for much larger thickness. \nHowever, this might be due to the fact that in this case the \nsuppression of the order parameter in t he superconductor layer \nwas not taken into account, unlike in case (a) . \nFrom Fig. 6(a), we see that there is a maximum of the \naverage magnetization on the spin mixing  angle  φ due to the \nfollowing factors. For a small magnetization of the \nferromagnet, the growth of the spin -mixing angle leads to the \nappearance of more triplet pairs, and the magnetization in a \nsuperconductor increases. However, at some point, the \nmagnetization of the ferromagnet reaches such a value that its \ngrowth already leads to a decrea se in the number of Cooper \npairs due to the suppression of superconductivity, and, as a \nconsequence, to a decrease in the number of singlet pairs from \nwhich triplet pairs are obtained. In addition, it is seen that with \nincrease in temperature, the magnetiz ation decreases. This is \ndue to the fact that the superconductivity is suppressed with \nincreasing temperature and, as a result, fewer  singlet and  \ntriplet Cooper pairs are generated  and contribute to the \nmagnetization. There is no maximum of the average ind uced \nmagnetization function IM(φ) at Fig.6 (b,c) for the S -FM \nbilayer. This is also a consequence of the transparency of the \ninterface.  \n4. Conclusion  \nWe have presented results for the  induced magnetization \nand superconducting correlations profile in the \nsuperconducting layer of a bilayer superconductor -\nferromagnet structure. We have  shown that  the induced \nmagnetization  monotonically  decreases with increasing \ntemperature  as a consequence of the inverse proximity effect \nwhich  vanishes at the critical temperat ure. The d ependence of \nthe induced magnetization on the coordinate and on the spin \nmixing angle, has a maximum for S -FI, and is monotonous for \nS-FM structures. The reason for this behaviour  is a decrease \nin the singlet component and a growth of the triplet  component \nwhen approaching the superconductor -ferromagnetic \ninterface. As the magnetization is proportional to the product \nof both, a local maximum in the profile appears.  \n  \nFigure  6. Average magnetization at (a) the S -FI bilayer, (b) the \nS-FM interface for the temperature close to the critical limit, \n(c) the S -FM interface for the weak proximity effect limit, \nversus the angle of spin mixing, φ, for various values of the \ntemperature, T. Here 𝜉= 93 nm. The transparency 𝑡=0.7 \n(b), 𝑡=0.1 (c). \n   \n 9  \n The reason for the local maximum in the averaged \nmagnetization as function of the spin mixing  angle  is the \ncompetition between singlet -to triplet conversion, provided by \nspin-mixing. A stronger spin mixing leads to suppression of \nsuperconductivity and, the refore, to a decrease of the induced \nmagnetization.  \nThe magnitude of the induced magnetization is also \ndifferent for S -FI and S -FM structures. In the S -FI case it is \nusually large r as a  consequence of the finite transparency of \nthe S -FM interface for elec trons.  \nSuperconductor -ferromagnet  structures  as discussed in this \npaper may have various applications, e.g. as  sensitive \nbolometers and thermometers [40,41] . \nAcknowled gements  \nPugach N. and Yagovtsev V. thank Prof. Ekomasov E. G. \nfor valuable discussions and the program Mirror Labs of \nNational Research University Higher School of Economics for \nsupport of  interrussian  cooperation.  \nThe calculations of inverse proximity effect in S -FI bilayers \nand estimations of the spin -mixing angle were funded by the \nRussian Ministry of Education and Science, Megagrant \nproject N 2019 -220-07-6383. 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Lee  \nMaterials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, \nTennessee 37831, United States  \n \nS. Yoon  \nDepartment of Physics, Gachon University, Seongnam, 13120, Republic of Korea  \n \nS. Song, S. Park  \nDepartment of Physics, Busan National University, Busan 46241, Republic of Korea  \n \nKeywords: ferromagnetic insulator, p erovskite LaCoO 3, crystallographic orientation , \nferromagnetic ordering, pulsed laser epitaxy  \n \nFerroma gnetic insulators play a crucial role in the develop ment of  low-dissipation quantum  \nmagnetic device s for spintronics. Epitaxial  LaCoO 3 thin film is a prominent ferromagnetic \ninsulator , in which the  robust ferromagnet ic ordering emerges  owing to epitaxial strain. \nWhe reas it is evident that strong spin-lattice coupling induces ferromagnetism, the reported \nferromagnetic properties of epitaxially strained LaCoO 3 thin films were highly consistent. For \nexample, even under largely modulated degree of st rain, the reported Curie temperatures of \nepitaxially strained LaCoO 3 thin films lie within 80 –85 K , without much deviation . In this \nstudy , substantial enhancement (~18%) in the  Curie temperature of epitaxial LaCoO 3 thin \nfilms is demonstrated via crystallog raphic orientation dependence . By changing the \ncrystallographic orientation of the films from (111) to (110),  the crystal -field energy  was \nreduced and the charge transfer between  the Co and O orbitals was enhanced. These  \nmodifications  led to a considerable  enhancement of the ferromagnetic properties  (including   \n2 \n \nthe Curie temperature and magnetization), despite the identical nominal degree of epitaxial \nstrain . The findings of this study provide  insights into facile tunability of ferromagnetic \nproperties via structural symmetry control  in LaCoO 3. \n \n1. Introduction  \nFerromagnetic insulator s (FM-Is) are essential component s in dissipation -less spintronic \ndevices with highly suppressed leakage currents.[1, 2] They  generat e pure spin currents by \nfilter ing charges and in troducing spins into adjacent nonmagnetic layer s via the magnetic \nproximity effect.[3-8] However , FM -Is are scarce in nature compared to more conventional FM \nmetals  because  the general superexchange  mechanism in insulators promotes \nantiferromagnetic (AFM) ordering.[9-12] Iron yttrium garnet (YIG), α-Fe2O3, spinel CoFe 2O4, \nEuS, EuO, and strained EuTiO 3 thin films  are some of examples of FM -I materials.[13-19] \nEpitaxially -strained LaCoO 3 (LCO) , e.g., epitaxially deposited on SrTiO 3 (STO) , \n(LaAlO 3)0.3(Sr2AlTaO 8)0.7 (LSAT), or YAlO 3 substrates, is an emerging FM-I with a  Curie \ntemperature above the  liquid N 2 temperature (TC = ~80 K), with a robust ferromagnetic \norigin.[20, 21] \n \nTable 1.  TC obtained in LCO thin films: A summary of reported TC of LCO thin films, \ndepending on the applied magnetic field ( H), type of substrates, crystallographic  orientation , \nand film thickness.  Note that a robust TC with few exceptions is found for the LCO thin films. \nIn contrast a substantial increase of TC (~ 18%) is observed in the current study.  \nTC (K) H [Oe] substrate  orientation  thickness [nm] reference  \n43.2 200 LAO (100)  100 \nPhys. Rev. B  77, 014434 (2008)  69.2 200 SLAO  (100)  100 \n71.6 200 SLGO  (100)  100 \n76.0 200 LSAT  (100)  100 \n77.6 200 STO (100)  100 \n80/87 200 LSAT  (100)  8/50 Phys. Rev. B  79, 024424 (2009)  \n86 2000  STO (100)  100 \nEur. Phys. J. B  76, 215 –219 (2010)  75 2000  LAO (100)  100 \n85 2000  LSAT  (100)  100 \n87 2000  PMN -PT (100)  100 \n84 2000  SLAO  (100)  100 \n80 2000  STO/LSAT  (100)  20-90 J. Appl. Phys.  105, 07E503 (2009)  \n76-83 50 LAO (100)  30-95 J. Appl. Phys.  109, 07D717 (2011)  \n75.9/81.2  50 STO/LSAT  (100)  15 Phys. Rev. B  91, 144418 (2015)  \n80 1000  STO/LSAT  (100)  30 Nano Lett.  12, 4966 -4970 (2012)  \n94 1000  LSAT  (110)  60 Phys. Rev. Lett.  111, 027206 (2013)  \n24 1000  LSAT  (111)  60 \nPhys. Rev. B  92, 195115 (2015)  \n92/90 500 STO/ LSAT  (111)/ (100)  35/38   \n3 \n \n85 500 STO/LSAT  (100)  30 AIP Advances  8, 056317 (2018)  \n85 500 STO (100)  12 PNAS  115, 2873 –2877 (2018)  \n78 500 STO (100)  70 Curr. Appl. Phys.  28, 87-92 (2021)  \n50 500 STO (111)  80 Phys. Status Solidi B , 2100424 (2021)  \n76.1-90.2 1000  STO (100), (110) and (111)  30 This work  \n \nLong-range FM ordering in epitaxial LCO thin film s is facilitated by strain -induced lattice \ndistortion s. Although  bulk LCO is nonmagneti c with  a rhombohedral crystal structure, the \ntensile  strain in the thin film is known to induce monoclinic and tetragonal phases, resulting in \na spin-ordered state.[20, 22, 23] The large and facile modification of the crystalline symmetry \noriginates from the ferroelastic nature of LCO , which favors  the formation of  twin-domain \nstructure s under mechanical pressure or epitaxial strain .[24-31] Such twin -domain structures \nhave been observed as dark stripe patterns in high -angle annular dark -field scanning \ntransmission electron microscopy  (HAADF STEM) imaging.[20, 23, 32, 33] Detailed \nmicrostructure analyses using HAADF STEM  on (100) -oriented LCO thin films enable \nclassif ication of  basic lattice units into compressed - (c-unit), tensile -stretched - (t-unit) , and \nbulk-units (b-unit).[23] Among the se, c-units with coexisting low (LS) and high  spin- (HS) \nstates of  the Co spin configuration were found to be responsible for the FM phase. It should \nbe noted that the FM phase , characterized by TC, in the (100) -oriented LCO  thin film was \nmostly  unchang ed. As summarized in  Table 1 , the reported values of TC of (100) -oriented \nLCO thin films are mostly in the range 80 –85 K,[20, 34-45] despite the large difference in the \ndegree of epitaxial strain.  Note that TC might decrease further below 80 K, if the crystalline  \nquality is not ideal .[43] Such unsusceptible behavior in TC suggests that the magnetically active \nc-units are highly resilient against orthogonal stress, likely  stemming from the ferroelastic \nnature of  LCO.[23] \n \nModifications in the crystallographic orientation lead  to disparate lattice symmetry, offering \nextra tunability in the functional properties of epitaxial thin films.  Even though an identical \nfilm/substrate system is used, implying the same degree of nominal epitaxial strain , the \nstructural symmetry of the thi n film can be modulated by the crystallographic orientation of \nthe substrate , leading to stress that is not orthogonal to the lattice . For instance, the FM TC of \nepitaxial SrRuO 3 thin film s can be increased from 147 to 157 K, a s the crystallographic  \norient ation changes  from  (100) to (111) , because of the  decreas ed spin dimensionality  within \nthe trigonal symmetry .[46] In epitaxial La0.7Sr0.3MnO 3 thin films, (110) and (111) -oriented thin \nfilms exhibit larger spin moment s than (100)-oriented thin film s.[47] As the crystallographic  \norientation of LaAlO 3 thin films on  KTaO3 substrate s changed from (100) to (110) or (111),   \n4 \n \nemergent superconductivity w as observed . The superconducting critical temperatures could \nbe further modulated between the  (110) and (111) -oriented thin films , with the values of  1 and \n2 K, respectively.[48-50] Because the crystallographic  orientation determines the \ncrystallographic symmetry of the lattice unit, it is expected to modify the robust FM -I phase \nof LCO epitaxial thin films as well. Th is strategy would provide fundamental insight into the \nstructure -property relationship of LCO, which would not have been possible using only (100) -\noriented thin films with different strain state s, but the same type of orthogonal stress. Indeed, \nstudies on LCO thin films with crystallographic  orientations other than (100) -orientation are \nhighly limit ed and report inconsistent  TC values , as shown in Table 1, and a systematic \ncomparison between (100) -, (110) -, and (111) -oriented LCO epitaxial thin films has not been \nconducted thus  far. \n \nIn this study, we demonstrate a large modulation of the FM phase in  LCO epitaxial thin films, \nincluding the substantial tuning of TC, by tailoring the crystalline symmetry via \ncrystallographic  orientation control.  We fabricated epitaxial LCO thin films on  (100)-, (110) - \nand (111) -oriented  STO substrates using  pulsed laser epitaxy (PLE) . The structural quality \nand crystallographic symmetry of the thin films have been determined by X -ray diffraction \n(XRD) measurements and HAADF STEM imaging. The FM phases for the (100) -, (110) - and \n(111) -oriented LCO thin films have  TCs of 81.5, 90.2, 76.1 K, respectively . The large \nmodulation of the FM TC, i.e., ~18% enhancement in the (110) -oriented thin film compared to \nthe (111) -oriented one, captures the effectiveness of the crystallographic symmetry control on \nthe lattice units. The physical origin of the TC enhancement was investigated by characteriz ing \nthe microscopic lattice unit s, electronic structure measurements , and theoretical calculations . \nIn particular, the reduced crystal -field splitting energy and stronger Co -O charge transfer in \nthe (110) -oriented thin film resulted in considerable enhance ment of the FM properties.  \n \n2. Results and Discussion  \nThe global lattice structures of the LCO thin films (30 nm) were characterized using XRD to \nreveal the systematic crystallographic  orientation dependence. Figure 1 a shows the XRD \nθ−2θ scans for the LCO thin films . Fine thickness fringes are commonly observed along with \nthe corresponding LCO peaks  (Figure 1a; Figure S1 a and b , Supporting Information) , \nindicating consistent crystalline phases with atomically smooth surfaces and interface s. In \naddition, atomic force microscopy images show atomically smooth surfaces of the epitaxial  \nLCO thin films  with low surface roughness of < 1 u.c. ( Figure S2, Supporting Information ).   \n5 \n \nFigure 1b shows the XRD reciprocal space maps around the (103), (120), and (212) Bragg \nreflections for the LCO thin films epitaxially grown on the (100)-, (110) -, and (111) -oriented \nSTO substrates, respectively. Despite the difference in the crystallographic  orien tations, all \nfilms were fully strained to the substrates. Although  the nominal lattice mismatch between the \nthin film and the substrate is identical , it is important to note that the degree of strain still \ndiffers owing to symmetry modification . The differ ence in the degree of strain is shown  in \nFigures 1c , in terms of the pseudocubic unit cell volume ( Vu.c.) and out -of-plane stress ( εoop \n(%) = 100 × (dfilm − dbulk) / dbulk, where dfilm and dbulk are the out -of-plane distance s between \nthe atomic planes of the thin film and the bulk, respectively) for distinctive  crystallographic  \norientation s. The s ame trends are shown for the 10- and 20 -nm-thick  LCO thin films (Figure \nS1c and d, Supporting Information) . Both Vu.c. and εoop were the largest for the (110) -oriented \nthin film, indicating the largest lattice distortion, and the smallest for the (111) -oriented film. \n  \nFigure 1.  Crystal structures  of epitaxial LCO thin films with various crystallographic \norientations:  a) XRD θ-2θ scans of epitaxial LCO thin films (#) grown on STO substrates  \nwith different orientations . b) XRD  reciprocal space maps (RSM) of the (100) -, (110) -, and \n(111) -oriented LCO thin films, shown around the (103), (120), and (212) Bragg reflections of \nthe STO substrates, respectively. Crystallographic orientation -dependent c) unit cell volume \n(Vu.c.) and  out-of-plane strain ( εoop). Z-contrast HAADF STEM images for LCO t hin films \ngrown on d) (100), e) (110), and f) (111) STO substrates. Dark stripe patterns are observed in \nall crystallographic orientations of the LCO thin films. Two distinct regions are marked with \nwhite horizontal lines. Region I near the interface is a “uniform region” without any dark \n0.20 0.210.740.760.78(212)\nLCO\nSTO\n0.17 0.18 0.190.540.56\nqx (Å-1)(120)\nLCO\nSTO58.058.1(c)\nVu.c. (Å3)\n(100) (110) (111)-0.5-0.4-0.3\neoop (%)\nOrientation0.24 0.260.760.780.80\nLCO(b)\nqz (Å-1)(103)\nSTO\n10 20 30 40 50 60 70 80##\n#(300)(220)\n(100)\n#\n#(200)\n(111)(a)\nIntensity (arb. units)\n2q (degrees)#(110)\n(e) (d) (f)\nLCO (100)\nSTO (100)III\n2 nm[100][001]\n[010]\nSTO (111)[01][111]\n[11]\n2 nmLCO (111)\nSTO (110)[00][110]\n[10]LCO (110)\n2 nm  \n6 \n \nstripe patterns. Region II away from the interface is an “ordered region” with dark stripe \npatterns.  \n \nFigure 1d–f show s high-resolution HAADF STEM images of epitaxial LCO thin films grown \non d) (100) -, e) (110) -, and f) (111) -oriented STO substrates, respectively . The STEM images \ncommonly exhibit a coherent alignment of the atomic columns without any defects and/or \nmisfit dislocations at the interfaces. Interestingly, distinctive dark stripe patterns  (with ~3 u.c. \nperiodicity) were observed , above a certain thickness of the interfacial layer.  We carefully  \nexclude the presence of oxygen vacancy ordering at the dark stripe pa tterns by electron \nenergy loss spectroscopy (EELS) measurement s (Figure S3, Supporting Information).  While \nthe dark stripe patterns have been frequently recognized for the (100) -oriented LCO thin \nfilms,[20, 23, 32, 33] we report similar patterns in  (110) - and (111) -oriented thin films with \nsystematic changes for the first time. The LCO thin films can be categorized into  two regions  \nconsidering the dark stripe pattern , i.e., Region I (near the interface) and Region II  (above the \ninterface) .[20] Region I is free of the  dark strip e patterns, suggesting a conventional fully  \nstrained state . The thickness of Region I is the thickest for the (110) -oriented thin film, and \nthe thinnest  for the (111) -oriented thin film , consistent  with the fact that the (110) -oriented \nthin film has the largest deviation in Vu.c. from the bulk value (55.604 Å3). On the other hand, \nRegion II presents twin -domain structures due to  the ferroelastic nature of the LCO thin film, \nas mentioned earlier.  For more detailed structural analyses, we measured the distance of La \ncolumns in Region II along the two orthogonal axes perpendicular to the HAADF STEM \nimage (Table S1, Supporting Information). The lattice structure of magnetically active c-units , \nwhich was robust in the (100) -oriented LCO thin films,[21] is distorted differently depending \non the substrate orientations, suggesting that the robust structure can be tuned by symmetry \nconstraints. The volume of c-unit is the largest in the (110) -orien ted LCO thin film, while it is \nthe smallest on the (111) -oriented film, which is consistent with the XRD results.  \n \nThe FM ordering in the LCO thin films is strongly coupled to the crystallographic symmetry . \nFigure 2 a show s the temperature -dependent magnetization, M(T), of the LCO thin films \nunder field -cooled (FC , 1000 Oe ) cooling along the in -plane direction. A clear FM transition \nwith a sudden upturn of M(T) is observed at 81.5, 90.2, and 76. 1 K for the (100)-, (110) -, and \n(111)-oriented LCO thin films, respectively. The inset of Fig ure 2a shows the linear \nextrapolation of M(T) for the TC determination . The m agnetic -field-dependent magnetization, \nM(H), measured at 2 K , demonstrates  a clear FM hysteresis loop, as shown in Figure 2b. The   \n7 \n \nvalues of saturation magnetization ( Ms) and remanent magnetization ( Mr) are summarized in \nFigure 2c and 2d, respectively, exhibiting consistent orientation -dependent trend s with  TC \n(Figure 2e) . Indeed, a substant ial enhancement in ferromagnetic properties, i.e., TC (~18%), \nMs (~36%), and Mr (~49%), is observed, by comparing the (110) -oriented thin film to the \n(111) -oriented one. The same trend was observed from the Curie -Weiss (C -W) law fitting of \nthe M(T) curves, in terms of the effective ratio of  the HS-state (Figure S4, Supporting \nInforma tion).[29] Within the prevailing spin configuration model of LCO epitaxial thin \nfilms,[23, 29, 33, 51] in which the LS - and HS -states coexist, the effective ratios of HS -state are \n68.4, 69.5, and 57.2%, respectively, for the (100) -, (110) -, and (111) -orien ted LCO thin films. \nX-ray photoelectron spectroscopy results consistently support that the (110) -oriented LCO \nthin film exhibits the largest HS state ratio (Figure S5, Supporting  Information) . \n \nFigure 2. Crystallographic  orientation  dependent magnetic properties : a) M(T) curves of \nLCO thin films  on (100), (110), and (111) STO substrates along the in -plane direction. M(T) \ncurves were obtained during field-cooled cooling at 100 0 Oe of magnetic field. The i nset of \nFigure 2a  show s TC’s defined by linear extrapolation . b) M(H) curves  of LC O thin films  on \n(100), (110), and (111)  STO substrates along the in -plane direction at  2 K. M(H) curves of \nLCO thin films show FM hysteresis loops. Crystallographic  orientation -dependent FM \nproperties i ncluding c) saturation magnetization ( Ms), d) remanent magnetization ( Mr), and e) \nCurie temperature ( TC) are summarized for the LCO thin films.  \n \nThe electronic structures of LCO thin films provide a microscopic mechanism f or FM \nmodulation with distinctive crystallographic  symmetries. In particular , we performed \nspectroscopic ellipsometry to capture minute changes in the electronic band structures of the \nLCO thin films . The real part of the optical conductivity , σ1(ω), of the  (100) -, (110) -, and \n(111) -oriented LCO thin film s is shown in Figure  3a, which close ly resembl es σ1(ω) of bulk \n50 100 150 200 250 3000.00.10.20.30.4(a)m0H // ab  @ 1000 Oe\nFCM (mB Co-1)\nT (K) (100)\n (110)\n (111)\n-4 -2 0 2 4-1.0-0.50.00.51.0(b)\nm0H // ab \n@ 2 KM (mB Co-1)\nm0H (T) (100)\n (110)\n (111) 0.60.8(c)Ms (mB Co-1)\n0.30.4(d)Mr (mB Co-1)\n(100) (110) (111)75808590 (e)TC (K)\nOrientation40 60 80 1000.00.10.20.3\n76.1 KM (mB/Co)\nT (K)90.2 K81.5 K  \n8 \n \nLCO . [20, 52] Below 4 eV , σ1(ω) consists of two Lorentz oscillators centered at ~1.5, and ~ 3.0 \neV, labelled as the β (d-d transition within Co 3 d orbitals, brown dotted lines) and γ peaks  \n(charge transfer transition, O 2p → Co 3 d, green dotted lines ), respectively, as shown in \nFigure 3b.[52] The quantitative values of the peak positions ( ωj) and  spectral weight s (SWj) of \nthe β and γ peaks are summarized  in Fig ure 3c and 3d. The (110) -oriented LCO thin film \nexhibit ed a lower ωj for both the β and γ transition s compared to  those of the (100 )- and \n(111) -oriented thin films. In addition , the (110) -oriented thin film show ed a larger SW j value \nfor both the β and γ transitions compared to those of the  (100) - and (111) -oriented thin  films.   \n \nFigure 3. Optical properties and electronic structures : a) σ1(ω) of LCO thin films on \ndifferently oriented  STO substrates. For comparison, we displayed  the reported σ1(ω) of LCO \nthin film (black dotted line) and bulk (grey dotted line). b) σ1(ω) can be deconvoluted by the \nLorentzian oscillators corresponding to β (Co d-d transition) and γ (O 2p to Co 3d charge \ntransfer transition) peaks. Each c) ωj and d) SWj of the individual Lorentzian oscillators is \nshown as a function of the crystallographic  orientation of the LCO thin films.  \nLCO thin film: Reproduced with permission.[20] Copyright 2012 , ACS Publications . Bulk \nLCO: Reproduced with permission.[52] Copyright 2014, Nature portfolio . \n \nThe strongly intertwined electronic structure and FM properties of the LCO epitaxial thin \nfilms m odulated by crystallographic  orientation are illustrated in Figure 4 . When we consider \nan HS configuration within LCO, which is necessary for FM ordering, the net spin moment \ncan be evaluated from the energy states of the electron orbitals within the atomic picture.  In \nparticular, th is tendency can be qualitatively captured by taking into account  the energy cost \n(Δ) of the spin-state transition from a n LS to HS configuration . A larger spin moment, and \nhence stronger FM behavior , is expected for a smaller Δ. Δ can be defined as, Δ = ΔCF – Δex – \nW/2, where  ΔCF and Δex are the crystal -field splitting and  Hund  energ ies, respectively, and W \n01000(b)\n g (O 2 p ® Co 3 d)\nb (Co d-d) (100)\n01000 (110)\ns1(w) (W-1 cm-1)\n0 1 2 3 401000 (111)\nw (eV)0 2 4 6010002000\ns1(w) (W-1 cm-1)\nw (eV) (100)\n (110)\n (111)\n LaCoO3 thin film\n Bulk LaCoO3(a)\n1.41.52.62.7(c)\nwj (eV) g\n b\n(100) (110) (111)451516(d)\nSWj (eV2)\nOrientation  \n9 \n \nis the bandwidth.[29] Thus, the competition between ΔCF, Δex, and W determines  Δ. Both  the \nbond length  (rCo-O, average distance between Co and O ions) and bond angle ( θ, Co-O-Co \nangle ) influence Δ and the energy scales,[51, 53, 54]  in which ΔCF ~ rCo-O–5 and W ~ rCo-O–3.5cos(π \n− θ).[29, 55] Apparently, ΔCF has a stronger dependence on rCo-O, and hence, is a dominant factor \nfor determin ing Δ. For the CoO 6 octahedra, ΔCF is determined by the energy difference \nbetween the Co t2g and eg states , as depicted in Figure 4a. Because ΔCF is directly related to ωβ \nand partially related to ωγ, the observation  of the lowest ωβ and ωγ in the (110) -oriented LCO \nthin film (Figure 3c) indicates that the film has the lowest ΔCF, leading to a stronger FM spin \nordering  (small Δ, Figure 2).[55, 56 ] We can directly infer this insight from  the lattice structure, \nas the (110) -oriented LCO thin film has the  largest Vu.c. among the films studied (Figures 1c \nand 1d;  Table S1, Supporting Information). We further note that the charge transfer transition \nfrom the O 2 p to Co 3 d orbital s, depicted by SWγ, is the largest for the (110) -oriented LCO \nthin film.  We expect  that (110) -oriented LCO thin film to have  θ value close r to 180° than the \nother orientations, as it has  the largest Vu.c., that would lead  to a stronger charge transfer \nbetween Co and  O orbitals.[34, 35] It has been reported that a stronger charge transfer transition \nleads to enhanced  magnetic ordering in LCO thin films.[34, 57] In general, the facile spin and \ncharge transfer between Co 3 d and O 2 p orbitals introduces a strong FM superexchange \ninteraction.[34, 58] \n   \n024(100)pDOS (state/eV f.u.) HS Co3+ t2g    LS Co3+ t2g\n HS Co3+ eg     LS Co3+ eg\n O 2p0246\nwg(110)\nwb\n-3 -2 -1 0 1 2 3024(111)\nEnergy (eV)-0.5 2.0 2.50.02.55.0total DOS\n(state/eV f.u.)\nEnergy (eV) (100)\n (110)\n (111)LCO ( 100)LCO ( 110)\nLCO ( 111)(b) ΔCF ~ rCo-O-5\nVu.c.= 58.10 Å3\nεoop= −0.26 %\nVu.c.= 58.03 Å3\nεoop= −0.38 %\nVu.c.= 57.96 Å3\nεoop= −0.49 %(a)\nΔCF\nΔCFeg\nt2g\neg\nt2g\negσ\negπ\na1gΔCF\n[010][001]\n[100]\n[00][110]\n[10]\n[01][111]\n[11]\nCo O\n  \n10 \n \nFigure 4. Origin of the FM modulation in the LCO thin films: a) Schematic diagram of \nthe crystallographic  orientation -dependent spin -state configuration. Crystallographic  \nsymmetry modulates the orbital energy level for the Co3+ state to determine ΔCF. b) The pDOS \nand total DOS (inset of bottom panel) of  LCO thin films under different symmetry obtained \nby DFT calculation . Optical transitions β (brown arrow) and γ (green arrow) indicate the Co \nd-d transition and Co -O charge transfer transition, respectively . \n \nThe total (total DOS) and partial  density of states (pDOS) calculated by ab initio  density \nfunctional th eory (DFT) calculations consistently support the modulation in  ΔCF and charge \ntransfer between  the Co and O orbitals , as shown in Figure 4b. We used  the experimental \nlattice parameters obtained from the HAADF STEM analys es (Table S1 , Supporting \nInformation ) for the calculation s. Previously, it was found that the rock salt-type HS/LS -FM \norder  within the c-unit is responsible for the magnetic properties of epitaxially strained \nLCO.[23, 54] Therefore, w e used the rock  salt-type HS/LS -FM order, which  is the most stable \nstate regardless the crystallographic  symmetry (Table S2, Supporting Information). The \nmodulated optical transitions β (brown arrow) and γ (green arrow)  from the DFT calculation s \nare systematically  reproduced , as shown in the inset of the bottom most  panel of Figure 4b. As \nthe crystallographic  orientation changes from (111) to (100) to (110), the separation between \nthe Co -O hybridized states and the Co 3 d states increases on the order of 0.1 eV, which is \nconsistent with the results of σ1(ω) shown in Fig ure 3. \n \nRecently, diffusion Monte Carlo calculations show ed that the magnetic ground state of the \nLCO thin film highly depends on the La -La distance , with the HS/LS -FM order becoming the \nmost stable state below a specific compressive strain.[59] In terms of the exchange mechanism, \nthis order mainly results from correlation superexchange , which refer s to the magnetic \ninteraction whereby two electrons of O2− mediate the exchange coupling between neighboring \nHS and LS Co3+. More specific ally, the double occurrence of AFM superexchange coupling \nalong the HS Co3+ dx2−y2 – O2− pσ – LS Co3+ dx2−y2 – O2− pσ – HS Co3+ dx2−y2 pathway \nfacilitate s the FM ordering ( Figure S6, Supporting  Information). In this mechanism, the net \nspin moment of LS Co3+ dx2−y2 (which would be empty in  the ideal LS Co3+ configuration) \nmay be a descriptor for the superexchange coupling  strength because it reflect s the spin -\npolarized charge transfer from  neighboring  O2− pσ orbitals.  \n   \n11 \n \nTo quantitatively assess the spin -polarized charge transfer, we analyzed the eigen -occupations \nof the d orbital density matrix of the HS and LS Co3+ stabilized in the c-units of (100) -, \n(110) -, and (111) -oriented LCO thin films (Table 2). The eigen -occupations are fictitiou s \nauxiliary physical quantities, which often provide important insight s into the orbital states of \nthe transition metal ions.[60-62] The HS Co3+ was confirmed to have an HS d6 orbital \nconfiguration with all majority -spin orbitals and the minority -spin dxy orbital fully occupied , \nbut the other minority -spin orbitals almost empty. The result for LS Co3+ shows  that the three \nlower -energy d orbitals,  i.e., dxy, dyz, and dzx orbital s, are almost fully occupied, which is \nconsistent with LS Co3+. In contrast , the two higher -energy d orbitals ( the eg orbitals),  i.e., \ndx2−y2 and d3z2−r2 orbitals , are expected to be empty in the ideal LS  Co3+ orbital configuration, \nbut as mentioned above, they are partially occupied because of the spin -polarized charge \ntransfer from O2−. Interestingly, the net spin moment of LS Co3+ dx2−y2, estimated by the \ndifference between the eigen -occupations of its majority - and minority -spins, was the largest \nin the  LCO  with the (110)  orient ation  (Table 2). This DFT calculation result directly implies  \nthat the spin -polarized charge transfer, which determines the superexchange coupling  \nstrength , occurs more actively  in the (110) -oriented LCO thin films than in the (100) - or \n(111) - oriented films.  In addition, local magnetic moments of HS Co3+ within HS/LS -FM \norder are summarized in Table S3. Upon comparison, the local magnetic moment was the \nlargest in the c-unit stabilized the (110) -orientation, followed by that in the (100) - and (111) -\norientations, which also matches with experimental result s. \n \nTable 2. Orbital occupations of the HS and LS states for LCO thin films: A summary for \noccupations of t2g (dxy, dyz and dzx) and eg orbital ( dx2−y2 and d3z2−r2) for HS and LS states.  \n   HS LS \n   majority -spin minority -spin majority – \nminority spin  majority -spin minority -spin majority – \nminority spin  \n(100) eg 3z2-r2 1.0211  0.2333  0.7878  0.4811  0.3258  0.1553  \nx2-y2 1.0635  0.4318  0.6317  0.4886  0.3359  0.1527  \nt2g xy 0.9726  0.9721  0.0005  0.9737  0.9741  -0.0004  \nyz 0.9943  0.2213  0.7730  0.9801  0.9459  0.0342  \nzx 0.9932  0.2175  0.7757  0.9797  0.9437  0.0360  \n(110) eg 3z2-r2 1.0215  0.2304  0.7911  0.4872  0.3322  0.1550  \nx2-y2 1.0601  0.4279  0.6322  0.4806  0.3244  0.1562  \nt2g xy 0.9733  0.9730  0.0003  0.9745  0.9747  -0.0002  \nyz 0.9932  0.2154  0.7778  0.9791  0.9466  0.0325  \nzx 0.9920  0.2167  0.7753  0.9776  0.9435  0.0341  \n(111) eg 3z2-r2 1.0250  0.2318  0.7932  0.4957  0.3376  0.1581  \nx2-y2 1.0725  0.4331  0.6394  0.4972  0.3502  0.1470  \nt2g xy 0.9732  0.9727  0.0005  0.9750  0.9751  -0.0001    \n12 \n \nyz 0.9957  0.2333  0.7624  0.9814  0.9417  0.0397  \nzx 0.9957  0.2332  0.7625  0.9814  0.9417  0.0397  \n \n3. Conclusion  \nWe studied the unexpectedly large modulation of ferromagnetic properties , including a \nsizable increase in TC, in crystallographic orientation dependent -LCO thin films. By \nsystematically controlling the crystallographic symmetry, we obtained LCO thin films with \nvarious structural parameters. The c lear FM phases strongly coupled to the crystallographic  \nsymmetry demonstrate d that the (110) -oriented LCO thin film exhibit ed highly enhanced FM \nproperties. Optical spectroscopy and DFT  calculation s revealed that LCO thin films with \nsmall  ΔCF and large spin -polarized charge transfer between the Co 3 d and O 2 p orbitals \nfacilitate FM ordering in the (110) -oriented LCO thin film. T his study implies  that the \nsymmetry constraint is an efficient tuning parameter for  the exchange coupling strength in \nferromagnetic insulator LCO thin films , in which ferromagnetism  is highly robust against \nconventiona l strain due to its  ferroelastic nature.  \n \n4. Experimental Section  \nEpitaxial thin film growth : High -quality LCO epitaxial thin films were fabricated using PLE on \nconventionally HF -treated STO substrates with various orientation s [(100), (110), and (111) ]. \nAn excimer KrF laser (λ = 248  nm, IPEX864; Lightmachinery) with a fluence of 0.9 J  cm−2 and \na repetition rate of 2 Hz was used. The thin films were synthesized at 500 ºC under an oxygen \npartial pressure of 100 mTorr . Each set of thin films  with distinctive orientations  (with the same \nnumber of laser shots) was simultaneously fabricated.  \nAtomic and crystal structure characterization : The crystalline structure and lattice parameters \nof the LCO thin films were characteri zed using  high-resolution X RD (PANalytical X’Pert \nPro). HAADF STEM measurements were performed on a Nion UltraSTEM200 microscope \noperated at 200  kV. The microscope was equipped with a cold field emission gun and a \ncorrector of third - and fifth -order aberra tions for sub -angstrom resolution. The collection \ninner half -angle for the HAADF STEM w as 65 mrad.  Cross -sectional TEM specimens were \nprepared via ion milling after conventional mechanical polishing.   \nSurface characterization:  Atomic force microscopy measurements were performed using a \ncommercial system (Park Systems, NX10) to examine the surface topography and roughness.  \nMagnetization measurements : A magnetic property measurement system (MPMS3;  Quantum \nDesign)  was used to characterize the i n-plane magnetic properties of the thin films . The   \n13 \n \ntemperature -dependent magnetization , M(T), was measured from 300 to 2 K under a magnetic \nfield of 1000 Oe. The magnetic field -dependent magnetization , M(H), was measured at 2 K.  \nChemical composition characterization : X-ray photoelectron spectroscopy (XPS; AXIS \nSUPRA, KRATOS Analytical) with Al Kα radiation was used to study the chemical state and \ncomposition of the LCO thin films. All the X -ray photoelectron spectra were calibrated using \nthe C -C bonding peak (284.5 eV).  The Co 2 p and valence state s of the LCO thin films were \ndeconvoluted using Gaussian -Lorentz curves.  \nOptical spectroscopy : The optical conductivit y of the LCO thin films w as measured  using \nspectroscopic ellipsometry (J. A. Woollam Co., Inc.). A wavelength range from mid -infrared \nto UV (0.6 –6.2 eV) with incident angle s of 60 ° and 65 ° was used. We employed a three -layer \nmodel analysis (surface roughness (50% of material and 50% of void s), LCO, and STO layers) \nto obtain dielectric function s and optical conductivities of the thin films.  \nTheoretic al calculation s: Ab initio  DFT calculations were performed  using the Vienna ab \ninitio  simulation package (VASP) code.[63] The Perdew –Burke –Ernzerhof plus Hubbard \ncorrection (PBE  + U + J) was used for the exchange -correlation functional ,[64] in which the \ndouble -counting interactions were corrected using the full localized limit (FLL).[65] The \nvalues used for the on -site direct Coulomb paramete r (U) and the anisotropic Coulomb \nparameter ( J) were 4.5 and 1.0 eV, respectively.[54] A plane wave basis set at a cutoff energy \nof 600 eV was used to expand the electronic wave functions, and the valence electrons were \ndescribed using projector -augmented wave potentials. All atoms were relaxed by the \nconjugate gradient algorithms until none of the remaining Hellmann –Feynman forces acting \non any atoms exceeded 0.02eV Å−1. \n \nSupporting Information   \nSupporting Information is available from the Wiley  Online Li brary or from the author . \n \nAcknowledgements  \nDongwon Shin and Sangmoon Yoon contributed equally to this work. The authors thank M. \nF. Chisholm for technical assistance on TEM work . This work was supported by the  Basic \nScience Research Programs through the National Research Foundation of Korea ( NRF -\n2021R1A2C2011340 and NRF -2020K1A3A7A09077715 ). The microstructural analysis work   \n14 \n \nat ORNL was supported by U.S. DOE, Basic Energy Sciences, Materials Sciences and \nEngine ering Division.  \n \nReceived: ((will be filled in by the editorial staff))  \nRevised: ((will be filled in by the editorial staff))  \nPublished online: ((will be filled in by the editorial staff))  \n \nReferences  \n[1] K. Uchida, J. Xiao, H. 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Zaanen, Phys Rev B Condens Matter  1995 , 52, \nR5467.  \n    \n18 \n \n \n \nSupporting Information  \n \n \nTunable Ferromagnetism in LaCoO 3 Epitaxial Thin Films \n \nDongwon Shin1,#, Sangmoon Yoon2,3,#, Sehwan Song4, Sungkyun Park4, Ho Nyung Lee2, and \nWoo Seok Choi1,* \n \n1Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea  \n2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, \nTennessee 37831, United States  \n3Department of Physics, Gachon University, Seongnam, 13120, Republic of Korea \n4Department of Physics, Pusan National University, Busan 46241, Republic of Korea  \n \n*Corresponding author: choiws@skku.edu  \n    \n19 \n \nThickness dependent crystal  structures of epitaxial LCO thin films  \nHigh -quality LCO thin films were epitaxially grown on STO substrat es using PLE as shown \nin Fig ure 1a (30 nm), Fig ure S1a (10 nm) and Fig ure S1b (20 nm). Figure s S1c and 1d show \nthat 10 - and 20 -nm-thick LCO thin films also exhibit the same orientation -dependent trend in \nVu.c. and εoop, as the 30 nm -thick films, as shown in Figures 1c and 1d. \n \nAbsence of oxygen vacancy ordering  \nFigure S3a shows HAADF STEM image for the film -interior region that EELS linescan is \nperformed. The EELS spectra of twin -wall and bulk -like domains are compared in Figure \nS3band c. First, there is no noticeable difference between two regions at Co -L2, 3 edge. It is \nwidely known that HS and LS Co3+ is hard to be distinguished in Co -L2, 3 edge because of \nsubtle changes. This results also exclude the presence of oxygen vacancy ordering in pristine \ntwin-wall domains, because the red -shift of Co -L3 edge is not observed in the tw in-wall \ndomains. Second, the prepeak (denoted by the black arrow) emerges at the O -K edge spectra \nof twin -wall domain. It would be the fingerprint that electronic structure, presumably spin \nstates, is different between twin -wall domain and bulk -like domain s. However, it is not the \ndirect experimental evidence that ferromagnetism forms in dark -striped regions. Our scenario \nis supported by theoretical results, and also explains the trend of unconventional magnetic \nproperties. Atomic -scale electron magnetic ci rcular dichroism at low temperature under \ndevelopment may provide direct experimental evidence to end this issue.  \n \nEstimation of spin moment and HS/LS ratio  \nTo characterize  the FM properties of the LCO thin films, we analyzed the M(T) curves using  \nthe Curie -Weiss (C -W) law. According to the C -W law, the C-W temperature, spin moment, \nand ratio of the HS-state were calculated.  The inverse of the susceptibility H/M as a function \nof T above TC can be fitted by the C-W law H/M = (T – θ)/C, where C is a const ant and θ is \nthe C -W temperature (Fig ure S4a). All the LCO thin films exhibit ed a robust FM phase with θ \nsimilar to the TC values obtained in Fig ure 2e (Fig ure S4b). From the C values (slope of the \nC-W law fitting), we used μ = (3kBC/NA)1/2 ≈ 2.84 C1/2, where kB is Boltzmann’s constant and \nNA is Avogadro’s number, to determine the spin moment.[1] Furthermore, considering a \nsimple model of μ = g√S(S+1), where g ~ 2 is the gyromagn etic ratio of a free electron and S \nis the spin value, we determine the ratio of the HS -state, as summarized in Fig ure S4c. \n   \n20 \n \nDetermining Co valence state and the ratio of HS state  \nWe performed the X -ray photoelectron spectroscopy (XPS) to examine the binding energy of \nCo, which leads to the estimation of valence state as shown in Fig ure S5a. Distinct peaks of \nCo 2 p3/2 and 2 p1/2 are observed  without any satellite peaks, indicating the Co3+ state within the \nLCO thin films. The absence of Co2+ indicates  the absence of oxygen vacancies in the thin \nfilms. In addition, the amounts of LS and HS states were estimated. In Fig ure S5b, the peaks \nnear ~0.235, 0.935, 2.335 , and 4.735 eV correspond to the contributions of Co3+ HS, LS, O \n2p, and Co3+ 3d, respectively.[2, 3] From the integration of the HS and LS peaks, we obtained \nthe HS (LS) ratio of (100) -, (110) - and (111) -oriented LCO thin films to be 63.07% (36.93%), \n65.08% (34.92%) , and 61.16% (38.84%), respectively.  \n \n \n \n \n \n  \nFigure S1.  Crystal structures of the epitaxial LCO thin films (10 and 20 nm) grown on \nSTO substrates.  XRD θ-2θ scans near the (002), (110), and (111) Bragg planes of a) 10 nm \nand b) 20 nm LCO thin films (#) on (100), (110), and (111) STO substrates planes (*), \nrespectively . Crystallographic  orientation -dependent unit cell volume ( Vu.c.), and out -of-plane \nstrain (εoop = 100 × (dbulk − dfilm) / dbulk) of the c) 10- and d) 20-nm-thick LCO thin films,  \nrespectively.  \n  \n30 35 40 45 50(a)Intensity (arb. units)\n2q (degrees)10 nm\n (100)\n (110)\n (111)\n30 35 40 45 50(b)Intensity (arb. units)\n2q (degrees)20 nm\n (100)\n (110)\n (111)57.357.6(c)Vu.c. (Å3)10 nm\n-1.8-1.5-1.2\neoop (%)\n57.858.0(d)Vu.c. (Å3)20 nm\n(100) (110) (111)-0.9-0.6\neoop (%)\nOrientation  \n21 \n \n \nFigure S2. Surface roughness of STO substrates and LCO thin films. The rms roughness \nare 0.118, 0.157, and 0.100 nm for the (100) -, (110) -, and (111) -oriented STO substrates, \nrespectively. In addition, the rms roughness are 0.141, 0.226, and 0.148 nm for ( 100)-, (110) -, \nand (111) -oriented LCO thin films, respectively.  \n \n \n \n \nFigure S3. STEM -EELS (electron energy loss spectroscopy) results.  a) HAADF STEM \nimage for the film -interior region that EELS linescan is performed. EELS b) O K-edge and c) \nCo L-edge spectra are respectively shown for the twin -wall and bulk -like domains.  \n1 μmSTO (100) LCO (100)\nSTO (110) LCO (110)\nSTO (111) LCO (111)(a)\n(b)\n(c)\n  \n22 \n \nc\n c\n c cctcc\nabc \nc-unit (100)  (110)  (111)  \na [Å] 3.86 3.85 3.85 \nb [Å] 3.905  3.905  3.85 \nc [Å] 3.68 3.7 3.62 \nV [Å3] 55.47  55.63  53.66  \n \nTable S1.  Structural prop erties of the c-units with different crystallographic symmetry \nin the FM state.  Lattice parameters of a, b, c, and V of the c-units, shown for LCO thin films \nwith different crystallographic orientations . The schematics show distorted c-units by the \ncrystal lographic  symmetry modulation.  \n \n \n \nFigure S4. FM properties of LCO thin films from the Curie -Weiss law fitting.  a) The \ninverse of the susceptibility H/M as a function of T above TC. The solid lines indicate Curie -\nWeiss law fitting H/M = (T − θ) / C, where C is constant, θ is the C -W temperature. The \ntemperature -dependent magnetic susceptibility provides spin moment and ratio of the HS state \nestimated by the C -W law. The summary of t he orientation -dependent FM properties; b) θ and \nc) HS state ratio.   \n40 60 80 100010203040 (100)\n (110)\n (111)H/M (T (mB Co-1)-1)\nT (K)(a)\nq707580(b)\nq (K)\n(100) (110) (111)0.60.7(c)HS (%)\nOrientation  \n23 \n \n \nFigure S5. Characterization of Co valence state and spin states.  a) XPS spectra of Co 2 p \nlevels. b) XPS spectra near the valence -band regio n. c) Summary of the LS and HS ratio in the \nLCO thin films with distinct crystallographic orientations.  \n \n \n \n \n \n \n \n   \nAFM (HS)   FM (HS/LS)   FM (HS)   NM (LS)   \nEnergy  \n[meV/f.u.]  (100)  0 -55 274 297 \n(110)  0 -50 -25 277 \n(111)  0 -129 227 185 \n \nTable S2.  Total energy calculations of the c-units with different symmetry in four different \nmagnetic configurations.  The magnetic configurations are schematically demonstrated in the \nfirst raw of the table, where HS (orange sphere) and LS (blue sphere) of Co atoms and spins \n(black arrow) are shown. The energy of each magnetic configuration is represented relative to \nthat of the HS AFM configuration, which is set to zero energy.  \n820 810 800 790 780 770\nBinding energy (eV)Co 2 p\n(111)(110)Intensity (arb. units)Co3+ 2p1/2Co3+ 2p3/2(100) (a) (c) (b)  Co3+ 3d HS    O 2p \n Co3+ 3d LS    Co 3+ 3d\n Peak sum(100)Intensity (arb. units)(110)\n8 4 0\nBinding energy (eV)(111)(100) (110) (111)34363840606264\nLSSpin state (%)\nOrientationHS\n  \n24 \n \n \nFigure S6. Proposed FM superexchange  mechanism for the epitaxial LCO thin film.  \nSchematic representation of FM HS Co3+ - Co3+ superexchange interaction promoted via empty \nLS Co3+ and filled O 2 p orbitals. HS Co3+ dx2−y2 (blue and yellow lobes), O2− pσ (purple lobes) \nand spin (black arrow) are shown.  \n \n \n \n \n \n \n (100)  (110)  (111)  \nMHS Co3+ \n(μB) 2.986  2.993  2.976  \n \nTable S3.  The local magnetic moment of Co3+ within the c-units.    \n \n \n \n \n \n \nReference s \n[1] M. Herlitschke, B. Klobes, I. Sergueev, P. Hering, J. Perßon , R. P. Hermann, Physical \nReview B  2016 , 93. \n[2] T. Saitoh, T. Mizokawa, A. Fujimori, M. Abbate, Y. Takeda, M. Takano, Physical \nReview B  1997 , 55. \n[3] A. G. Thomas, W. R. Flavell, P. M. Dunwoody, C. E. J. Mitchell, S. Warren, S. C. Grice, \nP. G. D. Marr, D. E. Jewitt, N. Khan, V. R. Dhanak, D. Teehan, E. A. Seddon, K. Asai, \nY. Koboyashi, N. Yamada, Journal of Physics: Condensed Matters  2000 , 3, 9259.  \n \n \ndx2-y2pσ\nxy\nx\ny\nx\ny\nxy\nxy\npσHS Co3+ HS Co3+ LS Co3+ O O\ndx2-y2 dx2-y2"    },    {        "title": "1305.3158v1.Interface_induced_room_temperature_ferromagnetism_in_hydrogenated_epitaxial_graphene.pdf",        "content": "1\nInterface induced room-temperature \nferromagnetism in hydrogenated epitaxial graphene \n \nA.J.M. Giesbers1,*, K. Uhlířová2,M. Konečný1, 4, E. C. Peters3, M. Burghard3, J. Aarts2, and \nC.F.J. Flipse1,† \n \n1Molecular Materials and Nanosystems, Eindhoven University of Technology, 5600 MB \nEindhoven, The Netherlands. \n2Magnetic and Superconducting Materials, Leiden Institute of Physics, 2333 CA Leiden, The \nNetherlands. \n3Max-Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart \nGermany. \n4CEITEC BUT, Technická 10, 616 69 Brno, Czech Republic \n \nContact information \n*A.J.M.Giesbers@tue.nl, †C.F.J.Flipse@tue.nl  \n   2\nDue to the predominantly surf ace character of graphene, it is highly suitable for \nfunctionalization with external atoms and/or molecules leading to a plethora of new and \ninteresting phenomena. Here we show ferromagnetic properties of hydrogen-\nfunctionalized epitaxial graphene on SiC. Ferromagnetism in such a material is not \ndirectly evident as it is inherently compos ed of only non-magnetic constituents. Our \nresults nevertheless show strong ferromagnet ism, which cannot be explained by simple \nmagnetic impurities. The ferromagnetism is unique to hydrogenated epitaxial graphene \non SiC, where interactions with the interfacial buffer layer play a crucial role. We argue \nthat the origin of the observed ferromagn etism is governed by electron correlation \neffects of the narrow Si-dangling-bond (S i-DB) states in the buffer layer exchange-\ncoupled to localized states in the hydroge nated graphene layer. This forms a quasi-\nthree-dimensional ferromagnet with a Curie temperature higher than 300 K.  \n \nOwing to its capability of ballistic transport over micrometer distances1, as well as its very \nlong spin relaxation time and spin relaxation length2, 3, graphene represents a close-to-ideal \nmaterial for spintronic applications4. In this context, considerable effort has recently been \ndirected to rendering graphene ferromagnet ic via chemical modi fication. Thus far, \nferromagnetic order in graphene has been attained through covalent functionalization, \ninvolving the linkage of radical species like the spin-bearing carbon atom of an organic molecule or hydrogen atoms to the graphene layer\n5-17. Along these lines, functionalization of \nepitaxial graphene by aryl radicals has been reported to yield disordered magnetism, \ncomprising a mixture of ferromagnetic, supe rparamagnetic and antiferromagnetic regions18. \nWith the aid of combined atomic and magnetic force microscopy, it could be proven that these randomly dispersed regions are constituted by the attached moieties. This lack of a \nperiodic functionalization pattern of the gra phene sheet prevents the achievement of long \nrange ferromagnetic order, thus limiting the use of such samples in spintronic devices. \nFurthermore, room temperature ferromagnetism has been detected in partially hydrogenated \nepitaxial graphene grown on silicon carbide (SiC), and attributed to hydrogen monomers bonded to the graphene sheet\n12. Despite these accomplishments, however, both the \nmechanism underlying the ferromagnetic ordering, and the role played by the SiC substrate \nused for the epitaxial graphene growth, has not yet been clarified.  Here, we experimentally \ndemonstrate that spin ordering within hydrogena ted epitaxial graphene critically depends on \nthe presence of the underlying buffer layer. In a ddition, it is shown that the created magnetic 3\nareas are distributed over the entire graphene sheet, thus enabling to effectively tune the \noverall magnetization through the density of attached hydrogen atoms. \n \nTo explore the ferromagnetism in epitaxial graphene, we use samples grown on insulating \n6H-SiC substrates following the procedure described in ref. 19 (all samples originate from \nthe same wafer). The atomic force micrograph (A FM) of the sample surface (Fig. 1a) reveals \nterrace steps originating from a slight miscut of the SiC substrate. The terraces are typically \n3-5 um wide and approximately 10 nm high and are overgrown with a continuous carpet of \ngraphene20, 21. The inset in Fig. 1a shows a schematic  cross-section of the layer sequence at \nthe surface with a graphene layer on top of an  interfacial carbon layer (buffer layer) partly \nbonded to the SiC substrate. On the terrace edges an unintentional region of bilayer graphene has formed under the current gowth conditions\n19, discernible as brighter areas in the \ncorresponding AFM phase image (Fig. 1b). The presence of a small bilayer area is confirmed \nby Raman microscopy and low energy electron microscopy (LEEM) investigations (see \nsupplementary information Fig. S1) and has the same coverage in all samples. After growth, \nthe graphene samples are hydrogenated by an atomic hydrogen source in an ultrahigh vacuum \nchamber, for different exposure times. Succ essful hydrogenation is testified by an \nenhancement of the sp3-defect associated Raman D-peak, whose intensity increases with \ntreatment time (Fig. 1c and d), as discussed in more detail in the supplementary information. Increasing hydrogen exposure also leads to a rising C-H signal in x-ray photo-absorption \nspectra (XPS) (see supplementary information). The inset in Fig. 1d illustrates the hydrogen \nbonded on the top graphene layer. \n The magnetic properties of the hydrogenated graphene samples are determined using a \ncommercial SQUID with a sensitivity of 5·10\n-8 emu. All measurements are performed at \nroom temperature unless stated otherwise. Figure 2a shows the magnetization of an epitaxial \ngraphene sample hydrogenated for 3 minutes. The linear background is related to the bulk \nSiC diamagnetism and can be subtracted by a li near fit to the high field part of the curve \nwhere all other forms of magnetism are assumed to be saturated. The resulting diamagnetic \nsusceptibility χ = 0M/Hm , with m = (1.97 ± 0.05)·10-5 kg the sample mass and 0 = 4·10-7 \nTm/A the vacuum permeability, is χSiC = -(4.1 ± 0.1)·10-9 m3/kg, within its error in good \nagreement with literature ( χSiC = -4.01·10-9 m3/kg). Consistent values for χSiC, within the error \nrange, were found for all samples used in this work. The data obtained after subtraction of the 4\nSiC diamagnetic background are shown in Fig. 2b for three different temperatures. The \ncurves show a clear ferromagnetic response fr om the hydrogenated epitaxial graphene. The \nhysteresis loop shows a saturation magnetization of Ms = ± 27·10-7 emu, a remanent \nmagnetization of Mr = ± 7·10-7 emu and a coercive field of Hc = ± 91 Oe at 300 K. Upon \ndecreasing the temperature a small increase in the high field magnetization occurs. A similar \ntrend is observed for the coercive field and th e remanent magnetization (inset Fig. 2b). The \nmeasured saturation magnetization at room temp erature corresponds to a value of about 0.9 B \nper unit cell. \n \nFigure 2c compares the ferromagnetic signal  for the 3 min hydrogenated sample under in-\nplane magnetic field, along ( θ = 0 deg) and perpendicular ( θ = 90 deg) to the terraces, as well \nas for out-of-plane (OofP) orientation (inset Fig. 2c). A notable anisotropy can be discerned, \nwith easier magnetization along the terrace steps (black curve), as compared to perpendicular \nalignment (red curve) and the out-of-plane dir ection (blue curve). This difference manifests \nitself in a lower saturation magnetization and in the case of the out-of-plane signal in a more stretched hysteresis loop. The preferred ma gnetization along the terrace edges might result \nfrom the predominant formation of double site hydrogen sites aligned along the zigzag \ndirection of graphene\n22.The double H-sites show elongated sh aped charge structures of 3 nm \nor more with 6-fold symmetry coinciding with the 6 fold symmetry of the graphene \nhoneycomb lattice. In atomic resolution STM it was shown that the armchair edge of the graphene layer coincides with the SiC terrace structure\n23. Combined, these results lead to \nanisotropy between the terrace edge and perpendicular to the edge direction which could \nexplain the observed anisotropy in the magn etization. The out of plane magnetization \ncontribution is probably due to a non collinear spin orientation in the buffer layer, similar as \nfor the √3x√3R30 6H -SiC(0001) structure of SiC24, which will be discussed later.  \n \nTo tune the ferromagnetic signal we can use the hydrogen coverage, as is shown in Fig. 2d for hydrogenation times between 0 and 120 minutes. While the pristine graphene (0 min, \nblack curve) displays no magnetic signal, a short hydrogen exposure (0.5 min, red curve) \nresults in a clear ferromagnetic signal. From the corresponding hysteresis loop, a coercive \nfield of H\nc = ± 65 Oe and remanent magnetization of Mr = ± 2.4·10-7 emu is extracted. At \nhigh fields ( H = 3000 Oe), the magnetization reaches a saturation value of 14·10-7 emu. This \nsaturation magnetization, Ms, increases up to a treatment time of 3 minutes (27·10-7 emu), 5\nwhich is followed by a decrease for longer treatments, finally resulting in Ms (120 min) = \n13·10-7 emu. The same trend is observed in the coer cive field and the remanent magnetization \nfor the different samples.  \n \nIn order to determine the origin of the ferromagnetic behavior, we have investigated the \nmagnetization properties of severa l control samples (Fig. 2c). Firstly, a sample prepared in \nthe same manner as the 3 minute sample, except that the hydrogen bottle is kept closed, is \nfound to exhibit no ferromagnetic signal (red curve). Secondly, the same procedure is applied \nto an untreated bare SiC sample, which likewise does not lead to ferromagnetic signatures \n(not shown). Thirdly, to test the influence of the underlying substrat e, a quasi-freestanding \nmonolayer of graphene24 (QFMG) is used as a third control sample (Fig. 3f shows a \nschematic). It is obtained by growing only a buffer layer24-26 on the SiC, followed by \nhydrogen intercalation to passivate the SiC subs trate and turn the buffer layer into a QFMG. \nOwing to the reduced substrate interaction, QFMG is of superior quality compared to \nepitaxial graphene24. Pristine QFMG samples not subjected to hydrogenation (exemplified by \ngreen curve) do not display ferromagnetism as expected for pure graphene (in total two \nsamples were studied). Remarkably, also afte r 3 minutes of hydrogenation, no ferromagnetic \nsignal at room temperature emerges for such samples (hQFMG, schematic in Fig. 3f) (blue \ncurve representative for one out of two samples). The above findings highlight that the hydrogenated graphene is not ferromagnetic at  room temperature and the buffer layer is \ncrucial to render the epitaxial graphene ferromagnetic. Finally, in a fourth control experiment \nusing two buffer layer samples (one shown, cyan curve) and three hydrogenated buffer layer \nsamples (one shown, magenta curve) no ferromagne tic signal is detected (schematics of the \nsamples are shown in Fig. 3f). This absence consolidates that hydrogenated epitaxial \ngraphene requires both the hydrogenated graphene  and the underlying buffer layer to become \nferromagnetic. At low temperatures, the linear background magnetization, observable for \nboth the buffer layer samples and the hQFMG, leads to a smaller χ\nSiC compared to the pure \nSiC substrates. This difference hints toward an unsaturated low temperature paramagnetic contribution in these samples, akin to fluorinated graphene laminates\n27.The presence of \nlocalized paramagnetic-like states in the buffer layer was recently also suggested from spin \ntransport experiments in epitaxial graphene28. In our preliminary high magnetic field \nmagnetization measurements the paramagnetism of the bufferlayer is indeed confirmed, \nsaturating at H/T  25 kOe/K (see supplementary information Fig S4).  6\n \nFurther insight into the ferromagnetic properties  of the hydrogenated graphene is gained by \ndetecting the remanent magnetization with th e aid of Magnetic Force Microscopy (MFM) \n(see Fig. 3). By placing the sample briefly on either the south pole (-B) or north pole (+B) of \na permanent magnet prior to MFM measurements, we can magnetize the sample in \nrespectively a negative or positive out-of-plane remanent magnetization state as we observed in the SQUID measurements of Fig. 2c. Figure 3a and b show the magnetic signal of the \nsame area for the two magnetization directions with their respective cross-sections in panel c. \nThe highlighted dirt particle is an artifact due to crosstalk with the topography\n29 and serves as \na position marker on the sample. The topology of the sample is similar to Figure 1a and b. \nThe clear difference in MFM contrast between the single (1L) and bilayer (2L) areas, indicate \ntheir different magnetization. This might be due to different hydrogen coverages30, 31, in \naccord with the lower overall D-peak intensity on the bilayer regions observed in Raman \nimages. Other possible contributions are the different electronic structure of the bilayer \ngraphene, as well as different interactions among the hydrogen sites, or in the specific case of \nthe bilayer graphene between hydrogen sites and the buffer layer due to the increased \ndistance between the buffer layer and the hydrogenated layer. In the SQUID measurements \nthe bilayer areas will reduce the overall saturation magnetization, however since the bilayer \ncoverage is similar for all samples the results above are not affected. The switching of the out-of-plane remanent magnetization direction is clearly visible in the MFM cross-sections in \nFig. 3c. Specifically, after positive B-field magn etization, the MFM signal is positive and the \nsignal from the single layer is slightly larger th an that from the bilayer.  After negative B-field \nmagnetization the MFM signal has reversed the sign and the response from the single layer is again highest. These changes show that the color inversion between panel a and b is due to a \ncomplete flip of the magnetization direction, while the signal from the single layer is always \nhigher than that from the bilayer. That the flip is not symmetric around zero indicates a \nconstant background phase shift, and might be  attributed to electrostatic interactions \nsimultaneously probed by the metallic tip. Electric field microscopy (EFM) confirmed this magnetic-field independent electrostatic background\n32 (see supplementary figure S3).  \n \nThe MFM measurements corroborate the ferromagnetism of the hydrogenated epitaxial \ngraphene sample and show that the signal originates from the whole surface. Together with \nthe observed variation of the ferromagnetic strength with hydrogen coverage, the magnetic anisotropy and control sample magnetic measurements these results form a conclusive set of 7\nobservations which rule out any possible magnetic contaminations as the origin of the \nobserved magnetic behavior.  \n \nThe observed ferromagnetism in our hydrogenated epitaxial graphene is best interpreted in \nterms of an exchange coupled interaction between localized electron states of the buffer-layer \nand either spin-polarized localized states or th e mid-gap states of the hydrogenated graphene \nlayer22. The overall paramagnetic behavior of the buffer-layer indicates the presence of \nlocalized magnetic moments, which are the localized defect states forming an insulating \nbehavior as has been shown by STM and STS experiments33, 34.These states are attributed to \nSi-dangling bond (DB) states which probably be have as Hubbard Coulomb repulsion driven \nNeel like states, described by a non-collinear spin density wave, similar as was shown by \nAnisimov35 for a smaller unit cell reconstructed surface, the √3x√3R30 6H -SiC(0001). The \nSiC buffer-layer has a 6 times larger unit cell, 6 √3x6√3R30 SiC(0001) surface structure with \na band gap of 1 eV, formed by localized Si-DB states33, 34. Upon hydrogen adsorption on top \nof the graphene layer, carbon-hydrogen bonds are created, forming a mid-gap state22.This \nlocalized mid-gap state can be spin-split in filled and unfilled localized states close to the \nFermi-level due to the Coulomb interaction of the Si-DB states of the buffer layer, forming a \nquasi 3-dimensional ferromagnetic state with a Curie temperature of 300 K or higher. A \nsecond option to explain the FM behavior at 300 K is that the hydrogenated graphene layer is intrinsically ferromagnetic, but with a much lower Curie temperature due to the two-\ndimensionality, which would become quasi-three dimensional if the paramagnetic buffer-\nlayer will exchange couple to it. \n To conclude, hydrogenated epitaxial graphene shows a ferromagnetic behavior with a Curie \ntemperature higher than 300 K and a magnetic moment of 0.9\nB per effective carbon hexagon \narea. We have shown that both the hydrogen coverage and the buffer-layer with the Si-\ndangling bonds, play a crucial role for the hi gh temperature ferromagnetic properties. To \nexplain the ferromagnetism in our graphene system at room temperature, we tentatively \npropose an exchange coupled interaction between the Coulomb induced localized Si-DB \nstates of the buffer-layer and the localized mid-gap state or the two-dimensional \nferromagnetic hydrogenated graphene layer. The buffer-layer stabilizes the ferromagnetic \nbehavior at room temperature and this quasi three-dimensional system can explain the relatively high Curie temperature, higher than 300 K. The high Curie temperature in 8\ncombination with a small coercive field (100 Oe ) and high spin relaxation time in graphene \nmakes hydrogenated epitaxial graphene a favor able material for spintronic applications. \n \n \n \n  \n \n \n  9\nReferences \n1. 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Kuila, T. et al., Chemical functiona lization of graphene and its applications, Progress in \nMaterials Science  57, 1061–1105 (2012). \n15. Wei, W. and Qu, X., Extraordinary Physical Properties of Functionalized Graphene, \nSmall  8, 2138 (2012). 10\n16. Papagno, M. et al., Large Band Gap Opening between Graphene Dirac Cones Induced by \nNa Adsorption onto an Ir Superlattice, ACS Nano  6, 199 (2012). \n17. Suk, J. W., Piner, R. D., An, J., Ruoff, R. S., Mechanical Properties of Monolayer \nGraphene Oxide, ACS Nano  4, 6557 (2010). \n18. Hong, J. et al., Effect of Nitrophenyl F unctionalization on the Magnetic Properties of \nEpitaxial Graphene, Small 7, 1175–1180 (2011). \n19. Emtsev, K. V. et al., Towards wafer-size graphene layers by atmospheric pressure \ngraphitization of silicon carbide, Nature Mat.  8, 203 (2009). \n20. Norimatsu, W., Kusunoki, M., Fromat ion process of graphene on SiC (0001), Physica E  \n42, 694 (2010). \n21. Poon, S. W., Chen, W., Wee, A. T. S., and Tok E. S., Growth dynamics and kinetics of monolayer and multilayer graphene  on a 6H-SiC(0001) substrate, Phys. Chem. Chem. Phys.  \n12, 13522 (2010). \n22. Scheffler, M. et al., Probing Local Hydrogen Impurities in Quasi-Free-Standing \nGraphene, ACS Nano  6, 10590 (2012). \n23. Hu, T. W., Ma, D. Y., Ma, F., Xu, K. W., Preferred armchair edges of epitaxial graphene \non 6H-SiC(0001) by thermal decomposition, Appl. Phys. Lett.  101, 241903 (2012). \n24. Speck, F. et al., The quasi-free-standing na ture of graphene on H-saturated SiC(0001), \nAppl. Phys. Lett.  99, 122106 (2011). \n25. Ferralis, N., Maboudian, R., and Carraro, C., Ev idence of Structural Strain in Epitaxial \nGraphene Layers on 6H-SiC(0001), Phys. Rev. Lett.  101, 156801 (2008). \n26. Emtsev, K. V., Speck, F., Seyller, Th., Ley, L., Riley, J. D., Interaction, growth, and \nordering of epitaxial graphene on SiC{0001} surfaces: A comparative photoelectron spectroscopy study, Phys. Rev. B  77, 155303 (2008). \n27. Nair, R. R. et al., Spin-half paramagnetism in graphene induced by point defects, Nature \nPhys.  8, 199 (2012).  \n28. Maassen, T. et al., Localized States Infl uence Spin Transport in Epitaxial Graphene, \nPhys. Rev. Lett.  110, 067209 (2013).  \n29. Schendel, P. J. A. van, Hug, H. J., Stiefel, B., Martin, S., and Güntherodt, H-J., A method \nfor the calibration of magnetic force microscopy tips, J. Appl. Phys.  88, 435 (2000). \n30. Luo, Z. et al., Thickness-dependent reversible hydrogenation of graphene layers, ACS \nNano  3, 1781-1788 (2009). \n31. Ryu, S. et al., Reversible Basal Plane Hydrogenation of Graphene, Nano Lett.  8, 4597–\n4602 (2008). 11\n32. Martínez-Martín, D., Jaafar, M., Pérez, R., Gómez-Herrero, J., and Asenjo, A., Upper \nBound for the Magnetic Force Gradient in Graphite, Phys. Rev. Lett.  105, 257203 (2010). \n33. Nie, A., Feenstra, R. M., Tunneling spectros copy of graphene and related reconstructions \non SiC(0001), J.Vac.Sci.Technol.  A27, 1052 (2009). \n34. Cervenka, J., Ruit van der, K., Flipse, C. F. J., Effect of local doping on the electronic \nproperties of epitaxial graphene on SiC, Phys.Status Solidi A  207, 595 (2010); Cervenka, J., \nRuit van der, K., Flipse, C. F. J., Giant inelastic tunneling in epitaxial graphene mediated by \nlocalized states, Phys. Rev. B  81, 205403 (2010). \n35. Anisimov, V. I. et al., SiC(0001): A surface Mott-Hubbard insulator, Phys Rev. B 61, \n1752 (2000). \n   12\nMethods \nFor the magnetization measurements we used  a Magnetic Properties Measurement System \nfrom Quantum Design with Reciprocating samp le magnetometer. The best resolution was \nobtained with an amplitude of 3 cm and a frequency of 2Hz with a system resolution of 5·10-8 \nemu. The samples were placed into standard ø 6 mm PE straws which exactly fixated the 4x4 \nmm sample without further adhesives.   \nThe magnetic signal in the MFM measurements is acquired by a magnetic Co-Cr coated tip \nwith a force constant of k = 2.8 N/m and a quality factor Q = 226 in constant lift mode of 50 \nnm and an oscillation amplitude of 30 nm is used to avoid short range and van der Waals \ninteractions. The MFM signal, or phase-shift \nθ, is directly related to the magnetic force F \nbetween the tip and substrate: \n   Δߠ ൎ െொ\nௗி\nௗ௭ ,    \nleading to a force in the order of nano-Newtons.  \n  13\nFigures: \n \nFigure 1| Epitaxial graphene on 6H-SiC.  (a) Atomic force micrograph of the typical SiC \nterrace structure on top of which the graphene is grown. The inset shows a schematic of the \nsurface structure with from top to bottom a graphene layer, a buffer layer and the SiC \nsubstrate. The gray spheres represent carbon, the blue spheres silicon, and the yellow ovals the silicon dangling bonds. (b) Phase image showing the single layer graphene areas on top of \nthe terraces and the narrow bilayer regions at  the terrace edges. (c) Raman spectra of \nhydrogenated graphene with a treatment time t = 0 min. (black), 30 min. (red) and 120 min. \n(blue). Clearly visible is the upcoming D-peak  intensity with increasing treatment time. (d) \nD-peak intensity as a function of treatment time (the line is a guide to the eye). The inset shows the schematic bonding of hydrogen (orange spheres) to the graphene layer. \n  \n4 m 4 m\n1500 2000 2500 3000 04 0 8 0 1 2 02D G\n Intensity (A.U.)\nRaman shift (cm-1) 120 min\n 30 min\n 0 minDD-Peak Intensity  (A.U.)\nt (min)\n0 180 nm 90\na b\ncd\n14\n \nFigure 2| Magnetization of hydrogenated epitaxial graphene. (a) Room temperature \nmagnetization as a function of the applied magn etic field for hydrogena ted epitaxial graphene \ntreated for 3 minutes. The red line shows the di amagnetic contribution of the SiC substrate. \n(b) Temperature dependence of the magnetiza tion after subtraction of the diamagnetic \nbackground showing a clear ferromagnetic hysteresis loop. The inset shows a zoom of the \ncoercive field and remanent magnetization (3 min. treatment). (c) Direction dependence of \nthe magnetization (3 min. treatment). (d) Ferromagnetic signal for different hydrogen \ntreatment times, t = 0, 0.5, 3, 10, 30, 90, and 120 minutes. (e) Magnetization for different \ncontrol samples. (f) Schematic representati ons of the various control samples.  \n  -1000 0 1000-30-1501530\n90o0oOofP\n-1000 0 1000-30-1501530\n-1000 0 1000-30-150153002 0 0 0 4 0 0 0-30-1501530\n-150 0 150-10010\n0 2000 4000-300-200-1000100\n \n 10 min 0 min\n 0.5 min\n 3 min  \n 30 min\n 90 min\n 120 min\n M (10-7 emu)\nH (Oe)\nM (10-7 emu)\nH (Oe) =0o\n = 90o\n OofPM (10-7 emu)\n hQFMG\n BL\n hBL 3 min\n SiC+G\n QFMG\n  \nH (Oe)M (10-7 emu)\n 4K\n 77K\n 300K\n \nH (Oe)M  (10-7 emu)\n  \nH (Oe) M (10-7 emu)\nH (Oe) 3 min\n SiC diam.\nT = 300 Kabc\nde\nQFMG hQFMG\nBL hBLf15\n   \nFigure 3| Magnetic Force Micrograp hs of hydrogenated graphene.  (a) Magnetic force \nmicrograph after applying a positive magnetic field to the sample showing high and low remanent magnetization for single and bilayer respectively (scale: +3.3 ± 0.2 deg). (b) \nInversion of the remanent magnetization after applying a negative magnetic field to the sample (scale: -14.4± 0.2 deg). (c) Cross-s ection of the positive (a) and negative (b) \nmagnetization. We repeated the switching be tween positive and negative magnetization \nseveral times yielding the same result.  \n  ac\nb3.23.33.4\n0123-14.5-14.4-14.3MFM signal (deg)\n \nx (m)-B+B2L 1L 2L 1L 2L\n2L 1L 2L 1L 2L\n1 m2L 1L 2L 1L 2L\n+B\n‐B 1 m2L 1L 2L 1L 2L16\nSupplementary information \n \nSamples \nThe epitaxial graphene samples used in th is research are grown on insulating 6H-SiC \nsubstrates from II-VI Inc. following the procedur e described in ref. [S1]. All samples in this \narticle originate from the same wafer from II-VI inc. (EI1037-07-EV) that was diced in 4x4 \nmm pieces by laser cutting. After growth the sample surface consists of 1-5 m wide terraces \nwith a step height of about 10 nm as was shown by AFM in the main text. The terraces are \ncovered with single layer graphene with a narrow strip of bilayer graphene at the edges. In \nthe main text this was visualized by the AFM phase image, figure S1 shows a typical Raman map of the surface confirming the alternating structure by plotting the 2D-peak area and 2D-\npeak position, respectively Fig. S1a and b. The dark red areas indicate single layer graphene \nand the bright yellow areas indicate bilayer graphene as can be seen from the complete \nspectra at the red and blue dot displayed in Fig. S1c with matching colors. Figure S1d shows \na low energy electron microscopy (LEEM) image of a similar sample again showing the alternating single and bilayer area. The respective intensity versus energy spectra taken on \nthe numbered areas are shown in Fig. S1e. The number of dips in the spectra corresponds to \nthe number of layers.   \n \nHydrogenation of graphene \nAfter growth the samples are covered with hydr ogen by an atomic hydrogen source in an \nultra high vacuum chamber. The hydrogenation is done at a pressure of 2·10\n-6 mbar and \ndifferent coverages are achieved by varying the exposure time. After hydrogenation the \nRaman signature of the sample has drastically changed. Figure 1c (main text) shows the \nRaman spectrum before and after hydrogenation with a treatment time of 30 minutes and 120 \nminutes. The main difference between the hydrogenated spectra and the non-hydrogenated \nspectra is the appearance of an additional peak at 1375 cm-1 [S2], the so-called D-peak. The \nD-peak is caused by breathing modes of sp2 rings and is activated by a defect [S3]. The \nchemisorption of hydrogen on grap hene is expected to create sp3 bonds. These sp3 bonds act \nas defects and allow sp2 breathing modes, which show up as a D-peak in the Raman \nspectrum. The intensity of the D-peak is dire ctly related to the amount of defects present \n[S3], in our case the amount of hydrogen. Figure  1d (main text) shows the evolution of the D-\npeak intensity as a function of treatment ti me, which clearly shows the increased hydrogen 17\ncoverage with treatment time. Recently it was phenomenologically shown [S4] that the \nintensity ratio of the D and D’ peak (at 1620 cm-1) for sp3 defects was ~10 or larger. In our \nSiC-graphene samples the SiC background [S1] impedes a proper analysis of the D’ peak, \nhowever we repeated the same hydrogen treatments on graphene flakes on SiO 2 which indeed \nshowed a I(D)/I(D’) ratio of 11, corroborating that we indeed have hydrogenated our \nsamples.  X-ray photo-absorption spectroscopy \nIn addition, the presence of hydrogen is confirmed by X-ray photo-absorption spectroscopy (XPS) measurements (see Fig. S2a-c) [S5-S7]. Figure 2a shows the XPS spectrum for an \nuntreated sample (black line). In accordance to ref. [S5] the spectra can be fitted with four \nmain peaks, one originating from the SiC substrate (red), two from the two differently \nbonded carbon atoms in the bufferlayer (S1: magenta, S2: cyan) and one from the graphene \nlayer (green). After the sample is treated with hydrogen the main left peak starts to shift to \nhigher binding energies. This shift can be accounted for by adding a new peak for the binding \nenergy of carbon-hydrogen bonds at 284.74 ± 0.10 eV (brown peak in Fig. S2b and c) [S6]. \nThe intensity of this C-H peak increases w ith increasing treatment time, whereas the C-C \npeak intensity decreases as expect ed for higher hydrogen coverage.  \n \nElectric Force Microscopy \nTo rule out any electrostatic contributions to the MFM signal in the main text these interactions between a metallic AFM tip and the hydrogenated graphene sample, are \nmeasured by electric force microscopy (EFM). Figure S3a shows the AFM topography of the sample measured simultaneously with the EF M signal. Panel b and c show the EFM signal \nafter the sample was place on respectively the south pole (-B) and the north pole (+B) of a \npermanent magnet. Neither of the EFM measurements shows a response to the magnetic field \ndirection. This indeed confirms that the magnetic field dependent MFM results are related to \nthe remanent magnetization of the hydrogena ted epitaxial graphene. The electrostatic \ninteractions with the metallic tip only lead to a constant offset in the MFM measurements as \nwas indeed shown in the main text.  \n Paramagnetism in the buffer layer \nTo confirm if the smaller diama gnetic background (smaller than χ\nSiC), measured for the \nbuffer layer, is related to a pa ramagnetic signal in this layer we perform a high magnetic field 18\nmagnetization measurement at low temperatures (see Fig. S4). After subtraction of the \ndiamagnetic sample substrate background ( χSiC= -(4.1 ± 0.1)·10-9 m3/kg) a clear paramagnetic \nsignal becomes visible. As a function of H/T both the 2.5 K and 4 K curve coincide and can \nbe fitted with a single Brillouin function \nܯൌߤܬ݃ܰ ቈ2ܬ1\n2ܬ݄݊ݐܿቆሺ2ܬ1ሻݖ\n2ܬቇെ1\n2ܬ݄݊ݐܿ൬ݖ\n2ܬ൰ \nwith z = gJBH/kBT, N the number of spins, g the g-factor, J the angular momentum quantum \nnumber, B the Bohr magneton, H the magnetic field, kB the Boltzman constant and T the \ntemperature. The best fit was obtained for J = 1/2 with N and g as fitting parameters leading \nto N = (3.7 ± 0.3)·1014 spins and g = 2.7 ± 0.2. An enhancement of the g-factor of the buffer \nlayer was already shown by Maassen et al. [S8] but has also been observed in graphene [S9]. \nThe extracted number of spins N on our 4 mm × 4 mm sample is equivalent to 1.2 B per \nhexagon area and can originate from the Silicon dangling bonds [S10] but also from carbon \nvacancies [S11] and other defect structures present in the unscreened buffer layer. \n \nReferences supplementary information \nS1. Emtsev, K. V. et al., Nature Mat.  8, 203 (2009). \nS2. Elias, D. C. et al., Science  323, 610-613 (2009). \nS3. Ferrari, A. C., and Robertson, J., Phys. Rev. B  61, 14095 (2000). \nS4.Eckmann, A. et al., Nano Lett.  12, 3925 (2012). \nS5. Emtsev, K. V., Speck, F., Seyller, Th., Ley, L., Riley, J. D., Phys. Rev. B  77, 155303 \n(2008). \nS6. Haberer, D. et al., Adv. Mater.  23, 4497 (2011). \nS7. Luo, Z. et al., ASC Nano  3, 1781 (2009). \nS8. Maassen, T. et al., Localized States Influence Spin Transport in Epitaxial Graphene, Phys. Rev. Lett.  110, 067209 (2013).  \nS9. Kurganova, E. V. et al., Spin splitting in graphene studied by means of tilted magnetic-\nfield experiments, Phys. Rev. B  84, 121407(R) (2011). \nS10. Anisimov, V. I. et al., SiC(0001): A surface Mott-Hubbard insulator, Phys Rev. B 61, \n1752 (2000). \nS11. Yazyef, O. V., Emergence of magnetism in graphene materials and nanostructures, Rep. \nProg. Phys.  73, 056501 (2010). \n 19\n \n  20\nSupplementary figures \n \n \nFigure S1|  Number of graphene layers.  Raman map image of the 2D-peak Area (a) and \nPosition (b) showing the alternating single (red) and bilayer (yellow) areas. (c) Full Raman \nspectra at the red and blue locations indicated in  (a) and (b). In blue the typical spectra for a \nsingle layer graphene and in red for the bila yer graphene. (d) LEEM image showing a clear \ncontrast difference between the single layer graphene in the middle of a terraces and bilayer \ngraphene on the terrace edges. (d) LEEM intens ity as a function of energy on the positions as \nindicated in (d). The number of dips corres ponds to the number of layers, corroborating the \nalternating single and bilayer areas with incidental multilayer patches.  \n \n21\n\nFigure S2| X-ray photoemission spectra for hydrogenated graphene.  (a) spectra for an \nuntreated epitaxial graphene samples with the S1 (magenta) and S2 (cyan) peaks originating from the bufferlayer, the SiC (red) peak from the substrate and the CC (green) peak from the \nsp\n2 graphene carbon atoms. (b) spectrum with a hydrogen treatment of 30 min showing the \nupcomming CH (brown) peak of sp3 carbon-hydrogen bonds. (c) Same spectra after 120 min \nhydrogen treatment.   288 286 284 282\nCHSiC CC\nS1\ncb\n120 min30 minI (A.U.)0 mina\nS2 I (A.U.) I (A.U.)\nB.E. (eV)22\n \nFigure S3|  Electric Force Microscopy of hydrogenated graphene.  (a) Atomic force \nmicroscope height image taken with a Pt/Ir conductive tip. (b) Electrical force image after the \nsample was placed on the south pole of a permanent magnet. (c) Electrical force image after \nthe sample was placed on the north pole of a permanent magnet. (d) Cross section of the \nimages in (b) and (c) after respective negativ e (-B) and positive (+B)  magnetization showing \nno magnetic field response.  \n  \n+B1 mSL DL\n-BSL DL\n1 m\nSL DL\n1 m\n024-6.5-6.0-5.5-5.0-BEFM signal  (deg) \nx (m)+Ba c b d23\n \nFigure S4|  Paramagnetism in bufferlayer on SiC.  Magnetization of the buffer layer for 2.5 \nK and 4 K. The curves overlap as a function of H/T after subtraction of the diamagnetic \nbackground and show a clear paramagnetic signal saturating at high magnetic fields. The red \nline is a fit to the Brillouin function for J = 1/2. \n 01 0 2 00153045\n 2.5 K\n 4 K \n Brillouin fitM (10-7 emu)\nH/T (kOe/K)"    },    {        "title": "1401.0878v1.Hybrid_paramagnetic_ferromagnetic_quantum_computer_design_based_on_electron_spin_arrays_and_a_ferromagnetic_nanostripe.pdf",        "content": "arXiv:1401.0878v1  [quant-ph]  5 Jan 2014Hybrid paramagnetic-ferromagnetic\nquantumcomputerdesignbased on\nelectron spin arrays anda ferromagneticnanostripe\nJérômeTRIBOLLET∗\nInstitutdeChimiedeStrasbourg,StrasbourgUniversity,U MR 7177(CNRS-UDS),\n4 rueBlaisePascal,CS 90032,F-67081StrasbourgCedex, Fra nce\nE-mail: tribollet@unistra.fr\nAbstract\nDesigning an assembly of quantum nano-objects which can int eract between themselves\nand be manipulated by external fields, while staying isolate d from their noisy environment is\nthe key for the development of future quantum technologies, such as quantum computers and\nsensors. Electron spins placed in a magnetic field gradient, interacting by dipolar magnetic\ncouplings and manipulated by microwave pulses represent a p ossible architecture for a quan-\ntum computer. Here, a general design for the practical imple mentation of such nanodevice is\npresented and illustrated on the example of electron spins i n silicon carbide placed nearby a\npermalloyferromagnetic nanostripe. Firstly,theconfined spinwaveresonance spectrum ofthe\nnanostripe and the properties of its magnetic field gradient are calculated. Then, one shows\nhow to avoid microwave driven electron spin decoherence. Fi nally, one shows that decoher-\nence due to ferromagnetic fluctuations is negligible below r oom temperature for spins placed\nfar enough from the nanostripe.\n∗To whomcorrespondenceshouldbeaddressed\n1Introduction\nDuring last decades, the development of nanotechnologies h as allowed the experimental demon-\nstration of entanglement between the quantum states of diff erent pairs of nanosystems,1–4a key\npreliminarysteptowardsquantuminformationprocessing. However,upscalingthoseexperiments\nto at least ten coupled quantum systems, called quantum bits or qubits, on a single chip remains\nextremely challenging. This difficulty is linked to the need to couplequantum bitsovernanoscale\ndistancestoperformconditionallogicaloperations,exce ptwhensomealternativestrategyforlong\ndistance entanglement is available.5,6In the context of quantum bits encoded on electron spins in\nsolids, the easiest solution to couple them is to use the dipo lar magnetic coupling,3,4,7which has\nhowevertwodrawbacks. Firstly,itbecomestooweakbeyondt ennanometersforefficientquantum\nstates entanglement. Secondly, it is a permanent coupling, which requires methods to effectively\nswitch on/off those couplings. The up scaling problem is thu s directly related in this context to\nthe difficulty to precisely positiontheelectron spins on th e chip and also to manipulatethem with\nnanoscale resolution. State of art nanolithography and ion implantation methods used to build\nsilicon8or diamond3,4,9based small scale quantum devices with few coupled electron spins are\nstill far from reaching the ten qubits scale. Building solid state arrays of dipolar coupled electron\nspin qubits thus requieres an alternative nanotechnology a nd also a method and a nanodevice al-\nlowing to switch on/offthosedipolar couplings. Transmiss ionElectron Microscope(TEM) based\nnanofabrication methods with nanometer scale precision we re recently demonstrated with carbon\nnanotubes10andgraphenematerials,11andcouldprovidetherequiredalternativenanotechnology .\nAlso, few years ago, it was shown that selectiveon/off switc hing of the dipolar coupling is possi-\nble using sequences of microwave pulses applied to dipolar c oupled electron spins submitted to a\nmagnetic field gradient.7The problem was thus shifted to the production of strong magn etic field\ngradient. Inthefieldofquantumcomputingwithnuclearspin s,itwasproposedtousedysprosium\nnanoferromagnet toproduce ahugemagneticfield gradient.12However,in thecontextofelectron\nspin qubits, the nanoferromagnet also produces unwanted ad ditional decoherence of the electron\nspins13dueeithertoincoherentthermalexcitationortocoherentm icrowaveexcitationofthespin\n2waves14confined in the nanoferromagnet.15,16Here I present a quantum register design contain-\ning a permalloy ferromagnetic nanostripe and nearby arrays of electron spins in silicon carbide\n(SiC). The spin qubits are silicon vacancies17,18,20,21,23that could be created in SiC using TEM\nnanofabricationmethods.10,11,18,22,23Usingmagnetostatictheory,12theLandauLifschitzequation\nof motion of magnetization in the nanoferromagnet,16a one dimensional Schrodinger equation\nsolved numerically by a transfer matrix method,24and the density matrix theory of spin qubit de-\ncoherence,13Ishowhowtodesignsimultaneouslythemagneticfieldgradie ntandtheconfinedspin\nwaves spectrum of the nanoferromagnet in order to avoid addi tional decoherence. The designed\nquantum register could be operated with microwave pulses38and read out with high sensitivity\nusing ensemble optical measurements of the paramagnetic re sonance of silicon vacancy electron\nspinsinSiC.17,18\nDesignandfabricationofananodevicewithelectronspinar rays\nnearbya ferromagneticnanostripe\nThegeneral designproposedhereforquantuminformationpr ocessingcouldbeapplied,inprinci-\nple,toallkindofelectronspinqubitslikephosphorousdon orsinsilicon,8,34nitrogendonorsorNV\ncenters in diamonds,3,4,9transition metals ions or shallow donors in Zinc Oxide,13,25–28N@C60\nmolecules,29paramagnetic defects in graphene,33paramagnetic dangling bonds arrays on hydro-\ngenpassivatedsiliconsurface,31orsupramolecularassembliesofparamagneticmoleculessu chas\nneutral radical molecules30or copper phtalocyanine molecules,32as long as regular dense arrays\nof them could be produced at precise positions nearby a ferro magnetic nanostripe. Without loss\nof generality and to be more quantitative, I focus here on a na nodevice made with silicon carbide\nand permalloy materials and build with top-down methods. Th e successive steps of two possible\nnanofabrication processes allowing to build such a nanodev ice are described on figure 1a and 1b.\n3The resulting nanodevice is shown on figure 1c. Each of the thr ee SiC polytypes, 4H, 6H, and\n/MT97/MT47\n/MT49/MT47 /MT50/MT47 /MT51/MT47 /MT52/MT47 /MT53/MT47 /MT54/MT47\n/MT99/MT47\n/MT120/MT121\n/MT122\n/MT87\n/MT84/MT100/MT87/MT47/MT50\n/MT79/MT76/MT47/MT50/MT66/MT48/MT122\n/MT66/MT49/MT120/MT40/MT116/MT41/MT80/MT121\n/MT54/MT72/MT32/MT45/MT32/MT83/MT105/MT67\n/MT32/MT119/MT97/MT102/MT101/MT114/MT54/MT72/MT45/MT83/MT105/MT67/MT98/MT47\n/MT49/MT47 /MT50/MT47 /MT51/MT47 /MT53/MT47 /MT54/MT47 /MT52/MT47\nFigure 1: a/ First process: 6H-SiC wafer / step in wafer produ ced by focused ion beam (FIB)\nmilling/ epitaxial growth of isotopically purified (withou t nuclear spins) 6H-SiC on patterned\nwafer/ nanotrench created by focused ion beam (FIB) milling method35nearby the edge of the\nstep: depth W, widthT, lengthL/ ion assistedtrench filling methods36used to fill thenanotrench\nwith Permalloy / TEM production10,18,22,23,32of identical and parallel linear arrays of silicon va-\ncancies (white dots) aligned along the x axis. b/ Second proc ess: it combines electron beam\nlithography and permalloy plating:37first steps as in 1a and then: 1b4: resist deposition and pla-\nnarization,1b5: nanotrenchproducedinresistbyebeamlit hography,furtherfilledwithPermalloy\n(red), 1b6: removal of the resist, and then TEM production of silicon vacancies. c/ Design of the\nproposed SiC-Py quantum register corresponding to nanofab rication method a/. In the following\ncalculations: L=100µm,T=100nm,W=800nm.\n3C, has some paramagnetic defects related to silicon vacanc ies which could potentially encode a\ngood electron spin quantum bit.17,18,20,21,23Some silicon vacancies defects in 6H-SiC has a spin\nS=3/2 with an isotropic g factor g=2.0032 and a small zero field splitting.18Their spin state\ncan also be prepared by optical alignment18and it can be readout by the highly sensitive ODMR\nmethod(opticallydetectedmagneticresonance).19Itseemsthushighlyrelevanttoencodeanelec-\n4tronspinqubitonthiskindofsiliconvacancydefectin6H-S iC(forexampleatXbandorathigher\nmicrowave frequency, using:/vextendsingle/vextendsingleS=3\n2,MS=−3\n2/angbracketrightbig\n≡ |0/angbracketrightand/vextendsingle/vextendsingleS=3\n2,MS=−1\n2/angbracketrightbig\n≡ |1/angbracketright). In the\nfollowing calculations, one simplifies the spin qubit model by using a simple pseudo spin s=1/2\nwithag factorg=2, withoutanylossofgenerality.\nDipolarmagneticfieldgradientoutsidethenanoferromagne tand\neffectiveIsingcouplingbetweenelectronspinqubits\nThepracticalimplementationofsingleandtwoqubitsquant umgatesonchainsofdipolarcoupled\nelectron spin qubits, as those shown on figure 1, requires app ropriate microwave pulse sequences\nand a strong magnetic field gradient along the direction of th e spin chains of the nanodevice to\ntransform their dipolar coupling into an effectiveIsing li ke coupling.7As here I also consider the\npossibility to perform ensemble measurements of the spin st ates of the spin qubits, for example\nusing the highly sensitiveODMR method,19the magnetic field gradient produced by the permal-\nloy ferromagnetic stripe shown on figure 1 has not only to be st rong enough along x, but also\nto be enough homogeneous along y and z. Strong enough along x m eans here that the energy\ndifference between one qubit j (at position xj) and the next one j+1 (at position xj+1) along the\nspinchain has to bemuch larger thantheenergy associated to thedipolarcouplingbetween them,\nwhich depends on their relative distance linter. In the nanodevice described on figure 1, linteris\nthus also the periodicity of each spin chain. The condition f or an effective Ising coupling trans-\nlates into/vextendsingle/vextendsinglegµB/parenleftbig\nB0+Bdip,z/parenleftbig\nxj+1/parenrightbig/parenrightbig\n−gµB/parenleftbig\nB0+Bdip,z/parenleftbig\nxj/parenrightbig/parenrightbig/vextendsingle/vextendsingle>>µ0µ2\nBg2\n4πl3\ninter, or approximately,\n/vextendsingle/vextendsingle/vextendsinglegµBdBdip,z(x)\ndxlinter/vextendsingle/vextendsingle/vextendsingle>>µ0µ2\nBg2\n4πl3\ninter, whereBdip,z(x)is the value of the z component of the dipolar\nmagnetic field produced by the ferromagnetic nanostripe at p osition x, assuming that z≈0 for\neach spinqubit,and thatthey coordinateof each spinqubitd oes not matterforthevalueof Bdip,z\napplied to the qubit. The irrelevance of the y coordinate is d ue to the approximate invariance by\ntranslation of the gradient produced by the long ferromagne tic nanostripe (infinite stripe model).\nThis approximation is valid as long as the spin chains are cre ated far from the edges of the ferro-\n5magnetic stripe along the y direction. Assuming a permalloy ferromagnetic stripe with a length\nL=100µmalongtheyaxis,awidth T=100nmalongthexaxis,andadepth W=800nmalong\nthe z axis, further assuming that the ferromagnetic stripe i s fully magnetized along the z axis by\nthe staticapplied magneticfield B0, one can calculate using magnetostatictheory12the properties\nofthedipolarmagneticfield producedby thisnanoferromagn et,as shownonfigure 2.\n−1000 −500 0 500 1000−500005000\nz (nm)Bz (z)  (G)\n  \n−800 −400 0 400 800−4000−200002000\nx (nm)C(x)  (G.nm)\n−1000 −600 −200 200 600 1000−1000−5000100\nx (nm)Bz(x)  (G)\n−800 −400 0 400 800−2−1012\nx (nm)dBz(x)/dx  (G/nm)z= 0 nm z= 0 nmfull:    x = 0 nm\ndash:  x= 230 nm\nc/ d/b/ a/\nFigure 2: a/ Bdip,z(z,x)for two different x position: x=0 (continuous black) and x=xoptim=\n230nm(dash red). b/ C(x) =/integraltext+100\n−100dz/parenleftbig\nBdip,z(x,z)−Bdip,z(x,0)/parenrightbig\n. One defines the position\nxoptimbyC(xoptim) =0. For the permalloy ferromagnetic nanostripe considered h ere (see figure\n1), one finds xoptim=230nm. This position xoptimis the position where the homogeneity of\nBdip,z(x,z)is optimalalong thezdirection. c/ Bdip,z(z=0,x). d/ Gradient along x of thedipolar\nmagnetic field produced by the ferromagnetic nanostripe, as suming z=0 and y=0:dBdip,z(x)\ndx. For\ntheferromagneticPermalloynanostripe,thesaturationma gnetizationis Msat,Py,withµ0Msat,Py=\nBsat,Py=11300G.\nFigure2showsthataslongasthespinqubitsencodedontosil iconvacanciesinSiCarecreated\nfar from the edges of the ferromagnetic stripe along the y dir ection and close enough to the z=0\nplane, then the relevant z component of the dipolar magnetic field produced by the ferromagnetic\nnanostripe can be considered as only dependent on x, and thus its gradient can also be approxi-\nmatelyconsideredasbeingonedimensionalalongx. Onefinds hereBdip,z(xoptim=230nm,0)≈\n−676GanddBdip,z(xoptim=230nm,0)\ndx≈1.44G/nm. From those results, one can thus estimate\n6whether thecondition givenaboveto obtain an effectiveIsi ng coupling between the spin qubits is\nsatisfied. Assumingthat most of the spin qubits are created i n SiC by TEM at x positionscloseto\nxoptim=230nmand z=0, and further assuming a distance linter=5nmbetween successive spin\nqubits along the x direction, one finds that/vextendsingle/vextendsingle/vextendsinglegµBdBdip,z(x)\ndxlinter/vextendsingle/vextendsingle/vextendsingleis more than hundred times larger\nthan thedipolarcouplingenergyµ0µ2\nBg2\n4πl3\ninter. One could thus,at first sight,concludethat theproposed\nnanodevice described on figure 1, satisfies the requirements for the practical implementation of\nquantum information processing with dipolar coupled elect ron spin qubits manipulated by appro-\npriatesequencesofmicrowavepulses.7However,inordertoavoidmicrowavedrivendecoherence,\nitisthusabsolutelynecessary todesigntheferromagnetic nanostripesuchthatno spectraloverlap\nbetween the precession frequencies of the spin qubits and th e precession frequencies of the spin\nwavesconfined insidetheferromagneticnanostripe14–16occurs.\nDesigning the field sweep confined spin wave resonance spec-\ntrumtoavoidcoherentlydrivenelectronspinqubitdecoher ence\nOneassumeshere thatthepermalloyferromagneticnanostri peis almostfullymagnetizedat equi-\nlibrium along z at sufficiently low temperature and sufficien tly high magnetic field (see figure 1).\nThe excitation of a ferromagnetic spin wave implies that the magnetization inside the ferromag-\nnetic nanostripe is no more uniform. In other words that mean s that the dipolar magnetic field\nacting on the nearby electron spin qubits outside the nanofe rromagnet will become fluctuating,\nleadingto spinqubitdecoherence.13,38As itcan beseen on figure 2a(in thecase x=0)and on fig-\nure 3a, all the spinscontained insidethe ferromagneticnan ostripeproduce a very inhomogeneous\ndipolar magnetic field inside the ferromagnetic nanostripe itself. This leads to the confinement of\nthe ferromagnetic spin waves15along the z axis, mainly on the edges of the ferromagnetic nan os-\ntripe. Here I have developed a method to calculate the confine d spin wave resonance spectrum of\na long ferromagnetic nanostripe fully magnetized. First, t he Landau Lifschitz equation of motion\nofmagnetizationinthenanoferromagnet16islinearized. Then,theonedimensionaleigenenergies\n70 200 400 600 800−80−60−40−20020406080\nz* (nm)1D eigenenergie (micro−eV)\n0 200 400 600 800−5−4−3−2−101x 105\nz* (nm)1D wavefunction amplitude (a.u.)a/ b/\nFigure3: Theonedimensionaleigenenergiesandeigenfunct ionsofthespinwavesconfinedinthe\ninhomogeneouseffectivepotentialexistinginsidethefer romagneticnanostripealongthedirection\nz of the static magnetic field applied: a/ one dimensional eig enenergies of the spin waves along\nz axis (horizontal lines) represented on top of the inhomoge neous effective confining potential\ninside the nanostripe; b/ one dimensional eigenfunctions o f the spin waves confined along z (the\nout of equilibrium magnetization component δmx(z)has been plotted here). Here the permalloy\nferromagnetic nanostripe has the dimensions: length L=100µmalong the y axis, width T=\n100nmalong thex axis, and depth W=800nmalong thez axis. It was furtherassumed that the\nferromagneticstripeisfullymagnetizedalongthezaxisby thestaticmagneticfieldapplied B0,and\nthe magnetization of Permalloy at saturation was taken equa l toµ0Msat,Py=Bsat,Py=11300G.\nNote also that for numerical calculations, the new variable z∗was used, defined by z∗=z+450\n(in nm).\nand eigenfunctions of the spin waves confined in the inhomoge neous effective potential existing\ninside the ferromagnetic nanostripe along z are calculated , as it is shown respectively on figure\n3a and figure 3b, using an effective Schrodinger equation sol ved numerically by a transfer matrix\nmethod.24Then, the three dimensional eigenenergies of the confined sp in waves are determined,\nand finally the resonance fields at which the microwave magnet ic field of fixed frequency can ex-\nciteaconfinedspinwavearedetermined,simplybywritingth eenergyconservationrule. Thisrule\nsays that the energy of the microwave photon absorbed should be equal to the three dimensional\neigenenergie of the confined spin wave excited. It is importa nt to note that the combinationof the\ntheoretical and of the experimental field sweep microwaveab sorption spectrum, providesan indi-\n8rect method to determine the dipolar magnetic field produced by the ferromagnetic nanostripe on\nthespinqubits,becauseitisthesamemagnetizationdistri butionthatproducesthedipolarmagnetic\nfield insideand outsidethenanostripe. The full field sweep s pectrum expected forthe nanodevice\ndescribed on figure 1 is plottedon figure 4a. It includes thefie ld sweep spectrum expected for the\n0.80.911.11.21.31.4\nx 10400.20.40.60.811.21.41.61.82\nBo (Gauss)mw abs (a.u.)\n1.25 1.26 1.27 1.28 1.29\nx 10400.20.40.60.811.21.41.61.82\nBo (Gauss)mw abs (a.u.)a/ b/\nB\nCBA\nC\nFigure 4: a/ Full field sweep microwave absorption spectrum o f the nanodevice designed on\nfigure1,includingthespinwavesresonancespectrum(bluer esonancelines)andtheelectron spin\nqubits paramagnetic resonance spectrum (red resonance lin es), as it could be measured at a fixed\nmicrowave frequency ν=34GHzand with a varying magnetic field (see text for more details).\nb/ Zoom in the high magnetic field range of the field sweep micro wave absorption spectrum of\nthe proposed nanodevice, showing that no spectral overlap b etween the ferromagnetic spin waves\nand the electron spin qubits occur with this design (only red resonance lines are present in this\nfield range). Note that the linewidth and the strength of the r esonances (oscillator strength) were\narbitrarily chosen here, that means, without taking into ac count the relative symmetry of the spin\nwavemodeandofthemicrowavephotonmodeinsidethemicrowa vecavity,whichdeterminesthe\noscillatorstrength,inordertoseeanyspinwaveresonance satisfyingtheenergyconservationrule\nalone.\nferromagnetic nanostripe (blue resonance lines with reson ance magneticfields occurring between\n0.80and1.40Tesla). Italsoincludesthefieldsweepspectru mexpectedfor16electronspinqubits\ncreated at positions x close to xoptim=230nm, with x values comprised between x=199nm\nandx=274nm(ensemble C of 16 red resonance lines with resonance magneti c fields occurring\naround 1.28 Tesla). In order to see more clearly that those el ectron spin qubits resonance lines\n9(C) are shifted towards high magnetic field values by the dipo lar magnetic field Bdip,z(x,0)pro-\nducedbytheferromagneticnanostripeandpreviouslycalcu lated,Ialsoaddedonfigure4athefield\nsweep spectrumexpected foran electron spinqubitcreated a t infinitedistancefrom theferromag-\nnetic stripe along the x axis (single A red resonance line wit h resonance magnetic field occurring\naround 1.215 Tesla) and also the field sweep spectrum expecte d for an electron spin qubit created\nat distance x=500nmfrom the ferromagnetic stripe (single B red resonance line w ith reso-\nnancemagneticfieldoccurringaround1.250Tesla). Themost importantresultshownonfigure4a\nand highlighted by figure 4b, is the possibility to design the electron spin based quantum register\nproposed here such that no spectral overlap occur between th e 16 resonances lines of the shifted\nelectron spin qubits and the resonance lines of the two confin ed spin waves of the ferromagnetic\nnanostripeoccurring at thehighest magneticfield values,t he so called edgespin wavemodes, oc-\ncurring here at Q band, at 1.227 Tesla and at 1.362 Tesla. This provides a magnetic field interval\nof1350 G free of anyspin waveresonance. Further assumingan inhomogeneouslinewidthbelow\n1 G for each silicon vacancy electron spin qubit in nuclear sp in free isotopicallypurified SiC, one\ncouldhopetobuildidenticalparallelarrays of675electro nspinqubitsalongthexaxis. However,\nstate of art pulse EPR spectrometers operating at Q band have a resonator bandwidth of around\n1 GHz, corresponding here to a magnetic field interval of arou nd 350 G, in which only around\n175 electron spin qubits could be manipulated using selecti ve microwave pulses.38It seems thus\npossible in principle to build a model quantum register of th is kind having tens of electrons spins\nqubitswith paramagneticresonance lines not overlappingt heconfined spin waveresonance lines,\nthusavoidinganycoherentlydrivenspindecoherence.\nInvestigation of the spin qubit decoherence process due to i nco-\nherentmagneticfluctuationsofthenearbynanoferromagnet\nTillnow,Ihavenotconsideredthefactthattheincoherentt hermalexcitationofferromagneticspin\nwaves can produce some time dependent fluctuating dipolar ma gnetic field outside the ferromag-\n10netic nanostripe, at the positions where the electron spin q ubits are created. Those fluctuations\nalways exist in a ferromagnetic nanostripe at equilibrium a t a non zero temperature T. As it is\nalso known from the standard density matrix theory of electr on spin decoherence,13any fluctuat-\ning magnetic field lead to electron spin decoherence. Thus he re, one expects that the fluctuating\ndipolarmagneticfieldactingonelectronspinqubitswillpr oduceanewspindecoherenceprocess,\nwhich must be added to the other electron spin decoherence pr ocesses, which are intrinsic to the\nsiliconvacanciesinthesiliconcarbidematrixalone. Thea nalysisofthisnewdecoherenceprocess\nrequires the knowledge of the quantum correlation function s of this time dependent fluctuating\ndipolar magnetic field outside the ferromagnetic nanostrip e.13This is a complicated problem not\naddressed to date. Here I present the results of a theoretica l study of this problem based on the\nformalismsofthesecondquantizationofspinwavesandofth edensitymatrix(detailswillbepub-\nlished elsewhere). The theory developed shows that this new electron spin decoherence process\ndepends mainly, on the saturation magnetization of the nano ferromagnet, on the temperature, on\nthespectraldetuningbetweentheparamagneticresonanceo ftheelectronspinqubitandtheferro-\nmagnetic resonance of the nanostripe, and also on the distan cex of the qubit to the ferromagnetic\nnanostripe, as long as the qubit has a position assumed far fr om the edges of the ferromagnetic\nnanostripealongthey andz axis(y«L,z«W and x>T/2). Result softhecalculation oftheelectron\nspin coherence time T2and of the electron spin longitudinalrelaxation time T1of an electron spin\nqubitofthenanodeviceshownonfigure1andoperatedatmicro waveQbandarepresentedonfig-\nure 5. At helium temperature, at microwaveQ band, and with an applied magneticfield ofaround\n1.28 Tesla, one finds: T1(x=230nm,2K) =3.4sandT2(x=230nm,2K) =6.4s. In those\nexperimental conditions, this new decoherence process for the electron spin qubitsshould thus be\nnegligible compared to other decoherence processes intrin sic to natural SiC. However, it could\nalso become the dominant decoherence process for electron s pin qubits embedded in a nuclear\nspinfreeisotopicallypurifiedSiCmatrix,assumingintrin sicspincoherencetimeinSiCsimilarto\nthosefoundinSi,intherangeoffewseconds.34Thissuggestthatquantuminformationprocessing\nwith tens of electrons spins qubits encoded onto silicon vac ancies in SiC, coherently manipulated\n1120025030035040045050055010−210−1100101102103\n x (nm)T1 or T2  (s)\n05010015020025030010−1100101\nT (K)T1 or T2  (s)\n  \n30 K\n300 K3 Kb/ a/\nx = 230 nm\nFigure 5: Effects of the thermal fluctuations of the dipolar m agnetic field produced by the ferro-\nmagnetic nanostripe on the electron spin coherence time T2 a nd on the electron spin longitudinal\nrelaxationtimeT1ofanelectronspinqubitofthenanodevic eshownonfigure1. a/T2(thinlines)\nand T1 (thick lines), as a function of the qubit position x for three different temperatures T= 3 K,\n30 K, and 300 K. b/ T2 (squarre) and T1 (cross), as a function of the temperature of the nanode-\nvice, assumingan electron spin qubit position x=xoptim=230nm, where its resonant magnetic\nfield at Q band (34 GHz) is around 1.28 Tesla, and also assuming y=z=0. Permalloy nanostripe\ndimensions considered: length L=100µmalong the y axis, width T=100nmalong the x\naxis, and depth W=800nmalong the z axis, as shown on figure 1. The estimated T2 and T1\nare valid as long as the shifted spin qubit resonance line do n ot overlap any confined spin wave\nresonanceline,asitisrequieredtousethenanodeviceshow nonfigure1. Itwasalsoassumedthat\ntheferromagneticresonance linewidthofthepermalloynan ostripeisequal to3 µeVatQ band.\nby microwave pulse and submitted to the strong dipolar magne tic field gradient of the permalloy\nferromagneticnanostripeshouldbepossible,atleastatcr yogenictemperatures. Also,atroomtem-\nperature one finds T1(x=230nm,300K) =25msandT2(x=230nm,300K) =47ms. Thus,\nthosetheoreticalestimates,combinedwiththerecentobse rvationofroomtemperaturespincoher-\nenceofsomesiliconvacanciesinSiCover80 µs,18suggestthataSiC-Permalloyquantumregister\nsuch as the one designed here should keep its quantum coheren ce properties at room temperature\nat a sufficiently high level to perform fundamental studies o f quantum entanglement over tens of\ndipolarcoupledelectrons spinsqubitsofsiliconvacancie sat roomtemperature.\n12Discussion\nSiC is a very attractive material for the practical implemen tation with a top-down approach of\nthe general quantum register design proposed here, firstly d ue to the long intrinsic electron spin\ncoherence time of its paramagnetic defects18similar to the one in silicon,34secondly due to the\npossibilitytocreatesiliconvacanciesarraysinSiCusing advancedTEMmethods,andthirdlydue\nto the possibilities for optical initialization and optica l readout of the spin state of some silicon\nvacancies in SiC.17,18Note that a dysprosium ferromagnetic nanostripe would prod uce a three\ntimes bigger magnetic field gradient than in the case of perma lloy. However, one expects with\ndysprosium an electron spin decoherence rate at least one or der of magnitude larger than in the\ncase of permalloy at cryogenic temperature due to its three t imes larger saturation magnetization\nand its muchlarger ferromagneticresonance linewidth. In a ddition,Dysprosiumisno moreferro-\nmagnetic at room temperature, contrary to permalloy, which suppresses any possibility for room\ntemperature quantum information processing. Thus the hybr id paramagnetic-ferromagnetic nan-\nodevicepresentedhere,basedontheSiCmaterialwidelyuse dforhightemperatureelectronicsand\non the permalloy material widely used for magnetic elements in computers , seems to be an ex-\ncellent choice for quantum information processing with dip olar coupled electron spins selectively\nand coherentlymanipulatedbyresonant microwavepulses.\nReferences\n(1) R. Blatt et al.,Nature 453,1008 (2008).\n(2) M.Neeley et al.,Nature 467,570 (2010).\n(3) F. 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Leeet al.,Nature 466, 845(2010).\n(17) A.L.Falk et al.,NatureCommunications 4, 1819(2013).\n(18) D.Riedel etal., Phys.Rev.Lett. 108,226402(2012).\n(19) H.Kraus et al., NaturePhysics 2826,1(2013).\n(20) H.Itoh et al.,IEEE TransactionsonNuclearScience 37, 1732(1997).\n(21) T.Wimbaueretal., Phys.Rev.B 56,7384(1997).\n(22) J.W.Steeds et al., DiamondRelat. Mater. 11,1923(2002).\n(23) J.Lefevreet al., J. ofAppl.Phys. 105,23520(2009).\n(24) C. Jirauschek,IEEE J. ofQuantumElectronics 45, 1059(2009).\n(25) J.Tribolletet al., EurophysicsLetters 84, 20009(2008).\n(26) S.T.Ochsenbein etal., NatureNanotechnology 6,112(2011).\n14(27) R.E. Georgeet al., Phys.Rev.Lett. 110, 27601(2013).\n(28) K.Benzid etal., EurophysicsLetters 104,47005(2013).\n(29) W.Harneit, Phys.Rev.A 65, 32322(2002).\n(30) A.Tamuliset al.,Chem. Phys.Lett. 436,144(2007).\n(31) S.R. Schofield etal., NatureCommunications 4,1649, 1(2013).\n(32) M.Warner et al.,Nature 503,504 (2013).\n(33) R.R. Nairet al., Naturephysics 8, 199(2012).\n(34) A.M.Tyryshkinet al., Naturematerials 11, 143(2012).\n(35) R. Menzel et al.,Appl.Surf. Science 136, 1(1998).\n(36) A.Cheryl, ElectronicDesign 47,38 (1999).\n(37) X.Yang et al., J.Vac. Sci. Technol.B 21, 3017(2003).\n(38) A. Schweiger and G. Jeschke, Principles of pulse electron paramagneticresonance (Oxford\nUniversityPress, Oxford, UK ;New York,2001).\nCompetingfinancialinterests\nTheauthordeclare thathehas nocompetingfinancial interes ts.\n15"    },    {        "title": "0704.3975v3.Spectroscopy_and_critical_temperature_of_diffusive_superconducting_ferromagnetic_hybrid_structures_with_spin_active_interfaces.pdf",        "content": "arXiv:0704.3975v3  [cond-mat.supr-con]  12 Nov 2007Spectroscopy and critical temperature of diffusive superco nducting/ferromagnetic\nhybrid structures with spin-active interfaces\nAudrey Cottet\nLaboratoire de Physique Th´ eorique et Hautes ´Energies, Universit´ es Paris 6 et 7,\nCNRS, UMR 7589, 4 place Jussieu, F-75252 Paris Cedex 05, Fran ce and\nLaboratoire de Physique des Solides, Universit´ e Paris-Su d,\nCNRS, UMR 8502, F-91405 Orsay Cedex, France\n(Dated: July 3, 2021)\nThedescriptionoftheproximityeffectinsuperconducting/ ferromagnetic heterostructuresrequires\nto use spin-dependent boundary conditions. Such boundary c onditions must take into account the\nspin dependence of the phase shifts acquired by electrons up on scattering on the boundaries of fer-\nromagnets. The present article shows that this property can strongly affect the critical temperature\nand the energy dependence of the density of states of diffusiv e heterostructures. These effects should\nallow a better caracterisation of diffusive superconductor /ferromagnet interfaces.\nPACS numbers: 73.23.-b, 74.20.-z, 74.50.+r\nI. INTRODUCTION\nWhen a ferromagnetic metal ( F) with uniform mag-\nnetization is connected to a BCS superconductor ( S),\nthe singlet electronic correlations characteristic of the\nSphase can propagate into Fbecause electrons and\nholes with opposite spins and excitation energiesare cou-\npled coherently by Andreev reflections occurring at the\nS/Finterface. Remarkably, the ferromagnetic exchange\nfield induces an energy shift between the coupled elec-\ntrons and holes, which leads to spatial oscillations of the\nsuperconducting order parameter in F1,2. This effect\nhas been observed experimentally through oscillations\nof the density of states (DOS) in Fwith the thickness\nofF3, or oscillations of the critical current I0through\nS/F/Sstructures4,5,6,7, with the thickness of For the\ntemperature. The oscillations of I0have allowed to ob-\ntainπ-junctions8, i.e. Josephson junctions with I0<0,\nwhich could be useful in the field of superconducting\ncircuits9,10. A reentrant behavior of the superconducting\ncritical temperature of S/Fbilayers with the thickness\nofFhas also been observed11. At last, some F/S/F\ntrilayers have shown a lower critical temperature for an\nantiparallelalignment ofthe magnetizationsin the two F\nlayers as compared with the parallel alignment12, which\nshould offer the possibility of realizing a superconducting\nspin-switch13,14.\nFor a theoretical understanding of the behavior of\nS/Fhybrid circuits, a proper description of the inter-\nfaces between the different materials is crucial. For a\nlong time, the only boundary conditions available in\nthe diffusive case were spin-independent boundary con-\nditions derived for S/normal metal interfaces15. Re-\ncently, spin-dependentboundaryconditionshavebeenin-\ntroduced for describing hybrid diffusive circuits combin-\ning BCS superconductors, normal metals and ferromag-\nnetic insulators16. These boundary conditions take into\naccount the spin-polarization of the electronic transmis-\nsion probabilities through the interface considered, but\nalso the spin-dependence of the phase shifts acquired byFIG. 1: a. Diffusive F/S/F trilayer consisting of a BCS\nsuperconductor Swith thickness dSplaced between two fer-\nromagnetic electrodes F1andF2with thickness dF. In this\npicture, the directions of the magnetic polarizations in F1and\nF2are parallel [antiparallel], which corresponds to the con-\nfiguration C=P[AP]. b.S/Fbilayer consisting of a BCS\nsuperconductor Swith thickness dS/2 contacted to a ferro-\nmagnetic electrode Fwith thickness dF.\nelectronsupon transmissionorreflectionby the interface.\nThe first property generates widely known magnetoresis-\ntance effects17. The second property is less commonly\ntakeninto account. However, the Spin-Dependence ofIn-\nterfacial Phase Shifts (SDIPS) can modify the behavior\nof many different types of mesoscopic circuits with ferro-\nmagneticelements, likethoseincludingadiffusivenormal\nmetal island18, a resonant system19,20, a Coulomb block-\nade system20,21,22, or a Luttinger liquid23. It has also\nbeen shown that the SDIPS has physical consequences\ninS/Fhybrid systems16,24,25,26. One can note that, in\nsome references, the SDIPS is called ”spin-mixing angle”\nor ”spin-rotation angle” (see e.g. Refs. 24,26). In the\ndiffusive S/Fcase, the spin-dependent boundary condi-\ntions of Ref. 16 have been applied to different circuit\ngeometries27,28,29,30,31but the only comparison to exper-\nimental data has been performed in Ref. 29. The authors2\nof this Ref. have generalized the boundary conditions of\nRef. 16 to the case of metallic S/Finterfaces with a su-\nperconducting proximity effect in F. They have showed\nthat the SDIPS can induce a shift in the oscillations of\nthe critical current of a S/F/SJosephson junction or of\nthe DOS of a S/Fbilayer with the thickness of F. Signa-\ntures of this effect have been identified in the Nb/PdNi\nhybrid structures of Refs. 3,5. Nevertheless, the problem\nof characterizing the SDIPS of diffusive S/Finterfaces\nhas raised little attention so far, in spite of the numerous\nexperiments performed.\nA good characterization of the properties of diffusive\nS/Finterfaces would be necessary for a better control\nof the superconducting proximity effect in diffusive het-\nerostructures. The present article presents other conse-\nquences of the SDIPS than that studied in Ref. 29, which\ncould be useful in this context. In particular, the SDIPS\ncan generate an effective magnetic field in a diffusive Sin\ncontact with a diffusive F, like found for a ballistic Sin\ncontact with a ferromagnetic insulator24. This effective\nfield can be detected, in particular, through the DOS of\nthe diffusive Flayer, with a visibility which depends on\nthe thickness of F. A strong modification of the varia-\ntions of the critical temperature of diffusive S/Fstruc-\ntures with the thickness of Fis also found. These effects\nshould allow to characterize the SDIPS of diffusive S/F\ninterfaces through DOS and critical temperature mea-\nsurements, by using the heterostructures currently fabri-\ncated. The calculations reported in this paper are also\nappropriate to the case of a diffusive Slayer contacted\nto a ferromagnetic insulator ( FI).\nThis paper is organized as follows: Section II presents\nthe initial set of equations used to describe the het-\nerostructures considered. The case of F/S/Ftrilayers\nis mainly addressed, but the case of S/F(orS/FI) bi-\nlayers follows straightforwardly. Section III specializes\nto the case of a weak proximity effect in Fand a super-\nconducting layer with a relatively low thickness dS≤ξS,\nwithξSthe superconducting coherence length in S. The\nspatial evolution of the electronic correlations in the S\nandFlayers is studied in Section III.A. The energy-\ndependent DOS of S/Fheterostructures is calculated in\nSection III.B. Section III.C considers briefly the limit of\nS/FIbilayers. Section III.D discusses SDIPS-induced\neffective field effects in other types of systems. Section\nIII.E compares the present work to other DOS calcu-\nlations for data interpretation in S/Fheterostructures.\nCritical temperatures of S/Fcircuits are calculated and\ndiscussed in Section III.F. Conclusions are presented in\nSection IV. Throughout the paper, I consider conven-\ntional BCS superconductors with a s-wave symmetry.\nII. INITIAL DESCRIPTION OF THE PROBLEM\nThis article mainly considers a diffusive F/S/F tri-\nlayer consisting of a BCS superconductor Sfor−dS/2<\nx < d S/2, and ferromagnetic electrodes F1forx∈{−dS/2−dF,−dS/2}andF2forx∈ {dS/2,dS/2+dF}\n(see Figure 1.a). The magnetic polarization of the two\nferromagnets can be parallel (configuration C=P) or\nanti-parallel (configuration C=AP), but the modu-\nlus|Eex|of the ferromagnetic exchange field is assumed\nto be the same in F1andF2. Throughout the struc-\nture, the normal quasiparticle excitations and the su-\nperconducting condensate of pairs can be characterized\nwith Usadel normal and anomalous Green’s functions\nGn,σ= sgn(ωn)cos(θn,σ) andFn,σ= sin(θn,σ), with\nθn,σ(x) the superconducting pairing angle, which de-\npends on the spin direction σ∈ {↑,↓}, the Matsubara\nfrequency ωn(T) = (2n+1)πkBT, and the coordinate x\n(see e.g. Ref. 32). The Usadel equation describing the\nspatial evolution of θn,σwrites\n/planckover2pi1DS\n2∂2θn,σ\n∂x2=|ωn|sin(θn,σ)−∆(x)cos(θn,σ) (1)\ninSand\n/planckover2pi1DF\n2∂2θn,σ\n∂x2= (|ωn|+iEexσsgn(ωn))sin(θn,σ) (2)\ninF1andF2, withDFthe diffusion constant of the fer-\nromagnets and DSthe diffusion constant of S. The self-\nconsistentsuperconductinggap∆( x) occurringin(1) can\nbe expressed as\n∆(x) =πkBTλ\n2/summationdisplay\nσ∈{↑,↓}\nωn(T)∈{−ΩD,ΩD}sin(θn,σ) (3)\nwith Ω Dthe Debye frequency of S,λ−1=\n2πkBTBCS\nc/summationtext\nωn(TBCS\nc)∈{0,ΩD}ω−1\nnthe BCS coupling\nconstant and TBCS\ncthe bulk transition temperature of\nS. I assume ∆ = 0 in F1andF2. The above equations\nmustbe supplemented with boundaryconditionsdescrib-\ning the interfaces between the different materials. First,\none can use\n∂θn,σ\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=±(dS/2+dF)= 0 (4)\nfor the external sides of the structure. Secondly, the\nboundary conditions at the S/Finterfaces can be cal-\nculated by assuming that the interface potential locally\ndominates the Hamiltonian, i.e. at a short distance it\ncauses only ordinary scattering (with no particle-hole\nmixing) (see e.g. Ref. 33). This ordinary scattering\ncan be described with transmission and reflection am-\nplitudes tS(F)\nn,σandrS(F)\nn,σfor electrons coming from the\nS(F) side of the interface in channel nwith a spin\ndirection σ. The phases of tS(F)\nn,σandrS(F)\nn,σcan be\nspin-dependent due to the exchange field EexinF1(2)\nand a possible spin-dependence of the barrier poten-\ntial between SandF1(2). Boundary conditions tak-\ning into account this so-called Spin-Dependence of In-\nterfacial Phase Shifts (SDIPS) have been derived for3\n|tS\nn,↑|2,|tS\nn,↓|2≪1 and a weakly polarized F16,29. When\nthere is no SDIPS, the boundary conditions involve the\ntunnel conductance GT=GQ/summationtext\nnTnand the magne-\ntoconductance GMR=GQ/summationtext\nn(|tS\nn,↑|2− |tS\nn,↓|2), with\n↑(↓) the majority(minority) spin direction in the Felec-\ntrode considered, GQ=e2/h, andTn=|tS\nn,↑|2+|tS\nn,↓|2.\nIn the case of a finite SDIPS, one must also use the\nconductances GF(S)\nφ= 2GQ/summationtext\nn(ρF(S)\nn−4[τS(F)\nn/Tn]),\nGF(S)\nξ=−GQ/summationtext\nnτS(F)\nnandGF(S)\nχ=GQ/summationtext\nnTn(ρF(S)\nn+\nτS(F)\nn)/4, withρm\nn= Im[rm\nn,↑rm∗\nn,↓] andτm\nn= Im[tm\nn,↑tm∗\nn,↓]\nform∈ {S,F}. In the following, I will focus on the ef-\nfects ofGF\nφandGS\nφ, and I will assume GMR,GF(S)\nξand\nGF(S)\nχto be negligible, like found with a simple barrier\nmodel in the limit Tn≪1 andEex≪EF29. In this\ncase, one finds that the boundary conditions for the S/F\ninterface located at x=xj= (−1)jds/2, withj∈ {1,2},\nwrite\nξF∂θF\nn,σ\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxj= (−1)jγTsin[θF\nn,σ−θS\nn,σ]+iγF\nφǫC,j\nn,σsin[θF\nn,σ]\n(5)\nand\nξF∂θF\nn,σ\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxj−ξS\nγ∂θS\nn,σ\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxj=/summationdisplay\nm∈{F,S}iγm\nφǫC,j\nn,σsin[θm\nn,σ]\n(6)\nwhere the indices SandFindicate whether θn,σ\nand its derivative are taken at the SorFside\nof the interface. These equations involve the re-\nduced conductances γT=GTξF/AσFandγF(S)\nφ=\nGF(S)\nφξF/AσF, the barrier asymmetry coefficient γ=\nξSσF/ξFσS, the superconducting coherence lengthscale\nξS= (/planckover2pi1DS/2∆BCS)1/2, the magnetic coherence length-\nscaleξF= (/planckover2pi1DF/|Eex|)1/2, the gap ∆ BCSfor a bulk S,\nthe normal state conductivity σF(S)of theF(S) mate-\nrial, and the junction area A. The coefficient ǫC\nn,jtakes\ninto account the direction of the ferromagnetic polar-\nization of electrode Fjin configuration C ∈ {P,AP}.\nOne can use the convention ǫP,j\nn,σ= (−1)jσsgn(ωn) and\nǫAP,j\nn,σ=σsgn(ωn), in which the factor sgn( ωn) arising\nfrom the definition of θn,σand the terms ( −1)jandσ\narising from the boundary conditions have been included\nforcompactnessoftheexpressions. Notethatinthepres-\nence of a finite SDIPS i.e. γF(S)\nφ/ne}ationslash= 0, the right hand side\nof equation (6) is not zero contrarily to what found in\nthe spin-degenerate case15. In the general case, γF\nφand\nγS\nφare different (see e.g. Appendix A). This implies that,\nwith the present approximations, each interface is char-\nacterized by three parameters: γT,γF\nφandγS\nφ. For the\nsake of simplicity, symmetric F/S/Ftrilayers are con-\nsidered, so that γT,γF\nφandγS\nφare the same for the two\nS/Finterfaces.\nBefore working out the above system of equations, it is\ninteresting to note that the angle θn,σcalculated in theparallel configuration C=Pforx >0 also corresponds\nto the angle θn,σexpected for a S/Fbilayer consisting of\na superconductor Sfor 0< x < d S/2, and a ferromag-\nnetic electrode Fforx∈ {dS/2,dS/2+dF}(Figure 1.b).\nIn practice, using a F/S/Fgeometry can allow one to\nobtain more information on spin effects, as shown below.\nIII. CASE OF A THIN SUPERCONDUCTOR\nAND A WEAK PROXIMITY EFFECT IN F\nA. Spatial variations of the pairing angle\nI will assume that the amplitude of the superconduct-\ning correlations in F1(2)is weak, i.e. |θn,σ| ≪1 for\nx∈ {−dS/2−dF,−dS/2}andx∈ {dS/2,dS/2 +dF}\n(hypothesis 1) so that one can develop the Usadel equa-\ntion (2) at first order in θn,σ. This leads to\n∂2θn,σ\n∂x2−/parenleftBigg\nkC,j\nn,σ\nξF/parenrightBigg2\nθn,σ= 0 (7)\nin the ferromagnet Fj, withj∈ {1,2},kC,j\nn,σ= (2[iηC,j\nn,σ+\n|ωn/Eex|)])1/2andηC,j\nn,σ= (−1)jǫC,j\nn,σ. Combining Eqs.\n(4) and (7), one finds in Fj\nθn,σ(x) =θF\nn,σ(xj)cosh/parenleftbigg/bracketleftbig\nx−(−1)j/parenleftbig\ndF+dS\n2/parenrightbig/bracketrightbigkC,j\nn,σ\nξF/parenrightbigg\ncosh/parenleftBig\ndFkC,j\nn,σ\nξF/parenrightBig\n(8)\nThis result together with boundary condition (5) leads\nto\nθF\nn,σ(xj) =γTsin(θS\nn,σ(xj))\nγTcos(θSn,σ(xj))+iγF\nφηC,j\nn,σ+BC,j\nn,σ(9)\nwithBC,j\nn,σ=kC,j\nn,σtanh[dFkC,j\nn,σ/ξF]. Thisallowstorewrite\nthe boundary condition (6) in closed form with respect\ntoθS\nn,σ, i.e.\nξS∂θS\nn,σ\n∂x/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nxj= (−1)j+1LC\nj,n,σsin[θS\nn,σ(xj)] (10)\nfor theS/Finterface located at x=xj, with\nLC\nj,n,σ\nγ=γT(BC,j\nn,σ+iγF\nφηC,j\nn,σ)\nγTcos(θSn,σ)+BC,j\nn,σ+iγF\nφηC,j\nn,σ+iγS\nφηC,j\nn,σ(11)\nInthe following, Iwill assume |Eex| ≫∆BCSlikeinmost\nexperiments, so that kC,j\nn,σ= 1+iηC,j\nn,σ. I will also assume\ndS/ξS≤1, so that one can use, for the Cconfiguration\nand−dS/2< x < d S/2,\nθn,σ(x) =/tildewideθC\nn,σ−αC\nn,σ(x/ξS)−βC\nn,σ(x/ξS)2(12)4\nwith/vextendsingle/vextendsingle/vextendsingleθn,σ(x)−/tildewideθC\nn,σ/vextendsingle/vextendsingle/vextendsingle≪1 (hypothesis 2 ). Note that al-\nthough experiments are often performed in the limit of\nthick superconducting layers dS> ξS, assuming dS≤ξS\nis not unrealistic since one can obtain diffusive supercon-\nducting layerswith athickness dS∼ξS(seee.g. Ref. 34).\nFurthermore, using relatively low values of dSis more\nfavorable for obtaining efficient superconducting spin-\nswitches35. Hypothesis 2 allows one to develop sin( θn,σ)\nandcos(θn,σ)atfirstorderwithrespectto θn,σ−/tildewideθC\nn,σinS.\nAccordingly, I will neglect the space-dependence of ∆( x)\nand assign to it the value ∆Cin configuration C. Note\nthat I do not make any assumption on the value of the\nangle/tildewideθC\nn,σ, which is not necessarily close to the bulk BCS\nvalue. The coefficient LC\nj,n,σis transformed into its conju-\ngate when the magnetic polarizationofelectrode Fjis re-\nversed. Therefore, I will note LP\n1,n,σ=LP(AP)\n2,n,σ=Lσ\nnand\nLAP\n1,n,σ= (Lσ\nn)∗. The aboveassumptionsleadto αP\nn,σ= 0,\nβP\nn,σ=4Lσ\nnsin(/tildewideθC\nn,σ)\n4δS+cos(/tildewideθCn,σ)δ2\nSLσn(13)\nαAP\nn,σ=iIm[Lσ\nn]4sin(/tildewideθC\nn,σ)−cos(/tildewideθC\nn,σ)δ2\nSβAP\n4+2Re[ Lσn]cos(/tildewideθCn,σ)δS(14)\nand\nβAP\nn,σ=/parenleftBig\n8Re[Lσ\nn]+4|Lσ\nn|2cos(/tildewideθC\nn,σ)δS/parenrightBig\nsin(/tildewideθC\nn,σ)\n8δS+6Re[Lσn]cos(/tildewideθCn,σ)δ2\nS+|Lσn|2cos2(/tildewideθCn,σ)δ3\nS\n(15)\nwithδS=dS/ξS. On the other hand, from (1), one finds\nβC\nn,σ=∆Ccos(/tildewideθC\nn,σ)−|ωn|sin(/tildewideθC\nn,σ)\n2∆BCS(16)\nThe comparison between equations (13), (15) and (16)\nallowsoneto find /tildewideθC\nn,σasa function of ∆C. Then, one has\nto calculate ∆Cby using the self-consistency relation (3).\nI will study below the DOS and the critical temperature\nfollowingfromthese Eqs., in alimit whichleads tosimple\nanalytical expressions.\nB. Low-temperature density of states of S/F\nheterostructures\nThe DOS of the ferromagnets F1andF2of Figure 1.a\ncan be probed at x=±(dF+dS/2) by performing tun-\nnelling spectroscopy through an insulating layer. So far,\nthis quantity has been less measured3,36,37,38than criti-\ncal temperatures or supercurrents. However, this way of\nprobing the superconducting proximity effect is very in-\nteresting because it allows one to obtain spectroscopic\ninformation. It has been shown that the zero-energy\nDOS of a Flayer in contact with a superconductor os-\ncillates with the thickness of F. For certain thicknesses,this zero-energy DOS can even become higher than its\nnormal state value N039,40,41,42, as shown experimen-\ntally in Ref. 3. Remarkably, the SDIPS can shift these\noscillations29. Although the energy dependence of the\nDOS of diffusive S/Fstructures has raised some theoret-\nical and experimental interest, the effect of the SDIPS on\nthis energy dependence has not been investigated so far.\nFor calculating analytically the low-temperature\nDOS of the structure of Fig. 1.a., one can as-\nsumeγ2\nTcos(θS\nn,σ(xj))/|(γTcos(θS\nn,σ(xj)) +iγF\nφηP,2\nn,σ+\nBP,2\nn,σ)(γT+iγS\nφηP,2\nn,σ)| ≪1 (hypothesis 3 ), which leads\ntoLσ\nn=γ/parenleftBig\nγT+iγS\nφσsgn(ωn)/parenrightBig\n. This hypothesis is e.g.\nvalid for dF≥ξFand any value of γS(F)\nφifγTis rel-\natively small (see e.g. Fig. 2). I will also assume\nthat the lowest order terms in δSprevail in the numera-\ntors and denominators of expressions (13) and (15), i.e.\nβP\nn,σ∼ Lσ\nnsin(/tildewideθC\nn,σ)δ−1\nSandβAP\nn,σ∼Re[Lσ\nn]sin(/tildewideθC\nn,σ)δ−1\nS\n(hypothesis 4 ). Taking into account hypothesis 3 and\nγ∼1, hypothesis 4 is valid provided γTand/vextendsingle/vextendsingle/vextendsingleγS\nφ/vextendsingle/vextendsingle/vextendsingleare\nrelatively small compared to 1. Importantly, hypotheses\n3 and 4 are less restrictive regarding the value of γF\nφ. Ac-\ncordingly, I will often use γT,/vextendsingle/vextendsingle/vextendsingleγS\nφ/vextendsingle/vextendsingle/vextendsingle≪/vextendsingle/vextendsingle/vextendsingleγF\nφ/vextendsingle/vextendsingle/vextendsinglein the Figs. of\nthis paper. Hypotheses 3 and 4 lead to\n/tildewideθC\nn,σ= arctan/parenleftbigg∆C\n|ωn|+ΩCn,σ/parenrightbigg\n(17)\nwith\nΩP\nn,σ= 2∆BCSγ(γT+iγS\nφσsgn(ωn))δ−1\nS(18)\nand\nΩAP\nn,σ= 2∆BCSγγTδ−1\nS (19)\nEquations (17) and (18) show that an effective magnetic\nfieldHeffappears in the Slayer in the Pconfiguration,\ndue toγS\nφ/ne}ationslash= 0. From Eqs. (17) and (18), Heffcan be\nexpressed as\ngµBHeff=/planckover2pi1vS\nF\nds2GS\nφℓS\ne\n3σSA= 2ES\nTHGS\nφ\nGS(20)\nHere,vS\nFandℓS\nedenote the Fermi velocity and mean\nfree path in S, andGS=σSA/dSandES\nTH=/planckover2pi1DS/d2\nS\ndenote the normal state conductance and the Thouless\nenergy of the Slayer of Fig. 1.a. From Equations (17)\nand (19), the effective field effect disappears in the AP\nconfiguration because the two contacts are assumed to\nbe symmetric, and therefore, their contributions to Heff\ncompensate each other in the APcase. Note that, in\nprinciple, the γF\nφterm can induce an effective field ana-\nlogue to Heffin theFlayer, but this effect is not rele-\nvant in the regime studied in this paper (see Appendix\nB). The effects of Heffon the DOS of the structure will\nbe investigated in next paragraphs. In order to calculate5\nFIG. 2: Zero energy density of states NF(ε= 0) at x=\ndF+dS/2 as a function of dF, for the F/S/Fstructure of\nFigure 1.a in the Pconfiguration, with γ= 1,γT= 0.12,\nanddS/ξS= 1. The different curves correspond to different\nSDIPS parameters: γF\nφ=γS\nφ= 0 (black dotted curve), {γF\nφ=\n−2.1,γS\nφ= 0}(black full curve), and {γF\nφ=−2.1,γS\nφ= 0.12}\n(red full curve). The low-temperature self-consistent gap ∆P\nfound for these three different cases is indicated at the bott om\nright of the Figure. Using a strong value for γF\nφallows one\nto change significantly the phase of the oscillations of NF\nσ(0)\nwithdF(effect similar to Ref. 29). One can check that the\nhypotheses 1 to 4 are valid for the parameters used in this\nFigure. Note that for the different cases considered here, th e\ncritical temperature TP\ncof the structure (see Sec. III.F) is\nsuch that 0 .615TBCS\nc< TP\nc<0.635TBCS\nc. The four red\npoints correspond to the red full curves shown in Fig. 3.\n∆C, one has to combine the self-consistency relation (3)\nwith Eq. (17), which gives, at low temperatures,\nRe[log(ΩC\nn,σ+/radicalBig/parenleftbig\nΩCn,σ/parenrightbig2+(∆C)2\n∆BCS)] = 0 (21)\nThis equation can be solved numerically. The resulting\n∆Cis independent from the values of nandσused in\nEq. (21). Then, the value of the pairing angle θn,σin\nthe ferromagnets can be found by using Eqs. (8), (9),\n(12) and (17). Note that for γ≪1, the above Eqs.\nare in agreement with formula (5) of Ref. 29, obtained\nwith rigid boundary conditions, i.e. θn,σequal to its bulk\nvalue at the Sside. The energy dependence of θn,σcan\nbe found by performing the analytic continuation ωn=\n−iε+ Γ and sgn( ωn) = 1 in the above equations. The\nrate Γ = 0 .05 is used to account for inelastic processes43.\nAt last, the density of states Nσ(x,ε) at position xfor\nthe spin direction σ∈ {↑,↓}can be calculated by using\nNσ(x,ε) = (N0/2)Re[cos[ θn,σ(x)]], where N0/2 is the\nnormal density of states per spin direction.\nIn the following, I will mainly focus on NF(ε) =/summationtext\nσ∈{↑,↓}Nσ(x=dF,ε). Figure 2 shows the variations\nof the zero energy density of states NF(ε= 0) as a func-\ntion ofdF, for interface parameters γT=γS\nφ= 0.12 and\nγF\nφ=−2.1. Importantly, the value γT= 0.12 seems real-FIG.3: Densityofstates NF(ε)atx=dF+dS/2asafunction\nof the energy ε, for the F/S/Fstructure of Figure 1.a with\nγ= 1,γT= 0.12,dS/ξS= 1,γF\nφ=−2.1 andγS\nφ= 0.12.\nThe different panels correspond to different values of dF/ξF.\nThe red full curves correspond to the Pconfiguration and the\nblue dashed curves to the APconfiguration. For finite values\nofγS\nφ, some characteristic ”double-gap” structures appear in\nNF\nσ(ε) in the Pcase, due to a SDIPS-induced effective field\nappearing in S. The visibility of this effect strongly depends\non the thickness dFof theFlayers. For the different cases\nconsidered here, one can check that hypotheses 1 to 4 are\nvalid and that the critical temperature TC\ncof the structure\n(see Sec. III.F) is such that 0 .62TBCS\nc< TC\nc<0.67TBCS\nc.\nNote that for the above parameters and γS\nφ= 0, the curves\nobtained in the PandAPcase would be identical and very\nclose to the APcurves shown here.\nistic, at least for the weakly polarized Nb/Pd 0.9Ni0.1bi-\nlayers used in Ref. 3, for which one finds γT∼0.15 (see\nRef. 44). In addition, a simple barrier model suggests\nthat the situation/vextendsingle/vextendsingle/vextendsingleGF\nφ/vextendsingle/vextendsingle/vextendsingle≫GT, withGF\nφ<0 can happen\n(see Appendix A). The value γF\nφ=−2.1 used in Fig. 2\nthus seems possible .One can see that γF\nφcan change6\nsignificantly the phase of the oscillations of NF(ε= 0)\nwithdF(this effect has already been studied in Ref. 29\nin the case of rigid boundary conditions but I recall it\nhere for the sake of completeness). In Fig. 2, using for\nγF\nφa strong negative value allows to get NF(0)< N0for\nthe lowest values of dF, like often found in experiments.\nNote that from Fig. 2, γS\nφcan also shift the oscillations\nofNF(ε= 0) with dF. Here, this effect is much weaker\nthan that of γF\nφ, but one has to keep in mind that the\nlimitγS\nφ≪γF\nφis considered.\nEquation (20) shows that a measurement of Heff\nshould allow to determine the conductance GS\nφof a diffu-\nsiveS/Finterface. In this context, studying the energy\ndependence of NF(ε) is very interesting, because it can\nallow to see clear signatures of Heff, as shown below.\nFigure 3 shows the energy dependence of NF(ε) in the\nPandAPconfigurations, for a finite value of γS\nφand\ndifferent values of dF. ForC=P,NF(ε) shows some\n”double-gap” structures which disappear if the device is\nswitched to the APconfiguration. These double struc-\ntures are an indirect manifestation of the effective mag-\nnetic field Heffoccurring in Sin thePconfiguration,\ndue toγS\nφ/ne}ationslash= 0. Although Heffis localized in the S\nlayer, the double-gap structure that this field produces\nin the DOS of Sis transmitted to the DOS of Fdue to\nAndreev reflections occurring at the S/Finterfaces, as\nshown by Eq. (9). Interestingly, Rowell and McMillan\nhave already observed that an internal property of a S\nlayer can be seen through the superconducting proximity\neffect occurringin a nearbynormal layer. More precisely,\nthese authors have found that the DOS of an Ag layer\ncan reveal the phonon spectrum of an adjacent supercon-\nducting Pb layer47. Remarkably, the visibility of Heffin\nNF(ε) is modulated by quantum interferences occurring\ninF. Indeed, Heffis more visible for certain values of\ndF(e.g.dF/ξF= 1.0 or 1.2 in Fig. 3) than others (e.g.\ndF/ξF= 2.1 in Fig. 3), due to the dF-dependence of\nEq.(9).\nIt is useful to note that the SDIPS-induced effective\nfieldHeffshould also occur in the S/Fbilayer of Figure\n1.b. In this case, the Thouless energy and normal state\nconductance of the Slayer correspond to /tildewideES\nTH= 4ES\nTH\nand/tildewideGS= 2GSrespectively, sothat one finds gµBHeff=\n/tildewideES\nTHGS\nφ//tildewideGS. Double gap structures strikingly similar to\nthose shown in Figure 3 were indeed measured very re-\ncently by P. SanGiorgio et al., at the ferromagnetic side\nof diffusive Nb/Ni bilayers, in the absence of any exter-\nnal magnetic field48. Remarkably, the visibility of the\nobserved double structures varies with dF, as predicted\nabove. Note that in S/Fbilayers, the field Heffshould\nalso be observable directly at the Sside by measuring\nNS(ε) =/summationtext\nσ∈{↑,↓}Nσ(x= 0,ε). However, for param-\neters comparable to those of Figure 3, this should not\nenhance the resolution on Heff(see figure 4, left). For\ncertain values of dF,Heffis even more visible in NF(ε)\nthan inNS(ε)(see Figure 3).\nBefore concluding this section, I would like to empha-FIG. 4: Energy dependentdensityofstates NS(ε)measurable\nat the S side ( x= 0) of the S/Fstructure of Figure 1.b. The\nleft panel corresponds to the case of a metallic Fcontact\nwith parameters corresponding to that of Figs. 2 and 3. In\nthis case, the DOS measured at the Sside does not allow to\nresolve the SDIPS-induced effective field Heffbetter than a\nDOS measurement at the Fside of the structure, as can be\nseen from a comparison with Fig. 3. For certain values of dF,\nHeffis even more visible in NF(ε) thanNS(ε). The right\npanel corresponds to the case in which Fis not a metal but\nan insulating ferromagnet, i.e. γT= 0 . In this case, one\ncan use the reduced SDIPS parameter λS\nφ=GS\nφξS/AσS. For\nλS\nφ/ne}ationslash= 0,NS(ε) shows strong signatures of the effective field\nHeffinduced by the FIlayer inS. One can check that the\nhypotheses 1 to 4 are valid for the parameters used in this\nFigure.\nsize that from Eqs. (17) and (19), in the APconfig-\nuration, the SDIPS-induced effective field Heffdisap-\npears for the F/S/Fstructure considered in this paper\nbecause the two contacts are assumed to be symmetric\nand have thus opposite contributions to Heffin theAP\ncase. In the case of a dissymmetric structure, this should\nnot be true anymore, but the SDIPS-induced effective\nfield should nevertheless vary from the Pto theAPcase.\nThis is one practical advantage of working with F/S/F\ntrilayers instead of S/Fbilayers.\nC. Low-temperature density of states of S/FI\nbilayers\nTwenty years ago, internal Zeeman fields were ob-\nservedin superconductingAllayerscontactedtodifferent\ntypes of ferromagnetic insulators ( FI) (see Refs. 49,50,\n51,52). Using a ballistic S/FIbilayer model, Ref. 24\nsuggested that the observed internal fields could be in-\nduced by the SDIPS24. However, the inadequacy of this\nballistic approach for modeling the actual experiments\nwas pointed out in Ref. 52. Most of the experiments\non Al/FIinterfaces were interpreted by their authors\nin terms of a diffusive approach with no SDIPS, and an\ninternal Zeeman field added arbitrarily in the Al layer\n(see Refs. 51,52,53). The calculations of Section III.B.\nprovide a microscopic justification for the use of such an\ninternalfield in the diffusive model. Indeed, using γT= 0\nin the above calculations allows one to address the case7\nof diffusive S/FIbilayers. One finds that the SDIPS-\ninduced effective field Heffof Eq. (20) can occur in a\nthin diffusive Slayer contacted to a FIlayer. This effec-\ntive field effect can be seen e.g. in the density of states\nNS(ε) of theSlayer atx= 0 (see Figure 4, right). Re-\nmarkably, it wasfound experimentally52thatHeffscales\nwithd−1\ns, in agreement with Eq. (20)54.\nD. SDIPS-induced effective fields in other types of\nsystem\nInterestingly, the SDIPS can induce effective field ef-\nfects in othertypes ofsystems. First, the caseof S/N/FI\ntrilayers with a thickness dNof normal metal Nhas\nbeen studied theoretically27,28. In this case, a con-\nductance GN\nφsimilar to GS\nφcan be introduced to take\ninto account the SDIPS for electrons reflected by the\nFIlayer. The Nlayer is subject to an effective field\ngµBH′\neff=ETHGN\nφ/GNwithGNtheconductanceof N.\nThe expression of H′\neffis analogue to that of Heff(see\nEq. 20), up to a factor 2 which accounts for the symme-\ntry of the F/S/Fstructure with respect to x= 0 in the\nPconfiguration. Secondly, an effective field H′′\neffdefined\nbygµBH′′\neff= (/planckover2pi1vW\nF/2L)(ϕ↑\nL+ϕ↑\nR−ϕ↓\nL−ϕ↓\nR) has been\npredicted for a resonantsingle-channel ballistic wire with\nlengthLplaced between two ferromagnetic contacts19,\nwithϕσ\nL(R)the reflection phase of electrons with spin σ\nincident from the wire onto the left(right) contact, and\nvW\nFis the Fermi velocity in the wire. The expression of\nH′′\neffalsoshowsstrongsimilaritieswith that of Heff(see\nEq. 20, middle term). In practice, signatures of the field\nH′′\neffcould be identified in a carbon nanotube contacted\nwith two ferromagnetic contacts22,56. The fields Heff,\nH′\neffandH′′\neffhave the same physical origin: the en-\nergies of the states localized in the central conductor of\nthe structure depend on spin due to the spin-dependent\nphase shifts acquired by electrons at the boundaries of\nthis conductor. In all cases, the DOS of the central con-\nductor reveals the existence of the SDIPS-induced effec-\ntivefieldonlyifitalreadypresentsastrongenergydepen-\ndence near the Fermi energy in the absence of a SDIPS.\nIn theF/S/Fcase, this energy dependence is provided\nby the existence of the superconducting gap in S57. In\ntheS/N/FI case, it is provided by the existence of a su-\nperconducting minigap in N. At last, in the case of the\nballistic wire, it is provided by the existence of resonant\nstates in the wire.\nE. Comparison between the present work and\nother models for data interpretation in S/F\nheterostructures\nForcharacterizingthe propertiesof S/Finterfaces, one\nhas to interpret the experimental data showing the oscil-\nlationsof the density ofstates NF(ε) inFwith the thick-nessdFofF(or the oscillations of the critical current I0\nof aS/F/SJosephson with dF). However, if one uses a\nsimple description with spin-degenerate boundary condi-\ntions, the amplitude and the phase of these signals are\nnot independent, which makes the agreement with ex-\nperimental results impossible in most cases. The SDIPS\nconcept can solve this problem since it produces a shift\nof the signals oscillations with respect to the GS(F)\nφ= 0\ncase. However, in many cases, the observed shifts were\nattributed to the existence of a magnetically dead layer\n(MDL) at the Fside of the S/Finterface (see e.g. Refs.\n3,58,59). In other cases, the discrepancy between the\ntheory and the data was solved by taking into account\nspin-scattering processes in the Flayer (see e.g. Ref. 6).\nIn order to have a better insight into superconducting\nproximity effect experiments, one must stress the impor-\ntance of estimating experimentally the MDL thickness\nand the spin-scattering rate. In principle, spin-scattering\nrates can be estimated experimentally, as was done for\ninstance for the CuNi alloy60which is frequently used\nin proximity effect measurements (see e.g. 6,58). An\nexperimental determination of the MDL thickness has\nalso been performed in a few structures used to measure\nTcorI061,62,63,64,65, but, so far, this parameter has not\nbeen used for a real quantitative analysis of the data. In\nsomesituations, amodel combiningtheSDIPSwithspin-\nscattering and/or a MDL may be necessary. In any case,\nit is important to point out that descriptions based on\nspin-degenerateboundaryconditionsare, inprinciple, in-\ncomplete since they do not account for the effective field\neffect described in section III.B.\nBefore concluding this section, it is interesting to note\nthat the effective field effect produced by GS\nφinSor the\nphaseshift ofthe spatialoscillationsofthe DOS provided\nbyGF\nφinFwill remain qualitatively similar when the\nsigns ofGS\nφandGF\nφare changed (not shown). The sign\nofGF\nφintheexperimentsofRefs.3,5couldbedetermined\nfrom a quantitative comparison between the theory and\nthe data29. Below, we present a study of the critical tem-\nperatureof S/Fstructureswhichcangiveinformationon\nthe signs of GS\nφandGF\nφthrough qualitative signatures.\nF. Critical temperature of S/Fheterostructures\nThe critical temperature of S/Fhybrid structures\nhas already been the topic of many theoretical (see\ne.g. Refs. 13,14,35,66,67) and experimental (see e.g.\nRefs. 11,12) studies, but the effects of the SDIPS on this\nquantity have raised little attention so far. I show below\nthat the SDIPS can significantly modify the critical tem-\nperatures of S/Fdiffusive structures. Calculating the\ncritical temperature TC\ncof the structure of Figure 1.a in\nconfiguration Crequires to consider the limit in which\nsuperconducting correlations are weak in Sas well as in\nF(hypotheses 1 and 2 are then automatically satisfied).\nEquations (13), (15) and (16), then lead to8\nFIG. 5: Critical temperature TP\ncfor theF/S/Fstructure of\nFigure 1.a in the Pconfiguration (top panels) and difference\n∆Tc=TAP\nc−TP\ncbetween the critical temperatures in the P\nandAPconfigurations (bottom panels), as a function of the\nthickness dFof theFlayers. The left panels show the effect of\na finiteγS\nφand the right panels the effect of a finite γF\nφ. The\nfour panels show the case γS\nφ=γF\nφ= 0 with full red lines,\nfor comparison. The other parameters used in the Figure\nareγ= 1,γT= 0.12 anddS/ξS= 1. The SDIPS modifies\nTC\ncand ∆Tcin a quantitative or qualitative way, depending\non the case considered. The type of effects produced by the\nSDIPS on TC\ncand ∆Tcstrongly depend on the signs of γF\nφ\nandγS\nφ.\n/tildewideθC\nn,σ=∆C\n|ωn|+2∆BCSbCn,σδ−1\nS(22)\nwith\nbP\nn,σ=4L0\nn,σ\n4+δSL0n,σ(23)\nbAP\nn,σ=8Re[L0\nn,σ]+4/vextendsingle/vextendsingleL0\nn,σ/vextendsingle/vextendsingle2δS\n8+6Re[ L0n,σ]δS+/vextendsingle/vextendsingleL0n,σ/vextendsingle/vextendsingle2δ2\nS(24)\nand\nL0\nn,σ\nγ=γT(BP,2\nn,σ+iγF\nφσsgn(ωn))\nγT+BP,2\nn,σ+iγF\nφσsgn(ωn)+iγS\nφσsgn(ωn) (25)\nThese Eqs. together with (3) lead to\nlog/parenleftbiggTBCS\nc\nTCc/parenrightbigg\n= Re/bracketleftBigg\nΨ/parenleftBigg\n1\n2+bC\nn,σ\nδSexp(Γ)TBCS\nc\nTCc/parenrightBigg/bracketrightBigg\n−Ψ/parenleftbigg1\n2/parenrightbigg\n(26)where Γ denotes Euler’s constant. The resulting TC\ncis\nindependent from the values of nandσused in Eq. (26).\nNotethatinthecase γS\nφ=γF\nφ= 0andδS→0, thisequa-\ntion is in agreement with Eqs. (18) and (19) of Ref. 35.\nPerforminga numerical resolution of Eq. (26) together\nwith (23) and (24), one obtains the results of Fig. 5,\nwhich shows the critical temperature TP\ncof the structure\nin thePconfiguration (top panels) and the difference\n∆Tc=TAP\nc−TP\nc(bottom panels) as a function of dF,\nfor different interface parameters68. Here, for simplic-\nity, I consider cases where TC\ncdoes not show a strongly\nreentrant behavior, i.e. a cancellation of TC\ncin a certain\ninterval of dF. Such a behavior can happen for instance\nfor larger values of γT(see e.g. Ref. 35) or γ(S)F\nφ, but\nthis corresponds to a minority69of the experimental ob-\nservations made so far. The four panels of Fig. 5 show\nthe case γS\nφ=γF\nφ= 0 with full red lines, for comparison\nwith the other cases, where the SDIPS is finite. I first\ncomment the results obtained for the TP\nc(dF) curves (top\npanels of Fig. 5). In some cases, the SDIPS can modify\nquantitatively the TP\nc(dF) curves, for instance by am-\nplifying the dip expected in TP\nc(dF) in the absence of a\nSDIPS (see e.g. dotted curve in the upper left panel, cor-\nresponding to γS\nφ>0). In some other cases the SDIPS\ncan modify qualitatively the TP\nc(dF) curves, for instance\nby transforming the minimum expected in TP\nc(dF) into a\nmaximum (see e.g. curve with circles in the upper right\npanel, corresponding to γF\nφ<0). I now comment the\nresults obtained for the ∆ Tc(dF) curves (bottom pan-\nels of Fig. 5). In the range of parameters studied in\nthis work, one always finds ∆ Tc>0 because the effects\nof the two Flayers on Sare partially compensated in\ntheAPcase. In the absence of a SDIPS and for a low\nγT, the ∆Tc(dF) curve presents a maximum at a finite\nvalue ofdF(see red full lines in bottom panels). In some\ncases, the SDIPS can modify quantitatively the ∆ Tc(dF)\ncurves, for instance by increasing the value of this max-\nimum (see dotted curve in the bottom left panel, cor-\nresponding to γS\nφ>0). In other cases, the SDIPS can\nmodify qualitatively the ∆ Tc(dF) curves, for instance by\nturning the maximum expected in ∆ Tc(dF) into a mini-\nmum (see dashed curve in the bottom left panel, corre-\nsponding to γS\nφ<0), or by increasing the complexity of\nthe variations of ∆ TcwithdF(see curve with circles in\nthe bottom right panel, corresponding to γF\nφ<0), or by\ntransforming ∆ Tc(dF) into a monotonically decreasing\ncurve with a reduced amplitude (see dot-dashed curve in\nbottom right panel, corresponding to γF\nφ>0). Remark-\nably, the type of effects produced by the SDIPS on the\nTC\nc(dF) and ∆Tc(dF) curves depend on the sign of γF\nφ\nandγS\nφ. Critical temperature measurements can thus be\nan interesting way to determine the values of γF\nφandγS\nφ.\nFor simplicity, I have shown here results for γF\nφ/ne}ationslash= 0 and\nγS\nφ/ne}ationslash= 0 separately. Nevertheless, the behavior predicted\nforγF\nφ/ne}ationslash= 0 and γS\nφ/ne}ationslash= 0 simultaneously remain highly in-\nformative on the values of γF\nφandγS\nφ. Note that if γF\nφ9\nandγS\nφare increased compared to the values used in Fig.\n5, theTC\nc(dF) curve gets some cancellation points like\nin Ref. 35, but the behaviors of TC\ncand ∆Tcremain, in\nmany cases, qualitatively dependent on the signs of γF\nφ\nandγS\nφ(not shown).\nBeforeconcluding, Inotethat withthe approximations\nused in the previous section (III.B), the electronic corre-\nlations inside Swere affected by the presence of the F\nelectrodes through the parameter γS\nφonly. Consequently,\nthe self-consistent gap ∆CoftheSlayerwas independent\nfromdF, and it was furthermore identical for the Pand\nAPconfigurations for γS\nφ= 0. In the present section, I\nhave not neglected the dependence of TP\ncandTAP\ncondF\nandγF\nφbecause I have considered parameters for which\nhypothesis 3 is not acceptable anymore. In particular, I\nhave used lower values for dF70.\nIV. CONCLUSION\nThis article shows that the Spin-Dependence of In-\nterfacial Phase Shifts (SDIPS) can have a large variety\nof signatures in diffusive superconducting/ferromagnetic\n(S/F) heterostructures. Ref. 29 had already predicted\nthat the SDIPS produces a phase shifting of the oscilla-\ntions of the superconducting correlations with the thick-\nness ofFlayers. This article shows that this is not the\nonlyconsequenceoftheSDIPSin S/Fcircuits. Inpartic-\nular, the SDIPS can produce an effective magnetic field\nin a diffusive Slayer contacted to a diffusive Flayer.\nThis effective field can be seen e.g. through the DOS\nof the diffusive Flayer, with a visibility which oscillates\nwith the thickness of F. The SDIPS can also modify\nsignificantly the variations of the critical temperature of\naS/Fbilayer or a F/S/Ftrilayer with the thickness of\nF, either in a quantitative or in a qualitative way, de-\npending on the regime of parameters considered and in\nparticular the sign of the conductances GS\nφandGF\nφused\nto account for the SDIPS of the S/Finterfaces. In the\ncase of a F/S/Fspin valve, this last result also holds for\nthe thickness-dependence of the difference between the\ncritical temperatures in the parallel and antiparallel lead\nconfigurations. These effects should help to determine\nthe parameters GS\nφandGF\nφof diffusive S/Finterfaces.\nThe calculations shown in this paper are also appropri-\nate to the case of thin diffusive Slayers contacted to\nferromagnetic insulators.\nI thank P. SanGiorgio for showing me his experimental\ndata prior to publication, which stimulated part III.B of\nthis work. I acknowledge discussions with T. Kontos and\nW. Belzig. This work was supported by grants from the\nSwiss National Science Foundation and R´ egion Ile-de-\nFrance.V. APPENDIX A: PARAMETERS OF A S/F\nINTERFACE FROM A DIRAC BARRIER MODEL\nFIG. 6: Conductances GT(full line),˛˛GS\nφ˛˛(dashed lines), and˛˛GF\nφ˛˛(dash-dotted lines) reduced by the number of channels\nn, as a function of the spin-averaged barrier strength Zof a\nS/Finterface modeled with a Dirac barrier (see text). The\ncurveswere obtained with typical Fermi energies EF\nf= 0.8 eV\nandES\nf= 12 eV in FandSrespectively, a spin-polarization\nP= 0.06 of the density of states in F, and a spin asymmetry\nα= 0.06 for the barrier. Note that for the set of parameters\nused this Figure, one finds GF\nφ<0 andGS\nφ<0. However, for\nother parameters (e.g. by using α <0), one can reverse the\nsigns ofGF\nφandGS\nφ, or obtain opposite signs for GF\nφandGS\nφ\n(not shown).\nThe exact values of the conductances GT,GF\nφandGS\nφ\nof aS/Finterface depend on the details of this inter-\nface and on the microscopic structure of the contacted\nmaterials. Nevertheless, it is already interesting to study\na simplified Dirac barrier model which shows that the\nparameters regime assumed in this paper is, in princi-\nple, possible. Neglecting the transverse part of the elec-\ntronsmotion, one finds rS(F)\nn,σ= (kσ\nS(F)−kσ\nF(S)−iZσ)/Dσ\nandtS(F)\nn,σ= 2(kσ\nSkσ\nF)1/2/DσwithDσ=kσ\nS+kσ\nF+iZσ.\nHere,kσ\nS(F)is the Fermi electronic wavevector in S(F)\nandZσis the strength of the Dirac barrier for elec-\ntrons with spin σ. I use /planckover2pi1kσ\nS= (2meES\nf)1/2, with\nmethe free electron mass and ES\nfthe Fermi level in\nS. ForF, I use an s-band Stoner model, in which\n/planckover2pi1kσ\nF= (2meEF\nf[1±2P/(1 +P2)])1/2, withPthe spin-\npolarization of the density of states in FandEF\nfthe\nFermi level in F. I assume that the barrier can have a\nspin dependence α= (Z↓−Z↑)/(Z↑+Z↓) due to the\nferromagnetic contact material used to form the inter-\nface. The results given by this approach are shown in\nFig. 6. The conductances GT,/vextendsingle/vextendsingle/vextendsingleGF\nφ/vextendsingle/vextendsingle/vextendsingleand/vextendsingle/vextendsingle/vextendsingleGS\nφ/vextendsingle/vextendsingle/vextendsingle, reduced\nby the number of channels n, are shown as a function of\nthe spin-averagedbarrierstrength Z= (Z↑+Z↓)/2. The\nthree types of conductances go to zero when Zgoes to\ninfinity with αconstant71. One can see that in the limit\nTn≪1 (i.e. here GTh/ne2≪1) andP≪1 in which the10\nboundary conditions (5,6) have been derived,/vextendsingle/vextendsingle/vextendsingleGF\nφ/vextendsingle/vextendsingle/vextendsinglecan\nbe significantly stronger than GT, as assumed in Figs. 2\nand 3. Note that for the set of parameters used in Fig.\n6, one finds GF\nφ<0 andGS\nφ<0. However, for other pa-\nrameters (e.g. by using α <0), one can reverse the signs\nofGF\nφandGS\nφ, or obtain opposite signs for GF\nφandGS\nφ\n(not shown). This suggests that there is no fundamental\nconstraint on the signs of GF\nφandGS\nφin the general case.\nVI. APPENDIX B: EFFECTIVE FIELD EFFECT\nIN AFLAYER\nThis appendix reconsiders the case of the S/Fbilayer\nof Fig.1.b, in the limit dF≪ξFwhereθn,σ(x) can be\napproximatedwithaquadraticforminthe Flayer. From\nEqs. (2) and (5), one finds, for x∈[dS/2,dS/2+dF],\nθn,σ(x) =/tildewideθF\nn,σ+bF\nn,σ/bracketleftBigg\ndS−2x\ndF+/parenleftbiggdS−2x\n2dF/parenrightbigg2/bracketrightBigg\n(27)\nwith\n/tildewideθF\nn,σ= arctan\nγTsin(/tildewideθS\nn,σ)\nγTcos(/tildewideθSn,σ)+2ξF\ndFΩn\nEF\nTH+iγF\nφηP,2\nn,σ\n\n(28)\nbF\nn,σ=Ωnsin(/tildewideθF\nn,σ)\nEF\nTH(29)\nΩn=|ωn|+iEexηP,2\nn,σand/tildewideθS\nn,σ=θS\nn,σ(x=dS/2). Here,\nGF=σFA/dFandEF\nTH=/planckover2pi1DF/d2\nFdenote the con-\nductance and Thouless energy of the Flayer (note that\ndF≪ξF⇐⇒Eex≪EF\nTH). From completeness, I also\ngive the analogous equations for x∈[0,dS/2]. Assuming\nthatθn,σ(x) can be approximated with a quadratic form\nin theSlayer, one finds, from Eqs. (1) and (6),\nθn,σ(x) =/tildewideθS\nn,σ+bS\nn,σ/bracketleftBigg\n2(2x−dS)\ndS+/parenleftbigg2x−dS\ndS/parenrightbigg2/bracketrightBigg\n(30)\nwith\n/tildewideθS\nn,σ= arctan\nγγTsin(/tildewideθF\nn,σ)+4ξS\ndS∆\neES\nTH\nγγTcos(/tildewideθFn,σ)+4ξS\ndS|ωn|\neES\nTH+iγγS\nφηP,2\nn,σ\n\n(31)and\nbS\nn,σ=|ωn|sin(/tildewideθS\nn,σ)−∆cos(/tildewideθS\nn,σ)\n/tildewideES\nTH(32)\nThe notations used in the above equations are the same\nas in Section III. In particular, the Thouless energy and\nthe normal state conductance of the Slayer with thick-\nnessdS/2 are denoted /tildewideGS= 2σSA/dSand/tildewideES\nTH=\n4/planckover2pi1DS/d2\nS, and one has ηP,2\nn,σ=σsgn(ωn). In the limit\n/tildewideθF\nn,σ≪1, Eq. (31) is in agreement with Eqs. (17)\nand (18). The analytic continuation of Eq. (31) shows\nthat the Slayer is subject to the effective field Heff=\n/tildewideES\nTHGS\nφ//tildewideGS, in agreement with Eq. (20). The analytic\ncontinuationofEq. (28) showsthat the Flayerissubject\nto an analogue effective field HF\neff, defined by\ngµBHF\neff=EF\nTHGF\nφ\nGF(33)\nIt is interesting to compareEqs. (28) and (9). First, note\nthat in the regime/vextendsingle/vextendsingle/vextendsingle/tildewideθF\nn,σ/vextendsingle/vextendsingle/vextendsingle≪1, Eq. (28) can be recovered\nfromthe low dF-limit ofEq. (9). The interestofEq. (28)\nis that it is valid beyond the regime of a weak supercon-\nducting proximityeffect studied in part III. In particular,\nEq. (28) indicates that, in principle, the field EF\nTHcan\noccur in a thin Flayer for any value of the tunneling\nconductance GTof theS/Finterface. Interestingly, in\nthe case of a thick FlayerdF≥ξF/2, from Eq. (9), the\ncontribution of GF\nφto the pairing angle θF\nn,σ(x) cannot\nbe put anymore under the form of an effective exchange\nfield, due to the non-linearity of the BC,j\nn,σterm. Thus,\nstrictly speaking, the concept of a SDIPS-induced effec-\ntive field is valid only in the limit of thin metallic layers.\nHowever, it might be possible, in principle, to observe\nreminiscences of the double-gap structure appearing in\nNF(ε) in the regime of intermediate layer thicknesses\ndF∼ξFat least, due to the continuity of the equations.\nThen, one can wonder why this effect does not appear\nin section III. This is due to the regime of parameters\nchosen: section III assumes Eex≫∆BCSlike in most\nexperiments, and it furthermore focuses on the typical\nenergy range probed in superconducting proximity effect\nmeasurements, i.e. |ε|/lessorsimilar2∆BCS. In such a regime, one\ncan neglect the term |ωn|compared to iEexσsgn(ωn) in\nEqs. (9) and (28), so that HF\neffdoes not emerge in the\nmodel. 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In Ref. 3, Kontos et al. have stud-\nied Nb/Pd 0.9Ni0.1bilayers with the same interface area as\nfor the previous Nb/Pd bilayers. The two types of bilay-\ners were realized with the same sources of Nb and Pd and\nsimilar fabrication processes. Considering that only 10%\nof Ni were added to the Pd material in the S/Fcase, the\nconductivity of Pd 0.9Ni0.1was probably of the same order\nas that of Pd, and the conductance of the Nb/Pd interface\nwas probably of the same order as that of Nb/Pd 0.9Ni0.1.\nThis leads to AσF/GT∼26.5 nm. Then, using the value\nξF∼4 nm found in Refs. 3,45 for dPd0.9Ni0.1= 8 nm, one\nfindsγT=GTξF/AσF∼0.15.\n45T. Kontos, Ph.D. thesis, Universit´ e Paris-Sud, Orsay,\nFrance, 2002.12\n46The width ofthe minigap appearing inthe densityofstates\nof aS/Nbilayer versus energy gives a direct access to the\nparameter γTof this interface, as explained in A. Golubov\nand M. Yu. Kuprianov, Sov. Phys. JETP 69, 805 (1989).\n47J. M. Rowell and W. L. McMillan, Phys. Rev. Lett. 16,\n453 (1966).\n48Paul SanGiorgio, private communication.\n49P. M. Tedrow, J. E. Tkaczyk, and A. Kumar, Phys. Rev.\nLett.56, 1746 (1986).\n50R. Meservey and P. M. Tedrow, Phys. Rep. 238, 173\n(1994).\n51J. S. Moodera, X. Hao, G. A. Gibson, and R. Meservey,\nPhys. Rev. Lett. 61, 637 (1988), X. Hao, J. S. Moodera,\nand R. Meservey, Phys. Rev. B 42, 8235 (1990).\n52X. Hao, J. S. Moodera, and R. Meservey, Phys. Rev. Lett.\n67, 1342 (1991).\n53J. A. X. Alexander, T. P. Orlando, D. Rainer, and P. M.\nTedrow, Phys. Rev. B 31, 5811 (1985).\n54The fact that Heffscales with d−1\nsis also in agreement\nwith Eq. (1) of Ref. 13, proposed by De Gennes from\nCooper’s argument55.\n55L. N. Cooper, Phys. Rev. Lett. 6, 689 (1961); P. G. de\nGennes, Rev. Mod. Phys. 36, 225 (1964).\n56S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Graber, A.\nCottet and C. Sch¨ onenberger, Nature Phys. 1, 99 (2005).\n57Without superconductivity, Heffwould not be not resolv-\nable anymore in the S/Fstructures considered in this pa-\nper, since it was assumed implicitly that the normal-state\nDOS of these structures is constant near the Fermi level.\n58V. A. Oboznov, V. V. Bol’ginov, A. K. Feofanov, V. V.\nRyazanov and A. I. Buzdin, Phys. Rev. Lett. 96, 197003\n(2006).\n59M. Weides, M. Kemmler, E. Goldobin, D. Koelle, and R.\nKleiner, H. Kohlstedt, A. Buzdin, Appl. Phys. Lett. 89,\n122511 (2006).\n60W. P. Pratt, Jr., private communication; see also S.-Y.\nHsu, P. Holody, R. Loloee, J. M. Rittner, W. P. Pratt, Jr.,\nand P. A. Schroeder, Phys. Rev. B 54, 9027 (1996).61Th. M¨ uhge, N. N. Garif’yanov, Yu. V. Goryunov, G. G.\nKhaliullin, L. R. Tagirov, K. Westerholt, I. A. Garifullin,\nand H. Zabel, Phys. Rev. Lett. 77, 1857 (1996).\n62J. Aarts and J. M. E. Geers, E. Br¨ uck, A. A. Golubov, R.\nCoehoorn, Phys. Rev. B 56, 2779 (1997).\n63J. W. A. Robinson, S. Piano, G. Burnell, C. Bell, M. G.\nBlamire, Phys. Rev. Lett. 97, 177003 (2006).\n64S. Piano , J. W. A. Robinson, G. Burnell, M. G. Blamire,\nEur. Phys. J. B 58, 123 (2007).\n65C. Bell, R. Loloee, G. Burnell, and M. G. Blamire, Phys.\nRev. B71, 180501(R) (2005).\n66E. A. Demler, G. B. Arnold and M. R. Beasley, Phys.\nRev. B55, 15174 (1997); M. G. Khusainov and Yu. N.\nProshin, Phys. Rev. B 56, R14283 (1997); L. R. Tagirov,\nPhysica C 307, 145 (1998); Yu. N. Proshin and M. G.\nKhusainov JETP 86, 930 (1998); Ya. V. Fominov, N. M.\nChtchelkatchev, and A. A. Golubov, Phys. Rev. B 66,\n014507 (2002); A. Bagrets, C. Lacroix and A. Vedyayev,\nPhys. Rev. B 68, 054532 (2003).\n67C.-Y. You, Ya.B. Bazaliy, J. Y.Gu, S.-J. Oh, L.M. Litvak,\nand S. D. Bader, Phys. Rev. B 70, 014505 (2004).\n68With the parameters of Fig. 5, the approximations bP\nn,σ=\nL0\nn,σandbAP\nn,σ= Re[L0\nn,σ] used in Ref. 35 would be suffi-\ncient for calculating the TP\ncandTAP\nccurves, but not the\ndifference ∆ Tc. Indeed, for the different cases studied in\nthis Figure, the relative difference between the results ob-\ntained with these approximations and those given by Eqs.\n(23) and (24) is smaller than ∼0.8% forTP\ncandTAP\nc, but\nit can reach ∼18% for ∆ Tc. This is why I have used the\ngeneral equations (23) and (24) for plotting Figure 5.\n69L. R. Tagirov, I. A. Garifullin, N. N. Garifyanov, S. Ya.\nKhlebnikov, D. A. Tikhonov, K. Westerholt, and H. Zabel,\nJ. Magn. Magn. Mater. 240, 577 (2002).\n70Note that both regimes dF≤ξF(see e.g. Ref 37) and\ndF≥ξF(see e.g. Ref. 67) seem to be accessible in practice.\n71ForZ→+∞,GF(S)\nφvanishes because arg( rF(S)\nn,σ)→πand\narg(tF(S)\nn,σ)→ −π/2 forσ∈ {↑,↓}."    },    {        "title": "2401.04219v2.Ferromagnetic_Materials_for_Josephson_π_Junctions.pdf",        "content": "Ferromagnetic Materials for Josephson πJunctions\nNorman O. Birge1,a)and Nathan Satchell2,b)\n1)Department of Physics and Astronomy, Michigan State University, East Lansing,\nMichigan 48824, USA\n2)Department of Physics, Texas State University, San Marcos, Texas, 78666,\nUSA\n(Dated: 12 March 2024)\nThe past two decades have seen an explosion of work on Josephson junctions contain-\ning ferromagnetic materials. Such junctions are under consideration for applications\nin digital superconducting logic and memory. In the presence of the exchange field,\nspin-singlet Cooper pairs from conventional superconductors undergo rapid oscilla-\ntions in phase as they propagate through a ferromagnetic material. As a result,\nthe ground-state phase difference across a ferromagnetic Josephson junction oscil-\nlates between 0 and πas a function of the thickness of the ferromagnetic material.\nπ-junctions have been proposed as circuit elements in superconducting digital logic\nand in certain qubit designs for quantum computing. If a junction contains two or\nmore ferromagnetic layers whose relative magnetization directions can be controlled\nby a small applied magnetic field, then the junction can serve as the foundation for a\nmemory cell. Success in all of those applications requires careful choices of ferromag-\nnetic materials. Often, materials that optimize magnetic properties do not optimize\nsupercurrent propagation, and vice versa. In this review we discuss the significant\nprogress that has been made in identifying and testing a wide range of ferromagnetic\nmaterials in Josephson junctions over the past two decades. The review concentrates\non ferromagnetic metals, partly because eventual industrial applications of ferromag-\nnetic Josephson junctions will most likely start with metallic ferromagnets (either in\nall metal junctions or junctions containing also an insulating layer). We will briefly\nmention work on non-metallic barriers, including ferromagnetic insulators, and some\nof the exciting work on spin-triplet supercurrent in junctions containing noncollinear\nmagnetic inhomogeneity.\na)Electronic mail: birge@msu.edu\nb)Electronic mail: satchell@txstate.edu\n1arXiv:2401.04219v2  [cond-mat.supr-con]  8 Mar 2024I. INTRODUCTION\nIt has been over 20 years since Ryazanov and co-workers first demonstrated a mag-\nnetic Josephson junction whose ground-state phase difference could be either zero or π\ndepending on the sample temperature.1,2That breakthrough was followed soon after by\nKontos et al. , who demonstrated the 0 −πtransition as a function of the thickness of\nthe ferromagnetic material in a series of magnetic junctions.3The evidence for the 0 −π\ntransition in those pioneering works was indirect in the sense that only the amplitude of\nthe critical current was measured; direct measurements of the junction phase were carried\nout soon thereafter.4–7The physical mechanism underlying these phenomena had been un-\nderstood already 20 years earlier;8–10nevertheless, the laboratory demonstrations of these\nphenomena sparked widespread interest in magnetic Josephson junctions and in π-junctions\nin general.11–13There have been many proposals to use π-junctions as circuit elements in\nsuperconducting digital logic14,15and in certain qubit designs for quantum computing,16,17\nand several of these ideas are being tested in the laboratory (see Section III of this review).\nAt about the same time as the first π-junction demonstrations, Bergeret, Volkov and\nEfetov predicted theoretically that magnetic Josephson junctions made from conventional\nspin-singlet superconductors could carry long-range spin-triplet supercurrent under specific\nconditions that include the presence of non-collinear magnetic inhomogeneities inside the\njunctions.18–21Unlike the demonstration of magnetic πjunctions, the Bergeret prediction\ncame as a total surprise to the superconductivity community. The idea that supercurrents\ncould be spin-polarized led to much excitement and the coining of the name “superconduct-\ning spintronics” for this new area of condensed matter physics.22–26\nPut together, the two breakthroughs mentioned above catalyzed a huge surge in interest\nin superconducting/ferromagnetic (S/F) hybrid systems. This review covers only a small\nsubset of the work that has been done since 2001, focusing mainly on S/F/S π-junctions\nand their variants, including S/I/F/S, S/I/s/F/S and S/FI/S junctions (where I = insulator\nand FI = ferromagnetic insulator). Our emphasis is on the wide range of magnetic materials\nthat have been used in such devices. The centerpiece of the review are the Tables in Section\nIV, which we hope provide a complete list of magnetic materials that have been used inside\nJosephson junctions, usually with the aim of making π-junctions. To our knowledge, the last\ntime such a compilation has been made was in 2006 by Kupriyanov, Golubov, and Siegel.27\n2That review covered both experimental and theoretical developments; ours focuses almost\nentirely on experimental results.\nThere has been a lot of work on other experimental signatures of S/F physics besides π-\nstate Josephson junctions, such as the critical temperature and critical field of S/F bilayers\nand multilayers. We do not discuss those results here, even though some of that work28–36\npredated the experimental demonstrations of πJosephson junctions discussed above.\nThe structure of this review is as follows: Section II discusses general considerations,\nincluding the physical parameters that describe Josephson junctions, ballistic vs diffusive\ntransport regimes, the different types of junctions, and how the junction response to a\nmagnetic field is modified when the junction contain ferromagnetic materials. Section III\ndiscusses applications of π-junctions in quantum and classical circuits. Section IV discusses\nmetallic S/F/S π-junctions and contains the Tables of magnetic materials mentioned above.\nIt includes a short subsection about “spin-valve” junctions whose critical current or phase\nstate are controllable by changing the magnetic configuration inside the junction. Section V\nmentions briefly the many different types of π-junctions that have been created in the past\nfew years in exotic materials other than metals. Section VI discusses spin-triplet Joseph-\nson junctions, including a very brief mention of so-called “ ϕandϕ0-junctions.” Finally,\nSection VII contains a discussion of some open challenges and directions for the future.\nII. GENERAL CONSIDERATIONS\nA. S/F physics, length scales, and theoretical predictions for IcRN\nThe physics of S/F systems has been reviewed many times;11–13,21,23we encourage readers\nnew to the topic to read the excellent pedagogical article by Demler et al.37Superconducting\nand ferromagnetic materials in contact or in close proximity to each other interact in two\ndistinct ways. Effects due to the magnetic field produced by F are generally referred to as\n“orbital effects”; they are nearly always present in experiments and should not be forgotten,\nbut they are not the main focus of this review. Here we focus on proximity effects, whereby\nelectron pair correlations are induced in a material in contact with a superconductor.\nWe start by defining several important length scales. The superconducting coherence\nlength is defined as ξbal\nS=ℏvF/∆ or ξdif\nS= (ℏDS/∆)1/2in the ballistic and diffusive limits,\n3respectively, where ∆ is the superconducting gap, and vFandDSare the Fermi velocity and\ndiffusion constant in S. The diffusion constant, D, is related to the electron mean free path,\nl, via D=1\n3vFl. In an S/N bilayer (N = normal metal), pair correlations penetrate N over\na distance called the “normal-metal coherence length”, which has the following forms in the\nballistic or diffusive limits: ξbal\nN=ℏvF/(2πkBT) orξdif\nN= (ℏDN/(2πkBT))1/2, where vFand\nDNare now in the normal metal, and Tis the temperature. That length can be quite long –\nup to several hundred nm in noble metals at dilution refrigerator temperatures. In contrast,\nin ferromagnetic materials the pair correlations oscillate in sign due to the exchange split-\nting between the majority and minority spin bands in F. (Those oscillations are sometimes\nreferred to as “FFLO oscillations” after Fulde-Ferrell38and Larkin-Ovchinnikov,39but we\nemphasize that the original FFLO prediction was for a bulk ferromagnetic superconductor\nin the ballistic limit, rather than a proximity system.) The period of the oscillations is 2 πξF,\nwhere ξFis sometimes called the “exchange length” or the “ferromagnetic coherence length”.\nIn the ballistic or diffusive limits with Eex>> k BT, the results are ξbal\nF=ℏvF/(2Eex) or\nξdif\nF= (ℏDF/Eex)1/2, respectively, where 2 Eexis the exchange splitting between the ma-\njority and minority spin bands. In addition to the oscillation, the pair correlations decay\nalgebraically in the ballistic limit or exponentially in the diffusive limit. Due to the large\nexchange energies in strong F materials, Eex≈1eV,ξFis less than 1 nm in those materials.\nThat is one reason why experimental progress in S/F systems lagged behind the theoretical\ndevelopments for so long. One of the keys to the experimental breakthroughs of 2001 was\nthe use of dilute ferromagnetic alloys: CuNi alloy in the case of the Ryazanov group2and\nPdNi alloy in the case of the Aprili group.3,40Diluting the Ni decreased Eexsubstantially\nthereby increasing ξFto values of several nm. In that limit one must include the contribution\nof the temperature to the exchange length ξF; in fact, Ryazanov’s first demonstration of a\nπ-junction was carried out by varying the temperature of a Josephson junction with fixed\nferromagnetic layer thickness.2\nWhen discussing the critical current, Ic, of Josephson junctions, we will quote results\nfor the IcRNproduct, where RNis the normal-state resistance of the junction, usually\ndetermined from the slope of the I–Vrelation for applied currents I >> I c.IcRNis a useful\nquantity because it is independent of junction area and because it can be compared with\nthe standard Ambegaokar-Baratoff result for short S/I/S junctions: IcRN=π∆/(2e).41\nCalculations of Icfor ferromagnetic Josephson junctions have been carried out in several\n4different regimes. In addition to the length scales defined above, one must also consider the\nelectron mean free path, lF, and the thickness dFof the F layer. The first calculation of\nIcRNwas performed for the purely ballistic or “clean” limit, defined by dF<< ξ S<< l F, lS\n– i.e. neglecting impurity scattering in both F and S.9The decay and oscillations of IcRN\nare given by the numerical maximum with respect to φof the ballistic limit supercurrent\nIS(φ),9\nIS(φ)RN=π∆α2\n2eZ∞\nαdy\ny3\"\nsin \nφ−y\n2!\ntanh \n∆ cosφ−y\n2\n2kBT!\n+ sin \nφ+y\n2!\ntanh \n∆ cosφ+y\n2\n2kBT!#\n,(1)\nwhere φis the phase difference across the junction, and α≡dF/ξbal\nF. In the limit of large α\nand for Tnear Tc, Eqn. 1 can be simplified to,9\nIcRN=π∆2\n4eT\f\f\f\f\fsin\u0000\ndF/ξbal\nF\u0001\n\u0000\ndF/ξbal\nF\u0001\f\f\f\f\f. (2)\nThe simplified form of the ballistic limit shows 0 −πoscillations at dF/ξbal\nF=nπ, but misses\nthe first transition, which occurs at dF/ξbal\nF≈0.33π.\nThe opposite limit is the diffusive or “dirty” limit, defined by lF, lS<< d F, ξS, ξF– i.e.\nthe limit where impurity scattering is very strong in both F and S. In that limit the Usadel\nequation is valid, and the original calculation produces the result:10,13\nIcRN=π∆2\n4eTc2x\f\f\f\fcos(x) sinh( x) + sin( x) cosh( x)\ncosh(2 x)−cos(2 x)\f\f\f\f, (3)\nwhere x=dF/ξdif\nF. In the limit dF>> ξdif\nFandTnear Tc, that result simplifies to:10,13\nIcRN=√\n2∆2\n4eTc \ndF\nξdif\nF!\nexp \n−dF\nξdif\nF!\f\f\f\f\fsin \ndF\nξdif\nF+π\n4!\f\f\f\f\f, (4)\nwhich predicts 0 −πoscillations at dF/ξdif\nF= (n+ 3/4)π.\nThere are several ways in which real Josephson junctions deviate from the strict assump-\ntions used to calculate the expressions above. We know of only one paper that incorporates\nrealistic models of the Fermi surfaces of the S and F materials into a calculation of Ic.42\nIn addition, strong F materials have different mean free paths for majority and minority\nelectrons, which are not captured in the usual theoretical formulations. In the presence of\n5spin-flip or spin-orbit scattering in F, the length scales governing the decay and oscillations\nare no longer equal,43,44hence one defines independent length scales ξF1andξF2for the two\nphenomena. In addition, it has been shown that the position of the first 0 −πtransition\nvaries depending on the boundary conditions at the S/F interface, including the presence of\nan insulating barrier or a normal-metal spacer layer.44–46\nIt is common to address some of the effects discussed above by utilizing a phenomeno-\nlogical expression for the diffusive regime:\nIcRN=IcRN(0) exp\u0012−dF\nξF1\u0013\f\f\f\fsin\u0012dF\nξF2+ϕ\u0013\f\f\f\f, (5)\nwhere ϕis not necessarily equal to π/4 and IcRN(0) is a fictitious zero thickness fit parameter.\nSpin-dependent scattering generally causes ξF1to decrease and ξF2to increase relative to\nξdif\nF, resulting in the relation ξF1< ξ F2, which is observed in junctions containing weakly-\nferromagnetic CuNi alloys, as seen in Table III.\nSeveral authors have pointed out that the strong elemental F materials, Ni, Fe, and Co, do\nnot satisfy the conditions of validity of the Usadel equation. In such materials the exchange\nenergy is very large, hence the length ξbal\nFis very short. Bergeret et al.47were the first to\ndiscuss the intermediate limit, defined by ξbal\nF<< l F<< ξbal\nS. This limit clearly requires\nthat Eex>>∆, which is true for the strong F materials. The expression for IcRNin the\nintermediate limit involves a sum over Matsubara frequencies, which we do not reproduce\nhere. (See Eqns. (19) and (20) in Ref. 47.) The dependence of IcRNondFhas a similar\nform to Eqn. (5), except now one finds that the oscillation period is governed by the ballistic\nexpression for the exchange length, ξF2=ξbal\nF, while the decay is governed by the mean free\npath, ξF1≈lF. Thus, in the intermediate limit one expects to observe ξF1> ξF2, which is\nindeed the case for many of the materials listed in Table I.\nWhen it comes to fitting experimental data, one requires data over a wide range of\nthicknesses to obtain accurate estimates of all the parameters in Eqn. (5). If only a single\n0−πtransition is observed in the measured thickness range, then the parameters ξF2and\nϕhave large uncertainties and are highly correlated. To remove that correlation, we prefer\nto use the following fitting form:\nIcRN=IcRN(0) exp\u0012−dF\nξF1\u0013\f\f\f\fsin\u0012dF−d0−π\nξF2\u0013\f\f\f\f, (6)\nwhere d0−πis the thickness of the first 0 −πtransition, which is often determined quite\n6accurately in the experiments. Expression (6) is the one we used when analyzing data from\nother groups, to extract the parameters shown in the Tables in Section IV.\nB. Josephson junction geometry: planar vs “sandwich-style”\nMicrofabricated Josephson junctions generally take one of two geometries. In planar or\nlateral junctions, the supercurrent flows in the plane of the substrate, while in “sandwich-\nstyle” junctions, the current flows perpendicular to the plane. Most S/I/S junctions are\nof the sandwich type due to the necessity of keeping the insulating barrier extremely thin.\nIn contrast, most S/N/S junctions studied nowadays are planar so that the supercurrent\nflows over a long distance in the N metal. Exceptions include junctions containing very\nhigh-resistivity metals such as silicides,48or junctions studied long ago on very thick alloy\nfilms.49Most S/F/S Josephson junctions have the sandwich structure due to the very short\nlength scale over which the critical current oscillates and decays in ferromagnetic materials.\nThe notable exceptions are junctions based on half-metallic CrO 250or on single-crystal\nferromagnetic nanowires,51where the supercurrent may propagate over very long distances.\nAll of the Josephson junctions analyzed in the Tables in Section IV have the sandwich\ngeometry.\nC. S/I/F/S vs S/F/S π-junctions\nJosephson junctions are characterized by several parameters. In addition to ∆ and IcRN,\nthese include the Josephson energy, EJ= (ℏ/2e)Icand the Stewart-McCumber parameter,\nβc= (2e/ℏ)IcR2\nNC, where Cis the junction capacitance. The value of βcdetermines whether\nthe junction dynamics are underdamped ( βc>1) or overdamped ( βc<1). Without an ex-\nplicit insulating tunnel barrier, all S/F/S Josephson junctions are overdamped. They can\nbe made underdamped by inserting an insulating barrier to make an S/I/F/S junction,3,52\nor by using a ferromagnetic insulator (FI) as the barrier material: S/FI/S.53–55The differ-\nence between overdamped and underdamped junction dynamics is crucial for all π-junction\napplications – both quantum and classical.\nA related issue relevant for applications is whether the π-junction is being used as an\nactive or passive device. In the former situation, the phase drop across the junction is an\n7important dynamical variable, whereas in the latter situation the π-junction acts as a passive\nphase shifter with the junction phase hardly deviating from πduring circuit operation. For\npassive applications, the π-junction is invariably paired up with conventional S/I/S junctions\nwhose critical currents are much smaller than that of the π-junction. That configuration is\ncommon in applications that require underdamped junction dynamics or large IcRNproduct\n– both of which are difficult to achieve with S/F/S junctions containing a metallic barrier.\nD. Dependence of critical current on applied magnetic field\nA common way to assess the quality of a Josephson junction is to measure the critical\ncurrent vs magnetic field applied in a direction perpendicular to the current flow, Ic(H).\nThe local supercurrent in a junction depends on the local vector potential; in the case of a\nuniform field, the vector potential varies linearly along the direction perpendicular to both\nthe current and applied field.56For the case of a small rectangular junction of width w, with\napplied magnetic field, H, parallel to one of the symmetry axes, the resulting interference\npattern follows the “sync” function familiar from the single-slit diffraction pattern in optics:\nIc(Φ) = Ic0\f\f\f\fsin(πΦ/Φ0)\n(πΦ/Φ0)\f\f\f\f(7)\nwhere Φ 0=h/2e= 2.07×10−15Tm2is the flux quantum for electron pairs, and Φ =\nµ0Hw(2λeff+dbarrier ) is the magnetic flux through the junction, where λeffis the effective\nLondon penetration depth of the S electrodes and dbarrier is the total thickness of the barrier.\nA small junction has wsmaller than the Josephson penetration depth and thus uniform\ncurrent distribution.56Because of the similarity with optics, a plot of such Ic(H) data is\ncolloquially referred to as the “Fraunhofer pattern.” The period of the Fraunhofer pattern\nin the applied field His inversely proportional to the width wof the junction in the direction\ntransverse to the applied field. For the case of circular junctions or elliptical junctions with\nthe field applied along a symmetry axis, the interference pattern follows an Airy function,56\nbut we still refer to the data as the Fraunhofer pattern out of habit.\nThe introduction of magnetic materials inside the junction leads to complications due to\nthe field produced by the magnetic material; the Fraunhofer pattern is modified and may\nbe severely distorted. In principle, if one knew the exact configuration of the magnetization\nMas a function of H, one could calculate the vector potential everywhere and deduce the\n8Ic(H) pattern.57,58The converse is not true, however, except in the simplest cases discussed\nbelow. Consider first the worst situation, which is an S/F/S Josephson junction of relatively\nlarge lateral size (more than a few µm) containing a thick layer of a strong F material such\nas Ni, Fe, Co, or Gd. Due to the complicated multi-domain magnetic configuration of the F\nmaterial, the Ic(H) pattern has a random-looking set of peaks and valleys with no discernible\nperiod and no clear central peak.59–61In such a situation it is neither possible to calculate the\ndomain structure from the Ic(H) data nor to deduce the maximum critical current density\nJcinside the junction.\nFortunately, there exist many situations that produce well-behaved Fraunhofer patterns\nfrom which one can extract key junction parameters – in particular the maximum value of\nJc. An illustrative example is the early work of Ryazanov on a Josephson junction contain-\ning a weakly-ferromagnetic CuNi alloy.62That material has a mild perpendicular magnetic\nanisotropy (PMA),63which is advantageous because the vector potential corresponding to\nan out-of-plane magnetization points in the plane and does not affect the Fraunhofer pat-\ntern. In addition, if both the magnetization and domain size are small, then the vector\npotential effectively undergoes a random walk throughout the sample and never gets very\nlarge in magnitude. For those reasons, Ryazanov’s CuNi virgin-state junction produced an\nexcellent Fraunhofer pattern centered at zero field. After the junction was magnetized in a\nlarge in-plane field, the Fraunhofer pattern was shifted in field with only minor distortion,\nindicating that the CuNi alloy had acquired a nearly-uniform in-plane component to its\nmagnetization. After demagnetizing the sample, the Fraunhofer pattern reverted back to\nits original shape centered at the origin.\nEven in junctions containing strong F materials, the Fraunhofer pattern is well-behaved\nif the magnetization of the material is uniform and parallel to the direction of the applied\nfieldH. If it stays uniform over the full measured field range, then Eqn.(7) is still valid but\nwith a total magnetic flux that includes the contribution from the F material:\nΦ =µ0Hw(2λeff+dF) +µ0M(1–Ndemag)wdF (8)\nwhere Mis the magnetization of the F material and Ndemag is the demagnetizing factor\ndetermined by the shape of the F layer. Most workers using Eqn.(8) ignore the demagnetizing\nfactor, which is small for thin oblong-shaped F layers. Note that Eqn.(8) neglects any\nmagnetic flux that returns inside the junction – a quantity that is difficult to calculate\n9FIG. 1. IcRNvsµ0Hfor Josephson junctions containing a PdFe alloy with 3% Fe with thicknesses\nlabeled in the figure. Junctions have elliptical shape and lateral dimensions of ≈1.3×0.55µm2.\nThe data acquired before the field at which the PdFe magnetization vector reverses direction (solid\nmarkers), and the corresponding fits (lines) show good agreement for both the positive (red, dashed)\nand negative (blue) field sweep directions. The hollow circles are the corresponding data points\nafter the PdFe magnetization switches. These Fraunhofer patterns display magnetic hysteresis\nand are increasingly shifted with larger dF. Adapted with permission from IEEE Trans. Appl.\nSupercond. 27, 1800205 (2017).64\naccurately. Setting the total flux Φ = 0 shows that the central peak of the Fraunhofer\npattern is shifted in field by an amount:\nHshift=−Md F\n2λeff+dF(9)\nwhere we have neglected Ndemag. Note that the denominator should also include the thick-\nnesses of any other non-superconducting layers inside the junction – e.g. the Cu spacers\nused in most of the junctions studied by the authors.\nWe illustrate Eqn. 9 with data from junctions containing a PdFe alloy with about 3% Fe,\nshown in Figs. 1 and 2. We have chosen this data set because this weakly-ferromagnetic alloy\nhas a large value of ξF(several nm, see Table III), hence thickness fluctuations are negligible\nand there is very little scatter in the data of IcRNvsdF. Nevertheless, the magnetization is\nlarge enough to shift the Fraunhofer patterns considerably. The PdFe thickness where IcRN\nhas its peak value in the πstate is about 24 nm. At that thickness, the Fraunhofer pattern\n101101000\n5 1 01 52 02 53 03 5051015200 \n IcRN (µV)a)b\n)π \n µ0Hshift (mT)d\nPdFe (nm)FIG. 2. Characterization of Josephson junctions containing a weakly ferromagnetic PdFe alloy. a)\nThe maximal IcRNproduct of the junctions versus PdFe thickness. The minimum indicates the\nPdFe thickness at which the junctions transition between the 0 and π-phase states. The solid line\nis a fit of the data to Eqn. 6. b) The field shift of the Ic(B) data (the “Fraunhofer pattern”) vs\nPdFe thickness. The field shift increases with dPdFedue to the increasing total magnetic moment\ninside the junctions and is fit to Eqn. 9. Reproduced with permission from IEEE Trans. Appl.\nSupercond. 27, 1800205 (2017).64Copyright 2017, IEEE.\nis shifted by about 14 mT, and the value of Icmeasured at zero field is significantly smaller\nthan the peak value extrapolated from the fit, Figure 1 (c). Hence this material would not\nbe ideal for π-junction applications unless the junctions were considerably narrower in their\nlateral dimension, with correspondingly wider Fraunhofer patterns.\nIII. APPLICATIONS OF π-JUNCTIONS\nSeveral applications of Josephson junctions involve superconducting loops threaded by\nhalf a quantum of magnetic flux, which has two degenerate ground states with supercurrent\ncirculating in opposite directions. The fundamental motivation for using a π-junction in\nplace of the flux bias is to improve the circuit performance in some way, e.g. by reducing\nnoise, reducing the size of the circuit (e.g. by eliminating the need for flux bias lines or a\nlarge geometrical inductance in the loop), or increasing the operating margins.\nEven before Ryazanov’s 2001 experimental breakthrough with S/F/S junctions, there\n11already existed experimental realizations of π-junctions without magnetic materials. The\nmost famous example involved the demonstration of the d-wave character of the high-T c\nsuperconductors, where it was shown that a phase shift of πis created by a Pb/YBCO/Pd\ndouble-junction based on two orthogonal crystal faces of a YBa 2Cu3O7single crystal,65or\nby three YBCO grain-boundary junctions in series.66π-junctions can also be realized in 4-\nterminal S/N/S junctions driven out of equilibrium by a control current flowing orthogonal\nto the supercurrent direction67–69or in 3-terminal junctions with the control current injected\ninto the junction.70–72While the nonequilibrium π-junctions demonstrate fascinating physics,\nwe would argue that they are not likely to be suitable for practical applications.\nA. π-junctions in superconducting qubits\nThe results from d-wave superconductors65,66inspired Ioffe and co-workers to propose\nusing π-junctions in superconducting phase or flux qubits for quantum computing.16,17\nA conventional flux qubit consists of a superconducting loop containing three Josephson\njunctions.73,74Under an applied flux bias of exactly Φ 0/2, states with clockwise and coun-\nterclockwise supercurrent flow are degenerate, and the symmetric and antisymmetric linear\ncombinations of those states can form the two states of a qubit. Replacing the Φ 0/2 flux\nbias with a π-junction is beneficial in several ways. In systems with many qubits the Φ 0/2\nflux bias can be provided either by a global magnetic field or by individual flux lines for\neach qubit. The former suffers from field inhomogeneity and fluctuations in loop area among\nthe qubits while the latter suffers from crosstalk between neighboring qubits and the cor-\nresponding need to calibrate every qubit flux precisely. Both suffer from fluctuations in dc\nsupply current. Replacing the flux bias with π-junctions alleviates those issues. In addition,\nremoving the large static flux bias means that the qubits can be made much smaller, thereby\nreducing sensitivity to magnetic field fluctuations and improving prospects for large-scale\nintegration. We note that individual flux bias lines are still needed for qubit operations, but\nthey supply only small, pulsed currents; there is no need for a large static dc current.\nThe idea to use π-junctions in various types of qubits has been developed further by\nseveral authors,75–81and is being pursued to this day.82–85While in the early proposals, the\nπ-junction was considered to be an active device,75–77later works emphasized the advantages\nof using the π-junction as a passive phase-shifter,78,80,81,85where the only constraint on the\n12critical current of the π-junction is that it be much larger than the critical currents of the\nconventional S/I/S junctions in the loop.\nA crucial issue confronting all types of qubits is decoherence. Kato et al. published the\nfirst calculation of the relaxation and dephasing rates of a flux qubit containing a ferro-\nmagnetic π-junction as a passive phase shifter.78Those authors state explicitly that only\nunderdamped π-junctions with very large critical current density have sufficiently long de-\nphasing times to be suitable for qubit applications. Nevertheless, the first demonstration\nof a ferromagnetic π-junction in a qubit, carried out by Feofanov et al. in 2010,80used\nan overdamped S/F/S π-junction in series with a standard underdamped S/I/S junction to\nform a phase qubit. The choice of using a phase qubit rather than a flux qubit was for device\nsimplicity. Feofanov et al. compared the properties of their π-junction qubit with a reference\nqubit without the π-junction. While there was no noticeable decrease in the decoherence\ntime of the π-junction qubit relative to the reference qubit, both qubits had coherence times\nof only a few nanoseconds. Calculation of the expected decoherence time using the theo-\nretical framework of Kato et al.78gave a result in the nanosecond range, but that theory\nwas not intended to describe the phase qubit of Feofanov et al. Later, Shcherbakova et al.81\nfabricated flux qubits with and without a π-junction and showed explicitly the shift of the\nmagnetic-field dependence of the qubit energy splitting by exactly half of a flux quantum,\nbut they did not measure the decoherence times of their qubits.\nVery recently, Kim et al. fabricated and measured flux qubits with and without an S/F/S\nπ-junction.85Theπ-junction was used as a passive phase shifter with a critical current much\nlarger than those of the conventional S/I/S junctions in the qubit loop. The coherence time\nof the standard qubit was more than 1,000 times larger than that of the phase qubit studied\nearlier by Feofanov et al.80, making this a much more stringent test of how the π-junction\naffects qubit performance. Kim et al. found that the energy relaxation time ( T1) of the\nflux qubit decreased from about 15 µsto 1.5µsupon insertion of the π-junction, while the\nvalue of T2decreased from about 11 µsto 1.5µs. The authors attribute the decreases to\ndissipation in the S/F/S junction due to quasiparticle excitations. While these results are\nsomewhat disappointing, they are not surprising given the insistence by Kato et al. that\nonly underdamped π-junctions are suitable for use in quantum circuits, even when the π-\njunction is used as a passive phase shifter.78Kim et al. suggest using a π-junction based on\na ferromagnetic insulator (FI) rather than a metallic ferromagnet, and we certainly agree.\n13Unfortunately, the only FI that has been studied extensively in Josephson junctions is GdN,\nwhere a number of complex behaviors have been observed. We delay our discussion of GdN\nJosephson junctions to Section V of this review. Another possibility would be to use an\nS/I/F/S junction, if the critical current density can be made large enough and the damping\nsmall enough to satisfy the requirements laid out by Kato et al.\nNowadays it is possible to fabricate so-called “fluxonium” qubits with coherence times as\nlarge as a millisecond.86Like traditional flux qubits, fluxonium qubits also have an optimal\nworking point when flux biased at Φ 0/2.87It would be very informative to repeat the ex-\nperiment of Kim et al. on a fluxonium qubit, but with an underdamped S/FI/S or S/I/F/S\nπ-junction in place of Kim’s S/F/S π-junction.\nB. π-junctions in superconducting digital logic\nA more promising application of π-junctions, in our opinion, is in superconducting digital\nelectronics.88,89There have been numerous proposals for using πphase shifters in supercon-\nducting digital circuits, starting with the pioneering papers by Terzioglu and Beasley in\n199814and Ustinov and Kaplunenko15in 2003, and continuing to the present.90–108The\ninitial motivation for introducing π-junctions into single-flux-quantum (SFQ) digital logic\ncircuits was to eliminate the need for the large geometrical loop inductances needed to store\nsingle flux quanta, thereby reducing the circuit size substantially.15Subsequently it was ar-\ngued that incorporating π-junctions could also alleviate the need for current bias lines and\nincrease circuit operating margins.90More recently, entire new families of superconducting\ndigital logic incorporating π-junctions are being explored, as we will see below.\nThe distinction between active and passive roles of the π-junction is crucial in this context\nof classical digital circuits, just as it was in qubits. The speed at which a Josephson junction\nswitches into the voltage state is limited by the time τswitch =ℏ/(eIcRN). Metallic S/F/S\nπ-junctions without an insulating barrier have very low values of IcRN, hence they are\nnot suitable as switching elements in high-speed circuits. The easiest way around that\nproblem is to use S/F/S junctions only as passive phase shifters embedded in circuits that\ncontain conventional S/I/S junctions.80The critical current of the S/F/S phase shifter must\nbe considerably larger than that of the S/I/S junctions, so that the S/F/S junction never\nswitches into the voltage state and its phase drop stays close to π. An alternative approach\n14is to develop π-junctions with a large IcRNproduct. That is not trivial; simply adding an\ninsulating barrier increases the value of RNby a large factor but simultaneously decreases\nIc– usually by a larger factor.52An ingenious solution is to insert a thin superconducting\nlayer between the insulating barrier and the F layer, to get S/I/s/F/S.109–112Depending on\nthe thickness of the “s” layer, the S/I/s/F/S behaves either as a single Josephson junction\nwith large IcRNproduct, or as a series combination of a standard S/I/S junction and an\nS/F/S junction.\nThe first laboratory demonstration of a single-flux-quantum (SFQ) digital logic circuit\ncontaining intrinsic πphase shifts was reported by Ortlepp et al.90using high-T c/low-T chy-\nbrid Josephson junctions.113Theπphase shifts were introduced into superconducting loops\nby using orthogonal facets of the YBCO a-b plane, as was done in the original measure-\nment of the d-wave symmetry in YBCO.65Ortlepp et al.90demonstrated proper functioning\nof a toggle flip-flop circuit, and they used circuit simulations to compare two identically-\nfunctioning circuits with and without πphase shifters. The circuit with the phase shifters\ncontained fewer current bias lines, took up considerably less area, and had wider operating\nmargins – i.e. better stability against variations in circuit parameters. Shortly thereafter,\nBalashov et al.114demonstrated an all-niobium toggle flip-flop with a πphase shifter based\non the “trapped flux” method.115Theπphase shifter replaces a large geometrical induc-\ntance, hence diminishes the circuit size substantially. Using the same type of trapped-flux\nπ-shifter, Wetzstein et al. compared the bit error rates of toggle flip-flops with and without\naπphase shifter, and concluded that the former had improved robustness against circuit\nnoise.93\nThe first demonstrations of all-Nb SFQ circuits containing ferromagnetic π-junctions were\nreported by Feofanov et al.80and by Khabipov et al.92in 2010. Those works represent a\nmajor milestone because all modern large-scale SFQ circuits are fabricated using only Nb-\nbased superconductors, and because they represent the first example of a scalable process\nfor incorporating π-junctions into SFQ circuits. In comparison to trapped-flux π-shifters,\nS/F/S junctions are more compact and do not require a delicate initialization procedure.\nBoth groups used the π-junction as a passive phase shifter in a circuit containing conventional\nS/I/S switching junctions. Both groups identified circuit size as the main advantage of using\nπ-junctions.\nThere has been a surge of interest in π-junctions in the past few years in Japan. The\n15Yoshikawa group at Yokohama University has shown through simulations that including\nπ-junctions in certain SFQ circuits reduces the junction count significantly, while the new\ncircuits have wider operating margins and lower static power dissipation that their conven-\ntional counterparts.95,98They and others propose similar advantages using π-junctions in\nultra-low-power adiabatic quantum flux parametron circuits.97,106,108The Fujimaki group at\nNogoya University recently invented a new family of superconducting logic called “half-flux-\nquantum” logic, where information propagates in the form of voltage pulses with area equal\nto Φ 0/2 rather than Φ 0as in standard SFQ circuits.94,96,100–104The basic unit of half-flux\nquantum logic was originally a 0 −πSQUID, where the 0-junction and π-junction in the\nSQUID must have similar critical currents and switching properties. The inventors soon\nrealized, however, that it is more practical to use a 0 −0−πSQUID instead, where the\nπ-junction is used as a passive phase shifter and never switches into the voltage state. Half-\nflux-quantum circuits offer several advantages compared to standard SFQ circuits, including\nsmaller bias currents, smaller circuit size, and lower power dissipation.\nAnother new logic family, called “phase logic”, was introduced very recently by the group\nat Moscow State University.105,107A major selling point of phase logic is the absence of induc-\ntors, which take up much of the space in standard SFQ circuits and cause crosstalk between\nneighboring cells, thereby being a major hindrance to scaling those circuits to smaller sizes.\nIn the original proposal, phase logic circuits were based on bistable Josephson junctions\nwith a dominant second harmonic in their current phase relation: Is(ϕ) =Icsin(2ϕ). The\nexistence of such junctions has been demonstrated in special situations,116,117and proposed\nto exist in others,118–121but we emphasize that they are not available as “off the shelf” com-\nponents ready for use in electronic circuits. Hence it was encouraging when Maksimovskaya\net al.107suggested a way to construct phase logic circuits using standard π-junctions. Phase\nlogic appears to be a promising avenue for scaling superconducting circuits to higher densi-\nties, and is being pursued also by circuit designers in the US.122–125\nWe note that the 0 −0−πSQUIDs that form the basic unit of the Nagoya group’s\nhalf-flux-quantum logic are equivalent to a single junction with a negative second harmonic\ncurrent-phase relation: Is(ϕ) =−Icsin(2ϕ).105The ground state of such a SQUID is doubly-\ndegenerate, with a phase shift of ±π/2 across the SQUID. In contrast, the basic unit of the\nMoscow group’s original phase logic proposal is a junction with a positive second harmonic\ncurrent-phase relation, which puts its ground state phase at either 0 or π. Those two\n16situations have very different consequences for circuit design.\nC. Controllable junctions: superconducting memory\nJosephson junctions with controllable properties are suitable for applications in super-\nconducting memory. The controllable property can be either the magnitude of Icor the\nphase state of the junction, but in the former case the junction must be active, i.e. it must\nswitch into the voltage state when the applied current surpasses Ic, which means it must\nhave a large IcRNproduct for high-speed operation, as discussed earlier.\nThere are several proposals for controlling the magnitude of Icin Josephson junctions\ncontaining a single magnetic layer. The Ryazanov group developed controllable S/F/S junc-\ntions containing a weakly-ferromagnetic PdFe alloy.126In a collaboration with the Hypres\nCorporation in the US, that idea was extended to controllable S/I/s/F/S junctions with a\nlarge IcRNproduct.109,110Control of Icwas achieved by the orbital effect – i.e. by shifting\nthe Fraunhofer pattern of the junction via partial magnetization of the PdFe. A disadvan-\ntage of that approach is that it does not scale well as the junction size is reduced. The\njunction size can be reduced in one in-plane dimension, but must remain large in the other\ndimension to achieve adequate Fraunhofer shift.127A recent proposal to control Icvia the\nstray field produced by the magnetic texture of a Co disk also relies on the orbital effect,\nbut the constraints on junction size are less severe because of the large magnetization of the\nCo.128\nSeveral other schemes for controlling Icof Josephson junctions have been proposed, and\nwe do not try to review all of them here. We mention one based on placing Abrikosov flux\nvortices in close vicinity to a Josephson junction,129one based on rotating the magnetization\nin a combination S/F/S and S/F/N/S device,130and another based on a multiterminal S/F\ndevice.131,132\nThe possibilities become richer when there are two independent F layers inside the junc-\ntion – one with fixed magnetization and the other with a magnetization that can be reori-\nented by application of a small external field. Either the magnitude of Icor the phase state\nof the junction can be controlled in such junctions, as predicted in a number of theoretical\nworks.133–141The first experimental verification of Icmodulation in such “spin-valve” junc-\ntions was performed by Bell et al. in 2004, using Co and Ni .80Fe.20(NiFe) as the fixed and\n17free F layer materials, respectively.142A similar experiment was revisited a decade later in a\nmuch more thorough study by Baek et al. ,143and also by Qadar et al. .144Those groups used\nNiFe-based alloys as the free layer because of their excellent magnetic switching properties,\neven though they are not so great for supercurrent transmission (see Table II). Baek et al.\nused Ni as the fixed layer material since Ni has the best supercurrent transport of any strong\nF material (see Table I in Section IV).\nControlling the phase state of the junction rather than the amplitude of Ichas two\nadvantages. First, there is no need for a large IcRNproduct, which is difficult to achieve in\nall-metal junctions. Second, the junction phase is intrinsically a digital rather than analog\nquantity; it can take only one of the two values zero or π, as long as the F-layer is far from\nthe thickness corresponding to the 0 −πtransition. The first demonstration of phase control\nin spin-valve junctions was carried out by one of the authors and his students,145,146using\nNi and NiFe for the fixed and free F-layer materials. Similar junctions formed the basis for\nNorthrop Grumman’s “Josephson Magnetic Random Access Memory” or JMRAM, which\nwas demonstrated on a 2 ×2 memory array in 2017.147A later demonstration on an 8 ×8\nmemory array revealed a difficulty with this approach, namely the inconsistent behavior of\nthe magnetic layers in different memory cells. The Ni fixed layers are multi-domain at the\nµm-scale lateral dimensions of the junctions, and require a very large initialization field to\nmagnetize completely.143And the switching fields of the NiFe free layers varied significantly\nfrom cell to cell. A great deal of materials research would be needed to turn this idea into\na successful memory technology.\nFinally, we mention proposals that involve intrinsic bistability of Icor of the junction\nphase state without requiring any change in the magnetic configuration.148,149\nIV. METALLIC π-JUNCTIONS\nThe Tables in this section are intended for the reader to make quick comparisons between\nmaterials and works. We attempt to include as much of the available literature as possible,\neven where fit parameters could not be extracted from the data; hence there are gaps in the\nTables. We present four Tables on single F layer junctions: I – pure elements, II – NiFe and\nNiFe-based alloys, III – weak ferromagnetic alloys, and IV – strong and other ferromagnetic\nalloys. Within each Table, materials are grouped together, and where there are multiple\n18works on the same material, the works are listed chronologically.\nThe methodology for putting together the Tables in this section is as follows. Fit param-\neters are either extracted from the original publications, or datasets are extracted from the\nplotted graphs, fit to Eqn. 6 and the best fit parameters are reported in the Table. We fit\nthe expression only to the nominal thickness of the ferromagnetic layer, ignoring magnetic\ndead layers and any role of additional buffer layers. The reported uncertainties in the orig-\ninal datasets are not taken into consideration for the fitting, so we quote all fit parameters\nto 2 significant figures without consideration of their uncertainties. For more rigorous data\nanalysis, the reader should refer to the published works, particularly for cases where Eqn. 6\ndoes not provide the best fit to the experimental data. We attempt to note in the main\ntext and figure captions as much as possible where subtleties exist in the experimental data\nwhich are not accounted for in the fitting here.\nThe parameters we report in the Tables include the fit parameters from Eqn. 6, corre-\nsponding to the decay and oscillation lengths ξF1andξF2, the thickness at which the first\n0-πtransition occurs, d0−π, and the fictitious zero thickness fit parameter, IcRN(0), which\nis necessary for the reader to reproduce the fits but doesn’t have a physical meaning. (Junc-\ntions without the ferromagnet will not have the extracted dF= 0 critical current as there\nare fewer interfaces in such devices.) We also extract the critical current at the peak of the\nπ-state, which is a useful parameter for device consideration and for comparing materials.\nWhen the range of thicknesses studied is not broad enough to cover multiple 0- πoscilla-\ntions (multiple minima in IcRN), the ξF2fitting parameter ambiguous, in which case that\nparameter will be missing from the Tables.\nThere are two common conventions for reporting the magnitude of critical Josephson\ncurrent. The first is the product of the critical current and normal state resistance, IcRN,\nquoted here in µV. The second convention is to report the critical current density, Jc,\nquoted here in kAcm−2. The main difference between IcRNandJcis that the junction\narea need not be known to report IcRN, which may be advantageous for a study where\nmultiple junction sizes are fabricated (intentionally or unintentionally). Where the original\npublications contain all the information necessary to perform the conversion, we report\nboth quantities at the peak of the π-state in the Tables. Otherwise, the reader can consider\nthe approximation that for metallic S/F/S junctions the product of area and normal state\nresistance, ARN≈10−14Ωm2, which arises primarily from the boundary resistances between\n19the S electrodes and the F layer.150Hence, for S/F/S junctions, one can estimate Jcby\ndividing IcRNbyARN(in the Tables, IcRNvalues quoted in µV can be multiplied by 10\nto get Jcin kAcm−2assuming ARN≈10−14Ωm2). For junctions containing an insulating\nlayer, such as the S/I/F/S geometry, the story is very different: ARNis typically several\norders of magnitude larger, and varies widely depending on the oxidation conditions used\nto produce the insulating barrier.\nA. Pure elements\nIn Table I, we highlight works studying Josephson junctions with pure elements.\nPure elemental Ni was among the first ferromagnetic barrier layers studied.151Ni has the\nlowest bulk magnetization of the transition metal ferromagnets (550 emu/cm3). Experimen-\ntally, it is found that Ni has favorable supercurrent carrying properties - the decay length is\nrelatively long for a strong ferromagnet and the IcRNis high. These favourable properties\nhave resulted in a number of reports on Ni Josephson junctions.143,147,151–157,160,162,163It is\nworthwhile noting that while Table I is obtained by fitting to Eqn. 6 which describes dif-\nfusive and intermediate limit transport, Ni junctions have been reported in the literature\nin both the diffusive154and ballistic155limits. While Ni may have favourable supercurrent\ncarrying properties, the magnetic properties of thin Ni layers are far from ideal. Ni layers are\nmagnetically ‘hard’, requiring a large initialization field to set the magnetization direction\nand a large switching field. It is well known that Ni films break up into small magnetic\ndomains, which sets a limit on single domain junction size, and can contribute stray fields\ninto the surrounding superconducting layers.\nFor determining the position of the 0 −πtransition for Ni from Table I, we guide the\nreader towards Refs. 155 and 147, which have the smallest increments in thickness around\nthe transitions. The data of Baek et al. covers both the first 0 −πandπ−0 transitions, which\noccur at 0.90 and 3.4 nm respectively.155The first π−0 transition thickness is confirmed by\nDayton et al. who observe it at 3.3 nm, again taking small increments in thickness around\nthe transition.147Most works on Ni do not study the very thin layers of Ni in the range\nof Baek et al. ’s observed 0 −πtransition, and we note that in some cases it may still be\npossible to model the data in those works assuming d0−π= 0.9 nm.\nCritical current oscillations have also been reported in pure elements of Co153,159,160,162\n20BarrierξF1\n(nm)ξF2\n(nm)d0−π\n(nm)IcRN(0)\n(µV)IcRN(π)\n(µV)Jc(π)\n(kAcm−2)dead layer\n(nm)Ref.Comments\nCu/Ni/Cu 3.8 0.98 2.6 2.8 1.0 10 151\nCu(Au)/Ni/Cu(Au) 1.7 0.45 152 ⃝\nNi 4.5 1.2 4.0 450 110 1.3 153\nAl2O3/Cu/Ni 0.66 0.53 3.0 3.7 3.4×10−32.3 154\nCu/Ni/Cu 2.4 0.79 0.90 330 ≥100 0.7 155\nCu/Ni/Cu 3.3⋆200 147 ⃝,†\nNi >30 >300 156 †\nNi & Cu/Ni 157 †\nCu/Co/Cu 158 †\nCo 2.1 0.29 1.0 130 65 0.8 153\nRh/Co/Rh 0.81 0.21 0.64 260 82 1600 0.0 159\nCo/Ru/Co 1.2 2.9 61 ‡\nCu/Co/Ru/Co/Cu 2.3 0.90 61 ‡\nFe 5.4 0.25 0.81 170 140 1.1 160\nAl/Gd/Al 1.2 20 59 ‡, 1 K\nHo 4.3 2300 2.0 161 ‡\nTABLE I. Extracted parameters for Nb based Josephson junctions with pure elements as the\nferromagnet in the barriers. The fitting methodology to Eqn 6 is given in the main text and best\nfit values are quoted to 2 significant figures without consideration to their uncertainty so should\nbe treated as approximations. The full barrier layers, including normal metals and insulators,\nare listed. Unless otherwise commented, the measurements were performed at 4.2 K. Comments:\n⃝The range of thickness values did not cover the thickness where the first 0- πtransition was\npredicted. †The range of thickness values studied was not sufficient for a meaningful fit to theory.\n‡No 0- πoscillations were observed or claimed, and the data are fitted here with only the exponential\ndecay component of Eqn 6. ⋆This thickness corresponds to the first π-0 transition.\nand Fe,160,164although they are less studied than Ni. Potential advantages of Co and Fe are\nthat thin films tend to be magnetically softer than Ni, with larger magnetic domains. An\nunexplored potential is that thin layers of Co can have perpendicular magnetic anisotropy\nwhen deposited on Pt or Pd normal metal spacer layers.\n21Finally we mention that the rare-earth metals Gd59and Ho161have been studied in\nJosephson junctions, including the thickness series to extract the decay length, however\nno critical current oscillations attributed to the 0- πtransition were observed. In the case\nof Gd, this may be due to the thickness increments (step size) between samples in that\nearly work being too large to observe the oscillation. Ho has been extensively studied as\na spin-triplet generator due to its intrinsic magnetic inhomogeneity,165which may make it\nunsuitable as a π-junction. Nb/Gd multilayers have been extensively studied for the obser-\nvation of critical temperature oscillations,33which can guide future junction experiments.\nIt is additionally noteworthy that Nb can seed rare-earth ferromagnet growth and that the\ninterface is considered to be sharp with minimal interdiffusion.166–168\nB. NiFe alloys\nIn Table II, we highlight works studying Josephson junctions with NiFe and NiFe-based\nalloys.\nNi.80Fe.20(hereafter NiFe) is a strong ferromagnetic alloy which is optimized for magnetic\nswitching at small magnetic fields, making it an excellent choice in spintronics research as\na soft magnetic layer. In many papers, NiFe is referred to as permalloy or Py. Josephson\nstudies of NiFe were well motivated by the spintronics community, and it has been used\nas the free layer of choice in superconducting spin valves.142,145,147Additional functionality\noffered by NiFe is the ability to set the direction of in-plane magnetic anisotropy by applying\na magnetic field during growth.\nFrom studying Table II, the reader will notice two possible positions of d0−πfor NiFe,\neither between 1 .3−1.8 nm or at 2.4 nm. We guide the reader towards the work of Dayton et\nal.,147which takes the smallest increments in thickness for NiFe around the 0 −πtransition\nand concludes that d0−π= 1.66 nm, falling in the middle of the 1 .3−1.8 nm range reported\nby other works.147\nAlthough the switching properties of NiFe at room temperature are excellent, the switch-\ning degrades at low temperatures relevant to Josephson junctions. To address this limita-\ntion, NiFe-based alloys with even lower coercivity have been developed174,175and tested in\nJosephson junctions, such as Ni .73Fe.21Mo.06(supermalloy), which we include also in Table\nII. Unfortunately, the decay length inside NiFe-based alloys is experimentally found to be\n22BarrierξF1\n(nm)ξF2\n(nm)d0−π\n(nm)IcRN(0)\n(µV)IcRN(π)\n(µV)Jc(π)\n(kAcm−2)dead layer\n(nm)Ref.Comments\nCu/NiFe/Cu 1.0 0.76 2.4 360 12 170 0.7 169 #\nNiFe 1.6 0.33 1.3 240 78 0.5 153\nNiFe 1.2 0.61 2.4 17 1.2 144\nCu/NiFe/Cu 1.5 0.58 1.8 69 12 160 0.0 170\nCu/NiFe/Cu 1.66 25 147 †\nAl/AlO x/Nb/NiFe 112 †\nNiFe 171 †\nCu/NiFe/Cu 0.71 0.74 1.5 250 10 100 172\nCu/Ni .73Fe.21Mo.06/Cu 0.48 0.96 2.3 150 0.20 1.8 0.2 173\nCu/Ni .65Fe.15Co.20/Cu 1.1 0.48 1.2 30 5 67 0.0 170\nCu/Ni/NiFe/Ni/Cu 0.64 0.78 0.92 800 50 530 172\nTABLE II. Extracted parameters for Nb based Josephson junctions with NiFe and NiFe-based\nalloys as the ferromagnet in the barriers. The fitting methodology to Eqn 6 is given in the main\ntext and best fit values are quoted to 2 significant figures without consideration to their uncertainty\nso should be treated as approximations. The full barrier layers, including normal metals and in\none case an additional ferromagnetic layer, are listed. Unless otherwise stated in the Table, the\ncomposition was Ni .80Fe.20and the measurements were performed at 4.2 K. Comments: # The\nbottom Nb electrode and barrier layers were epitaxial. †The range of thickness values studied was\nnot sufficient for a meaningful fit to theory.\neven shorter than NiFe, which reduces considerably the potential critical Josephson current\nof aπ-junction based on these materials. This is likely due to the short electron mean free\npath in these alloys.\nOther NiFe-based alloys which have been explored for use in Josephson junctions include\nNi.70Fe.17Nb.13, which was used by Baek et al. as the free layer in a spin-valve device.143\nIn later works, the same group used a more dilute non-magnetic concentration of the alloy\nas a normal metal spacer layer in junctions containing ferromagnetic Ni.155,163Qader et al.\ncharacterized Cu 1-x(Ni.80Fe.20)xas a possible barrier layer, however single layer Josephson\njunctions with this alloy have not been reported.176\nThe work of Bell et al. is noteworthy because the bottom Nb electrode and barrier layers\n23were grown epitaxially.169This work may, therefore, not be directly comparable to the other\nworks in Table II. As far as we are aware, this is the only work listed in any of the Tables to\nstudy an epitaxial barrier. Epitaxy plays a role in improving the mean free path of electrons\nthrough metals, which may improve the supercurrent carrying properties of junctions with\nepitaxial barriers. Epitaxy has not yet been systematically studied for π-junctions and may\nprovide fertile ground for future exploration.\nC. Weak ferromagnetic alloys\nIn Table III, we highlight works studying Josephson junctions with weak ferromagnetic\nalloys.\nIn the context of ferromagnetic Josephson junctions, weak ferromagnet refers to materials\nwith a low Curie temperature, and therefore low exchange energy. The structural and\nmagnetic properties of weak ferromagnetic alloys was reviewed in detail by Kupriyanov,\nGolubov, and Siegel.27\nFrom Table III, the reader will notice that although CuNi alloys have been widely studied,\nthey are far from an ideal material as a π-junction. The CuNi alloys are firmly in the diffusive\nlimit where ξF1< ξ F2, which in comparison to other materials presented in the Tables is\nsomewhat of a rare case. Superconducting junctions with weak ferromagnets are a system\nwhere the theoretical Usadel equations are valid, and therefore it is possible in these systems\nto directly test predictions of that theoretical framework. Oboznov et al. apply the Usadel\nequations to model their data, considering that their junctions showed strong spin dependent\nscattering, attributed to magnetic inhomogeneities in the barrier caused by the formation\nof Ni-rich clusters.178Further work studying Nb/CuNi bilayers using scanning tunneling\nspectroscopy support this idea, finding significant spatial variation in the thickness and\ncomposition of Ni in the CuNi layer.184In comparison to CuNi, the Pd-based alloys appear\nmuch more promising as potential π-junctions due to the larger critical Josephson currents\nobserved in the π-state.\nOther works on weak ferromagnetic alloys that are not included in Table III but which\nwe wish to highlight include an experiment where CuNi junctions were fabricated where the\nbarrier layer has a thickness step, such that junction has both 0 and πparts.185Further work\non CuNi includes observation of a positive second harmonic in the current-phase relation in\n24BarrierξF1\n(nm)ξF2\n(nm)d0−π\n(nm)IcRN(0)\n(µV)IcRN(π)\n(µV)Jc(π)\n(kAcm−2)dead layer\n(nm)Ref.Comments\nCu.48Ni.52/Cu 2 ▷ ◁\nCu.52Ni.48 177 ▷ ◁\nCu.47Ni.53/Cu 1.3 3.7 11 2200 0.15 1 4.3 178\nAl2O3/Cu .40Ni.60 0.78 1.35 5.2 400 5.0×10−33.1 52 2.11 K\nAlO x/Cu .40Ni.60 1.2 5.8 1.5×10−2179\nAlO x/Nb/Cu .40Ni.60 1.2 7.2 6.5×10−2179\nCu.47Ni.53 1.3 4.3 7.4 3200 2.0 20 180\nNbN/Cu .40Ni.60/NbN 1.8 2.0 5.6 160 2.2 8.8 181\nCu.48Ni.52/Al 1.3 3.2 15⋆>4.2 >60 182 ⃝\nAl/Al 2O3/Pd .88Ni.12 2.9 2.3 6.5 450 18 3 1.5 K\nPd.88Ni.12 7.7 4.4 7.4∗100 60 ⃝\nNbN/Pd .89Ni.11/NbN 4.4 3.6 8.5 540 ≥18 ≥71 183\nPd.89Ni.11/Al/AlO x 22 2.2×10−2103 †\nAl/AlO x/Nb/Pd .99Fe.01 109 †\nPd.97Fe.03 16.2 7.2 16 100 21 190 2.8 64\nPtxNi1-x 157 ▷ ◁\nTABLE III. Extracted parameters for Nb based Josephson junctions with weak ferromagnetic alloy\nbarriers. In Refs. 181 and 183, the superconductor was NbN, as indicated in the Table. The fitting\nmethodology to Eqn 6 is given in the main text and best fit values are quoted to 2 significant\nfigures without consideration to their uncertainty so should be treated as approximations. The full\nbarrier layers, including normal metals and insulators, are listed. Unless otherwise commented,\nmeasurements were performed at 4.2 K. Comments: ▷ ◁The main focus was to study the temper-\nature and/or compositional variation of the Josephson effect instead of the thickness dependence.\n†The range of thickness values studied was not sufficient for a meaningful fit to theory. ∗This\nvalue was extrapolated from measurements on junctions outside of this thickness. ⃝The range of\nthickness values did not cover the thickness where the first 0- πtransition was predicted. ⋆This\nthickness corresponds to the first π-0 transition.\njunctions with a CuNi thickness close to the 0 −πtransition.116\n25BarrierξF1\n(nm)ξF2\n(nm)d0−π\n(nm)IcRN(0)\n(µV)IcRN(π)\n(µV)Jc(π)\n(kAcm−2)dead layer\n(nm)Ref.Comments\nAl/Al 2O3/Ni3Al 4.6 0.45 ≥1.5⋄5 - 8 186 ⃝\nAl/AlO x/Cu/Fe .75Co.25 0.22 0.79 1.9∗>1000 <0.10 0.6 187 ⃝\nPt/Co .68B.32/Pt 0.28 0.20 0.30 56 7 35 0.0 188 1.8 K\nCu/Co .56Fe.24B.20/Cu 0.10 0.80 19 189\n[Nb/Cu] ML/Co .56Fe.24B.20/Cu 0.13 0.80 9.6 189\nCo.40Fe.40B.20 0.93 440 190‡, 1.6 K\nCo.47Gd.53 0.16 0.21 1.2 1.9 191\nNbN/GdN/NbN 0.40 80000 192‡, 0.3 K\nNbSe 2/Cr2Ge2Te6/NbSe 2 2.5 2.4 8.4 940 11 4.0×10−3193 1.6 K\nTABLE IV. Extracted parameters for Nb based Josephson junctions with strong and other fer-\nromagnetic alloy barriers. In Ref. 192, the superconductor was NbN and in Ref. 193, NbSe 2, as\nindicated in the Table. The fitting methodology to Eqn 6 is given in the main text and best fit\nvalues are quoted to 2 significant figures without consideration to their uncertainty so should be\ntreated as approximations. The full barrier layers, including normal metals and insulators, are\nlisted. Unless otherwise commented, measurements were performed at 4.2 K. Comments: ⃝The\nrange of thickness values did not cover the thickness where the first 0- πtransition was predicted. ‡\nNo 0- πoscillations were observed or claimed, and the data are fitted here with only the exponential\ndecay component of Eqn 6. ⋄This lower limit value is taken at the first observed oscillation, which\nis unlikely the first π-state due to the thickness range studied. ∗This value was extrapolated from\nmeasurements on junctions outside of this thickness.\nD. Strong and other ferromagnetic alloys\nIn Table IV, we highlight works studying Josephson junctions with strong and other\nferromagnetic alloys.\nIt is noteworthy that while most alloy barriers we have included in the Tables show\nsupercurrent transport in the diffusive or intermediate limits, the Ni 3Al barrier was found\nto be in the ballistic limit.186The long ξF1decay length and short ξF2oscillation lengths\nmeant that multiple 0- πoscillations in this material were accessible in the thickness range\n10-18 nm. The thickness range studied was unlikely to cover the first 0- πoscillation, so\nwe do not report a d0−πparameter for this material, and the reported Jc(π) corresponds to\n26the peak of the first oscillation in the measured thickness range, so a higher Jc(π) may be\npossible in thinner barriers. However, a potential downside of Ni 3Al is that the magnetic\ndead layer in this material was very large compared to others.\nThe amorphous Co-based alloys, CoFeB and CoB, are popular in spintronics and mag-\nnetic memory research due to their lack of crystalline anisotropy and weaker pinning of\nmagnetic domain walls due to the reduced density of grain boundaries.188,194For application\nin MRAM, they have been found to have large values of TMR when placed in magnetic tun-\nnel junctions.195In addition, thin layers can have perpendicular magnetic anisotropy when\nplaced on spacer layers such as Pt or Pd.188,194CoGd is an amorphous ferrimagnet, where\nthe composition can be tuned to have zero net magnetization while retaining spin-polarized\ntransport.196In Josephson junction studies, the general conclusion from Table IV is that\nthe amorphous alloys have very short ξF1decay length, most likely related to the short\nelectron mean free path in amorphous metals. This limitation may make them unsuitable\nasπ-junctions. Komori et al. showed that Jccan be enhanced in CoFeB by thermally\nannealing the junctions - which has been shown to crystallize the layer.190,197\nE. Advantages of weak vs strong ferromagnets (length scales)\nAn important consideration for constructing a π-junction is to consider the ratio of\nd0−π/ξF1. Since the supercurrent decays exponentially with ξF1, if the 0- πtransition oc-\ncurs at too great of a thickness with respect to ξF1, the resulting π-junctions will have small\ncritical Josephson currents.\nThe materials that we can describe as strong ferromagnets in Tables I, II, and IV have\nd0−π/ξF1ratios which can be ≤1. While this is great for optimizing the critical Josephson\ncurrent, one of the difficulties of fabricating π-junctions with strong ferromagnetic barriers\nis that the length scales ξF1andξF2are both short – particularly in the amorphous alloys\nin Table IV, where ξF1andξF2are<1 nm. As a result, very precise control over film\nthickness during deposition is needed to create a thickness series of samples with a small\nenough increments (step size) to observe the 0- πoscillations. For integrating junctions into\nan industrial process, the layer thickness margin of error for reproducibility of the π-junction\nwill be small, although sub-nm thick magnetic layers are routinely used in MRAM, so while\ndifficult, this level of control on an industrial scale is not impossible.\n27Weak ferromagnets offer the advantage that ξF1andξF2can be much longer, up to several\nnm in the Pd-based weak alloys (Table III), which similarly offer the d0−π/ξF1ratio≤1.\nLonger length scales in the Josephson junctions offer the advantage that the parameters of\naπ-junction, such as Jc, should in principle be more robust against small sample-to-sample\nvariations in the thickness of the barrier. In the weak ferromagnets, CuNi is somewhat of an\noutlier having a particularly large ratio of d0−π/ξF1, causing the critical Josephson current\nin the πstate to be small.\nF. Magnetic dead layers and normal metal spacer layers\nFerromagnetic materials are often modelled as slabs with uniform magnetization, however\nin reality they are rarely so simple. At the interface of a thin magnetic film, it is often ob-\nserved that the moments of the first few atomic layers may not align, reducing the observed\nmagnetization from the expected one. These are commonly referred to as magnetic dead\nlayers. Furthermore, the presence of such magnetic dead layers can degrade key properties\nsuch as the coercive field of the layer. Nb/ferromagnet interfaces are well known for having\nmagnetic dead layers160and much effort has been dedicated to studying the related inter-\nfacial roughness at Nb/ferromagnet interfaces.198,199Where possible, we have included the\nthickness of magnetic dead layers in the Tables.\nThe most reliable method to determine the presence of dead layers is to measure several\nsamples of varying ferromagnet thickness. The magnetometer returns a measurement of\nthe total magnetic moment of the sample which, so long as due care has been taken in the\nhandling of the sample and the data corrections, can be used to report the area normalized\nmagnetic moment.200If the moment/area is linear with ferromagnet thickness, then the total\ndead layer thickness ( di) can be determined according to moment/area = M(dF−di).\nAdditional normal metal layers between the Nb and ferromagnet layer are often intro-\nduced in the junctions, and we have included such layers as part of the “barrier” in the\nTables. These additional layers are commonly referred to in the literature as either buffers,\nseeds, spacers, or interlayers. Here we will use the term spacer layers to describe any addi-\ntional normal metal layers. The motivation for adding the spacers is often to provide better\nlattice matching for the ferromagnet to be deposited.\nConsidering the example of Co, the fcc spacer layers Cu, Pt and Rh have been studied.\n28The predicted mechanism is that the fcc spacer layers promote fcc growth of Co, which\notherwise may have a mixed fcc/hcp phase. The two desirable benefits that Co grown on\nthe fcc spacer layers gain compared to Co grown directly on Nb is that the magnetic dead\nlayer is reduced or removed completely, and that the improved Co crystal structure can\nimprove supercurrent carrying properties.61,159,201\nAs a counter example, the inclusion of Cu spacer layers in junctions containing Ni and\nCuNi alloys may be detrimental. Cu and Ni alloy very readily which can cause the Ni to\ninterdiffuse out of the intended layer and form interfacial alloy layers. Bolginov et al.180argue\nthat early works on CuNi, Refs. 4 and 178, contain large magnetic dead layers because of\ninterdiffusion of the Ni out of the CuNi layer into an adjacent Cu layer (which was added as\na necessary step in the fabrication process). By improving the fabrication process to exclude\nthe extra Cu layer, the resulting junctions without interdiffusion are shown in Table III to\nhave an order of magnitude improvement in Jc(π).180Kapran et al. describe a similar effect\ncomparing single layer Ni and bilayer Cu/Ni junctions, where the former junctions have an\norder of magnitude higher Jcfor the same thickness of Ni.157In addition to this Ni specific\nissue, a known downside of adding normal metal spacer layers is that every interface in the\nbarrier may reduce the critical Josephson current. In particular, normal metals with known\nlarge interface resistances should be avoided.202\nA further motivation for adding spacer layers is that they can provide a smoother surface\nfor growing the very thin ferromagnetic layers on compared to Nb. Thin film Nb has a\ncolumnar growth of grains which can result in a surface roughness comparable to the thick-\nness of the ferromagnetic barrier layer in a π-junction. Certain metals including Al, Au,\nand Cu have been shown to act as planarization layers which significantly reduce the surface\nroughness. We highlight two methods making use of this property as they are employed\nin works featured in the Tables. The first is to replace the bottom Nb electrode with a\nsuperlattice of Nb and a thin normal metal, where the superlattice will have lower surface\nroughness.203–205Several works in the Tables use this method, however they are not distin-\nguished in the Tables on the assumption that the use of a thick superlattice is unlikely to\naffect the reported junction parameters. The second approach is to place a much thinner\nsuperlattice on the surface of the single layer Nb electrode in place of the normal metal\nspacer layer, see Ref. 189 in Table IV.\n29BarrierIcRN(max)\n(µV)Jc(max)\n(kAcm−2)∆Ic(P vs AP)\n(%)Phase\ndetectionRef.Comment\nCo/Cu/NiFe 2 20 50 142\nNiFe/Cu/Co 3.5 25 206\nCu/Ni .70Fe.17Nb.13/Cu/Ni/Cu 4 50 500 143\nCu/Cu .70(NiFe) .30/Cu/NiFe/Cu 1 2.5 380 144\nNi/Cu/Ni 66 860 210 207\nCu/NiFe/Cu/Ni/Cu 11 110 93 yes 145\nNiFe/Cu/Ni yes 147 §\nCu/NiFe/Cu/Ni/Cu 9 90 800 208\nCu/NiFe/Cu/Ni/Cu 6.5 65 340 yes 146\nPt/Co/Pt/Co .68B.32/Pt 0.5 5 60 209 1.8 K\nTABLE V. Summary of literature on Nb based spin-valve Josephson junctions. The full barrier\nlayers, including normal metal spacer layers are listed. For single junction measurements, IcRN\n(max) corresponds to the largest critical Josephson current reported and ∆ Ic(P vs AP) is (the\ndifference in Icbetween P and AP)/(the lesser of Icin P or AP) expressed as a percentage,\nwhere P is the parallel and AP the antiparallel magnetic alignment. Unless otherwise commented,\nmeasurements were performed at 4.2 K. Comments: §The magnetic junctions were passive and\ntherefore the switching parameters were not directly measured.\nG. Spin-valve junctions\nIn Table V we highlight works studying controllable spin-valve junctions.\nFor spin-valve junctions we extract the parameters that may be useful for potential ap-\nplications, including the maximum critical Josephson currents, IcRNandJc, achieved in the\nwork, and what was the largest observed difference in critical current between the parallel\nand antiparallel magnetic alignments. The first quantity is an important consideration where\nthe junction is passive and does not switch (as the critical current must be larger than the\nactive junctions in the circuit) and the second is useful for memory schemes where readout\nis achieved by measuring directly the state of the spin-valve junction (superconducting or\nnormal) or any other application where large differences between the magnetic alignments\nis desirable.\n30It is possible that for the entries in Table V the two magnetic alignments correspond also\nto a change in phase difference between 0 and π. This can be inferred based on the 0 −π\ntransition of single layer junctions reported in the other Tables. Direct detection of the\nphase was achieved in spin-valve junctions by placing the junctions into a phase sensitive\nSQUID circuit.145–147\nThe highest reported critical current in Table V was for a junction where both magnetic\nlayers were pure Ni, which is not surprising from studying the presented Tables for single\nlayer junctions. While using NiFe as the free layer is magnetically beneficial (NiFe has much\nimproved switching properties compared to Ni) the resulting junctions have a much reduced\ncritical current.\nV. NON-METALLIC AND EXOTIC π-JUNCTIONS\nFerromagnetic insulators (FIs) have been predicted to show 0- πtransitions.54,77,210\nProgress in this area has been more difficult due to the limited availability of FI mate-\nrials, where GdN and Eu chalcogenides have been the most studied for potential integration\ninto superconducting devices. Of the available studies, the only FI material to show a\nJosephson current in S/FI/S junctions is GdN.55GdN Josephson junctions have shown\na range of interesting phenomena, suggesting that the physics inside these junctions is\ncomplex.55,83,117,192,211–215Complex behaviors of GdN junctions includes observation of a\npure second harmonic current-phase relation.117We also note an experiment on hybrid\njunctions containing InAs barriers coated in the FI EuS, where switching between 0 and π\ncoupling is attributed to the magnetic domain structure of the EuS.216\nCaruso et al. studied the thickness and temperature dependence of supercurrent in GdN\njunctions.192The thickness dependence at 0.3 K resulted in an exponential decay, where\nthe fit parameters are shown in Table IV. By studying the temperature dependence of the\njunctions, the authors characterize an incomplete 0- πtransition with temperature accom-\npanied by spin-triplet physics.192Follow up work reports on the possibility to quantitatively\ndescribe and eventually control spin-triplet transport through external magnetic fields which\nis correlated to an “extended” 0 −πtransition in temperature in these GdN FI barriers.215\nThe van der Waals family of materials have only recently been explored in the context of\nπ-junctions. In this case, both the superconductor and magnetic layer must be compatible\n31with the van der Waals fabrication method, so typically NbSe 2, ans-wave superconductor,\nis used. Reports on junctions and SQUIDs with a barrier of Cr 2Ge2Te6(which is described\nas either a ferromagnetic semiconductor or ferromagnetic insulator) showed a coexistence of\n0 and πphase in the junction region.217,218Another work studied the thickness dependence\nof S/F/S Josephson junctions with the Cr 2Ge2Te6barrier showing a pronounced minimum\nin supercurrent at a thickness of 8.4 nm, attributed to the 0 −πtransition.193Fitting Eqn 6\nto the thickness dependence yields the fit parameters shown on the final line of Table IV. In\na planar junction, a long ranged supercurrent was shown across a Fe 3GeTe 2van der Waals\nbarrier.219\nThe 0 −πtransition can occur in Josephson junctions without magnetic barriers if the\ngeometry of the junction allows for Zeeman splitting in a large enough magnetic field. The\nidea is that in a S/N/S junction, the normal metal in a large enough magnetic field will\nresemble a weak ferromagnet as the up and down spin bands are displaced by the Zeeman\nenergy.220,221An important factor here is that the applied magnetic field must be parallel to\nthe direction of supercurrent propagation, such that it does not contribute to the Fraunhofer\npattern discussed previously. Although an early attempt to observe a so-called Zeeman π-\njunction with the normal metal Ag by one of the authors was unsuccessful,72later works\nwith non-metallic barriers were successful. Zeeman π-junctions have been realized with\nbarriers of the Dirac semimetal Bi 1-xSnx,222the 2D material graphene,223the topological\ncrystalline insulator SnTe,224the two-dimensional electron gas system InSb,225,226and in a\nquantum dot.227In some of the successful implementations, the large g-factors and spin-orbit\ncoupling in these materials contributed along with the Zeeman effect to the observation of\n0-πtransitions. In the InSb junctions, it was possible to apply a gate voltage which could\ninduce the 0- πtransition at a set applied magnetic field.225\nElectron transport across a quantum dot is heavily influenced by the strong Coulomb\ninteraction, including the possibility of observing single electron tunneling.228A Josephson\njunction can be formed across a quantum dot when pairs can coherently tunnel. In a\nJosephson quantum dot junction, 0 −πtransitions are possible by tuning the occupancy\nlevels. Experimental realizations of π-junctions in quantum dots were achieved in an InAs\nnanowire229and in carbon nanotubes.230–232\nSeveral examples in this section explore planar junctions, as opposed to the sandwich\ngeometry of all the works presented in the Tables. Planar junctions open additional possi-\n32bilities for experimental control, such as adding voltage gates and manipulating strain by\nuse of piezoelectric substrate. Another possibility in planar junctions is to control the 0 −π\ntransition using only the geometry of the junction by introducing curvature.233,234\nVI. SPIN-TRIPLET JUNCTIONS\nThe data in the Tables show clearly that the length scales governing the decay and period\nof the 0- πoscillations, ξF1andξF2, are both very short – typically less than 1 nm in the\nstrong F materials and up to several nm in the weak F materials. In the late 1990’s, however,\nthere were some experimental hints of a long-range proximity effect in S/F systems based on\nmeasurements of the electrical resistance of long F wires connected to an S electrode.235–237\nThose results foreshadowed the 2001 theoretical breakthrough by Bergeret et al. mentioned\nin the Introduction. Further experimental support for the theory appeared in 2006,238,239\nbut the experimental breakthroughs that finally convinced the skeptics in the community\ndid not appear until 2010.50,165,240,241Those works all showed that the supercurrent in an\nS/F/S junction decayed over a distance much longer than ξFif the F layer contained the\nappropriate kind of noncollinear magnetization needed to convert spin-singlet Cooper pairs\nfrom the S electrodes into spin-triplet pairs in the central region of the junction. In the case\nof the “half-metal” CrO 2, a supercurrent can propagate hundreds of nanometers through the\nF material.50,242Since that time, there have been several reviews of spin-triplet proximity\neffects in S/F systems;21–24,243we do not review that large body of work here.\nA. Controllable π-junctions\nVery few experimental studies of spin-triplet supercurrents address the issue of the\nground-state phase difference across the junction. The theoretical papers, however, make\nspecific predictions about this issue.20,244–248Here we discuss only the sample geometry sug-\ngested by Houzet and Buzdin in 2007,244which was realized in many experiments. In that\ntheoretical work, the sequence of layers in the junction is S/F′/F/F′′/S, where the directions\nof the magnetizations of the three F layers can be controlled independently. (In experimen-\ntal devices, normal spacer layers are placed between adjacent F layers to prevent exchange\ncoupling between them.) In particular, the magnetizations Mof any two adjacent layers\n33must be non-collinear; let us choose adjacent layers to have orthogonal magnetizations to\nmaximize generation of spin-triplet supercurrents. If we define the direction of M′as the\nz-direction and the direction of Mas the x-direction, and if we keep all three magnetizations\ncoplanar, then the direction of M′′may be either plus or minus z. According to theory, the\nformer case produces a π-junction whereas the latter produces a 0-junction.20,244–246The\nonly experimental work that tests this prediction, to our knowledge, is the one by Glick\net al. in 2018.249To achieve the required orthogonal magnetizations, those authors used a\nPd/Co multilayer with perpendicular anisotropy as the central F layer. (Actually, two such\nmultilayers were placed back-to-back separated by a Ru spacer layer to achieve a synthetic\nantiferromagnet. That does not change the 0 −πphysics, according to theory.245,246) Those\nauthors used Ni as the fixed F′′layer and NiFe as the free F′layer. Seven out of eight\njunctions showed robust πphase switching when the direction of the NiFe magnetization\nwas reversed by a small applied field.\nFrom a practical perspective, spin-triplet junctions appear less attractive than the simpler\nspin-valve junctions discussed in Section III C. They do have one advantage, however: the\nconstraints on the thicknesses of the F layers needed for switching between the 0 and π\nstates of the junctions are less stringest in the case of spin-triplet junctions than they are\nfor spin-valve junctions.250\nB. ϕand ϕ0-junctions\nThe spin-singlet physics discussed in the majority of this article would appear to allow\nonly junctions with ground-state phase differences of 0 or π. That is not true, however.\nA spatially-extended junction containing only a single F layer with varying thickness, such\nthat some parts of the junction prefer the 0-state while others prefer the π-state, can have\ndegenerate ground states at phases ±ϕ.251–255Such junctions are called “ ϕ-junctions,” where\nthe phase ϕcan take any value between 0 and π. Such junctions still obey the conventional\ntime-reversal symmetry relation, Is(−φ) =−Is(φ), where φis the phase difference between\nthe superconducting condensates in the two S electrodes.\nSpin-triplet junctions containing three F-layers have an additional possibility, if the three\nmagnetizations are non-coplanar. In that case, time-reversal symmetry is broken, and cannot\nbe restored by a simple rotation of the entire spin system.256Such junctions may have a\n34ground-state phase difference of ϕ0, which can take any value between 0 and 2 π, and is\nnondegenerate on that interval. Such junctions are now called ϕ0-junctions, to distinguish\nthem from the ϕ-junctions discussed in the previous paragraph. The existence of ϕ0-junctions\nwas predicted in a number of theoretical works starting in 2007,256–266and two reviews\nhave appeared recently.267,268Because of the broken symmetries, ϕ0-junctions do not obey\nIs(−φ) =−Is(φ), hence this phenomenon is also referred to as the “anomalous Josephson\neffect”, or AJE. Since 2016, the AJE has been experimentally observed in a wide variety of\nexotic systems with strong spin-orbit interaction218,269–275and in planar out-of-equilibrium\ndevices,276but so far it has not been observed in metallic sandwich junctions.\nVII. OPEN CHALLENGES AND OUTLOOK\nWe hope that the presented Tables act as both a record of current literature and inspi-\nration for the future direction of the field. While magnetic π-junctions are somewhat of\na mature field, we feel from the presented Tables that a Goldilocks junction material that\npasses a large supercurrent in the π-state while having well controlled magnetic properties\nis still to be discovered. Such a material is particularly needed to realize practical spin-valve\njunctions. In this section, we outline some open challenges and provide an outlook on the\nfield.\nA. Spread in junction parameters and influence of fabrication method\nFrom examination of the Tables in this work, the reader may find themselves overwhelmed\nwith seemingly contradictory information on the same material - a thickness of ferromagnet\nthat provides a π-state from one work may not directly translate to the π-state in another\nwork. For Ni and NiFe, the two most studied barriers, the authors have directed the reader\ntowards what they consider to be the most reliable value for the thickness d0−π.\nIn some cases, the explanation for spread in the Tables may be as simple as differences\nin the way the ultra-thin layer thicknesses were calibrated, or that the range of thicknesses\nstudied unintentionally missed a 0 −πtransition. In other cases, the explanation may be\nmore complex. The decay length inside the ferromagnet, ξF1, is shown in the Tables to vary\nconsiderably between works. This parameter is sensitive to the exact layer morphology,\n35which itself is influenced by the presence or absence of normal metal spacer layers. We also\nmust consider what role, if any, does the junction fabrication process have on the measured\nparameters in the Tables. Bolginov et al. provide a detailed account of improvements made\nto their fabrication process, which resulted in a significant difference to the parameters,\nparticularly the thickness of d0−π, in junctions measured with CuNi barriers.180\nTwo common fabrication methods are based around either the ion beam etching or focused\nion beam techniques, which are described in detail elsewhere.204,277Here, we highlight a\nsignificant difference in the two methods which may affect junction parameters. The ion\nbeam etching process typically requires that the top Nb electrode be deposited separately\nto the bottom Nb electrode and barrier layers, post-fabrication and etching. To protect the\nbarrier layers from oxidation during fabrication, a normal metal capping layer must be added\nabove the barrier layers. In focused ion beam fabrication, both bottom and top electrodes as\nwell as the barrier layers can be deposited in the same vacuum cycle, meaning no additional\ncapping layers are required by the processing. We speculate that this difference may account\nfor the generally higher critical Josephson currents observed in focused ion beam junctions,\nalthough there has been no systematic study comparing the two techniques to date.\nBased on the discussion in this section, the authors highly recommend that while the\nTables in this work can be used to help choose materials to implement as π-junctions in\nan application, that researchers don’t assume that parameters provided in the Tables will\nexactly reproduce in their own process. It is therefore necessary to conduct a full systematic\nstudy of a material using the deposition and fabrication methods needed for the ultimate\napplication to determine the exact parameters for junctions produced by that process.\nB. Magnetic initialization\nIn the as-grown state, the direction of the F layer’s magnetization in a junction may be\nuncontrolled and could have broken up into magnetic domains. In a single F layer junction,\nthe physics driving the 0 −πoscillations should not depend on the direction of magnetiza-\ntion. However, for in-plane magnetization, the Fraunhofer pattern of an individual junction\ncan be strongly influenced by the direction and domain state of the F layer, as discussed in\nSection II D. Since determining the critical Josephson current in the junction relies on inter-\npretation of the Fraunhofer response, it may be preferable to initialize the magnetization to\n36a known direction and to minimize domains. Another case where setting the magnetization\ndirection of the ferromagnet in the junction is crucial are spin-valve junctions, where it is\nessential to know and control the relative orientations of the magnetic layers. For junctions\nwith perpendicular magnetic anisotropy, the magnetization is perpendicular to the applied\nfield so the Fraunhofer pattern should be less influenced and therefore magnetic initialization\nmay not be necessary.\nTo aid control of the in-plane magnetization direction in a junction, it is usual to create\na shape anisotropy by defining the junction area as an ellipse, where it is favourable for the\nmagnetization to point in the long axis. Even with a shape anisotropy, for a single domain F\nlayer, there are now two directions that the magnetization could point. When the magnetic\ndomains are small the situation is worse, as the magnetic layer may break up into domains\npointing in many different directions, and stray magnetic fields may emerge from the domain\nwalls.\nThe standard procedure, to set the direction of magnetization and minimize the number\nof domains, is to initialize the magnetic layer in a large magnetic field. The magnetic field\nmust be applied while the circuit is far below the Curie temperature – which might be below\nroom temperature in the case of very thin magnetic layers – but above the superconducting\ncritical temperature of Nb to avoid flux trapping. For Ni, where the small domain size is\nparticularly problematic, it is typical to apply a global initialization field of ≥350 mT to\nsaturate the Ni magnetization.155A superconducting digital logic circuit is unlikely to be\nimplemented in a cryogenic environment which includes the ability to apply a large global\nfield, hence it would be favourable to have a magnetic layer which forms a single magnetic\ndomain and does not require an initialization field procedure.\nIn some materials, such as NiFe, magnetic anisotropy can be set by a growth field during\ndeposition, which may be enough to set the direction of magnetization without the need for\nadditional initialization fields. Another mechanism which can set the magnetization is the\nexchange bias effect between a ferromagnetic and antiferromagnetic material.278However, it\nhas been found that supercurrent is severely suppressed in Josephson junctions containing\nmetallic antiferromagnetic barriers.279–281\n37C. Materials compatibility with Nb or CMOS fab (MRAM example)\nMRAM is an example where integrating the magnetic junctions into the traditional\nCMOS wafer fabrication has been extremely difficult. Generally, the magnetic layers of\nthe MRAM are incompatible with the CMOS process. One of the reasons for this is that\nelevated temperatures used for the CMOS process would cause the thin magnetic layers to\ninterdiffuse. The solution is that the standard CMOS wafer is manufactured in one front-\nend-of-the-line facility, then shipped to a separate facility for backend-of-the-line processing,\nwhere the MRAM is built. This approach has several technological disadvantages, including\nseverely limiting circuit design possibilities.\nFor superconducting electronics, although a multi-facility approach is possible without\ncompromising the overall transport properties of individual junctions,92,109,110,282,283in our\nopinion it would be best to avoid the multi-facility approach so that circuit design is not\nlimited. In 2019, MIT Lincoln Laboratory announced a new fabrication process for su-\nperconductor electronics called PSE2, which integrates Ni π-junctions - an important step\ntowards using π-junctions in logic circuits.156But we are not aware of any published circuit\ndemonstrations using that process. As we have previously motivated, pure Ni may not be\nthe ideal choice of material due to the high initialization field required and small magnetic\ndomains.\nThe Tables presented here might motivate the reader to revisit pure Fe or to pursue\nother Fe containing alloys (NiFe, PdFe, etc.) as potential π-junction materials. However,\nwe would urge caution on exploring further materials that include Fe when considering\nincorporation into an integrated fabrication process. Fe may be incompatible with scaled\nfabrication processes due to the potential detrimental effect Fe contamination would have\non the rest of the process.\nOur conclusion from a materials compatibility perspective is that Ni based alloys are well\nmotivated for further work, particularly those that are in the limit Kapran et al. describe as\n“strong-but-clean”, meaning strong ferromagnetism with ballistic transport properties.157\nWe also mention that it may be possible to explore alloys and compounds for magnetic\nbarriers where the component materials are otherwise non-magnetic, such as PdMn,27some\nHeuslers,241,284andL10-ordered MnAl285or MnGa.286Such an approach may negate any\nconcerns about ferromagnetic particle contamination in a Nb process.\n38D. Materials and read/write operations for spin-valves\nWe identify some outstanding challenges specific to spin-valve junctions. Firstly, to\nachieve a spin-valve junction with high critical current, it seems necessary to replace the\nNiFe free layer. Secondly, we must remove the requirement for global magnetic field for\nread/write operations and move towards on-chip operations.\nThe materials for spin-valve junctions have several requirements and important consid-\nerations: the fixed layer must be robust enough against the switching of the free layer, both\nlayers should be single domain so that the supercurrent and phase difference across the\njunction is uniform, both layers must be sufficiently thin to allow a significant supercurrent\nto propagate through the junction, magnetic dead layers (which can dominate in very thin\nlayers) must be minimized, and the 0 −πoscillations of each component layer must be well\ncharacterized. It is not obvious from the Tables presented in this work that there is an\nimmediate replacement for either NiFe as the free layer or Ni as the fixed layer.\nOn the prospect of on-chip writing, we note that there are adventurous proposals for\nswitching magnetic layers in the superconducting state of S-F systems,287–290although there\nis not yet experimental realization. We therefore suggest that established mechanisms used\nin the field of spintronics and technologies for MRAM can also be applied in the normal\nstate of a superconducting spin-valve. Spin transfer torque switching, where a current pulse\nis passed through the junction and induced magnetization dynamics causes the free layer\nto switch, was demonstrated by Baek et al. by applying a current density to their junction\nwhich greatly exceeded Jc.207Spin orbit torque switching, where a current pulse is passed\nthrough a layer with large spin Hall angle adjacent to the free layer causing it to switch, has\nshown promising results in room-temperature magnetic tunnel junctions, and promises faster\nswitching speeds than spin transfer torque.291Raytheon BBN Technologies have proposed a\nlow temperature superconductor-ferromagnet hybrid memory (not based on the Josephson\neffect) where magnetic switching is achieved by spin Hall effect.292\nE. Altermagnets\nSolids with an intrinsic magnetic phase are traditionally classed as having either fer-\nromagnetic or antiferromagnetic ordering. Recently, there has been great interest in the\n39fundamental study and potential applications of an emerging third class of magnetic solid,\nwhere the magnetic phase is known as altermagnetism.293Altermagnets have properties that\nblur the lines between the traditional ferromagnetic and antiferromagnetic orderings. On the\none hand, these materials have electronic properties consistent with ferromagnetism, they\nshow an anomalous Hall effect and have spin-polarized conduction bands. On the other\nhand, altermagnetic materials have antiparallel magnetic crystal order and zero net magne-\ntization, consistent with antiferromagnetism. Altermagnets are expected to be abundant in\nnature.293\nThe presence of a spin-polarized conduction band suggests that much of the rich physics\npresent in ferromagnet-superconductor hybrid systems are also present when the ferromagnet\nis replaced by an altermagnet, along with additional unique physics in the new altermagnet\nsystem. Notably in the altermagnet, the finite pair momentum appears in the absence of a\nnet magnetization.294Zero net magnetic moment junctions offers the significant advantages\nover ferromagnet systems that the junctions produce either zero or negligibly small stray\nmagnetic fields, and they don’t require any magnetic initialization.\nOf particular interest to the topic of this review are predictions of 0- πoscillations in\nJosephson junctions containing altermagnetic barriers.294–297To date, there are no experi-\nmental studies of altermagnet-superconductor hybrid systems. The most studied candidate\naltermagnet is RuO 2, a metal with a bulk room-temperature resistivity of ρ≈35µΩ-cm.298\nF. Neuromorphic and novel analogue computing\nMagnetic Josephson junctions have been proposed as artificial synapses for neuromorphic\ncomputing schemes.299–301Neuromorphic computing is seen as a potential solution to the\nproblem of poor energy efficiency when running neural networks on traditional computing\narchitectures. The Tutorial of Schneider et al. covers the topic of neuromorphic systems\nbased on magnetic Josephson junctions in detail.302Here we focus on the materials and\nrequirements for this application.\nThe requirements in this potential application is for the junction to provide a continuously\ntunable (analogue) response, provide a degree of plasticity/memory, and be scalable and\nenergy efficient enough for the architecture required to interconnect many such devices\nneeded for neuromorphic application.\n40So far, synapse behavior in magnetic Josephson junctions has been achieved in magnetic\nnanoclusters. The magnetic clusters do not spontaneously order, but can be aligned by\nsubsequent current pulses. The magnetic order can be tuned continuously from the dis-\nordered to ordered magnetic states, and the degree of order affects the critical current of\nthe junction.302–305Similar to the π-junctions presented in the Tables, there will be a wide\nvariation of performance in metrics such as the IcRNof barriers with different magnetic\nnanoclusters. We are aware of two magnetic nanocluster barriers which have been stud-\nied for this application, Mn clusters embedded in Si303and Fe clusters embedded in Ge.304\nCurrent limitations are that the IcRNproducts in these barriers are quite a low and the\nMn clusters in Si required an annealing step. A potential alternative to nanoclusters are\nultra-thin magnetic layers, sometimes called dusting layers, which can also be close to the\nsuperparamagnetic transition.\nVIII. CONCLUSIONS\nReaders new to this field may be overwhelmed by the sheer number of materials listed in\nthe Tables in Section IV, as well as the wide variability in parameters extracted by different\nresearch groups. To avoid that parting impression, we finish the review by expressing a few\nof our own opinions.\nFirst, if we had to choose a material from the Tables to make highly-reproducible π-\njunctions that can tolerate small variations in thickness, we would choose a PdNi alloy with\na Ni concentration in the range of 10 - 15%. Pham et al. report values of JcandIcRN\nin the π-state as large as 70 kA/cm2and 18 µV, respectively, in Pd 89Ni11junctions with\nNbN superconducting electrodes.183Because of its mild perpendicular anisotropy, junctions\ncontaining PdNi may not need magnetic initialization. The only drawback of this material\nis the cost of Pd; but we remind readers that only very small amounts of the material are\nneeded.\nSecond, if one wants the highest possible critical current density in a π-junction, then Ni\nis the obvious choice. But Ni junctions require magnetic initialization in a rather large mag-\nnetic field to establish a reproducible magnetized initial state. Without that initialization,\nthe Fraunhofer pattern will vary randomly from sample to sample, yielding non-reproducible\nvalues of Jc.\n41Third, if one wants a magnetically-soft free layer for a spin-valve junction, then NiFe is\nthe best choice so far. Our own attempts to dilute NiFe to lower its magnetization (and\nhence the magnetic switching energy) have been largely unsuccessful; in the case of doping\nwith Mo or Cr, the resulting material has a short mean free path, leading to a steep decay\nofJcwith thickness.\nAnd finally, we lament the heavy use of CuNi alloy by groups new to the field. We\nacknowledge the many “firsts” achieved by the Ryazanov group and their collaborators using\njunctions containing CuNi, but the high rate of spin-flip scattering in the material causes\nJcto decay very steeply with thickness, leading to very small values of JcandIcRNin the\nπ-state. The main advantages of CuNi are: i) ξFis long due to the small magnetization, so\nthickness fluctuations are tolerated; ii) the perpendicular anisotropy avoids distortions of the\nFraunhofer pattern; and iii) the material is less expensive than PdNi. We would recommend\nusing that material only when junction size is not an issue, so that the small value of Jcis\ntolerated. But we much prefer PdNi alloy.\nIn summary, we believe that π-junctions have an important role to play in the devel-\nopment of superconducting digital electronics. We hope that this review will help workers\nnavigate the vast literature as they explore new materials for π-junctions.\nACKNOWLEDGMENTS\nN.O.B. wishes to thank all the students who have worked with him on superconduct-\ning/ferromagnetic hybrid systems over the past two decades. Special thanks also go to B.\nBi, D. Edmunds, R. Loloee, and W.P. Pratt, Jr., without whom none of this work would\nhave been possible, and to the JMRAM team at Northrop Grumman Corporation, especially\nA.Y. Herr, D.L. Miller, O. Naaman, N. Rizzo, T.F. Ambrose, and M.G. Loving. N.S. wishes\nto thank G. Burnell and everyone who has worked with him on this topic, and acknowledges\nsupport from new faculty startup funding made available by Texas State University. We ac-\nknowledge helpful suggestions from F.S. Bergeret, A.I. Buzdin, A.A. Golubov, S. Jacobsen,\nM. Kupriyanov, O. Mukhanov, Z. Radovic, V.V. Ryazanov, I.I. Soloviev, and A.F. 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Lett.\n121, 240501 (2022).\n69"    },    {        "title": "1310.3590v1.Unveiling_the_Origin_of_the_Insulating_Ferromagnetism_in_LaMnO3_Thin_Film.pdf",        "content": "Unveiling the Origin of the Insulating Ferromagnetism in LaMnO 3 \nThin Film  \nY . S. Hou , H. J. Xiang*, and X. G. Gong* \nKey Laboratory of Computational Physical Sciences (Ministry of Education), State \nKey Laboratory of Surface Physics, and Department of Physics, Fudan University, \nShanghai 200433, People’s Republic of  China  \n \nBy combining  genetic algorithm  optimizations , first -principles calculations and the \ndouble -exchange model studies, we have unveiled that the exotic  insulating \nferromagnetism  in LaMnO 3 thin film originates from  the previously unreported \nG-type \n2 2 2 23z r x ydd orbital ordering. An insulatin g gap opens as a result of both the \norbital ordering and the strong electron -phonon coupling. Therefore , there exist two \nstrain induced phase transition s in the LaMnO 3 thin film, from the insulating A -type \nantiferromagnetic phase to the insulating ferromag netic phase  and then to the metallic \nferromagnetic phase. Th ese phase transition s may be exploited in tunneling  \nmagnetoresistance  and tunneling electroresistance related devices.  \n \n \n \nPACS number(s):  31.15.E -, 75.50.Dd , 75.70.Ak , 73.50. -h \n \n \n \n  Perovskite LaMnO 3 (LMO), a fundamental s trongly correlated electron system and \nthe parent compound of the  colossal magnetoresistance manganites , displays a \ncomplex correlation among structural, orbital, magnetic, and electronic degrees of  \nfreedom  and has been extensi vely studied o ver the past decades  [1]. Recently, LMO \nthin films attract considerable attentions not only because ma ny interesting and \nemerging  phenomena  are discovered in LMO thin film related superlattices [2-5], but \nalso because LMO thin films display exotic behavior different from bulk . \nExperiment s discovered  a surprising insulating ferromagnetic phase in LMO thin \nfilms  epitaxial ly grown on the square -lattice SrTiO 3 (STO) substrate [2, 7-13]. The \norigin of this puzzling insulating FM phase is unclear. The unexpected \nferromagnetism was considered to be extrinsic  and deficiency of La [10, 14] was \nsuggested as a possible explanation to it . However, it is not consistent with the fact \nthat FM phase in the LMO thin films with cation deficien cy tend s to be metallic [7, 15, \n16]. Previous models [17, 18] and recent first-principles studies  [19, 20] on LMO thin \nfilms predicted the metallic FM phase, instead of the experimentally observed \ninsulating FM phase. It is important to address how the prototypical antiferromagnetic \nMott insulator  transforms to an insulating ferromagnet .   \nTo answer such an intriguing question, we carry out  a comprehensive theoretical  \nstudy on the  LMO thin film by combining genetic algorithm  (GA) optimizations , \nfirst-principles calculations and the orbital -degenerate double -exchange (DE) model \nstudies. Our e xtensive GA simulations  give an insulating FM phase of the LMO thin \nfilm, which  crystallize s in the monoclinic P21/n  structure. This monoclinic P21/n  \nstructure has a peculiar arrangement pattern  of Jahn -Teller (JT) distortions of MnO 6 \noctahedra , giving rise to the previously unreported three -dimensional ly \n2 2 2 23alternatedz r x ydd\n orbital order. According to the Goodenough -Kanamori \nrules [21], this kind of orbital orders induces  three -dimensional ferromagnetism. \nFirst-principles calculations show that the band gap of the monoclinic P21/n  FM \nphase is 0.16 eV , compatible with the experimentally observed excitation energy [2] \n(\n0.14 eV ). By virtue of the orbital -degenerate DE model, it is revealed that the band gap opens as a result of both the orbital ordering and the strong electron -phonon \ncoupling. We also show that LMO thin film transform s from the insulating A -type \nantiferromagnetic  (A-AFM) phase to the insulating ferromagnetic phase , and then to  \nthe metallic ferromagnetic phase when the lateral lattice constant decreases.   \nIn this work, the widely adopted global optimization technique  GA [22-25] is \nused to search the ground state structure of the  LMO thin film. The GA method we \nemployed here is  similar to that described  in Ref. [25] with the exception  that the spin \ninformation of the magnetic ions is explicitly kept in the crossover operat ion and a \nmutation operation related to the spin direction is also added . A \n2 2 2  \nperovskite supercell containing  four formula units (20 atoms ) is used. In GA searches, \nin order to match the experimentally epitaxial strain induced by the square -lattice \nsubstrate STO,  the two in -plane lattice vectors are fixed at \n2LMO LMO STO a b a  \nwhere \naSTO  is the lattice constant (\n3.905Å ) of the cubic STO substrate. However, \nboth the length and the direction of the out -of-plane lattice vector  \nLMOc and the \ninternal ionic coordinate s are fully optimized. First-principles calculations based on \ndensi ty functional theory  are performed using the generalized gradient approximation \nwith the  PW91 parameterization plus on -site Coulomb  repulsion U method (GGA+U) \nas implemented in Vienna  Ab Initio Simulation  Package  (V ASP)  [26-30]. Because the \nexperimentally measured [31] \n3.5 eVU  describe s well the bulk LMO (see Sec. I \nof the Supplemental Material) [32] and gives a band gap (0.16 eV) of the LMO thin \nfilm close  to the experimentally observed  excitation energy [2] (\n0.14 eV ), it is \napplied to Mn 3 d electrons in the present work . A further detailed study showed that \nour main results remain correct if a reasonable U value is used (see Sec. II of the \nSupplemental Material) [32]. \nOur calculations  show that , Pbnm  LMO thin film strained  on STO is metallic \nFM, consistent with the recent work of Lee et al . [20]. By ex tensive GA search es, \nhowever, we find an insulating FM phase, which has a total energy lower than the metallic Pbnm  FM phase by \n6.9 meV f.u.  (Fig. 1). This insulating FM phase \ncrystallizes  in a slightly distorted monoclinic P21/n  structure with \n090.74 . Two \nkinds of Mn ions, denoted as Mn -A and Mn -B, are found in the monoclinic P21/n  \nstructure (Fig. 2b) while all Mn ions  are equivalent in the orthorhombic Pbnm  \nstructure (Fig. 2a). Mn-A has a MnO 6 octahedron elongated along  the c axis (Fig. 2c) \nwhile Mn-B has a MnO 6 octahedron stretched  in the ab plane (Fig. 2d). What is  most \nsignificant is that these two different kinds of Mn atoms are arranged in a \ncheckerboard G -type manner  (Fig. 2b). This is rather different from the bulk LMO \nwhere the relevant  \n3Q modes [1] of th e JT distortion s of all MnO 6 octahedra are \nwith their principal axes lying within  the ab plane and these axes are alternatively \narranged in this  plane .     \nThe band structure of the majority spin  (up) of the monoclinic P21/n  FM phase \nis plotted  in Fig. 3a.  Obviously, the monoclinic P21/n  FM phase is insulating. It has \nan indirect band gap of 0.16 eV , compatible with the experimentally observed  \nexcitation  energy [2] \n( 0.14 eV ) . Moreover , its bandwidth is narrower than that of \nthe metallic Pbnm  FM phase (Fig. 3c). We expect that the insulating P21/n  FM phase  \ncan be confirmed by the a ngle resolved photoemission spectroscopy  (ARPES) \nexperiment.  \nWe now examine the orbital order in the insul ating  P21/n  FM phase. Since the \nMnO 6 octahedron  of Mn -A is elongate d along the c axis, the \n223zrd  orbital is lower \nin energy than the \n22xyd  orbital  and thus the  single \nge  electron occupies the \n223zrd  \norbital, which can be evidenced by the PDOS (Fig. 4a). The notable  split between the \npeak of the \n223zrd  orbital and that of the \n22xyd  orbital  is consistent with the large \nJT distortion [1] \n22\n23 Å 0.27AQ Q Q   . In contrast, the orbital occupation is just \nopposite for Mn -B. The \n22xyd  orbital is lower in energy as a result of the in -plane \nstretch of MnO 6 octahedron and thus the  single \nge  electron mainly occupies the 22xyd orbital, which can also be verified by the PDOS (Fig. 4b) . Likewise, its \nrelative weak split between the peak of \n223zrd  orbital and that of \n22xyd  orbital is \ndue to the small JT distortion \n22\n23 Å0.14BQ Q Q   . Fig. 3b shows the \nge  charge \ndensity integrated from -1.5 eV  to the Fermi level, which displays the orbital order  of \nthe insulating P21/n  FM phase . In this energy interval, the spectral density of Mn-A \nhas predominantly  \n223zrd character  while Mn-B has predominantly \n22xyd  character . \nTherefore, we firstly report that the insulating P21/n  FM phase has a type of \nthree -dimensional ly \n2 2 2 23alternatedz r x ydd  orbital order, which is different from \nthe \n2 2 2 23 x z r ydd  type [18] (Fig. 3d) and completely different from the \n2 2 2 233/x r y rdd\n type [33].             \nBased on the established  orbital order, the magnetic interactions among the Mn \natoms in the  insulating P21/n FM phase can be deduced according to the \nGoodenough -Kanamori rules [21]. As a result of the three -dimensional ly  \n2 2 2 23alternatedz r x ydd\norbital order, the half-filled \nbond  \n223zrd  orbital  from \nMn-A overlaps with the empty \nbond  \n223zrd  orbital from Mn -B along the cubic \n[001] axis through the middle \nO2zp  orbital, giving rise to strong ferromagnetic \ninteractions. Besides , the half -filled \nbond  \n2gt  orbitals will give rise to an \nantiferromagnetic  interaction between Mn -A and Mn -B. Since the overlap of the \nbonding\n electrons is greater than that of the \nbonding  electrons, the \nferromagnetic interaction turns out to be stronger than the antiferromagnetic one. \nTherefore the superexchange interaction between Mn -A and Mn -B along the cubic \n[001] axis is ferromagnetic.  This mechanism also applies to the magnetic  interactions \nbetween Mn -A and Mn -B in the ab plane, where the mutually  interacting orbitals are \nthe half -filled \nbonding  \n22xyd  orbital of Mn -B and the empty \nbonding  \n22xyd\n orbital of Mn -A. To sum up, Mn -A and Mn -B interact ferromagnetically along [100], [010] and [001] axes (see Sec. III of the Supplemental Material) [32]. The \ndeduced magnetic interactions among Mn atoms are verified by density functional \ntheory (DFT) calculations by means of the four-states mapping method [34]. The \nconsidered nearest neighbor (NN) magnetic interaction paths  \nccJ , \n1abJ  and \n2abJ  are \nshown in Fig. 2b. As expected, our calculations find \n9.73 meVccJ , \n13.91 meVabJ\n and \n25.94 meVabJ . All of them are ferromagnetic. The next \nnearest neighbor (NNN) magnetic interactions a re found to be  much weaker than the \nNN ones. Therefore, the magnetic ground state is ferromagnetic. With the DFT \ncalculated magnetic exchange constants , our MC simulations (see Sec. IV of the \nSupplemental Material) [32] lead to a transition temperature  \n446 KCT , higher than \nexperimentally measured ones [7, 11] ranging from 115K to 240K. A possible \nexplanation to this discrepancy  is that the epitaxial strain is  gradually  relaxed from t he \nsubstrate to the film surface, thus t he FM  phase  locate s just  near the strained \nfilm-substrate interface  while the bulk -like A -AFM phase dominates at the film \nsurface, similar to the case of the s train-induced ferromagnetism in the \nantiferromagnetic LuM nO3 thin film [35].    \nIn order to unveil the mechanism of the insulating ferromagnetism, we employ \nthe orbital -degenerate DE model  with one \nge  electron per Mn3+ ion. An infinite \nHund coupling limit  [1], i.e., \nHJ , is adopted  in our study. With this useful \nsimplification, the DE model Hamiltonian reads       \n   \n 1 1 2 3\n2 2 2\n1 2 3H Ω ..\n12 (1)2.aa\nij i j AF i j i i i xi i zi\nij ij i\ni i i\nit d d H C J S S Q n Q Q\nQ Q Q\n     \n         \n\n  \n\n \nA detailed description of this Hamiltonian is given in the Sec. V of S upplementary \nMaterials  [32]. Note that the second term, the NN antiferromagnetic superexchange  \ninteraction  between Mn \n2gt  spins, is left out of considerations in the present work \nsince both the P21/n  and the Pbnm  phases are FM. The DFT relaxed structures of the P21/n  FM and the Pbnm  FM phases  are used.  Because  the breathing modes (\n1Q) are \nmuch smaller than the JT modes (\n2Q and \n3Q) in both phases, they are ignored in our \nstudy. Lastly  \n00.52 eVt  and \nλ1.4  (\nλ usually  estimated [1] between 1.0 and \n1.6) are used [36, 37]. This set of parameters results in  an energy difference between \nthe P21/n FM and the Pbnm  FM phases close to that obtained from the DFT \ncalculations. By exactly  diagonaliz ing the DE Hamiltonian, we show the band \nstructures  of the P21/n  FM and Pbnm  FM phases in Fig. 3a and Fig. 3c, respectively. \nAs expected, the band structure of P21/n  FM phase is characteristic of insulator and \nthat of Pbnm  FM phase  is characteristic of metal. The profiles of both band structures \nobtained from the model well reprod uce that obtained from the DFT calculations, \nindicating that the parameters involved  in the model are appropriate ly selected.  \nNow let ’s address why the P21/n  FM phase  is insulating while the Pbnm  FM \nphase is metallic.  For the P21/n  FM phase, as the three -dimensionally \n2 2 2 23alternatedz r x ydd\n orbital order has almost no overlap along the c axis \n(\n0zz\nab batt ) between the occupied orbitals, its bandwidth \n21/WPn  is determined \nmainly by the in -plane overlap (\n03\n4ab baxxt t t  or \n03\n4ab bayyt t t ) between the \nlower -energy occupied orbital \n223zrd  of Mn -A and the lower -energy occupied orbital \n22xyd\n of Mn -B. So  its bandwidth \n21/WPn  is proportional  to \n03t . Likewise , the \nbandwidth \nWPbnm  of Pbnm  FM phase determined by the  \n2 2 2 23typez r x ydd  \norbital order is proportional  to \n03t . Thus , P21/n  FM phase has a narrower bandwidth \nthan Pbnm  FM phase. On the other hand , the onsite energy  splitting of the two \nMnge\n orbitals due to the electron -phonon coupling  in an isolated MnO 6 \noctahedron  is proportional to the strength of the JT distortions [19]: \n22\n23 (2). QQ \n \nThus , the onsite energy splitting  in P21/n  FM phase is much larger than that in Pbnm  FM phase since the JT distortion s in the former are much severe r than that in the latter. \nTherefore, P21/n  FM phase possesses  a finite band gap  as a result  of the narrow  \nbandwidth and the large on -site \nge  level splitting  (Fig. 4a and Fig. 4b). For the \nPbnm  FM phase, however, its bandwidths of Mn \nge  bands are so wide and the \nelectron -phonon coupling is so weak that no band gap opens (Fig. 4c) . Furthermore, \ndecompositions  of the total energies of both phases (Table I) show that, the \nelectron -phonon coupling causes a much larger energy lowering in the P21/n  FM \nphase than the  Pbnm  FM phase  although  P21/n  FM phase has higher hopping energy \nand lattice elastic energy than the Pbnm  FM phase. In conclusion, it is revealed that \nthe orbital order and the electron -phonon cooperatively make the P21/n  FM phase \ninsulating while  the Pbnm  FM phase metallic and that the electron -phonon coupling \nplays a vit al role in stabilizing  the insulating P21/n  FM phase as the ground state of \nthe LMO thin film strained  on STO.  \nFinally, using the same strategy applied to the LMO thin film strained on STO, \nwe have systematically  investigated the  property dependence of LMO  thin films on \nlattice constants , which can be experimentally tuned by selecting different \nsquare -lattice substrates . Here, only the P21/n  and Pbnm  space groups are considered. \nAs usual , several common magnetic orders  in perovskite  are considered , namely, FM, \nA-AFM , C-type AFM (C -AFM) and G-type AFM (G -AFM) spin ordering s. The \nresults are shown in Fig. 1. As the C -AFM and G -AFM spin orders do not appear as \nthe lowest energy phase in the considered lattice constant window, they are not shown \nin Fig. 1 and not further discussed. Our results show that LMO thin films with lattice \nconstant ranging from 3.88 Å to 4.03 Å are insulating FM with the monoclinic P21/n  \nstructure and the three -dimensionally \n2 2 2 23alternatedz r x ydd  orbital order. \nHowever, LMO thin films with lattice constant s smaller than 3.88 Å are metallic FM \nwith the Pbnm  structure and \n2 2 2 23typez r x ydd  orbital order . When the lattice \nconstant is larger than 4.04 Å, the ground state of the LMO thin films becomes the \nPbnm  structure  with an insulating A -AFM order and \n2 2 2 233/ typex r y rdd  orbital order . The metallic Pbnm  FM and the insulating Pbnm  A-AFM phases are consistent \nwith the work of Lee et al. [20] on LMO thin film with large compressive and tensile \nepitaxial  strain. The phase transitions from the insulating P21/n  FM ( P21/n  FM I) \nphase to the metallic Pbnm  FM ( Pbnm  FM M) phase and then to the insulating Pbnm  \nA-AFM ( Pbnm  A-AFM I) phase are intuitively  illustrated by the obvious \ndiscontinuity  of the lattice constant \nc , shown in the upper  panel of Fig. 1. The phase \ndiagram shown in Fig. 1 is also confirmed by our GA optimizations . \nIn summary, the physical  origin of the well -known and puzzling insulating \nferromagnetism experimentally observed in the LaMnO 3 thin film grown on the \nsquare -lattice SrTiO 3 substrate has been investigated . We find that the insulating \nferromagnetic phase is intrinsically fr om strain induced orbital ordering, instead of \nextrinsic reasons  such as defects.  It crystallizes in a monoclinic P21/n structure , which  \nhas two different kinds of MnO 6 octahedra: one is elongated along the \nc  axis and \nthe other one is stretched  in the \nab  plane. They are arranged in a  checkerboard \nG-type manner , giving rising to a previously unreported three -dimensional ly \n2 2 2 23alternatedz r x ydd\n orbital order, which naturally leads to the ferromagnetism. \nDouble Exchange  model reveals that the band gap opens due to  both the orbital \nordering and the strong electron -phonon coupling. Finally, we find that epitaxially \nstrained LaMnO 3 thin film transform s from the insulating A -type antiferrom agnetic \nphase to the insulating ferromagnetic phase , and then to  the metallic ferromagnetic \nphase when the lateral lattice constant decreases. If LMO thin film is epitaxially \ngrown on some specified piezoelectric material s, an electric -field-induced \nmetal -insulator transition and an electric field control of the magnetism can be \nrealized experimentally at \nFM M 21 FM I Pbnm P n   and \n21 FM I A AFM IP n Pbnm \n phase boundaries, respectively.  These electric -field \ninduced phase transition s may be exploited in tunneling  magnetoresistance  (TMR ) \nand tunneling electroresistance (TER ) related devices.  \nWork was partially supported by NSFC, the Special Funds for Major State Basic \nResearch, Foundation for the Author of National Excellent Doctoral Dissertation of China, The Program for Professor of Special Appointment at Shanghai Institutions of \nHigher Learning, Research Program of Shanghai municipality  and MOE.  We thank \nfor H. R. Liu, X. Gu and J. H. Yang, X. F. Zhai and S. Dong for valuable discussion s.  \n \n \n \n \n \n \nReferences  \n[1] E. Dagotto, T. Hotta, and A. Moreo, Phys. 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Dagotto, Physical \nRevie w B 78, 201102 (2008).  \n[34]  H. Xiang, C. Lee, H. -J. Koo, X. Gong, and M. -H. Whangbo, Dalton . Trans . 42, \n823 (2013).  \n[35] J. S. White M. Bator, Y . Hu, H. Luetkens, J. Stahn, S. Capelli, S. Das, M. Dö beli, \nTh. Lippert, V . K. Malik, J. Martynczuk, A. Wokaun , M. Kenzelmann, Ch. \nNiedermayer, and C. W. Schneider, Phys. Rev. Lett. 111, 037201 (2013).  \n[36]  Z. Popovic and S. Satpathy, Phys. Rev. Lett. 84, 1603 (2000).  \n[37]  K. H. Ahn and A. J. Millis, Phys . Rev. B 61, 13545 (2000).  \n[38]  J. Rodriguez -Carvajal, M.  Hennion, F. Moussa, A. H. Moudden, L. Pinsard, and \nA. Revcolevschi, Phys . Rev. B 57, R3189 (1998).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figures  \n \nFigure  1 (Color online) Calculated Phase diagram of LaMnO 3 thin films.  With in the \nconsidered lattice constant window, LMO thin films transform from metallic  (M) \nPbnm  FM to insulating (I) P21/n  FM, then to the insulating Pbnm  A-AFM phases. \nLMO thin films  with the nearest neighbor Mn -Mn di stance close (or equal ) to that of \nbulk LMO [38] (\n3.99a Å) do not take the bulk structure because  in this case LMO \nthin film  is slightly stretch ed along the a axis and seriously compressed along the b \naxis. The evolution of the lattice constant \nc  of the ground state is given in the upper \npanel. The two red straight line s are guides to the eyes. The magnetic  and orbital \norders of each ground state are depicted in the middle panel . Blue arrows represent \nspins. Bottom panel gives the total energies o f the P21/n  A-AFM, P21/n  FM, Pbnm  \nA-AFM and Pbnm  FM phases as a function of the square -lattice substrate lattice \nconstant  \na. Black arrow indicates the lattice constant of the SrTiO3 substrate. Note \nthat P21/n  FM phase merges with the Pbnm  FM phase when the lattice constant \nbecomes smaller than 3.88 Å and that P21/n  structure merges with the Pbnm  structure \nwhen the lattice constant becomes larger than 4.03 Å. \n \n \n \nFigure  2 (Color online) Structure of LaMnO 3 thin films strained on SrTiO 3. (a) \nStructure of the metallic Pbnm  FM phase. (b) Structure and magnetic  interaction  paths  \nccJ\n, \n1abJ  and \n2abJ  of the insulating P21/n  FM phases . MnO 6 octahedra of Mn -A (c) \nand Mn -B (d) of  the insulating P21/n  FM phase  are shown . The ir local coordinate \nsystem s shown between  (c) and (d) are chosen in such a way that the local z axis is \nnearly along the c axis and xy plane nearly in the ab plane. Crystallographic axes are \ngiven by a, b and c . All numbers give Mn -O bond lengths in units of  Å.  \n \nFigure  3 (Color online)  Band structure and charge density of LaMnO 3 film strained on \nSrTiO 3. Band structures of the insulating P21/n  FM and the metallic Pbnm  FM phases  \nare shown in (a) and (c), respectively. Black solid ( red dashed) lines in both (a) and (c) \nare from the DFT  calculations (orbital -degenerate double -exchange model). The \nFermi level is set to zero. DFT calculated \nge  charge density distributions of the \ninsulating P21/n  FM and the metallic Pbnm  FM phases in the energy window of 1.5 \neV widths just below the Fermi level  are plotted in (b) and (d), respectively .  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure  4 (Color online)  Partial density of states (PDOS) and onsite energy splitting of \nLaMnO 3 thin film strained on SrTiO 3. PDOS of Mn -A, Mn -B of the insulating P21/n  \nFM phase and the Mn atom of the metallic Pbnm  FM phase are shown in the upper \npanels of the (a), (b) and (c), respectively. The Fermi level is indicated by the black \nvertical lines and set to be zero. D iagrammatic sketch es of the onsite energy splitting  \nof the two \nMnge  orbitals shown in bottom panels of (a), (b) and ( c) correspond to  \nMn-A, Mn -B of the insulati ng P21/n  FM phase and the Mn atom of the metallic Pbnm  \nFM phase, respectively. The parameter \n  is the energy difference between the \n223zrd\n and the \n22xyd  orbitals. The black (red) rectangular  box represents the band \nderived from the  \n223zrd  (\n22xyd ) orbital and the green horizontal line s indicate the \nFermi level.  \n \n \n \n \n \n \n \n \nTable I. Decompositions of the total energies (\ntotE ) of the insulating P21/n  FM and \nthe metallic  Pbnm  FM phases (20 atoms, namely, four formula units) into the hopping \nenergy (\nhoppingE ), electron -phonon coupling  (\nele phE ) and l attice elastic energy  \n(\nlatticeE ).   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary Materials for  \nUnveiling the Origin of the Insulating Ferromagnetism in LaMnO 3 \nThin Film  \n \n  Y. S. Hou, H. J. Xiang*, and X. G. Gong*  \n \nKey Laboratory of Computational Physical Sciences (Ministry of Education), State \nKey Laboratory of Surface Physics, and Department of Physics, Fudan University, \nShanghai 200433, P. R. China  \n \n1. A detailed study on the bulk LaMnO 3  \nWith the experimentally measured  [1] \n3.5 eVU  and \n0.9 eVJ  applied \nto the Mn 3 d electrons, our full structural optimizations  successfully  reproduce the \nA-type AFM (A -AFM) magnetic ground state in the bulk L aMnO3 (LMO) with a \nlower total energy than the FM state by amount of 8.5 meV/formula unit (f.u.), \nalthough it is not a trivial task to achieve this within  DFT framework  [2]. The \ncalculated direct gap  is 1.2 eV , consistent with the optical measurements  [3]. \nBesides, the relevant  magnetic exchange constants  are calculated to be \n2.09 meVabJ\n and \n1.13 meVcJ , respectively . They are consistent with \nexperimentally measured  [4] \n1.85 meVabJ  and \n1.1 meVcJ . With these \ncalculated magnetic exchange constants , Monte Carlo  (MC) simulations (Fig. S1)  \nreveal  that the transition temperature  \n116NTK , well consistent with the \nexperimentally observed  one [5] \n140 KNT . Intriguing ly, our systematical \nstudies (Fig. S2)  clearly  show  that either  too small or too  large U cannot correctly \nobtain the A -AFM magnetic ground state . We find that U should be between 2.5 \neV and 4. 0 eV to reproduce the A -AFM ground state.   \nFig. S1 The dependence of the energy difference  between the A -AFM and FM \n(upper panel), the Jahn -Teller distortions of the A -AFM state (middle panel) and \nthe FM state (bottom panel) on the U. Jahn -Teller Q is defined as \n22\n23 Q Q Q .   \n \n    \nFig. S2 Specific heat capacity of bulk LaMnO 3 as a function of temperature from \nMC simulation s with different lattice sizes (up to 39200  sites). A sharp \n  peak  \nof the specific heat capacity locates  at \n116 KNT . \n \n2. A detailed study on the impact of U upon the epitaxial ly strained  LaMnO 3 film \nwith SrTiO 3 lattice constant  \n Detailed study show that the monoclinic P21/n state is always more stable \nthan the Pbnm  state when a reasonable U between 2.5 and 4. 0 eV that describes \nwell bulk LaMnO 3 is adopted  (Fig. S3) . It is found that for U less than 2.0 eV , the \nmonoclinic P21/n state cannot be stabilized because  the small U significantly  \nunderestimates  the JT distortions. In other words, only the Pbnm  structure  is \nobtained. For the U larger than 3. 0 eV , the  P21/n FM phase becomes insulating \nand has a n appreciably  lower total energy than the metallic Pbnm  FM phase . \nMoreover, both the band gap of the insulating P21/n  FM phase and the total \nenergy difference between the insulating P21/n  FM and the metallic Pbnm  FM \nphases increase with U. It is worthy of noting that any U ranging from 1.0 eV to \n6.0 eV can not open a band gap in the Pbnm  FM phase.  \n \nFig. S3 Dependence of the relative stability between  the P21/n FM and the \nmetallic Pbnm  FM phases , the band gap and the Jahn -Teller distortions of  the \nP21/n  FM phase on the U. \n \n3. Origin of ferromagnetism in the insulating P21/n  phase  \n    In order to investigate  the magnetic interactions  in the insulating P21/n  phase, \nferromagnetic (FM) and antiferromagnetic (AFM) spin co nfigurations  between \nMn-A and Mn -B (Fig. S4) are considered. If Mn -A and Mn -B ferromagnetically \ninteract along the [001] axis, the occupied \n223zrd  orbital of Mn -A can strongly \nhybridize (\ntype ) with the unoccupied \n223zrd  orbital of Mn -B, and thereby \nthe total energy  is lowe red significantly (upper panel of Fig. S4). However, if they \ninteract  antiferromagnetically, the hybridization between the occupied \n223zrd  \norbital of Mn -A and the unoccupied \n223zrd  orbital of Mn -B are almost negligible  \nbecause  these two orbital has a large energy difference due to the Hubbard \nrepulsion U (upper panel of Fig. S4). Besides, the hybridizations  (\ntype ) \nbetween \n2gt  orbitals of Mn -A and Mn -B are also negligible. Thus, there is only a \ntiny energy gain for the antiferromagnetically  interacting case. Therefore, we find \nthat the  FM interaction between Mn -A and Mn -B along the [001] axis is preferred \nto the AFM interaction. A similar  mechanism applies to the magnetic in-plane \ninteractions between Mn -A and Mn -B except that the interacting orbitals are the \nunoccupied \n22xyd  orbital  of Mn -A and the occupied \n22xyd  orbital  of Mn -B \n(bottom panel of Fig. S4). To sum up, Mn -A and Mn -B interact ferromagnetically \nalong [100], [010] and [001] axes .   \nFig. S4 The orbital interaction s between Mn -A and Mn-B with FM and AFM spin \nconfigurations , respectively . Upper panel is for the Mn pair along the [001] axis \nwhile the bottom panel is for the Mn pair along the  [100] and [010] axes.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n4. Monte Carlo simulation of the transition temperature \nCT  of the epitaxially \nstrained LaMnO 3 film with SrTiO 3 lattice constant  \n \n   Fig. S5 Specific heat capacity of LaMnO 3 thin film grown on STO as a function of \ntemperature from MC simulation  with different lattice sizes (up to 39200  sites). A \nsharp \n  peak  of the specific heat capacity locates  at \n446 KCT . \n \n \n \n5. A detailed description of the orbital -degenerate double -exchange model  \nHamiltonian  \nThe model Hamiltonian in our present work reads  \n   \n 1 2 3\n2 2 2\n1 2 3..\n12 (1),2aa\nij i j AF i j i i i xi i zi\nij ij i\ni i i\niH t c c H c J S S Q n Q Q\nQ Q Q\n     \n           \n    \n\n \n   where \nid  \n()id  is the creation (annihilation) operator for the \nge  electron on \nthe orbital \n22α|x y a    and \n22β|3z r b   , with its spin parallel to the \nlocalized \n2gt  spin \niS\n ; \n,, ax y z\n  is the direction of the link connecting the two \nnearest neighbor  (NN) Mn3+ sites; Berry phase \nΩ2 2 2 2iji jj ii\nijcos cos sin sin e  \n arises due to the infinite Hund coupling, \nwhere \nθ  and \n  are the polar and azimuthal angles of the \n2gt  spins, \nrespectively. In the model Hamiltonian, the first term is the standard DE  \ninteraction . The  hopping parameters are  \n033 3 34x x x x\naa ab ba bbtt t t t    , \n033 3 34y y y y\naa ab ba bbtt t t t   \n, \n0z z z\naa ab bat t t    and \n0z\nbbtt . The second term is \nthe NN \n2gt  spins interaction through the antiferromagnetic superexchange  \n0AFJ\n. The third term is the electron -phonon coupling, where \nλ  is a \ndimensionless constant and the \norbitalge  operators are  \ni ia ia ib ibn d d d d , \nxi ia ib ib iad d d d\n, and \nzi ia ia ib ibd d d d . The last term is the lattice elastic \nenergy.  JT modes (\n2Q  and \n3Q ) and breathing mode (\n1Q ) are defin ed in Ref. [6]. \n \n \n \n \n \n \n \n[1] J. H. Park, C. T. Chen, S. W. Cheong, W. Bao, G. Meigs, V . Chakarian, and Y . U. \nIdzerda, Phys. Rev. Lett.  76, 4215 (1996).  \n[2] T. Hashimoto, S. Ishibashi, and K. Terakura, Phys . Rev. B 82, 045124 (2010).  \n[3] T. Arima, Y . Tokura, and J. B. Torra nce, Phys . Rev. B 48, 17006 (1993).  \n[4] R. J. McQueeney, J. Q. Yan, S. Chang, and J. Ma, Phys . Rev. B 78, 184417 \n(2008).  \n[5] E. O. Wollan and W. C. Koehler, Phys . Rev. 100, 545 (1955).  \n[6] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep.  344, 1 (2001).  \n \n \n \n "    },    {        "title": "1012.0195v1.The_second_order_dense_ferromagnetic_ferromagnetic_phase_transition.pdf",        "content": "arXiv:1012.0195v1  [cond-mat.stat-mech]  1 Dec 2010The second order dense ferromagnetic-ferromagnetic phase\ntransition\nAycan¨Ozkan, B¨ ulent Kutlu\nGazi¨Universitesi, Fen -Edebiyat Fak¨ ultesi, Fizik B¨ ol¨ um¨ u , 06500\nTeknikokullar, Ankara, Turkey,\ne-mail: aycan@gazi.edu.tr, bkutlu@gazi.edu.tr\nThe fcc spin-1 Ising (BEG) model has a dense ferromagnetic ( df) ground\nstate instead of the ferromagnetic ground state at low temperat ure region and\nexhibits the dense ferromagnetic ( df) - ferromagnetic ( F) phase transition for\nd=D/J= 2.9,k=K/J=−0.5,ℓ=L/J= 0 and h=H/J= 0. The critical\nbehavior of the dense ferromagnetic ( df) - ferromagnetic ( F) phase transition\nhas been investigated using the cellular automaton cooling and heatin g algo-\nrithms. The universality class and the type of the dense ferromagn etic (df) -\nferromagnetic ( F) phase transition have been researched within the framework\nof the finite - size scaling, the power law relations and the probability d istri-\nbution. The results show that the dense ferromagnetic- ferroma gnetic phase\ntransition is of the second order and the model shows universal se cond order\nIsing critical behavior at d= 2.9 parameter value through k=−0.5 line.\nPACS number(s): 05.10.-a, 05.50.+q, 64.60.-i\n1I . INTRODUCTION\nIn recent years, some of the studies indicated that the spin-1 Isin g model\nhas a ground state ordered structure which is named the dense fe rromagnetic\n(df) [1−4]. In absence of the dfordered structure, the phase diagrams were\nformed for some phase transitions which has been assumed the wea k first order\ntransition instead of the second order transition. The presence o f thedfordered\nstructure can be caused to some changes on the global phase diag rams [4]. This\ncase clarifies the differences among the results of the previous stu dies for the\n(kTC/J,d) phase diagram through the kline. [5−11]. For example, the phase\ndiagram has exhibited a tricritical point ( TCP) instead of a critical end point\n(CEP) fork=−0.5. While the MFA [7] and RG [8] studies exhibited a critical\nend point ( CEP) for k=-0.5 on the BEG model global phase diagram, CA [5 ,6],\nMCRG [9] ,TPCA [10] and CVM [11] studies show that there is the tricritical\npoint (TCP). Through the kline, the df-Fphase transition is very important\nfor determining the type of the phase boundary. The purpose of t his study is\nto define the dfordered phase and is to investigate the nature of df−Fphase\ntransition at d= 2.9 value through k=−0.5 line. This point creates the type\nof the special point of the ( kTC/J,d) phase space for k=−0.5. Therefore,\nwe have found the df-Fphase transition worthy of investigation in depth.\n2Furthermore, the universality class of the df-Fphase transition has not been\ninvestigated so far. The critical temperature and the statical cr itical exponents\nare estimated by analyzing the data within the framework of the finit e - size\nscaling theory and the power law relations.\nThe spin-1 Ising model, which is known as the generalized Blume-Emery -\nGriffiths (BEG) model, can be used to simulate many physical systems . The\nmodel firstly has been presented for describing phase separation and superfluid\norderingin He mixtures [12]. The versionsofthe model havebeen ap plied to the\nphysical systems such as the solid-liquid-gas systems [13], the mult icomponent\nfluids [14], the microemulsions [15], the semiconductor alloys [16 −18], He3-He4\nmixtures [12 ,19] and the binary alloys [20].\nThe BEG model Hamiltonian is defined as\nHI=−J/summationdisplay\n<ij>SiSj−K/summationdisplay\n<ij>S2\niS2\nj+L/summationdisplay\n<ij>(S2\niSj+SiS2\nj)+D/summationdisplay\niS2\ni+h/summationdisplay\niSi\n(1)\nwhich is equivalent to the lattice gas Hamiltonian under some transfor mations\n[21−23].∝angbracketleftij∝angbracketrightdenotes summation over all nearest-neighbor (nn) pairs of sites\nandSi=−1,0,1. The parameters J,K,L,Dandhare bilinear, biquadratic,\ndipole-quadrupole interaction terms, the single-ion anisotropy con stant and the\n3field term. The BEG model for k≥0 has been studied by mean field ap-\nproximation (MFA) [12 −14], the transfer matrix method [24], series expan-\nsion method [25], the constant coupling approximation [26], the pos ition-space\nrenormalization method [27], cluster variation method (CVM) [1], linea r-chain\napproximation[2], Monte Carlomethod (MC) [3] and Cellular Automaton (CA)\n[4−6,28,29].\nIn this paper, the fcc BEG model for d=D/J= 2.9,k=K/J=−0.5,\nℓ=L/J= 0 andh=H/J= 0 is simulated using cooling and heating algorithm\nimproved from Creutz Cellular Automaton. In the previous papers, the Creutz\ncellular automaton (CCA) algorithm and its improved versions have be en used\nsuccessfully to study the properties of the critical behaviors of t he Ising model\nHamiltonians [4 −6,28−50]. The CCA algorithm, which was first introduced\nby Creutz [51], is a microcanonical algorithm interpolating between t he conven-\ntional Monte Carlo and the molecular dynamics techniques. The Creu tz cellular\nautomaton (CCA) is faster than the conventional Monte Carlo met hod (MC).\nThe CCA does not need high quality random numbers and it is a new and a n\nalternative simulation method for physical systems. It has anothe r advantage\nallowing the specific heat to be computed from internal energy fluct uations.\nOur previous studies showed that the heating and the cooling algorit hms im-\n4proved from the Creutz Cellular Automaton algorithm are effective t o study\nthe phase space and the critical behavior of the Blume Emery Griffith s model\n[4−6,28−30,35,36].\nII . RESULTS AND DISCUSSION\nThe CA algorithm of spin-1 Ising model is a microcanonical algorithm. T he\ntotal energy H, which is conserved, is given by\nH=HI+HK (2)\nwhereHIis Ising energy which is given by equation1 and HKis kinetic energy.\nThe kinetic energy HKis an integer, equal to the change in the Ising spin\nenergy for any spin flip and its value lie in the interval (0, m).mis equal to\n24Jford=D/J= 2.9,k=K/J=−0.5,ℓ=L/J= 0 and h=H/J= 0 on\nfcc lattice. For a site to be updated, its spin is changed to one of the other two\nstates with 1 /2 probability. If this energy is transferable to or from the kinetic\nenergy variable of the site, such that the total energy H is conser ved, then this\nchange is done and kinetic energy is appropriately changed. Otherw ise the spin\nis not change [30 ,34−36].\nAt the heating and the cooling algorithms, the simulation consist of tw o\nparts, the initialization procedure and the computation of the ther modynamic\n5quantities. The initial configuration for heating and cooling algorithm s can be\nset in different shapes. In this study, the initial configurations are obtained at\nthree different shapes for heating algorithm during 20.000 CA steps . Firstly, all\nthe spins are up ( S= +1) at the absolute zero temperature for both algorithms.\nThe initial configuration of the heating algorithm has been obtained a t low\ntemperature ordered phase ( df) adding kinetic energy which is equal to the\nmaximum change in the Ising spin energy for the any spin flip to the spin\nsystem for set I and II. The another initial configuration (set III ) has been\nobtained flipping 8% of the spins to S= 0 state. The heating rate is realized by\nincreasing of 8% in the kinetic energy ( Hk) of 15% of the fcc lattice for two sets\nat the computation of the thermodynamic quantities. At set III, t he heating\nrate is realized by increasing of 8% in the kinetic energy ( Hk) of all site of the\nfcc lattice.\nOn the other hand, the initial configuration for the cooling algorithm is\nobtained adding energy to the 70% of the spin system for getting th e disordered\nphase (P) at high temperature. During the cooling cycle, the cooling rate is\nrealized by decreasing of 8% in the kinetic energy ( Hk) from 25% of the spin\nsystem. Theinitialconfigurationsarerunduringthe20.000CellularA utomaton\ntime steps. Instead of resetting the starting configuration at ea ch energy, it\n6was used the final configuration at a given energy as the starting p oint for\nthe next at both heating and cooling algorithms. The computed value s of the\nthermodynamic quantities (the order parameters ( M,Q), the susceptibility ( χ),\nthe Ising energy ( HI) and the specific heat ( C)) are averages over the lattice\nand over the number of time steps (2 .000.000 ) with discard of the first 100 .000\ntime steps during the cellular automaton develops [4 −6,28,29].\nThey have been computed on the fcc lattice with L= 8, 9, 10, 11 and 12\n(The total number of sites is N= 4L3) for periodic boundary conditions. The\nfcc lattice was formed in a simple cubic (sc) lattice. (The total numbe r of sites\nisN= 4L3= 6912 for L= 12 fcc lattice, this total site number equals to\nL= 19 for the simple cubic lattice). The presented figures are set III (heating\nalgorithm) results.\nThe order parameters, the Ising energy, susceptibility and specifi c heat are\ncalculated from\nM=1\nN/summationdisplay\niSi, Q=1\nN/summationdisplay\niS2\ni (3)\nUI= (−J/summationdisplay\n<ij>SiSj−K/summationdisplay\n<ij>S2\niS2\nj+D/summationdisplay\niS2\ni)/U0 (4)\nχ=N/angbracketleftbig\nM2/angbracketrightbig\n−∝angbracketleftM∝angbracketright2\nkT(5)\n7CI/k=N/angbracketleftbig\nU2\nI/angbracketrightbig\n−∝angbracketleftUI∝angbracketright2\n(kT)2(6)\nwhereU0is the ground state energy at kT/J= 0.\nThe ferromagnetic ( F) and the paramagnetic ( P) phases can be determined\nwith the average occupation of the states ∝angbracketleftP±1,0∝angbracketright. As the projectors for the\nstatesS= +1,−1and0are P+1=1\n2S(S+1),P−1=1\n2S(S−1)andP0= 1−S2,\nthe average occupation of the states are ∝angbracketleftP+1∝angbracketright=1\n2(Q+M),∝angbracketleftP−1∝angbracketright=1\n2(Q−M)\nand∝angbracketleftP0∝angbracketright= 1−Q, respectively. With considering the average occupation of the\nstates, another ferromagnetic phase can be determined as the d ense ferromag-\nnetic phase ( df).\nFerromagnetic ( F):∝angbracketleftP+1∝angbracketright ∝negationslash=∝angbracketleftP−1∝angbracketright ∝negationslash=∝angbracketleftP0∝angbracketright ∝negationslash= 0, (M∝negationslash=Q∝negationslash= 0)\nDense ferromagnetic ( df):∝angbracketleftP−1∝angbracketright →0;∝angbracketleftP+1∝angbracketright ∝negationslash=∝angbracketleftP0∝angbracketright ∝negationslash= 0, (M∼=Q∝negationslash= 0)\nParamagnetic ( P):∝angbracketleftP−1∝angbracketright=∝angbracketleftP+1∝angbracketright ∝negationslash=∝angbracketleftP0∝angbracketright ∝negationslash= 0, (M= 0,Q∝negationslash= 0).\nII .1 Temperature Variations of Thermodynamic Quantities f or the\ndF−F−PPhase Transitions\nThetemperaturevariationoftheorderparameters( M,Q), thesusceptibility\n(χ), theIsingenergy( HI) andthespecificheat( CI)areillustratedin figure1for\nexhibiting the general aspect of the successive df−F−Pphase transitions at\nd= 2.9 parameter value through k=−0.5 line. As it is seen in figure 1(a) and\n8figure 1(c), the order parameters and the Ising energy appear c ontinuously for\ndf−FandF−Pphase transitions. Therefore both phase transitions are of the\nsecond order as functional behavior. The susceptibility ( χ) and the specific heat\n(CI,C)exhibit two peaks at TC1andTC2temperatures corresponding to df−F\nandF−Pphase transitions(Figure 1(b), 1(d) and figure 2). It can be seen from\nfigure 1(c), the functional change of the Ising energy from orde r to order ( df\n−F) phase transition is different from the view of the order to disorder (F−P)\nphasetransition. For df-Fphase transition, the Isingenergydifference (∆ U) is\ngreater than for the F-Pphase transition. The estimated critical temperature\nfrom susceptibility and specific heat maxima is compatible with each oth er for\nF-Pphase transition. But the critical temperature values for df-Fphase\ntransition are not compatible (Figure 1(b) and 2(a)). Therefore s pecific heat\n(C) has been recalculated for only spin-spin interaction energy (Figur e 2(b)).\nThe spin-spin interaction energy Uis determined as\nU= (−J/summationdisplay\n<ij>SiSj)/U0 (7)\nThe specific heat calculated from Ucan show a sharp peak for df−Fphase\ntransition. Because, the first sum ( U) in Ising energy ( UI) distinguish the S=\n+1 and−1 states. Indeed, the obtained infinite critical temperature ( TC1(∞)=\n91.52±0.04)from the susceptibility ( χ) and specific heat ( C) peak temperatures\nare compatible with each other. TC2(∞) is obtained from the susceptibility ( χ)\nand specific heat ( CIandC) peak temperatures as 3 .20±0.02.\nThe temperature variations of the ∝angbracketleftP+1∝angbracketright,∝angbracketleftP−1∝angbracketrightand∝angbracketleftP0∝angbracketrightare given in figure\n3. The initial configuration for the heating algorithm is created as all spins are\nup (S= +1) at absolute zero temperature ( T= 0). If the enough energy is\nadded to the spin system, the S= 0 begins to arise. The excitation energy of\nthe singlespin flippingfrom S= +1toS= 0is 3.1Jwhile it is 24 Jfor the single\nspin flipping from S= +1 to S=−1. Therefore, at low temperature region,\nthe rising probability of the S=−1 state has to be lower than S= 0 state. It\ncan be seen in figure 3 that the spin system includes S= +1 and S= 0 states\npredominantly. The value of ∝angbracketleftP+1∝angbracketrightis about 1 and ∝angbracketleftP0∝angbracketrightis different from zero,\nwhile∝angbracketleftP−1∝angbracketrightappears almost zero indicating the dfphase for T < T C1(L). At the\nsame time,. So, Mis almost equal to the Q(M∼=Q∝negationslash= 0). As ∝angbracketleftP−1∝angbracketrightincreases\naboveTC1(L), thedfordered phase changes to the Fordered phase. At high\ntemperature region, the ferromagnetic - the paramagnetic phas e transition ( F\n-P) occurs . ∝angbracketleftP+1∝angbracketrightis equal to ∝angbracketleftP−1∝angbracketrightand the system is in the paramagnetic\n(P) disordered phase above TC2(L) temperature ( M∝negationslash=Q∝negationslash= 0). Therefore the\nphase space is divided into three regions ( df,FandP). It is obvious that to\n10follow the temperature variation of ∝angbracketleftP−1∝angbracketrightis a useful way to prove the existence\nof thedfordered phase.\nII .2 Probability Distribution of Order Parameter\nThe another useful procedure to distinguish the phase transition type is to\ncalculate the probability distributions of the order parameter ( P(M)). In our\nstudy the probability distribution is calculated by\nPL(M) =NM\nNCCAS(8)\nwhereNMis the number of times that magnetization Mappears, and NCCASis\nthe total number of the cellular automaton steps. The histogram w ith 200 bins\nare used for plotting the probability distribution of the magnetizatio n [24, 31].\nThe probability distribution of the order parameter ( P(M)) near the phase\ntransition temperature shows two peaks in the second phase tran sitions.\nThe probability distributions of the order parameter ( P(M)) are shown for\ndifferent temperature values in figure 4. The peaks of the order pa rameter\nprobability distribution exhibits minimum with increasing temperature a t the\nlow temperature region. This minimum corresponds to the second or derdf−F\nphase transition at the Tχ\nC1(L= 12) = 1 .479. Although the phase transition is\nof the second order, the probability distribution shows the single pe ak near the\n11phase transition temperature TC1(L). Because the system has S= +1 and 0\nspins below TC1(L) and the transition is from order ( df) to order ( F). However,\ntheprobabilitydistributionsinthe F−Pphasetransitionregionexhibitthetwo\npeaks with the contribution of the S=−1 state near the TC2(L= 12) = 3 .187.\nForT > T C2, there is a single peak focused to M= 0 indicating the disordered\n(P) phase.\nII.3 Finite - Size Scaling Analyses and the Statical Critica l Expo-\nnents\nThe values of the statical critical exponents ( ν,β,γ,α) are estimated within\nthe frameworkof the finite - size scaling theory and the power laws. The infinite\nlatticecriticaltemperature TC(∞)hasbeenobtainedfromthesusceptibilityand\nthe specific heat peak temperatures for the successive second o rderdf−F−P\nphasetransitionsandfromtheintersectionpointofBindercumulan tcurves( UL)\nforthesecondorder F−Pphasetransition[52]. Thefinite-sizescalingrelations\nof the Binder cumulant ( UL), the order parameter ( M), the susceptibility ( χ)\nand the specific heat ( C) are given by\nUL=G◦(εL1/ν) (9)\n12M=L−β/νX◦(εL1/ν) (10)\nkTχ=Lγ/νY◦(εL1/ν) (11)\nC=La/νZ◦(εL1/ν) (12)\nFor large x=εL1/ν, the finite lattice critical behaviors must be asymptoti-\ncally reproduced, that is,\nX◦(x)∝Axβ(13)\nY◦(x)∝Bx−γ(14)\nZ◦(x)∝Cx−α(15)\nAccording to the finite size scaling theory, the data for the finite - s ize\nlattices of the thermodynamic quantities should lie on a single curve fo r the\ntemperatures both below and above TC(∞) with universal critical exponents.\nThe critical exponents β,γandαhave been obtained from the Log-Log plots\nof the asymptomatic functions.\n13The temperature variation of the Binder cumulant is shown for the d ifferent\nlattice sizes in figure 5. The Binder cumulant curves intersect at the TUL\nC2(∞) =\n3.20±0.02correspondingto the F−Pphasetransition(Figure 5(a)). This value\nis compatible with TC2(∞) which is extrapolated according to the finite size\nscaling theory from the susceptibility and the specific heat peak tem peratures\n(TC(L)) of the finite lattices, respectively.\nTC(L) =TC(∞)+aL−1/ν(16)\nIt can be seen in the inset of the figure 5(a) that there is no interse ction at the\nBinder cumulant for the data of the df−Fphase transition region. However,\nthe Binder cumulant curves exhibit a plateau near the infinite lattice c ritical\ntemperature ( TC1(∞)) which is obtained from the susceptibility ( χ) and the\nspecific heat ( C) peak temperatures as Tχ\nC1(∞) = 1.52±0.04. In figure 5(b),\nthe scaling data of the Binder cumulants are shown for the second o rder phase\ntransition from FtoP. Near the TUL\nC2(∞), Binder cumulant curves have been\nscaled well for TC=TC2(∞) withν= 0.64. It can be seen in the inset of the\nfigure 5(b) that the data corresponding to dfordered phase could not be scaled\nwith the TUL\nC2(∞) critical temperature value. However, the scaling data of the\nbinder cumulant corresponding to the dfordered phase lie on a single curve for\n14ε=(T−TC1(∞))/TC1(∞) atT < TC1(∞) region using ν= 0.64 (Figure 5(c))\nand the finite size scaling relations validate for the dfordered region.\nInfigure6(a),thescalingdataoftheorderparameterisillustrate datthesuc-\ncessivedf−F−Pphase transitions for L= 8,9,10,11and 12 at TC=TC2(∞).\nThe order parameter data lie on the two different curves with slope= β/ν= 0.31\nandβ′/ν=−0.55 for the temperatures both below and above TC2(∞) respec-\ntively except for dfordered region with β= 0.31 andν= 0.64. As it is seen\nin the inset of the figure 6(a), the data of df−Fphase transition region have\nnot been scaled with TC=TC2(∞). However, the data corresponding to the\ndf−Fphase transition region have been scaled well with TC=TC1(∞) for\nT < TC1(∞) usingβ= 0.31 andν= 0.64 in figure 6(b).\nThe scaling data of the susceptibility have been shown in figure 7 with t he\nstraight lines describing the theoretically predicted behavior for lar gex(Equa-\ntion 8). The susceptibility data for the temperatures both below an d above\nTC2(∞) agrees with the asymptotic form except for the df-Fphase transition\nregion and so with the TC=TC2(∞),γ=γ′= 1.25 andν= 0.64 in figure 7(a)\nand 7(c). However, the data of dfordered phase ( T < T C1) have been scaled\nwith the TC=TC1(∞) usingγ= 1.25 andν= 0.64 in figure 7(b).\nThe finite size scaling data of the singular portion of the specific heat (C)\n15have been exhibited in figure 8. The data of F−Pphase transition region\nofCIhave been scaled well both below and above TC\nC2(∞) usingα= 0.12,\nν= 0.64 and the correction terms, b−=−70 andb+=−8 (Figure 8(a) and\n8(b)). Althoughthe dataofthe dfphaseregioncould notbeen scaledwith TC=\nTC\nC2(∞) in figure 8(a), the data lie on single curves with slope= −α/ν=−0.12\nat the both side of the TC\nC1(∞), using α= 0.12,ν= 0.64 and the correction\nterms,b−=−1 andb+=−0.3 in figure 8(c) and (d). On the other hand,\nthe specific heat ( CI) data calculated from UIscales well at df-F-Pphase\ntransitions using α= 0.12,ν= 0.64 forT < Tχ\nC1(∞)andT < TCI\nC2(∞) and\nT > TCI\nC2(∞).\nTheM,χandCdata have been analyzed within the framework of the finite\nsizescalingtheoryfor the successive df−F−Pphasetransitions. The estimated\nvalues of the statical critical exponents arein good agreementwit h the universal\nvalues (α= 0.12,β= 0.31,γ= 1.25,ν= 0.64) for the df−Fand theF−P\nphase transitions.\nII.4 Power Law Relations and the Infinite Lattice Statical Cr itical\nExponents\nOn the other hand, the critical exponent values for df−Fphase transition\ncan be obtained using the following power law relations [53].\n16M(L) =εβ(L)(17)\nχ(L) =ε−γ(L)(18)\nC(L) =ε−α(L)(19)\nwhereε=(T−TC(L))/TC(L). The finite lattice critical exponents β(L),β′(L),\nγ(L),γ′(L),α(L) andα′(L) of the order parameter ( M), susceptibility ( χ) and\nthe specific heat ( C) quantities are obtained from the slope of the log-log plot\nof the power laws relations for each finite lattices in the interval 0 .05≤ε≤0.2.\nTheinfinitelatticecriticalexponentsareobtainedusinglinearextra polationand\ntheir values are given in Table I. The estimated values for cooling algor ithm and\nthree simulation sets of heating algorithm are in good agreement with the finite\nsize scaling critical exponent estimations and the universal values f or 3dIsing\nmodel (β= 0.31,γ= 1.25,α=α′= 0.12 andν= 0.64).\nTable 1. The estimated values of the infinite lattice critica l exponents and the\ncritical temperatures ( α,β,γandTχ,C\nC1(∞)) using linear extrapolation.\n17df-Fphase transition\nHeating Cooling\nSet I Set II Set III Average of sets\nTχ\nC1(∞)1.50±0.021.50±0.031.52±0.031.51±0.031.52±0.04\nTC\nC1(∞) − − 1.52±0.01 − 1.52±0.01\nβ(T<TC1)0.31±0.010.31±0.010.30±0.010.31±0.010.31±0.01\nγ(T<TC1)1.25±0.011.23±0.031.23±0.021.24±0.031.23±0.02\nα(T<TC1)0.12±0.010.12±0.010.11±0.010.12±0.010.12±0.01\nα′(T>TC1)0.12±0.010.12±0.010.12±0.010.12±0.010.12±0.01\nIII. SUMMARY\nThe (kTC/J,d) phase diagrams is illustrated in figure 9 for the presence of\ndforder. The type of special point is determined by the d= 2.9 parameter. The\ncalculations show that model exhibits the phase transition from ord er to order\nford= 2.9 and the first order phase transition from order to disorder in the\n3≤d <4 parameter region. If the model doesn’t exhibit the dfordered phase\ninstead of the Fordered phase in the low temperature region, the continuous\nphase transition from Forder to Forder (F−F) is considered as the weak\nfirst order. This constitutes the part of the first order phase tr ansition line\nwhich creates the critical end point ( CEP) [7,8]. However, CA results show\nthat, the model has a dfordered phase for the parameters in the 2 .9≤d <4.0\nregion. The spin system contains S= +1 and S= 0 states. As a result of this,\nthe order parameters MandQare almost equal each other ( M∼=Q∝negationslash= 0) at\nlow temperatures. With increasing temperature, the dense ferro magnetic ( df)\nordered phase changes continuously to ferromagnetic ( F) ordered phase at the\n18d= 2.9,k=−0.5withtheenoughcontributionof S=−1state. Therefore,near\nthed= 2.9, the first order phase transition line have been changed to secon d\norder phase transition line, and there occurs the tricritical point TCP(Figure\n9). In order to determine the universality class of the successive df−F−P\nsecond order phase transitions, the static critical exponents ( α,β,γandν) are\nestimated within the frameworkofthe finite - size scaling theory. Th e estimated\nvalues of the critical exponents ( α= 0.12,β= 0.31,γ= 1.25 andν= 0.64)\nnear the TC1andTC2temperatures are in good agreement with the theoretical\nvalues for three sets. The df−Fphase transition is analyzed with the power\nlaws for comparing with the critical exponent values estimated from the finite\n- size scaling theory. The obtained values are in compatible with the fin ite -\nsize scaling analyze results and the universal values for the 3 dIsing model. The\nobtained results have shown that the df−Fphase transition is of the second\norder and it is compatible with the universal Ising critical behavior fo rd= 2.9\nparameter value through k=−0.5 line. As a result of this, the definition of the\ndfphase changes the phase transition type and the special point typ e in the\nphase space for the BEG model. This result will lead to reexamine the s tructure\nof phase spaces.\nACKNOWLEDGEMENT\n19This work is supported by a grant from Gazi University (BAP:05/200 3-07).\nReferences\n[1] Keskin M, Ekiz C, Yal¸ cın O, 1999 Physica A 267392\n[2] Albayrak E, Keskin M, 2000 J. Magn. Magn. Mater. 203201\n[3] Ekiz C, Keskin M, 2002 Phys. Rev. B 66054105\n[4]¨Ozkan A, Kutlu B, 2010 Int. J. of Mod. Phys. B accepted to publish\n[5] Sefero˘ glu N, Kutlu B, 2007 Physica A 374165\n[6]¨Ozkan A, Kutlu B, 2007 Int. J. of Mod. Phys. C 181417\n[7] Hoston W, Berker A N, 1991 Phys. Rev. Lett .671027\n[8] Netz R R, Berker A N, 1993 Phys. Rev. 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B 43119\n[53] Huang K, 1987 Statistical Mechanics John Wiley & Sons 396\nFigure Captions\nFigure 1. For ( d= 2.9,k=−0.5), the temperature dependence of (a) the\norder parameters ( M,Q), (b) the susceptibility ( χ), (c) the Ising energy ( HI)\nand (d) the specific heat ( CI/k).\nFigure 2. For ( d= 2.9,k=−0.5), the specific heat ( C/k) calculated from\nU.\nFigure 3. For ( d= 2.9,k=−0.5), the temperature dependences of the\n∝angbracketleftP∝angbracketright.∝angbracketleftP+1∝angbracketright,∝angbracketleftP−1∝angbracketrightand∝angbracketleftP0∝angbracketrightcorrespond to the S= +1,−1 and 0 spin states,\nrespectively.\nFigure 4. The probability distribution of the Mfor (d=−0.5,k= 0.9) on\nL= 12.\nFigure 5. For ( d= 2.9,k=−0.5), (a) the temperature dependence of the\nBinder cumulant ( UL), (b) the finite - size scaling of the Binder cumulant near\n23thedf−F−Pphase transition with TUL\nC2(∞), (c) the finite - size scaling of\nthe Binder cumulant near the df−Fphase transition with TC1(∞).\nFigure 6. For ( d= 2.9,k=−0.5), the finite - size scaling plots of (a) the\norder parameter with TC2(∞), (b) the order parameter with TC1(∞) near the\ndf−Fphase transition for T < TC1(∞).\nFigure 7. For ( d= 2.9,k=−0.5), the finite - size scaling plots of the\nsusceptibility (a) with ε=(T−TC2(∞))/TC2(∞) forT < TC2(∞), (b) near the\ndf−Fphase transition with ε=(T−TC1(∞))/TC1(∞) forT < TC1(∞), (c)\nwithε=(T−TC2(∞))/TforT > TC2(∞).\nFigure 8. For ( d= 2.9,k=−0.5), the finite - size scaling plots of the specific\nheat (a) for T < TC2(∞) withTC2(∞), (b) for T > TC2(∞) withTC2(∞), (c)\nforT < TC\nC1(∞) withTC\nC1(∞), (d) for T > TC\nC1(∞) withTC\nC1(∞).\nFigure 9. The phase diagram for k=−0.5. The phase space with dfphase\ncontains a TCPatd= 2.9.\n240.0 0.2 0.4 0.6 0.8 1.0 \n0 1 2 3 4 5\nkT/J M, Q \nL=8 \nL=9 \nL=10 \nL=11 \nL=12 d=2.9 \nk=-0.5 (a) \nFigure 1(a) 0246810 12 \n0 1 2 3 4 5\nkT/J χ\nTC1 TC2 (b) \nFigure 1(b) 0510 15 \n0 1 2 3 4 5\nkT/J HI(c) \nFigure 1(c) 010 20 30 40 50 60 70 \n0 1 2 3 4 5\nkT/J CI/k TC1 \nTC2 (d) \nFigure 1(d) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 \n0 1 2 3 4 5\nkT/J C/k L=8 \nL=9 \nL=10 \nL=11 \nL=12 TC1 \nTC2 \nFigure 2 Figure 3 0.0 0.2 0.4 0.6 0.8 1.0 \n0.5 2.5 4.5 6.5 \nkT/J <P> S=+1 \nS=0 \nS=-1 L=12 \nHeating \nAlgorithm \nTC2 TC1 F df PFigure 4 0.0 0.1 0.2 0.3 0.4 0.5 \n-0.5 0.0 0.5 1.0 \nMP(M) Tc1(L=12)=1.479 \nTc2(L=12)=3.187 \nTC1 (L) \nTC2 (L) -2.0 -1.5 -1.0 -0.5 0.0 \n1 2 3 4\nkT/J ULL=8 \nL=9 \nL=10 \nL=11 \nL=12 \n(a) \nTC2 (∞)=3.20±0.02 -2.00 -1.99 -1.98 \n0.5 1.0 1.5 2.0 Tχ\nC1 (∞)=1.52 \nFigure 5(a) 4(a) \n)=3.20±0.02 \nFigure 5(a) -2.0 -1.5 -1.0 -0.5 0.0 \n-20 -10 0 10 20 \nεL1/ν ULν=0.64 \nTC=T C2 (∞)=3.2 (b) \n-2.00 -1.98 \n-40 -30 -20 -10 dF region \nFigure 5(b) -2.00 -1.95 \n-20 -10 0 10 20 \nεL1/ν ULν=0.64 \nT<T C1 (c) \nTC=T C1 (∞)=1.52 \nFigure 5(c) slope= β/ν  = 0.31 \nslope =- β/ν = -0.55 \n-1.4 -0.9 -0.4 0.1 0.6 \n-0.5 0.0 0.5 1.0 1.5 \nLog( εL1/ν )Log(ML β/ν )\nL=8 \nL=9 \nL=10 \nL=11 \nL=12 T<T C2 \nT>T C2 df region (a) \nTC=T C2 (∞)ν=0.64 \nβ=0.31 \nFigure 6(a) slope= β/ν  = 0.31 \n-0.5 0.0 0.5 1.0 \n-0.5 0.0 0.5 1.0 1.5 \nLog( εL1/ν )Log(ML β/ν )\nT<T C1 (b) ν=0.64 \nβ=0.31 \nTC=T C1 (∞)\nFigure 6(b) slope= −γ/ν =-1.25 \n-2.0 -1.5 -1.0 -0.5 \n-1 0 1 2\nLog( εL1/ν )Log(kT χL-γ/ν )\ndf region \nT<T C2 (a) \nν=0.64 \nγ=1.25 \nTC=T C2 (∞)\nFigure 7(a) slope= −γ/ν = -1.25 \n-2.5 -2.0 -1.5 -1.0 \n-1 0 1 2\nLog( ε L1/ν )Log(kT χL-γ/ν )\nT<T C1 (b) \nν=0.64 \nγ=1.25 \nTC=T C1 (∞)\nG:\\df-F\\son\\set2sons Figure 7(b) slope= −γ/ν = -1.25 \n-2.0 -1.5 -1.0 -0.5 \n-1 0 1 2\nLog( ε/L1/ν )Log(kT χL-γ/ν )\nT>T C2 (c) ν=0.64 \nγ=1.25 \nTC=T C2 (∞)\nFigure 7(c) slope= −α/ν  = -0.12 \n1.5 2.0 \n-0.5 0.0 0.5 1.0 1.5 \nLog( εL1/ν )Log((C-b -)L −α/ν )\nL=8 \nL=9 \nL=10 \nL=11 \nL=12 α=0.12 \nν=0.64 \nb-=-70 T<T C2 df region (a) \nTC=T C2 (∞)\nFigure 8(a) slope= −α/ν  = -0.12 \n1.0 1.5 \n-0.4 0.1 0.6 1.1 1.6 \nLog( εL1/ν )Log((C-b -)L −α/ν )\nα=0.12 \nν=0.64 \nb+=-8 T>T C2 (b) \nTC=T C2 (∞)\nFigure 8(b) slope =- α/ν = -0.12 \n-0.3 0.2 \n-0.5 0.0 0.5 1.0 1.5 \nLog( εL1/ν )Log((C-b -)L −α/ν )\nT<T C1 α=0.12 \nν=0.64 \nb-=-1 (c) \nFigure 8(c) slope =- α/ν = -0.12 \n-0.7 -0.2 \n-0.5 0.0 0.5 1.0 1.5 \nLog( εL1/ν )Log((C-b +)L −α/ν )\nT>T C1 α=0.12 \nν=0.64 \nb+=-0.3 (d) \nFigure 8(d) 0.0 0.1 0.2 0.3 0.4 0.5 \n0.0 1.0 2.0 3.0 4.0 5.0 \ndkT/J SOPT \nFOPT \nF\ndf P\nTCP \nFigure 9 "    },    {        "title": "1412.6960v2.Possibility_of_ferromagnetic_neutron_matter.pdf",        "content": "Preprint typeset in JHEP style - HYPER VERSION OU-HET-844\nRIKEN-MP-104\nPossibility of ferromagnetic neutron matter\nKoji Hashimoto\nDepartment of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan\nMathematical Physics Lab., RIKEN Nishina Center, Saitama 351-0198, Japan\nE-mail: koji(at)phys.sci.osaka-u.ac.jp\nAbstract: We study ferromagnetism at high density of neutrons in the QCD hadron\nphase, by using the simplest chiral e\u000bective model incorporating magnetic \felds and\nthe chiral anomaly. Under the assumption of spatial homogeneity, we calculate the\nenergy density as a function of neutron density, with a magnetization and a neutral\npion condensation a laDautry and Neyman. We \fnd that at a high density the energy\nof the ferromagnetic order is lower than that of the ordinary neutron matter, and\nthe reduction e\u000bect is enhanced by the anomaly. Compared to the inhomogeneous\nphase with the alternating layer structure, our ferromagnetic phase turns out to be\nunfavored. However, once an axial vector meson condensation is taken into account\nin our simplest model, the ferromagnetic energy density is lowered signi\fcantly, which\nstill leaves some room for a possible realization of a QCD ferromagnetic phase and\nferromagnetic magnetars.\nKeywords: Magnetization, Neutron, High density QCD .arXiv:1412.6960v2  [hep-ph]  13 Jan 2015Contents\n1. Motivation: QCD and ferromagnetism 1\n2. Toy model of dense neutral fermions 3\n2.1 Fermions, magnetization and constant magnetic \feld 3\n2.2 Ferromagnetism at higher density 5\n2.3 Favoring ferromagnetic phase 8\n2.4 Similarity to Nambu-Jona-Lasinio model 10\n3. Chiral model of neutrons with pion condensation, magnetic \feld\nand anomaly 11\n3.1 Dense neutrons and pions with axial anomaly 11\n3.2 Spontaneous magnetization and the pion condensation 14\n3.3 Anomaly enhancement and comparison to the ALS phase 16\n3.4 Inclusion of axial vector meson condensation 18\n3.5 AdS/CFT treatment with a large Ncapproximation 20\n4. Summary and discussion 22\n1. Motivation: QCD and ferromagnetism\nFerromagnetic order in nature always attracts interest for study as it manifests micro-\nscopic structure of matter and materials. Among observed magnetic \felds in nature,\nperhaps the strongest stable magnetic \feld is on the surface of magnetars, which\ngoes up to 1015[G] and more [ 1,2,3]. The mechanism for generating such a strong\n\feld is yet to be uncovered, and it is natural to resort the origin to the high density\nof neutrons of which the neutron stars consist. In fact, after the discovery of pulsars,\nthe possibility of ferromagnetism at neutron stars was proposed [ 4,5,6,?]. However,\nnumerical simulations of neutron matter with realistic inter-nucleon potentials have\nnot shown the ferromagnetic phase [ 7]. So the possibility of the ferromagnetic phase\nat high density neutron matter, if exists in nature, waits for a new mechanism of the\nspontaneous magnetization.\nIn this paper, we study the possibility of the ferromagnetic phase at high density\nof neutrons, by using the simplest but general chiral e\u000bective action. Low energy dy-\nnamics of neutrons is governed by the chirally symmetric interactions through pions\n{ 1 {and the spin-magnetic coupling with magnetic \felds. Our model consists of dense\nneutrons coupled with neutral pions and magnetic \felds, together with the chiral\nanomaly term. These are indispensable ingredients, and we will see the outcome for\nthe magnetic phase from this minimal model.\nThe reason for choosing the neutral pion is simply for the realization of the\nferromagnetism, as other pion condensations such as charged pion condensation [ 8,9]\nhave not been shown to exhibit a ferromagnetism. In addition, with a neutral pion\ncondensation of the form \u0005 0(x)/sink\u0001x, a neutron lattice is formed with an\nalternating layer structure (ALS) [ 10,11,12,13], then the neutron spins cancel each\nother, and macroscopic magnetization would not emerge. In this paper, instead, we\nanalyze a neutral pion condensation of the di\u000berent form \u0005 0(x) =q\u0001xfollowing\nDautry and Neyman [ 14]1, and generalize the study to include magnetic \felds and\nQCD anomaly.\nOur study is motivated by the earlier work [ 16] in which, together with M. Eto\nand T. Hatsuda, the author proposed a mechanism for a ferromagnetic phase at\nhigh density of neutrons. The mechanism utilizes a neutral pion domain wall [ 17]\ncoupled to the magnetic \feld through the QCD chiral anomaly [ 18]. A spontaneous\nmagnetization was shown in [ 16] in the approximation of a single wall and one-loop\nneutrons. In this paper, we generalize the idea, and study in the simplest chiral\nmodel the Fermi energy of the dense neutrons and its back-reaction due to the pion\ncondensation and the magnetic \felds. A successive array of the domain walls can be\napproximated by the linear pion condensation of Dautry and Neyman.\nLet us describe what we will \fnd in this paper.\n\u000fToy model of neutral fermions.\nFirst we provide a toy model of a neutral fermion with a Zeeman coupling to\nmagnetic \felds. Under the assumption of the spatial homogeneity, we calcu-\nlate the energy density of the ferromagnetic phase and show that it is favored\ncompared to the ordinary fermion matter. (Sec. 2)\n\u000fSimplest chiral model and ferromagnetic order.\nThe toy model of the neutral fermions is the essential part of the chiral model\nof neutrons and pions. We analyze the simplest chiral e\u000bective model of dense\nneutrons and neutral pions, together with the magnetic \feld coupling and the\nQCD anomaly. We \fnd that the neutral pion condensation of form proposed\nby Dautry and Neyman is precisely in the same place as the magnetization,\nunder the assumption of the spatial homogeneity. The energy density of the\nferromagnetic-pion-condensation phase is lower than the ordinary neutron mat-\nter at high density around \u001a >5\u001a0where\u001a0is the standard nuclear density.\n1For a recent review on the condensation (called chiral density wave), see [ 15].\n{ 2 {Furthermore, the chiral anomaly term actually helps the ferromagnetic order.\nThe generated magnetic \feld is \u001840 [MeV]\u0018O(1017)[G]. (Sec. 3.1, 3.2)\n\u000fComparison to ALS.\nWe compare our energy density with that of the inhomogeneous ALS phase\n(which does not exhibit a magnetization), and \fnd that the ALS phase is\nfavored. The energy gain of the ALS is by several times greater than that of\nour ferromagnetic phase. (Sec. 3.3)\n\u000fAxial vector meson condensation.\nTo seek for the possibility of the ferromagnetism, we look at the axial vector me-\nson condensation accompanied by our model. Indeed, any axial vector meson\nplays the same role as the neutral pions, and the axial vector meson condensa-\ntion further reduces the energy density of the ferromagnetic phase signi\fcantly.\nIncorporation of a higher vector meson tower and its condensation is studied\nby using the AdS/CFT correspondence. (Sec. 3.4)\nIn summary, we analyze the ferromagnetic order of our simplest chiral model of dense\nneutrons with magnetic \felds and the QCD anomaly. We \fnd that our ferromagnetic\norder, as its simplest form, is not favored compared to the ALS phase. We further\n\fnd that the axial vector meson condensation and the QCD anomaly, together with\nthe pion condensation of Dautry and Neyman, signi\fcantly helps the reduction of the\nenergy density, which suggests a necessity for further investigation for a realization\nof the ferromagnetic phase. The analysis in this paper is for the minimal model\nas we have emphasized above, so the result should be understood only qualitatively.\nIncorporation of realistic nuclear forces and nucleon contact terms, and also inclusion\nof electrons and protons, would be important for a further progress for realizing the\nQCD ferromagnetic phase at high density of neutrons.\nThe organization of this paper is as follows. In Sec. 2, we provide the toy model\nof neutral fermions and study a ferromagnetism at high density. In Sec. 3.1 and\nSec. 3.2, we analyze the simplest chiral model of neutrons with the pions, magnetic\n\felds and the chiral anomaly. In Sec. 3.3, we present our result on the energy plot\nand a comparison to the ALS phase is made. In Sec. 3.4 and 3.5, incorporation\nof the axial vector meson condensation is studied, with a help of the AdS/CFT\ncorrespondence. Sec. 4 is for a summary and discussions.\n2. Toy model of dense neutral fermions\n2.1 Fermions, magnetization and constant magnetic \feld\nWe are interested in the e\u000bect of spin and its magnetization, for a general fermion\nsystem. The magnetization is a condensation of a spin operator of fermions. Since\n{ 3 {for relativistic systems the spin operator of a fermion  (x) is given by a spatial\ncomponent of an axial current,\nSi(x) =1\n2\u0016 \ri\r5 ; (2.1)\nwe can systematically write an action for the fermion with the spin magnetic coupling.\nWe consider the following general system of a neutral fermion. It is a system of a\nfree neutral fermion  (x) with a mass min 4 spacetime dimensions, with a Zeeman\ncoupling under a dynamical magnetic \feld Bi,\nLfermion =\u0016 (i@\u0016\r\u0016\u0000m+i\r0\u0016) +\u000b\u0016 \ri\r5 Bi\u00001\n2B2\ni: (2.2)\nA chemical potential \u0016for the fermion number is introduced such that we can treat\nthe fermion density \u001a. Since the magnetization is a back-reaction to the spacetime\nmagnetic \feld, we have included the kinetic term for the magnetic \feld Bi. The\nsecond term in ( 2.2) is the Zeeman coupling \u000bbetween the spin of the fermion and\nthe magnetic \feld Bi. The Zeeman coupling is a part of a so-called Pauli term.2\nOur fermion does not have a charge, since we are interested in e\u000bects induced\nparticularly by the spin magnetic interaction.3So in our model there is no standard\ncanonical coupling between the gauge \feld for the magnetic \feld and the fermion\n . Normally, for a charged spin-1 =2 fermion with an electric charge e, the Zeeman\ncoupling is measured in the unit of a Bohr magneton, as\n\u000b=g\n2e\n2m(2.3)\nwheregis the \\g-factor\" and e=2mis the Bohr magneton. Our fermion does not\nhave the electric charge, so we shall treat \u000bas a general spin-magnetic coupling. The\nrelation ( 2.3) can be thought of as a reference, for example for a neutron which will\nbe treated in section 3.\nUnder a constant magnetic \feld Bi, we consider the behavior of the dense neu-\ntral fermions. We shall quantize the spin of the fermion along the direction of the\nmagnetic \feld. Then there are two Fermi seas, one is for the up spin and the other\nis for the down spin. In the presence of the background magnetic \feld Bi, due to the\nspin-magnetic Zeeman coupling, we have a Zeeman splitting for the Fermi energy for\nspin up and down states.\n2One would notice that the interaction term added in ( 2.2) does not respect the Lorentz invari-\nance, as only the spatial index iis summed. However, since we need the density for our analysis,\nthe chemical potential term already broke the Lorentz invariance in ( 2.2), so we need not worry\nabout it.\n3For charged fermions, the magnetic \feld provides Landau levels which may change the story\nquite a bit, and will bring an interesting outcome. We shall come back to the charged fermion case\nelsewhere.\n{ 4 {It is easy to evaluate the free energy of each spin sector. In the non-relativistic\nfermions where mass mis large compared to the Fermi energy of the fermions, we\nobtain\nF\"=\u0000(2m)3=2\n15\u00192(\u0016\u0000m\u0000\u000bB)5=2F#=\u0000(2m)3=2\n15\u00192(\u0016\u0000m+\u000bB)5=2:(2.4)\nThe di\u000berence is just the sign of the Zeeman coupling, due to the spins. We have\ndenotedBas the magnitude of the magnetic \feld Bi. The total free energy of the\nsystem, including the magnetic \feld energy is given by\nF=\u0000(2m)3=2\n15\u00192h\n(\u0016\u0000m\u0000\u000bB)5=2+ (\u0016\u0000m+\u000bB)5=2i\n+1\n2B2: (2.5)\nIn the following of this section, we analyze this free energy and study the ferromag-\nnetism.\n2.2 Ferromagnetism at higher density\n2.2.1 Complete polarization of the spins\nA large magnetic \feld is expected to correspond to a high density of the fermion. For\na large magnetic \feld, one of the two terms for the spins in the free energy becomes\nill-de\fned; the expression ( 2.5) is valid only when\n\u0016\u0000m\u0000j\u000bjB > 0: (2.6)\nFor a large magnetic \feld, this condition is not met. In that case, we need to use\nthe following expression for the free energy\nF=\u0000(2m)3=2\n15\u00192(\u0016\u0000m+j\u000bjB)5=2+1\n2B2: (2.7)\nThe spins are fully aligned (see Figure 1Left).\nTo turn this free energy (as a function of the chemical potential) to the energy (as\na function of the fermion density), let us make a Legendre transform. The fermion\nnumber density is given by\n\u001a\u0011\u0000@F\n@\u0016=(2m)3=2\n6\u00192(\u0016\u0000m+j\u000bjB)3=2: (2.8)\nThen the energy is given by\nE\u0011F+\u0016\u001a=m\u001a+35=3\u00194=3\n21=351\nm\u001a5=3\u0000j\u000bjB\u001a+1\n2B2: (2.9)\nThe interpretation of each term is quite clear. The \frst term is the fermion mass\nenergy, as\u001ais the number density of the fermion. The second term is the Fermi\n{ 5 {Fermion energy\t\rFermion energy\t\rSpin up Spin down Spin up Spin down Figure 1: The fermi surface and polarization of spins. The spin-magnetic coupling modi-\n\fes the depth of the dispersion relation according to the fermion spins. Left: all the spins\nare polarized. Right: there remains some density of the opposite component of the spin.\nenergy. The third term is due to the spin magnetic coupling. And the last term is\nfor the magnetic \feld self energy.\nWe would like to \fnd an energy minimum for a given fermion density \u001a. It is\nquite straightforward, since the last two terms in the energy can be written as a\nperfect squared,\nE=m\u001a+35=3\u00194=3\n21=351\nm\u001a5=3+1\n2(B\u0000j\u000bj\u001a)2\u00001\n2\u000b2\u001a2: (2.10)\nSo, to minimize the energy, a spontaneous magnetization should take place,\nB=j\u000bj\u001a; (2.11)\nat which the energy density is given by\nE=m\u001a+35=3\u00194=3\n21=351\nm\u001a5=3\u00001\n2\u000b2\u001a2: (2.12)\n2.2.2 Co-existence of both spins\nThe magnetization at the high density in the description above assumes the complete\npolarization of the fermions. At not-so-high density of the fermions, we expect that\n{ 6 {not all the fermions are polarized (see Figure 1Right). Let us see indeed this is the\ncase.\nFrom the original total free energy ( 2.5), the equilibrium condition @F=@B = 0\nis\n0 =\u0000(2m)3=2\n15\u001925\n2\u000bh\n\u0000(\u0016\u0000m\u0000\u000bB)3=2+ (\u0016\u0000m+\u000bB)3=2i\n+B: (2.13)\nNote that this can be always satis\fed at B= 0. Therefore, no magnetization is\nalways a possibility of the equilibrium, and we need to compare if magnetized phase\nhas a lower energy density to conclude the ferromagnetism. As we shall see, for\nlower density there is no magnetization, while for a high density the ferromagnetism\nis preferred.\nTo see in more detail the density dependence, we calculate the density \u001aas\n\u001a\u0011\u0000@F\n@\u0016=(2m)3=2\n6\u00192h\n(\u0016\u0000m\u0000\u000bB)3=2+ (\u0016\u0000m+\u000bB)3=2i\n: (2.14)\nWe can eliminate \u0016by using the equilibrium condition ( 2.13), to obtain the equilib-\nrium condition in terms of the density,\n\u0012\n\u001a+B\n\u000b\u00132=3\n\u0000\u0012\n\u001a\u0000B\n\u000b\u00132=3\n=4\n(3\u0019)2=3m\u000bB: (2.15)\nThis equation determines the magnitude of the spontaneous magnetic \feld B, once\nthe density \u001ais given. (Again, B= 0 is an alternative solution satisfying this\nequation.)\nWe notice here that at a density\n\u001a=1\nj\u000bjB; (2.16)\nthe equation ( 2.15) can make one term vanish. This is nothing but the point when\nwe make a transition to the fully-polarized phase which we considered earlier. Sub-\nstituting ( 2.16) into ( 2.15), we obtain the threshold density\n\u001a2=32\u00194\n241\nm3\u000b6: (2.17)\nIf the density is above this value, \u001a > \u001a 2, the system in fully polarized and the\nanalysis reduces to what we have considered earlier.\nThere is another condition for which the equation ( 2.15) can have a non vanishing\nsolution for B. Using the following expansion\n(1 +\u000f)2=3\u0000(1\u0000\u000f)2=3=4\n3\u000f+8\n81\u000f3+O(\u000f5); (2.18)\n{ 7 {we notice that ( 2.15) can have a solution only when the slope around B\u00180 can\nsatisfy the following inequality\n\u001a2=3\u00014\n31\nj\u000bj\u001a<4mj\u000bj\n(3\u00192)2=3: (2.19)\nThis condition is rephrased as\n\u001a1\u0011\u00194\n31\nm2\u000b6<\u001a: (2.20)\nWhen\u001a\u0014\u001a1, we \fnd no solution for ( 2.15), other than B= 0. So, as is expected,\nfor low density there is no ferromagnetic phase.\nIn summary, we \fnd the following possible phases in our system;\n8\n<\n:\u001a\u0014\u001a1: B= 0\n\u001a1<\u001a<\u001a 2:B= 0 orB= nontrivial solution of ( 2.15), spin mixed\n\u001a2\u0014\u001a:B= 0 orB=j\u000bj\u001a(spins fully polarized)\n2.3 Favoring ferromagnetic phase\nTo study whether this ferromagnetic order can actually occur in the system of our\nconcern, let us compare the resultant energy ( 2.12) with the energy with no magnetic\n\feld (no magnetization).\nPuttingB= 0 reduces the system to that of the ordinary free fermion, and in\nthe non relativistic case, once given the density \u001a, we know the total energy\nEB=0=m\u001a+35=3\u00194=3\n10m\u001a5=3: (2.21)\nThe \frst term is the energy contribution from the fermion mass, and the second term\nis the fermion kinetic energy integrated to the Fermi surface.\nWe compare this EB=0with the total energy density with the fully polarized\nspins ( 2.12), to have\nE\u0000EB=0=35=3\u00194=3\n10m(22=3\u00001)\u001a5=3\u00001\n2\u000b2\u001a2: (2.22)\nIt is easy to show that this is always negative for the density \u001a\u0015\u001a2which is the\ncondition for the spin full polarization, see ( 2.17). So, we conclude that indeed the\nferromagnetic phase is preferred at the high density \u001a\u0015\u001a2.\nIt is also straightforward to show that even in the range \u001a1< \u001a < \u001a 2, the\nferromagnetic phase B6= 0 is preferred. To show this, we need numerical calculations\nsince the energy for this phase is not expressed in an analytic form.\nFinally, let us see the value of the chemical potential corresponding to the ferro-\nmagnetic phase, to \fnd some consistency conditions; \frst, a thermodynamic stability\ncondition, and second, the validity of the non-relativistic approximation. When all\n{ 8 {the spins are polarized, we have ( 2.11) which can be substituted to the relation\nbetween the density and the chemical potential ( 2.8), to \fnd\n\u0016=m+(6\u00192)2=3\n2m\u001a2=3\u0000\u000b2\u001a: (2.23)\nThe thermodynamics stability condition is\n@\u0016\n@\u001a>0 (2.24)\nwhich tells just the fact that larger chemical potential provides a higher density.\nUsing our relation ( 2.23) at the high density ferromagnetic phase, we have the ther-\nmodynamic stability condition\n\u001a<\u001a 3\u001122\u00194\n31\nm3\u000b6: (2.25)\nThe value\u001a3is larger than \u001a2, so the ferromagnetic phase is stable for \u001a2<\u001a<\u001a 3.\nSecond, we check the non-relativistic approximation. If we substitute the typical\nvalue\u001a=\u001a2for the ferromagnetic phase to the relation ( 2.23), we \fnd\n\u0016\u0000m+j\u000bjB= 2\u000132\u001941\nm\u000b2: (2.26)\nThis is the hight of the Fermi sea as measured from the bottom of the dispersion\nrelation, so the non relativistic approximation is valid when this value is much smaller\nthan the mass m,\n2\u000132\u00194\nm2\u001c\u000b2: (2.27)\nSo, our non relativistic approximation is valid when this condition is met for the\nspin-spin interaction coe\u000ecient \u000b2.\nTo gain more insight on the relation ( 2.27), let us adopt hypothetically the\nexpression of the magnetic moment for a charged fermion ( 2.3) (although our fermion\nis neutral). Using ( 2.3), the relation ( 2.27) is written as\n25=23\u00192\u001cjgje: (2.28)\nFor example, the observed values for electrons are jgj\u00182 ande2=4\u0019\u00181=137, so\nthis non-relativistic condition is not met. Note however that in this paper we are\ninterested in a neutral fermions, not the electron which has a minimal coupling to\nthe magnetic \feld. In the next section, we study neutrons in more details. We will\n\fnd that, although the g-factor for the neutrons is not so large, the non-relativistic\napproximation is valid: in addition to the magnetic \feld coupling, there appears a\npion coupling which plays the same role, and the approximation is valid for the total\ninteractions. The pion condensation is the main subject of the next section.\n{ 9 {2.4 Similarity to Nambu-Jona-Lasinio model\nIn the previous subsections, we have seen that the ferromagnetic phase is preferred\ncompared to the free neutral fermions, when the density is large enough. Let us brie\ry\ndiscuss the reason why the simple model ( 2.2) is expected to favor a ferromagnetic\nphase as for homogeneous phases. Indeed, we \fnd an interesting relation to the\nfamous Nambu-Jona-Lasinio (NJL) model [ 19,20] in the following. We can naturally\nassume that the phases under consideration is spatially homogeneous, therefore there\nis no electric \feld generated. In that case, the \feld Biserves as an auxiliary \feld\nand we can integrate it out in our system ( 2.2).4The resultant Lagrangian is\nLfermion =\u0016 (i@\u0016\r\u0016\u0000m+i\r0\u0016) +1\n2\u000b2\u0000\u0016 \ri\r5 \u00012: (2.29)\nImmediately we can see a resemblance to the NJL model, the renowned model for a\nspontaneous chiral symmetry breaking. The NJL model is characterized by a four-\nfermion interaction ( \u0016  )2, which can be thought of as a squared of chiral condensate\n\u0016  . The four-fermion interaction governs the condensation of the operator \u0016  . Our\nmodel can be considered as a generalization of the NJL model by replacing the\n(\u0016  )2coupling with the spin-spin interaction5SiSi. Since fermions possess spins,\nonce we turn on a nonzero density for the fermions, the spin-spin interaction may\ncause a spontaneous magnetization, as in the case of the NJL model. In fact, the\nspin-spin interaction is a popular interaction in condensed matter physics. When\nthe coe\u000ecient \u000b2of the last term is positive, the system is expected to favor a\nspontaneous magnetization, i.e.a ferromagnetic phase.\nFor the phase to be realized, a high density would be necessary so that the\nneighboring fermions can interact.6Therefore we also expect a phase transition from\nthe normal phase to the ferromagnetic phase as we increase the density, and the\ncritical density should be a function of the coupling \u000band the mass of the fermion\nmsince these are the only parameters of our system. This is what we have seen in\nthis section, and the similarity to the NJL model allows us to intuitively understand\nthe origin of the ferromagnetism.\nBefore ending this section, we should note one thing. Our analysis in this section\nassumes the homogeneity in space. Normally one can allow inhomogeneous pro\fle\nof the matter, which results in a spontaneous emergence of a spatial modulation. A\nmodulated phase would have smaller energy density compared to the ferromagnetic\n4Note that this integration is not allowed normally, but here we ignore the electromagnetic\npropagation. However, for a discussion of only a homogeneous phase, one can make the integration\nand it provides an intuitive picture.\n5To recover the Lorentz invariance of the interaction term of the system, one can add the axial\ndensity squared term\u0000\u0016 \r0\r5 \u00012so that the interaction recovers the Lorentz invariance. It is not\nour scope of this paper.\n6Note that in the NJL model, in contrast, the fermion density is not necessary for the conden-\nsation, and the phase is unique.\n{ 10 {phase studied in this section. In the analysis in this section, we treated only a\nconstant magnetic \feld Bi. However, normally the integration of Bias a constant\nauxiliary \feld is not allowed, because photons propagate and Biis a part of the\nphoton kinetic term. Once one integrates out the electromagnetic \feld properly, one\n\fnds a non-local action of fermions. The integrated nonlocal action can be used for\nanalyses of inhomogeneous phases of the fermions, see [ 21] for example. In this paper\nwe consider a homogeneous ferromagnetic phase, and whether it is realized or not\nshould be determined by a comparison with inhomogeneous phases. As for the QCD\napplication, we shall discuss this problem later in the next section.\n3. Chiral model of neutrons with pion condensation, magnetic\n\feld and anomaly\nWe saw in the previous section that a generic neutral fermion system, with the simple\nZeeman coupling, is shown to exhibit a ferromagnetism, under the assumption of the\nspatial homogeneity. As a concrete example, in this section we investigate a neutron\nmatter at a high density. Neutrons interact with each other not only via the mag-\nnetic \feld and the spin-magnetic interaction but also a pion exchange. Interestingly,\nthe two interactions have the same structure, under a simple pro\fle for a pion con-\ndensation. The pion condensation part is a laDautry and Nyman [ 14]. In addition,\nQCD has an axial anomaly term which relates the two condensations | the magnetic\n\feld and the pion condensation, and in fact enhances each other. The enhancement\nmakes the total free energy decrease. We evaluate the total energy density of the\nferromagnetic phase. Finally we compare the resultant ferromagnetic phase with the\nwell-studied ALS (Alternating layer structure) phase for pion condensation.\n3.1 Dense neutrons and pions with axial anomaly\n3.1.1 Axial anomaly for the pion Lagrangian\nLow energy action of QCD is given by the standard Lagrangian of the linear sigma\nmodel dictated by the breaking of the chiral symmetry,\nL=\u0016 (i@\u0016\r\u0016\u0000g(\u001b+i\r5\u001c\u0001\u0019)) \n+1\n2(@\u0016\u001b)2+1\n2(@\u0016\u0019)2\u0000m2\n\u0019f\u0019\u001b\u0000V(\u001b2+\u00192): (3.1)\nHere = (p;n)Tis the nucleon \feld, and \u001band\u0019are sigma model \felds leading to\npions.f\u0019is the pion decay constant, and m\u0019is the pion mass. The global symmetry\nis the chiral symmetry U(2)L\u0002U(2)R. The chiral symmetry is broken due to the\nchiral condensate, \u001b2+\u00192=f2\n\u0019, which is realized by the potential term V. Once\nthe sigma model \feld obtains the expectation value, the nucleons acquire a mass,\n{ 11 {gf\u0019=MN. In this paper we do not consider the di\u000berence of the masses for protons\nand neutrons.\nIn the ideal case with no proton, and no charged pions, the Lagrangian is\nL\u001b=\u0016 n(i@\u0016\r\u0016\u0000g(\u001b\u0000i\r5\u00193)) n\n+1\n2(@\u0016\u001b)2+1\n2(@\u0016\u00193)2\u0000m2\n\u0019f\u0019\u001b\u0000V(\u001b2+\u00192\n3); (3.2)\nwhere nis the neutron \feld. Since we want to deal with \fnite density of neutrons,\nwe include a chemical potential term for the neutron,\nLn=i\u0016 n\r0\u0016n n: (3.3)\nIn the presence of the magnetic \feld with which the neutrons interact through their\nmagnetic moment, we add the following Lagrangian,\nLB=\u00001\n2B2\ni+1\n2\u0016 n\ri\r5 ngne\n2MNBi: (3.4)\nThe \frst term is the energy of the magnetic \feld. The second term is the Pauli term\nfor the interaction between the magnetic moment of the neutron and the magnetic\n\feld. Note that the spin density of the fermion is given by1\n2\u0016 n\ri\r5 n, andgnis the\nneutrongfactor.\nIn the presence of the magnetic \feld and the neutral pion condensation which is\nspatially dependent, there exists an axial anomaly term,\nLanom=\u0000ie\n4\u00192f2\n\u0019\u0016em\u0014\n(\u001b+i\u00193)y@i(\u001b+i\u00193)\u0015\nBi: (3.5)\nHere\u0016emis the electromagnetic chemical potential. This term is relevant for, for\nexample, the neutral pion decay \u00190!2\rvia the axial anomaly, as is seen from\nthe fact that the electromagnetic chemical potential can be thought of as a constant\nbackground electrostatic potential A(em)\n0. So, our total Lagrangian is\nL=L\u001b+Ln+LB+Lanom: (3.6)\n3.1.2 Neutral pion condensation\nWe consider a neutral pion condensate, following Dautry and Nyman [ 14],\n\u001b+i\u00193=f\u0019exp(iq\u0001x); \u0019 1+i\u00192= 0: (3.7)\nThis corresponds to a speci\fc condensation of the neutral pion in the nonlinear\nrepresentation, since the relation between the linear and non-linear representation is\n\u001b+i\u00193\u0018f\u0019exp(i\u00050(x)) where \u0005 0(x) is the physical neutral pion excitation. The\ncondensation ( 3.7) corresponds to \u0005 0(x)\u0018q\u0001x, a linear pro\fle in space. This can\n{ 12 {be regarded as a dense parallel domain walls which was considered in the context of\nanomaly-enhanced pion condensation in [ 16].\nIn this paper, we shall generalize the study of Dautry and Neyman ( 3.7), to\ninclude the magnetization and the QCD anomaly. With this condensation ( 3.7), the\nanomaly termLanom is given simply as\nLanom=e\n4\u00192\u0016emqiBi: (3.8)\nIn the following, without losing generality, we can turn on only the x3components\nof the magnetic \feld and q3, which will be denoted as Bandq.\nAccording to Dautry and Nyman, if the condensation qis large, the neutron\nspins are fully polarized. In the non-relativistic approximation for the neutron Fermi\nmomentum, the free energy for the free neutrons in the background pion condensation\nand the magnetic \feld is derived from the Lagrangian above,\nFn=\u0000(2MN)3=2\n15\u00192\u0012\n\u0016n\u0000MN+1\n2gAq\u0000gneB\n4MN\u00135=2\n: (3.9)\nHere we have introduced the axial coupling gAwhich is, at the tree level, equal to gin\nthe\u001b-model Lagrangian L\u001b. The total Free energy including the pion condensation\nand the magnetic \feld is\nF=Fn+f2\n\u0019m2\n\u0019+1\n2f2\n\u0019q2+1\n2B2\u0000e\n4\u00192\u0016emqB: (3.10)\nThe second term is from the pion mass term together with the pion condensation\n(3.7). The third term is from the pion kinetic term with ( 3.7). The last term is the\naxial anomaly term.\n3.1.3 Hamiltonian and the neutron density carried by the anomaly\nOur interest is the core of the neutron star where we have the \f-equilibrium. In\naddition to the neutrons, there exist protons and electrons. The electromagnetic\nchemical potential is given by\n\u0016em=1\n2(\u0016p\u0000\u0016e) (3.11)\nwhere\u0016pand\u0016eare proton and electron chemical potential, respectively.7Assuming\nthe\f-equilibrium, we impose \u0016n=\u0016p+\u0016e. And since we approximate the system by\nthe pure neutron matter for simplicity, we also impose the charge neutrality condition\nin a trivial manner, \u001ap=\u001ae= 0, which is equivalent to have \u0016p=MN. Then the\nanomaly term in the free energy ( 3.10) is written as\n\u0000e\n4\u00192\u0016emqB=\u0000e\n4\u001921\n2(\u0016p\u0000(\u0016n\u0000\u0016p))qB=\u0000e\n4\u00192\u0012\nMN\u00001\n2\u0016n\u0013\nqB: (3.12)\n7The factor 1 =2 should be there, because the total Free energy given with the number density\nshould be\u0016p\u001ap+\u0016e\u001ae, and the total electric charge is \u001ap\u0000\u001ae. Its canonical conjugate is1\n2(\u0016p\u0000\u0016e).\n{ 13 {We evaluate the energy density of the system as a function of the neutron density\nand the condensation qand the magnetic \feld B. The neutron density is computed\nas\n\u001an=\u0000@F\n@\u0016n=(2MN)3=2\n6\u00192\u0012\n\u0016n\u0000MN+1\n2gAq\u0000gneB\n4MN\u00133=2\n+e\n8\u00192qB: (3.13)\nThe last term is the anomaly-induced baryon charge. Using the expression, the \fnal\nresult for the energy density is given by\nE=F+\u0016n\u001an\n=35=3\u00194=3\n21=351\nMN\u0010\n\u001an\u0000e\n8\u00192qB\u00115=3\n+\u0012\nMN\u00001\n2gAq+gneB\n4MN\u0013\u0010\n\u001an\u0000e\n8\u00192qB\u0011\n+f2\n\u0019m2\n\u0019+1\n2f2\n\u0019q2+1\n2B2\u0000e\n4\u00192MNqB: (3.14)\nFor a comparison, we write the expression of Dautry and Nyman [ 14]:\nE=35=3\u00194=3\n21=351\nMN(\u001an)5=3+\u0012\nMN\u00001\n2gAq\u0013\n\u001an\n+f2\n\u0019m2\n\u0019+1\n2f2\n\u0019q2: (3.15)\nThis is obtained from ( 3.14) by just putting B= 0. The di\u000berence from just the pion\ncondensation is obvious. Let us look at the second term of ( ??), which in fact exhibits\nthe nature of our model explicitly. In the absence of the pion condensation and the\nmagnetic \feld, the second term is simply \u001anMN. This is the cost of the energy due to\nthe mass of the neutron. Now, the cost for each neutron can be reduced by the pion\ncondensation due to the axial coupling, and by the magnetic \feld times the neutron\nmagnetic moment, as MN!MN\u00001\n2gAq+gneB\n4MN. Furthermore, the anomaly term\ncan reduce e\u000bectively the density of the neutrons, \u001an!\u001an\u0000e\n8\u00192qB. In addition, the\nlast term of ( 3.14) is for the QCD anomaly, and it makes the total energy decrease\nfurther.\n3.2 Spontaneous magnetization and the pion condensation\nFor a given density \u001anof the neutrons, we can minimize the energy E(3.14). Later\nwe present our numerical results. But here, to explain the intrinsic behavior of the\nsystem, we evaluate the minimization of the energy in the absence of the anomaly\nterm. Without the anomaly term, the energy density is simpli\fed as\nE=35=3\u00194=3\n21=351\nMN\u001a5=3\nn+\u0012\nMN\u00001\n2gAq+gneB\n4MN\u0013\n\u001an\n+f2\n\u0019m2\n\u0019+1\n2f2\n\u0019q2+1\n2B2: (3.16)\n{ 14 {The energy is quadratic in qandB, so we can analytically \fnd the minimum of the\nenergy. In fact, the energy density expression ( 3.16) can be brought to the following\nform with the perfect squared,\nE=E0+1\n2f2\n\u0019\u0012\nq\u0000gA\n2f2\n\u0019\u001an\u00132\n+1\n2\u0012\nB+gne\n4MN\u001an\u00132\n; (3.17)\nwhere the minimum energy is\nE0=35=3\u00194=3\n21=351\nMN\u001a5=3\nn+MN\u001an\u0000g2\nA\n8f2\n\u0019\u001a2\nn\u0000g2\nne2\n32M2\nN\u001a2\nn: (3.18)\nThe last term in the minimum energy density E0is due to the magnetic \feld. The\n\frst three terms are that of Dautry and Neyman [ 14], and compared to that, our\nenergy is smaller by the last term.\nThe minimization of the energy is achieved when the perfect squares in ( 3.18)\nvanish,\nB=\u0012\u0000gn\n4MN\u0013\ne\u001an; (3.19)\nq=gA\n2f2\n\u0019\u001an: (3.20)\nWe have obtained the spontaneous magnetization of the neutron matter. The gener-\nated magnetic \feld is a monotonic function of the density, and in particular in this\ncase of the absence of the anomaly, it is a linear function in the density.\nThe free energy Fnfor the neutrons, ( 3.9), is for fully polarized neutrons. Let\nus check if this can be achieved for qandBwhich we obtained above for given \u001an.\nThe condition that the opposite spin state is absent is\n\u0016n\u0000MN\u00001\n2gAq+gneB\n4MN<0: (3.21)\nThis means that the Fermi sea for the opposite spin state is below the conducting\nband. At the minimum of the energy, we obtain \u001andependence of the chemical\npotential\u0016nfrom ( 3.13) with the solution ( 3.20) and ( 3.19) as\n\u0016n=(6\u00192)2=3\n2MN\u001a2=3\nn+MN\u0000\u0012g2\nA\n4f2\n\u0019+g2\nne2\n16M2\nN\u0013\n\u001an: (3.22)\nIn terms of \u001an, the condition of the full polarization is equivalent to\n(6\u00192\u001an)2=3<4MN\u0014g2\nA\n4f2\n\u0019+g2\nne2\n16M2\nN\u0015\n\u001an: (3.23)\nSubstituting values as MN= 938 [MeV], e2=4\u0019= 1=137,f\u0019= 95 [MeV], gA= 1,\ngn=\u00003:8 andm\u0019= 135 [MeV], we obtain\n\u001an>0:39 [fm\u00003]: (3.24)\n{ 15 {02468101214859095100105110\u001an=\u001a0(E0\u0000MN)=\u001an[MeV]\nFigure 2: A plot of the energy per a neutron, as a function of the neutron density \u001an.\nStraight line: ordinary neutron matter without the pion condensation. Thick curved line:\nour result with both pion condensation qand magnetization Bwith the QCD anomaly term.\nThin curved line: the result of Dautry and Neyman [ 14] with only the pion condensation\nq. Dashed line: the energy with both qandBbut without the QCD anomaly term.\nThis shows that for the density around twice of the standard nuclear density, all the\nspins are polarized.\nAnother constraint comes from a thermal stability condition. At any thermal\nequilibrium, we need to make sure\n@\u0016n\n@\u001an>0: (3.25)\nThe condition ( 3.25) can be evaluated as\n\u001an<57:6 [fm\u00003]: (3.26)\nThe bound is extremely high density and unrealistic, so this thermodynamic insta-\nbility region is far above realistic neutron density.\n3.3 Anomaly enhancement and comparison to the ALS phase\nIn the previous subsection, we found that even without the anomaly term the total\nenergy density is lowered by the magnetic \feld. In fact, the magnetic coupling works\nin the same manner as the pion coupling. Now, let us see how the anomaly term can\nhelp the condensation. The full expression for the total energy density including the\nanomaly term was given in ( 3.14), and we can \fnd the minimum energy con\fguration\nby varying qandB. Analytic analysis is not easy since the energy is not quadratic\ninqandB, so we perform a numerical analysis to \fnd the energy minimum. The\nnumerical results are summarized in Fig. 2.\nFig.2is a plot of the energy per a neutron, as a function of the neutron density\n\u001an. The neutron density is normalized by \u001a0which is the standard nuclear density.\n{ 16 {46810121410203040506070\u001an=\u001a0p\neB[MeV]\nFigure 3: A plot of the magnetic \feld spontaneously generated, as a function of the\nneutron density \u001an. The neutron density is normalized by \u001a0(the standard nuclear density).\nThe evaluation is with the fully-polarized neutrons.\nThe thick line is our result with the anomaly term. We observe that for a larger\ndensity, the energy per a neutron decreases.\nIn Fig. 2, for a comparison, we show a thin curved line which is the result\nof Dautry and Nyman [ 14] (that is, with no magnetic \feld Bbut with the pion\ncondensation q). The dashed line in Fig. 2is the energy density with both qandB\nbut without the anomaly term. We can see that the anomaly term makes the energy\nper a neutron decrease. The straight line on the left is for free neutrons without the\npion condensation qand without B. So, as a comparison to the ordinary neutron\nmatter, we see that the ferromagnetic phase is preferred at high density.\nWe plot the magnetic \feld as a function of \u001an, in Fig. 3. It is a monotonic func-\ntion of the neutron density. We \fnd that the magnitude of the generated magnetic\n\feld isO(102) [MeV] and thus it is of the QCD scale.p\neB\u001840[MeV] corresponds\ntoO(1017)[G].\nNow, let us discuss whether our ferromagnetic phase is favored or not, in reality.\nThe famous phase for a pion condensation is the ALS (Alternating layer structure)\nphase [ 10,11,12], and we can compare the result of the ALS phase with ours. See\nthe result of the ALS (Fig. 4) and compare it with our result (Fig. 2). Since already\naround\u001an=\u001a0\u00185 the energy reduction of the ALS phase compared to the ordinary\nneutron matter is 70 [MeV] (see Fig. 4), while for our ferromagnetic phase the energy\nreduction is only 10 [MeV]. So, from this comparison, we conclude that the ALS phase\nis favored against our ferromagnetic phase.\nOur analysis in this paper is with the simplest model of neutrons, and we have\nnot included full nuclear forces. Once we include them in addition to our neutral\npion coupling and the Zeeman coupling, the total free energy may change. In fact, in\nthe following subsections, we include axial vector meson condensation and it makes\nthe total energy further decrease drastically.\n{ 17 {Figure 4: An energy plot taken from [ 12], comparing the ALS phase and the Fermi gas\n(ordinary neutron matter). The horizontal axis is the neutron density \u001ain the unit of\nthe standard nuclear density \u001a0, while the vertical axis is the energy gain per a neutron.\nThe upper curve is for the Fermi gas and the lower curve is for the ALS. Solid lines are\nfor neutrons, and dashed lines are for symmetric nuclear matter. The arrows indicate the\nenergy reduction by the ALS.\nIn summary, here our observation is that the ferromagnetism is closely related\nto the neutral pion condensation, and the axial anomaly can help the total energy\nto decrease and enhance the magnetic \feld. The ferromagnetic phase has an energy\ndensity smaller than that of the ordinary neutron matter. But the energy of the ALS\nphase is smaller, in the approximation presented.\n3.4 Inclusion of axial vector meson condensation\nIn the previous subsections, we considered only the neutral pion \feld for the coupling\nto the spins of the neutrons. The spin operator of the neutron is nothing but the\nspatial components of the axial current, and in QCD, we expect in\fnite number\nof quark bound states which can couple to the axial current. They are axial vector\nmesons whose spectrum starts at the lowest with a1(1260) meson. In this subsection,\nwe shall add the contribution of this lowest axial vector meson and will \fnd that it\nwill further make the free energy decrease, together with the full spin polarization.\nThe main system which we treat in this paper is described by ( 2.2). On the\nother hand, the e\u000bective Lagrangian for the axial vector meson a\u0016(x) coupled to the\naxial current and the neutrons is\nL=\u0016 (i@\u0016\r\u0016\u0000m+i\r0\u0016) \n+1\n2gaNN\u0016 \r\u0016\r5 a\u0016+1\n4(@\u0016a\u0017\u0000@\u0017a\u0016)2\u0000m2\na\n2a2\n\u0016: (3.27)\nHere,gaNN is the coupling of the axial vector meson to the neutron axial current,\n{ 18 {andmais the mass of the meson. For the lowest a1(1260) meson, the measured value\nisma= 1230\u000640 [MeV].\nAs before, we concentrate on a homogeneous phase, and let us assume a constant\nvacuum expectation value of the spatial component of the axial vector meson,\nhaii= const.6= 0: (3.28)\nThen, re-writing bi\u0011aima, the e\u000bective Lagrangian is now\nL=\u0016 (i@\u0016\r\u0016\u0000m+i\r0\u0016) +gaNN\n2ma\u0016 \ri\r5 bi\u00001\n2b2\ni: (3.29)\nWe observe that this Lagrangian has precisely the same form as ( 2.2), so the same\nmechanism of lowering the free energy by spin alignment can work.\nThis addition of the axial vector meson to the pion system modi\fes the total\nfree energy ( 3.10) a little bit. The resultant free energy is\nF=\u0000(2MN)3=2\n15\u00192\u0012\n\u0016n\u0000MN+1\n2gAq\u0000gne\n4MNB+gaNN\n2mab\u00135=2\n+f2\n\u0019m2\n\u0019+1\n2f2\n\u0019q2+1\n2B2+1\n2b2\u0000e\n4\u00192\u0016emqB: (3.30)\nHere we determined the orientation of biin space such that it may strengthen the\nspin polarization, and denote bas the magnitude of bi. Note that the axial vector\nmesonbenters exactly in the same manner as that of the pion condensation qand the\nmagnetic \feld Bexcept for the anomaly term (the last term in ( 3.30)). So, basically\nthe addition of the axial vector condensation enhances the spin polarization of the\nneutrons, and further reduces the energy density.\nAs before, to gain an intuition of the behavior of the system, we \frst analyze\nthe system without the anomaly term. Then the total energy is\nE=35=3\u00194=3\n21=351\nMN\u001a5=3\nn+\u0012\nMN\u00001\n2gAq+gneB\n4MN+gaNN\n2mab\u0013\n\u001an\n+f2\n\u0019m2\n\u0019+1\n2f2\n\u0019q2+1\n2B2+1\n2b2: (3.31)\nCompared to ( 3.16), we \fnd that we have additional terms\n\u0001E=gaNN\n2mab\u001an+1\n2b2; (3.32)\nwhich is independent of the other variables qandB. So we can minimize it indepen-\ndent of the other terms, and \fnd\n\u0001E0=\u0000g2\naNN\n8m2\na\u001a2\nn (3.33)\n{ 19 {024681020406080100\u001an=\u001a0gaNN\u00189\ngaNN\u001818(E0\u0000MN)=\u001an[MeV]\nFigure 5: A plot of the energy per a neutron, as a function of the neutron density \u001an.\nOn the previous \fgure for the pion condensation, here we added two thick dashed lines,\nshowing the axial vector condensation. The upper thick dashed line is for gaNN = 9 and\nthe lower is for gaNN\u001818.\nwith an axial vector condensation\njhaiij=b\nma=gaNN\n2m2\na\u001an: (3.34)\nTo evaluate the energy \u0001 E0in (3.33), we need the value of the axial vector\ncouplinggaNN. We refer to a generic argument of the chiral symmetry by regarding\nthe axial vector meson as a gauge boson of the symmetry [ 22,23,24],\ngaNN\nma=2gA\nm\u0019: (3.35)\nA naive substitution gA\u00181 and the mass for the pion and the a1meson provides\ngaNN\u001818. Another estimate is as follows. We take care of one of the other equations\ncoming from the chiral symmetry argument [ 22,23,24],ma=p\n2m\u001awhich is not\nwell satis\fed by the physical masses of the \u001ameson and the a1meson. So instead\nof using the a1meson mass in the chiral symmetry formula ( 3.35) we may use the\n\u001ameson mass m\u001a= 770 [MeV]. Then we obtain gaNN\u001816. However, a lattice\nsimulation with an axial vector dominance provides gaNN\u00189 (see for example [ 25]),\nso there is uncertainty for the coupling.\nIn our numerical estimate of the energy density, we choose two typical values,\ngaNN\u001818 and 9. Our result is shown in \fg. 5. We \fnd that the energy per a nucleon\ndrastically reduces further. Compared to the ALS phase, the case with gaNN\u001818\nhas a lower energy and thus favored. The case with gaNN\u00189 is almost at the same\norder with the ALS phase.\n3.5 AdS/CFT treatment with a large Ncapproximation\nThe AdS/CFT correspondence [ 26,27,28] is a well-estabilished tool for analyzing\nstrongly coupled gauge theory in a certain limit, and its application to QCD-like\n{ 20 {gauge theories were widely studied. However the AdS/CFT tools for strongly coupled\ngauge theories work practically for large Ncgauge theories and at the limit of strong\ncoupling, so it would not be suitable for precision analysis such as the energy gain\nvia the condensation which is our interest in this paper. Nevertheless, it is important\nto \fnd what kind of couplings among hadrons and the magnetic \feld is present in\nQCD, and what is the order of magnitude of the couplings. The AdS/CFT approach,\ncalled holographic QCD, is suitable for that purpose, and in this short subsection we\nshall investigate it.\nWe use Sakai-Sugimoto model [ 29,30] which is the stringy setup closest to QCD\nat present. The nucleon meson couplings were obtained in [ 31,32,33,34], and the\nQCD anomaly term was calculated in [ 30].\nBasically in holographic QCD we have a tower of mesons, and this is true for the\na1mesons. We have in\fnite number of axial vector mesons. On the other hand, we\nhave only a single pion (that is, in the model there does not appear excited resonances\nof the pion).\nIt is easy to read from [ 30] that the axial vector mesons does not participate in\nthe QCD anomaly term, so the only contribution to the anomaly term is the pion\ncoupling which we considered in this paper. So we do not need to take care of all\nthe mixing between the axial vector mesons and the magnetic \feld in the anomaly\nterm, at the leading large Ncexpansion and at the strong coupling limit.\nOn the other hand, the contribution of the axial vector meson to the nucleon\nspins, which we considered in the previous subsection, comes to a concern. Since we\nhave in\fnite number of axial vector mesons, all piles up as a sum and would cause\npossibly a tremendous contribution. We shall discuss the issue in the following.\nFirst, in the AdS/CFT correspondence, the axial vector mesons are gauge \felds\nat higher dimensions, and their interaction terms are basically given by the Yang-\nMills action in the higher dimensions. We need to excite only the \u001c3component of the\nisospin, while the Yang-Mills action contains only a commutator-type interaction, so\nthe direct interaction among the constant axial vector mesons vanish. This means\nthat we does not need to consider the inter-level interaction of the axial vector meson\ntower.\nWe have seen in the previous subsection that a single axial vector meson reduces\nthe total energy by ( 3.33), so when there exists a tower of the axial vector mesons\nwe have an energy reduction\n\u0001E0=\u0000\u001a2\nn1X\ni=1r(i); r(i)\u0011\u0012ga(i)NN\n8ma(i)\u00132\n(3.36)\nwhereiis the label of the resonances, and i= 1 corresponds to the lowest a1(1260).\nFrom this expression, we observe that all axial vector mesons contribute additively,\nand the issue is the magnitude of the ratio ga(i)NN=ma(i)wheniincreases.\n{ 21 {The ratio can be calculated analytically by the AdS/CFT correspondence [ 31].\nHowever, the approximation of large \u0015is not good, so here we provide only the\nresulting numbers for a reference. The method developed in [ 31] can be generalized\neasily for higher axial vector mesons, and we \fnd\nr(2)=r(1)'1:06; r(3)=r(1)'1:07 (3.37)\nat the large t'Hooft coupling limit. So the ratio does not decrease for larger i. This\nwould be natural from the original idea of gauged chiral symmetry by Wess and\nZumino [ 24] which derived the relation ( 3.35). Therefore, the e\u000bect of the inclusion\nof the higher axial vector mesons is important, and it has an e\u000bect of further reducing\nthe total energy density.\nAt largeNclimit, all the axial vector meson tower reasonably contribute since\nthe meson width is narrow, and one would imagine the tremendous amount of energy\nreduction by introducing all the axial vector meson tower. However, it is unnatural\nand an artifact of the large Nclimit, since in reality the meson width gets broader\nfor higher resonances and the higher mesons participate with higher energy but also\nwith more involved chiral interactions. So, here we just point out that axial vector\nmeson condensation has a tendency to further reduce the total energy density, and\nthe contribution from the tower of the resonances would not be negligible.\n4. Summary and discussion\nFor searching a QCD ferromagnetism at high density of neutrons, we studied the\nsimplest chiral Lagrangian ( 3.6) which accommodates neutrons at high density, the\npion condensation, the constant magnetic \feld with its self energy and the QCD\nanomaly. The pion condensation is a linear spatial pro\fle of the neutral pion ( 3.7)a\nlaDautry and Neyman [ 14] which generates a neutron spin alignment.\nWe solved a self-consistent equation for the total energy density for a given neu-\ntron density, by considering the neutron Fermi energy, the pion self energy and also\nthe self energy of the constant magnetic \feld. We have shown that the minimization\nof energy under the assumption of spatial homogeneity leads to the ferromagnetic\norder preferred compared to the ordinary neutron matter without the pion conden-\nsation, at the neutron density \u001a > 5\u001a0where\u001a0is the standard nuclear density.\nThe result is summarized in Fig. 2. The generated magnetic \feld (see Fig. 3) isp\neB\u001840[MeV] which is around O(1017)[G].\nHowever, a comparison to the ALS (alternating layer structure) phase [ 10,11,12],\nwhich is with another neutral pion condensation providing a spatially alternating spin\norder, shows that our ferromagnetic order has a larger energy density and thus is not\nfavored (see Sec. 3.3).\nWe further included axial vector mesons in our model, since the axial vector\nmeson condensation has the same coupling as the Dautry-Neyman neutral pion con-\n{ 22 {densation. We found that the axial vector meson enhances the energy reduction of\nthe ferromagnetic phase signi\fcantly (see Sec. 3.4). In QCD there exists a tower\nof axial vector meson resonances, and inclusion of the tower further enhances the\nreduction, which we roughly evaluated with the use of the AdS/CFT correspondence\n(Sec. 3.5).\nWe can summarize our results as follows:\n\u000fThe simple chiral model with the linear neutral pion condensation and magnetic\n\feld accommodates a ferromagnetic order.\n\u000fThe QCD anomaly term lowers the ferromagnetic energy.\n\u000fThe axial vector meson condensation further reduces the energy signi\fcantly.\n\u000fOur analysis is among spatially homogeneous phases, and needs to be compared\nin more detail with inhomogeneous phases such as the ALS.\nOur study is based on the simple chiral model ( 3.6), so the numerical results pre-\nsented in this paper is not suitable for a detailed comparison. For example, inclusion\nof realistic nuclear forces and nucleon contact terms would give more corrections.\nNevertheless, in our analysis, in particular the axial vector meson condensation is an\ninteresting and novel feature, and a further consideration would be of worth. In the\ncondensed phase of the axial vector mesons, low energy propagation modes are of\ninterest, in view of recent progress [ 35] in non-relativistic Nambu-Goldstone theorem\n[36,37].\nIn this paper, we concentrated on the hadron phase,8not a quark phase such\nas the color superconductivity. It would be interesting to extend our calculation, if\npossible, to a hadron-quark mixed phase. For a quark matter, the possibility of the\nferromagnetism was studied in [ 41,42], while the quark-hadron mixture phase was\nstudied in the context of neutron stars [ 43,44]. 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Suppl., 2121, 6 (2014),\nhttp://www.physics.mcgill.ca/ pulsar/magnetar/main.html (McGill Online\nMagnetar Catalog).\n{ 27 {"    },    {        "title": "0811.4430v1.Anomalous_Ferromagnetism_in_TbMnO3_Thin_Films.pdf",        "content": "Anomalous Ferromagnetism in TbMnO3 Thin FilmsB. J. Kirby,1,a),D. Kan,2 A. Luykx,2 M. Murakami,2 D. Kundaliya,2 and I. Takeuchi21Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland, 20899, USA2 Department of Materials Science and Engineering, University of Maryland, College Park, Maryland, 20742, USAABSTRACTMagnetometry, x-ray, and neutron scattering have been used to study the structural and magnetic properties of a TbMnO3 thin film grown on a [001] SrTiO3 substrate by pulsed laser deposition.  Although bulk TbMnO3 is a low temperature antiferromagnet, magnetometry measurements indicate the presence of low temperature ferromagnetism.  Depth profiling by x-ray and polarized neutron reflectometry reveals a net sample magnetization that is commensurate with the film thickness, indicating that the observed ferromagnetism is not due to an altered surface phase (such as Mn3O4), or external impurities that might give rise to an artificial magnetic signal.  Instead, these results show that the ferromagnetism is an intrinsic property of the TbMnO3 film.     \r1     Multiferroic materials, which feature simultaneous coupled ordering of magnetic moments and electric polarization, are the subject of widespread research interest.  Multiferroics hold the promise of novel devices in which the the magnetic properties are electrically controlled, and vice versa.  One example is that of an antiferromagnetic multiferroic thin film used to produce a preferred magnetization orientation in an adjacent ferromagnetic thin film via exchange bias [1,2] coupling.  For such a structure, it is possible to alter the nature of the exchange coupling (and therefore the magnetic orientation of the ferromagnet) by applying an electric field to change the interfacial magnetic domain structure of the multiferroic antiferromagnet.  For example, it has recently been demonstrated that the magnitude of the low temperature exchange bias field in YMnO3 / permalloy multilayers changes as a function of the electric field applied during cooling.[3]  \r To further pursue the technological possibilities of exchange biasing multiferroics, other materials need to be explored.  In bulk, TbMnO3  is a multiferroic material which in bulk exhibits a complex magneto-electric phase diagram, including antiferromagnetism below 30 K - making it a candidate for exchange-bias applications.[4-6]  However, scaling down manganite crystals into thin film form causes strain that can drastically affect the magnetic properties,[7] and relatively little work has been done to study the magnetic properties of TbMnO3  thin films.  As a first step towards determining the utility of TbMnO3,for exchange-bias applications, we have produced a thin film of the orthorhombic phase of TbMnO3, and studied it with magnetometry, x-ray, and neutron scattering measurements.   These measurements show that the TbMnO3  film exhibits an intrinsic ferromagnetic order. \r A TbMnO3 film was laser ablated from a  stoichiometric TbMnO3 target, onto a [001] oriented SrTiO3 substrate.  Figure 1 shows results of θ-2θ x-ray diffraction (XRD) scans, performed with a commercial diffractometer.[8]  The fabricated film exhibits the [00l] peak of the perovskite structure, with an out-of-plane lattice parameter c = 0.372 nm.  No peaks identifiable as originating from \r2additional phases were observed.  X-ray reciprocal space mapping measurements around the SrTiO3 [103] Bragg reflection (not shown) reveal that the in-plane lattice parameter of the TbMnO3 film closely matches that of the cubic SrTiO3 substrate (0.391 nm).  Bulk TbMnO3 has an orthorhombic perovskite structure which (in the pseudo-cubic convention) has lattice parameters of 0.393 nm in-plane, and 0.370 nm out-of-plane.[9]  Therefore, we conclude the fabricated film is compressively strained by the substrate, resulting in the tetragonally distorted orthorhombic phase of TbMnO3. \r Net magnetization measurements of the sample were performed using superconducting quantum interference device (SQUID) magnetometry.  Figure 2 shows the field H dependent magnetic moment m at temperature T  = 5 K after cooling from 100 K in zero field.[10]  Hysteretic behavior is observed, clearly indicating ferromagnetism - a drastic departure from the antiferromagnetic order observed for bulk TbMnO3.  m(T) in a 1.5 T field, is shown in the Fig. 1 inset, and reveals that a net magnetization persists above 100 K.    However, standard magnetometry measurements alone cannot determine whether the observed ferromagnetism is present throughout the film, or if it is instead due to a departure from the TbMnO3 stoichiometry in a small region of the film.     \r To determine if a net magnetization is truly intrinsic to the TbMnO3 film, we determined the structural and magnetic depth profiles via x-ray and polarized neutron reflectometry [11] (XRR and PNR, respectively) measurements at the NIST Center for Neutron Research.  XRR measurements were conducted at room temperature with wavelength λ = 0.154 nm x-rays (Cukα), and PNR measurements were conducted at cryogenic temperatures with λ = 0.475 nm cold neutrons.  The neutron beam was polarized by Fe/Si supermirror and a Mezei spin-flipper to be alternately spin-up (+) or spin-down (-) relative to an in-sample-plane magnetic field H, and was incident on the sample.  The reflected beam was intercepted by an analyzer supermirror/flipper assembly so that all four PNR cross sections (non spin-flip: R+ +, R- -, and spin-flip:  R- +, R+ -) could be measured with a 3He pencil detector. \r3\r In reflectometry, depth (z) dependent sample properties are determined from the scattering vector (Q) dependent reflectivities.[12] The x-ray and neutron beams are both highly penetrating, and probe the entirety of the TbMnO3 thin film. XRR is sensitive to the sample’s x-ray scattering length density ρX(z) (a function of electron density),[13] while PNR is sensitive to the sample’s nuclear scattering length density ρN(z) and volume magnetization M(z).[14,15]  Specifically, the non spin-flip scattering R+ + and R- - depend on ρN(z) and the component of M(z) parallel to H, while the spin-flip scattering R+ - and R- + depend solely on the component of M(z) perpendicular to H.  Thus, the structural and magnetic depth profiles of thin film structures can be determined by model-fitting XRR and PNR spectra.  For this work, model fitting was done using the NIST GA_REFL software package.[16]  \r Fitted Cukα XRR data are shown in Figure 3.  Clear oscillations are observed, indicating a strong sensitivity to the interface between the TbMnO3 film and the SrTiO3 substrate. PNR measurements of the same sample were conducted in a 0.55 T field, after cooling the sample from above 290 K to 6 K in a 5.5 T field.  The spin-flip channels were measured only at 6 K, and scattering was found to be at background levels, indicating no detectable magnetic component perpendicular to H.  The fitted non spin-flip scattering data are shown in Figure 4.  At 6 K (a), R+ + and R- - exhibit clear oscillations that are roughly 180 degrees out of phase from one another.  In the absence of nuclear spin polarization,[18] the only way these two spin states can differ is if the sample possesses a net magnetization parallel to H.  Further, the common frequency of the R+ + and R- - oscillations is essentially the same as that of the XRR oscillations shown in Figure 3, suggesting that the magnetized film thickness is similar to the total film thickness.   At 100 K (b), the splitting between R+ + and R- -  persists, indicating that the sample is still magnetized.  The higher frequency oscillations observed in panel a) and Fig. 3 are much weaker at 100 K.  This indicates that while neutrons are very sensitive to the magnetic interface between the film and the substrate, they are not very sensitive to the nuclear interface.  With the high frequency oscillations \r4damped, a very weak lower frequency oscillation becomes apparent, suggesting some depth dependence of ρN within the TbMnO3 film.     \r Results of model fitting the XRR and PNR data are shown in Figure 5.  The structural profiles are shown in panel a) (ρN  and both the real and imaginary components of ρX).  The x-ray profiles feature a distinct substrate/film interface, and show that the TbMnO3 film is 70 nm thick.  The nuclear profile features a slight decrease in ρN from top to bottom of the film, possibly due to a small variation in film strain.  The magnetic profiles are shown in Fig 5 b), and confirm that the TbMnO3 possess a net magnetization parallel to H that is essentially constant across the entirety of the 70 nm film.  Given the lattice parameters determined from XRD, the profiles shown in Fig. 5 correspond to magnetic moments of 0.6 and 0.1 μB per unit cell at 6 K and 100 K, respectively.  \r These results show that the ferromagnetism observed with SQUID is an intrinsic property of the TbMnO3  thin film.  While we cannot completely rule out a uniform distribution of impurities as the origin of ferromagnetism (magnetic clusters, for example), we have definitively shown that the ferromagnetism is not due to surface impurities.  Specifically, observation of a uniform sample magnetization at 100 K confirms that a surface layer of ferrimagnetic Mn3O4 (TC = 42 K) [20] is not the source of the ferromagnetic signal observed with SQUID.  Instead, it seems that the ferromagnetic behavior is indeed an intrinsic property of strained orthorhombic TbMnO3 films.  Very recent unpublished work by Rubi et al. also shows that thin TbMnO3 films on SrTiO3 substrates can exhibit ferromagnetism, and they present evidence suggesting that the ferromagnetism arises from in-plane compressive strain. [21]  This is a likely explanation for the ferromagnetism we have observed, as our film is also compressively strained in-plane. Determining how to tune the growth parameters to produce fully antiferromagnetic TbMnO3 films for exchange bias applications remains an important avenue of future research. In summary, our results reveal additional complexity to the already rich TbMnO3 \r5magneto-electric phase diagram, and have important implications for the role of TbMnO3 as a biasing multiferroic antiferromagnet.This work was supported by NSF-MRSEC under grant No. DMR 0520471, NSF DMR 0603644, and ARO W9IINF-07-1-0410.  We thank J. A. Borchers of NIST for valuable discussions.   \r6REFERENCES[1] W. H. Meiklejohn and C. P. Bean, Phys. Rev. 105, 904 (1957).[2] J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999).[3] V . Laukhin, V . Skumryev, X. Marti, D. Hrabovsky, F. Sanchez, M. V . Garcia-Cuenca, C. Ferrater, M. Varela, U. Luders,  J. F. Bobo, and J. Fontcuberta, Phys. Rev. Lett. 97, 227201 (2006).[4] T. Kimura, G. Lawes, T. Goto, Y . Tokura, and A. P. Ramirez, Phys. Rev. B 71, 224425 (2005).  [5] T. Goto, T. Kimura, G. Lawes, A. P. Ramirez, and Y . Tokura, Phys. Rev. Lett. 92, 257201 (2004).[6] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y . Tokura, Nature 426, 55 (2003).[7] D. Rubi, Sriram Venkatesan, B. J. Kooi, J. Th. M. De Hosson, T. T. M. Palstra, and B. Noheda1, Physical Review B 78, 020408(R) (2008). [8] c = λ / 2sinθ, where λ is the x-ray wavelength of 0.154 nm, and θ is the scattering angle.[9] J. A. Alonso, M. J. Martinez-Lope, M. T. Casais, and M. T. Fernandez-Diaz, Inorganic Chemistry 39, 917 (2000).[10]  Magnetometry measurements of similarly prepared TMO samples show that field cooling vs. zero field cooling has no observable effect on the resulting hysteresis loop.[11] See http://www.ncnr.nist.gov/instruments/ng1refl.[12] For monochromatic radiation (as for the cases described here), Q = 4π sinθ / λ, where θ is the scattering angle, and λ is the radiation wavelength.[13] L. G. Parratt, Physical Review 95, 359 (1954).  [14] G. P. Felcher, Phys. Rev. B 29, 1268 (1984).[15] C. F. Majkrzak, Physica B 221, 342 (1996).[16] P. A. Kienzle, M. Doucet, D. J. McGillivray, K. V . O’Donovan, N. F. Berk, and C. F. Majkrzak, see  http://www.ncnr.nist.gov/reflpak for documentation.\r7[17] M.R. Fitzsimmons, et al., Phys. Rev. B 76, 245301 (2007).[18] Kirby Dwight and Norman Menyuk, Phys. Rev. 119, 1470 (1960). [19] D. Rubi, C. deGraaf, C. J. M. Daumont, D. Mannix, R. Broer, B. Noheda, preprint available at http://ArXiv.org, arXiv:0810.5137v1. \r8FIGURESFigure 1:  The SrTiO3 and TbMnO3 [001] reflections observed using x-ray diffraction.  Error bars indicate +/- 1 sigma. \n\r9Figure 2:  SQUID magnetometry results.  Ferromagnetism is observed  at 5 K, and a net magnetization persists above 100 K (inset).  \n\r10Figure 3:  Fitted XRR data for a typical TbMnO3 film.  Some higher Q data is omitted for clarity.  Error bars indicate +/- 1 sigma.\n\r11Figure 4:  Fitted PNR.  Spin-dependent oscillations are present at 6 K (a), and at 100 K (b, inset).  Error bars indicate +/- 1 sigma.\n\r12Figure 5:  Models used to fit the data in Fig. 2-3.  a)  Structural profiles:  real and imaginary components of the x-ray scattering length density, and the nuclear scattering length density (multiplied 10).  b)  Magnetic profiles at 6 K and 100 K.\n\r13"    },    {        "title": "1808.03966v2.Resonant_Driving_induced_Ferromagnetism_in_the_Fermi_Hubbard_Model.pdf",        "content": "Resonant Driving induced Ferromagnetism in the Fermi Hubbard Model\nNing Sun,1Pengfei Zhang,1and Hui Zhai1, 2\n1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China\n2Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China\n(Dated: September 18, 2018)\nIn this letter we consider quantum phases and the phase diagram of a Fermi Hubbard model under periodic\ndriving that has been realized in recent cold atom experiments, in particular, when the driving frequency is\nresonant with the interaction energy. Due to the resonant driving, the e ffective Hamiltonian contains a correlated\nhopping term where the density occupation strongly modifies the hopping strength. Focusing on half filling, in\naddition to the charge and spin density wave phases, large regions of ferromagnetic phase and phase separation\nare discovered in the weakly interacting regime. The mechanism of this ferromagnetism is attributed to the\ncorrelated hopping because the hopping strength within a ferromagnetic domain is normalized to a larger value\nthan the hopping strength across the domain. Thus, the kinetic energy favors a large ferromagnetic domain\nand consequently drives the system into a ferromagnetic phase. We note that this is a di fferent mechanism\nin contrast to the well-known Stoner mechanism for ferromagnetism where the ferromagnetism is driven by\ninteraction energy.\nRecently, with the help of quantum gas microscope for\nfermions [1–6], tremendous experimental progresses have\nbeen made on quantum simulation of the Fermi Hubbard\nmodel. These progresses include the observation of equilib-\nrium properties such as short-range antiferromagnetic corre-\nlations [7–9], hidden antiferromagnetic correlations [10], in-\ncommensurate spin correlations [11], canted antiferromag-\nnetic correlations [12] and pairing correlations [13] in sev-\neral di fferent circumstances. In particular, the antiferromag-\nnetic quasi-long-range order has been successfully observed\nthrough entropy engineering [14]. These progresses also in-\nclude the study of non-equilibrium transport behaviors such as\nthe measurement of optical conductivity [15], and the spin and\ncharge transport behavior in the strongly interacting regime\n[16, 17].\nStudying Fermi Hubbard model with cold atoms also al-\nlows us to open up new avenue beyond the traditional con-\ndensed matter paradigm. One of such examples is the period-\nically driven Fermi Hubbard model [18, 19]. Since the typ-\nical parameters of a Hubbard model is the hopping strength\nJand the on-site interaction U, both of which are of the or-\nder of electron volt in strongly correlated solid-state materi-\nals, it is therefore hard to drive a solid-state material with fre-\nquency resonant with any of these two energy scales. How-\never, in cold-atom optical lattice realization of the Fermi Hub-\nbard model, the typical energy scales for these two parameters\nare both of the order of thousand Hertz, and it is quite easy\nto drive the optical lattices with such a frequency. When the\ndriving frequency is resonant with the interaction parameter\nU, the driving can strongly modify the Fermi Hubbard model.\nAs observed in a recent experiment from the ETH group, the\nshort-range antiferromagnetic correlation can be reduced, or\nenhanced, or even switch sign to become ferromagnetic cor-\nrelation [20]. Similar experiment has also been performed by\ndriving the Hubbard model with two-photon Raman transition\n[21]. Hence, by combining such a resonant driving with the\nquantum gas microscope, it is very promising to study novel\nphysics induced by periodic driving that cannot be accessed\nJ\nUShaking:A,!\n<latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit>Shaking:A,!\n<latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit><latexit sha1_base64=\"jNB8lmmHF+FSax+mbSJwe4mw6IA=\">AAACA3icbVDLSgNBEJz1GeNr1ZteBoPgQcKuCIqnqBePEc0DkhBmJ51kyOzsMtMrhiXgxV/x4kERr/6EN//GyeOgiQUNRVU33V1BLIVBz/t25uYXFpeWMyvZ1bX1jU13a7tsokRzKPFIRroaMANSKCihQAnVWAMLAwmVoHc19Cv3oI2I1B32Y2iErKNEW3CGVmq6u3WEB0xvu6wnVOecDujFEa1HIXRY0815eW8EOkv8CcmRCYpN96veingSgkIumTE134uxkTKNgksYZOuJgZjxHutAzVLFQjCNdPTDgB5YpUXbkbalkI7U3xMpC43ph4HtDBl2zbQ3FP/zagm2zxqpUHGCoPh4UTuRFCM6DIS2hAaOsm8J41rYWynvMs042tiyNgR/+uVZUj7O+17evznJFS4ncWTIHtknh8Qnp6RArkmRlAgnj+SZvJI358l5cd6dj3HrnDOZ2SF/4Hz+ANWqlvk=</latexit>FIG. 1: A schematic of the Fermi Hubbard model on a two-\ndimensional square lattice. Jdenotes the hopping strength. Uis the\non-site interaction. The system is time-periodically modulated by\nshaking the lattice in a sinusoidal way with frequency ωand ampli-\ntude A. Arrows indicate the direction of shaking. Balls with di fferent\ncolors and arrows indicate fermions with di fferent spins.\nin a static system. The goal of this letter is therefore to pre-\ndict quantum phases and phase diagram of the resonant driven\nFermi Hubbard model that is newly realized in cold atom ex-\nperiments.\nModel. We consider a two-dimensional square lattice un-\nder similar driving as realized in experiments [20, 28]. The\nlattice is periodically modulated along the ˆ xand ˆydirections\nwith a frequency ωand an amplitude A, whose single particle\nHamiltonian can be written as\nˆH0(t)=p2\n2m+ˆV(x+Acos(ωt),y+Acos(ωt)) (1)\nwhere mis the mass of an atom. We can now preform a unitary\ntransformation [22]\nˆU(t)=exp(ip·r0(t)), (2)arXiv:1808.03966v2  [cond-mat.quant-gas]  17 Sep 20182\nwhere r0(t)=−Acos(ωt)(1,1) are two-dimensional vectors.\nThis unitary transformation transfers position r0(t) into the\ncomoving frame where the lattice becomes static but an extra\ntime-dependent gauge field is introduce. The resulting Hamil-\ntonian is written as\nˆH0(t)=(p−A(t))2\n2m+ˆV(r), (3)\nwith A(t)=mr0(t). In principle, the Hamiltonian Eq. 3 is\nequivalent to the one Eq. 1 but it is more convenient for later\npurpose.\nNow we consider a single-band tight-binding model with\nthe nearest neighboring tunneling coe fficient Jand on-site\nHubbard interaction strength U. The Hamiltonian in a sec-\nond quantized form can be written by the Peierls substitution\nas\nˆH(t)=−J/summationdisplay\n/angbracketlefti,j/angbracketright\nσ=↑,↓eidij·A(t)ˆc†\niˆcj+U/summationdisplay\ni/parenleftBigg\nˆni↑−1\n2/parenrightBigg/parenleftBigg\nˆni↓−1\n2/parenrightBigg\n.(4)\nwhere ˆ ciσ(ˆc†\niσ) is the fermionic annihilation (creation) opera-\ntor on site iwith spinσ, ˆniσis the density operator on site i\nwith spinσ,/angbracketleft.../angbracketrightdenotes the nearest neighboring sites, and\ndij=di−djwithdibeing the position of the ith lattice site.\nThroughout this work we focus on the half-filling case and the\nchemical potential is set to zero.\nIf the modulation frequency ωis the largest energy scale\nof the problem, one can make a high-frequency expansion\nto obtain an e ffective time-independent Hamiltonian [23, 24].\nThe e ffective Hamiltonian takes the same form as the normal\nHubbard model and the only modification is that the tunnel-\ning coe fficient is renormalized by the oscillating gauge field\nas˜J=JB0(A), where we useBlto denote the lth Bessel\nfunction andA=mAωdis the normalized shaking amplitude\nhereinafter. dis the distance of two Wannier wave packets in\nthe nearest neighboring lattice sites.\nHowever, this expansion falls down when the modulation\nfrequencyω, or lth multiple of it, is comparable to one of\nthe energy scale of the problem, say, the Hubbard interaction\nstrength U. That is to say, l/planckover2pi1ω≈U, and we call it the lth\nresonance. Note that in this case, because U−l/planckover2pi1ωis a small\nenergy scale, we should apply another unitary transformation\nˆR(t)=exp(i/summationdisplay\njlωtˆnj↑ˆnj↓), (5)\nwhich alters the interaction strength to an e ffective one ˜U=\nU−l/planckover2pi1ω. Moreover, since ˆR(t) does not commute with the\nhopping term, it introduces an additional density dependence\nto the hopping term, and e ffectively it changes the gauge field\nto a spin and density dependent one as\n˜Ai j,σ(t)=A(t)−lωt\nd2dij((1−ˆni¯σ)ˆnj¯σ−(1−ˆnj¯σ)ˆni¯σ)).(6)\nNow the high frequency expansion can be safely applied, and\nto the lowest order it again results in a time-independent ef-fective Hamiltonian written as\nˆHeff=/summationdisplay\n/angbracketlefti,j/angbracketright,σ−ˆJ/angbracketlefti j/angbracketright\neff,σˆc†\niσˆcjσ+˜U/summationdisplay\ni/parenleftBigg\nˆni↑−1\n2/parenrightBigg/parenleftBigg\nˆni↓−1\n2/parenrightBigg\n.(7)\nHere ˆJ/angbracketlefti j/angbracketright\neff,σis defined as\nˆJ/angbracketlefti j/angbracketright\neff,σ=J0ˆai j¯σ+J1ˆbi j¯σ, (8)\nwhere ¯σdenotes the complement of σ,J0=JB0(A),J1=\nJBl(ηi jA) (ηi j=±1 for ( ix,iy)=(jx±1,jy) or ( ix,iy=\njx,jy±1)), and\nˆai jσ=(1−ˆniσ)(1−ˆnjσ)+ˆniσˆnjσ, (9)\nˆbi jσ=(−1)l(1−ˆniσ)ˆnjσ+ˆniσ(1−ˆnjσ). (10)\nIn above, the site dependence of J1is made implicitly. Note,\nhowever, that for even lthe Bessel function Blis an even func-\ntion, in which case ηi jcan be simply dropped and J1becomes\na constant. Compared to the o ff-resonance case, now the hop-\nping strength depends on site occupation of fermions. As we\nwill show below, this correlated hopping plays a key role in\nthe emergent new mechanism for ferromagnetism phase.\nSymmetry. Before we discuss how to solve this e ffective\nHamiltonian, let us first comment on the symmetry of this\nproblem. Note that the original Hubbard model possesses a\nSO(4) symmetry [25], which is composed of a spin SU(2),\ngenerated by ˆSz=(1/2)/summationtext\niˆc†\ni↑ˆci↑−ˆci↓ˆci↓,ˆS+=/summationtext\niˆc†\ni↑ˆci↓\nand ˆS−=ˆS†\n+, and a charge SU(2), generated by ˆLz=\n−(1/2)/summationtext\niˆc†\ni↑ˆci↑+ˆc†\ni↓ˆci↓+Ns/2,ˆL+=/summationtext\ni(−1)iˆci↑ˆci↓and ˆL−=\nˆL†\n+.Nsis the total number of sites. The spin SU(2) ensures\nthat the direction of spin-density-wave (SDW) order parame-\nter can be taken along any direction, while the charge SU(2)\nensures the degeneracy of a charge-density-wave (CDW) or-\nder and the fermion pairing order (P).\nIn the presence of periodic modulation, considering the\ntime-dependent Hamiltonian Eq. 4, it is straightforward to\nshow that the spin SU(2) symmetry stays, yet the charge\nSU(2) symmetry no longer holds because ˆLzdoes not com-\nmute with the/summationtext\ni,σfi(t)ˆniσterm. However, considering the\ntime-independent e ffective Hamiltonian Eq. 7, one can show\nthat the charge SU(2) symmetry is recovered for even lcase\nthough not for odd lcase [26]. Hereafter we focus only on the\neven lcase which possesses the same SO(4) symmetry as the\noriginal Hubbard model. In addition, the e ffective Hamilto-\nnian also possesses particle-hole symmetry at half-filling.\nPhase Diagram. We present our results on the phase dia-\ngram following from a standard mean-field treatment based on\nthis e ffective Hamiltonian Eq. 7, which is known to be quali-\ntatively reliable for a normal Hubbard model [26, 27]. Thanks\nto the SO(4) symmetry, we can choose SDW along ˆ zdirection\n(i.e. si=/angbracketleftˆni↑−ˆni↓/angbracketright) and CDW (i.e. ci=/angbracketleftˆni↑+ˆni↓/angbracketright−1) as the\norder parameters in our mean-field theory. Note that when we\nobtain the CDW order, it means that the system can have either\nCDW order or fermion pairing order, or an arbitrary combina-\ntion of them, as the order parameter of the degenerate ground3\n-� -� -� � � � �-�-���\n�/����/��\n���/� ������/�\n+����� FM1\n2\n3\nFIG. 2: Phase diagram for the e ffective Hamiltonian Eq. 7 of a\nresonantly driven Fermi Hubbard model with even lat half filling.\nThe phase diagram is controlled by two dimensionless parameters,\nJ1/J0and ˜U/J0. “CDW” and “SDW” denote charge and spin den-\nsity wave order. “P” denote fermion pairing order. “FM” denotes\nferromagnetism, “PS” denotes phase separation into high and low\ndensity regimes. Red lines denote the first order transition and the\nblue dashes lines denote the second order transition. Dashed arrows\nmark the lines along which the order parameters are plotted in FIG.\n3.\nstates. Higher order e ffect will break the degeneracy between\nCDW and fermion pairing order, but it is beyond the scope of\ncurrent work.\nThe phase diagram is shown in Fig. 2. Setting J0as the\nenergy unit, the phase diagram is controlled by two param-\neters of J1/J0and ˜U/J0, both of which can be easily tuned\nfrom positive to negative via control of ωandA. As breach-\nmark of our calculation, first of all, note that when J1/J0=1,\nbecause ˆ ai jσ+ˆbi jσ=ˆI, the kinetic energy term becomes\n−J0/summationtext\n/angbracketlefti j/angbracketright,σˆc†\niσˆcjσand the Hamiltonian recovers the usual Hub-\nbard model. In this case (labeled by 2 in Fig. 2), the result\nfor the normal Hubbard model is retrieved where we obtain a\nCDW order of ( π,π) with attractive interaction ( ˜U<0) and a\nSDW order of ( π,π) with repulsive interaction ( ˜U>0). Ex-\nplicitly, the order parameters are chosen as si=(−1)ix+iysand\nci=(−1)ix+iyc, and s(c) gradually vanishes as ˜Uapproaches\nzero from the positive (negative) side. As a result, a second or-\nder phase transition occur at ˜U=0 (gray dot in FIG. 2). This\ncan be seen from the order parameters plotted as a function of\n˜U, shown as green curves in Fig. 3.\nSince both CDW and SDW are ordered phases, the more\ngeneric situation should be either a first order transition or a\nphase co-existence regime in between. As marked by the red\nsolid lines in Fig. 2, the phase boundary of CDW and SDW\n��/��=�����/��=�����/��=-���\n-�-�-� �������������������\n�/������� ���������\nSolid: SDW\nDahsed: CDWFIG. 3: The CDW order parameter c(dashed lines) and the SDW\norder parameter s(solid lines) as a function of ˜Ufor three represen-\ntative cases labeled as 1-3 in Fig. 2, with J1/J0=1.5, 1.0 and−2.0,\nrespectively.\nat large|J1/J0|is a first order transition. The order parame-\nters are shown with orange lines in Fig. 3 for a representative\ncase (labeled by 1 in Fig. 2), where the CDW or SDW order\nparameter jumps from a finite value to zero at ˜U=0. At cer-\ntain regime of J1/J0, a CDW and SDW co-existence regime\nshows up in between as displayed by the yellow regime in Fig.\n2. The order parameters are shown with blue lines in Fig. 3\nfor a representative case (labeled by 3 in Fig. 2).\nThe most notable feature in Fig. 2 is the green region. In\nthis region a mean-field ansatz of CDW or SDW orders with\nordering vector at ( π,π) may not yield any ordered solution.\nHowever, when we consider the case of enlarged 2 ×2, 3×3,\nup to L×Ldomains, and within each domain the CDW and\nSDW order parameters are uniformly chosen as candswhile\nin its neighboring domain they are taken as −cand−s, the\nmean-field ansatz does yield ordered solutions. It can be seen\nfrom Fig. 4 which shows that the mean-field ground state en-\nergy decreases monotonically as Lincreases. It indicates that\nthe ground state will form large domains with opposite order\nparameter values. Moreover, minimizing ground state energy\nyields c=0 and s/nequal0 at positive ˜Uandc/nequal0 and s=0\nat negative ˜U. Hence, the system at positive ˜Upossesses\nspin order, and the increasing of the domain size means the\ndecreasing of the spin ordering wave vector. Eventually, the\nwave vector decreases toward zero, and the ground state be-\ncomes a ferromagnetic state. In another word, as the domain\nsize becomes larger and larger, the system is essentially made\nof ferromagnetic domains. For negative ˜Uthe system tends\nto phase separation with high density in one domain and low\ndensity in its neighboring domain. The transition between the\nferromagnetic phase to the SDW phase, as well as the transi-\ntion from phase separation to CDW, is a first order transition.\nIt should be emphasized that both the ferromagnetism and\nthe phase separation regime occur at small ˜U. In fact, it is\npurely due to the correlated hopping e ffect in the e ffective4\nFIG. 4: The mean-field ground state energy as a function of the do-\nmain size L. The insets show the configuration for 2 ×2, 3×3 and\n4×4 blocks where the CDW and SDW order parameters are uni-\nformly distributed within a domain and take opposite values between\nneighboring domains. For comparison, the solid line is the mean-\nfield energy when order parameters are all zero.\nHamiltonian Eq. 7, which originates essentially from the res-\nonant driving. Considering the mean-field configurations as\nshown in the insets of Fig. 4, let us look at the mean-field\nvalue of the e ffective hopping strength /angbracketleftˆJ/angbracketlefti j/angbracketright\neff,σ/angbracketrightwhich quanti-\nfies how the particle occupations a ffect the hopping strength\nand thus a ffect the bandwidth. We define Jintra\neff,σas/angbracketleftˆJ/angbracketlefti j/angbracketright\neff,σ/angbracketrightwith\nboth iand jin the same domain, and Jinter\neff,σas/angbracketleftˆJ/angbracketlefti j/angbracketright\neff,σ/angbracketrightwith i\nand jacross two neighboring domains. It is straightforward\nto write down both Jintra\neff,σandJinter\neff,σas\nJintra\neff,σ=1\n2/parenleftBig\nJ0[1+(c∓s)2]+J1[1−(c∓s)2]/parenrightBig\n, (11)\nJinter\neff,σ=1\n2/parenleftBig\nJ0[1−(c∓s)2]+J1[1+(c∓s)2]/parenrightBig\n, (12)\nwhere∓corresponds to di fferent spin component. One can\nshow that when|J1|<|J0|,|Jinter\neff,σ|is always smaller than |Jintra\neff,σ|.\nHence, the size of the domain tends to increase such that\nthere are more intra-domain links than inter-domain links, and\ntherefore the e ffective bandwidth on average becomes larger.\nFor a given filling, a larger bandwidth leads to more kinetic\nenergy gain.|J1|<|J0|is precisely the regime where we find\nferromagnetism or phase separation in the phase diagram of\nFig. 3 with arbitary weak interaction. This regime can be\neasily accessed when Ais small.\nAn alternative way to understand the emergence of this fer-\nromagnetism is to consider a uniform system with ˜U=0,\nwhere the Hamiltonian contains only the correlated tunneling\nterm. Substitute ˆJ/angbracketlefti j/angbracketright\neff,σby its mean-field value, it is straight-\nforward to compute the kinetic energy of this uniform sys-\ntem that depends on ni↑andni↓, where n↑=(n+sz)/2 and\nn↓=(n−sz)/2. We plot the kinetic energy as a function of\nnforsz=0 in Fig. 5(a) and as a function of szforn=1\nin Fig. 5(b) for two representative cases with J1/J0=−0.8\nand−1.8. One can see from Fig. 5(a) that for J1/J0=−0.8,\nthere are two local minima with one at n>1 and the other at\nn<1, who locate symmetrically on two sides of n=1, while\n��/��=-�����/��=-���\n��� ��� ��� ��� ���-���-���-���-���-���-���-������\n�������� �������(� ↑+�↓)��������� ������ℰ(�)/� �\n��� ��� ��� ��� ���-���-���-���-���-���-���-������\n�������� �������(� ↑+�↓)��������� ������ℰ(�)/� �FIG. 5: The mean-field energy with ˜U=0 (a) as a function of total\ndensity nwith sz=0 fixed, and (b) as a function of szwith n=1\nfixed. Solid line is for J1/J0=−0.8 and the dashed line is for J1/J0=\n−1.8.\nforJ1/J0=−1.8 there is only one minimum located at n=1.\nSimilarly, in Fig. 5(b) for J1/J0=−0.8, there are two lo-\ncal minima with one at positive szand the other at negative sz,\nsymmetrically distributed around sz=0, and for J1/J0=−1.8\nthere is only one minimum at sz=0. Thus, when the system\nis constrained with the average n=1 and sz=0, for the\ncase with J1/J0=−0.8, it will actually phase separate into\ndomains with either di fferent nor different sz, corresponding\nto phase separation and ferromagnetism, respectively. The\nchoice is made by the sign of ˜Uwhen a small but finite ˜U\nperturbation is turned on.\nConclusion and Outlook. The most significant finding of\nthis work is to provide an alternative mechanism for the on-\nset of ferromagnetism in the model with correlated hopping,\nwhich roots in the cooperation between the spin order and the\ncorrelated hopping. It is driven by the kinetic energy term, and\noccurs in the weakly interacting regime. This is in contrast to\nthe well-known Stoner ferromagnetism mechanism which is\ndriven by large interaction energy and occurs when the inter-\naction strength is beyond certain critical value. This is also\ndifferent from the ferromagnetism due to the super-exchange\nprocesses discussed in the experiment of Ref. [20] which re-\nquires ˜Uto be negative.\nFinally we shall comment on the experimental observation\nof this ferromagnetism. First of all, the system itself has been\nrealized with cold atoms, and moreover, a very recent experi-\nment shows that the heating is insignificant in the presence of\ndriving and the life time of the system can be about one sec-\nond [28]. Second, because this ferromagnetism is driven by\nthe kinetic energy, and as one can see from Fig. 5, the energy\ngain is of the order of bandwidth, thus one expects this fer-\nromagnetism to be observed when temperature is of the order\nof bandwidth, which can be accessed now by cold atom ex-\nperiments. Thirdly, the quantum gas microscope techniques\nmentioned at the beginning can be used to detect real space\nferromagnetic domains. Hence, it is quite promising to verify\nthis theory experimentally in very near future.\nAcknowledgment. This work is supported MOST un-\nder Grant No. 2016YFA0301600 and NSFC Grant No.5\n11734010.\n[1] E. Haller, J. Hudson, A. Kelly, D. A. Cotta, B. Peaudecerf, G.\nD. Bruce and S. Kuhr, Single-atom imaging of fermions in a\nquantum-gas microscope , Nature Physics 11, 738 (2015).\n[2] L. W. Cheuk, M. A. Nichols, M. Okan, T. Gersdorf, V . V . Ra-\nmasesh, W. S. Bakr, T. Lompe, and M. W. Zwierlein, Quantum-\nGas Microscope for Fermionic Atoms , Phys. Rev. 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Esslinger, Floquet dynamics in driven Fermi-\nHubbard systems , arXiv:1808.00506.Supplementary Material: Resonant Driving induced Ferromagnetism in the Fermi Hubbard Model\nNing Sun,1Pengfei Zhang,1and Hui Zhai1, 2\n1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China\n2Collaborative Innovation Center of Quantum Matter, Beijing, 100084, China\n(Dated: September 18, 2018)\nIn this supplementary material, we first discuss the symmetry of the resonantly driven Fermi Hubbard model.\nThe SO(4) symmetry [1] of the e ffective Hamiltonian for even lis proved. Then we establish the mean-field\ntheory adopted in this work for even l, which is consistent with the text-book results [2] for J0=J1.\nSYMMETRY\nIn this section we dicuss the symmetry of the resonantly driven Fermi Hubbard model. The symmetry is of vital significance\nsince it greatly simplifies the formulation of the mean-field description.\nSO(4) symmetry.— A usual bipartite Fermi-Hubbard model, written as ˆH=−J/summationtext\n/angbracketlefti,j/angbracketright,σˆc†\niσˆcjσ+˜U/summationtext\ni(ˆni↑−1/2)(ˆni↓−1/2),\npossesses SO(4) symmetry [1]. The SO(4) symmetry is resolved into two SU(2) symmetries, the spin SU(2) and the charge\nSU(2). The generators of spin SU(2) are given by:\nˆSz=1\n2/summationdisplay\niˆc†\ni↑ˆci↑−ˆci↓ˆci↓,ˆS+=/summationdisplay\niˆc†\ni↑ˆci↓, (1)\nand\nˆS−=ˆS†\n+,ˆSx=ˆS++ˆS−\n2,ˆSy=ˆS+−ˆS−\n2i, (2)\nwho satisfy the commutation relation of the SU(2) algebra. Due to the contraction between spin indices, the very beginning\ntime-dependent Hamiltonian ˆH(t) (Eq.(4) in the main text) is invariant under this SU(2) symmetry operations, is also the unitary\ntransformations ˆR(t) (Eq.(5) and (6) in the main text). Since the time average does not alter this attribute, one arrives at the\nconclusion that the e ffective static Hamiltonian ˆHeffin Eq.(7) of the main text also possess this spin SU(2) symmetry no matter l\neven or odd. This can also be verified directly by checking/bracketleftBigˆHeff,ˆSα/bracketrightBig\n=0 for anyα=x,y,z. The spin rotational symmetry hence\nallows us to automatically get a spin-balanced system without any additional magnetic field.\nHowever, regarding the charge SU(2) symmetry, the time-dependent Hamiltonian lacks it for the appearance of the time-\ndependent onsite energy term (the last term in Eq.(4) of the main text). As a result we do not expect the SO(4) symmetry in\ngeneral. Nevertheless, it emerges in the e ffective Hamiltonian with leven. We introduce the charge SU(2) as follows [1].\nˆLz=−1\n2/summationdisplay\niˆc†\ni↑ˆci↑+ˆc†\ni↓ˆci↓+1\n2N, (3)\nˆL+=/summationdisplay\ni(−1)iˆci↑ˆci↓=/summationdisplay\niexp(iQ·xi)ˆci↑ˆci↓ (4)\nwhere Q=(π,π),Nis the total number of lattice sites hereinafter, and\nˆL−=ˆL†\n+,ˆLx=ˆL++ˆL−\n2,ˆLy=ˆL−−ˆL−\n2i. (5)\n{ˆLz,ˆLx,ˆLy}forms an SU(2) algebra. It can be verified straightforwardly that the e ffective Hamiltonian ˆHeffwith leven commutes\nwith all these generators. This fact can also be seen from the following transformation\nˆP: ˆci↑→(−1)iˆc†\ni↑,\nwho maps ˆHeffwith leven of interaction +˜Uto the same class but of e ffective interaction−˜U, exchanging the role played by {Li}\nand{Si}. As a result, the spin SU(2) invariance for a model with −˜Uindicates the charge SU(2) symmetry for +˜U. And vice\nversa. In addition, this transformation also implies the phase diagram should be symmetric under +˜U↔− ˜Utogether with the\ninterchange of charge and spin order. However, in the lodd cases, this mapping falls down because of the additional minus sign\ninˆbi jσ.arXiv:1808.03966v2  [cond-mat.quant-gas]  17 Sep 20182\nParticle-hole symmetry.— The half-filled e ffective Hamiltonian ˆHeffholds particle-hole symmetry no matter leven or odd.\n(i) When lis even, define particle-hole transformation ˆC: ˆciσ→(−1)iˆc†\niσ. The Hamiltonian is invariant under ˆC. [ˆC,ˆHeff]=0.\n(ii) When lis odd, we further define the bipartite transformation ˆS, which switches the A/Bsublattices. Then the Hamiltonian\nis invariant under the combination of ˆCand ˆS: [ˆCˆS,ˆHeff]=0. This symmetry allow us to fix chemical potential µ=0 during\nthe mean-field calculation at half-filling situation.\nMEAN-FIELD TREATMENT\nIn this section we establish the mean-field theory adopted in this work. As explained in the main text, assuming no canted\norder, we can consider only the charge density wave (CDW) order and spin density wave (SDW) order in the zdirection because\nof the SO(4) symmetry. We focus on the half-filled spin-balanced system, in which case the total charge density and total\nmagnetic momentum is given by\n/angbracketleftˆn/angbracketright=1,/angbracketleftˆS/angbracketright=0. (6)\nPath integral approach.— Here we provide a derivation of the mean-field Hamiltonian Eq. (19) via a path integral approach.\nIn the path integral language, the real-time partition function of the system is given by:\nZ=/integraldisplay\nDψD¯ψexp(i/integraldisplay\ndtL) (7)\nL=/summationdisplay\ni,σi¯ψiσ∂tψiσ+/summationdisplay\n/angbracketlefti,j/angbracketright,σ/parenleftBig\nJ0ai j¯σ(¯ψψ)+J1bl\ni j¯σ(¯ψψ)/parenrightBig¯ψiσψjσ\n−˜U/summationdisplay\ni¯ψi↑ψi↑¯ψi↓ψi↓. (8)\nHereψcorresponds to the fermionic field operators, and ai j¯σandbl\ni j¯σare given by replacing the operators in ˆ ai j¯σandˆbl\ni j¯σwith\nfields. Since the e ffective Hamiltonian (Eq. (7) in the main text) contains six-fermion terms, the traditional decoupling based on\nHubbard-Stratonovich transformation does not applied directly. However, an alternative way of decoupling based on the similar\nspirit of Hubbard-Stratonovich transformation should be adopted here. Specifically, by inserting a delta function, we directly\nintroduce the auxiliary bosonic field:\nZ=/integraldisplay\nDψD¯ψDn/productdisplay\niσδ(ni,σ−¯ψi,σψi,σ) exp( i/integraldisplay\ndtL) (9)\nL=/summationdisplay\ni,σi¯ψiσ∂tψiσ+/summationdisplay\n/angbracketlefti,j/angbracketright,σ/parenleftBig\nJ0ai j¯σ(n)+J1bl\ni j¯σ(n)/parenrightBig¯ψiσψjσ\n−˜U/summationdisplay\nini↑ni↓. (10)\nThanks to the delta function inserted, one could replace all ψ†ψbynin the action. As an equivalence check, if we integrate out\nnfields first, we get our original action back. Now we introduce another field ηto absorb the delta function into an integral:\nZ=/integraldisplay\nDψD¯ψDnDηexp(i/integraldisplay\ndtL) (11)\nL=/summationdisplay\ni,σi¯ψiσ∂tψiσ+/summationdisplay\n/angbracketlefti,j/angbracketright,σ/parenleftBig\nJ0ai j¯σ(n)+J1bl\ni j¯σ(n)/parenrightBig¯ψiσψjσ\n−˜U/summationdisplay\nini↑ni↓−/summationdisplay\niσηiσ(ni,σ−¯ψi,σψi,σ) (12)\nAs a result, the fermion becomes quadratic. In general, one can integrate out all the fermions to get an e ffective action of wholly\nbosonic degrees of freedom. Whereas, the mean-field approximation is to say that all the bosonic fields will be replaced by their\nsaddle point solutions.\nMean-field Hamiltonian.— By doing a Legendre transformation and replacing the fermion bilinears with its expectation\nvalues, we obtain the mean-field Hamiltonian\nˆH=/summationdisplay\n/angbracketlefti,j/angbracketright,σ−/parenleftBig\nJ0ai j¯σ(n)+J1bl\ni j¯σ(n)/parenrightBigˆψ†\niσˆψjσ+˜U/summationdisplay\nini↑ni↓+/summationdisplay\niσηiσ(ni,σ−ˆψ†\ni,σˆψi,σ) (13)3\nFIG. 1: Schematic of the two cases. Left panel: case 1. Right panel: typical examples of case 2 when L=2 and L=3, respectively, as labeled.\nLattice sites of the same color have the same local particle density ciand local spin density si.\nwhere the following replacement is carried out:\nˆai jσ=(1−ˆniσ)(1−ˆnjσ)+ˆniσˆnjσ,→(1−niσ)(1−njσ)+niσnjσ (14)\nˆbi jσ=(−1)l(1−ˆniσ)ˆnjσ+ˆniσ(1−ˆnjσ)→(−1)l(1−niσ)njσ+niσ(1−njσ) (15)\nFor the half-filled spin-balanced situation, we focus on the following two cases here. Notice that, in a half-filled spin-balanced\nsystem, the local particle density can be expressed in terms of local charge density and local spin density as\nni↑=1\n2(1+ci+si),ni↓=1\n2(1+ci−si) (16)\nwhere/summationtext\nici=0 and/summationtext\nisi=0 . The two cases are then:\n1.ci=(−1)ix+iyc,si=(−1)ix+iys\n2.ci=(−1)⌈ix/L⌉+⌈iy/L⌉c,si=(−1)⌈ix/L⌉+⌈iy/L⌉s\nwhere candsare CDW and SDW order parameters introduced, and ⌈x⌉denotes the ceiling function of x. Substituting these two\nexpressions into the original mean-field Hamiltonian Eq.(13) will yield the mean-field Hamiltonian in each case.\nCase 1. The first case is actually the ( π,π) order, where the lattice is naturally divided into A/Bsublattices and the unit cell\nof the lattice consists of 2 sites, with one from Aand the other from B. The schematic is shown in FIG. 1 (Left).\nTo be specific, we explicit show here the workflow in the first case. By substitution,\nnA≡/angbracketleftˆni∈A/angbracketright=1+c,sA≡/angbracketleftˆSz\ni∈A/angbracketright=s/2, (17)\nnB≡/angbracketleftˆni∈B/angbracketright=1−c,sB≡/angbracketleftˆSz\ni∈B/angbracketright=−s/2, (18)\nand the mean-field Hamiltonian is explicitly written as:\nˆHmf=/summationdisplay\n/angbracketlefti,j/angbracketright,σ{−J0[(1−nA¯σ)(1−nB¯σ)+nA¯σnB¯σ]−J1[(1−nA¯σ)nB¯σ+nA¯σ(1−nB¯σ)]}ˆcA†\niσˆcB\njσ+H. C. +˜UN\n2(nA↑nA↓+nB↑nB↓)\n+ηc/bracketleftBigg\nc−(ˆnA↑+ˆnA↓)−(ˆnB↑+ˆnB↓)\n2/bracketrightBigg\n+ηs/bracketleftBigg\ns−(ˆnA↑−ˆnA↓)−(ˆnB↑−ˆnB↓)\n2/bracketrightBigg\n(19)\nwhere ˆ cA\niσ(ˆcB\niσ) are the annihilation operators of isite and spin σonA(B) sublattice, ˆ nµσ=1\nN/2/summationtext\ni∈µˆniσ,µ=AorB, and\nnµσ=/angbracketleftˆnµσ/angbracketright. Hereηcandηsare Lagrangian multipliers of the CDW and SDW orders, respectively, who are also variational\nparameters to optimize the ground-state energy of ˆHmfto get a mean-field solution.\nAfter some substitution and simplification, the mean-filed Hamiltonian in momentum space is written as\nˆHmf=/summationdisplay\nkσ−Pσ(c,s)Q(k)ˆcA†\nkσˆcB\nkσ+H.c.+˜UN\n2(1+c2−s2)\n+ηc/bracketleftBigg\nc−(ˆnA↑+ˆnA↓)−(ˆnB↑+ˆnB↓)\n2/bracketrightBigg\n+ηs/bracketleftBigg\ns−(ˆnA↑−ˆnA↓)−(ˆnB↑−ˆnB↓)\n2/bracketrightBigg\n(20)4\nwhere ˆ cA\nkσ(ˆcB\nkσ) are the annihilation operators on A(B) sublattice of quasi-momentum kand spinσ,Q(k)=/summationtext\niexp(ik·di),{di}\nare the lattice vectors, and\nP↑(c,s)=J0\n2/bracketleftBig\n1−(c−s)2/bracketrightBig\n+J1\n2/bracketleftBig\n1+(c−s)2/bracketrightBig\n(21)\nP↓(c,s)=J0\n2/bracketleftBig\n1−(c+s)2/bracketrightBig\n+J1\n2/bracketleftBig\n1+(c+s)2/bracketrightBig\n(22)\nThe summation of kin Eq.(20) is over the first Brillioun zone. Minimization the energy for all parameters {c,s,ηc,ηs}yields a\nset of mean-field equations that is solved by numerical iteration in this work.\nCase 2. The second case actually includes a series of circumstances of Lbeing 2,3,4,...,∞. Typical examples of L=2\nandL=3 are shown in the right panel of FIG. 1. In this case, for each independent L, doing the similar substitution as above and\nsolving optimization problem in a numerical way returns us a set of self-consistent mean-field solutions in each L. The ground\nstate should be the one with the lowest ground state energy.\nFIG. 2, 3, 4, 5 in the main text are based on the mean-field numerics of above two cases.\nCheck the formulation for J 0=J1.— Finally, we explain why our mean-field theory reduces to a traditional one appearing\nin common text books, e.g. [2]. In fact, the Pσin this case is just a constant with no candsdependence. Thus the variation of c\nandsyields:\nηc=−˜UNcηs=−˜UNs. (23)\nUsing this relation, we see the mean-field Hamiltonian reduces to a familiar one that appears in textbooks [2].\n[1] C. N. Yang and S. C. Zhang, SO4symmetry in a Hubbard model , Mod. Phys. Lett. B 4, 759 (1990).\n[2] A. Altland, and B. D. Simons. Condensed matter field theory, second edition, Cambridge University Press, 2010."    },    {        "title": "0808.2358v1.Canted_Magnetization_Texture_in_Ferromagnetic_Tunnel_Junctions.pdf",        "content": "arXiv:0808.2358v1  [cond-mat.str-el]  18 Aug 2008Canted Magnetization Texture in Ferromagnetic Tunnel Junc tions\nIgor Kuzmenko and Vladimir Fal’ko\nDepartment of Physics, Lancaster University, Lancaster, L A1 4YB, United Kingdom\n(Dated: November 15, 2018)\nWe study the formation of inhomogeneous magnetization text ure in the vicinity of a tunnel\njunctionbetweentwoferromagnetic wires nominallyinthea ntiparallel configurationanditsinfluence\non the magnetoresistance of such a device. The texture, depe ndent on magnetization rigidity and\ncrystalline anisotropy energy in the ferromagnet, appears upon an increase of ferromagnetic inter-\nwire coupling above a critical value and it varies with an ext ernal magnetic field.\nPACS numbers: 72.25.Mk, 75.47.De, 75.60.Ch\nSpin polarized transport in multilayer ferromagnet-\nmetal-ferromagnet system and magnetic tunnel junc-\ntion is a subject of intense theoretical and experimental\nstudies.1,2The majority of such studies addresses the in-\nvestigationofthetunnelingmagneto-resistance3,4,5(MR)\nand giant MR6,7,8,9,10effects, which consist of a switch\nfrom lower to higher conductivity when polarization of\nleads in a MR device change from an antiparallel to par-\nallel configuration. In tunnel junctions, the MR effect is\nthe result of a difference between rates of tunneling of\nelectrons majority and minority bands on the opposite\nsides of a junctions, and it’s the strongest when magne-\ntization switching by an external magnetic field changes\nfrom an ideally antiparallel magnetization state in the\ntwo wires to a parallel one.11,12,13,14Any deviations of\nthe magnetization near the interface (where tunneling\ncharacteristics of the device are formed) from perfectly\nparallel/antiparallel orientation would reduce the size of\nthe effect.\nIn this paper we investigate a possibility of formation\nof inhomogeneous magnetization texture in the vicinity\nof a highly transparent tunnel junction caused by fer-\nromagnetic coupling of magnetic moments on the op-\nposite sides carried by tunneling electrons. We find\nthat a canted magnetization state can form if such fer-\nromagnetic tunneling coupling, t′, exceeds some critical\nvaluetcdetermined by the interplay between crystallineanisotropy and magnetization rigidity in the ferromag-\nnet. This means that a tunnel junction with t′< tccan\nbe viewed as an atomically sharp magnetic domain wall,\nwhereas the increase of the junction transparency above\ntcgradually transforms it into a broad texture typical\nfor bulk ferromagnetic material. For t′> tc, we study\nthe evolution of the texture upon application of an ex-\nternal magneticfield and construct a parametricdiagram\nfor distinct magnetization regimes. As an example, we\nconsider a device consisting of a tunnel junction between\ntwo easy-axis ferromagnetic wires magnetically biased at\nthe ends as illustrated in Fig.1. We show that when the\nmagnetic field exceeds somecritical value, Bc, the barrier\nbetween the states with polarization parallel/antiparallel\nto the field disappears. As a result, both wires will be\npolarized parallel to the magnetic field and the domain\nwall will be pushed away toward the end of the wire.\nWhen a magnetic field is swept back (to B < B c) and its\nsign changes, the domain wall returns back to the tunnel\njunction. The resulting hysteresis in the magnetization\nstate of the device leads to a hysteresis loop in its MR,\nwhich we analyze taking into account the formation of\nthe texture near the tunnel junction with t′> tc.\nA quasi-1D magnetization texture, S(z), near the tun-\nneljunctionbetweentwowiresoflength L,whichchanges\nslowly on the scale of the lattice constant acan be de-\nscribed using the energy functional,\nE=J\n2/parenleftBigg−0/integraldisplay\n−L+L/integraldisplay\n+0/parenrightBigg\ndz/parenleftBig\n∂zS(z)/parenrightBig2\n−w2\na3L/integraldisplay\n−Ldz/bracketleftbigg\nξ1/parenleftBig\nSz(z)/parenrightBig2\n+ξ2/parenleftBig\nSy(z)/parenrightBig2/bracketrightbigg\n+\n+gdw3\n2a3L/integraldisplay\n−Ldzdz′\nw2V(z−z′)/braceleftbigg\nSx(z)Sx(z′)+Sy(z)Sy(z′)−2Sz(z)Sz(z′)/bracerightbigg\n−µBw2\na3L/integraldisplay\n−LdzSz(z)+E′.(1)\nHereJ∼tw2/ais the magnetization rigidity (where tis\nthe exchange coupling between the neighboring atoms, w\nis the wire thickness). The crystalline anisotropy param-\neters are ξ1> ξ2>0, whereas gd=γ2/a3parameterizesdipole-dipole interaction of magnetic moments, with\nV(z) =1\n2w/integraldisplay\nd2ρd2ρ′2z2−(x−x′)2−(y−y′)2\n/parenleftbig\n(x−x′)2+(y−y′)2+z2/parenrightbig5/2,\nwhere the integration is carriedout overthe cross-section\nofthewire, d2ρ=dxdy,d2ρ′=dx′dy′. For|z|> w,V(z)2\ndecreases as w3/z3, and for |z| ≪w,V(z)∝ln(w/|z|).\nFor a smooth magnetization texture varying at a length\nscale longer than the wire thickness, we approximate the\nnon-local dipole-dipole interaction as V(z)≈V0wδ(z),\nV0=/integraltextdz\nwV(z).Finally,µis the magnetic moment per\natom, and B=Bezis an external magnetic field.\n \neV\n2\n\u0001\u0000\u0002\n\u0003\u00042\n\u0005\u0006\u0007\n\b\t\n\n\u000b (b)(a)\n()0\f \r\n()0 \u000eS ()L \u000fS()0\u0010 \u0011\n()0 \u0012S ()\u0013 \u0014S\neV\n2\n\u0015\u0016\u0017\n\u0018\u00192\n\u001a\u001b\u001c\n\u001d\u001e\n\u001f \neV\n2\n!\"#\n$%2\n&'(\n)*\n+, (b)(a)\n()0- .\n()0 /S ()L 0S()01 2\n()0 3S ()4 5S()06 7\n()0 8S ()L 9S()0: ;\n()0 <S ()= >S\nFIG. 1: (a)Ferromagnetic wires. Polarizations of the wire\nendsz= 0 and z=±LareS(±0) andS(±L), respectively.\nThe angle between S(±0) and the axis zisθ(±0).(b)Band\nalignment of the majority and minority bands for electrons i n\nthe vicinity of the left and right-hand side of a biased ferro -\nmagnetic junction in the antiparallel configuration. Direc tion\nof spin quantization on either side of the junction is deter-\nmined by the orientation of S(±0) of magnetization near the\ninterface.\nThe inter-wire coupling energy, E′, includes the ex-\nchange interaction due to the penetration of the polar-\nized electron wave function through the barrier, from one\nferromagnetic metal into another. For the ferromagnetic\nsign of the inter-wire coupling,\nE′=−t′w2\na2/parenleftBig\nS(+0)·S(−0)/parenrightBig\n.\nIn the following, we will focus on the magnetiza-\ntion texture formed near the tunnel junction between\ntwo ferromagnetic metals with antiparallel polarization\nS(±L) =±ezfixed by magnetic reservoirs at the dis-\ntant ends of the wires. Without coupling between wires\n(t′= 0), this would determine homogeneous magnetiza-\ntionS(z >0) =ezandS(z <0) =−ez. The exchange\ninteraction between the wires may give rise to the for-\nmation of canted magnetization texture in the vicinity of\nthe junction, with boundary values of spin S(±0) deter-\nmined by the interplay between the magnetization rigid-\nity, crystalline anisotropy, Zeeman energy and the ex-\nchange inter-wire interaction. In the case when the easy\nmagnetization direction is along zaxis, we parameterize\nS(z) =ezcos(θ(z))+eysin(θ(z)), (2)\nwithθ(−L) =πandθ(L) = 0. Then, the total energy ofthe interacting wires takes the form\nE=J\n2/parenleftbigg−0/integraldisplay\n−L+L/integraldisplay\n+0/parenrightbigg\ndz/braceleftbigg\n(θ′(z))2+α2sin2(θ(z))+\n+2α2λB/bracketleftBig\n1−cos(θ(z))/bracketrightBig/bracerightbigg\n−t′cos(θ0),(3)\nwhereθ′(z) =dθ(z)/dz,θ0=θ(−0)−θ(+0). Relevant\nparameters present in Eq.(3) are\nα=/radicalbigg\nw2\na32(ξ1−ξ2)+3gdV0\nJ, λB=2µBw2\nJa3α2≡B\nBc,\nwhere\nBc=Ja3α2\n2µw2.\nThe meaning of the parameter Bcis the following. When\nB=Bc, the energy barrier between the states of\nnoninteracting wires ( t′= 0) with polarizations paral-\nlel/antiparallel to the external magnetic field disappears\nand the polarizationantiparallelto the magneticfield be-\ncomes absolutely unstable, leading to a jump of the mag-\nnetic domain wall from the junction toward the magnet-\nically biased end of the wires. Below we assume that Bc\nis much less than the field required to switch the polar-\nization of the left/right hand side leads. The parameters\nαandλBcharacterize a typical domain wall width in an\ninfinite wire. For example, α−1is the domain wall width\nforB= 0 for a long wire with Lα≫1. An external mag-\nnetic field B=Bezmakes the domain wall asymmetric.\nIt compresses the domain wall width on the side where\nmagnetizationisalignedwith Btoα−1\n+,α+=α√1+λB,\nand it stretches the domain wall width on the side where\nSis antiparallel to Btoα−1\n−,α−=α√1−λB.\nTo minimize the energy in Eq. (3), we employ the\nfollowing procedure. First, we fix the boundary values\nθ(±0) and find the optimal form of θ(z) for given θ(±0).\nThen we minimize E(θ(+0),θ(−0)) versus the canting\nanglesθ(±0). The first step of such a procedure requires\nsolving the optimum equation\nθ′′(z) =α2\n2/bracketleftBig\nsin(2θ(z))+2λBsin(θ(z))/bracketrightBig\n.(4)\nThe latter shows that that θ(z >0) [π−θ(z <0)] takes\nits maximal value θ(+0) [π−θ(−0)] atz= 0, and that it\ndecreases as exp( −α+z) [exp(α−z)] for|z| ≫α−1. Also\nthe differential equation (4) has the first integral v±(θ) =\n−θ′(z),\nv±(θ) =α/radicalbig\n(1±λB)2−(cos(θ)+λB)2,(5)\nwhere the sign ±corresponds to positive/negative z.\nSubstituting v±into energy in Eq. (3) and changing the\nintegration variable from ztoθwe arrive at\nE=J/parenleftBiggθ(+0)/integraldisplay\n0v+(θ)dθ+π/integraldisplay\nθ(−0)v−(θ)dθ/parenrightBigg\n−\n−t′cos(θ0), (6)3\nwhich represents the explicit dependence of energy\nE(θ(+0),θ(−0)) on the cantingangles on eachside ofthe\njunction. Minimizingtheenergywithrespecttothesean-\ngles, we identify possible regimes for the magnetization\ntexture.\nFirst, we consider the magnetization texture in the\nabsence of an external magnetic field, B= 0. In this\ncase the texture is symmetric, θ(−z) =π−θ(z), so that\nv+(θ) =v−(θ) =αsin(θ) and\nE= 2Jα(1−cos(θ(+0)))+ t′cos(2θ(+0)).\nThe latter result indicates the existence ofa critical value\ntc=Jα/2 of the coupling constant t′, such that for t′<\ntcthe energy reaches its minimum when θ(+0) =π−\nθ(−0) = 0 and magnetization is homogeneous in each\nof two wires, whereas for t′> tcthe energy minimum\ncorresponds to the magnetization texture in the vicinity\nof the tunnel junction, with\nθ(+0) =π−θ(−0) = arccos( tc/t′).\nIn the presence of external magnetic field, the magne-\ntization texture becomes asymmetric. In this case, we\ndetermine canting angles θ(±0) numerically from the set\nof two equations,\n∂θ(±0)E=Jv±(θ(±0))−t′sin(θ(−0)−θ(+0)) = 0 .\n(7)\nThe results of numerical analysis of Eqs.(7) are gath-\nered in the parametric diagram in Fig. 2, where we indi-\ncate the six parametric intervals separated by lines B1,\nB′\n1,t1, and the axis B= 0 corresponding to different\nregimesofmagnetizationtexture. Below, wedescribe the\nevolution of the magnetization state of the junction upon\nsweeping the magnetic field from −Bcto +Bc(in com-\nparison to an inverse sweep from + Bcto−Bc). When\nthe exchange coupling constant t′and the magnetic field\nBare not strong enough (parametric interval I(I′) in\nFig.2), the energy functional has two minima. The first\nof them corresponds to θ(+0) =π−θ(−0) = 0 and the\nmagnetization is homogeneous in each of two wires. The\nsecondcorrespondsto the statewith both wirespolarized\nparallel to the external magnetic field and the domain\nwall pushed away towards the end z=−L(z=L) of\nthe wire. As a result, the polarizationsof the wires in the\ntunnelingareaareantiparallel/parallelonetoanotherde-\npending on whether we increase the magnetic field from\nB= 0 or decrease the field from B1(decrease the field\nfromB= 0 or increase the field from B′\n1). When one\ndecreases the magnetic field from B1(increases from B′\n1)\nso that the field changes its sign, the state with homo-\ngeneous polarization of both wires and the domain wall\nplaced at the end z=−L(z=L) of the wire becomes\nunstable, and the domain wall is pushed towardsthe tun-\nneling area z= 0.\nChanging t′and/orBandcrossingthe line t1resultsin\ncontinuous variation of the canting angles from θ(+0) =\nπ−θ(−0) = 0 on the left from the line t1to finite values\non the right from the line t1(parametric interval II(II′)inFig.2)andformationofthe magnetizationtexturenear\nthe tunnel junction. The values of the canting angles are\nfound from equations (7). The second state corresponds\ntothesituationwhenboththewiresarepolarizedparallel\nto the external magnetic field and the domain wall is\npushedawaytowardstheend z=−L(z=L)ofthewire.\nAt the line B1(B′\n1) the barrier between metastatic and\nground states disappears and there is just one minimum\nof the energy functional corresponding to the state with\nboth wires polarized parallel to the external magnetic\nfield and the domain wall pushed away towards the end\nz=−L(z=L) of the wire. As a result, the polarization\nin the vicinity ofthe tunnel junction is homogeneous,i.e.,\nθ(−0) =θ(+0) = 0 ( θ(−0) =θ(+0) =π). \n1II\nII′cB B\n05 . 0\n5 . 0−1\n1−5 . 0 5 . 1 2ct t′I\nI′III\nIII′X\n'X1B\n1B\n1B′1B′1t\nFIG. 2: Parametric diagram t′-B. There are four lines, B1,\nB′\n1,t1, andtheaxis B= 0separatingphases onefrom another\n(see the text for details). The lines B1andt1(B′\n1=−B1and\nt1) touch one another at the point X(X′). The boundary\nvaluesθ(±0) as a function of the external magnetic field are\ncalculated for t′= 0.5tc, 0.95tcand 1.2tc(dashed lines).\nTypical dependencies of θ(+0) on a magnetic field B\nare illustrated in Fig. 3 for t′= 0.5tc,t′= 0.95tc, and\nt′= 1.2tc, respectively. The dependencies of θ(−0) onB\ncan be obtained from Fig. 3 by the transformation\nθ(−0,B) =π−θ(+0,−B).\nThe formation of canted magnetization in the vicinity\nof a tunnel junction with high transparency would affect\nthe magnetoresistance characteristic of such a junction.\nThe latter can be modelled using the tunnel Hamilto-\nnian approach. When the domain wall width α−1is\nsufficiently larger then the elastic mean free path, the\ntunneling Hamiltonian can be written as\nH=H0+HT, H 0=/summationdisplay\nν=±/summationdisplay\nk,σǫσ(k)c†\nνkσcνkσ,\nwhereǫσ(k) =ǫ(k)−σ∆. Here H0is the Hamiltonian\nof the isolated ferromagnetic wires, HTis the tunneling\nHamiltonian, cνkσandc†\nνkσareannihilation and creation\noperators of electron propagating in the left ( ν=L) or\nright (ν=R) wire with wave vector kand spin parallel\n(σ= 1/2 or↑) or antiparallel ( σ=−1/2or↓) to the wire\npolarization S(±0) (2). In the following we will assume\nthat the Fermi momenta for the majority and minority4\n \n-1.2 -0.8 -0.4 0 0.4 0.8 1.2()0+θ\ncB Bπ\n2π\n-0.8 -0.4 0 0.4 0.8()0+θ\ncB Bπ\n2π\n-0.8 -0.4 0 0.4 0.8()0+θ\ncB Bπ\n2π()a\n()b\n()c\nFIG. 3: Canting angle θ(+0) as functions of λBfort′= 0.5tc\n(panel (a)), t′= 0.95tc(panel (b)), and t′= 1.2tc(panel (c)).\nbands are sufficiently larger than α±and therefore treat\nconduction band electrons as three dimensional.\nHT=/summationdisplay\nσ,σ′/summationdisplay\nk,k′/bracketleftbigg\ntσ′σ(k′,k)c†\nRk′σ′cLkσ+h.c./bracketrightbigg\n,\nwhere the tunneling matrix elements tσ′σ(k′,k) describe\nthe transfer of an electron with wave vector kand spin\nstateσfromtheleftwiretothestatewith k′andσ′inthe\nright wire and the quantization axes for the conduction\nband electrons are directed along the magnetization vec-\ntorsS(±0) (2) which are not necessarily collinear so that\nthe transitions between the majority/minority bands of\none wire and majority/minority bands of the other are\npossible even for the nominally antiparallel configuration\nofS(±L). We consider the model in which electrons tun-\nnelfromonewiretoanotherwithoutspinflipping. Inthis\ncasethe spin dependence of tσ′σ(k′,k) is determined by a\nsingle parameter, the angle θ0=θ(−0)−θ(+0) between\nthe vectors S(±0),\ntσσ′(k′,k) =˜t/vextendsingle/vextendsingle/vextendsingle/vextendsingle2π2/planckover2pi12vz(k)vz(k′)\nL2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1/2\n×\n×/bracketleftBig\ncos(θ0/2)δσσ′+isin(θ0/2)τx\nσσ′/bracketrightBig\n,\nwhereτxis the Pauli matrix, and vz(k) is a component\nof electron velocity v(k) =∇kǫσ(k)//planckover2pi1perpendicular to\nthe interface.\nWhen vectors S(+0) and S(−0) are antiparallel (para-\nmetric intervals IandI′in Fig.2), electrons can tunnelonly from the majority band of one wire to the minor-\nity band of the other,15so that conductance of such a\njunction is\nG↑↓=2πe2|˜t|2\n/planckover2pi1/parenleftbigg2π/planckover2pi1\nL/parenrightbigg2\nN↑vz\n↑N↓vz\n↓,\nwhereN↑andN↓are the densities of states in the major-\nity and minority bands at the Fermi level, and vz\n↑/vz\n↓are\nthe average value of |vz(k)|over the majority/minority\nFermi surface,\nvz\nσ=1\nNσ/summationdisplay\nk/vextendsingle/vextendsingle/vextendsinglevz(k)/vextendsingle/vextendsingle/vextendsingleδ/parenleftBig\nǫ−ǫ(k)/parenrightBig\n.\nWhen the ends z=±0 of the wires have parallel magne-\ntization, i.e., θ(±0) = 0 or π, the conductance is\nG↑↑=2πe2|˜t|2\n/planckover2pi1/parenleftbigg2π/planckover2pi1\nL/parenrightbigg2(N↑vz\n↑)2+(N↓vz\n↓)2\n2,\nwhich determines the magneto-resistance (MR) ratio\nδ0=G↑↑−G↑↓\nG↑↑.\n \n()a\n()b\n()c\n-1 -0.5 0 0.5 1G\n↑↑G\n↑↓G\ncB B\n-1 -0.5 0 0.5 1G\n↑↑G\n↑↓G\ncB B-1 -0.5 0 0.5 1cB BG\n↑↑G\n↑↓G\n-1 -0.5 0 0.5 1G\n↑↑G\n↑↓G\ncB B-1.5 -1 -0.5 0 0.5 1 1.5cB BG\n↑↑G\n↑↓G\n-1.5 -1 -0.5 0 0.5 1 1.5cB BG\n↑↑G\n↑↓G\nFIG. 4: Magnetoresistance for a positive (left) and negativ e\n(right)magnetic fieldsweep, showingtheeffectofcantedmag -\nnetization texture on the form of hysteresis in the magnetic\ntunnel junction, for: (a)t′= 0.5tc,(b)t′= 0.95tc, and(c)\nt′= 1.2tc.\nTheformationofacantedmagnetizationtextureinthe\nvicinity of a junction with angle θ0between magnetiza-\ntion directions of the opposite sides of the tunnel barrier5\nproduces conductance G(θ0) and the reduced observable\nMR ratio, δ,\nG(θ0) =G↑↑cos2(θ0/2)+G↑↓sin2(θ0/2),\nδ=G↑↑−G(θ0)\nG↑↑=δ0sin2(θ0/2). (8)\nThe results of calculations for the conductance in MR\ndevices with various junction transparencies (and pos-\nitive and negative magnetic field sweep in the magne-\ntoresistance) are gathered in Fig.4. The conductance\ndip at small magnetic fields indicates the regime when\na ferromagnetic domain wall in the device in Fig.1 is\npinned to the tunnel junction. The detailed structure\nof the hysteresis loop depends on whether the magneti-\nzation texture is formed near the junction or not which\nofcoursedepends onthe value ofthe ferromagneticinter-\nwire coupling. For small inter-wire coupling, in Fig.4(a),the jump between parallel and antiparallel polarizations\nin the vicinity of the tunnel junction give rise to jumps in\nthe conductance between G↑↑and flat conductance min-\nimum equal to G↑↓. With increasing the inter-wire cou-\npling towards the critical value tc(determined by the in-\nterplaybetweencrystallineanisotropyandmagnetization\nrigidityin ferromagnet),anintervalofmagneticfieldsap-\npears where the conductance (8) increases continuously,\ndue to the formation of the canted magnetization tex-\nture and its change by a magnetic field, Fig.4(b). For\ninter-wire coupling above a critical value tc, Fig.4(c), the\nminimum of the conductance exceeds G↑↓even in the\nabsence of a magnetic field, indicating the formation of\na broad texture, which also suggests a reduction of MR\nratio in a high-transparency junction.\nThe authors thank to G. Bauer and E. McCann for\nuseful discussions. This project has been funded by EU\nSTREP Dynamax and ESFCRP “Spin current”.\n1G.A. Prinz, Phys. Today 48(4), 58 (1995); Science 282,\n1660 (1998), and references therein.\n2A. Brataas, G.E.W. Bauer, P.J. Kelly, Physics Reports,\n427, 157 (2006).\n3M. Julliere, Phys. 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Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n13J. Mathon and A. Umerski, Phys. Rev. B 63, 220403\n(2001).\n14Y. Li, B.Z. Li, W.S. Zhang, and D.S. Dai, Phys. Rev. B\n57, 1079 (1998)\n15E. McCann and V.I. Fal’ko, Phys. Rev. B 66, 134424\n(2002).\n16J. Shi, K. Pettit, E. Kita, S.S.P. Parkin, R. Nakatini, and\nM.B. Salamon, Phys. Rev. B 54, 15273 (1996).\n17E.Yu. Tsymbal, D.G. Pettifor, J. Shi, and M.B. Salamon,\nPhys. Rev. B 59, 8371 (1999)."    },    {        "title": "1108.1294v1.Ferromagnetic_spin_coupling_of_2p_impurities_in_band_insulators_stabilized_by_intersite_Coulomb_interaction__Nitrogen_doped_MgO.pdf",        "content": "arXiv:1108.1294v1  [cond-mat.mtrl-sci]  5 Aug 2011Ferromagnetic spin coupling of 2 p-impurities in band insulators stabilized by intersite\nCoulomb interaction: Nitrogen-doped MgO\nI. Slipukhina,1,∗Ph. Mavropoulos,1S. Bl¨ ugel,1and M. Leˇ zai´ c1\n1Peter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, D-52425 J¨ ulich, Germ any\n(Dated: September 15, 2018)\nFor a nitrogen dimer in insulating MgO, a ferromagnetic coup ling between spin-polarized 2 p-holes\nis revealed by calculations based on the density functional theory amended by an on-site Coulomb\ninteraction and corroborated by the Hubbard model. It is sho wn that the ferromagnetic coupling is\nfacilitated by a T-shaped orbital arrangement of the 2 p-holes, which is in its turn controlled by an\nintersite Coulomb interaction due to the directionality of thep-orbitals. We thus conjecture that\nthis interaction is an important ingredient of ferromagnet ism in band insulators with 2 pdopants.\nPACS numbers: 75.50.Pp, 75.30.Hx, 71.70.-d, 71.15.Mb\nFerromagnetic (FM) insulators offer the potential for\nuse as active barrier materials in novel spintronic tun-\nneling devices. In the past years we have witnessed a\nnew vista in engineering FM insulators not by doping\nnonmagnetic insulators with transition-metal ions as is\ntraditionally accomplished in diluted magnetic semicon-\nductors [1, 2], but by doping with sp-elements. The\nnovel magnetic materials design was encouraged by un-\nexpected experimentalobservationsofroom-temperature\nferromagnetism in defective wide-gap oxide semiconduc-\ntors and insulators [3–7]. This phenomenon (referred to\nassp- ord0-magnetism)wasassociatedwith the partially\nfilledp-statesof the intrinsic defects like cation/anionva-\ncancies [3, 6, 8] or first-row (2 p) dopants [9].\nShortly after, numerous cation-deficient or N- and C-\ndoped oxides and sulfides were theoretically predicted\nto be FM at room temperature [10–14]. Among those,\nMgO is distinct as it is certainly the most important bar-\nrier material for magnetic tunnel junctions. Recently, N-\ndoped MgO films were experimentally shown to exhibit\nferromagnetic properties upon thermal annealing [15].\nIt is commonly concluded that the FM interaction be-\ntween the defects in d0-magnets is due to partially oc-\ncupied spin-polarized defect states, that are sufficiently\nextended to provide an exchange interaction via the\ndouble-exchange mechanism. Most likely, however, at\nlow concentrations the impurity 2 pelectrons experience\na strong on-site Coulomb repulsion Ubecause of their\nspatial localization [16], leading to insulating behavior\nthat changes the mechanism of exchange interaction and\ncould weaken or change the sign of magnetic coupling.\nCorrelation effects in defect-free oxides with partially\nfilled oxygen p-shells were studied in Refs. [17–19].\nThe importance of electron correlations in the impu-\nrityp-states was recently investigated using density func-\ntional theory (DFT) approaches amended by an on-site\nCoulomb repulsion [11, 20] for the example of N-doped\nMgO.Jahn-Teller-like(JT)distortions,capturedby’+ U’\n[21]orself-interactioncorrection[22]schemes,werefound\ntoevokeanenergysplittingbetweentheoccupiedandun-occupied nitrogen2 p-states, increasingthe localizationof\na spin-polarized hole at one of the 2 p-orbitals by pushing\nit deeper into the band gap and thus breaking the initial\nsymmetry of the electronic state.\nIn this Letter we present new insights into the role\nof electron correlations in N-doped MgO by carrying\nout DFT calculations where strong correlations are ac-\ncounted for by the GGA (generalized-gradient approx-\nimation [23])+ Uapproach, that are conceptualized on\nthe basis of a minimal Hubbard model. We demonstrate\nthat the symmetry breaking and the subsequent metal-\ninsulator transition occurs even without JT distortions,\nand is thus electronically driven. More importantly we\nfind for the insulating state of N-dimers, which are likely\ntobeformedintheannealingprocess,aFMspincoupling\nandaT-shaped2 p-orbitalarrangement(OA). Employing\na Hubbard model we show that a weak intersite Coulomb\nrepulsion, that is well accounted for by the DFT, com-\nbined with the on-site Coulomb interaction is responsi-\nble for this result. While in periodicstrongly correlated\n3dsystems the favored spin- and orbital arrangement is\nusually well explained by the Kugel-Khomskii model [24]\nwhich does not resort to the intersite repulsion, similar\narguments applied to N-N dimers in MgO yield an anti-\nferromagnetic(AFM) coupling, in contradiction with our\nfirst-principlesresults. The nearest-neighborFMinterac-\ntion is not sufficient to explain the long-range FM order,\nbut it is a necessary condition. Our aim is to shed some\nlight on the physics driving this interaction.\nWe utilized the full-potential linearized augmented\nplane wave code FLEUR[25] in our GGA+ Ucalculations.\nRelaxations were performed with an on-site Coulomb en-\nergyU=4.6 eV and Hund’s exchange J=1.2 eV applied\nonthe 2p-statesofOand N. The totalenergieswerecom-\npared[26] for a set of U-values that rangesbetween 3 and\n6 eV, reported for O 2 p-states in transition-metal oxides\nfrom photoemission and Auger experiments [27], and for\ntwovaluesof J, 0.6and1.2eV(fromHund’sexchangefor\natomic N and O [26], we expect the physically relevant\nJ-values in a solid to fall within this range). Supercells2\nof 64 atoms of host MgO, with a Brillouin-zone 2 ×2×2\nk-mesh are used to study a single N impurity and N-N\ndimers at O sites. N-O-N “dimers” are treated in a 144-\natomic supercell and sampled at the Γ-point. Different\nOAs are introduced by initiating a specific occupation of\nthep-orbitals of N in the GGA+ Udensity-matrix [28].\nCalculations are performed for all possible types of OA\nat FM and AFM spin alignment (12 different spin-orbital\nconfigurations in total) for N-N and N-O-N dimers.\nWe firstconsider the caseof asingle Nimpurity substi-\ntuting O in MgO to check whether the symmetry break-\ning and the metal-insulator transition upon applying a\nfiniteUis driven electronically or by local lattice distor-\ntions. Figure 1(a) shows the N density of states (DOS) of\nthe structurally unrelaxed supercell. Within the GGA,\nN introduces two triply degenerate states, an occupied\nspin-upp-state in the valence band and a partially unoc-\ncupied spin-down state at the Fermi level, EF. A single\nspin-polarized hole is evenly distributed among the three\nNp-orbitals. The situation changes drastically upon ap-\nplyingU: the minority 2 p-state splits into a doubly de-\ngenerateoccupied and a non-degenerateunoccupied level\n(Fig.1(b)). This symmetry breaking occurs even without\nlattice distortions, which clearly demonstrates that it is\nan electronically driven effect. Similar behavior was no-\nticed for the cation vacancy in ZnO [29]. This is different\nfromthepreviousreportsonMgO:N[20], wherethemain\neffect of both + Uor self-interaction corrections resulted\nin symmetry breaking via a JT distortion.\nNext, we examine the relative spin and p-orbital ar-\nrangement for two substitutional N impurities in MgO.\nRecent LSDA studies [30] have shown that N-N impurity\npairing in a structurally relaxed MgO supercell leads to a\nnon-magnetic insulating ground state with a fully occu-\npied bonding ppσ(as well as ppπ) and completely empty\nantibonding ppσ∗states. However, we show that this is\nno longer the case in the presence of strong correlations.\nWe consider the following relative positions of the two N\natoms: (i) at the nearest neighbor (n.n.) sites, at a dis-\ntancedN-N= 2.97˚A – a N-N dimer; (ii) connected via an\nO atom ( dN-N= 5.83˚A ) – a N-O-N dimer, and model,\ncorrespondingly,the1stand2ndn.n.interactionbetween\nN atoms in the oxygen sublattice of MgO. We omit the\nN-Mg-N configuration from the analysis because we find\nthe magnetic interactions to be negligibly small [31].\nWe define a coordinate system with the pxandpyor-\nbitals pointing toward neighboring N (or O) atoms in\nthexyplane (Fig.1(c)), while the pzorbital points out of\nplane, toward Mg. In the considered cases of N-N and N-\nO-N dimers, both N atoms are situated along the xaxis,\nthus the largest hopping is of ppσtype between neigh-\nboringpxorbitals, while pyandpzorbitalsareessentially\nnon-bonding. The sign and strengthofthe magneticcou-\nplingbetweenthetwospin-polarizedholesisfoundbythe\ntotal energy difference ∆ EAFM−FMbetween the AFM\nand FM state, calculated within GGA+ Ufor eachp-hole\nFigure 1: (color online) Spin- and orbital-resolved DOS for a\nsingle N in MgO calculated in GGA (a) and GGA+ U(b). (c)\nMinority p-holes spin-density distribution in xy-plane around\nthe N-N dimer, with ( xx), T-shaped ( xy), and (yy) OA. Blue,\nred, and grey spheres denote Mg, O, and N, respectively.\n(d) Spin- and orbital-resolved DOS of N-atom 1 (2) (solid\n(dashed) lines) of the N-N dimer for the depicted OAs, with\nFM spin alignment; red, blue, green line: px-,py-,pz-orbitals.\nOA. To be able to disentangle the electronic mechanism\nof OA from the JT one, for the N-N dimer we performed\ncalculations with and without structural relaxations.\nThe results are summarized in Table I. For each OA\nthe spin configuration (FM or AFM) with the lowest en-\nergy is shown. Clearly, there are substantial energy dif-\nferencesbetweendifferenttypesofOA.ComparingtheN-\nN and N-O-N dimers, the rapid decrease of ∆ EAFM−FM\nwith the N-Ndistance confirmspreviousstudies[31]. Im-\nportantly, for both dimers the FM T-shaped OA (( xy) or\n(xz)) is of lowest energy, moreover for any choice of U\nandJparameters (see Supplemental Material [26]).\nIt was suggested [11, 20] that the introduction of U\nwouldreducethe magneticcouplingin d0-magnets. How-\never, a precise physical value of the parameter Uis not\nknown. To check the persistence of the magnetic cou-\npling, we repeated the calculations for the N-N dimer\nvaryingUinthe rangebetween 0-6eV(neglectingatomic3\nTableI:TheGGA+ Utotal energies ∆ Erelativetothelowest-\nenergy state and exchange interaction ∆ EAFM−FMboth in\nmeV/supercell, calculated for different OA for N-N and N-\nO-N dimers in MgO. Positive (negative) ∆ EAFM−FMcorre-\nsponds to FM (AFM) coupling.\nN-N N-N, relaxed N-O-N\nOA∆E∆EAFM−FM∆E∆EAFM−FM∆E∆EAFM−FM\n(xx)156 −129 99−189 35 −8\n(xy)4 26 0 34 7 2\n(xz)0 24 13 31 0 2\n(yz)185 1 258 1 19 0\n(yy)192 2 280 1 28 0\n(zz)198 5 294 3.6 12 0\nrelaxations), with J/U=0.26. At U=J= 0, the system\nis half-metallic and ferromagnetic with ∆ EAFM−FM=\n265 meV. However, at U >2 eV, it becomes insulating\nand the interaction through the double-exchange mecha-\nnism is no longer possible. Thus, the magnetic coupling\nis much weaker and we expect AFM coupling due to ki-\nnetic exchange interaction. Instead, orbital arrangement\nsets in and despite the insulating state, the interaction\nbetween the spin-1/2 holes remains FM with total mag-\nnetic spin moment of 2 µB. In the following we provide a\nqualitative understanding of this unexpected finding.\nIn the first step we show that kinetic exchange alone\nfavors an AFM ground state. We consider a minimal\nmultiorbital Hubbard model that describes the system of\nstronglycorrelatedopen-shell2 p-electronsintermsofthe\nkinetic energy, t, of spin-conservinghoppingsbetween or-\nbitals of largest overlap (n.n. ppσ-orbitals, see Fig.2(a)),\nthe on-site Coulomb repulsion energy Uand the interor-\nbitalexchange(Hund’srule)coupling J(seealsoEq.(1)).\nFor a given OA, the favored magnetic interaction is\ndetermined by hopping of the electrons within the N-\nN-dimer. A substantial magnetic coupling occurs only\nfor those OAs that allow hopping between the two px-\norbitals of the N-dimer, where at least one of these or-\nbitals is half-filled: the ( xx) and the T-shaped ( xy) and\n(xz) OAs (see Table I). The relative coupling strength\ncan be qualitatively understood by considering that each\nspin-conserving hopping tinto half-filled orbitals lowers\nthe energy by t2/(U−J) (in 2nd order perturbation\ntheory) if the intermediate virtual atomic state satisfies\nHund’s rule for one of the atoms, and t2/Uotherwise.\nIn the (xx) OA, this gives EAFM=−2t2/U,EFM= 0,\ni.e.an AFM ground state (the factor 2 arises because\nthere is one half-filled orbital per atom participating in\nthe hopping). In the T-shaped OA there is only one half-\nfilled orbitalparticipating; then we have EAFM=−t2/U,\nEFM=−t2/(U−J),i.e., a FM ground state.\nAmong all possible types of OA, the T-shaped and\nAFM (xx) states are the ones that have the lowest en-\nergy, because these are the only ones that allow hopping\nbetween the N atoms. The hopping between O and Natoms provides a smaller total energy gain, because of (i)\nthe separation between the on-site energy levels of O and\nN, that differ by 0.5-1 eV, and (ii) the delocalization of\ntheOstatesthat formanitinerantband. Fromtheprevi-\nousdiscussion followsthat amongthe twoOAs, the AFM\n(xx) one should be lower in energy by 2 t2/U−t2/(U−J)\nas long as J < U/2. Thus, the arguments of the kinetic\nexchange interaction favor the AFM ( xx) state, contra-\ndicting our DFT finding of a FM T-shaped ground state.\nObviously, a qualitative description of the relative sta-\nbility of spin- and orbitally-arranged states in N-doped\nMgO requests an extension of the minimal Hubbard\nmodel beyond the conventional electron hopping tand\non-siteCoulomb repulsion U. The N-N dimer breaks\nthe cubic symmetry of the MgO lattice. Considering\nthe directional nature of the 2 p-orbitals, this leads to\nan orbital-dependent intersite Coulomb repulsion, that\nis well accounted for within the GGA/LDA. In terms of\nthe Hubbard model it is expressed by an additional effec-\ntive intersite Coulomb repulsion energy Vbetween n.n.\nppσ-orbitalsof N-N and N-O pairs. To capture this effect\nby such an extended Hubbard model we study an impu-\nrity cluster of 2 N and 6 n.n. O atoms (Fig.2(a)), with\n1 (2)p-orbital on each O (N) site that form n.n. ppσ-\norbitals. Thus, we restrict ourselves to 10 p-orbitals that\ncanbe occupied by18electronsand 2holesofspins ↑and\n↓. The Hamiltonian for such a system can be written as:\nH=/summationdisplay\nm,sǫmnms−t/summationdisplay\n<m,m′>/summationdisplay\ns[c†\nmscm′s+h.c.]\n+U/summationdisplay\nmnm↑nm↓+V/summationdisplay\n<m,m′>nmnm′ (1)\n+ (U′−J)/summationdisplay\n≪m,m′≫,snmsnm′s+U′/summationdisplay\n≪m,m′≫,snmsnm′−s\nHere<m,m′>indicates the pair of n.n. orbitals of ppσ-\noverlap, ǫmdenotes the energy level, c†\nms(cms) creates\n(annihilates) a particle of spin s=↑,↓of orbital m, while\nnms,nm=nm↑+nm↓are the spin-number and num-\nber operators, respectively. Since we have two orbitals\nat the N atoms, U′accounts for the inter-orbital on-site\nCoulomb interaction; ≪m,m′≫stands for the two on-\nsite orbital pairs (1,2) and (3,4). We neglect on-site spin-\nflip and pair-hopping terms as they are non-zero only for\nthe states with an empty and a fully occupied orbital at\nthe same site, which turned out to be of much higher\nenergy. The matrix elements of (1) are evaluated in the\n18-electron-statebasis(190 basis functions in total). The\neigenvalues of the model are obtained numerically by ex-\nactly diagonalizing (1) for a given set of parameters. We\nimpose the relations U > U′> J >0,U′=U−2J\n(for a derivation see the Supplemental Material [26]) and\nwe taket= 0.75 eV (extracted from the splitting of the\nDOS on Fig.1(d)). As the p-levels of N atoms are higher\nin energy than those of O, we take ǫm= 0.5 eV for N and\nzero for O. The pz-orbitals which are omitted from the4\nmodel are considered to be always filled and have only\nan effect of a constant shift on the on-site energies ǫm.\nFigure 2: (color online) (a) A schematic view of the clus-\nter ofp-orbitals, used in the model Hamiltonian (1). The\ndashed blue line depicts px- andpy-orbitals of the n.n. nitro-\ngen atoms (N 1and N 2), while the solid black line depicts the\ncorresponding orbitals of the O atoms. The possible hopping\nprocesses tbetween two orbitals on neighboring sites are also\nshown. (b) The ( U−2J) vs.Uphase diagram of the model\nforV= 0 and V= 0.2 eV. Orange circles denote the Uand\nJvalues, considered in our ab initio calculations [26], outlin-\ning the physically relevant area (in green). (c) The phases I ,\nII, III and IV, corresponding to different sequences of sever al\norbitally-arranged states in the order of energetic stabil ity.\nFigure 2(b) shows a ( U−2J) vs.Uphase diagram, re-\nsulting from Hamiltonian (1) for V= 0 and V= 0.2 eV.\nTheblacklinesdenotetheboundariesbetweenthephases\nI, II, III and IV, corresponding to different sequences\nof energetic stability of OAs (see Fig. 2(c)). We find\nthat the FM ( xy) state is the lowest-energy one for the\nphases II, III and IV. However, for almost all UandJ\nvalues, the AFM ( xx) state (phase I) is the ground state\nifV= 0, contradicting our ab initio results [26]. More-\nover, atV= 0 the sequence of DFT states from Table I\n(phase III) is not reproduced. In contrast, at finite Vthe\nwhole sequence of DFT states is reproduced for a wide\nrange of physically relevant values of UandJ. Hence,\nthe intersite Coulomb interaction V, which defines the\nrepulsion of electrons on n.n. p-orbitals directed towards\neachother,iscrucialforstabilizingtheFMstateindoped\ninsulators with partially occupied p-orbitals like MgO:N.\nIn summary, we have shown that the physics of spin-\npolarized holes of N-doped MgO is largely determined by\np-electron correlation effects. For a single impurity, the\nsplitting of the N 2 p-states at finite Uis mainly an elec-\ntronic effect, enhanced by the lattice distortions. More-\nover, the p-electron correlations lead to a spin and or-\nbital arrangement of the 2 p-hole states of the N-N dimer,\nresulting in a FM coupling between T-shaped orbitally-\narrangedspin-polarized p-holes. The FM state is realized\nat reasonable values of UandJ, if anintersite Coulomb\nrepulsion Vis accounted for. Considering the small valueofV≈0.2 eV, necessary to drive the phase transition\nfromanAFM toaFM groundstate, andthefact thatthe\ndirectional nature of the p-orbitals controls the strength\nof the intersite Coulomb repulsion, we conjecture that\nour finding affects all oxides where magnetism is due to\np-electrons or defects in the p-electron systems.\nWe appreciate valuable discussions with K. Rushchan-\nskii and A. Liebsch, and the support of the J¨ ulich Super-\ncomputing Centre. I.S. and M.L. gratefully acknowledge\nthe support of the Young Investigators Group Program\nof the Helmholtz Association, Contract VH-NG-409.\n∗Corresponding author: i.slipukhina@fz-juelich.de\n[1] K. Sato, L. Bergqvist, J. Kudrnovsk´ y, P.H. Ded-\nerichs, O. Eriksson, I. Turek, B. Sanyal, G. Bouzerar,\nH. Katayama-Yoshida, V.A. Dinh, T. Fukushima,\nH. Kizaki, R. Zeller, Rev. Mod. 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B 80,\n184403 (2009)."    },    {        "title": "1709.09461v1.Effects_of_geometrical_frustration_on_ferromagnetism_in_the_Hubbard_model_on_the_Shastry_Sutherland_lattice.pdf",        "content": "arXiv:1709.09461v1  [cond-mat.str-el]  27 Sep 2017Effects of geometrical frustration on\nferromagnetism in the Hubbard model on the\nShastry-Sutherland lattice\nPavol Farkaˇ sovsk´ y\nInstitute of Experimental Physics, Slovak Academy of Scien ces\nWatsonova 47, 040 01 Koˇ sice, Slovakia\nAbstract\nThe small-cluster exact-diagonalization calculations an d the projector\nquantum Monte Carlo method are used to examine the competing effects of\ngeometrical frustration and interaction on ferromagnetis m in the Hubbard\nmodel on the Shastry-Sutherland lattice. It is shown that th e geometrical\nfrustration stabilizes the ferromagnetic state at high ele ctron concentrations\n(n/greaterorsimilar7/4), where strong correlations between ferromagnetism and t he shape\nof the noninteracting density of states are observed. In par ticular, it is\nfound that ferromagnetism is stabilized only for these valu es of frustration\nparameters, which lead to the single peaked noninterating d ensity of states\nat the band edge. Once, two or more peaks appear in the noninte racting\ndensityof states at thebandegdetheferromagnetic stateis suppressed. This\nopens a new route towards the understanding of ferromagneti sm in strongly\ncorrelated systems.\n11 Introduction\nSince its introduction in 1963, the Hubbard model [1] has become, on e of the most\npopularmodelsofcorrelatedelectronsonalattice. Ithasbeenuse dintheliterature\nto study a great variety of many-body effects in metals, of which fe rromagnetism,\nmetal-insulator transitions, charge-density waves and supercon ductivity are the\nmost common examples. Of all these cooperative phenomena, the p roblem of fer-\nromagnetism in the Hubbard model has the longest history. Althoug h the model\nwas originally introduced to describe the itinerant ferromagnetism in narrow-band\nmetals like Fe,Co,Ni and others, it soon turned out that the single-band Hubbard\nmodel is not the canonical model for ferromagnetism. Indeed, th e existence of sat-\nurated ferromagnetism has been proven rigorously only for very s pecial limits. The\nfirst well-known example is the Nagaoka limit that corresponds to the infinite-U\nHubbardmodel withoneholein ahalf-filledband [2]. Another example, w here sat-\nurated ferromagnetism has been shown to exist, is the case of the one-dimensional\nHubbard model with nearest and next-nearest-neighbor hopping at low electron\ndensities [3]. Furthermore, several examples of the fully polarized g round state\nhave been found on special lattices as are the bipartite lattices with sublattices\ncontaining a different number of sites [4], the fcc-type lattices [5, 6], the lattices\nwith long-range electron hopping [7, 8, 9, 10, 11], the flat bands [12, 13, 14, 15] and\nthe nearly flat-band systems [16, 17, 18, 19]. This indicates that th e lattice struc-\nture, which dictates the shape of the density of states (DOS), pla ys an important\nrole in stabilizing the ferromagnetic state.\nMotivated by these results, in the current paper we focus our att ention on\nthe special type of lattice, the so-called Shastry-Sutherland latt ice (SSL). The\nSSL represents one of the simplest systems with geometrical frus tration, so that\nputting the electrons on this lattice one can examine simultaneously b oth, the ef-\n2fect of interaction as well as the effect of geometrical frustratio n on ground-state\nproperties of the Hubbard model. This lattice was first introduced b y Shastry and\nSutherland [20] asaninteresting example ofa frustrated quantum spin system with\nan exact ground state. It can be described as a square lattice with the nearest-\nneighbor links t1and the next-nearest neighbors links t2in every second square\n(see Fig. 1a). The SSL attracted much attention after its experim ental realization\n(a)\nt1t2(b)\nt1t2t3\nFigure 1: (a) The original SSL with the first ( t1) and second ( t2) nearest-neighbor\ncouplings. (b) The generalized SSL with the first ( t1), second ( t2) and third ( t3) nearest-\nneighbor couplings.\nin theSrCu2(BO3)2compound [21]. The observation of a fascinating sequence of\nmagnetization plateaus (at m/ms=1/2, 1/3, 1/4 and 1/8 of the saturated magne-\ntizationms) in this material [22] stimulated further theoretical and experimen tal\nstudies of the SSL. Some time later, many other Shastry-Sutherla nd magnets have\nbeen discovered [23, 24]. In particular, this concerns an entire gr oup of rare-earth\nmetal tetraborides RB4(R=La−Lu). These materials exhibit similar sequences\nof fractional magnetization plateaus as observed in the SrCu2(BO3)2compound.\nFor example, for TbB4the magnetization plateau has been found at m/ms=2/9,\n1/3, 4/9, 1/2 and 7/9 [23] and for TmB4atm/ms=1/11, 1/9, 1/7 and 1/2 [24].\nTo describe some of the above mentioned plateaus correctly, it was necessearry to\ngeneralize the Shastry-Sutherland model by including couplings bet ween the third\n3and even between the forth nearest neighbors [25]. The SSL with th e first, second\nand third nearest-neighbor links is shown in Fig. 1b and this is just the lattice that\nwill be used in our next numerical calculations.\nThus our starting Hamiltonian, corresponding to the one band Hubb ard model\non the SSL, can be written as\nH=−t1/summationdisplay\n/angbracketleftij/angbracketright1,σc+\niσcjσ−t2/summationdisplay\n/angbracketleftij/angbracketright2,σc+\niσcjσ−t3/summationdisplay\n/angbracketleftij/angbracketright3,σc+\niσcjσ+U/summationdisplay\nini↑ni↓,(1)\nwherec+\niσandciσare the creation and annihilation operators for an electron of\nspinσat siteiandniσis the corresponding number operator ( N=N↑+N↓=\n/summationtext\niσniσ). The first three terms of (1) are the kinetic energies correspon ding to\nthe quantum-mechanical hopping of electrons between the first, second and third\nnearest neighbors and the last term is the Hubbard on-site repulsio n between two\nelectrons with opposite spins. We set t1= 1 as the energy unit and thus t2(t3)\ncan be seen as a measure of the frustration strength.\nTo identify the nature of the ground state of the Hubbard model o n the SSL\nwe have used the small-cluster-exact-diagonalization (Lanczos) m ethod [26] and\nthe projector quantum Monte-Carlo method [27]. In both cases t he numerical\ncalculations proceed in the folloving steps. Firstly, the ground-sta te energy of the\nmodelEg(Sz) is calculated in all different spin sectors Sz=N↑−N↓as a function\nof model parameters t2,t3andU. Then the resulting behaviors of Eg(Sz) are used\ndirectly to identify the regions in the parametric space of the model, where the\nfully polarized state has the lowest energy.\n2 Results and discussion\nToreveal possible stabilityregionsoftheferromagneticstateinth eHubbardmodel\non the SSL, let us first examine the effects of the geometrical frus tration, repre-\nsented by nonzero values of t2andt3, on the behavior of the non-interacting DOS.\n4The previous numerical studies of the standard one and two-dimen sional Hub-\nbard model with next-nearest [3] as well as long-range [9, 10, 11] hopping showed\nthat just this quantity could be used as a good indicator for the eme rgence of\nferromagnetism in the interacting systems. Indeed, in both models the strong cor-\nrelation between ferromagnetism and the anomalies in the nonintera cting DOS are\nobserved. Inthefirstmodeltheferromagenticstateisfoundat lowelectronconcen-\ntrations and the noninteracting DOS is strongly enhanced at the low -energy band\nedge, while in the second one the ferromagnetic phase is stabilized at the high elec-\ntron concentrations and the spectral weight is enhanced at the h igh-energy band\nedge. This leads to the scenario according to which the large spectr al weight in\nthe noninteracting DOS that appears at the low (high) energy band edges allows\nfor a small kinetic-energy loss for a state with total spin S/negationslash= 0 in reference to one\nwithS= 0. At some finite value of interaction U, the Coulomb repulsion paid\nfor the low-spin states overcomes this energy loss and the high-sp in state becomes\nenergetically favored. The key point in this picture is the assumption that the\nshape of the DOS is only weakly modified as the interaction Uis switched on, at\nleast within its low (high) energy sector.\nThe noninteracting DOS of the U= 0 Hubbard model on the SSL of size\nL= 200×200, obtained by exact diagonalization of H(forU= 0) is shown in\nFig.2. Theleftpanelscorrespondtothesituationwhen t2>0andt3= 0,whilethe\nright panels correspond to the situation when both t2andt3are finite. One can see\nthat once the frustration parameter t2is nonzero, the spectral weight starts to shift\nto the upper band edge and the noninteracting DOSbecomes stron gly asymmetric.\nThus taking into account the above mentioned scenario, there is a r eal chance\nthat the interacting system could be ferromagnetic in the limit of high electron\nconcentrations. To verify this conjecture we have performed ex haustive numerical\n5-5 0 500.050.1DOSt2=0\nt3=0\n-5 0 500.050.1DOSt2=0.2\nt3=0\n-5 0 500.050.1DOSt2=0.4\nt3=0\n-5 0 500.050.1DOSt2=0.6\nt3=0\n-5 0 500.050.1DOSt2=0.8\nt3=0\n-5 0 5\nE00.050.1DOSt2=1\nt3=0-5 0 500.050.1DOSt2=1\nt3=0\n-5 0 500.050.1DOSt2=1\nt3=0.2\n-5 0 500.050.1DOSt2=1\nt3=0.4\n-5 0 500.050.1DOSt2=1\nt3=0.6\n-5 0 500.050.1DOSt2=1\nt3=0.8\n-5 0 5\nE00.050.1DOSt2=1\nt3=1\nFigure 2: Non-interacting DOS calculated numerically for different va lues oft2andt3\non the finite cluster of L= 200×200 sites.\nstudies of the model Hamiltonian (1) for a wide range of the model pa rameters\nU,t2andnatt3= 0. Typical results of our PQMC calculations obtained on finite\ncluster of L= 6×6 sites, in two different concentration limits ( n≤1 andn >1)\nare shown in Fig. 3. There is plotted the difference ∆ E=Ef−Eminbetween the\nferromagnetic state Ef, which can be calculated exactly and the lowest ground-\nstate energy from Eg(Sz) as a function of the frustration parameter t2. According\n60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6\nt200.20.40.60.811.21.41.61.82∆ E(a)\nn=1/4\nn=1/2\nn=1\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6\nt200.10.20.30.40.50.60.70.8∆ E(b)\nn=5/4\nn=3/2\nn=7/40 0.4 0.8 1.2 1.6\nt200.20.40.60.8∆ En=5/4\nn=7/4L=36\nL=64\nFigure 3: The difference ∆ E=Ef−Eminbetween the ferromagnetic state Efand\nthe lowest ground-state energy from Eg(Sz) as a function of the frustration parameter\nt2calculated for n≤1 (a) and n >1 (b) on the finite cluster of L= 6×6 sites\n(U= 1,t3= 0). The inset shows ∆ E, calculated for two different electron densities on\nclusters of L= 6×6 andL= 8×8 sites.\nto this definition the ferromagnetic state corresponds to ∆ E= 0. It is seen that for\nelectron concentrations below the half filled band case n= 1, ∆Eis the increasing\nfunction of t2, and thus there is no sign of stabilization of the ferromagnetic stat e\nforn≤1, in accordance with the above mentioned scenario. The situation lo oks\nmore promising in the opposite limit n >1. In this case, ∆ Eis considerably\nreduced with increasing t2, however, this reduction is still insufficient to reach the\nferromagnetic state ∆ E= 0 for physically reasonable values of t2(t2<1.6) that\ncorrespond to the situation in the real materials. To exclude the fin ite-size effect,\nwe have also performed the same calculations on the larger cluster o fL= 8×8\nsites, but againno signs of stabilization the ferromagnetic state ha ve been observed\n(see inset to Fig. 4b).\nFor this reason we have turned our attention to the case t2>0 andt3>0.\nThe noninteracting DOS corresponding to this case is displayed in Fig. 1 (the right\npanels). Thesepanelsclearlydemonstratethatwiththeincreasing valueofthefrus-\ntration parameter t3, still a more spectral weight is shifted to the upper band edge.\nA special situation arises at t3= 0.6, when the spectral weight is strongly peaked\n7at the upper band edge. In this case the nonintercting DOS is pract ically identical\nwith one corresponding to noninteracting electrons with long-rang e hopping [9].\nSince the long-range hopping supports ferromagnetism in the stan dard Hubbard\nmodel for electron concentrations above the half-filled band case [9, 10, 11], we\nexpect that this could be true also for the Hubbard model on the SS L, at least for\nsome values of frustration parameters t2andt3. Therefore, we have decided to per-\nform numerical studies of the model for a wide range of t3values at fixed t2,Uand\nn(t2= 1,U= 1,n= 7/4). To minimize the finite-size effects, the numerical calcu-\nlations have been done on two different finite clusters of L= 6×6 andL= 8×8\nsites. The results of our calculations for ∆ Eas a function of t3are displayed in\nFig. 4a. In accordance with the above mentioned assumptions we fin d a relatively\n0.2 0.4 0.6 0.8 1\nt300.010.020.030.040.05∆ E(a)\nU=1\nn=7/4\nt2=1L=36\nL=640 2 4\nU0.40.50.60.70.8t3 ferroL=36\nt2=1, n=7/4\n0.2 0.4 0.6 0.8 1 1.2 1.4 1.6\nt20.40.50.60.70.80.91t3(b)\nferro\nn=7/4\nU=1L=36\n1.75 1.8 1.85\nn0.150.350.550.75t3\nt2=1, U=1ferro\nPQMC\nexact\nFigure 4: (a) The difference ∆ E=Ef−Eminas a function of the frustration parameter\nt3calculated for U= 1,t2= 1,n= 7/4 and two different finite clusters of L= 6×6\nandL= 8×8 sites. The inset shows the ground-state diagram of the mode l in thet3-U\nplane. (b) The comprehensive phase diagrams of the model in t het3-t2andt3-nplane.\nwide region of t3values around t3= 0.6, where the ferromagnetic state is stable.\nIt is seen that the finite-size effects on the stability region of the fe rromagnetic\nphase are negligible and thus these results can be satisfactorily ext rapolated to\nthe thermodynamic limit L=→ ∞. Moreover, the same calculations performed\nfor different values of the Hubbard interaction Ushowed that correlation effects\n8(nonzero U) further stabilize the ferromagnetic state and lead to the emerge nce of\nmacroscopic ferromagnetic domain in the t3-Uphase diagram (see inset to Fig. 4a).\nThis confirms the crucial role of the Hubbard interaction Uin the mechanism of\nstabilization of ferromagnetism on the geometrically frustrated lat tice. In Fig. 4b\nwe have also plotted the comprehensive phase diagrams of the mode l in thet3-n\nas well as t3-t2plane, which clearly demonstrate that the ferromagnetic state is\nrobust with respect to doping ( n/greaterorsimilar7/4) and frustration.\nTo check the convergence of PQMC results we have performed the same calcu-\nlations by the Lanczos exact diagonalization method. Of course, on such a large\ncluster, consisting of L= 6×6 sites, we were able to examine (due to high memory\nrequirements) only several electron fillings near the fully occupied b and (N= 2L).\nThe exact diagonalization and PQMC results for the width of the ferr omagnetic\nphase obtainedonfinite cluster of L= 6×6sites, for threedifferent electron fillings\nfrom the high concentration limit ( N= 66,67,68), are displayed in the inset to\nFig. 4b and they show a nice convergence of PQMC results.\nLet us finally turn our attention to the question of possible connect ion between\nferromagnetism and the noninteracting DOS that has been discuss ed at the begin-\nning of the paper. Figs. 4a and 4b show, that for each finite Uandnsufficiently\nlarge (n/greaterorsimilar7/4), there exists a finite interval of t3values, around t3∼0.6, where\nthe ferromagnetic state is the ground state of the model. To exam ine a possible\nconnection between ferromagnetism and the noninteracting DOS, we have calcu-\nlated numerically the noniteracting DOS for several different values oft3from this\ninterval and its vicinity. The results obtained for U= 1,n= 7/4 andt2= 1 are\ndisplayed in Fig. 5. Comparing these results with the ones presented in Fig. 4a for\nthe stability region of the ferromagnetic phase at the same values o fU,nandt3,\none can see that there is an obvious correlation between the shape of the noninter-\n91 2 300.050.1DOSt2=1\nt3=0.5\n1 2 300.050.1DOSt2=1\nt3=0.54\n1 2 300.050.1DOSt2=1\nt3=0.58\n1 2 300.050.1DOSt2=1\nt3=0.6\n1 2 300.050.1DOSt2=1\nt3=0.62\n1 2 3\nE00.050.1DOSt2=1\nt3=0.64\n1 2 3\nE00.050.1DOSt2=1\nt3=0.68\n1 2 3\nE00.050.1DOSt2=1\nt3=0.72\nFigure 5: Non-interacting DOS calculated numericallly for t2= 1 and different values\noft3(neart3= 0.6) on the finite cluster of L= 200×200 sites.\nacting DOS and ferromagnetism. Indeed, the ferromagnetic stat e is stabilized only\nfor these values of frustration parameters t2,t3, which lead to the single peaked\nnoninterating DOS at the band edge. Once, two or more peaks appe ar in the\nnoninteracting DOS at the band egde (by changing t2ort3), ferromagnetism is\nsuppressed.\nInsummary, thesmall-clusterexact-diagonalizationcalculationsan dthePQMC\n10method were used to examine possible mechanisms leading to the stab ilization of\nferromagnetism in strongly correlated systems with geometrical f rustration. Mod-\nelling such systems by the Hubbard model on the SSL, we have found that the\ncombined effects of geometrical frustration and interaction stro ngly support the\nformation of the ferromagnetic phase at high electron densities. T he effects of\ngeometrical frustration transform to the mechanism of stabilizat ion of ferromag-\nnetism via the behaviour of the noninteracting DOS, the shape of wh ich is deter-\nmined uniquely by the values of frustration parameters t2andt3. We have found\nthat it is just the shape of the noninteracting DOS near the band ed ge (the single\npeaked DOS) that plays the central role in the stabilization of the fe rromagnetic\nstate. Since the same signs have been observed also in some other w orks (e.g., the\nHubbard model with nearest and next-nearest neighbor hopping, or the Hubbard\nmodel with long range hopping), it seems that such a behaviour of th e noninter-\nacting DOS near the band edge should be used like the universal indica tor for the\nemergence of ferromagnetism in the interacting systems.\nThis work was supported by the Slovak Research and Development A gency\n(APVV) under Grant APVV-0097-12 and ERDF EU Grant under the c ontract\nNo. ITMS26210120002 and ITMS26220120005.\n11References\n[1] J. Hubbard, Proc. R. Soc. London A 276, 238 (1963).\n[2] Y. Nagaoka, Phys. Rev. 147, 392 (1966).\n[3] E. M¨ uller-Hartmann, J. Low. Temp. Phys. 99, 342 (1995).\n[4] E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989).\n[5] M. Ulmke, Eur. Phys. J. B 1, 301 (1998).\n[6] S. Pandey and A. Singh, Phys. Rev. B 75, 064412 (2007).\n[7] P. Pieri, Mod. Phys. Lett. B 10, 1277 (1996).\n[8] M. Salerno, Z. Phys. 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Takigawa, M. Horvatic, C. Berthier, H. Kageyam a, Y. Ueda,\nS. Miyahara, F. Becca, and F. Mila, Science 298, 395 (2002).\n[23] S. Yoshii, T. Yamamoto, M. Hagiwara, S. Michimura, A. Shigekawa , F. Iga,\nT. Takabatake, and K. Kindo, Phys. Rev. Lett. 101, 087202 (2008).\n[24] K. Siemensmeyer, E. Wulf, H. J. Mikeska, K. Flachbart, S. Gaba ni, S. Matas,\nP. Priputen, A. Efdokimova, and N. Shitsevalova, Phys. Rev. Lett .101,\n177201 (2008).\n[25] K.ˇCenˇ carikov´ a and P. Farkaˇ sovsk´ y, Physica Status Solidi b 252 , 333 (2015).\n[26] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).\n[27] M. Imada and Y. Hatsugai, J. Phys. Soc. Jpn. 58, 3752 (1989) .\n13"    },    {        "title": "2112.14515v1.A_unified_theory_of_ferromagnetic_quantum_phase_transitions_in_heavy_fermion_metals.pdf",        "content": "arXiv:2112.14515v1  [cond-mat.str-el]  29 Dec 2021A unified theory of ferromagnetic quantum phase transitions in heavy fermion metals\nJiangfan Wang1and Yi-feng Yang1,2,3,∗\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Science, Beijing 1 00190, China\n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n3Songshan Lake Materials Laboratory, Dongguan, Guangdong 5 23808, China\n(Dated: December 30, 2021)\nMotivated by the recent discovery of a continuous ferromagn etic quantum phase transition in\nCeRh 6Ge4and its distinction from other U-based heavy fermion metals such as UGe 2, we develop a\nunified explanation of their different ground state properti es based on an anisotropic ferromagnetic\nKondo-Heisenberg model. We employ an improved large- NSchwinger boson approach and predict\na full phase diagram containing both a continuous ferromagn etic quantum phase transition for large\nmagnetic anisotropy and first-order transitions for relati vely small anisotropy. Our calculations re-\nveal three different ferromagnetic phases including a half- metallic spin selective Kondo insulator\nwith a constant magnetization. The Fermi surface topologie s are found to change abruptly between\ndifferent phases, consistent with that observed in UGe 2. At finite temperatures, we predict the devel-\nopment of Kondo hybridization well above the ferromagnetic long-range order and its relocalization\nnear the phase transition, in good agreement with band measu rements in CeRh 6Ge4. Our results\nhighlight the importance of magnetic anisotropy and provid e a unified theory for understanding the\nferromagnetic quantum phase transitions in heavy fermion m etals.\nKeywords: Ferromagnetic Kondo lattice; Magnetic anisotropy; Quantu m phase transitions;\nKondo hybridization\nINTRODUCTION\nHeavy fermion or Kondo lattice materials are proto-\ntype strongly correlated electron systems that exhibit\nrich ground states due to competing interactions [1, 2].\nAn indirect Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction is typically induced and oscillates with dis-\ntance between neighboring Kondo impurities. Its re-\nsulting ferromagnetic (FM) or antiferromagnetic (AFM)\nlong-range orders can often be suppressed by pressure,\nmagnetic field, or chemical substitution and tuned to\na heavy Fermi liquid (HFL) with an enhanced effec-\ntive mass of Landau quasiparticles due to the Kondo\nscreeningby conduction electrons. However,unlike AFM\nKondo lattices, whose magnetic quantum phase transi-\ntions (QPTs) and associated quantum criticality have\nbeen intensively studied in the past decades [3, 4], the\nFM Kondo lattices are relatively less explored [5–9]. It\nhas been believed that FM QPT in clean metallic sys-\ntems should typically be of first order if not interrupted\nby other instabilities [10] because of non-analyticities in\nthe free energy induced by certain soft modes related\nto the Fermi surfaces [11, 12]. Evidences may be found\nfrom experiments on many U-based intermetallics such\nas UGe 2[13].\nThe situation, however, has been changed lately when\na pressure-tuned FM quantum critical point (QCP)\nand the associated strange metallic behaviors were ob-\nserved in the stoichiometric Kondo lattice compound,\nCeRh6Ge4[14]. Similar observations had been re-\nported previously in the non-stoichiometric compound\nYbNi4(P1−xAsx)2[15], but might be naively attributed\nto disorder effects. These observations violate theprevailing wisdom and have stimulated rapid theoreti-\ncal progress and debates on the underlying mechanism\n[14, 16]. Analogous to the AFM quantum criticality, a\nKondobreakdownscenariohasalsobeenpredictedacross\nthe FM QCP [14]. But this picture cannot explain the\nlatest observations of angle-resolved photoemission spec-\ntroscopy(ARPES)[17]andultrafastopticalspectroscopy\n[18], where strong hybridization effects have been de-\ntected at finite temperatures above the FM transition.\nCompared to CeRh 6Ge4, the intensively studied U-\nbased compound, UGe 2, exhibits very different ground\nstate properties under pressure [13]. At low pressures, it\nhastwodifferent FM phaseswith strong(FM2) andweak\n(FM1) spin polarizations, respectively, and enters a HFL\nupon further increasing pressure. Both FM2-FM1 and\nFM1-HFL transitions are of first order [19], with drastic\nchanges of Fermi surfaces reported by quantum oscilla-\ntion experiments [20, 21]. A superconducting dome oc-\ncursinside the FM phaseand the maximalsuperconduct-\ning transition temperature occurs right at the FM2-FM1\ntransition point [13].\nGiven these distinct properties, one may doubt if there\nshould exist a single mechanism for both compounds\nwithout resorting to microscopic details. In this work,\nwe propose such a unified theory and show that their\nvery different properties can in fact be explained solely\nbased on lattice anisotropy. The Ce atoms of CeRh 6Ge4\nexhibits a quasi-one-dimensional structure [14], while\nthe nearest neighbor U-U separations in UGe 2are more\nisotropic [22]. Such structural anisotropy on atomic dis-\ntances may have a sensitive influence and cause a mag-\nnetic anisotropy in the RKKY interaction, as reflected in\nan anisotropic Kondo-Heisenberg model. We will solve2\nthe model using an improved Schwinger boson large- N\napproachtakingintoconsiderationthespatialcorrelation\neffect. The method has lately been applied to the AFM\nKondo lattice [23], yielding a global phase diagram and\na unified explanation of the AFM QCP in YbRh 2Si2and\nthe quantum critical non-Fermi liquid phase (a metallic\nspin liquid) recently observed in the frustrated Kondo\nlattice CePdAl [24]. Here we further develop this ap-\nproach for the FM phase, predict a full phase diagram\nof the anisotropic FM Kondo lattice, and give a unified\nexplanation of the rather complicated and qualitatively\ndifferent behaviors of CeRh 6Ge4and UGe 2. Our meth-\nods and results are reported in detail in the following\nsections.\nMETHODS\nWe start with the Hamiltonian:\nH=/summationdisplay\n/angbracketleftij/angbracketrighttijc†\niaαcjaα+JK/summationdisplay\niSi·si−/summationdisplay\ni,δJδ\nHSi·Si+δ,(1)\nwherec†\niaαcreates a conduction electron of spin αand\nchannel (orbital) a= 1,2,···,Kon siteiof a square\nlattice (summations over repeated indices are implied),\nsi=1\n2/summationtext\naαβc†\niaασαβciaβis its spin operator, and Si\ndenotes the local spin formed by either Ce 4 f-electrons\nor U 5f-electrons beyond the Hill limit [25]. δ=x,y\ndenotes the unit vectors along the two spatial direc-\ntions. In principle, the Kondo couping will induce an\nanisotropic RKKY interaction if the electron hopping tij\nis anisotropic. Here, we apply directly an anisotropy on\nthe FM Heisenberg interaction ( Jx\nH/negationslash=Jy\nH>0) while\nkeeping the electron hopping isotropic to emphasize the\neffect of magnetic anisotropy alone.\nTo deal with FM and Kondo effect on the same ba-\nsis, we use the Schwinger boson representation of local\nspins,Si=1\n2/summationtext\nαβb†\niασαβbiβ, together with the con-\nstraintnb(i)≡/summationtext\nαb†\niαbiα= 2S, and perform large- N\ncalculationsby extending the spin groupSU(2) to SU( N)\nwithα= 1,···,Nand a fixed ratio κ≡2S/N=\nK/Nfrom the perfect Kondo screening condition [26].\nThe constraint is implemented by a Lagrange multiplier,\nλ/summationtext\ni(nb(i)−2S). The Kondo and Heisenberg terms can\nbe decomposed by two auxiliary fields:\nJKSi·si→1√\nNb†\niαciaαχia+H.c.+|χia|2\nJK,\n−Jδ\nHSi·Si+δ→∆δb†\ni+δ,αbiα+H.c.+N|∆δ|2\nJδ\nH,(2)\nwhereχ†\niacreates a spinless fermion with positive charge\ncalled holon, and ∆ δ=−Jδ\nH/summationtext\nα/angbracketleftBig\nb†\niαbi+δ,α/angbracketrightBig\n/Ndenotes\nthe hopping amplitude of bosonic spinons along the δ-\ndirection. We will use the mean-field ratio ∆ x/∆yto\nmeasure the degree of magnetic anisotropy.In the large- Nlimit, one only needs to consider the\none-loop self-energies of spinons and holons:\nΣb(p,iνn) =−κ\nβV/summationdisplay\nkmgc(p−k,iνn−iωm)Gχ(k,iωm),\nΣχ(p,iωm) =1\nβV/summationdisplay\nkngc(k−p,iνn−iωm)Gb(k,iνn),(3)\nwheregc(p,iωm) = (iωm−εc\np)−1isthebareGreen’sfunc-\ntion of conduction electrons with dispersion εc\np,Gband\nGχare the self-consistent Green’s functions of spinons\nand holons, ωm(νn) are the fermionic (bosonic) Matsub-\nara frequencies, βis the inverse temperature, and Vis\nthe total number of lattice sites.\nThe Green’s functions satisfy:\nGb(p,iνn) =1\niνn−εbp−Σb(p,iνn),\nGχ(p,iωn) =1\n−J−1\nK−Σχ(p,iωn)−b2\niωn+εc\n−p,\nGα\nc(p,iωn) =1\niωn−εcp+b2δα,1\n−J−1\nK−Σχ(−p,−iωn),(4)\nwhereεb\np≡λ−2(∆xcospx+∆ycospy) is the barespinon\ndispersion. The FM long range order is accounted for by\nthe condensation of bosonic spinons on one of its spin\ncomponent (chosen to be α= 1 here), /angbracketleftbi,1/angbracketright=√\nNb/negationslash= 0,\nso that/angbracketleftSz/angbracketright ≡N−1/summationtext\nαsgn(1+N\n2−α)/angbracketleftb†\niαbiα/angbracketright=b2[27].\nIn this formalism, the Kondo effect at large JKis\nsignaled by the negative effective Kondo coupling (or\nholon energy level), J∗\nK(k)−1≡J−1\nK+ ReΣ χ(k,0)<0,\nwhich energetically favors formation of Kondo bound\nstates. It then leaves an energy gap around the Fermi\nenergy for both holons and spinons, producing a large\nelectron Fermi surface due to the generalized Luttinger\nsum rule [23, 28]. As one increases Jδ\nH, the spinon\n(holon) gap shrinks and finally vanishes at some point\nwhere the bosonic spinons start to condense due to the\ndevelopment of long-range FM order. Inside the FM\nphase, the condensation (magnetization) provides an ef-\nfective hybridization field between the conduction elec-\ntrons (cia,α=1) and the holons, as may be directly seen\nfrom the decomposed Kondo term in Eq. (2).\nIn practice, Eqs. (3)(4) are solved self-consistently\nunder two constraints corresponding to minimizing the\nfree energy with respect to λandb. Previous studies\nof the Kondo lattice models using Schwinger boson ap-\nproach have only focused on the paramagnetic phase and\nadopted a local (or momentum-independent) approxima-\ntion for the self-energies [5, 29, 30]. Here we use the\nfast Fourier transform of the self-consistent equations\nand solve the nonlocal (or momentum-dependent) self-\nenergies efficiently in the coordinate space [23]. More\ndetails can be found in Supplementary Materials.3\n/s32/s33/s34/s35/s32/s33/s34/s36/s32/s33/s34/s32/s32/s33/s32/s37/s32/s33/s32/s38/s32/s33/s32/s35/s32/s33/s32/s36/s32/s33/s32/s32/s39/s40/s41/s42/s40/s43\n/s32/s33/s34/s32 /s32/s33/s32/s37 /s32/s33/s32/s38 /s32/s33/s32/s35 /s32/s33/s32/s36\n/s39/s44/s45/s42/s46/s47/s43FM2 FM1FM0\nHFL\ncoexisting \n   region /s32/s33/s34/s34/s33/s35/s34/s33/s36/s34/s33/s37/s34/s33/s38/s34/s33/s34/s39/s40/s41/s42/s40/s43\n/s34/s33/s38/s34/s34/s33/s32/s44/s34/s33/s32/s34/s34/s33/s34/s44/s34/s33/s34/s34\n/s39/s45/s46/s42/s47/s48/s43(a)\n(b) (c)CeRh 6Ge 4\nUGe 2\n/s32\n/s33\n/s34\n/s35/s36/s37/s38/s39/s40\n/s33/s41/s35/s34/s41/s42/s34/s41/s35/s35/s41/s42/s35/s41/s35\n/s43/s37/s38/s44/s45/s46/s40/s32/s33\n/s34/s33\n/s35/s33\n/s33/s36/s37/s38/s39/s40\n/s35/s41/s33/s42/s41/s43/s42/s41/s33/s33/s41/s43/s33/s41/s33\n/s44/s37/s38/s45/s46/s47/s40FM1 \nFM2crossover \nFM Strange Metal\nHFL HFLCeRh6Ge 4UGe 2\nFIG. 1: Phase diagram of the anisotropic ferromag-\nnetic Kondo lattice. (a) Zero temperature phase diagram\nof the model described by Eq. (1). The heavy Fermi liq-\nuid (HFL) phase at large TK/Jy\nHends at the red solid line,\nwhile the ferromagnetic (FM) phases at small TK/Jy\nHend\nat the blue solid line. Between the red and blue lines (light\ngreen area), HFL and FM solutions coexist. The three differ-\nent FM phases, FM0, FM1 and FM2, are separated by two\ncontinuous transition lines (the black and grey dashed line s).\nThe two orange arrows correspond to increasing pressure of\nCeRh 6Ge4and UGe 2. The inset shows the phase diagram\nwithin the full range of anisotropy. (b)(c) The experimen-\ntal temperature-pressure phase diagram of CeRh 6Ge4and\nUGe2, respectively. Black lines are continuous transitions,\nshort dashed lines are first-order transitions, and long das hed\nlines are crossover lines. The small yellow region of (c) de-\nnotes the superconducting phase. The experimental phase\ndiagrams are reproduced based on Refs. [14] and [13].\nRESULTS\nWe first summarize the predicted zero temperature\nphase diagram in Fig. 1a in terms of the anisotropy ra-\ntio ∆x/∆yand the Doniach ratio TK/Jy\nH, whereTK=\nDe−2D/JKis the characteristic Kondo energy scale and\nDis the electron half bandwidth. The HFL phase exists\non the right hand side of the red line (larger TK/Jy\nH),\nand the FM phases appear on the left hand side of the\nblue line (smaller TK/Jy\nH). The two lines merge together\nfor large anisotropy (small ∆ x/∆y), indicating a contin-\nuous quantum phase transition and a single FM QCP.While for smaller anisotropy (larger ∆ x/∆y), there is\na coexisting region (light green area) between the two\nlines, indicating a first-order transition between FM and\nHFL. It should be mentioned that the Schwinger boson\nlarge-Ntheory suffers from the pathology of predicting\nartificial first-order transitions [31], which can be alle-\nviated via introducing a biquadratic interaction term,\n−ζ/summationtext\ni,δJδ\nH(Si·Si+δ)2[5, 14, 23]. In our case, such a\nterm with a reasonable choice of ζonly slightly shifts the\ntricritical point from ∆ x/∆y≈0.02 to 0.03 (see Sup-\nplementary Materials) and has no effect on the overall\npicture and qualitative properties.\nThe FM phase is further divided by another two con-\ntinuous transition lines into three different regions: FM0,\nFM1, and FM2. The FM1-FM0 transition line merges\nwith the FM phase boundary (the blue line) at large\n∆x/∆y, so the FM0 phase only exists at small ∆ x/∆y\n(large anisotropy) and has the smallest magnetization\namong the three FM phases. It can be suppressed con-\ntinuously by increasing TK/Jy\nHand becomes a HFL.\nAs shown in Fig. 1b, FM0 may correspond to the\nobserved FM phase of CeRh 6Ge4at ambient pressure,\nwhich also exhibits a severely reduced magnetic moment\n(0.28µB/Ce[14]ascomparedtotheestimated1.28 µB/Ce\nfrom crystalline electric field analysis [32]) and is turned\ncontinuously into a HFL through a FM QCP by apply-\ningpressure[14]. TheleftmostFM2phasehasthe largest\nmagnetization, whichdecreasesgraduallywith increasing\nTK/Jy\nH, while the intermediate FM1 phase is featured\nwith a constant magnetization insensitive to the value\nofTK/Jy\nH. Both FM2-FM1 and FM1-FM0 transitions\nare accompanied by a change of the Fermi surface topol-\nogy. For relatively larger ∆ x/∆y, the FM2-FM1 and\nFM1-HFL transitions (with the light green area collaps-\ning into a first-order transition line in reality) resemble\nthe experimental observations in UGe 2under pressure as\nreproduced in Fig. 1c, in particular the observed Fermi\nsurface changes at the two transitions [20, 21] and the\nmagnetization plateau in between [19]. For very large\n∆x/∆y(inset), where the HFL covers the whole FM1 re-\ngion and penetrates into the FM2 phase, one might find\na first-order transition directly between FM2 and HFL.\nThe FM phase transition in URhAl under pressure may\ncorrespond to this type [33, 34].\nTo see more details on these different FM phases, we\nplotinFig. 2variousquantitiesasfunctionsof TK/Jy\nHfor\ndifferent anisotropy ratios. The holon phase shift is de-\nfined as δχ≡ −1\nV/summationtext\npImln[−Gχ(p,0)−1]. The quantity\nδχ/πmeasuresthe fraction oflocal spins being effectively\nKondo screened, which reaches unity in the HFL phase,\nindicating full Kondo screening. The spinon gap ∆ sis\ndetermined by the renormalized spinon energy at the Γ\npoint, satisfying∆ s−εb\nΓ−ReΣb(Γ,∆s) = 0. In the quasi-\none dimensional case ∆ x/∆y→0,δχ/πdecreases con-\ntinuously from unity at the FM0-HFL transition point,\nmarked by the blue dashed line in Fig. 2a. Right at this4\n/s32/s33/s34\n/s32/s33/s35\n/s32/s33/s36\n/s32/s33/s32/s37/s38/s39/s40/s41/s42/s43/s44/s38/s42/s43/s45/s40\n/s32/s33/s32/s46/s32/s33/s32/s47/s32/s33/s32/s48/s32/s33/s32/s49/s32/s33/s32/s50\n/s51/s52/s53/s54/s55/s56/s32/s32/s33/s34\n/s32/s33/s35\n/s32/s33/s36\n/s32/s33/s32/s37/s38/s39/s40/s41/s42/s43/s44/s38/s42/s43/s45/s40\n/s32/s33/s32/s46/s32/s33/s32/s47/s32/s33/s32/s48/s32/s33/s32/s49\n/s50/s51/s52/s53/s54/s55/s32<S z+s z>\n(n b,1 +Knc,1)/N <S z+s z>\n(n b,1 +Knc,1 )/N Δx/Δ y=0.03 Δx/Δ y=0.1 /s32/s33/s34\n/s34/s33/s35\n/s34/s33/s36\n/s34/s33/s37\n/s34/s33/s38\n/s34/s33/s34/s39/s40/s41/s42\n/s34/s33/s34/s35 /s34/s33/s34/s43 /s34/s33/s34/s36\n/s44/s45/s41/s46/s47/s48/s32/s34\n/s35\n/s36\n/s37\n/s38\n/s34/s49/s50/s51/s50/s52/s32/s34/s53/s50/s32/s33/s34\n/s34/s33/s35\n/s34/s33/s36\n/s34/s33/s37\n/s34/s33/s38\n/s34/s33/s34/s39/s40/s41/s42\n/s34/s33/s34/s43/s34/s33/s34/s35/s34/s33/s34/s44/s34/s33/s34/s36/s34/s33/s34/s45\n/s46/s47/s41/s48/s49/s50/s32/s34\n/s35\n/s36\n/s37\n/s38\n/s34/s51/s52/s53/s52/s54/s32/s34/s55/s52Δx/Δ y=0.1 Δx/Δ y=0 (a) (b)\n(c) (d)\nFM2 FM1 FM0 FM2 FM1 \nFIG. 2: Holon phase shift and magnetization. (a) The\nholon phase shift ( δχ) and the spinon gap (∆ s) as functions\nofTK/Jy\nHfor ∆ x/∆y= 0. (b) Same plot as (a), but for\n∆x/∆y= 0.1. Multiple solutions exist between the red and\nblue dashed lines. (c) Evolution of the total magnetization\n/angbracketleftSz+sz/angbracketrightand the quantity ( nb,1+Knc,1)/Nwith respect\ntoTK/Jy\nHinside the FM phases. The parameters used are\n∆x/∆y= 0.03,κ= 2S/N = 0.2 andnc/N= 0.5. (d) Same\nplot as (c), but for ∆ x/∆y= 0.1.\npoint, ∆ svanishes and the spin susceptibility diverges\n(see Supplementary Materials), indicating formation of\ntheFMlongrangeorder. WehavethusacontinuousFM-\nHFL transition and a FM QCP as observed in CeRh 6Ge4\n[14]. The situation for ∆ x/∆y= 0.1 is significantly dif-\nferent. Within a certain range of TK/Jy\nHbetween the\nred and blue dashed lines of Fig. 2b, both the HFL and\nFM solutions can exist at zero temperature, so the true\nground state should correspond to a first-order transi-\ntion with a jump of δχ/πfrom 1 to 0 .5 somewhere inside\nthe coexisting region. The constant δχ= 0.5πis asso-\nciated with the half-metal nature of the FM1 phase, as\nwill be explained later. Further reducing TK/Jy\nHdrives\nthe holon phase shift away from the 0 .5πplateau and the\nsystem enters the FM2 phase for small TK/Jy\nH.\nThe distinctions between the three FM phases can\nbe further elucidated by comparing their total magne-\ntization Mz=/angbracketleftSz+sz/angbracketright, shown in Fig. 2c and 2d for\n∆x/∆y= 0.03 and 0 .1, respectively. We see that Mz\ndecreases with increasing TK/Jy\nHin the FM2 phase, but\nkeeps constant in the FM1 phase. In contrast to the wide\ncoexisting region at ∆ x/∆y= 0.1, there exists a stable\nFM0 phase next to the FM1 phase at ∆ x/∆y= 0.03,\nwhereMzfurther decreases with increasing TK/Jy\nH. We\nfind that the constant magnetization of FM1 satisfies\nMz=κ(1−nc/N), where nc=/summationtext\nαnc,αis the electron\nconcentration per channel (see Fig. S3 of SupplementaryMaterials for different κ). As shown in Figs. 2c and 2d,\nour numerical calculations also find another relation of\nthe FM1 phase, nb,1+Knc,1= 2S, which is analytically\nrelated to the magnetization plateau (see Supplementary\nMaterialsSection IV). The physical meaning ofthis iden-\ntity can be understood more directly in the case of N= 2\nandK= 2S= 1, where it becomes nb,↑+nc,↑= 1. This\nleads to the commensurability nc,↑=nb,↓due to the\nconstraint nb,↑+nb,↓= 2S= 1. Since c↑andb↓are\nboth minority components in the FM phase, they can\nfully participate in the Kondo spin-flip scattering and\nthe commensurability implies a full screening of this spin\ncomponent. Meanwhile, the remaining majority compo-\nnents (c↓,b↑) are not involved in the Kondo scattering\nand contribute to the plateau in the total magnetization.\nThese different screening properties have important\nconsequences on their respective band structures, which\ncan be best seen in Fig. 3 showing the evolution of the\nelectron Femi surface and dispersion relation inside the\nFM phases. Due to the hybridization between conduc-\ntion electrons and holons, there exist two quasiparticle\nbands for the minority component ( α= 1), which we de-\nnote asE±(p). Inside the FM2 phase (see Fig. 3a), the\nchemicalpotential intersectswith the E+(p) band, sothe\nFermi surface contains two electron-like pockets. Inside\nthe FM0 phase (see Fig. 3c), the chemical potential in-\ntersectswith the E−(p) band, givingrise to four hole-like\npockets in the Fermi surface. Right in between, the FM1\nphase is featured with an indirect energy gap separating\ntheE+(p) andE−(p) bands, and the chemical potential\nlies just within the gap. We have thus a spin selective\nKondo insulator [7–9, 35], in which the holon phase shift\ncan be calculated from the quasiparticle dispersions via\nδχ/π=1\nV/summationtext\np[1+θ(−εc\np)−θ(−E+(p))−θ(−E−(p))] =\n1\nV/summationtext\npθ(−εc\np), giving the constant δχ/π= 0.5 in Fig. 2b.\nQualitatively, the above evolution of E±(p) can be ex-\nplained by the dispersion relation given by the hybridiza-\ntion between conduction electrons and holons near the\nFermi energy:\nE±(p)≈1\n2/parenleftBig\nεc\np−εχ\n−p±/radicalBig\n(εcp+εχ\n−p)2+4Zχ\n−pb2/parenrightBig\n,(5)\nwhereεχ\np=Zχ\np/J∗\nK(p) is the holon disper-\nsion without spinon condensation, and Zχ\np=\n[−∂ωReΣχ(p,ω)|ω=0]−1>0 is its quasiparticle residue.\nAsTK/Jy\nHincreases, εχ\npgradually evolves from being\nfully above the Fermi energy, to being entirely below\nthe Fermi energy in the HFL phase. This is a general\nfeature of Kondo systems that accounts for the stability\nof forming Kondo bound states [23, 36]. As a result,\n−εχ\n−p, and hence E±(p), evolves in the opposite way,\ncausing the evolution of the Fermi surfaces from electron\nto hole pockets. In addition, because εχ\npbecomes\nincreasingly flat approaching the HFL phase boundary,\nthe quasiparticles become more and more heavy, giving\nrise to the sharp quasiparticle peak at the Fermi energy5\n/s32/s33/s34/s33 /s35/s36/s37\n/s38 /s39 /s38/s40 /s41 /s39\n/s32/s33/s34/s33 /s35/s36/s37\n/s38 /s39 /s38/s40 /s41 /s39\n/s32/s33/s34/s33 /s35/s36/s37\n/s38 /s39 /s38/s40 /s41 /s39\n/s32/s33/s34/s33/s42/s43/s36/s44\n/s32/s33 /s34 /s33\n/s42/s45/s36/s44\n/s32/s33/s34/s33/s42/s43/s36/s44\n/s32/s33 /s34 /s33\n/s42/s45/s36/s44\n/s32/s33/s34/s33/s42/s43/s36/s44\n/s32/s33 /s34 /s33\n/s42/s45/s36/s44/s33/s46/s34 /s34/s46/s47 /s34/s46/s34\n/s48/s49\n/s33/s46/s34 /s34/s46/s47 /s34/s46/s34\n/s48/s49\n/s33/s46/s34 /s34/s46/s47 /s34/s46/s34\n/s48/s49\n/s32/s33/s34/s33/s42/s43/s36/s44\n/s32/s33 /s34 /s33\n/s42/s45/s36/s44\n/s32/s33/s34/s33/s42/s43/s36/s44\n/s32/s33 /s34 /s33\n/s42/s45/s36/s44\n/s32/s33/s34/s33/s42/s43/s36/s44\n/s32/s33 /s34 /s33\n/s42/s45/s36/s44/s33/s46/s34 /s34/s46/s47 /s34/s46/s34\n/s48/s49\n/s33/s46/s34 /s34/s46/s47 /s34/s46/s34\n/s48/s49\n/s33/s46/s34 /s34/s46/s47 /s34/s46/s34/s48/s49\n/s32/s34/s46/s34/s50/s34/s46/s34/s34/s34/s46/s34/s50 /s35/s36/s37\n/s38 /s39 /s38/s40 /s41 /s39\n/s32/s34/s46/s34/s50/s34/s46/s34/s34/s34/s46/s34/s50 /s35/s36/s37\n/s38 /s39 /s38/s40 /s41 /s39\n/s32/s34/s46/s34/s50/s34/s46/s34/s34/s34/s46/s34/s50 /s35/s36/s37\n/s38 /s39 /s38/s40 /s41 /s39ΓXX' M\nE+\nE-(a) \n(b) \n(c)(d) \n(e) \n(f)\nFIG. 3: Fermi surface evolution with TK/Jy\nH.(a)-(c) The conduction electron Fermi surface, dispersion , and density of\nstates for the α= 1 component, at ∆ x/∆y= 0.03 and different TK/Jy\nH. Two quasiparticle bands can be observed, denoted as\nE+andE−. The dashed curve in (b) is a guide to the eye. (d)-(f) Same plo ts as (a)-(c), but for the α= 2,···,Ncomponents.\nA finiteN= 4 is taken to include the conduction electron self-energy i n (d)-(f). The values of TK/Jy\nHare: (a)(d) 0.044 (FM2\nphase); (b)(e) 0.061 (FM1 phase); (c)(f) 0.071 (FM0 phase).\nin the density of states shown in Fig. 3c.\nFor comparison, we also show the Green’s function\nof conduction electrons for other spin components α=\n2,···,N, whose self-energies are the order of O(1/N).\nAs an example, we take N= 4 and show its evolution\nin Figs. 3d-3f with the same values of TK/Jy\nHtaken in\nFigs. 3a-3c. Unlike that of the α= 1 component, the\nFermi surfaces for α= 2,···,Ndo not disappear in the\nFM1phase, revealingthe half-metallicnatureofthe FM1\nphase. On the other hand, we see that their Fermi sur-\nface also expands slightly with TK/Jy\nH, which may be\nattributed to the band bending caused by the precursor\neffect of Kondo resonance in the self-energy. Both Figs.\n3c and 3f suggest that the Kondo effect has partly taken\nplaceintheFM0phase, whichexplainsitshighlyreduced\nmagnetic moment. The FM0-to-HFL phase transition is\ntherefore not of a Kondo-breakdown type. Rather, our\ntheory predicts the coexistence of localized moments and\nhybridization effect within FM0, which may be respon-\nsible for the discrepancy between the measured Fermi\nsurfaces and those from density functional theory (DFT)\ncalculations with fully localized f-electrons [37].\nTo compare with spectroscopic measurements, we plot\nin Fig. 4 the Fermi volume ( VFS) evolving with tem-\nperature above the FM0 ground state. At high tem-\nperatures, the Fermi volume is small and nearly half of\nthe Brillouin zone, consistent with a half-filled conduc-\n/s32/s33/s34/s32/s35/s34\n/s32/s36/s34/s37/s38/s39\n/s37/s38/s40\n/s37/s38/s41/s42/s43/s44\n/s33/s37/s45/s46/s33/s37/s45/s47/s33/s37/s45/s33\n/s48/s49/s50/s45/s51/s52/s32/s48/s34\n/s32/s33/s34/s32/s47/s34\n/s32/s47/s34\nFIG. 4: Fermi volume evolution with temperature.\n(a) Temperature dependences of the electron Fermi volume\n(VFS=1\nV/summationtext\npθ(−˜εc\np)) above the FM0 ground state for\n∆x/∆y= 0.03. The renormalized electron dispersion ˜ εc\npis\ndetermined by the electron Green’s function with self-ener gy\nincluded at N= 4. The dashed line indicates a ln( T) behavior\nofVFSat intermediate temperatures. (b) The electron Fermi\nsurface at two different temperatures marked as (1) and (2)\nin (a).6\ntion band with negligible Kondo effect. As the temper-\nature decreases, the Fermi volume increases logarithmi-\ncally, reaches a maximal value, and then decreases while\napproaching the FM0 ground state. The logarithmic in-\ncrease of the Fermi volume indicates the development of\nKondo hybridization at intermediate temperatures, con-\nsistent with the measurements of ARPES [17] and ultra-\nfast optical spectroscopy [18], while its decrease above\nthe FM0 ground state results from a suppression of the\nKondo hybridization due to enhanced spin fluctuations,\nconfirming the “relocalization” phenomenon observed in\nmany heavy fermion materials [2, 38].\nDISCUSSION AND CONCLUSION\nTo summarize, we have predicted a phase diagram of\nthe anisotropic FM Kondo-Heisenberg model with both\ncontinuous and first-order FM quantum phase transi-\ntions. Our results provide a unified explanation of the\nvery rich and qualitatively different experimental obser-\nvations in two different types of FM Kondo lattice com-\npounds CeRh 6Ge4and UGe 2with extremely large or rel-\natively small magnetic anisotropy, respectively. More\nspecifically, for large anisotropy, we find a continuous\nquantum phase transition between the FM and HFL\nphases, consistent with that observed in the quasi-one-\ndimensional CeRh 6Ge4. As the magnetic anisotropy is\nreduced, we find a coexisting region of FM and HFL, im-\nplying afirst-orderFM-HFL phase transitionasobserved\nin UGe 2. In addition, there exist three different types of\nFM ordered phases, among which the intermediate FM1\nphase shows half-metallic property and is identified as\na spin selective Kondo insulator. Our findings suggest\nthe important role of magnetic anisotropy on interpret-\ning the experimental observations in FM Kondo lattice\nmaterials.\nIt should be noted, however, that our model studies\nhaveignoredsomepeculiarmicroscopicdetailsinrealistic\nmaterials. Forexample,CeRh 6Ge4hasastrongmagnetic\neasy-planeanisotropy,whichcorrespondstoanXXZtype\nHeisenberg interaction [14]. This easy-plane anisotropy\nmight favor a triplet resonating-valence-bond state [39],\ncause a small discontinuity of the holon phase shift at\nthe FM QCP, and further reduce the magnetic moment\nin the FM phase [14]. In UGe 2, the strong spin-orbit\ncoupling can result in FM fluctuations of Ising character,\nwhich may be important for its superconducting proper-\nties [13]. In a more realistic study, one may also want\nto consider the variation of conduction electron concen-\ntration with external parameters, which might tune the\nFM1-FM2 transition into a first-order one [40] or alter\nthe sign ofthe RKKYinteraction[41]. These details may\nresult in some additional interesting features but will not\nchange our overall theoretical picture. It is straightfor-\nward to extend our theory to include these details infuture studies.\nCONFLICT OF INTEREST\nThe authors declare that they have no conflict of in-\nterest.\nACHNOWLEDGMENTS\nThis work was supported by the National Key Re-\nsearch and Development Program of China (Grant No.\n2017YFA0303103), the National Natural Science Foun-\ndation of China (Grants No. 12174429, No. 11774401,\nand No. 11974397), and the Strategic Priority Research\nProgram of the Chinese Academy of Sciences (Grant No.\nXDB33010100).\nAUTHOR CONTRIBUTIONS\nYang YF conceived and supervised the project. 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B\n2015;91:081108.\n[41] Bernhard BH, Lacroix C. Coexistence of magnetic order\nand Kondo effect in the Kondo-Heisenberg model. Phys.\nRev. B 2015;92:094401.arXiv:2112.14515v1  [cond-mat.str-el]  29 Dec 2021A unified theory of ferromagnetic quantum phase transitions in heavy fermion metals\n- Supplementary Materials -\nJiangfan Wang1and Yi-feng Yang1,2,3\n1Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Science, Beijing 1 00190, China\n2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n3Songshan Lake Materials Laboratory, Dongguan, Guangdong 5 23808, China\nI. Self-consistent equations\nWithin the SU( N) Schwinger boson representation, the anisotropic ferromagnet ic Kondo lattice model (Eq. (1) in\nthe main text) has the following Lagrangian:\nL=/summationdisplay\npaαc†\npaα(∂τ+εc\np)cpaα+/summationdisplay\npαb†\npα(∂τ+εb\np)bpα+/summationdisplay\npa|χpa|2\nJK\n+1√\nVN/summationdisplay\npkaαb†\npαckaαχp−k,a+c.c.+VN/parenleftBigg\n∆2\nx\nJx\nH+∆2\ny\nJy\nH/parenrightBigg\n−2VλS, (S1)\nwhereεb\np=λ−2(∆xcospx+∆ycospy) is the bare spinon dispersion, and λis the Lagrange multiplier associated with\nthe spinon constraint. The τ-dependence of each field is implied. In the paramagnetic phase, the Green’s functions\nof spinon, holon and conduction electron are:\nGb(p,iνn) =1\niνn−εbp−Σb(p,iνn),\nGχ(p,iωn) =1\n−J−1\nK−Σχ(p,iωn),\nGc(p,iωn) =1\niωn−εcp−Σc(p,iωn). (S2)\nBoth Σ band Σ χare order of unity, while Σ cis order of 1 /N:\nΣc(p,iωn) =1\nN/summationdisplay\nkmGχ(k−p,iνm−iωn)Gb(k,iνm). (S3)\nTherefore, one can ignore Σ cin the large- Nlimit, and simply use the bare form of Gcin the self-energy equations (Eq.\n(3) in the main text). At finite- N, one needs to include Σ cto calculate the renormalized dispersion and Fermi surface\nof conduction electrons. Minimizing the free energy with respect to λand ∆ δgives rise to the following constraints:\nκ=−1\nβV/summationdisplay\npnGb(p,iνn),\n∆δ\nJδ\nH=−1\nβV/summationdisplay\npnGb(p,iνn)cospδ. (S4)\nIn practice, we choose different values of ∆ δas input parameters, and determine the value of Jδ\nHusing the second\nconstraint of Eq. (S4) after self-consistency has been achieved .\nTo study the FM ordered phase, we allow the spinons to have a static uniform condensation on the α= 1 spin\ncomponent:\nbiα(τ)→/braceleftbigg√\nNb α= 1\nbiα(τ)α= 2,···,N. (S5)2\nThe Lagrangian can then be separated into two parts, L=L1+LN−1, with\nL1=/summationdisplay\npa(c†\npa,1,χ−p,a)/parenleftbigg∂τ+εc\np−b\n−b−J−1\nK/parenrightbigg/parenleftbiggcpa,1\nχ†\n−p,a/parenrightbigg\n+VNb2(λ−2(∆x+∆y)),\nLN−1=/summationdisplay\npa,α>1c†\npaα(∂τ+εc\np)cpaα+/summationdisplay\np,α>1b†\npα(∂τ+εb\np)bpα+1√\nVN/summationdisplay\npka,α>1b†\npαckaαχp−k,a+c.c.\n+VN/parenleftBigg\n∆2\nx\nJx\nH+∆2\ny\nJy\nH/parenrightBigg\n−2VλS. (S6)\nThe condensation provides a hybridization field for the holons and co nduction electrons of the α= 1 spin flavor, with\nthe following Green’s functions:\nGχ(p,iωn) =1\n−J−1\nK−Σχ(p,iωn)+b2\n−iωn−εc\n−p,\nGα=1\nc(p,iωn) =1\niωn−εcp+b2\n−J−1\nK−Σχ(−p,−iωn). (S7)\nThe Green’s functions of the Schwinger boson and conduction elect ron of the α >1 spin flavors remain the same as\nin Eq. (S2). Minimizing the free energy with respect to λ, ∆δandbgives the following constraints:\nκ=b2−N−1\nN1\nβV/summationdisplay\npnGb(p,iνn),\n∆δ\nJδ\nH=b2−N−1\nN1\nβV/summationdisplay\npnGb(p,iνn)cospδ,\n0 =λ−2(∆x+∆y)+ReΣ b(0,0), (S8)\nwhere (N−1)/N→1 in the large- Nlimit. The last constraint requires that the renormalized spinon spec trum\ntouches zero energy at the Γ point, consistent with our assumptio n of a uniform condensation. Again, only the first\nand the third constraints are solved self-consistently, while the se cond constraint is used to find Jδ\nHat the end of\ncalculation.\nII. The effects of biquadratic spin interaction term\nIn order to alleviate the artificial first-order transition of the Sch winger boson mean-field theory, we introduce a\nbiquadratic spin interaction term into the Hamiltonian, so that the “H eisenberg” term becomes\nHH=−/summationdisplay\niδJδ\nH/bracketleftbig\nSi·Si+δ+ζ(Si·Si+δ)2/bracketrightbig\n=−/summationdisplay\niδJδ\nH(1−ζ/2)Si·Si+δ+C, (S9)\nwhere we have used the identity ( Si·Si+δ)2= 3/16−(Si·Si+δ)/2, andCis a constant. The condition that the\neffective Heisenberg coupling Jδ\nH(1−ζ/2) remains ferromagnetic requires ζ <2. At large- N, the Heisenberg term is\nwritten as\nHH=−/summationdisplay\niδJδ\nH\nN/parenleftbigg\nTiδ+2ζ\nN2T2\niδ/parenrightbigg\n, (S10)\nwhereTiδ=/summationtext\nαβSαβ\niSβα\ni+δ, andSαβ\ni=b†\niαbiβ−κδαβis the SU( N) generator. It reduces to Eq. (S9) in the case of\nN= 2. Using the mean-field expansion Tiδ=/angbracketleftTiδ/angbracketright+δTiδ=/angbracketleftTiδ/angbracketright+(Tiδ−/angbracketleftTiδ/angbracketright), one has\nT2\niδ=−/angbracketleftTiδ/angbracketright2+2/angbracketleftTiδ/angbracketrightTiδ+O(δT2\niδ), (S11)3\n/s32/s33/s34/s35/s32/s33/s34/s36/s32/s33/s34/s32/s32/s33/s32/s37/s32/s33/s32/s38/s32/s33/s32/s35/s32/s33/s32/s36/s32/s33/s32/s32/s39/s40/s41/s42/s40/s43\n/s32/s33/s34/s32/s32/s33/s32/s44/s32/s33/s32/s37/s32/s33/s32/s45/s32/s33/s32/s38/s32/s33/s32/s46\n/s39/s47/s48/s42/s49/s50/s43/s39/s39/s51/s52/s32\n/s39\n/s39/s39/s51/s52/s36\n~\nFIG. S1: Effects of the biquadratic term. The phase diagram with ( ζ= 2) or without ( ζ= 0) the biquadratic term. The\nshadowed areas denote the coexisting regions.\nso that\nHH=−/summationdisplay\niδJδ\nH\nN/parenleftbigg\n1+4ζ\nN2/angbracketleftTiδ/angbracketright/parenrightbigg\nTiδ+C, (S12)\nwhereCis a constant. The above equation defines a modified Heisenberg cou pling\n˜Jδ\nH≡Jδ\nH/parenleftbigg\n1+4ζ\nN2/angbracketleftTiδ/angbracketright/parenrightbigg\n=Jδ\nH/parenleftBigg\n1+4ζ\n(˜Jδ\nH)2∆2\nδ/parenrightBigg\n, (S13)\nwhich suppressesthe artificialfirst-ordertransition. Fig. S1sho wsthe phasediagraminterms ofthe modified Doniach\nratioTK/˜Jy\nHforζ= 0 and ζ= 2. The size of the coexisting region decreases as one includes the b iquadratic term,\nand the tricritical point is shifted slightly from ∆ x/∆y= 0.02 to 0.03.\nIII. Quantum criticality in the 1D limit\nIn Fig. S2, we showphysical quantities asfunctions of T/TKat ∆x= 0 and different ∆ y. The uniform susceptibility\nof impurity spin is calculated as\nχu=2\nV/summationdisplay\nk/integraldisplaydω\nπ1\neβω−1ImGb(k,ω)ReGb(k,ω). (S14)\nIt follows the Curie law T−1at high temperatures, becomes constant in the heavy Fermi liquid p hase (red) at low\ntemperatures, and diverges with another power law T−γon the ferromagnetic side of the QCP (blue). The QCP\nlocates at ∆ y= 0.014 (orTK/Jy\nH= 0.067), with a critical power index γ≈0.67. The power index γincreases with\n∆yon the FM side due to the weakening of Kondo screening effect, as su ggested by the decreasing holon phase shift.\nIV. Derivation of some identities in FM1\nIn the SU( N) representation, we define the impurity spin polarization along the zdirection as\nMS\nz≡1\nV/summationdisplay\ni/angbracketleftSz(i)/angbracketright ≡1\nVN/summationdisplay\ni\n/summationdisplay\nα≤N/2/angbracketleftBig\nb†\niαbiα/angbracketrightBig\n−/summationdisplay\nα>N/2/angbracketleftBig\nb†\niαbiα/angbracketrightBig\n. (S15)4\n~T -1 ~T -γ\nFIG. S2: Quantum criticality at 1D limit. Temperature dependence of (a) the holon phase shift and (b) t he uniform\nimpurity spin susceptibility at ∆ x= 0 and different ∆ y. The black solid line indicates T−1Curie behavior at high temperature,\nwhile the dashed lines indicate T−γdivergence of the susceptibility at low temperature.\nThisdefinitioncorrectlygives MS\nz= 0intheparamagneticphase,sincethespinonconcentration nb,α≡1\nV/summationtext\ni/angbracketleftBig\nb†\niαbiα/angbracketrightBig\nis identical for all spin flavors. In the FM phase, the α= 1 spin flavor develops condensation, while all the other spin\nflavors remain equivalent. This gives us\nMS\nz=1\nN(nb,1−nb,α/negationslash=1) =1\nN(Nb2−O(1)) =b2+O(1/N). (S16)\nThe spin polarization of conduction electrons can be defined in the sa me way\nms\nz≡1\nVN/summationdisplay\nia\n/summationdisplay\nα≤N/2/angbracketleftBig\nc†\niaαciaα/angbracketrightBig\n−/summationdisplay\nα>N/2/angbracketleftBig\nc†\niaαciaα/angbracketrightBig\n\n=K\nN(nc,1−nc,α/negationslash=1), (S17)\nwherenc,α≡1\nV/summationtext\ni/angbracketleftBig\nc†\niaαciaα/angbracketrightBig\nis the electron concentration of spin flavor α, which is independent of the channel\nindex. As discussed in the main text, the FM1 phase also satisfies ano ther identity, nb,1+Knc,1= 2S, which results\ninKnc,1= (N−1)nb,α/negationslash=1when combined with the constraint/summationtext\nαnb,α= 2S. This consistently leads to the plateau\nof the total magnetization:\nMz≡MS\nz+ms\nz=nb,1+Knc,1−nb,α/negationslash=1−Knc,α/negationslash=1\nN\n=2S−K\nN−1(nc,1+(N−1)nc,α/negationslash=1)\nN\n=2S−K\nN−1nc\nN, (S18)\nwherenc=/summationtext\nαnc,αis the total electron concentration per channel. In the large- Nlimit, Eq. (S18) reduces to\nMz=κ(1−nc/N).\nWe now derive the following relation in the FM phases:\nδχ\nπ=1\nV/summationdisplay\np[1+θ(−εc\np)−θ(−E+(p))−θ(−E−(p))]. (S19)\nUsing Eq. (S7), we have\nln[−Gχ(p,ω+)−1] = ln[(J−1\nK+Σχ(p,ω+))(ω++εc\n−p)+b2]−ln[ω++εc\n−p], (S20)\nwhereω+≡ω+i0+. Near the Fermi energy, one has\nΣ′\nχ(p,ω+) = Σ′\nχ(p,0+)+∂ωΣ′\nχ(p,ω+)|0ω, (S21)5\n(a) (b) \nFIG. S3: Effects of κ.(a) Comparison of the coexisting region (light green areas) at different values of κ. The dashed lines are\nextrapolations. (b) The total magnetization and the quanti tynb,1+Knc,1as functions of TK/Jy\nHat ∆x/∆y= 0.1,nc/N= 0.5\nand different values of κ.\nwhere Σ′\nχ(Σ′′\nχ) denotes the real (imaginary) part of self-energy. Defining the q uasiparticle residue Zχ\np≡\n[−∂ωΣ′\nχ(p,ω+)|0]−1and the holon’s dispersion relation εχ\np≡Zχ\np(J−1\nK+Σ′\nχ(p,0)), we have\nln[−Gχ(p,ω+)−1] = ln[(ω−εχ\np−iZχ\npΣ′′\nχ(p,ω+))(ω++εc\n−p)−Zχ\npb2]−ln[ω++εc\n−p]−ln[−Zχ\np]. (S22)\nAt low temperature, one has lim ω→0Σ′′\nχ(p,ω) = 0−andZχ\np>0, therefore lim ω→0(−iZχ\npΣ′′\nχ(p,ω+)) =i0+. The pole\nof the holon Green’s function determines the quasiparticle spectru m:\n0 = (ω−εχ\np)(ω+εc\n−p)−Zχ\npb2\n→ω=εχ\np−εc\n−p±/radicalBig\n(εχ\np+εc\n−p)2+4Zχ\npb2\n2=−E∓(−p). (S23)\nNote that E±(p) is defined as the quasiparticle pole of Gα=1\nc(p,ω), which is exactly the pole of Gχ(p,ω) after the\nparticle-hole transformation p→ −p,ω→ −ω. Therefore, we have\nδχ\nπ=−1\nπV/summationdisplay\npImln[−Gχ(p,0+)−1]\n=−1\nπV/summationdisplay\np/parenleftbig\nImln[E+(p)+i0+]+Imln[ E−(p)+i0+]−Imln[εc\np+i0+]/parenrightbig\n+1\n=1\nV/summationdisplay\np[1+θ(−εc\np)−θ(−E+(p))−θ(−E−(p))], (S24)\nwhere we have assumed the inversion symmetry E±(p) =E±(−p),εc\np=εc\n−p.\nV. The effects of κ\nThe influence of κon the phase diagram is shown in Fig. S3(a). For smaller κ, the FM phases are suppressed by the\nHFL phase, so that the coexisting region becomes narrower and is s hifted towards smaller values of TK/Jy\nH. Roughly\nspeaking, this is because a small value of κenhances the quantum zero motion of spin, thus suppresses the F M order\nand favors the Kondo effect. We also find that the tricritical point s hifts to larger values of ∆ x/∆yasκbecomes\nlarger. Fig. S3(b) compares the plateaus of magnetization and the quantity ( nb,1+Knc,1)/Nat different values of κ,\nwith ∆ x/∆y= 0.1 andnc/N= 0.5. One can see that the two relations Mz=κ(1−nc/N) and (nb,1+Knc,1)/N=κ\nalways hold inside the FM1 phase."    },    {        "title": "0707.0029v1.Zooming_into_the_coexisting_regime_of_ferromagnetism_and_superconductivity_in_ErRh4B4_single_crystals.pdf",        "content": "arXiv:0707.0029v1  [cond-mat.str-el]  30 Jun 2007Zooming into the coexisting regime of ferromagnetism and su perconductivity in\nErRh 4B4single crystals\nRuslan Prozorov,∗Matthew D. Vannette, Stephanie A. Law, Sergey L. Bud’ko, and Pa ul C. Canfield\nAmes Laboratory and Department of Physics & Astronomy, Iowa State University, Ames, IA 50011\n(Dated: 29 June 2007)\nHigh resolution measurements of the dynamic magnetic susce ptibility are reported for ferro-\nmagnetic re-entrant superconductor, ErRh 4B4. Detailed investigation of the coexisting regime\nreveals unusual temperature-asymmetric and magnetically anisotropic behavior. The supercon-\nducting phase appears via a series of discontinuous steps up on warming from the ferromagnetic\nnormal phase, whereas the ferromagnetic phase develops via a gradual transition. A model based\non local field inhomogeneity is proposed to explain the obser vations.\nPACS numbers: 74.25.Dw; 75.50.Cc; 74.25.Ha; 74.25.-q; 74. 90.+n\nThe coexistence of the long-range magnetic order and\nsuperconductivity was first discussed even before the ap-\npearance of the microscopic theory of superconductivity\n[1]. Since then this topic remains one of the most inter-\nesting and controversial in the physics of superconduc-\ntors with many reviews and books devoted to the subject\n[2, 3, 4, 5, 6]. Despite significant effort in new materi-\nals design and discovery, there are only few, confirmed,\nferromagnetic superconductors. Local, full-moment fer-\nromagnetic superconductors: ErRh 4B4[7] (TFM≈0.9\nK,Tc≈8.7K), Ho xMo6S8[8] (TFM≈0.7 K,Tc≈1.8\nK); the weakly ferromagnetic ErNi 2B2C [9] (TFM≈2.3\nK,TAFM≈6 K,Tc≈11 K); and more recent itin-\nerant superconducting ferromagnets, UGe 2(TFM≈33\nK,Tc≈0.95 K at P≈1.3 GPa) [10] and URhGe\n(TFM≈9.5 K,Tc≈0.27 K). Whereas all these ma-\nterials are very interesting on their own, the coexistence\nof growing, large, local moment ferromagnetism and su-\nperconductivity is most clearly presented in ErRh 4B4,\nwhich is the subject of the present Letter. In particular,\nwe are interested in the details of the narrow tempera-\nture interval ( ∼0.3 K) where the two phases coexist and\ninfluence each other.\nErRh4B4was extensively studied over past 30 years\n[2, 3, 4, 5, 6]. The ferromagnetic phase is primitive\ntetragonal with the c−axis being the hard and a−axis\nbeing the easy magnetic axes. Detailed measurements of\nanisotropic magnetization and upper critical filed, Hc2\nwere done by Crabtree et al.[11, 12], who found that\nHa\nc2(alonga−axis) peaks at 5.5 K due to large param-\nagnetic spin susceptibility in that direction [3, 11]. In the\ncontrast, Hc\nc2collapses near the onset of the long-range\nferromagnetic order.\nNeutron diffraction studies have established the ex-\nistence of a modulated ferromagnetic structure at the\nlengthscale of ∼10 nm [13, 14]. In single crystals, results\nsuggested that coexisting phases consists of a mosaic of\nnormal FM domains and SC regions larger than ∼200\nnm in size. The SC regions contain modulated FM mo-\nment with a period of ∼10 nm. These regions could be\nregular domains, spontaneous vortex lattices or laminarstructures with ≥200 nm periodicity and modulated SC\ndomains in between [14]. Thermal hysteresis is observed\nboth in the normal Bragg peak intensity and the small-\nangle peaks. For the small-angle peaks, the intensity is\nhigher on cooling than on warming. This is opposite to\nthe behavior of the regular Bragg peaks from the FM\nregions [13]. Furthermore, the first-order transition, ob-\nserved in satellite peaks temperature dependence [14], is\nconsistent with the spiralstate of Blount and Varma [15].\nHowever, a continuos transition was reported in other\nneutron diffraction [13, 16] and specific heat experiments\n[17]. Such a transition can be realized in a modulated\nstructure or via spontaneous vortex phase.\nTheoretically, some striking features of the coexist-\ning phase include an inhomogeneous, spiral, FM struc-\nture [15, 18] or a fine domain, ”cryptoferromagnetic”\nphase [2, 19], a vortex - lattice modulated spin struc-\nture [20], type-I superconductivity [2, 20, 21], a gapless\nregime and possibly, an inhomogeneous Fulde-Ferrell-\nLarkin-Ovchinnikov (FFLO) state [2]. Another interest-\ning possibility is the development ofsuperconductivity at\nthe ferromagnetic domain walls [22, 23].\nIn this Letter we report precision measurements of the\ndynamic magnetic susceptibility of ErRh 4B4with an em-\nphasisonthenarrowtemperatureregionwhereferromag-\nnetism and superconductivity coexist. We find that the\ntransition is highly asymmetric when FM →SC (heat-\ning) and SC →FM (cooling) data are compared. The\nFM↔SC transition proceeds via a series of discrete steps\nfrom FM to SC phase upon warming and proceeds via a\nsmooth crossover from the SC to FM state upon cooling.\nWith this new information we analyze relevance of some\npredictions made over years for the coexisting phase.\nSingle crystalsof ErRh 4B4were grownat high temper-\natures from molten copper flux as described in [24, 25].\nResulting samples were needle shaped with crystallo-\ngraphicc-axisalongthe needle. Transportmeasurements\ngave residual resistivity ratio of about 8, consistent with\nprevious reports. The anisotropic Hc2(T) curves (see in-\nset to Fig.3 below) are consistent with earlier reports as\nwell [11, 12].2\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s32/s52\n/s84/s32/s40/s75/s41/s72/s32/s61/s32/s48/s72/s61/s52/s48/s48/s48/s32/s79/s101/s72/s32 /s32/s99\nFIG. 1: Dynamic magnetic susceptibility in ErRh 4B4single\ncrystal measured along the magnetic easy axis (perpendicu-\nlar to the needle-shaped sample). Each curve corresponds to\na fixed value of the applied dc magnetic field in the range\nindicated in the figure. (color online)\nTheACmagnetic susceptibility, χ, was measured with\na tunnel-diode resonator (TDR) which is sensitive to\nchanges in susceptibility ∆ χ∼10−8. Details of the mea-\nsurement technique are described elsewhere [26, 27, 28].\nIn brief, properly biased tunnel diode compensates for\nlosses in the tank circuit, so it is self-resonating on its\nresonant frequency, ω= 1/√\nLC∼10 MHz. A sample\nis inserted into the coil on a sapphire rod. The effective\ninductance changes and this causes a change in the res-\nonant frequency. This frequency shift is the measured\nquantity and it is proportional to the sample dynamic\nmagnetic susceptibility, χ[26, 27, 28]. Knowing geomet-\nricalcalibrationfactorsofourcircuit, weobtain χ(T,H).\nAdvantages of this technique are: very small AC excita-\ntion field amplitude ( ∼20 mOe), which means that it\nonly probes, but does not disturb the superconducting\nstate; high stability and excellent temperature resolution\n(∼1 mK), which allowed detailed study of the coexist-\ning region, which is only ∼500 mK wide. Normal-state\nskin depth is largerthan the sample size, so we probe the\nentire bulk in the coexisting region, but when supercon-\nducting phase becomes dominant, there is a possibility\nthat some FM patches still exist, but are screened.\nFigure1showsthefulltemperaturescalemagneticsus-\nceptibility, χ, in single crystal ErRh 4B4for an applied\nfield oriented along the easy axis and perpendicular to\nthe needle-shaped crystal. The peak in χ(T) at the fer-\nromagnetic to superconducting boundary below 1 K is/s48/s46/s57 /s49/s46/s48 /s49/s46/s49 /s49/s46/s50/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s72/s32 /s32/s99/s52\n/s84/s32/s40/s75/s41/s72/s32/s61/s32/s48/s32/s45/s32/s51/s32/s107/s79/s101/s72/s32/s61/s32/s48\nFIG. 2: Magnetic susceptibility in a ferromagnet-\nsuperconductor transition region measured at different ap-\nplied fields. Note the temperature scale and highly assymet-\nric character of the FM-SC and SC-FM transitions. (color\nonline). Blue - warming, red - cooling.\na common feature observed in local moment ferromag-\nnets [29]. Clearly, superconductivity is fullysuppressed\nin the ferromagnetic phase. Note that at elevated fields,\nthe response is nonmonotonic on the SC side close, to\nthe FM boundary, indicative of enhanced diamagnetism\n(larger, negative χ), which may be due to suppressed\nmagnetic pairbreaking or entering into another phase,\nsuch as FFLO [2].\nFigure 2 zooms into the SC ↔FM transition region.\nMeasurements were taken after zero-field cooling, apply-\ning external field and warming up (ZFC-W) above Tc\nand then cooling back to the lowest temperature (FC-C).\nThere is striking asymmetry of the transitions - when the\nsuperconducting phase develops out of the FM state, the\nresponse proceeds with jumps in the susceptibility, which\nare clearly associated with the appearance of supercon-\nducting regions of finite size. The steps are present up\nto the largest field at which superconductivity survives.\nDecreasing temperature shows a completely different re-\nsult: the transition is smooth and gradual and proceeds\nto lower temperatures.\nTo better understand the dynamics of the transition,\nthe top frame of Fig. 3 shows measurements at H= 0 for\ndifferent temperature ramp rates. Temperature variation\nis shown in the inset. These data clearly demonstrate\nthatthishysteresisisinsensitivetoheating/coolingrates.\nIt should be noted that all the other data presented in\nthis Letter were taken with the slowest cooling rate of 60\nµK/s.\nSimilar hysteresis and steps are also present in the an-3\n/s48/s46/s57 /s49/s46/s48 /s49/s46/s49 /s49/s46/s50/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s52\n/s84/s32/s40/s75/s41/s72/s32/s61/s32/s48/s48 /s49/s48/s48/s48/s48 /s50/s48/s48/s48/s48 /s51/s48/s48/s48/s48 /s52/s48/s48/s48/s48/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s50/s120/s49/s48/s45/s51\n/s32/s75/s47/s115\n/s49/s120/s49/s48/s45/s51\n/s32/s75/s47/s115\n/s50/s120/s49/s48/s45/s52\n/s32/s75/s47/s115\n/s54/s120/s49/s48/s45/s53\n/s32/s75/s47/s115/s84/s32/s40/s75/s41\n/s116/s105/s109/s101/s32/s40/s115/s41/s114/s97/s109/s112/s32/s114/s97/s116/s101/s115/s58\nFIG. 3: Hysteresis of the transition at zero applied field mea -\nsured at different ramp rates. Temperature sweep profiles are\nshown in the inset. Clearly, this hysteresis and jumps are no t\ndynamic effects.\n/s48/s46/s57 /s49/s46/s48 /s49/s46/s49 /s49/s46/s50 /s49/s46/s51 /s49/s46/s52/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s52\n/s84/s32/s40/s75/s41/s50/s56/s48/s48/s32/s79/s101\n/s48/s46/s57 /s49/s46/s48 /s49/s46/s49 /s49/s46/s50 /s49/s46/s51 /s49/s46/s52/s45/s49/s46/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48\n/s84/s32/s40/s75/s41/s72/s61/s48\n/s72/s32/s124/s124/s32/s99/s50 /s52 /s54 /s56/s50/s52/s54/s56/s49/s48\n/s78/s111/s114/s109/s97/s108 /s83/s85/s80/s69/s82/s67/s79/s78/s68/s85/s67/s84/s73/s78/s71/s72/s32/s124/s124/s32/s99\n/s32/s32/s72/s32/s40/s107/s79/s101/s41\n/s84/s32/s40/s75/s41/s108/s105/s110/s101/s115/s32/s45/s32/s84/s68/s82\n/s115/s121/s109/s98/s111/s108/s115/s32/s45/s32/s116/s114/s97/s110/s115/s112/s111/s114/s116\n/s72/s32 /s32/s99/s70/s77\nFIG. 4: Transition region measured for magnetic field ori-\nented along the needle and c-axis. Inset: summary phase di-\nagram for two orientations measured by transport (symbols)\nand TDR.\nother orientation, when external magnetic field is applied\nalongthe c−axis. Thisis showninFig. 4. Note that peak\ninχ(T) at the FM boundary is not present, which is\nconsistent with the behavior of anisotropic local-moment\nferromagnet [29]. The inset to Fig. 4 shows the phase di-\nagram obtained from resistivity and TDR measurements\nfor both orientations. There is excellent agreement be-\ntween the two techniques and, as noted earlier, this dia-/s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s49/s48\n/s57\n/s55\n/s54/s44/s56/s52\n/s51/s44/s53/s50/s52\n/s84/s32/s40/s75/s41/s72/s32/s61/s32/s54/s54/s48/s32/s79/s101\n/s49/s48/s46/s49/s32/s75\n/s72/s32/s124/s124/s32/s99/s52\n/s84/s32/s40/s75/s41/s72/s32/s61/s32/s48\nFIG. 5: Details of the hysteresis with partial scans as ex-\nplained in the text. Inset shows anothe partial loop on a\ncooling part of the curve.\ngram is consistent with previous reports [11, 12].\nFinally, Fig. 5 shows so called minor hysteresis loops\n(not as function of field, but temperature). The labels\nshow the evolution of the susceptibility. It starts from\nlow temperature at (1) when sample was warmed up to\nfirst signs of superconductivity that appeared as small\njump at (2), then warmed further and reaching almost\nfull superconductivity at (3), but then cooled back down\nto (4) as indicated by arrow and warmed back to (5).\nNote that along (3) →(4)χis significantly different\nfrom (4) →(5). Another similar minor cooling-warming\nloop follows (6) →(7)→(8) after which the sample was\ncooleddowntoreturnto(10) = (1)via(9). Interestingly,\nthere are no steps or jumps observed on the minor loops\nevenonwarming. Also, theslope dχ/dtissimilaroncool-\ning and warming and is very different from the original\nsteep slope (2) →(3). This is consistent with the present\nof vortices, probably pinned by the modulated FM/SC\nstructure. The inset in Fig. 5 shows a small minor loop\non a cooling part. This loop has small slope comparable\nto the larger loops described in the main frame.\nLet us now turn to the interpretation of these results.\nClearly, the FM →SC transition proceeds via a series of\njumps in diamagnetic screening due to formation of su-\nperconducting regions of macroscopic volume, roughly\n5%−20% of the sample volume depending on the ap-\nplied field and temperature. Indeed, each observed step\nmay be a result of simultaneous formation of many in-\ndividual superconducting domains of similar size. These\nsteps in χare present both for H||candH||aaxes, al-\nthough in the latter case the steps are smaller and are\nmore pronounced, possibly due to magnetic and shape4\nanisotropies. The number of steps increases with the in-\ncreasingfield and the firststep (the first signofsupercon-\nductivity) occurs at a higher temperature for larger ap-\nplied field. Overall, FM →SC transition is apparently of\nthefirstorderandexhibitsbehaviorconsistentwith type-\nI superconductivity as predicted theoretically [2, 20, 21].\nFrom our point of view, the first jump occurs at a tem-\nperature where internal field is equal to a supercooling\nfield of a type-I superconductor. (Note that apparent su-\nperheating of the FM →SC transition [17] corresponds to\nsupercooling of a regular type-I superconductor [17, 30],\nbecause we enter normal phase on cooling). When first\nsuperconductingdomains appear, effective magneticfield\naround them increases due to the flux expulsion and in-\nternal field becomes more inhomogeneous. This net in-\ncrease in the internal field in the remaining FM regions\nstabilizes them to higher temperatures. The system now\nneeds to get farther away from the initial FM boundary,\ndeeper into the SC state to produce more superconduct-\ningpatches. Inthisscenario,theobservedjumpsin χcor-\nrespond to a cascade of supercooling transitions. If the\ntemperature is lowered before the transition is complete\ndomains stay stable to lowertemperatures due to physics\nsimilar to superheating of a type-I superconductor. It is\nalso quite possible that superconducting domains have\nthe modulated spin structure seen in neutron scattering\n[14]. Finally, itseemsthatferromagneticdomainsarenot\ndirectly related to the observed steps, because at higher\nfields, the number of these domains decrease and domi-\nnant domains (along the applied field) grow in size.\nIn a striking contrast with FM →SC transition, the\nSC→FMtransitionissmoothandproceedstomuchlower\ntemperatures. Yet the transition is hysteretic as evident\nfrom the minor loops shown in Fig. 5. It is possible\nthat Abrikosov vortices are being spontaneously created\nas the temperature is lowered and the systems crosses\nover into the normal state when vortex cores overlap.\nThe vortex state is also compatible with long-range co-\nherence observed in neutron scattering experiments [14].\nAt the same time ferromagnetic modulation with a pe-\nriod of∼10 nm may also develop between the vortices\n[2, 15, 18, 19]. This would be also be similar to FFLO\nstate in the presence of vortices [31]. Moreover, this\nwould explain different intensities of small-angle satel-\nlite peaks, because coherence volume in the vortex state\nmust be much larger compared to domain-like state on\nwarming. We alsonote that unusual enhancement of dia-\nmagnetism in the vicinity of the FM boundary from the\nSC side, could be due to an FFLO pocket as predicted by\nBulaevskii for ErRh 4B4[2]. If we plot temperature of a\nminimum in χ(T) as function applied field, and also Hc2,\nwe obtain phase diagram remarkably similar to Fig.7 of\nRef.[2].\nOverall,weconcludethat wearewitnessinganunusual\ntransition. It is definitely first order on warming with\nsigns of type-I superconductor ”supercooling”. However,it is smooth upon cooling and exhibits smooth minor\nloops similar to a type-II superconductor. A second or-\nder transition occurs between normal and FFLO phases\nas well as between normal and SC for type-II supercon-\nductor. However, transition from SC to FFLO state is\nfirst order as well as from SC to spiral state. It is possi-\nble that size of the new phase nuclei is so small that we\ncannot resolve it upon cooling.\nDiscussions with Lev Bulaevskii, Alexander Buzdin,\nVladimir Kogan and Roman Mints are appreciated.\nWork at the Ames Laboratory was supported by the De-\npartment of Energy-Basic Energy Sciences under Con-\ntract No. DE-AC02-07CH11358. R. P. acknowledges\nsupport from NSF grant number DMR-05-53285 and the\nAlfred P. Sloan Foundation.\n∗Electronic address: prozorov@ameslab.gov\n[1] V. L. Ginzburg, Soviet Phys. JETP 4, 153 (1957).\n[2] L. N. Bulaevskii, A. I. Buzdin, M. L. Kuli, and S. V.\nPanjukov, Adv. Phys. 34, 175 (1985).\n[3] O. Fischer, Magnetic superconductors , vol. 5 (Elsevier,\n1990).\n[4] M. L. Kulic, Comptes Rendus Physique 7, 4 (2006).\n[5] M. B. Maple, Physica B 215, 110 (1995).\n[6] K. P. Sinha and S. L. Kakani, Magnetic Superconduc-\ntors: Recent Developments (Nova Science Publishers,\nNew York, 1989).\n[7] W. A. Fertig, D. C. Johnston, L. E. DeLong, R. W. Mc-\nCallum, M. B. Maple, and B. T. Matthias, Phys. Rev.\nLett.38, 987 (1977).\n[8] M. Ishikawa and O. Fischer, Solid State Commun. 23, 37\n(1977).\n[9] P. C. Canfield, S. L. Bud’ko, and B. K. Cho, Physica C\n262, 249 (1996).\n[10] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche,\nR. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R.\nWalker, S. R. 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Matsumoto, and H. Umezawa, Phys. Rev.\nB20, 1915 (1979).\n[21] K. E. Gray, J. Zasadzinski, R. Vaglio, and D. Hinks,\nPhysical Review B 27, 4161 (1983).\n[22] A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, Zh.\nExp. Teor. Fiz. 87, 299 (1984).\n[23] A. I. Buzdin and A. S. Mel’nikov, Phys. Rev. B 67,\n020503/1 (2003).\n[24] S. Okada, K. Kudou, T. Shishido, Y. Satao, and\nT. Fukuda, Jpns. J. Appl. Phys., Part 2: Letters 35,\nL790 (1996).\n[25] T.Shishido, J.Ye,T.Sasaki, R.Note, K.Obara, T.Taka-\nhashi, T. Matsumoto, and T. Fukuda, J. Solid State\nChem.133, 82 (1997).\n[26] R. Prozorov, R. W. Giannetta, A. Carrington, and F. M.Araujo-Moreira, Phys. Rev. B 62, 115 (2000).\n[27] R. Prozorov, R. W. Giannetta, A. Carrington,\nP. Fournier, R. L. Greene, P. Guptasarma, D. G. Hinks,\nand A. R. Banks, Appl. Phys. Lett. 77, 4202 (2000).\n[28] R. Prozorov and R. W. Giannetta, Supercond. Sci. Tech-\nnol.19, R41 (2006).\n[29] M. D. Vannette, A. Safa-Sefat, S. Jia, S. A. Law,\nG. Lapertot, S. L. Bud’ko, P. C. Canfield, J. Schmalian,\nand R. Prozorov, J. Mag. Mag. Mater. in print (2007).\n[30] J. Feder, S. R. Kiser, and F. Rothwarf, Phys. Rev. Lett.\n17, 87 (1966).\n[31] M. Ichioka, H. Adachi, T. Mizushima, and K. Machida,\nJ. Magn. Magn. Mater. 310, 593 (2007)."    },    {        "title": "0804.1713v1.Nonergodic_thermodynamics_of_disordered_ferromagnets_and_ferroelectrics.pdf",        "content": "arXiv:0804.1713v1  [cond-mat.dis-nn]  10 Apr 2008Nonergodic thermodynamics of disordered ferromagnets and ferroelectrics\nP. N. Timonin∗\nPhysics Research Institute at Southern Federal University , 344090, Rostov - on - Don, Russia\n(Dated: November 4, 2018)\nPhenomenological thermodynamic theory describing the pro perties of metastable states in dis-\nordered ferromagnets and ferroelectrics with frustrative random interactions is developed and its\nability to describe various nonergodic phenomena in real cr ystals is demonstrated.\nPACS numbers: 75.10.Nr, 77.80.-e\nI. INTRODUCTION\nMany types of disorder in ferromagnets and ferro-\nelectrics can induce spin- or dipole-glass states if emer-\ngent random interactions cause frustration and make the\nground state of a crystal highly degenerate1. Besides\nthe classic example of the solid solution (alloy) of fer-\nromagnet with antiferromagnet2it can be the random\nanisotropy3, simple dilution in the case of competing\nnearest and next-nearest interactions4or in the presence\nof strong dipole-dipole ones5as well as structural defects\nsuch as dislocations6. Now it is widely recognized that\nthe unusual nonergodic properties of disordered ferro-\nmagnets and ferroelectrics in their glassy phases result\nfrom the appearance of a number of metastable states\neach having more or less random directions of local spon-\ntaneous electric or magnetic dipole moments1. Using the\ndifferent protocols of changing temperature and external\nfield (e. g. field-cooled, FC, zero-field cooled, ZFC, etc.)\none may arrive at the metastable states with different\nnet magnetization (polarization) and differences in other\nthermodynamic functions.\nFirst adequate theoretical description of such history\n- dependence of system properties in glassy phases was\nachieved in Refs. [7, 8] using numerical simulations in\nthe local mean-field theory of random-bond magnets. It\nwas explicitly shown that various regimes of temperature\nand field variations do bring system to different minima\nof thermodynamic potential thus providing the history\n- dependence of thermodynamic functions. Under the\nassumption of the possibility of sufficiently slow (quasi-\nstatic) variations of external parameters making the pre-\ndictionsofstaticthermodynamictheoryvalidthenumber\nof nonergodic features of glassy phases were described in\nRefs. [7, 8]: the temperature dependencies of FC and\nZFC functions, thermal and isothermal remanent mag-\nnetizations, the form of hysteresis loops, FC heat capac-\nity in different fields. These results are in reasonable\nagreement with the experimental data for a series of ran-\ndom magnets. Later the method of the local mean-field\nsimulations were used to reveal the presence of noner-\ngodic reentrant spin-glass phase in 2d XY random-bond\nmodel9.\nSubsequent studies of nonergodic phenomena in spin\nsystems with random interactions were mainly focused\non numerical simulations of hysteresisloops. The smallerloops inside the main one and evolution via macroscopic\nspin avalanches on the main loop are shown to exist in\nEdwards-Anderson10and Sherrington- Kirkpatrick11,12\nspin-glass models. The existence of hysteresis loops is\nestablished in random - anisotropy model13. The new\nhard-spin mean-field method is developed for frustrated\nrandom - bond systems14,15,16which provides the evi-\ndencesfor the existenceofmultiple metastablestatesand\ncandescribeanumberofnonergodicphenomenawithless\nsimulation efforts.\nInspiteofdefinitesuccessesinnumericalsimulationsof\nnonergodic effects in glassy phases these methods are not\ndestined to provide some general picture for systemati-\nzation and qualitative explanation of these effects. It is\nrather difficult to find from the existing numerical results\nthe possible interrelations between a variety of history -\ndependent phenomena observed in different experimen-\ntal regimes, see Refs. [17-25]. Yet it seems that they\nshould be necessary present as, e. g., the hysteresis loops\nand ZFC - FC magnetization differences have the com-\nmon origin. Then one may hope to obtain the unified\npicture of such phenomena using analytical calculations\nin some simplified models. But nowadays there are no\nsimple enough microscopic models in which nonergodic\neffects in finite field could be described by the analytical\nmethods.\nThus in present situation some phenomenological ap-\nproach may help to achieve general understanding of the\nquasi-static large field response in glassy phases and the\nrole played in it by metastable states. Apparently, it\nshould be based on some realistic mechanism underlying\nthe appearance of such states. Such mechanism for the\nrandom-interaction systems has been described in Ref.\n[26]. It consists in the subsequent condensations of nu-\nmerous sparse fractal modes defined by the eigenvectors\nof the random matrix of pair-wise interactions (exchange\nmatrix in magnets or the matrix of harmonic interac-\ntion of polar atomic displacements in ferroelectrics) at\ntheir localization threshold27. In random ferroelectrics\n(relaxors) these modes play the role of the notorious\n”softmode”astheyarethedelocalizedeigenmodesofpo-\nlar atomic oscillation with the lowest frequencies. Their\namplitudes throughout the disordered crystal can be ob-\ntained by the numerical methods of Refs. [27]. The con-\ndensation (freezing) of these modes in relaxors results in\nappearance of stable polar atomic displacements same\nas in the case of ordinary soft mode albeit these dis-2\nplacements are highly inhomogeneous and sparse. Their\nanalogsinrandomIsingmagnetscanberelatedtothede-\nlocalized relaxational magnetic modes which freeze first\nat a macroscopic spin glass transition.\nIt is important to realize that only the freezing of\nthese macroscopic modes can result in emergence of sta-\nblespontaneouslocalmomentsataglasstransition. Such\nobjectsasnano-domainswillneverhavespontaneousmo-\nments stable for a macroscopic time so this notion is use-\nless for the description of the true thermodynamic tran-\nsition into the glass phase. Thus Burns temperature in\nrelaxors at which experiments reveal the appearance of\n(fluctuating) polar nano-domains marks the onset of the\nparaelectricGriffiths’phaseinwhichonlynon-analyticity\nin the field dependence of polarization28and slow non-\nexponential relaxation29appear.\nThe suggested mechanism of a macroscopic spin glass\ntransition is in sharp contrast with that of phase transi-\ntion in homogeneouscrystals. The last takesplace due to\nthe condensation of just one periodic or uniform in space\neigenmode which correspondsto the largest eigenvalue of\nexchange matrix (in magnets) or the lowest eigenvalue of\nthe matrix of effective harmonic interaction (soft mode\nin ferroelectrics). In ideal crystal it is sufficient to sta-\nbilize the other modes with eigenvalues close to the crit-\nical one throughout some low-symmetry phase regions.\nThis stabilization of the rest of near-critical modes re-\nsults from nonlinear mode couplings which are always\npresent if there is some finite spatial overlap between the\ncritical mode and near-critical ones. As all eigenmodes\nof an ideal crystal is just the plane waves strongly over-\nlapping throughout the whole sample, the condensation\nof just one critical mode makes the system stable.\nIn a strongly disordered crystal with random interac-\ntions the situation changes principally. The randomness\nof pair-wise interactions makes the eigenmodes near the\nboundary of the spectrum localized27so the transition\nmay start only with the condensation of the first de-\nlocalized eigenmode at the localization threshold some-\nwhat far from the lower boundary of the spectrum. Ow-\ning to the sparse fractal structure of the modes near\nthis threshold27, the condensation of the first delocal-\nized mode can not stabilize the other sparse near-critical\nmodes which do not overlapessentially with the first one.\nSo they proceed to condense at slightly lower tempera-\ntures until almost all crystal sites acquire spontaneous\nmoments. If the average fractal dimension of these con-\ndensing modes is df< d, the average number of sites\nparticipating in each mode is of the order N1=Ndf/d.\nSo the number ofthem needed to coverthe N-site crystal\nisN0=N/N1=N1−(df/d).\nThus strongdisorder can transformthe ordinaryphase\ntransition into the macroscopic number of them spread-\ning in some temperature interval. As each mode after\ncondensation can have (at least) two stable states with\nthe reverse directions of local moments, we may end up\nwith up to 2N0metastable states. Note that each of\nthese numerous subsequent transitions is still a macro-scopic one as the macroscopic number of local sponta-\nneous moments of order N1appears at it. Apparently,\nthe mean-field estimate of the potential barrier between\nmetastable states would also give the macroscopic value\nproportional to N1.\nHerewemaynotethatabovepicturedescribesonlythe\npossibility ofemergenceofnumerousmetastablestates in\ncrystal with random interactions. Do it actually realizes\ndepends on the details of pair-wise interaction between\nthe sparse fractal modes. If this interaction tends to ori-\nent the net moments of the modes in parallel, we may\nhave ordinary ferromagnetic (ferroelectric) phase with or\nwithout some metastable states. Principally, sketched\nabove mechanism allows for phenomenological descrip-\ntion of the thermodynamics of metastable states in the\nspirit of Landau’s theory of phase transitions. The case\nwith purely glass transition was considered in Ref. [26].\nHere we present the simplified and more general deriva-\ntion of the phenomenological Landau’s potential for the\ngeneric case of competing ferromagnetic (ferroelectric)\nand glassyinteractions in disordered crystals. For a large\nclassofsuchrandomsystemsitispossibletoobtainsemi-\nanalytical description of the properties of the emergent\nmetastable states thus providing the comprehensive ac-\ncount of possible nonergodic effects in random ferromag-\nnets and ferroelectric relaxors.\nII. LANDAU’S POTENTIAL FOR RANDOM\nFERROMAGNETS AND FERROELECTRICS\nFurther we mainly resort to the magnetic terminol-\nogy and designations just to be specific. Let us consider\nthe ferromagnet with the second-order transition and\none-component order parameter where some disorder of\nrandom-interaction type presents tending to destroy the\nferromagnetic order. According to above considerations\nwe can introduce in this case the Landau potential for\nsome specific realizationof disorderdepending on the net\nmagnetizations of sparse fractal modes, mi,i= 1,...,N0.\nIf the temperature interval ( Tg,Tg−∆T) at which\nthe condensation of these modes takes place is narrow,\n∆T≪Tgthen we can consider the small minearTgand\nexpand the potential F(m) in powers of mi. It would\ncontain only even powers of these order parameters due\nto the globalinversionsymmetry. Alsowe canretain in it\nthe terms with no more than two different modes as one\ncan suggest that their sparse spatial structure makes the\nsimultaneous interactions of three or more modes negli-\ngibly small.\nFurtherwemaynotethat∆ T≪Tgisactuallythenec-\nessary condition when one consider the frustrating disor-\nder causing the (near) degeneracy of condensing types\nof magnetic (dipole) arrangements. This means that in\nsuch case the condensing modes have nearly the same\nthermodynamic potentials, close transition temperatures\nand thermodynamic parameters. The simplest way to\nimitate this near degeneracy is to suppose that all ther-3\nmodynamic properties of all modes are identical. Then\nF(m)shouldbesymmetricunderallpermutationsof mi.\nTheimmediateeffectofthis assumptionwillbethe merg-\ning of actually subsequent transitions to just one point,\nTg. In this way we avoid the consideration of narrow\ntemperature interval ( Tg,Tg−∆T) and gain the sig-\nnificant simplification of the model which still preserve\nthe essential features of nonergodic transitions caused by\nfrustrating disorder. We may say that model with per-\nmutation symmetry is the ”minimal” one and, probably,\nthe simplest model allowing for more or less adequate ac-\ncount of real nonergodic effects in large class of random\nferromagnets and ferroelectric relaxors.\nAt last, the quadratic terms of the potential should\nreflect the competition between ferromagnetic (ferroelec-\ntric) and glassy orders. Thus we arrive at the unique\nform of potential up to forth order in miobeying the\nabove criteria,\nF(m) =τf\n2N0/parenleftBiggN0/summationdisplay\ni=1mi/parenrightBigg2\n+τg\n4N0N0/summationdisplay\ni,j=1(mi−mj)2\n+a\n4N0/summationdisplay\ni=1m4\ni+b\n4N0/parenleftBiggN0/summationdisplay\ni=1m2\ni/parenrightBigg2\n−hN0/summationdisplay\ni=1mi(1)\nHerehis external field conjugate to the net magnetiza-\ntion of a sample\n¯m=N−1\n0N0/summationdisplay\ni=1mi\nNote that this relation is the consequence of the adopted\nprincipleofmodeequivalencewhichincludes thesupposi-\ntion ofequal numbers ofsites participatingin each mode.\nN0in the denominators of Eq. (1) ensure the equal order\nof different terms’ contributions at large N0≫1. Note\nalso the absence in (1) the term\nc\nN0N0/summationdisplay\ni=1miN0/summationdisplay\nj=1m3\nj\nallowedformally by the abovecriteria. It can be removed\nby the linear transform of miresulting in rescaling of\nhandτfand appearance of two forth-order terms with\nthree-andfour-modeinteractionswhichmustbedropped\ndue to assumed smallness of such interactions. Thus the\npresence in Fof this term amounts just to the handτf\nrescaling and it can be omitted.\nThe coefficients aandbinFare some constants spe-\ncific for a given disorder realization, while τfandτgare\nlinear decreasing functions of temperature Tchanging\ntheir signs at temperatures TfandTgcorrespondingly\nalso being disorder dependent. Yet in the ferroelectric\nand ferromagnetic solid solutions all potential parame-\nters must be the self-averaging quantities. It means that\nthey depend only on the impurities’ concentrations in ac-\ncordance with the experimental data showing no notice-\nable variations of the properties of different samples withthe same composition. Here we should note that this is\ntrue only for the solid solutions with deeply frozen disor-\nder. In the relaxors with annealing mediated ordering of\ncomponents such as PSN the parameters of the potential\nwill alsodepend on the degreeof orderingachievedat the\nhigh-temperature annealing.\nGenerally, the crystal with negligible disorder should\nhaveTg≪Tfas in this case we may have just the or-\ndinary transition into homogeneous ferro-phase without\nany traces of disorder. So the growth of disorder will\nresult in the increase of Tgalong with the lowering of Tf.\nThe difference between TfandTgmust be small to\nmake the expansion of Fin small mimeaningful. Ac-\ncording to Eq. (1) below Tfferromagnetic order may set\nin, while below Tgglassy states described by the N0−1\n- dimensional order parameter composed of the indepen-\ndentmi−mjcomponents can appear favoring the an-\ntiparallel miorientations. What actually results from\nthe competition of glass- and ferro-order depends on the\nrelation between TfandTgvalues as well as the rates\nofτfandτgdecreasing. Finding the evolution of F(m)\nminima for different τfandτgrelations we can get the\npossible variants of phase sequences in crystals with ran-\ndom interactions. In spite of huge amount of glassy min-\nima that F(m) may have, to find all of them is rather\nsimple task in the present model which can be fulfilled\nby semi-analytical methods.\nIII. THERMODYNAMICS OF COMPETING\nGLASS- AND FERRO-STATES IN ZERO FIELD\nUsing the notations/bracketleftbig\nmk/bracketrightbig\n=N−1\n0N0/summationtext\ni=1mk\ni, we can rep-\nresent the potential density f(m) =F(m)/N0in the\nsimple form\nf(m) =τg\n2/bracketleftbig\nm2/bracketrightbig\n+τf−τg\n2¯m2+a\n4/bracketleftbig\nm4/bracketrightbig\n+b\n4/bracketleftbig\nm2/bracketrightbig2−h¯m\nDifferentiating f(m) with respect to miwe get the equa-\ntions of state\n/parenleftbig\nτg+b/bracketleftbig\nm2/bracketrightbig/parenrightbig\nmi+am3\ni=h+(τg−τf) ¯m(2)\nThe solutions of these equations are minima of f(m)\ndescribing possible (meta)stable states if they render the\npositive definiteness of the matrix\nGi,j≡∂2f(m)\n∂mi∂mj=δi,j/parenleftbig\nτg+b/bracketleftbig\nm2/bracketrightbig\n+3am2\ni/parenrightbig\n+N−1\n0(τf−τg+2bmimj).\nLet us consider first the homogeneously magnetized\nferro-state with the equal mi=m0. From Eq. (2) we\nhave\nτfm0+(a+b)m3\n0=h (3)4\nThus ath= 0 the ferro-state appears at τf<0 ifa+b >\n0. It is stable for\nτg>3a+b\na+bτf (4)\nTurning to the possible glassy solutions of Eq. (2) with\nunequalmi, we note that all miobey the same equation\nsosuchsolutionsexistifthereareseveralrealrootstoEq.\n(2). Being of third order with respect to miit can have\none or three real roots. In the last case mican acquire\nonly two of the root values with the largest modules: one\npositive, m+>0, and one negative, m−<0, as only\nthese two can make the coefficient at δi,jin matrix ˆG\npositive which is necessary for the stability. Thus every\nglassy state is defined by the number, N+, ofmihaving\nm+values (or the number of m−ones,N−=N0−N+).\nHence in the glassy states\n/bracketleftbig\nmk/bracketrightbig\n=n+mk\n++n−mk\n−,¯m=n+m++n−m−\nn±≡N±/N0, n ++n−= 1\nand to find a glass state with a given n+(n−) we need\nto obtain the corresponding roots, m±, of Eq. (2) which\nnow becomes\n/parenleftbig\nτg+b/bracketleftbig\nm2/bracketrightbig/parenrightbig\nm±+am3\n±=h+(τg−τf) ¯m(5)\nThus the present model may generally have the quasi-\ncontinuous set of states (local minima) defined by the\nparameter n+which changes in the interval 0 < n+<1\nin the infinitesimal steps ±N−1\n0. Note that there are/parenleftbigg\nN0\nN+/parenrightbigg\nstates with the same N+=n+N0which differ\nby the permutations of miand have the identical ther-\nmodynamic parameters due to the adopted permutation\nsymmetry.Introducing the variable\nx=−m−/m+\nwe get from Eq. (5) the following equations\n/parenleftbigg\n1−τf\nτg/parenrightbigg\n(n+−xn−)R(x,n+,β) =\nx(x−1)R(x,n+,β)3+h\na/parenleftbigga\n−τg/parenrightbigg3/2\n(6)\nm+=/radicalBig\n−τg/aR(x,n+,β) (7)\nR(x,n+,β)≡/bracketleftbig\n1−x+x2+β/parenleftbig\nn++n−x2/parenrightbig/bracketrightbig−1/2\nHereβ≡b/a.\nThus we need to solve only one Eq. (6) for x=\nx(n+,τg,τf,h) to obtain using Eq. (7) and xdefini-\ntion full description of the thermodynamic properties of\na crystal in the metastable states with a given n+. So we\nhave for the net spontaneous moment in such states\n¯m= (n+−xn−)/radicalBig\n−τg/aR(x,n+,β) (8)\nand for the Edwards-Anderson glass order parameter\nq=/bracketleftbig\nm2/bracketrightbig\n−¯m2= (−τg/a)n+n−(1+x)2R(x,n+,β)2.\nFurther we consider the simplest case a >0,b >0 so the\nglassy metastable states appear at τg<0 .\nThen we have for the susceptibility χ=∂¯m\n∂h\nχ−1=τf−τg−τgR(x,n+,β)2(1+x)(2x−1)(2−x)+2β/bracketleftbig\n(2x−1)n++x2(2−x)n−/bracketrightbig\n(2x−1)n++(2−x)n−+2βn+n−(1+x)(9)\nThe equilibrium value of the potential for a given state\nis\nfeq(n+,τg,τf,h) =/parenleftbig\nτgq+τf¯m2−3h¯m/parenrightbig\n/4\nHere we omit the term arising from the degeneracy of\nglassy states\n−T\nNSconfig=−T\nNln/parenleftbigg\nN0\nN+/parenrightbigg\n∼N0\nN∼N−df/das it vanishes at large N. Then we get for a given\nmetastable state the entropy\nS=−∂feq(n+,τg,τf,h)\n∂T=−1\n2/parenleftBig\nτ′\ngq+τ′\nf¯m2/parenrightBig\n,\nτ′\nf,g≡∂τf,g\n∂T,\nand the heat capacity5\nC\nTgχ=/parenleftbig\nτ′\nf−τ′\ng/parenrightbig2¯m2+2τ′\ng/parenleftbig\nτ′\nf−τ′\ng/parenrightbig¯mm+[n+(2x−1)−n−x(2−x)]\nn+(2x−1)+n−(2−x)+2n+n−β(1+x)\n+(τ′\ng)2m2\n+/bracketleftbig\nn+(2x−1)+n−x2(2−x)/bracketrightbig\n+n+n−(1+x)a−1(τf−τg)\nn+(2x−1)+n−(2−x)+2n+n−β(1+x)(10)\nFora >0,b >0 the stability conditions for the so-\nlutions of Eq. (6) providing the positive definetness of\nmatrixˆGare represented by the inequalities\n1/2< x <2, χ > 0 (11)\nNote that the condition χ >0 ensures the stability with\nrespect to the ferromagnetic fluctuations while at the\nboundaries of the first inequality the glassy instability\noccurs.\nTurning to the case h= 0 we can see that simple solu-\ntions of the equation of state (6) exist for n+→1,0 and\nn+=n−= 1/2. In the first case Eq. (6) tends to the\nzero-field limit of Eq. (3) and we have\n¯m=/radicalBig\n−τf\na+b,χ−1=−2τf,q= 0,feq=τf¯m2/4,\nS=−τ′\nf¯m2/2,C=Tg(τ′\ng)2\n2(a+b)\nand Eqs.(11) defines the sector on the ( τf,τg)-plane in\nwhich nearly homogeneous states with n+→1,0 are\nstable\n3+β\n1+βτf< τg<3+4β\n4(1+β)τf<0 (12)\nNote that such states exist only in the part of the region\nEq. (4) where the homogeneously magnetized ferro-state\nis stable. This just means that the set of metastable\nstates to which the states with n+→1,0 belong has\nmore narrow region of existence.\nFor the state with fully disordered moments ( n+=\nn−= 1/2 ) we have x= 1 from Eq. (6) at h= 0 so\n¯m= 0,χ−1=τf−τg3+β\n1+β,q=−τg\na+b,feq=τgq/4,\nS=−τ′\ngq/2,C=Tg(τ′\ng)2\n2(a+b)\nandχ >0 requires that\nτg<1+β\n3+βτf (13)\nalong with τg<0. It can be shown that the sector of\nstability of fully disordered states is the largest one and\nthe stability regions of all other states belong to it, the\nmost narrow sector being that of nearly homogeneous\nstates (12).\nFurther we find\n∂feq\n∂n+=−a\n4(m++m−)(m+−m−)3,Thusfeq(n+,τg,τf,h) as function of n+has one ex-\ntremum at\nm++m−= 0 ( x= 1),\nand we have at this point\n∂2feq\n∂n2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nx=1= 8am4\n+χ(τf−τg)1+β\n1+4βn+n−.\nSo it is minimum at τg< τfand maximum at τg> τf.\nThus the states with n+=n−= 1/2 have the lowest\npotential at τg< τfwhile the global minimum is at the\nstates with n+→1,0 atτg> τf.\nHence the first-order transition from the fully disor-\ndered glass to ferro-state takes place at τg=τf. One\ncan see from the above equations that potentials of these\nstates indeed become equal at τg=τf. The phase dia-\ngram of the model is shown in Fig. 1. The topology of\nthe phase stability regions in it is the same that is found\nin the various microscopic models of disorderd magnets,\ncf. Refs. [13], [30-32].\nIn a specific crystal there is some linear relation be-\ntweenτfandτgboth being linear functions of T. It\ndefines a straight line on ( τf,τg) plane which crystal fol-\nlows under Tvariations and, hence, the phase sequence\nproper for a given crystal. For example, if the relation\nτf=τg+τ0holds,τ0being some constant, the crystal\nundergoesjustonesecond-ordertransition-intotheordi-\nnary ferro-phase if τ0<0 or in the glass phase if τ0>0.\nIn the first case there is another qualitative change of\nthermodynamic properties (not a phase transition) - the\nsubsequent appearance of numerous metastable states in\nadditiontoferro-statebeginswhen crystalreachesthe re-\ngions of their stability belonging to the sector (13). Yet\nin these cases the application of field causes no drastic\nchanges in the temperature dependencies of ¯ m,χand\nCapart from the usual differences in their FC and ZFC\nvalues.\nMuch more pronounced field effects occur when the\npath on ( τg,τf) plane intersects the line τg=τf. Then\nglass or ferro-phase can appear as intermediate one in\nsome finite temperature interval neighboring para-phase\nand first-order transition between them takes place at\nτg=τf. Yet there are also different temperature behav-\nior when path crosses the region of coexistence of ferro-\nand glass states (the sector in Fig. 1 bounded by the\ndashed lines) and when it always stays in this region.\nThese four principally different paths are shown in Fig.\n1. Dotted linesin itcorrespondtopaths τf= 2.5τg+1.5a6\nτf/a\nτg/aA\nB glass\nferropara-phase\n0(a) τf/a\nτg/aA1\nB1\nferroglasspara-phase\n0(b)\nFIG. 1: (Color online) Phase diagram of the model for β= 5 (a) and β= 0.1 (b). Thick lines denote the phase transitions\nbetween the phases. Upper dashed lines show the stability bo undaries for the ferro-state (Eq. 4) and lower dashed lines s how\nthat of fully disordered states with n+= 0.5 (Eq. 13). Dotted lines correspond to paths τf= 2.5τg+ 1.5a(A andA1) and\nτf= 0.5(τg−a) (B and B1).\n(A andA1) andτf= 0.5(τg−a) (B and B1) which we\nuse further to present some typical temperature varia-\ntions of thermodynamic variables which differ essentially\non them.\nThe spin-glass phase on paths B and B1is often called\n”reentrant” as crystal returns again on cooling to the\nequilibrium state with zero net magnetization. The pos-\nsible microscopic mechanism of such reentrance is con-\nsidered in Ref. [9].\nWe must note that here the first-order transition does\nnot mean that at τg=τfthe crystal do jump from one\nstate to another on the laboratory time scale. Actually\nit must traverse the rough potential landscape and to\novercome on the way a number of macroscopic barriers\nbetween the localminima(partiallyorderedstates)to ar-\nrive to the global minimum. It takes an infinite time and\nthe standard thermodynamics indeed predicts formally\nwhat will happen after infinite time when system will be\nfully relaxed. But actually one has just finite laboratory\ntime to measure the thermodynamic quantities.\nIt is quite usual situations in all nonergodic systems (i.\ne. those with metastable states) - they all have infinite\n(Arrhenius) relaxation time for the decay of metastable\nstatesintostableoneirrespectiveoftheirnumber-totra-\nverse just one macroscopic barrier also needs an infinite\ntime the same as to traverse many of them. This cir-\ncumstance causes no difficulties in the application of the\nresults of the equilibrium thermodynamics to the known\nnonergodicsystemssuchasordinaryferroelectricandfer-\nromagnets in their ferro-phases. The simple example is\nthe description of hysteresis loop. Here the infinite-time\nthermodynamic prediction that spontaneous magnetiza-\ntionshouldreverseitsdirectionjustwhenthefieldchange\nits sign is easily modified with due account of the long-\nlivingmetastablestates(inwhichfieldandmagnetization\nhave opposite signs). As they can not relax to the global\nminimum at the laboratory time and persist until they\nbecome unstable (at the coercive fields) one just includethem in the description of the response to sufficiently\nslow (quasi-static) field thus obtaining upper and lower\nbranches of hysteresis loop. Apparently, to apply the re-\nsults of static thermodynamic to the dynamic process of\nfield variation we must be sure that during it the sys-\ntem stays close to a local minimum. Hence, in this case\nthe quasi-static condition means that the characteristic\ntime of field variations is greater than the magnetization\nrelaxation time in a local minimum.\nIn the same way we can get the description of quasi-\nstatic responses from the thermodynamic results for the\npresent model. To do this one must assume that quasi-\nstatic regimes can be attained experimentally, i. e. that\nsufficiently slow temperature or field variations can be\nachieved on the laboratory time scale which ensure the\nlocation ofsystem near some minimum during these vari-\nations. Then we just should take into account that once\nthe system entered the metastable state it can not leave\nit in such quasi-static processes until that state becomes\nunstable.\nHence, the validity of this picture of the temperature\nevolution in which crystal is trapped in a local minimum\nand has the obtained above thermodynamic parameters\ncrucially depends on the rate of cooling or heating. It\nshould be sufficiently small to provide the quasi-static\nevolution of the system, that is the characteristic time of\ntemperature variations, t0=/parenleftbigdlnT\ndt/parenrightbig−1, must be greater\nthan the largest magnetic relaxation time in the specific\nminimum. The same is true for characteristic time of\nfield variations. One may expect that when we are not\nclose toTgnearly homogeneouslymagnetized states with\n¯m∼m0would have rather moderate relaxation times\ncomparable to those the crystal has deep in para-phase.\nAlso far from Tgthe magnetic susceptibility of highly\ndisordered states with ¯ m≪m0may be very small mak-\ning the amplitude of magnetization relaxation in them\nsmall too. Then we would observe their nearly static pa-7\nrameters even for t0less than relaxation times which can\nspread up to huge macroscopic values in such states. If\nχreaches large values close to Tgin the states with low\n¯m≪m0their thermodynamic description would become\nuseless due to the strong aging effects in them. Then one\nshould necessarily resort to the dynamic theory to get\nthe adequate account of their (dynamic) properties.\nProvided the above quasi-static conditons are fullfilled\nanother question inherent to such highly nonergodic sys-\ntems arises - to which of numerous stable states the sys-\ntem will go after its present state becomes unstable?\nStrictly speaking, one should again turn to the dynamics\nto answer it, yet, as we will see, there is actually no vast\nchoice of possibilities in the framework of the heuristic\nassumption that system adopts the most smooth varia-\ntions of its average magnetization.\nThus crystal can stay in metastable state for a very\nlarge (Arrhenius) time and leave it only when it becomes\nunstable. Before this the only way to bring the noner-\ngodic crystal to some other state is to apply the external\nfield which could make the present state unstable. This\ncircumstance made former investigators to believe that\nferroelectric relaxors have no phase transitions exhibit-\ning only ”field - induced” ferroelectricity17.\nAll these considerations must be taken into account\nin the interpretation of the temperature dependences of\nmetastable parameters described by the above equations\nand shown in Figs. 2 - 5. Note that to plot them one\ndoes not need to solve the equation of state (6) as it\nprovides along with the expression for a thermodynamic\nvariable the parametric representation of its temperature\ndependence, xbeing the parameter varying in the region\nof stability of a given state.\nThe temperature dependencies of metastable states’\nmagnetizations in Figs. 2 - 5 are bounded by the lines\nat which states become unstable. In the glass phase they\ncorrespond to x= 1/2,2 so their equations follow from\nEqs. 6, 8 at these values of x\n¯m2=\n1+3\n2β±/radicalBigg/parenleftbigg\n1+3\n2β/parenrightbigg2\n+2τg\nβ(τf−τg)\n3\n×τf−τg\n2a.(14)\nHere minus and plus signs are for (Figs. 2, 4) and (Fig.\n3) correspondingly. In the ferro-phase this relation is not\ngenerally valid as some states seize to exist due to the\nferromagnetic instability ( χ→ ∞). In this case it is\nhard to get the compact analog of Eq. (14)\nAs crystal leaves its state only when it becomes unsta-\nble, it is ratherevident that the quasi-staticheating (Fig.\n2) or cooling (Fig. 3) of crystal being in the ferro-state\nbeyond its stability point in the glass phase makes it to\ngo through the succession of metastable states on the\nboundary of their stability regions. So the self-organized\ncriticality shows up in these processes as system is per-\nmanently unstable in finite temperature intervals and itsevolution proceeds via small but macroscopic avalanches\nof spin upturns.\nEqs. (14) describe the temperature dependencies of\nmagnetization in these regimes. In particular, for the\nvanishing of magnetization on heating in Figs. 2, 4 we\nget atτg→0\n¯m=2a\n|τf|/parenleftbigg−τg\n3a+2b/parenrightbigg3/2\n.\nAlso the specific temperature hysteresis occurs in the\ncooling-heating-cooling cycles (Fig. 2) and cooling-\nheating ones (Fig. 3) starting on the paths A and B from\nthehigh-temperaturephases(inthepresenceofinfinitesi-\nmalpositive field tohave ¯ m >0). Thesecyclesareshown\nby the directed lines on the temperature dependencies\nof thermodynamic parameters. Their remarkable prop-\nerty is that they allow to enter a variety of metastable\nstates (actually all in the range 0 .5< n+<1 in the\ncase of intermediate glass phase in Fig. 2 and those with\n0.628< n+<1 having the limited stability regions in\nFig. 3). For example, in the case of Fig. 2 we can stop\nheating at some Twhen crystal traverses the stablility\nboundaries of metastable states and start cooling it thus\ntrapping crystal in the state it was in at the stop.\nNote the sharp rise of the susceptibilities and heat ca-\npacities of metastable states in Figs. 2, 3. Actually this\ndenotesthedivergenceof χandC(cf. Eq.(10))duetothe\nferromagnetic instability metastable states experience in\nthe ferro-phase at τf> τg(τg/a <−1 in Fig. 2 and\nτg/a >−1 in Fig. 3).\nOn the path A1the appearance of intermediate glass\nphase in zero field is marked by the jumps in heat capac-\nity andχtemperature derivative (Fig. 4) while on the\npathB1the intermediate ferro-phase (Fig. 5) has usual\nCurie - Weiss anomaly of χand the usual jump in heat\ncapacity (not shown). In these cases the appearance of\nmetastablestatescannotbeobservedinzero-fieldexperi-\nments - at all Tthe crystal will stay in the state it enters\nfirst on cooling. But the presence of metastable states\ncan be easily revealed in the glass phase. Here stopping\nat some Tand applying for some time the field (above\nsome threshold one needed to make the former state un-\nstable) onecan end up in everymetastablestate choosing\nthe value of the applied field. In the further temperature\nvariations the crystal will stay in the so prepared state\nuntil it vanishes.\nIV. THERMODYNAMICS IN FINITE FIELD\nFurther weexamine the propertiesofmetastable states\nin finite fields. In the glass phase at τg< τfall\nmetastable states always have χ >0 so they have not\nferromagnetic instabilities here. Again, substituting x=\n1/2 andx= 2 in Eqs. (6, 9) we get the parametric\nrepresentation of two lines on (¯ m,h)-plane bounding the\nregion where these states exist (0 < n+<1 being the8\nn+\n1\n0.95\n0.9\n0.85\n0.8\n0.7\n0.62\n0.57\n0.52\n-0.5 -100.20.4\nτg/am\n-1.513\nχ\n-1 0τg/a-1 -0.5τg/a01\nC/Tn+\n1\n0.95\n0.9\n0.85\n0.75\n0.7n+\n0.7\n0.75\n0.85\n0.9\n0.95\n1\nFIG. 2: (Color online) Temperature dependencies of metasta ble states’ magnetization, susceptibility and heat capaci ty in zero\nfield for β= 5 corresponding to path A in Fig.1(a). Dotted lines show the stability boundaries and dashed ones represent the\nparameter’s variations predicted by the equilibrium therm odynamics (those of the states with n+= 1/2 in the present case).\nDirected lines show the evolution of magnetization in real- time quasi-static regime.\n-3 -1χ4\n0n+\n0.8\n0.85\n0.9\n0.95\n0.99\n1\nC/T0.2\n0\n-3 -1n+\n0.8\n0.85\n0.9\n0.95\n0.99\n1n+\n1\n0.95\n0.9\n0.85\n0.8\n0.7\n0.65\n0.6\n0.550.4\n0.2m\n0 -2 -4 τg/a 1 τg/a τg/a00.6\nFIG. 3: (Color online) The same as in Fig. 2 for path B in Fig. 1( a).\nn+\n1\n0.95\n0.875\n0.8\n0.75\n0.7\n0.65\n0.6\n0.55m\nτg/a0\n-3.502.51\n0\n1 0 -1χ\nC/T\nτg/a\nFIG. 4: (Color online) The same as in Fig. 2 for path A1in Fig.1(b). Dash-dotted line corresponds to C/T.χandCare\nshown only for the n+= 1/2 state as others are inaccessible at h= 09\nn+\n1\n0.95\n0.875\n0.8\n0.75\n0.7\n0.65\n0.6\n0.55m\n-3 10\nτg/a1\nFIG. 5: (Color online) Temperature dependencies of\nmetastable states’ magnetization in zero field for β= 0.1\ncorresponding to path B1in Fig. 1(b). Dashed directed line\nshow the decay of nearly homogeneous state’s magnetization\nin the infinitesimal field. The other lines have the same mean-\ning as in Fig.2.\nparameter). Excluding n+we get the equation two real\nroots of which define the couple of these lines\n3a+b/bracketleftBigg\n2+ ¯m/parenleftbigg2a\nh+(τg−τf) ¯m/parenrightbigg1/3/bracketrightBigg\n+τg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2a\nh+(τg−τf) ¯m/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/3\n= 0 (15)\nThese roots have h+ (τg−τf) ¯m >0 andh+\n(τg−τf) ¯m <0 with the interval 0 < n+<1 be-\ning swept when hvaries correspondingly in the intervals\n−hf< h < h e,−he< h < h f\nhf=/radicalbigg\n−τg\n3a+b/parenleftbigg\nτf−τg1+β\n3+β/parenrightbigg\n, (16)\nhe=/radicalbigg\n−τg\n3a+4b/parenleftbigg\nτf−4τg1+β\n3+4β/parenrightbigg\n(17)\nThe magnetization curves ¯ m= ¯m(h,n+) of metastable\nstates defined by Eqs. (6, 8) fill the region bounded by\nthe line segments of Eq. (12) as Figs. 6, 7 show. Again\nthe self-organized criticality appears in these segments\nas system is permanently unstable in them and its evo-\nlution proceeds via the avalanches of spin upturns. Such\navalanches and small magnetization jumps are shown\nto exist in hysteresis loops of Sherrington-Kirkpatrick\nmodel11,12.\nAt|h|> heall metastable states vanish leaving the\nonly stable homogeneous state with magnetization curve\nm0(h) given by Eq. (3). As it was discussed above\nthe crystal will stay in a local minimum until it van-\nishes so Fig. 6 represents possible forms of hysteresis\nloops for quasi-static periodic field. If amplitude of such\nfield|h|> hethe form of quasi-static loop is completely\ndescribed by Eqs. (3, 12). For smaller amplitudes theinner loops inside the outer one will be observed formed\nby the boundaries of Eq. (12) and ¯ m= ¯m(h,n+) and\n¯m= ¯m(h,1−n+) curves with some n+defined by the\nfield amplitude.\nWe can find the analog of the coercive field hcfor the\nglassy loops putting ¯ m= 0 in Eq. (15),\nhc= 2a/parenleftbigg|τg|\n3a+2b/parenrightbigg3/2\nThe inclined hysteresis loop, see Figs. 6(b,c), 7(a),\nis the hallmark of glass phase; such loops are ubiqui-\ntous in random magnets and relaxors. They are found\nin the numerical simulation of spin glass phases of 2 d\nshort-range random-bond Ising model10, Sherrington-\nKirkpatrick model11,12, random anisotropy model13as\nwell as in the local mean-field simulations7,8. Qualita-\ntively the transition at τf=τgmanifest itself by the\nchange of loop form, inclined in the glass phase it be-\ncomes the ordinary one with the vertical sides in the\nferro-phase as disordered metastable states can not be\nreached in the quasi-static periodic field, see Fig. 6(a).\nItisshownthatsuchchangeofhysteresislooptakesplace\nin Sherrington-Kirkpatrickmodel with the growth of fer-\nromagnetic exchange11.\nAlso smaller loops vanish in the ferro-phaseas the field\nevolution of all metastable states shown on Fig. 6(a),\n7(b,c) ultimately ends by the transition into the homo-\ngeneous one. For b= 0 it happens strictly at τf=τg\nwhile for finite b >0 inclined loop transforms into rect-\nangular one already in the glass phase slightly before the\nthermodynamic transition into the ferro-phase. Gener-\nally this change takes place when hcbecomes less than\n−hf, that is at\nτf=/parenleftBigg\n1+β\n3+β+2/radicalBigg\n3+β\n(3+2β)3/parenrightBigg\nτg\nIt is also easy to get for all hthe equilibrium magne-\ntizationmeqcorresponding to the states with the lowest\npotentialhaving x= 1inglassphaseaswasshownabove.\nFrom Eqs. (6), (8) we have\n¯meq=h\nτf−τg(18)\nfor the equilibrium states with\nneq\n+=1\n2/parenleftbigg\n1+h\nhAT/parenrightbigg\nhAT=√−τg(τf−τg)√\na+b\nThe Almeida-Thouless field hATmarks the thermody-\nnamic transition between the completely ordered equi-\nlibrium phase at |h|> hATto the sequence of disordered\nones at|h|< hAT.10\nH -2 21.5\n-1.5/c109(b)\nhe\nH -1 11.5\n-1.5/c109(a)\nhf\nH -3 31.5\n-1.5/c109(c)\nhAT hc\nFIG. 6: (Color online) Field dependencies of metastable sta tes’ magnetizations for β= 0.1 on the path A1,µ≡¯mp\n−a/τg,\nH≡hp\n−a/τ3g, (a) -τg=−1.1a, (b) -τg=−0.75a, (c) -τg=−0.5a\n.\n(b)\nHµ\nhf he(a)\nHµhAT\n-0.15 0.15-0.50.5\n-0.50.5\n-0.2 0.2(c)\nHµ\n-0.60.6\n-0.3 0.3\nFIG. 7: (Color online) The same as in Fig. 6 for β= 5 and path B. (a) - τg=−1.5a, (b) -τg=−0.65a, (c) -τg=−0.5a\nFor the disordered states ¯ meqis represented in the in-\nteriors of hysteresis loops in Figs. (6, 7) by dashed lines\nwith tangents defined by the equilibrium susceptibility\nχeq=∂¯meq\n∂h=1\nτf−τg(19)\nand ending at h=±hAT.\nGenerally χeqis unobservable quantity as to follow the\nrelation in Eq. (18) under the field variations the crystal\nhastoovercomethemacroscopicbarriersinseriesoffirst-\norder phase transitions (stepping in them from the state\nwith some n+to that with n+±N−1\n0) which are not\nshown in Figs. 6, 7 (see Fig.1 in Ref. [26)]. As we\ndiscussed above, this would need a very long time and\ncrystal will stay in the local minimum on the laboratory\ntime scales. So the only susceptibility measurable under\nfield variations is that given by Eq. (9) which defines the\ntangents of the magnetization curves ¯ m= ¯m(h,n+) of\nthe specific metastable state at a given field. It is always\nless than χeq, thus in the equilibrium states with x= 1\nχ−1=τf−τg/bracketleftbigg\n1+2h2\nAT\nh2\nAT+β(h2\nAT−h2)/bracketrightbigg\nwhile at the right ( x= 1/2) and left ( x= 2) boundaries\nof hysteresis loop in glass phase we have\nχ−1=τf−τg/bracketleftbigg\n1+6β\n(3+4n±β)[3+β(1+3n±)]/bracketrightbiggwith plus and minus sign correspondingly.\nYet in the case b≪a(β≪1)χeqcan be determined\njust from the shape of hysteresis loop. Indeed, Eq. (15)\ngives then for its boundaries the almost straight lines\nwith tangent equal to χeq, see Figs. 6(b, c). But for\nlargeβthis relation is lost owing to the intricate form\nof the loop as Figs. 7(a) show. For small βthe shape\nof hysteresis loop in the glass phase is mostly defined\nby the fields heandhf. In the ferro-phase ( τf< τg)\nthe loop boundaries are no longer given by Eq. (15) as\nhere some states vanish due to ferromagnetic instability\nbefore reaching the glassy one at x= 1/2 orx= 2. Also\ninthe ferro-phasethe expressionforthe field hf, Eq. (16)\nchanges to\nhf=−2\n3/radicalBigg\n−τ3\nf\n3(a+b)(20)\nin the interval\n3+β\n3(1+β)τf< τg<3+4β\n12(1+β)τf.\nThe field in Eq. (20) is just the coercive field for the\nhomogeneous ferro-state. Note that at\nτg=3+4β\n12(1+β)τf11\nFIG. 8: (Color online) Temperature dependencies of charac-\nteristic fields; (a) - β= 0.1 on the path A1, dash-dotted line\nrepresents h= 0.2a, (b) -β= 5 and path B. Dashed lines\n-hAT, dotted ones - hc. Solid lines show the modules of hf\nandhe.\nthe field hfandhemerge at this value so at\nτg>3+4β\n12(1+β)τf\ndisordered metastable states no longer exist at all hand\nwe have the ordinary ferromagnetic loop. The temper-\nature dependencies of the loop’s characteristic fields for\npathsA1and B are shown in Fig. 8, in the ferro-phase\nhcandhewere found numerically.\nNow we can turn to the temperature dependencies of\nmetastable states’ parameteres in a constant field. In\ngeneral these states gradually vanish at larger fields, first\nthe most disordered ones, as one may expect. Fig. 9\nillustrates this process for the path A. Temperature de-\npendencies of ¯ m,χandCin constant field are closely\nrelated to that of hfandhe. Thus for paths A and\nA1hffirst grows on cooling and then diminishes after\nreaching the maximum, see Fig. 8(a) for path A1where\nhf,max≈0.28ais attained at τg≈ −0.24a. Alsohe> hf\nexhibits the similar behavior. This results in the essen-\ntial changes in χandCdependencies for field-cooling\nand field-heating in Fig. 10 and, especially, in Fig. 11\nas compared to those in Figs. 2, 4 for h= 0. In the\ninsets of Figs. 10, 11 there are slight anomalies of these\nparameters when crystal enters on cooling the interval\nwhereh < hfand leaves the homogeneous state to join\nthe sequence of glassy ones with n+<1 until it reaches\nthe first state being stable throughout all this interval.It varies for different paths and fields. In Figs. 10, 11\nthey are those with n+= 0.916 and n+= 0.921 corre-\nspondingly. This processis nothing else than the descend\nthroughout the upper branch of the growing hysteresis\nloop. In further cooling the crystal stays in this state un-\ntil it vanishes due to ferromagnetic instability near the\npoint where h≈he(at anotherbranch ofthe loop having\nnow the rectangularform, see Fig. 6(a)) for path A1). So\nχandCdiverge at this point according to Curie-Weiss\nlaw and crystal returns to the ferro-state. This is new\nfeature for path A1which is absent in zero field, Fig. 4.\nOnheating from this state n+againstartsdiminishing at\nthe lowerboundaryof the interval where h < hfreaching\nn+= 0.916(0.921) (upper branch of the loop). Further\nheating in this state ends up at the upper boundary of\nh < hfinterval in the sequence of states with n+growing\nup to 1 (lower branch of the loop). This cause the slight\nhigh-temperature spikesin χandCmanifesting the glass\ninstabilities of these states.\nHere we may note that Fig. 11 explains the origin\nof additional pronounced peak in χappearing in relaxor\nPMN in FC regime18,19. It is the consequence of the\ninstability of nearly homogeneous state the crystal en-\nters on field-cooling. Moreover, the behavior of PMN\ndielectric susceptibility observed in all field-cooling-field-\nheating cycle18,19resembles qualitatively that of χin\nFig. 11. The appearance of additional spikes of\nmagnetic susceptibility in FC regime was also regis-\ntered in random ferromagnets PrNi0.3Co0.7O320and\n(Fe0.17Ni0.83)75P16B6Al321.\nWeshouldalsonotethatmagnetizationsintheregimes\nof cooling from the para-phase shown in Figs. 10(a),\n11(a) are the field-cooled ones, mFC. These figures rep-\nresent also the zero-field-cooled magnetizations, mZFC,\nobtained at turning on the field after cooling the sam-\nple in zero field. They are the metastable curves with\nn+= 1/2 (if they exist) or the stability boundaries cor-\nrespondingto the lowerbranchof inclined hysteresisloop\nor to the homogeneous states lines when the loop is rect-\nangular as in Figs. 10(a). Indeed, in zero-field cooling\nprocess on paths AandA1the system is trapped in the\nn+= 1/2 state, see Figs. 2(a), 4(a), and stays in it af-\nter turning on the field or goes to the lower loop branch\nor to the n+= 1 state depending on the loop form and\nfield strength. So in Fig. 10(a) mZFCcorresponds to the\nlowerstabilitylinewhile in Fig. 11(a) it follows n+= 1/2\ncurve and then jump to n+= 1 curve when former seized\nto exist.\nHereweshouldnotethatofteninexperimentsandsim-\nulations another definition of mZFCis used, namely, the\nmagnetization registered in field-heating after zero-field-\ncoolingprocess. AsFig. 10(a)showsthisistheill-defined\nquantity as it will depend in this case on the temperature\natwhichtheturnfromcoolingtoheatingtakesplace. Yet\nwhen the field at the turning point leaves the n+= 1/2\nstate stable as for path A1in Fig. 11(a) the field-heating\nafter zero-field-cooling protocol gives a unique mZFCthe\nsame as in simple ZFC process described above. The de-12\n-1.4 0\n/c116g/a-0.7 -1.4 -0.2\n/c116g/a-0.80.5\n-0.5m0.5\n-0.5m(a) (b)\nFIG. 9: (Color online) Temperature dependencies of metasta ble states’ magnetizations for β= 5 corresponding to the path A\nin the fields h= 0.03a- (a) and h= 0.07a- (b).\nτg/a0.6\n0\n0 -0.5C/T\nn+\n1\n0.985\n0.967\n0.95\n0.935\n0.916-0.03 -0.120.11\n0.02\nτg/an+\n0.916\n0.935\n0.95\n0.967\n0.985\n1\n0 -0.50.41.6\nχ0.71\n0.65\n-0.1 -0.040.4\n0.2m\n-0.9 0 τg/an+\n1\n0.98\n0.95\n0.916\n0.85\n0.750.102\n0.09\n-0.13 -0.04\nFIG. 10: (Color online) Temperature dependencies of metast able states’ magnetizations, susceptibility and heat capa city in a\nfieldh= 0.13aforβ= 5 corresponding to the path A. Insets show the high-tempera ture anomalies.\ntailed picture of mZFCandmFCbehavior for path A1\nis shown in Fig.12. The qualitativly similar temperature\ndependences of these magnetizations are found in local\nmean-field simulations7and in field-heating after zero-\nfield-cooling experiments in PMN18.\nThe applied field cause also some peculiarities in tem-\nperature dependencies of ¯ m,χandCwhen ferro-phase\nis intermediate between para- and glassy ones (path B).\nAs Fig. 13 shows they appear in field-heating regime of\ncrystal being initially in the most disordered state with\nn+= 1/2. After reaching the limit of stability of this\nstate the crystal enters the sequence of states with larger\nn+on the boundaries of their existence regions until it\njoins the fully homogeneous ferro-state with n+= 1.\nThis process is accompanied by the spikes in χandC.\nAlong with it the reverse process of the demagnetizationof almost homogeneous initial state on heating is possi-\nble in the infinitesimal field. It is shown in Fig. 5 by the\ndirected dashed lines. The presence of some small field\nhere is necessary to remove the ferromagnetic instability\nof disorderedstates, without the field the inhomogeneous\nstates will jump right to the ferro-state due to this insta-\nbility.\nTheexhaustivestudyofboththeseprocesseswasmade\nin the graphite-intercalated magnet Cu0.93Co0.07Cl2\nhaving the phase sequence of path B22. The similarity of\nthe experimental data to ¯ mshown in Figs. 13, 5 is quite\nimpressive as authors has revealed the existence of nu-\nmerous metastable states using multiple cooling-heating\ncycles, see the example in Fig. 13. Also in Ref.[23]\n”thermal remagnetization” similar to that in Fig. 13\nwas observed in disordered ferromagnets Nd3(Fe,Ti)2913\nC/T\n-1.51.5\nm\n-2 0 τg/an+\n1\n0.9\n0.75\n0.65\n0.5\n0.4\n0.3\n0.2\n0.1\n0211.2n+\n1\n0.985\n0.967\n0.95\n0.935\n0.921\nχ\n0.1 -1.20.3 0.650.74\n-0.1 0τg/a-1.2τg/a0.103.5\nn+\n1\n0.985\n0.967\n0.95\n0.935\n0.9210 -0.100.55\nFIG. 11: (Color online) Temperature dependencies of metast able states’ magnetizations, susceptibility and heat capa city in a\nfieldh= 0.2aforβ= 0.1 corresponding to the path A1. Insets show the high-temperature anomalies.\nτg/a1.2\n0\n-1.5 0.5mn+\n1\n0.9\n0.8\n0.6\n0.5\n0.4FC\nZFC(A)\nZFCFC(B)\n-1.50.5 τg/a1.2\n0mn+\n1\n0.9\n0.8\n0.6\n0.5\n0.4\nFIG. 12: (Color online) Temperature dependencies of mZFCandmFCon path A1in fieldh= 0.3a(A)andh= 0.35a(B).\n-1.52\n-6 40m\nτg/a02.5\n-2.6 -1.4n+\n1\n0.9\n0.8\n0.65\n0.5\n0.4\n0.3\n0.2\n0.1\n0n+\n0.5\n0.6\n0.7\n0.8\n0.95\n1χ\nτg/a -2.6 -1.4 τg/a2.8\n0C/Tn+\n0.5\n0.6\n0.7\n0.8\n0.95\n1\nFIG. 13: (Color online) Temperature dependencies of metast able states’ magnetizations, susceptibility and heat capa city in a\nfieldh=aforβ= 0.1 corresponding to the path B.14\nandNd3(Fe,Re)29. The depolarization and polariza-\ntion on heating were also seen in PLZT ferroelectric\nceramics24,25.\nAt last we should mention the example of the relaxor\nthe thermodynamics of which is definitely distinct from\nthat of the present model. It is PMN-PT with lead ti-\ntanate (PT) content between 0.06 and 0.2 in which the\ntemperature evolution of hysteresis loop shows the grad-\nual transition from the inclined to the rectangularform33\ninstead of the sharp one as in Fig. 6. This means that\nbetween glass and ferro-phase in this compound there\nare the sequence of mixed phases where the most stable\nstates are those with partially ordered dipole moments.\nProbably the model with the three- and four - mode in-\nteractions can describe such intermediate mixed phases.\nV. DISCUSSION AND CONCLUSIONS\nThus the present thermodynamic theory can describe\nin a unified manner a wealth of nonergodic phenomena\nin disordered ferromagnets and ferroelectrics. It demon-\nstrates what the effects of numerous metastable states\ncan actually be and how they can manifest themselves\nin various regimes of real-time quasi-static experiments.\nAlso it shows what a form the results of rigorous statis-\ntical mechanics should have to describe the phase tran-\nsitions in crystals with random interactions.\nThe important conclusion can be made on the rela-\ntions between the predictions of eqiulibrium thermody-\nnamics valid in the infinite-time limit and finite-time ex-\nperiments. Contrary to naive expectations that there are\nno such relations the present study shows that they canberevieledinfinite-fieldquasi-staticexperiments. Inpar-\nticular, they are straightfordly manifested in the change\nof hysteresis loop form closely related to the form of the\nstatic thermodynamic potential. Yet its form and ther-\nmodynamic path for a given crystal can be obtained in\nsome other sufficiently full set of finite-field quasi-static\nexperiments. ’Sufficiently full’ means here that a wide\nspectrum of inhomogeneous metastable states should be\nexplored to determine the regions of their existence and\ntheir thermodynamic properties. The exellent examples\nof such studies providing the valuable data for the future\ntheory are given in Refs. [18, 22].\nApparantly the present phenomenology can be im-\nproved and expanded in many ways. Thus one may\nconsider a multicomponent order parameter as, say, in\nHeisenberg ferromagnet or cubic ferroelectric and less\nsparse modes having three- and four-mode couplings in\nLandau’s potential. Also one may include in it higher-\norder terms in mito expand the results to larger Tand\nhregions. Eventually it may be worth to explore the\nmodels with the broken in some way permutation sym-\nmetry. It seems also important to consider the role of\nfluctuations (i. e. the modes which do not condense)\nin the present approach. The quantitative description of\nthe nonergodic thermodynamics of PMN - type relaxors\nneeds also the consideration of random-fields.\nAcknowledgments\nI gratefully acknowledge useful discussions with M.P.\nIvliev, I.P. Raevski, V.B. Shirokov, V.I. Torgashev, E.D.\nGutlianskii, S.A. Prosandeev, V.P. Sakhnenko.\n∗Electronic address: timonin@aaanet.ru\n1K. Binder and A. P. Young , Rev. Mod. Phys. 58, 801\n(1986).\n2S. F. Edwards and P. W. Anderson , J. Phys. F 5, 965\n(1975).\n3D. R. Denholm and T. J. Sluckin , Phys. Rev. B 48,\n901 (1993).\n4S. Khmelevskyi , J. Kudrnovsky , B. L. Gyorffy , P.\nMohn , V. Drchal and P. Weinberger , Phys. Rev. B\n70, 224432 (2004).\n5M. J. Stephen and A. Aharony , J. Phys. C 14, 1665\n(1981).\n6P. N. Timonin , JETP 89, 525 (1999).\n7C. M. Soukoulis , K. Levin , and G. S. Grest , Phys.\nRev. B 28, 1495 (1983).\n8C. M. Soukoulis , G. S. Grest , and K. Levin , Phys.\nRev. B 28, 1510 (1983).\n9W. M. Saslow , and G. Parker , Phys. Rev. Lett. 56,\n1074 (1986).\n10H. G. Katzgraber , F. Pazmandi , C. R. Pike , Kai Liu\n, R. T. Scalettar , K. L. Verosub , and G. T. Zimanyi ,\nPhys. Rev. Lett. 89, 257202 (2002).\n11F. Pazmandi , G Zarand , and G. T. Zimanyi , Phys.Rev. Lett. 83, 1034 (1999).\n12V. Banerjee , S. Das and S. Puri , Phys. Rev. E 71,\n026105 (2005).\n13E. Vives and A. Planes , Phys. Rev. B 63, 134431\n(2001).\n14B. Yucesoy and A. N. Berker , Phys. Rev. B 76, 014417\n(2007).\n15H. Kaya and A. N. Berker , Phys. Rev. E 62, R1469\n(2000).\n16A. Kabakcioglu , A. N. Berker , and M. C. Yalabik\nPhys. Rev. E 49, 2680 (1994).\n17G. A. Smolenskii, Sov. Phys. Sol. St. 23, 784 (1981).\n18Z.- G. Ye and H. Schmid, Ferroelectrics 145, 83 (1993).\n19R. Sommer , N. K. Yushin and J. J. van der Klink ,\nPhys. Rev. B 48, 13230 (1993).\n20M T. Escote , C. H. Westphal and R. F. Jardim , J.\nAppl. Phys. 87, 5908 (2000).\n21Y. Yeshurun , M. B. Salamon , K. V. Rao and\nH. S. Chen , Phys. Rev. B 24, 1536 (1981).\n22I. S. Suzuki and M. Suzuki , Phys. Rev. B 73, 094448\n(2006).\n23S. J. Collocott , J. B. Dunlop and P. A. Watterson ,\nPhys. Rev. B 76, 054434 (2007).15\n24D. Viehland , J. F. Li , S. J. Jang , L.E. Cross and\nM. Wuttig , Phys. Rev. B 46, 8013 (1992).\n25Z. Kutnjak , C. Filipic and A. Levstik , J. Eur. Ceram.\nSoc.21, 1313 (2001).\n26P. N. Timonin, JETP 92, 1038 (2001); cond-mat/0004168.\n27M. Janssen , Int. J. Mod. Phys. B 8, 943 (1994); Phys.\nRep.295, 1 (1998); F. Evers and A. D. Mirlin , cond-\nmat/0707.4378.\n28R. B. Griffiths , Phys. Rev. Lett. 23, 17 (1969).\n29A. J. Bray , Phys. Rev. Lett. 59, 586 (1987).30J. A. Hertz, D. Sherrington and Th. M. Nieuwenhuizen,\nPhys. Rev. E 60, R2460 (1999).\n31D. H. Kim , G. J. Rodgers, B. Kahng, and D. Kim ,\nPhys. Rev. E 71, 056115 (2005).\n32S. Niidera , S. Abiko and F. Matsubara , Phys. Rev. B\n72, 214402 (2005).\n33S. M. Emelyanov , F. I. Savenko , Yu. A. Trusov ,\nV. I. Torgashev and P. N. Timonin , Phase Transitions\n45, 251 (1993)."    },    {        "title": "1907.12306v1.Magnetization_Dynamics_in_Holographic_Ferromagnets__Landau_Lifshitz_Equation_from_Yang_Mills_Fields.pdf",        "content": "Magnetization Dynamics in Holographic\nFerromagnets:\nLandau-Lifshitz Equation from Yang-Mills Fields\nNaoto Yokoi1;2\u0003, Koji Sato2y, and Eiji Saitoh1;2;3;4;5z\n1Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n4Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\nAbstract\nWe introduce a new approach to understand magnetization dynamics in ferro-\nmagnets based on the holographic realization of ferromagnets. A Landau-Lifshitz\nequation describing the magnetization dynamics is derived from a Yang-Mills\nequation in the dual gravitational theory, and temperature dependences of the\nspin-wave sti\u000bness and spin transfer torque appearing in the holographic Landau-\nLifshitz equation are investigated by the holographic approach. The results are\nconsistent with the known properties of magnetization dynamics in ferromagnets\nwith conduction electrons.\n\u0003yokoi@spin.t.u-tokyo.ac.jp\nykoji.sato.c5@tohoku.ac.jp\nzeizi@ap.t.u-tokyo.ac.jparXiv:1907.12306v1  [hep-th]  29 Jul 20191 Introduction\nThe Landau-Lifshitz equation [1] is the fundamental equation for describing the dynamics\nof magnetization (density of magnetic moments) in various magnetic materials. It has been\nalso playing a fundamental role in the development of modern spintronics [2]: For instance,\nits extension to the coupled systems of localized magnetic moments and conduction electrons\nhas led to the concepts of spin transfer torque [3, 4] and spin pumping [5]. So far, the\nsymmetries and reciprocity in electronic systems have been the guiding principles to develop\nsuch extensions. In this article, we introduce another guiding principle to explore the new\nextensions and magnetization dynamics on the basis of the holographic duality.\nThe holographic duality is the duality between the quantum many body system de\fned\nind-dimensional space-time and the gravitational theory (with some matter \felds) which\nlives in (d+1)-dimensional space-time [6, 7, 8].1We constructed a holographic dual model of\nthree-dimensional ferromagnetic systems, which exhibits the ferromagnetic phase transition\nwith spontaneous magnetization and the consistent magnetic properties at low temperatures\n[10].2In the holographic duality, \fnite temperature e\u000bect in ferromagnetic systems can\nbe incorporated as the geometrical e\u000bect of black holes in higher dimensional bulk gravity,\nand the Wick rotation at \fnite temperatures is not required for the analysis in the dual\ngravitational theory. Thus, the novel analysis for real-time dynamics of quantum many\nbody systems in nonequilibrium situations can be performed using the holographic approach\n(for a review, see [14]). In addition, the holographic duality is known to be a strong-weak\nduality, which relates strongly correlated quantum systems to classical gravitational theories.\nFrom these viewpoints, the holographic approach can provide new useful tools to analyze\nnonequilibrium and nonlinear dynamics of magnetization in ferromagnets.\nIn ferromagnets, spin currents are generated by magnetization dynamics. From the holo-\ngraphic dictionary between the quantities of ferromagnets and gravitational theory [10], the\nspin currents in ferromagnets correspond to the SU(2) gauge \felds in the dual gravitational\ntheory. This correspondence indicates that the dynamics of spin currents, consequently the\ndynamics of magnetization, can be described by the Yang-Mills equation for SU(2) gauge\n\felds [15] in the holographic dual theory. In the following, we derive a Landau-Lifshitz\nequation for magnetization dynamics from the Yang-Mills equation within the holographic\nrealization of ferromagnets. This derivation can provide novel perspectives for magnetization\ndynamics from the non-abelian gauge theory.\n1See [9] for a recent review on the applications of the holographic duality to condensed matter physics.\n2Other holographic approaches to ferromagnetic systems have been also discussed in [11, 12, 13].\n2This article is organized as follows. In Section 2, we summarize the results of the mag-\nnetic properties obtained from the holographic realization of ferromagnets in thermodynamic\nequilibrium. An extension to nonequilibrium situation including the \ructuations of magne-\ntization and spin currents is discussed in the dual gravitational theory, and the holographic\nequation of magnetization dynamics is derived in Section 3. In Section 4, temperature depen-\ndences of the parameters in the resulting holographic equation are investigated by numerical\ncalculations. Finally, we summarize the results in Section 5.\n2 Holographic Dual Model of Ferromagnets\nWe begin with a brief summary on the holographic dual model of ferromagnets [10]. The\ndual model is the \fve-dimensional gravitational theory with an SU(2) gauge \feld Aa\nMand a\nU(1) gauge \feld BM, whose action is given by\nS=Zp\u0000gd5x\u00141\n2\u00142(R\u00002\u0003)\u00001\n4e2GMNGMN\u00001\n4g2Fa\nMNFaMN\n\u00001\n2(DM\u001ea)2\u0000V(j\u001ej)\u0015\n: (1)\nHere,Ris the scalar curvature of space-time, and the \feld strength is de\fned by Fa\nMN=\n@MAa\nN\u0000@NAa\nM+\u000fabcAb\nMAc\nNandGMN=@MBN\u0000@NBM, respectively. The index alabels\nspin directions in the SU(2) space (a= 1\u00183), the index Mlabels space-time directions in\n\fve dimensions ( M;N = 0\u00184), and\u000fabcis a totally anti-symmetric tensor with \u000f123= 1.\nThe model also includes a triplet scalar \feld \u001eawith the covariant derivative DM\u001ea=\n@M\u001ea+\u000fabcAb\nM\u001ec, and theSU(2)-invariant scalar potential V(j\u001ej) with the norm j\u001ej2=\nP3\na=1(\u001ea)2. Note that the scalar \feld is neutral under the U(1) gauge transformation. In\norder to guarantee asymptotic Anti-de Sitter (AdS) backgrounds, the negative cosmological\nconstant \u0003 =\u00006=`2is introduced. The \feld-operator correspondence in the holographic\nduality [7, 8] leads to the following holographic dictionary between the \felds of the dual\ngravitational theory and the physical quantities of ferromagnets:\nDual gravity Ferromagnet\nScalar \feld \u001ea() Magnetization Ma\nSU(2) gauge \feld Aa\nM() Spin current Ja\ns\u0016\nU(1) gauge \feld BM() Charge current J\u0016\nMetricgMN() Stress tensor T\u0016\u0017\nTable 1: Holographic dictionary between the dual gravitational theory and ferromagnets.\n32.1 Black Hole as Heat Bath\nIn order to establish the holographic dictionary, thermodynamical properties of the physical\nquantities of ferromagnets should be calculated in the dual gravitational theory. In Ref. [10],\nthe temperature dependences of magnetic quantities and the behavior of ferromagnetic phase\ntransition are thoroughly discussed. In the context of the holographic duality, \fnite tem-\nperature e\u000bects in the ferromagnets can be incorporated by introducing the black holes into\nthe dual gravitational theory as the background space-time. Indeed, the dual gravitational\ntheory has the charged black hole solution which is a solution to the Einstein, Yang-Mills,\nand Maxwell equations derived from the action (1):\nRMN+\u0012\n\u0003\u00001\n2R\u0013\ngMN=\u00142\n2e2\u0012\n2GKMGKN\u00001\n2GKLGKLgMN\u0013\n+\u00142\n2g2\u0012\n2Fa\nKMFaKN\u00001\n2FaKLFaKLgMN\u0013\n; (2)\nrMFaMN+\u000fabcAb\nMFcMN= 0;rMGMN= 0; (3)\nwhererMis the covariant derivative for the a\u000ene connection, and the space-time indices\nM;N are raised or lowered by the bulk metric gMN. Here, we neglect the contribution from\nthe scalar \feld and set \u001ea= 0 for the background. The metric of the black hole3is given by\nds2=gMNdxMdxN=r2\n`2\u0000\n\u0000f(r)dt2+dx2+dy2+dz2\u0001\n+`2\nf(r)dr2\nr2; (4)\nwith the radial function,\nf(r) = 1\u0000(1 +Q2)\u0010rH\nr\u00114\n+Q2\u0010rH\nr\u00116\n: (5)\nHere, we de\fne the parameter Q:\nQ2=2\u00142\n3\u0012\u00162\ne\ne2+\u00162\ns\ng2\u0013\n: (6)\nTheU(1) charge\u0016eandSU(2) charge\u0016sof the black hole are supported by the time com-\nponents of the gauge \felds,\nB0=\u0016e\u0010rH\n`\u0011\u0012\n1\u0000r2\nH\nr2\u0013\nandA3\n0=\u0016s\u0010rH\n`\u0011\u0012\n1\u0000r2\nH\nr2\u0013\n: (7)\nNote that the black hole solution (4) is asymptotically AdS at r!1 , and has the (outer)\nhorizonr=rH.\n3This type of non-abelian black holes has been discussed in the context of the holographic duality, in the\nliterature such as [11, 16].\n4For the following discussion, we make a coordinate change of the radial coordinate rinto\nubyu= 1=r, and the black hole metric becomes\nds2=1\nu2\u0012\n\u0000f(u)dt2+dx2+dy2+dz2+du2\nf(u)\u0013\n; (8)\nand the transformed function f(u) is given by\nf(u) = 1\u0000\u0000\n1 +Q2\u0001\nu4+Q2u6; (9)\nwhere we have set the coupling parameters e=g= 1 and the black hole parameters rH=\n`= 1, for simplicity.\nIn the holographic dual model, the black hole (8) plays the role of the heat bath; due to\nthe Hawking radiation, the black hole temperature is given by\nT=2\u0000Q2\n2\u0019; (10)\nand the calculations on the black hole background lead to the thermodynamical properties\nof the corresponding ferromagnet. Since we focus only on the dynamics of magnetization\nand spin current, the background space-time is \fxed to be the black hole metric (8) in the\nfollowing.\n2.2 Thermodynamics of Ferromagnets from Scalar Dynamics on Charged\nBlack Hole\nIn order to investigate the thermodynamics of magnetization, we examine the equation of\nmotion for the scalar \feld \u001ea, which is also derived from the action (1):\n1p\u0000g@M\u0000p\u0000gDM\u001ea\u0001\n+\"abcAb\nMDM\u001ec=@V\n@\u001ea: (11)\nHere, we consider a static and homogeneous solution in the boundary coordinates, x\u0016=\n(t;x1;x2;x3), which corresponds to the homogeneous magnetization in ferromagnets. With-\nout loss of generality, the ansatz for such a scalar \feld, which is invariant under the transla-\ntions on the boundary, is given by\n\u001e1=\u001e2= 0; \u001e3= \b(u)6= 0: (12)\nInserting this ansatz, the metric (8), and the gauge \felds (7) into the equation (11), we obtain\nthe following equation for \b( u):\nu2f(u)d2\b\ndu2+\u0012\nu2df(u)\ndu\u00003uf(u)\u0013d\b\ndu=@V\n@\b: (13)\n5This equation governs the thermodynamics of magnetization in the dual gravitational theory.\nWe can analyze the solution to this equation numerically with a simple quartic potential\nV(j\u001ej) =\u0015\u0000\nj\u001ej2\u0000m2=\u0015\u00012=4, and the asymptotic behavior of the numerical solution near the\nboundaryu\u00180 (orr\u00181) is obtained:\n\b(u)'H0u\u0001\u0000+M(T)u\u0001+\u0010\n\u0001\u0006= 2\u0006p\n4\u0000m2\u0011\n: (14)\nAccording to the standard recipe in the holographic duality [17, 18], the coe\u000ecients H0and\nM(T) in the asymptotic expansion correspond to an external magnetic \feld and a magne-\ntization at temperature T(underH0), respectively. In Ref. [10], the resulting temperature\ndependences of magnetization, magnetic susceptibility, and speci\fc heat have been shown\nto reproduce the ferromagnetic phase transition in the mean \feld theory. Furthermore, the\ntemperature dependences at low temperatures are also consistent with the existence of the\nspin wave excitations (magnons) and conduction electrons in low-temperature ferromagnets.\nFor later convenience, we also comment on the solutions for the gauge \felds. Assuming\nthe translational and rotational invariance on the boundary, equilibrium solutions for the\ngauge \felds are given by the following form:\nB0=b(u) andA3\n0=a3(u); (15)\nwhere all the other components vanish. Inserting this ansatz and (12), the Maxwell and\nYang-Mills equations on the black hole are reduced to the following simple forms:\nd\ndu\u00121\nudb\ndu\u0013\n= 0 andd\ndu\u00121\nuda3\ndu\u0013\n= 0: (16)\nThe general solutions are given by the forms (7) in terms of u,\nb(u) =\u0016e\u0000\n1\u0000u2\u0001\nanda3(u) =\u0016s\u0000\n1\u0000u2\u0001\n: (17)\nHere, we impose the boundary conditions B0= 0 andA3\n0= 0 at the horizon ( u= 1), which\nguarantee the regularity of the gauge \felds on the horizon. The remaining integral constants,\n\u0016eand\u0016s, correspond respectively to the electrochemical potential of underlying electrons\nand the spin chemical potential (or spin voltage), through the holographic dictionary.\nTo summarize, the solutions (14) and (17) on the charged black hole describe the ther-\nmodynamical property of the holographic dual ferromagnets in the equilibrium.\n3 Magnetization Dynamics in Holographic Ferromagnets\nIn this section, we extend the holographic analysis in the equilibrium, summarized in the\nprevious section, to more general situations including the dynamics of magnetization and\n6spin currents. In order to discuss the dynamics of magnetization and spin currents, the\nstatic and homogeneous ansatze for the scalar \feld (12) and the gauge \felds (15) need to be\ngeneralized. Here, we focus on the dynamics with the long wave length in the ordered phase\n(symmetry broken phase) below the Curie temperature, where various phenomena in modern\nspintronics are intensively studied.\n3.1 Generalized Ansatz and E\u000bective Equations of Motion\nFor the scalar \feld, following the standard derivation of the equation for magnetization\ndynamics, we consider the generalized ansatz for the scalar \feld as a factorized form:\n\u001ea(u;t;x ) = \b(u)na(t;x) with3X\na=1nana= 1; (18)\nwhere \b(u) is a solution of the equation (13) with the asymtotic behavior (14). Note that,\nsince we focus only on the dynamics of spontaneous magnetization, we \fx H0= 0 throughout\nthis article. In this ansatz, na(t;x) corresponds to the (local) direction of magnetization in\nferromagnets.\nIn ferromagnetic systems, the magnetization dynamics generates various dynamics of spin\ncurrents [2]. In the holographic dual theory, the scalar dynamics is also expected to induce\nthe dynamics of the corresponding SU(2) gauge \feld, and thus we generalize the static and\nhomogeneous ansatz for the SU(2) gauge \felds to the following factorized forms:\nAk\n0(u;t;x ) = (1\u0000u2)ak\n0(t;x);\nA?\n0(u;t;x ) = (1\u0000u2)a?\n0(t;x);\nAk\ni(u;t;x ) =Gk(u)ak\ni(t;x);\nA?\ni(u;t;x ) =G?(u)a?\ni(t;x) (i= 1\u00183); (19)\nwhere we set the radial component Aa\nu\u00110 by using the gauge degrees of freedom. Due to the\nnontrivial scalar solution \b( u), corresponding to the spontaneous magnetization, the SU(2)\ngauge symmetry is broken to U(1). The gauge \felds can be correspondingly decomposed into\nan unbroken component Ak\n\u0016and two broken components A?\n\u0016, which are de\fned by Ak\n\u0016/na\nandn\u0001A?\n\u0016= 0, respectively. As in the case of the static solutions, the time components\nof gauge \felds should satisfy the horizon boundary condition, Aa\n0= 0 atu= 1, for the\nregularity. Although the spatial components Aa\niare not required to vanish on the horizon,\nthe regularity (or \fniteness) at the horizon is required. The asymptotic solutions to the\nlinearized Yang-Mills equation near the boundary ( u\u00180) give the asymptotic expansions for\n7the radial functions Gk(u), andG?(u),\nGk(u) = 1\u0000\u001bk\nsu2+O(u4);\nG?(u) = 1 +\u001b?\nsu2+O(u4): (20)\nWe discuss the concrete numerical solutions of Ga(u) and their physical implications in the\nnext section.\nSince the scalar \feld \u001eadoes not have the U(1) charge, the \ructuation (or dynamics) of\n\u001eadoes not induce further dynamics for the U(1) gauge \feld, which implies the solution for\nB\u0016in (7) is unchanged, and we can neglect the dynamics of B\u0016.\nAt \frst, we consider the equation of motion for the scalar \feld \u001ea. Inserting the generalized\nansatz (18) into the equation (11), we obtain the following equation for na:\n\u0014\nu5@u\u0000\nu\u00003f(u)@u\b\u0001\n\u0000@V\n@\b\u0015\nna=\u0014u2\nf(u)DtDtna\u0000u2DiDina\u0015\n\b: (21)\nHere, we have used the gauge condition Aa\nu= 0, and the gauge covariant derivative is de\fned\nasD\u0016na=@\u0016na+\"abcAb\n\u0016nc. The left-hand side of the equation (21) is proportional to the\nequation (13), and thus vanishes for the solution \b( u). Since \b( u) is a non-trivial solution,\nwhich is not identically zero, we have the e\u000bective equation of motion for na:\nf\u00001DtDtna\u0000DiDina= 0: (22)\nNext, the equation of motion for the gauge \felds is considered. The Yang-Mills equation\nfor theSU(2) gauge \feld Aa\nMis derived by the variation of the holographic action (1) and\ngiven by\n1p\u0000g@N\u0000p\u0000gFNMa\u0001\n+\u000fabcAb\nNFNMc=JMa; (23)\nwhere theSU(2) current is de\fned as\nJa\nM=\"abc\u001ebDM\u001ec=\"abc\u001eb\u0010\n@M\u001ec+\"cdeAd\nM\u001ee\u0011\n: (24)\nUnlike the static case, the generalized ansatz (18) and (19) give the non-vanishing currents:\nJa\n\u0016= \b2\u0010\n\"abcnb@\u0016nc+\"abc\"cdenbad\n\u0016ne+O\u0000\nu2\u0001\u0011\n: (25)\nNote that the radial component of the currents still vanishes, Ja\nu\u00110, due to the gauge \fxing\nconditionAa\nu\u00110. With this current, we can explicitly write down the Yang-Mills equations\non the charged black hole (8), in the boundary direction :\nJa\n0=u3f@u\u0000\nu\u00001Fa\nu0\u0001\n+u2(DiFa\ni0); (26)\nJa\ni=u3@u\u0000\nu\u00001fFa\nui\u0001\n\u0000u2f\u00001(D0Fa\n0i) +u2\u0000\nDjFa\nji\u0001\n; (27)\n8where the gauge covariant derivative for the \feld strength is de\fned as D\u0016Fa\n\u0017\u001a=@\u0016Fa\n\u0017\u001a+\n\"abcAb\n\u0016Fc\n\u0017\u001a. Inserting the ansatz (19), the Yang-Mills equations give the equations for naand\naa\n\u0016. In summary, using the generalized ansatze, we have obtained the coupled equations of\nmotion for naandaa\n\u0016, (22), (26), and (27).\n3.2 Landau-Lifshitz Equation from Yang-Mills Equation\nSince it is di\u000ecult to \fnd the general solutions for the coupled non-linear partial di\u000berential\nequations, we seek simple trial solutions for naandaa\n\u0016to obtain the e\u000bective equations of\nmotion. At \frst, instead of looking for general solutions to the equation (22), we consider\nthe solutions to the simpler equations:\nDtna= 0 and Dina= 0; (28)\nwhich are explicitly given by\n@\u0016na+\u000fabcab\n\u0016nc= 0 +O(u2): (29)\nThese equations lead to the ground state solutions for the e\u000bective Hamiltonian for na:\nHe\u000b=f\n2(\u0019a)2+1\n2(Dina)2; (30)\nwhere the conjugate momentum is de\fned by \u0019a=f\u00001Dtna. In this article, we wish to\ndiscuss the dynamics of magnetization and spin currents in the boundary ferromagnetic\nsystem, which is given by the leading terms in the asymptotic expansions at u\u00180. Hence,\nthe higher order terms in the expansion with respect to uare irrelevant, and we neglect them\nin the following. Dropping the O(u2) term, we can easily obtain the solution to (29) for aa\n\u0016\nin terms of na,\naa\n\u0016=C\u0016na\u0000\"abcnb@\u0016nc; (31)\nwhere we have introduced a vector \feld C\u0016which is arbitrary at this stage. This solution\ndemonstrates the clear separation of the gauge \felds:\nak\n\u0016=C\u0016naanda?\n\u0016=\u0000\"abcnb@\u0016nc: (32)\nThe relation for the broken components, a?\n\u0016, is nothing but a non-abelian analogue of the\nrelation between the gauge \feld and the quantum phase of Cooper pair, A\u0016=@\u0016\u0012, in super-\nconductivity, and also corresponds to the Maurer-Cartan one-form of G=H\u0018SU(2)=U(1)\nin terms of the Nambu-Goldstone modes na[19, 20]. Requiring the matching condition to\n9the static solution (17), aa\n0=\u0016s\u000ea3andaa\ni= 0 forna= (0;0;1), the vector \feld C\u0016should\nsatisfy the condition:\nC0=\u0016sandCi= 0; (33)\nin the static and homogeneous limit. Note that the relation (31) and the ansatz (18) do not\ninduce new contributions of the scalar \felds to the energy-momentum tensor TMNin the\nEinstein equations, and consequently the analysis in the probe approximation remain intact.\nNext, we consider the e\u000bective Yang-Mills equations, (26) and (27). It is not di\u000ecult to\nshow that the relation (31) leads to vanishing currents Ja\n\u0016up toO(u2), using the explicit\nform (25). Furthermore, the ansatz for gauge \felds (19) with Aa\nu= 0 implies\n@u\u0000\nu\u00001Fa\nu0\u0001\n= 0;and@u\u0000\nu\u00001fFa\nui\u0001\n= 0 +O(u4): (34)\nDropping the higher order terms such as O(u4), the remaining Yang-Mills equations reduce\nto\nDiFa\ni0= 0;andD0Fa\n0i+DjFa\nji= 0: (35)\nFrom the viewpoint of the boundary theory (on the ferromagnet side), the \frst equation\ncorresponds to a non-abelian version of Gauss's law, and the second corresponds to a non-\nabelian version of Ampere's law without source and currents, for the spin gauge \felds [21].\nUsing the relation (31), we obtain the SU(2) \feld strength,4\nFa\n\u0016\u0017=nah\n(@\u0016C\u0017\u0000@\u0017C\u0016)\u0000\"bcdnb@\u0016nc@\u0017ndi\n\u0011naf\u0016\u0017: (36)\nNote that a component of the \feld strength, f\u0016\u0017, parallel to the magnetization naonly\nremains. With the \feld strength (36), the e\u000bective Yang-Mills equations (35) and the Bianchi\nidentity for the SU(2) gauge \feld are reduced to the following equations:\n@\u0016f\u0016\u0017= 0 and \u000f\u0016\u0017\u001a\u001b@\u0017f\u001a\u001b= 0: (37)\nThe above equations are the same form as the Maxwell equations, and the terms depending\nonnain the gauge \feld f\u0016\u0017actually corresponds to the so-called spin electromagnetic \feld\ndiscussed in the study on ferromagnetic metals [22, 21]. The gauge \feld (36) also corre-\nsponds to the unbroken U(1) gauge \feld upon the symmetry breaking from SU(2) toU(1),\nwith a space-dependent order parameter, which is frequently discussed in the context of\nsolitonic monopoles in non-abelian gauge theories [23]. Since the unbroken gauge \felds in\n4We used the relation \"abc@\u0016nb@\u0017nc=na\"bcdnb@\u0016nc@\u0017nddue toP\nanana= 1.\n10the holographic dual theory are identi\fed as the (exactly) conserved currents in the bound-\nary quantum system, the gauge \feld C\u0016is naturally identi\fed as the spin current with the\npolarization parallel to the magnetization na, which originates from conduction electrons.\nSo far, we have obtained the relation between the gauge \feld aa\n\u0016and the (normalized)\nmagnetization na, which implies that the gauge \feld dynamics can be solely reduced to the\ndynamics of the magnetization and the spin electromagnetic \feld C\u0016. Finally, we consider\nthe remaining Yang-Mills equation in the radial u-direction, in the holographic dual theory :\n1p\u0000g@\u0016\u0000p\u0000gF\u0016ua\u0001\n+\u000fabcAb\n\u0016F\u0016uc=Jua: (38)\nThis equation is derived by the variation of the radial u-component of the SU(2) gauge \felds\nand speci\fes the dynamics of the gauge \felds in the \fve-dimensional bulk; this equation\ncannot be seen in the ferromagnetic system on the boundary. With the ansatz (18), the\nradial component of the current also vanishes ( Ja\nu\u00110), and the gauge \fxing condition\nAa\nu\u00110 leads to the simple SU(2) \feld strength Fa\n\u0016u=\u0000@uAa\n\u0016such as\nFk\n0u= 2uak\n0(t;x); F?\n0u= 2ua?\n0(t;x);\nFk\niu= 2u\u001bk\nsak\ni(t;x); F?\niu=\u00002u\u001b?\nsa?\ni(t;x); (39)\nwhere we used the ansatz (19) and discarded the irrelevant O(u3) terms. From these forms,\nthe second term in the left-hand side of (38) automatically vanishes due to \"abcab\n\u0016ac\n\u0016= 0.\nInserting the forms of \feld strength (39) and the relation (31), the equation can be recast as\nthe following form:\n@0\u0010\nC0na\u0000\"abcnb@0nc\u0011\n\u0000@i\u0010\n\u001bk\nsCina+\u001b?\ns\"abcnb@inc\u0011\n= 0; (40)\nwhere the subleading terms are neglected. Here, we can write down the e\u000bective equation of\nmotion for the magnetization nain our holographic dual model:\nC0_na\u0000\"abcnbnc\u0000\u001b?\ns\"abcnbr2nc\u0000\u001bk\nsCi@ina= 0; (41)\nwhere the dot denotes the time-derivative and r2=@i@i.5Here, we consider the condition,\n@0C0\u0000\u001bk\ns@iCi= 0, on the unbroken gauge \feld due to the constraintP\nanana= 1. This\ncondition implies the conservation of the spin current of conduction electrons, which corre-\nsponds to the unbroken gauge \feld C\u0016, as seen below. Note that, since the Maxwell equations\n(37) forC\u0016is gauge invariant, this condition can be consistently imposed as a gauge \fxing\ncondition.\n5A simlar analysis on e\u000bective equations at the linearized level in two-dimensional magnetic systems has\nbeen also discussed in [11].\n11Considering the matching condition (33), we decompose C0intoC0=\u0016s+~C0. Finally,\nwe obtain the holographic equation for magnetization dynamics:\n\u0016s_na\u0000\"abcnbnc\u0000\u001b?\ns\"abcnbr2nc+~C0_na\u0000\u001bk\nsCi@ina= 0: (42)\nHere, we take the spin chemical potential to be \u0016s=\u0000Ms=\rwith the magnitude of spon-\ntaneous magnetization Msand the gyromagnetic ratio \r(>0),6and also identify the spin\ncurrent and spin accumulation due to conduction electrons as Jk\nsi=\u0000\u001bk\nsCiand \u0001\u0016s=~C0,\nusing the holographic dictionary. Then, the holographic equation becomes the same form as\nthe Landau-Lifshitz equation (without damping terms),\nMs\n\r_na+\"abcnbnc=\u0000\u001b?\ns\"abcnbr2nc+ \u0001\u0016s_na+Jk\nsi@ina: (43)\nThe last two terms in the right-hand side can be interpreted as the well-known terms from spin\ntransfer torque, which describes the transfer of spin angular momentum between localized\nmagnetic moments and conduction electrons [21]. Furthermore, the holographic Landau-\nLifshitz equation (42) also naturally incorporate the spin inertia term proportional to the\nsecond time-derivative of the magnetization, which is discussed in metallic ferromagnets [24].\nIt should be noted that the holographic Landau-Lifshitz equation automatically incor-\nporates the spin transfer torque due to conduction electrons without introducing the cor-\nresponding \felds to electrons in the dual gravitational theory. This is consistent with the\nthermodynamical results at low temperatures, which was obtained in the previous paper [10].\n4 Phenomenology of Holographic Magnetization Dynamics\nIn the isotropic ferromagnets su\u000eciently below the Curie temperature ( T\u001cTc), the dynamics\nof magnetization vector (or density of magnetic moments), Ma, is described by the Landau-\nLifshitz equation [1, 26]:\n@Ma\n@t=\u0000\u000b\u000fabcMbr2Mcwith3X\na=1MaMa=M(T)2= const. (44)\nIn the following discussion, the external magnetic \feld and the damping term (or relaxation\nterm) are ignored for simplicity. From the quadratic constraint, the magnetization vector\ncan be represented as Ma(x;t) =M(T)na(x;t) with the unit vector na(x;t). In terms of\nna(x;t), the Landau-Lifshitz equation becomes\nM(T)@na\n@t=\u0000\u000bM(T)2\u000fabcnbr2nc: (45)\n6The negative sign is introduced due to the negative value of the gyromagnetic ratio for electrons.\n12Note that the equation has two parameters, the magnitude of spontaneous magnetization,\nM(T), at the temperature T, and the spin sti\u000bness constant, \u000b.\nComparing the holographic equation (42) with the Landau-Lifshitz equation (45), we \fnd\nthat the spin chemical potential, \u0016s, in the gauge \feld solution (17) should be proportional\nto the magnitude of magnetization, and the spin sti\u000bness constant is given by the coe\u000bcients\n\u001b?\nsin the gauge \feld solution (20) in the following way :\n\u0016s/\u0000M(T) and \u001b?\ns/\u000bM(T)2: (46)\nIn our holographic dual model, the magnitude of magnetization, M(T), at the temperature\nTis given by the static solution of the scalar \feld \b( u) through the formula (14). The\n\frst relation between the magnitude of magnetization and the spin chemical potential in\nferromagnets is well-known, and frequently used as the starting point to analyze the various\nspintronic phenomena [2].\nAlthough the spin chemical potential in the equilibrium, \u0016s, is an integration constant,\nthe coe\u000ecient, \u001b?\ns, is the derived quantity from the gauge \feld equation, and thus the second\nrelation in (46) on the spin sti\u000bness constant is a nontrivial consequence in the holographic\ndual model. In order to obtain the coe\u000ecient, \u001b?\ns, we consider the linearized equation of\nmotion for gauge \felds on the background solution, with the static and homogeneous ansatz,\nA?\ni=kG?(u), wherek= const.7Inserting this ansatz into the Yang-Mills equation (27),\nwe have the following linearized equation for G?(u):\nu3d\ndu\u0012f(u)\nu\u0012dG?\ndu\u0013\u0013\n+\u0000\nua3(u)\u00012\nf(u)G?= 0; (47)\nwhere the metric (8) and the SU(2) gauge \feld (17) are assumed to be the background.\nNote that this is a linear equation for G?, and the constant kis irrelevant. Here, we\nimpose the \frst relation in (46), \u0016s=\u0000M(T)=M(0), which is the magnetization normal-\nized by the saturated magnetization, M(T= 0).8Using the numerical results of the holo-\ngraphic spontaneous magnetization, M(T) in [10], which is obtained using the scalar poten-\ntialV(j\u001ej) =\u0015\u0000\nj\u001ej2\u0000m2=\u0015\u00012=4 with\u0015= 1 andm2= 35=9, we can numerically solve the\nequation (47) and obtain the asymptotic expansion (20) near the boundary ( u\u00180). The\nnumerical results of temperature dependences of the spin-wave sti\u000bness, D(T)'\u001b?\ns=M(T),\nwhich appears in the dispersion relation of spin-waves, !=D(T)k2, and the spin sti\u000bness\nconstant,\u000b(T)'\u001b?\ns=M(T)2, are shown in Figure 1.\n7The nontrivial pro\fle A?\nx(u) on the background does not contribute to the energy-momentum tensor in\nthe Einstein equation at the linearized level.\n8The proportionality constant is chosen for convenience in numerical calculations.\n13Figure 1: Temperature dependence of the spin-wave sti\u000bness is shown in Figure ( a): The\ndots are numerical results for D(T)=D(0), and the bold line is the magnetization curve,\nM(T)=M(0). Temperature dependence of the spin sti\u000bness constant, \u000b(T)=\u000b(0), is shown in\nFigure (b). All the results are calculated with the parameters, \u0015= 1 andm2= 35=9.\nThe results on the spin-wave sti\u000bness in Figure 1( a) clearly show that D(T)/M(T),\nwhich is consistent with the relation (46) based on the Landau-Lifshitz equation (44). Fur-\nthermore, the results in Figure 1( b) imply the slight temperature dependence of the spin\nsti\u000bness constant, \u000b=\u000b(T), which can be attributed to the nonlinear spin-wave e\u000bects [25].\nA similar argument also holds for the unbroken (or parallel) component of the gauge\n\felds,Ak\ni, and we can obtain the coe\u000ecient \u001bk\ns, which leads to the spin torque term in the\nholographic Landau-Lifshitz equation (42). The nontrivial pro\fle of gauge \feld, Ak\nx(u), which\nis the parallel component to the spin chemical potential, Ak\n0, leads to the non-vanishing o\u000b-\ndiagonal contribution in the right-hand side of the Einstein equation (2), and thus induces the\n\ructuation of the metric gtx(u) =htx(u)=u2, wherehtx(u) parameterizes the \ructuation \fnite\non the boundary. At the linearized level, two \ructuations, Ak\nx(u) andhtx(u), form the closed\nequations, which come from the Yang-Mills equation and Einstein equation, respectively\n[27, 28]:\nud\ndu \nf(u)\nu \ndAk\nx\ndu!!\n+\u0012da3(u)\ndu\u0013d\ndu\u0000\nu2htx\u0001\n= 0; (48)\nu\u00002d\ndu\u0000\nu2htx\u0001\n+ 2\u0012da3(u)\ndu\u0013\nAk\nx= 0: (49)\nDeleting the metric \ructuation, we can obtain the equation for Gk(u):\nud\ndu \nf(u)\nu \ndGk\ndu!!\n\u00002u2\u0012da3(u)\ndu\u00132\nGk= 0: (50)\nWe can numerically solve the equation, and obtain the coe\u000ecient \u001bk\nsfrom the asymptotic\nexpansion of the solution in (20). The resulting temperature dependence of the spin torque\ncoe\u000ecient, \u001cs(T) =\u001bk\ns=M(T), is shown in Figure 2.\n14Figure 2: Temperature dependence of the spin torque coe\u000ecient: The dots are numerical re-\nsults for\u001cs(T)=\u001cs(0), and the bold line is the magnetization curve M(T)=M(0). The dashed\nline is the \ftting curve, \u001cs(T)=\u001cs(0) =c(1\u0000T=Tc)2=5withc'1:41, near the Curie temper-\nature.\nThe results on the magnitude of the spin transfer torque, \u001cs(T), show that the spin torque\ne\u000bect is approximately constant at low temperatures (in comparison with magnetization\ncurve), and is vanishing towards the Curie temperature as \u001cs(T)/(1\u0000T=Tc)2=5. This\nproperty at low temperatures is consistent with the phenomenological form of the spin transfer\ntorque, ( Jk\ns\u0001rna)=M(T), whose magnitude is independent of the norm of magnetization due\ntojJk\nsj/M(T) at the leading order [21]. In addition, the \fnite spin torque coe\u000ecient is\na consequence of the both \ructuations of the gauge \feld and metric. In accordance with\nthe holographic dictionary [27], the metric \ructuation htxcorresponds to the temperature\ngradient,rxT=T, in the ferromagnetic system. This calculation implies that the e\u000bect of\nspin transfer torque appears only in the nonequilibrium situations, where spin transfer is\naccompanied by heat (or entropy) transfer.\n5 Summary and Discussion\nWe have discussed a novel approach to understand magnetization dynamics in ferromagnets\nusing the holographic realization of ferromagnetic systems. The Landau-Lifshitz equation de-\nscribing magnetization dynamics was derived from the Yang-Mills-Higgs equations in the dual\ngravitational theory. This holographic Landau-Lifshitz equation automatically incorporates\nnot only the exchange interaction but also the spin transfer torque e\u000bect due to conduction\nelectrons. Furthermore, we numerically investigated the temperature dependences of the\nspin-wave sti\u000bness and the magnitude of spin transfer torque in the holographic dual theory,\nand the results obtained so far are consistent with the known properties of magnetization\ndynamics in ferromagnets with conduction electrons.\n15This holographic approach to magnetization dynamics can be applied to more generic sit-\nuations. For instance, the holographic Landau-Lifshitz equation can incorporate the damping\nterm by considering more generic metric \ructuations, which correspond to phonon dynamics\nin the boundary ferromagnets. Moreover, the holographic dual theory may provide geomet-\nric approaches to spin caloritronics [29], where magnetization dynamics is considered under\ntemperature gradients, from higher dimensional perspectives. We thus believe that the holo-\ngraphic approach provides useful tools to analyze nonequilibrium and nonlinear dynamics\nof magnetization in ferromagnets, and also leads to new perspectives in spintronics from\ngravitational physics.\nAcknowledgement\nThe authors thank M. Ishihara for collaboration at the early stage of this work, and also K.\nHarii and Y. Oikawa for useful discussions. The works of N. Y. and E. S. were supported\nin part by Grant-in Aid for Scienti\fc Research on Innovative Areas \"Nano Spin Conversion\nScience\" (26103005), and the work of K. S. was supported in part by JSPS KAKENHI Grant\nNo. JP17H06460. The works of N. Y. and E. S. were supported in part by ERATO, JST.\nReferences\n[1] L. D. Landau and E. M. Lifshitz, \\On the theory of the dispersion of magnetic perme-\nability in ferromagnetic bodies,\" Phys. Z. Sowjet. 8, 153 (1935).\n[2] S. Maekawa, S. O. Valenzuela, E. Saitoh and T. 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Van Wees, \\Spin caloritronics,\" Nature materials, 11\n(5), 391 (2012).\n18"    },    {        "title": "1810.07831v1.Out_of_plane_auto_oscillation_in_spin_Hall_oscillator_with_additional_polarizer.pdf",        "content": "arXiv:1810.07831v1  [cond-mat.mes-hall]  17 Oct 20181\nOut-of-plane auto-oscillation in spin Hall oscillator\nwith additional polarizer\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Tech nology (AIST), Spintronics Research Center, Tsukuba,\nIbaraki 305-8568, Japan\nAbstract —The theoretical investigation on magnetization dy-\nnamics excited by the spin Hall effect in metallic multilaye rs\nhaving two ferromagnets is discussed. The relaxation of the\ntransverse spin in one ferromagnet enables us to manipulate\nthe direction of the spin-transfer torque excited in anothe r\nferromagnet, although the spin-polarization originally g enerated\nby the spin Hall effect is geometrically fixed. Solving the\nLandau-Lifshitz-Gilbert-Slonczewski equation, the poss ibility to\nexcite an out-of-plane auto-oscillation of an in-plane mag netized\nferromagnet is presented. An application to magnetic recor ding\nusing microwave-assisted magnetization reversal is also d iscussed.\nIndex Terms —Landau-Lifshitz-Gilbert-Slonczewski (LLGS)\nequation, microwave-assisted magnetization reversal (MA MR),\nspin Hall effect, spin torque oscillator (STO), spintronic s.\nI. I NTRODUCTION\nSPIN-transfer torque [1,2] originated from the spin Hall\neffect [3,4] in nanostructured ferromagnetic/nonmagneti c\nbilayers enables us to manipulate the magnetization withou t\ndirectly injecting the electric current into the ferromagn et.\nMotivated by the demand to develop practical devices, such a s\nmagnetic memory, microwave generators, and neuromorphic\ncomputing, the magnetization switching and oscillation by the\nspin Hall effect have been experimentally demonstrated and /or\ntheoretically investigated [5-15].\nThe variety of the magnetization dynamics excited by the\nspin Hall effect is, however, limited because of the geometr ical\nrestriction of the direction of the spin-transfer torque [1 6], i.e.,\nwhen the electric current flows in the nonmagnet along the\nx-direction and the ferromagnet is set in the z-direction, the\nspin polarization of the spin current generated by the spin H all\neffect is fixed to the y-direction. Due to the restriction, the spin\nHall effect, for example, cannot excite an out-of-plane aut o-\noscillation of a magnetization in a ferromagnet in the absen ce\nof an external magnetic field [16], which is a problematic\nissue in current magnetics. This is a disadvantage of the spi n\nHall geometry to giant magnetoresistive (GMR) system and\nmagnetic tunnel junctions (MTJs), where the torque directi on\ncan be changed by controlling the magnetization direction i n\nthe pinned layer [17-22], and the out-of-plane auto-oscill ation\nof the in-plane magnetized system has already been reported\nboth experimentally and theoretically [23-32].\nThe purpose of this paper is to show that an insertion\nof another ferromagnet having a tilted magnetization to the\nspin Hall geometry enables us to manipulate the spin-transf er\ntorque direction and excite an out-of-plane auto-oscillat ion of\nan in-plane magnetized ferromagnet. The reason for such aN1F1NF2N2\npspin-transfer torque\nJsJexyzm\nFig. 1. Schematic view of the ferromagnetic/nonmagnetic mu ltilayer. The\nelectricJeand spin Jscurrent densities flow along the x- andz-directions,\nrespectively. The spin polarization of Jsflowing from N 1to F1layer points\nto they-direction. Passing through the F 1layer, however, the direction of the\nspin polarization becomes parallel or antiparallel to the m agnetization p.\nphenomenon is because the relaxation of the transverse spin\nin this additional ferromagnet results in the modification o f\nthe direction of the spin polarization. Solving the Landau-\nLifshitz-Gilbert-Slonczewski (LLGS) equation, it is foun d that\nan out-of-plane auto-oscillation of the magnetization can be\nexcited in an in-plane magnetized free layer. An applicatio n\nto magnetic recording using microwave-assisted magnetiza tion\nreversal (MAMR) is also discussed.\nII. O UT-OF-PLANE AUTO -OSCILLATION IN IN -PLANE\nMAGNETIZED FREE LAYER\nThe system we consider is schematically shown in Fig.\n1, where two ferromagnets, F 1and F2, are sandwiched by\ntwo nonmagnets, N 1and N 2. There is another nonmagnet,\nN, between the F 1and F2layers. From hereafter, we use the\nsuffixies F k, Nk, (k= 1,2) and N to distinguish quantities\nrelated to these layers and their interfaces. The unit vecto rs\npointing in the magnetization direction of the F 1and F2\nlayers are denoted as pandm, respectively. The F 1layer\nacts as the polarizer whereas the F 2layer is the free layer, as\ndescribed below. An electric current is applied to the N 1layer\nalongx-direction. The spin-orbit interaction in the N 1layer\nscatters the spin-up and spin-down electrons to the opposit e\ndirections, generating spin current flowing along z-direction\nand polarizing along y-direction. The electric and spin current\ndensities flowing along the x- andz-directions in the N 1layer\nare given by [33]\nJe=σN1Ex+ϑN1σN1\n2e∂zey·µN1, (1)2\nJs=/planckover2pi1ϑN1σN1\n2eExey−/planckover2pi1σN1\n4e2∂zµN1, (2)\nrespectively, where Exis the external electric field in the\nx-direction. The conductivity and the spin Hall angle are\nσ= 1/ρandϑ, respectively, where ρis the resistivity. The\nspin accumulation is denoted as µ, whereas eyis the unit\nvector in the y-direction. We note that the vector notation by\nboldface represents the direction of the spin polarization . The\nspin current given by Eq. (2) creates the spin accumulations\nin each layer, which obey the diffusion equation with the spi n\ndiffusion length ℓ.\nLet us explain the central idea of this paper. The spin\npolarization of the spin current flowing in the z-direction\ngenerated in the N 1layer via the spin Hall effect points to\nthey-direction [3,4,33]. However, in the F 1layer, the spin\ntransverse to the local magnetization pprecesses due to the\nexchange interaction and relaxes rapidly. As a result, only the\nspin polarization parallel or antiparallel to pdirection survives\nduring the transport through the F 1layer. When phas a finite\nz-component pz, the spin current having a finite z-component\nis injected into the F 2layer, and move its magnetization m\nto thez-direction, resulting in an excitation of an out-of-\nplane precession, as schematically shown in Fig. 1. We shoul d\nemphasize here that the magnetization palso has a finite y-\ncomponent pybecause, if py= 0, the spin polarization injected\nfrom the N 1to F1layer relaxes at the N 1/F1interface, and\nthe net spin polarization emitted from the F 1to F2layer\nbecomes zero. Therefore, the magnetization pshould have\nbothy- andz-components. In other words, pshould be tilted\nin theyzplane to maximize the efficiency of the spin injection\nhaving the finite z-component of the spin polarization. This\nis an important difference from GMR or MTJ, where the y-\ncomponent of the polarizer is unnecessary to excite an out-o f-\nplane auto-oscillation [23-32]. We note that the tilted mag netic\nanisotropy has been investigated by making use of a higher-\norder anisotropy or an interlayer exchange coupling betwee n\ntwo ferromagnets [34,35].\nWe show the spin-transfer torque formula acting on the\nmagnetization min the F 2layer. The spin current density\nat the F 2/N interface flowing from the F 2to N layer is given\nby\nJF2→N\ns=1\n4πS/bracketleftbigg(1−γ2)g\n2m·(µF2−µN)m\n−g↑↓\nrm×(µN×m)−g↑↓\niµN×m/bracketrightBig\n,(3)\nwhereg=g↑↑+g↓↓is the sum of the conductances for\nspin-up and spin-down electrons, and γ= (g↑↑−g↓↓)/g\nis its spin polarization. The conductance gis related to the\ninterface resistance rasg/S=h/(e2r), whereSis the cross-\nsectional area. The real and imaginary parts of the mixing\nconductance are denoted as g↑↓\nrandg↑↓\ni, respectively. Since\ng↑↓\nr≫ |g↑↓\ni|for typical ferromagnetic/nonmagnetic interfaces\n[36,37], we neglect the terms related to g↑↓\niin the following\ncalculations. The spin currents at the other F k/Nkand F1/N\ninterfaces are obtained in a similar manner. We assume that\nthe thickness of the N layer is sufficiently thinner than the\nspin diffusion length, and therefore, the spin current in th islayer is conserved. The spin-transfer torque excited in the\nF2layer is, according to the conservation law of the angular\nmomentum, the transverse spin current ejected from this lay er,\ni.e.,τ= [γ0/(MF2dF2)]m×[(JF2→N\ns+JF2→N2s)×m],\nwhereγ0,M, anddare the gyromagnetic ratio, saturation\nmagnetization, and thickness, respectively. Using Eqs. (1 )-(3)\nand the spin diffusion equation, the spin-transfer torque i s\ngiven by\nτ=−γ0/planckover2pi1ϑN1g↑↓\nr(F2/N)g′\nF1/NσN1Ex\n2e˜gF1[g↑↓\nr(F2/N)+g′\nF1/N]MF2dF2pym×(p×m)\n1−λ1λ2(m·p)2.\n(4)\nHere, we define\n1\ng′\nF1/N=2\n(1−γ2\nF1/N)gF1/N+1\ngsd(F1)tanh(dF1/ℓF1)\n−g′\nF1/N1\n[gsd(F1)sinh(dF1/ℓF1)]2,(5)\n1\ng′\nF1/N1=2\n(1−γ2\nF1/N1)gF1/N1+1\ngsd(F1)tanh(dF1/ℓF1)\n+1\ngsd(N1)tanh(dN1/ℓN1),\n(6)\n1\n˜gF1=g′\nF1/N1tanh[dN1/(2ℓN1)]\ngsd(F1)gsd(N1)sinh(dF1/ℓF1), (7)\ngsd(F1)\nS=h(1−β2\nF1)σF1\n2e2ℓF1, (8)\ngsd(N1)\nS=hσN1\n2e2ℓN1, (9)\nλk=g↑↓\nr(Fk/N)−g′\nFk/N\ng↑↓\nr(Fk′/N)+g′\nFk/N, (10)\n[(k,k′) = (1,2),(2,1)], respectively. The spin polarization of\nthe conductivity in ferromagnet is β= (σ↑−σ↓)/(σ↑+σ↓).\nThe spin-transfer torque excited in the F 1layer can be cal-\nculated in a similar manner [38]. The spin-transfer torques in\nsystems having two ferromagnets with different geometries\nare discussed in Refs. [39,40]. In the following, we use\ntypical values of the parameters found in experiments and fir st-\nprinciples calculations in the spin Hall geometry; ℓF= 12 nm,\nℓN= 2.5nmρF= 1600 Ω nm,ρN= 3750 Ω nm,β= 0.56,\nr= 0.28kΩnm2,γ= 0.70,ϑ= 0.1,g↑↓\nr/S= 15 nm−2,\ndF= 1 nm,dN= 4 nm,γ0= 1.764×107rad/(Oe s), and\nM= 1250 emu/c.c. [14,36,37,41,42].\nWe study the magnetization dynamics in the F 2layer under\nthe effect of the spin-transfer torque given by Eq. (4). The\nmagnetization in the F 1layer,p, is assumed to be pinned.\nThe LLGS equation of the magnetization min the F 2layer\nis given by\ndm\ndt=−γ0m×H−γ0/planckover2pi1ηjpym×(p×m)\n2e[1−λ1λ2(m·p)2]MF2dF2+αm×dm\ndt,\n(11)3\nwhere, according to Eq. (4), j=σN1Exand\nη=ϑN1g↑↓\nr(F2/N)g′\nF1/N\n˜gF1[g↑↓\nr(F2/N)+g′\nF1/N]. (12)\nUsing the values of the parameters mentioned above, η=\n0.043andλ= 0.878. The magnetic field,\nH=−4πMF2mzez, (13)\nconsists of the demagnetization (shape anisotropy) field al ong\nthez-direction. The Gilbert damping constant is α, and we use\nα= 0.01in this paper. Figure 2(a) shows typical trajectories\nof the magnetization dynamics in steady states obtained fro m\nthe numerical simulation of Eq. (11), where j=±100×106\nA/cm2andp= (0,1/√\n2,1/√\n2). This figure indicates that the\nout-of-plane precession can be excited by the spin Hall effe ct.\nThis is the main result in this paper. We should also emphasiz e\nthat the oscillation direction, i.e., clockwise or counter clock-\nwise around the z-axis, can be changed by changing the sign\nofj. The magnetization moves to the positive (negative) z-\ndirection by the negative (positive) current, as shown in Fi g.\n2(a).\nWe can obtain the analytical formulas revealing the relatio n\nbetween the current and the oscillation frequency by solvin g\nthe LLGS equation averaged over the constant energy curve\n[43]. The energy density in the present system is E=\n−MF2/integraltext\ndm·H= 2πM2\nF2m2\nz. An auto-oscillation is excited\nwhen the spin-transfer torque compensates with the damping\ntorque, and therefore, the field torque ( −γ0m×H) principally\ndetermines the magnetization dynamics. Since the field torq ue\nconserves the energy density E, the auto-oscillation can be\napproximated as occuring on a constant energy curve. As a\nresult, we can estimate current density necessary to excite an\nauto-oscillation on a constant energy curve of Efrom the\nequation/contintegraltext\ndt(dE/dt) = 0 [16], where the time integral is\nover the precession period. To simplify the discussion we\nuse the cone angle of the magnetization, θ= cos−1mz,\ninstead of E, to identify a constant energy curve in the present\ncase. Assuming that plies in the yz-plane (i.e., px= 0 and\np2\ny+p2\nz= 1), the current density necessary to excite the steady\nprecession of the magnetization with the cone angle θis found\nto be\nj(θ) =−2αeMF2dF2\n/planckover2pi1ηP(θ)4πMF2. (14)\nHere,P(θ)is given by\nP(θ) =[gz(θ)pzsin2θ+gy(θ)cosθ]py\nsin2θcosθ, (15)\nwheregz(θ)andgy(θ)are given by\ngz(θ) =1\n2\n1/radicalBig\nc2\n+−a2+1/radicalBig\nc2\n−−a2\n, (16)\ngy(θ) =1\n2√λ1λ2\nc+/radicalBig\nc2\n+−a2−c−/radicalBig\nc2\n−−a2\n, (17)withc±= 1±√λ1λ2pzcosθanda=√λ1λ2pysinθ. On the\nother hand, the precession frequency at the cone angle θis\nf(θ) =γ0\n2π4πMF2|cosθ|. (18)\nWe confirm the validities of Eqs. (14) and (18) by comparing\nthem with the numerical simulations. We add the random\ntorque,−γ0m×h, due to thermal fluctuation to Eq. (11), for\nthe sake of generality. The components of the random field\nhsatisfy the fluctuation-dissipation theorem, /angbracketlefthi(t)hj(t′)/angbracketright=\n[2αkBT/(γ0MV)]δijδ(t−t′), where the temperature Tand\nthe volume of the F 2layerVare assumed to be T= 300 K and\nV=πr2dF2withr= 50 nm. We solve Eq. (11) numerically\nfor0≤t≤1µs with the initial condition m(0) = + ex.\nRepeating such calculation 104times with random h, the\naveraged Fourier spectra of the x-component of m,|mx(f)|,\nare obtained, as shown in Fig. 2(b). As shown, each spectrum\nhave one peak at a certain frequency, which increases with\nincreasing the current magnitude. The dependence of the pea k\nfrequency on the current obtained from such simulation is\nshown by red circles in Fig. 2(c). The analytical relation\nbetween the current and frequency obtained from Eqs. (14)\nand (18) is also shown in Fig. 2(c) by the black (solid) line. W e\nobtain good agreement between the simulation and analytica l\nformulas, indicating the validity of Eqs. (14) and (18).\nIn this paper, we consider an excitation of an auto-\noscillation in an in-plane magnetized free layer by inserti ng\nan additional ferromagnet between the free layer and the\nnonmagnet having the spin Hall effect. We also notice that\nanother situation is possible [43], where an auto-oscillat ion of\na perpendicularly magnetized free layer is excited by addin g\nanother ferromagnet on the other side of the nonmagnet.\nIII. A PPLICATION TO MICROWAVE ASSISTED\nMAGNETIZATION REVERSAL\nAt the end of this paper, let us discuss the application of\nthe above result to magnetic recording. A spin torque oscill ator\n(STO) consisting of an in-plane magnetized free layer and a\nperpendicularly magnetized pinned layer [23-26] is a candi date\nfor the recording head of a hard disk drive using MAMR [28-\n32,45]. An oscillating magnetic field generated from the STO\nacts as a microwave field on the recording bit, and reduces the\nrecording field [45]. In the original design of MAMR, an STO\nhaving a current-perpendicular-to-plane (CPP) structure was\nassumed [45], where the current flows parallel to the recordi ng\nmedia, as shown in Fig. 3(a). In this case, only a linearly\nporalized field with regard to the recording bit can be obtain ed\nin the microwave field emitted from the STO.\nThis design, however, does not make full use of the idea\nembodied in MAMR. An interesting idea in MAMR is the\nchirality matching between the STO and the recording bit. As\nshown in Fig. 2(a), the magnetization in an STO consisting\nof in-plane magnetized free layer shows oscillation with bo th\nclockwise and counterclockwise chiralities, depending on the\nsign of the current. The magnetization in the recording bit\nalso has an oscillation chirality, depending on the magneti zed\ndirection. It was shown that the recording field in MAMR is\nsignificantly reduced when the chirality of the STO matches4\nmz1\n-10\n1-1\n01\n-10\ncurrent density (106 A/cm2)frequency (GHz)\n-300 -200 -100 0 100 200 300010203040\nfrequency (GHz)0 2 4 6 8 10|mx(f)| (arb. unit)\n00.51.01.5\nj=10, 20, 30 (×106 A/cm2)(a) (b)\nT=300 (K)\nmxmy(c) j=-100 (×106 A/cm2) T=0 (K)\n:theory\n:simulation\nj=100 (×106 A/cm2)\nFig. 2. (a) Steady state trajectories of the magnetization o btained by numerically solving Eq. (11). The current densit ies arej= 100 and−100×106\nA/cm2for red (mz<0) and blue ( mz>0) lines, respectively. The precession directions are indic ated by arrows. The temperature is 0 K. (b) The Fourier\ntransformations of mx(t),|mx(f)|, forj= 10 (red),20(green), and 30(blue)×106A/cm2. The temperature is 300 K. (c) The comparison between the\npeak frequencies of |mx(f)|(red circles) and the analytical solution (black line).\nCPP-STO(a)\nLP-MWSH-STOCPP-STO(b)\nCP-MW(c)\nCP-MW\nrecording media recording media recording mediacurrent\ncurrentcurrent\nFig. 3. (a) Schematic picture of recording method using MAMR . The\nmicrowave (MW) emitted from the CPP-STO acts as a linearly po larized (LP)\noscillating field on the magnetizations in the recording med ia. (b) Another\nstructure of recording method using MAMR in Ref. [46]. The mi crowave\nemitted from the CPP-STO acts as a circularly polarized (CP) oscillating\nfield. (c) A new recording method using MAMR proposed in this w ork. The\nSTO is driven by the spin Hall (SH) effect, as shown in Fig. 1.\nwith that of the recording bit [44]. According to this princi ple\nof the chirality matching, Kudo et al. proposed a concept of\nresonant switching [30,46], where the current in the STO flow s\nalong the direction perpendicular to the recording media; s ee\nFig. 3(b). In this case, the microwave field emitted from the\nSTO is a circularly polarized with regard to the recording bi t.\nTherefore, by changing the current direction in the STO, the\noscillating field having the chirality of both the clockwise and\ncounterclockwise can be generated, and applied in MAMR by\ntaking considerations of chirality matching into account.\nThe results shown in this paper may add another vital\nadvantage in designing the recording head in MAMR. For\nexample, let us consider the system shown in Fig. 3(c),\nwhere the current in the recording head flows parallel to the\nrecording media. In this structure, the current does not flow\nin the free layer, in contrast to the structure shown in Fig.\n3(a). Instead, the spin Hall effect excites an oscillation i n the\nfree layer, as schematically shown in Fig. 1. The circularly\npolarized microwave field is then emitted from the STO, as\nin the case shown in Fig. 3(b). Therefore, the structure in\nFig. 3(c) satisfies the condition to achieve MAMR combined\nwith chirality matching. In addition, the vital advantage, whichresides in this structure, is that the recording head can in\nprinciple be placed closer to the media than in the case of\nFig. 3(b) because of the absence of the electrode between the\nfree layer and the recording media. Therefore, the STO using\nthe spin Hall effect will be an interesting candidate for the\nrecording head of next generation.\nIV. C ONCLUSION\nIn conclusion, it was shown that the spin Hall effect can\nexcite the out-of-plane precession of the magnetization in a\nferromagnet by inserting another ferromagnet having a tilt ed\nmagnetization between the nonmagnetic heavy metal and the\nferromagnet. The phenomenon is due to the relaxation of the\ntransverse spin in this additional ferromagnet. Although t he\nrelaxation of the spin in the additional layer induces a loss\nof spin polarization, it enables us to manipulate the direct ion\nof the spin-transfer torque excited on the free layer. Using\nthe spin-transfer torque formula derived from the diffusiv e\nspin-transport theory and solving the LLGS equation both\nnumerically and analytically, the relation between the cur rent\nand the precession frequency was obtained. It was also shown\nthat the chirality of the precession can be reversed by rever sing\nthe current direction. An application to magnetic recordin g\nusing microwave assisted magnetization reversal was also\ndiscussed.\nNote added : After our submission, we were notified that Dr.\nSuto in Toshiba proposed another solution based on a nonloca l\ninjection of spin current to generate a circularly polarize d\nmicrowave from a spin-valve with a current flowing parallel\nto the recording media (unpublished).\nACKNOWLEDGMENT\nThe author is grateful to Takehiko Yorozu, Yoichi Shiota,\nHitoshi Kubota, and Shingo Tamaru for valuable discussions .\nThe author is also thankful to Satoshi Iba, Aurelie Spiesser ,\nHiroki Maehara, and Ai Emura for their support and encour-\nagement. This work was supported by JSPS KAKENHI Grant-\nin-Aid for Young Scientists (B) 16K17486.5\nREFERENCES\n[1] J. C. Slonczewski, ”Current-driven excitation of magne tic multilayers”,\nJ. Magn. Magn. Mater. vol. 159, p.L1, 1996.\n[2] L. Berger, ”Emission of spin waves by a magnetic multilay er traversed\nby a current”, Phys. Rev. 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Institute of Nanosurface Science and Engineering, Shenzhen University, Shenzhen 518060, China ; \n2. Electron Microscopy Center of Shenzhen University, Shenzhen 5180 60, China ; \n3. National Key Laboratory of Radio Frequency Heterogeneous Integration , Shenzhen University, Shenzhen \n518060, China ; \n4. Department of Applied Physics, The University of Tokyo, Tokyo 113 -8656, Japan ; \n5. Quantum Materials and Applications Research Center, National Institutes for Quantum Science and \nTechnology (QST), Tokyo 152 -8550, Japan ; \n6. Institute for AI and Beyond, The University of Tokyo, Tokyo 113 -8656,  Japan ; \n7. WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan ; \n8. National Synchrotron Radiation Laboratory, and School of Nuclear Science and Technology, University of \nScience and Technology of China, Hefei, Anhui 230029, China ; \n9. International Centre for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for \nPhysical Sciences at the Microscale (HFNL), and Synergetic Innovation Center of Quantum Information and \nQuantum Physics, University of Science and Technol ogy of China, 230026 Hefei, China ; \n10. Key Laboratory of Education Ministry for Modern Design and Rotor -Bearing System, Xi’an Jiaotong \nUniversity, Xi'an 710049, China . \nE-mail address:  dfdiao @szu.edu.cn . \n \nAbstract:   \nThe emergence of bulk carbon ferromagnetism is long -expected over years.  At nano scale, carbon \nferromagnetism was detected in graphene nanoribbon segments1–3, graphene triangulene dimers4,5, and other \ngraphene -like molecules6–8 by analyzing their magnetic edge states via scanning tunneling microscop y(STM )9, \nand its origin can be explained by  local redistribution of electron wave function3,5,8. Recently, there are \nexperimental efforts on discovering orbital ferromagnetism in twisted bilayer graphene10,11, which is still \nlimited in STM  region . In larger scale , carbon ferromagnetism can be created by deliberately producing defects \nin graphite12–14, and detect ed by macroscopic technical magnetization.  Meanwhile, it becomes crucial to \ndetermine that the detected magnetization is originated from carbon rather than from magnetic impurities. One \nsolution to eliminate this uncertainty is X -ray magnetic circular dichroism  (XMCD ). A contrast of dichroism \nat * absorption energy was observed in the scanning x -ray microscope image of proto n-irradiated graphite15. \nNonetheless , a reproducible , full section of  XMCD spectr um across  C-1s absorption  energy has not appeared  \nyet, which should be decisive for assuring the indisputable existence of bulk carbon ferromagnetism.  Besides , \nthe lack of direct observation on the atomic structure  of the ferromagnetic carbon leaves the structural origin \nof its ferromagnetism  still in mist.  In this work, for detecting bulk carbon ferromagnetism , we manage d to \ngrow all -carbon film consisting of vertically aligned graphene multi -edge  (VGME ), which wove into a three -\ndimensional hyperfine -porous network. Magnetization (M-H) curves and XMCD s pectra co -confirmed bulk \ncarbon ferromagnetism of VGME  at room temperature , with the average unit magnetic momentum of ~0.0006 \nB/atom. The influence of magnetic impurities on magnetization was excluded by both absorption spectra and \ninductively coupled plasma mass spectrometry measurements. The spin transfer behavior also verified the \nlong-range and robust feature of the bulk carbon ferromagnetism . Our work provides direct evidence of \nelementary resolved bulk carbon ferromagnetism at room temperature and clarifies its origin from -electrons \nat graphene edge s.  \n \nText:  \nThe emergence of  bulk carbon ferromagnetism is long -expected over years. In nanoscale, there are  \ntheoretical underpinning s for carbon ferromagneti sm such as  the zigzag  arrangement at graphene edges1–3, \nvacancies16, and adatoms17,18, where the redistribution of electron wave function occurs and local \nferromagnetism  is generated . In experimental, local ferromagnetism was detected in graphene nanoribbon \nsegments1–3, graphene triangulene dimers4,5, and other graphene -like molecules6,7. In addition, l ocal \nmagnetization in graphene was modulated by adatoms and doping19,20. Recently, there are experimental efforts \non discovering orbital ferromagnetism in twisted bilayer graphene11,21. These cases of  low-dimensional carbon \nferromagnetism are at cluster scale with several hexagons in -plane, and a few layers out -of-plane. The \nferromagnetic order s were detect ed under scanning tunneling  microscop y by measuring local magnetization \nor electric polarization  via analyzing their edge states.  \nCarbon ferromagnetism in larger scale can be created b y deliberately produc ing defects in graphite through  \nirradiation12–14, oxidation22, or mechanical exploiting23. This type of ferromagnetism is detectable in \nmacroscopic technical magnetization.  The hysteresis loop in magnetization curve  is ascribed to magnetic \nmoments of edge states at the defected positions. Meanwhile, it becomes crucial to determine that the detected \nmagnetization is originated from  carbon rather than from  magnetic impurities, because carbon ferromagnetic \nsignals are several orders of magnitude lower than magnetic element , and M-H curve  has no elemental \nresolution. Just out of this fact, the experimental observations of macroscopic ferromagnetis m in carbon \nmaterials were always challenged on the issue of impurities, or at least not well recognized , since it is difficult \nto rule out the influence of impurities . There are even reports against carbon ferromagnetism after careful \nexperiments24,25. \nOne solution  to eliminate th is uncertainty is X-ray magnetic circular dichroism  (XMCD ), which can \nrecognize carbon  ferromagnetism from the change of its absorption spectra under opposite magnetic field . A \ncontrast of dichroism at 284 eV was observed in scanning X-ray microscope image of proto n-irradiated \ngraphite15. Nonetheless, a repr oducible, full section of XMCD spectrum across C-1s absorption energy has  \nnot appeared yet , which should be decisive for assuring the indisputable existence of bulk carbon \nferromagnetism.  \nFor ferromagnetic carbon in any form, the atomic structure is the source of its unique electronic structure \nthat produces ferro magneti c order . In the studies of low -dimensional ferromagnetic carbon allotropes, the \nvisualization of their atomic structures significantly improved the interpretation on the origin of local carbon \nferromagnetism. However , in the case of bulk ferromagnetic carbon , the lack of direct observation on the \natomic structure leaves the structural origin of its ferromagnetism still in mis t. \nIn this work, we manage d to grow all -carbon film consist ing of vertically aligned graphene edges, which \nwove into a three -dimensional hyperfine -porous network . The vertical graphene multi -edge  (VGME)  \nstructur al feature  was identified by high resolution transmission electron microscop y (HRTEM) . M-H curves \nand XMCD spectra co -confirmed the room temperature ferromagnetism of VGME. Carefully repeated  \nmeasurements were carried out to reduce the noise, and the average unit magnetic momentum of ~0.0006 \nB/atom was determined . The influence of magnetic impurities on magnetization was excluded by absorption \nspectra and inductively coupled plasma mass spectrometry measurements. Angular dependent \nmagnetor esistance of VGME was measured in room temperature , and the s pin transfer behavior  verified the \nlong-range and robust feature of ferromagnetism . Our work provides direct evi dence of elementary resolved \nbulk carbon ferromagnetism at room temperature and clarifies its origin from -electrons at graphene edges.   \nThe VGME structure  was produced in electron cyclotron resonance (ECR) plasma via a physical vapor \ndeposition (PVD) approach with a low -energy -electron -irradiation technique, which was explained in detail \nin our previous work26,27. This approach allows us to produce VGME structure  at wafer scale on variable \nsubstrate surface s include but not limited to SiO 2. Fig. 1a-c depict a sequence of plan -view high -angle annular \ndark-field (HAADF) images of the  as deposited film consist of  VGME structure (VGME film) , progressively \nzooming in. Fig. 1a  reveal s the porous architecture of the VGME film as black void regions are distributed \nthroughout the film. The pores are of irregular shapes, and the ir average  size generally does not exceed 2 nm  \naccording to statistical measurements (see F ig. S1). Bright regions of the images represent the solid part of \nVGME structure, which consists of turbostratic graphene. The binding configuration of the turbostratic \ngraphene was confirmed by Raman and XPS spectra (see F ig. S2). The curving and crosslinking nature of the \ngraphene layers can be clearly observed by the line -shaped projections in Fig.1b.  The dangled edges around \nsuperfine pores are highlighted in purple dashed lines. Theses vertically dangled grap hene edges are believed \ncrucial for the emergence of ferromagnetic moment. Fig.1c magnifie s the graphene layers around one of the \npores. The interlayer spacing of 0.35 nm is revealed as labeled by black arrows in the image, corresponding \nto the van der Waals distance between graphene layers. Inside each graphene layer exists the spacing of 0.21 \nnm which corresponding to the distance between neighboring zigzag edges, as labeled by yellow arrows and \nillustrated by the inset sketch. According to the viewing direction and the appearance of the zigzag edge \nprojection, it is verified that the zigzag di rection of the graphene layer is in parallel with viewing direction , \ni.e., vertical to the substrate surface. The side view TEM image of the VGME film in Fig. 1d confirms that \nmost  graphene layers orient vertically to the film/substrate interface. Fig. 1e further highlights the features of \ncrosslinking, splitting, and merging between neighboring graphene layers by colored dash lines. Such features \nsuggesting the observed pores are not perpendicular linear tunnel through the thickness of the film , instead, \nthey are discontinuously distributed inside  the whole film . The blue square part of the image is magnified in \nFig. 1f, in which a line -shaped projection of the graphene layer is observed with the inner spacing of 0.12 nm, \ncorresponding to the spacing  between two nearest atom ic planes  which are along armchair direction, as labeled \nby yellow lines and illustrated by the inset sketch. The spacing of 0.12 nm in side view co -demonstrate that \nthe vertically aligned edges are zigzag type, as Fig. 1c does.  This stru cture thereby is named as vertical \ngraphene multi -edge (VGME) structure, which consist of multiple graphene layers aligned parallel to the film \ngrown direction. These graphene layers further wove into a three -dimensional hyperfine -porous network.   \n \nFig. 1 Atomic image of vertical graphene multi -edge (VGME) structure. a, Plan view HADDF  image  in which the \nsuperfine pores in the structure are observed, as highlighted in the image by dashed circles. The bright lines in the figure \nrepresents the projections of curved graphene layers. A square area b is selected for zoom -in observation. b, zoom -in image \nof the selected area in a . The curve, dangle, and crosslink features of the graphene layers can be observed clearly, as guided \nby dashed lines. A square area c is selected for further zoom -in observation.  c, zoom -in image of the selected area in b. The \natomic structural detail of the graphene layer  can be identified , as indicated by dashed lines . The spacing of 0.35 nm between \ngraphene layers and 0.21 nm between zigzag edges are labeled.  d-f, Side view TEM images  of the VGME  film, progressively \nzooming in . The vertical orientation of the graphene layers can be seen in d; the crosslink, merge and split features can be \nseen in e; the spacing of 0.35 nm between graphene layers and 0.12 nm between nearest atom lin es along armchair direction \nare labeled.  \nThe magnetization (M-H) curve of the above VGME structure is shown in Fig.  2a in blue color . The \ncurve was measured at the temperature of 300 K, and the saturation magnetic field H of 0.7 T. The direction \n \nof magnetization is vertical to the VGME film, that is parallel to the graphene edges. A hysteresis loop with \nsaturation feature is clearly shown in the M -H curve. The magnetization Ms at 0.7 T, the residue magnetization \nat 0 T M R, and the coercive magnetic field H c (at 0 magnetization) are listed inside the figure. The M s value \nof VGME film  at 0.7 T  is 4.7 emu/g, converting to unit magnetiz ation of ~0.000 8 B per atom. This is three \norder of magnitude higher than reported in de fected HOPG23. Furthermore, the M -H curve shows rapid \nincrement at low field, and reaches 4.3 emu/g at ±0.2 T. When magnetic field further increases, the \nmagnetization slowly  saturates . For the need of compare study, we prepared an amorphous carbon film (a -C) \nwith the same technique, during which the irradiating energy of electrons were at a rather low level so that \nVGME structure was not formed.  Compar ing measurement results of HOPG and a-C samples  are also given  \nin Fig . 2a in grey and yellow curves . These curves are magnified for better resolution  in Fig. 2b and c. The  \nMS value of HOPG sample is 1.1 ×10-3 emu/g and the H c value is 25 Oe, close to the earlier reported value 23, \nimplying that the results of our hysteresis measurement are reasonable. The M s value of a -C film is 1.4×10-4 \nemu/g, four orders of magnitude lower than that of VGME structure. Since the VGME and a -C samples are \nprepared by the same technique, in can be  inferred that potential contribution to M s from impurities could  not \nexceed  1.4×10-4 emu/g. Therefore, t he detected hysteresis loop and large saturation magnetization  of VGME \nstructure could be ascribed to bulk carbon ferromagnetism. To verify this interpretation, XMCD measurements \nwere introduced  for elementa l-resolved magnetic detecti on.  \n \nFig. 2 Magnetization curves  of VGME , HOPG and a -C structures at 300 K.  a, the hysteresis loop of VGME with  magnetic \n \nfield H perpendicular to film surface , as indicated inside the figure . The saturation magnetization Ms, residue Magnetization \nMR and coercive field H c derived from the loop are listed. The Ms reaches 4.7 emu/g, equal s to 0.0008 B/atom.  M-H curves \nof and a -C are presented in the same scale in the figure. b, magnified hysteresis loop of HOPG with H perpendicular to \ngraphene layer, as indicated inside the figure. c, magnified  hysteresis loop of a -C with H perpendicular to the film surface.  \nFig.3a summarizes  the X -ray absorption near edge spectra (XANES) collected from VGME and a-C film \nas well as HOPG in vertical and horizontal  positioning directions . The near -edge detailed spectra are \nhighlighted in yellow background, and the X -ray inject direction are presented above each spectrum. All \nsample s show  clear C -1s K -edge absorption  signals . The XANES of the VGME sample shows a strong peak \nat * position  of ~284.5 eV , which is more similar to that of vertical HOPG than to horizontal  HOPG, \nsuggesting the strong X -ray absorption from 2p orbital. This is due to the reason that the  incident direction of \nthe X-ray was  perpendicular  to the film surface, and the graphene layers in VGME are vertically grown as \nwere observed in TEM images. It should be po inted out that the * peak fro m VGME sample is extremely \nstronger than from  the other three. Such feature can be ascribed to the vertical graphene multi -edge structure , \nwhich produces abundant edge states those are more active under X-ray excitation . In order t o check whether \nour results were  affected by magnetic impurities, t he absorption spectra at the energy ranges of L -edge of Fe, \nCo, and Ni were acquired , and no signals were detected, as can be seen in Fig 2b -d. This indicates that the \nobserved magnetization signal of the sample does not originate from magnetic impurities other than carbon.  \nIn addition, the trace of metallic impurities in VGME film were analyzed by inductively coupled plasma mass \nspectrometry (ICP -MS), and the results showed that the total content of metallic impurities (mainly Al, Cr and \nFe) are less than 30 ppm (see Fig. S 3). Such content level coincides with the above interpretation. Therefore, \nthe influence of metallic impurities on the electronic structure of total VGME sample, for example, via \nalloying or proximity effect , can be neglected.   \n \nFig.3 X-ray absorption spectroscopy of the VGME, HOPG and a -C structures. a,  C-1s K-edge absorption spectra of \nVGME, a -C and HOPG (vertical and horizontal ) samples. The near -edge detailed spectra are highlighted in yellow \nbackground, and the X -ray incident direction are presented beside  each spectrum.  b-d, absorption spectra of VGME sample \nat the energy ranges of L -edge of Fe, Co, and Ni , respectively . No signal  is observed, suggesting the influence of magnetic \nimpurities can be neglected.  \nFig. 4 compa res the XMCD spectra of VGME and a -C structure, which are detected at 300 K with the \nexternal , out-of-plain magnetic field of 0.7 T. In order to guarantee the reproducibility and increase signal -to-\nnoise ratio, the spectra were repeatedly taken for 10 times, and the average  signals from  increasing scanning  \ntimes are exhibited in Fig 4a -f. As sh own in the figure s, the dichroism of * peak at ~284.5 eV  becomes more \nprofound  after multiple repeating m easurements on VGME sample , and the noise level is effectively reduced, \nso that the dichroism peak can be ide ntified at rather low amplitude.  The profound feature of the  dichroism  at \n* excitation en ergy of VGME structure provides a direct evidence of bulk carbon ferromagnetism originated \nfrom 2p orbital. Since  the dichroism appears at the same energy as in XA NES spectra, its origin can be ascribed \nto the edge states in the VGME structure, as mentioned above. The amplitude of VGME -XMCD peak is \n~0.005 and stabled after repeatedly mea surements, corresponding to the effective magnetic moment of \n~0.000 6 B per atom  by relative area ratio metho d (see description in method section) . This value is close to \n \nits M s that calculated from the hysteresis loop.  On the other hand, after 10 times of repeating  measurements, \nthe XMCD spectrum of a -C film only shows fluctuations at the amplitude of ~0.002 , such amplitude is \ncomparable to background noise level of the measurement s, as indicated  in each panel  with parallel dashed \nlines . We note that  for the L -edge absorption spectra of transition metals, spin and orbital angular momentum \ncan be separate d by introducing sum rule28. Yet so far, this methodology is not suitable for the K -edge \nabsorption spectra as in the case of carbon. Recent breakthrough in the analysis of K-edge spectra  shed light \non extracting the orbital contribution to the s -p dichroism signal29.  Therefore, we anticipate the XMCD of C -\n1s K -edge to unveil more information about the origin of bulk carbon ferromagnetism and benefits  deeper  \nunderstanding of carbon electronic structures.  \n \nFig. 4 XMCD spectra of VGME and a -C structure s detected at 300 K with the external out -of-plain magnetic field of \n0.7 T. a -g, the average results of XMCD spectra of VGME structure with increasing scan times from 1 to 10 scans. The \n \ndirhosim peak s at ~28 4.5 eV are indicated by solid curve s in each spectrum. h, the average result XMCD spectra of a -C \nstructure after 10 scans. The noise range s between ±0.002 a.u. are indicated by parallel dashed lines  in each panel . \nFor a bulk material with long -range ferromagnetic order, one should expect multiple magnetic dynamic \nbehaviors other than a classic hysteresis loop. For such purpose, we investigated the spin trans port behavior \nin the VGME film. Fig. 5a-b illustrate the principle of the measurement setup. A platinum layer Pt is coated \non top of the VGME or a -C film to  build  a double layer junction, being referred as Pt/VGME or Pt/a -C.  When \nan electric current J c is injected through Pt layer along X-axis, a spin current J s is naturally formed along Z-\naxis due to spin -Hall effect  (SHE) , in which the spin polarized electrons diffuse to the interface of the junction. \nSimultaneously, a reversed bounced back spin current J’s1 at the interface is generated, and transfer in to J c due \nto inverse spin -Hall effect  (ISHE) , keeping the overall J c barely changed.  Under this condition, an external \nmagnetic field B is further applied along different directions, by which the magneti c moment in VGME is \ndetermined if B is large enough. When B is along Y -axis as shown in Fig . 5a, the magnetic moment in VGME \nM1 is with the same direction as in the spin polarized electron, therefore, no spin moment is absorbed and J c \nis unchanged. When B is along X -axis as Fig . 5b illustrates, the magnetic moment in VGME M2 is \nperpendicular to that of the spin polarized electron, the spin moment will be absorbed by VGME and the \nbounced back spin current decreases to J’ s2. Therefore, the overall charge current after ISHE is also lowered \nto J’ c. Such angular dependence of can be detected by measuring the change of longitudinal resistance xx in \nPt/VGME or Pt/ a-C with respect to the rotation angle of  B. Fig. 5c-e summarize the angular dependences of \nxx of Pt/VGME and Pt/ a-C on the rotation of magnetic field B in X-Y ( angle), Y-Z( angle) and X-Z \n( angle) plane s, respectively. A periodical response of xx with the angular change is clearly observed in \nPt/VGME , whereas in  Pt/a-C the resistance barely ch anges  (xx maintains  near zero). It can be inferred that \nthe angular dependent magnetoresistance  is relate d to VGME film. The xx along  angle shows a cosine \nfeature, where the resistance is lower when B||Y-axis, and higher when B ⊥Y-axis, in accordance to the \nbehavior of spin Hall magnetoresistance. We also notice that when  B rotates in  and  angle, same angular  \ndependence appears with larger amplitude.  This could be from the negative magnetoresistance of the VGME \nfilm, which is not our focus here . By subtracting  xx along   angle from xx along  angle, we obtained a \nchange of  xx exactly equals to xx along  angle.  The angular dependent magnetoresistance  behavior \nalong -angle imply s a classic spin Hall magnetoresistance ( SMR ) process30,31. The observation of such SMR \nbehavior at 300 K further proves the robust of the magnetization in VGME structure.  \n \nFig. 5 Spin Hall magnetore sistance in VGME structure. a -b, The principle  of current J c variation in response to external \nmagnetic field B in parallel and perpendicular to Y axis . Js represents the spin current generated in Pt layer via spin Hall \neffect. J’ s1 and J’ s2 represent the bounced spin current when B||Y and B ⊥Y, respectively. M 1 and M 2 represent the \nmagnetization moment of VGME by B  in different directions.  c-e, the change of longitudinal resistance  xx in response to  \nrotati ng B  of 9 T around ,  and  angles  at 300 K . The definition of each angle is illustrated inside each figure. Blue \ncurves represent  xx of Pt/VGME bilayer, and orange c urves represent xx of Pt/a-C bilayer . \nIn conclusion, w e produced an all-carbon film consisting of vertically aligned graphene edges, which \nwove into a three -dimensional hyperfine -porous network.  A reproducible, full section of XMCD spectr um \nacross  C-1s K-edge absorption  energy was detected for the first time, and the average amplitude of dichroism  \nat 28 4.5 eV reaches ~0.0 05 at the 300 K, equals to uni t magnetic moment of ~0.000 6 B per atom. The \nmagnetization mea sured by XMCD is co -confirmed by  hysteresis loop  of M -H curve . The influence of \nmagnetic impurities on magnetization was excluded  by XA NES and ICP -MS measurements. Spin transfer \n \nbehavior of SMR was observed in the VGME film, which verified the long -range and robust features of the \nbulk ferromagnetic order at room temperature . Our work provides direct evidence of elementary resolved bulk \ncarbon ferromagnetism at room temperature and clarifies its origin from -electrons at graphene edges.  \n \nMethods:  \nSample preparation:  \nA silico n wafer of 0.5 mm thickness with a 300 nm oxide layer was used as the substrate for the film growth. \nThe thickness of the VGME was controlled via deposition time. For spin transport  measurement s, Pt layers \nwith the thickness of 5 nm were sputtered via thermal evaporation.  \nTEM characterization:  \nFor TEM characterization, the carbon film with 20nm thickness was transferred from Si substrate to a lacey \ncarbon TEM grid for plan view characterization. The side view lamella was produced by a Focused \nIon/Electron Dual beam system (Thermo -Fisher Scios). Both plan and side view characterization were \nperformed by a double aberration -corrected electron microscope (Thermo -Fisher, Titan Themis G2) with \nCeta2 CMOS camera and HAADF detector in TEM and HAADF STEM modes. The acceleration voltage of \nelectron beam w as set to 80kV to minimize the beam damage to the carbon film.  \nRaman spectroscopy : \nFor Raman  measurement, the carbon film was deposited on Si substrate with the thickness of 20 nm. The \nspectra were obtained on H ORIBA  HR Evolution. An excitation laser with the wavelength of 532 nm was \nused with the emitting power of 1 µW, and the intensity coefficient of 3.2% was used for measurement. The \nnumerical hole -diameter of the project lens was 1  m. The measurement on each sample was repeated three \ntimes at different locations randomly selected on the film surface. The irradiating time of focused laser bea m \non each detecting area was less than 5 min.  \nM-H curves:   \nFor M -H measurement, the carbon film was deposited on Si substrate with the thickness of 300 nm. The \nsample containing carbon film and S i substrate was cut into the in -plane size of 2 ×4 mm2, and weighted on a \ndigital weighing balance. The result was marked as m 1. The sample was glued onto a plastic sample holder \nin the position that the film surface was perpendicular to the external magnetic line. Hysteresis curve of the \nsample was obtained with a physical property measurement system (on Quantum Design, MPMS3). T he raw \nmagnetization result was  recorded  as M 1. The Si substrate without carbon film was measured with the same \nmethod. The weight and magnetization of the Si  was recorded as m 2 and M 2. Since the thickness of the \ncarbon film (300 nm) was much smaller than that of the substrate (0.5 mm), the contribution of the carbon \nsample to m 2 can be neglected (less than 1‰). Therefore, the magnetization response of carbon film can be \ncalculated as M 1- m1×M2/m2. \nLaser ablation inductively coupled plasma mass spectrometry (LA -ICP-MS): \nFor LA-ICP-MS measurement s, the carbon film was deposited on Si substrate with the thickness of 300 nm. \nThe as -deposited carbon film on Si substrate was cleaned by ultrasonic cleaning in acetone bath and dried \nnaturally. The cleaned sample was mounted flatly in the sample chamber o f laser -ablation chamber  \n(NWR193HE ) which was coupled with the ICP-MS system . The chamber was closed and Ar was introduced \nas background. A pulse laser with the frequency of 5 Hz was introduced  to evaporate the film. The sampling \ntime at each position was 40 s, and the process repeated at 24  different positions in total. A NIST -610 sample \nwas used for concentration calibration of the most possible metallic impurities, including Al, P, Ti, V , Cr, Mn, \nFe, Co, Ni, Cu, Zn, and Pb.  \nX-ray photon spectroscopy (XPS):  \nFor XPS measurement, the carbon film was deposited on Si  substrate with the thickness of 20 nm. The as \ndeposited sample was cleaned in acetone bath and dried naturally before measurement. The film surface was \ntested firstly, followed by Ar plasma etching for 2 min to remove the part that contacted with atmosphere, and \nthe same test was carried out on the fresh -exposed surface. An X -ray source was used, and the whole spectr um  \nwere collected in the energy range from  0 to 1,350 eV . O 1s and C 1s spectra was magnified to check the \ncomposition difference between film body and surface.  \nX-ray magnetic circular dichroism  \nFor XMCD measurement, 10 times of repeatedly scan were carried out continuously. Each time a MXCD \ncurve was derived from a pair of XA NES spectra taken under opposite magnetic fields. The integrated area of \nXMCD curve AXMCD  was calculated by summing up the normalized amplitude of all data points. The area of \nXANES spectra AXANES was derived with the same process. The ratio of the area of  XMCD  to the area of \nXANES was calculated , and the value r=AXMCD\nAXANES is considered as the effective magnetic moment from each \ncarbon atom.  \nDevice fabrication:  \nFor transport measurement, the carbon film was deposited on Si substrate with the thickness of 20 nm. The \nsample was cut into pieces (in-plane size 4 × 4 mm2) with a diamond pencil, followed by 3 min  ultrasonic \ncleaning in acetone bath. For Hall measurements, a Hall -bar shaped photoresist coating was fabricated on the \ncarbon film by lithography method, and Ar etching was introduced to remove the photoresist and the \nuncovered carbon film, what left was a Hall -bar shaped carbon film. The line width of the Hall bar is 100 m \n(see Fig. S4) . The electrodes were connected to a standard sample puck with  Pt wires (diameter of 20 m) by \nusing a wiring bonding machine. For  spin transport  measurements, a  Pt film w as sputtered on the whole \nsurface of the carbon film by  DC sputtering in Ar atmosphere, followed by Hall -bar fabrication process as for \nHall measurements, with the  Ar etching time slightly longer.  \nWe also prepared the devices by directly attaching Pt wire on carbon and Pt/carbon film surfaces with silver \npaste instead of fabricating the Hall -bar sample. The results showed that the influence of different techniques \non measurement results can be neglected . \n  \nData availability : \nThe data that support the findings of this study are available from the  corresponding author on reasonable \nrequest.  \nReferences:  \n1. Son, Y . -W., Cohen, M. L. & Louie, S. G. Half -metallic graphene nanoribbons. Nature  444, 347 –349 (2006).  \n2. 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Lett.  75, 152 –155 (1995).  \n29. N’Diaye, A. et al.  Interplay between Transition -Metal K -edge XMCD and Magnetism in Prussian Blue \nAnalogs. ACS Omega  7, 36366 –36378 (2022).  \n30. Avci, C. O. et al.  Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers. Nat. \nPhys.  11, 570 –575 (2015).  \n31. Oyanagi, K. et al.  Paramagnetic spin Hall magnetoresistance. Phys. Rev. B  104, 134428 (2021).  \n \nAcknowledgements:  \nThe authors would like to thank Prof.  Kai Chen  for discussion on  XMCD  technique , and thank Dr.  Duo Zhao  \nfor discussion on SMR measurement. We also thank BL12B beamline of National Synchrotron Radiation \nLaboratory (NSRL, China) and BL07U  beamline of Shanghai Synchrotron Radiation Facility (SSRF).  The \nwork is supported by National Natural Science Foundation of China  (grant No. 51875364 ) \nAuthor contributions:  \nC.W, D.D and E.S. designed the research. N.J. , M.Y . and X. Z. performed the transmission electron \nmicroscopy analysis. Z.Y. and X. M. prepared the sample and assisted with all measurements. T.K. and S. D.  \ndesigned the M -H and SMR measurements,  Q. L, W. Y . and D. H. assisted with XMCD  measurements. L.Y . \nassisted with M -H measurements. D. D. and E.J. supervised the research. C.W.  and D.D. co -wrote the paper. \nAll authors discussed the results and commented on the manuscript.  \nCompeting interests:  \nThe authors declare no competing interests.  \n "    },    {        "title": "0710.5477v1.Thermoelectrical_manipulation_of_nano_magnets.pdf",        "content": "arXiv:0710.5477v1  [cond-mat.mes-hall]  29 Oct 2007Thermoelectrical manipulation of nano-magnets\nA. M. Kadigrobov,1,2R. I. Shekhter,1M. Jonson,1,3and V. Korenivski4\n1Department of Physics, G¨ oteborg University, SE-412 96 G¨ o teborg, Sweden\n2Theoretische Physik III, Ruhr-Universit¨ at Bochum, D-448 01 Bochum, Germany\n3School of Engineering and Physical Sciences, Heriot-Watt U niversity, Edinburgh EH14 4AS, Scotland, UK\n4Nanostructure Physics, Royal Institute of Technology, SE- 106 91 Stockholm, Sweden\n(Dated: October 29, 2018)\nWe propose a device that can operate as a magneto-resistive s witch or oscillator. The device\nis based on a spin-thermo-electronic control of the exchang e coupling of two strong ferromagnets\nthrough a weakly ferromagnetic spacer. We show that the loca l Joule heating due to a high con-\ncentration of current in a magnetic point contact or a nanopi llar can be used to reversibly drive the\nweak ferromagnet through its Curie point and thereby exchan ge-decouple the strongly ferromag-\nnetic layers, which have an antiparallel ground state. Such a spin-thermionic parallel-to-antiparallel\nswitching causes magnetoresistance oscillations where th e frequency can be controlled by proper\nbiasing from essentially DC to GHz.\nManipulation of the magnetic state on the nanome-\nter scale is the central problem of applied magneto-\nelectronics. The torque effect1,2, which is based on the\nexchange interaction of electrons injected into a ferro-\nmagnetic region, is one of the key phenomena leading\nto current-induced magnetic switching. Current-induced\nprecession and switching of the orientation of magnetic\nmoments due to this effect has been observed in many\nexperiments3,4,5,6,7,8,9,10,11,12,13.\nTheefficiencyofcurrent-inducedswitchingis,however,\nlimited by the necessity to work with high current densi-\nties. Anaturalsolutiontothisproblemistouseelectrical\npoint contacts (PCs). This is because now the current\ndensity is high only near the PC, where it can reach14,15\nvalues of order 109A/cm2. The characteristic energies\nsupplied to the electronic system are determined by the\nvoltage drop V, and are for V∼0.1÷1 V (which is eas-\nily reached in experiments) comparable to the exchange\nenergy in magnetic materials.\nOne of the characteristic features of such high current-\ndensity states is the possibility to create local heating\nin regions with a size of a few nanometers. The energy\nsupplied to the electronic system in this restricted region\nresults in an enhanced local temperature, which to a high\ndegree of accuracy can be controlled by the applied volt-\nage. Electrical manipulation of nanomagnetic conduc-\ntors by such controlled Joule heating is a new principle\nfor current-induced magnetic switching. In this Letter\nwe discuss one of the possibilities of the thermoelectri-\ncal magnetic switching causedby a non-linearinteraction\nbetween the spin-dependent electrical transport and the\nmagnetic sub-system of the conductor due to the Joule\nheating effect. We predict that a special design of mag-\nnetic PC can provide both a fast switching and a smooth\nchange of the magnetization direction in nano-size re-\ngions of the magnetic material controlled by the voltage.\nWe also predict temporal oscillations of the magnetiza-\ntion direction in such regions (which are accompanied\nby electrical oscillations) under an applied DC voltage.\nThese phenomena are potentially useful for microelec-\ntronic applications such as memory devices and voltagecontrolled oscillators.\nEquilibrium magnetization distribution in a magnetic\nstack.The system under consideration is shown in Fig.1\nwhere three ferromagnetic layers are coupled to a non-\nmagnetic conductor. We assume the following conditions\nto be satisfied: The Curie temperature T(1)\ncof region 1\nis smaller than the Curie temperatures T(0,2)\ncof regions\n0 and 2; in region 2 there is a magnetic field directed\nopposite to the magnetization of the region, which can\nbe an external field, the fringing field from layer 1, or a\ncombination of the two. We require this combined mag-\nnetostatic field to be weak enough so that at low temper-\natures the magnetization of layer 2 is kept parallel to the\nmagnetization of layer 0 due to the exchange interaction\nbetween them via region 1 (we assume the magnetization\ndirection of layer 0 to be fixed). In the absence of an ex-\nternalfieldandifthetemperature T > T(1)\nc,theproposed\nF/f(N)/F tri-layer is similar to the antiparallel spin-flop\n’free layers’ widely used in memory applications16.\nAsTapproaches T(1)\ncthe magnetic moment of layer\n1 decreases and the exchange coupling between layers 0\nand 2 weakens. This results in an inhomogeneous distri-\nbution of the stack magnetization. Euler’s equation for\nthe magnetization distribution that minimizes the free\nenergy of the system can be written as follows (see, e.g.,\nRef. 17):\nd\ndx/parenleftbigg\nα(x)M2(x)dθ\ndx/parenrightbigg\n−β\n2M2sin2θ+HM\n2sinθ= 0; (1)\nHere the x- andz-axes are directed along the stack and\nthe magnetization direction in the region 0, respectively;\nθ(x) is the angle between the magnetic moment /vectorM(x) at\npointxand the z-axis;M=|/vectorM|;α∼I/aM2, where\nI∼kTcandais the exchange energy and the lattice\nspacing, respectively, kis the Boltzmann constant; βis\nthe dimensionless constant of the anisotropy energy. Be-\nlow we assume the lengths L1,2of regions 1 and 2 to be\nsmaller than the domain wall lengths in these regions.\nUsing Eq. (1) and the boundary condition at the inter-\nfaceSbetween magnetic region 2 and the non-magnetic2\nL2H\nL10 1 2 3\nFigure 1: Orientation of the magnetic moments in a stack of\nthreeferromagnetic layers (0, 1, 2); thepresence ofamagne tic\nfield directed opposite to the magnetization is indicated by an\narrow outside the stack.\nFigure 2: Magnetic moment orientations in the stack at tem-\nperatures T1< T < T(1)\nc.\nregion 3 dθ/dx|S= 0 (see, e.g., Ref. 18) one finds the\nmagnetization in region 1 to be inhomogeneous θ1(x) =\nθ2x/L1,0≤x≤L1, while the magnetic moments in\nregion 2 are approximately parallel with the accuracy\nα1M2\n1(T)/α2M2(T)≪1 (the subscripts 1 and 2 refer to\nthe region 1 and 2, respectively); the dependence of their\ntilt angle θ2onHandTis given by the equation\nθ2=D(H,T)sinθ2;D(H,T) =L1(L2HM2(T))\n4α1M2\n1(T)(2)\nWhile writing Eq.(2) we neglected the anisotropy energy\nin comparison with the magnetic energy in region 2.\nAt low temperatures ( D(T,H)<1) Eq.(2) has the\nsingle root θ2= 0 and hence a parallel orientation of all\nthe magnetic moments in the stackis thermodynamically\nstable. At temperatures T≥T1(hereT1is the tempera-\nture at which D(T,H) = 1) one has D≥1 and a second\nrootθ2/negationslash= 0 appears in addition to the root θ2= 0. The\nparallel magnetization corresponding to θ2= 0 is now\nunstable and the magnetization direction of region 2 tilts\nwith an increase of T(see Fig.2) until at T≥T(1)\ncthe\nexchange coupling between layers 0 and 2 vanishes and\ntheir magnetic moments become antiparallel.\nCurrent-voltage characteristics of the stack under\nJoule heating. If the stack is Joule heated by the cur-\nrentJits temperature T(V) is determined by the heat-\nbalance condition\nJV=Q(T), J=V/R(θ2) (3)\nand Eq. (2) that determines the temperature dependence\nofθ2(T(V)). Here Q(T) is the heat flux from the stack,\nR(θ2) is the stack resistance. Eqs. (2) and (3) define the\ncurrent-voltage characteristics (IVC) of the stack J=\nR(θ2(V))Vin a parametric form.\nAs follows from the previous paragraph, in the whole\nintervals T(V)< T1andT(V)> T(1)\ncthe stack re-\nsistances are R(0) and R(π), respectively, that is IVC1.2\n1.41.4\n1.20.8\n0.80.4\n0.40J/J\nV/VC\nC..aa’\nbb’\nFigure 3: Numerical results for the I−Vcharacteristics\n(IVC) of the magnetic stack. Here R(θ2) =R+−R−cosθ2,\nR−/R+= 0.2,D0= 0.2;Jc=Vc/R+. The branches 0 −a′\nandb−b′of the IVC correspond to the parallel and antipar-\nallel orientations of the stack magnetization, respective ly; the\nbrancha−bcorresponds tothe inhomogeneous magnetization\ndistribution shown in Fig.2\nbranches J=R(0)VandJ=R(π)Vare linear at\nV < V 1=/radicalbig\nR(0)Q(T1) andV > V c=/radicalBig\nR(π)Q(T(1)\nc),\nrespectively. At V1≤V≤Vcthe stack temperature is\nT1≤T(V)≤T(1)\nc, and the magnetization direction of\nregion 2 changes with a change of Vand hence IVC is\nnon-linear there. Differentiating Eqs.(2,3) with respect\ntoVone finds that inside this interval the differential\nconductance Rd(V) =dJ/dVis negative if d[G(θ2)(1−\nD0sinθ2/θ2)]/dθ2<0(hereD0=D(H,T= 0)); e.g., for\nthe stack resistance of the form R(θ2) =R+−R−cosθ2)\n(hereR±= (R(π)±R(0))/2) one has dJ/dV < 0 if\nD0(H)<3r/(1+3r) wherer=R−/R+, and hence IVC\nof the stack is N-shaped as shown in Fig.3.\nSelf-exciting electric, thermal and magnetic direction\noscillations . The set of equations describing thermal and\nelectrical processes in the stack connected in series with\nan inductance Land DC voltage drop ¯V(see Fig.4) is\nCVdT\ndt=J2R(θ2)−Q(T);LdJ\ndt+JR(θ2) =¯V(4)\nwhereCVistheheatcapacity. Forthe casethat themag-\nnetic moment relaxation is the fastest process the tem-\nperature dependence of θ2=θ2(T(t)) is given by Eq.(2).\nThe set of equation Eq.(4) has always a steady-state\nsolutionEqs.(2,3). Astudy ofitsstability with respectto\nsmall perturbations shows that inside the interval V1≤\n¯V≤Vcthis solution is unstable if L>Lcr=tv|Rd(¯V)|\nwheretv∼(TCVR+/¯V2) isthe characteristictime ofthe\nvoltageevolutionand Rd=dV/dJ < 0isthe stackdiffer-\nential resistance. As a result periodic oscillations of the\ncurrentJ(t) and the voltagedrop on the stack V(t) spon-\ntaneous arise: the current J(t) and the voltage drop on\nthe stack V(t) periodically move along the limiting cycle\nin the plane ( J,V) as is shown in Fig.5. The temper-\nature of the stack T(t) and the magnetization direction3\nFigure 4: Equivalent circuit for a Joule-heated magnetic\nstack. A differential resistance Rd(V) biased by a fixed DC\nvoltage¯Vis connected in series with an inductance L;V(t) is\nthe voltage drop over the stack and J(t) is the total current\n1.2J/J\nV/V\n0.90.95 0.9 1.05 1.11.1\ncc\n. .\nFigure 5: Numerical results showing spontaneous oscillati ons\nofthecurrent J(t)andthevoltagedrop V(t)overthestackfor\nthe parameters R−/R+= 0.2,D0= 0.1 and (L−L cr)/Lcr=\n0.25;Jc=Vc/R+. The time development of the current and\nthe voltage drop over the stack follows the dashed line or\nthe thin solid line towards the limiting cycle (thick solid l ine)\ndepending on the initial state.\nθ2(t) adiabatically follows these electric oscillations ac-\ncording to the following equalities: Q(T(t)) =V(t)J(t)\nandθ2(t) =θ2(T(t)) (see Eq.(2)) For ( L−Lcr)/Lcr≪1\nthe oscillation frequency is ω= 1/tv.\nWith afurther increaseofthe inductance the character\nof the periodic motion changes its character: at L ≫Lcrthe current and the voltage slowly move along the\nbranches of the IVC 0 aandbb′with the velocity ˙J/J≈\n1/tj(heretJ=L/R+is the characteristic time of the\ncurrent evolution) fast switching between them at points\naandbwith the velocity ∼1/tv. Therefore, in this case\nthe stack periodically switches between the parallel and\nantiparallel magnetic states.\nIn conclusion , we have shown that Joule heating of\nthe magnetic stack presented in Fig.1 allows electrical\nmanipulations of the mutual magnetization orientation\nof layers 0 and 2 as follows:\n1) In the regime of a controlled current there is a hys-\nteresis loop: with an increase of the current along the\na-a’ IVC branch the magnetic directions are parallel up\nto point a. At this point the stack is heated up to the\ntemperature T1at which D= 1 and the parallel ori-\nented state looses its stability. As a result the system\nswitches to the b−b′branch of IVC at which T > T(1)\nc\nand the magnetic moment /vectorM2switches to the antiparal-\nlel orientation. With a further change of the current the\nstack remains on the b-b’ branch of IVC until the current\nreaches point bwhere the stack is cooled below T(1)\ncand\nsystem switches again to the a−a′branch at which the\nmagnetic moments are parallel.\n2) At the regime of a controlled voltage and a low\ninductance one may smoothly re-orient the magnetiza-\ntion directions from the parallel to the antiparallel (see\nFig.2) changing the applied voltage inside the interval\nV1≤V≤Vcthat is moving along a−bbranch of IVC.\n3) An increase of the inductance of the circuit under a\nfixed DC voltage(see Fig.4) allowsto excite electrical pe-\nriodic oscillations accompanied by periodical switchings\nof the magnetization directions in the stack from parallel\nto the antiparallel orientation.\nAcknowledgement. FinancialsupportfromtheSwedish\nKVA, VR, and SSF is gratefully acknowledged.\n1J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996);\nibid.195, L261 (1999).\n2L. Berger, Phys. Rev. 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Lett. 84, 945\n(2004).\n13M.Kerekes, R.C. Sousa, I.L. Prejbeanu, O. Redon, U.\nEbels, C. Baraduc, B. Dieny, J-PNozeres, P.P. Freitas, and\nP. Xavier, J. Appl. Phys. 97, 10P501 (2005); R.C. Sousa,\nM.Kerekes, I.L. Prejbeanu, O. Redon, J. Appl. Phys. 99,\n08N904 (2006). B. Dieny, J.P Nozeres,\n14A.V. Khotkevich and I.K. Yanson, Atlas of Point Contact\nSpectraof Electron-Phonon Interactions in Metals, Kluwer\nAcademic Publishers, Boston/Dordrecht/London (1995)\n15J.J. Versluijs, M.A. Bari, and M.D. Coey, Phys. Rev. Lett.\n87, 026601 (2001).\n16D. C. Worledge, IBM Journal of Research and Develop-4\nment50, 69 (2006).\n17L.D. Landau, E.M. Lifshits, and L.P. Pitaevski, Elec-\ntrodynemics of Continuous Media, 43, Butterworth-\nHeinemann, Oxford, Pergamon Press 1998.18A.I. Akhiezer, V.C. Bar’yakhtar, S.V. Peletminskii, Spin\nWaves, 5.4, p.42, North-Holland Publishing Company -\nAmsterdam John Wiley & Sons, INC. - New York, 1968."    },    {        "title": "1402.4346v1.The_Complexity_of_Ferromagnetic_Two_spin_Systems_with_External_Fields.pdf",        "content": "arXiv:1402.4346v1  [cs.CC]  18 Feb 2014The Complexity of Ferromagnetic Two-spin\nSystems with External Fields\nJingcheng Liu1, Pinyan Lu2, and Chihao Zhang1\n1Shanghai Jiao Tong University, {liuexp, chihao.zhang }@gmail.com\n2Microsoft Research. pinyanl@microsoft.com\nAbstract. We study the approximability of computing the partition\nfunction for ferromagnetic two-state spin systems. The rem arkable algo-\nrithmbyJerrumandSinclairshowedthatthereisafullypoly nomial-time\nrandomized approximation scheme (FPRAS) for the special fe rromag-\nnetic Ising model with any given uniform external field. Late r, Goldberg\nand Jerrum proved that it is # BIS-hard for Ising model if we allow incon-\nsistent external fields on different nodes. In contrast to the se two results,\nwe prove that for any ferromagnetic two-state spin systems e xcept the\nIsing model, there exists a threshold for external fields bey ond which the\nproblem is # BIS-hard, even if the external field is uniform.\n1 Introduction\nSpin systems are well studied in statistical physics and applied proba bility. We\nfocus on two-state spin systems in this paper. An instance of a spin system is\ndescribed by a graph G(V,E), where vertices are particles and edges indicate\nneighborhood relation among them. A configuration σ:V→{0,1}assigns one\nof the two states to every vertex. The contribution of local inter actions between\nadjacent vertices is quantified by a matrix A=/bracketleftbigg\nA0,0A0,1\nA1,0A1,1/bracketrightbigg\n=/bracketleftbigg\nβ1\n1γ/bracketrightbigg\n, where\nβ,γ≥0. The contribution of vertices in different spin states is quantified b y\na vector b=/bracketleftbiggb0\nb1/bracketrightbigg\n=/bracketleftbiggµ\n1/bracketrightbigg\n, where µ >0. Thisµis also called the external\nfield of the system, which indicates a priori preference of an isolate vertex. The\npartition function Z(β,γ,µ)(G) of a spin system G(V,E) is defined to be the\nfollowing exponential summation:\nZ(β,γ,µ)(G) =/summationdisplay\nσ∈{0,1}V/productdisplay\nv∈Vbσv/productdisplay\n(u,v)∈EAσu,σv.\nWe call such a spin system parameterized by ( β,γ,µ). If the parameters are\nclear from the context, we shall write Z(G) for short. Although originated from\nstatistical physics, the spin model is also accepted in computer scie nce as a\nframework for counting problems. For example, with β= 0,γ= 1 and µ= 1,\nZ(β,γ,µ)(G) is the number of independent sets (or vertex covers) of the gra phG.Given a set of parameters ( β,γ,µ), it is a computational problem to com-\npute the partition function Z(β,γ,µ)(G) where graph Gis given as input. We\ndenote this computation problem as Spin(β,γ,µ) and want to characterize their\ncomputational complexity in terms of β,γandµ. Forexactcomputation, poly-\nnomial time algorithms are known only for the very restricted settin gs that\nβγ= 1 or ( β,γ) = (0,0), and for all other settings the problem is proved to\nbe #P-hard [2]. Therefore, the main focus is to study their approxim ability.\nFor any given parameter ε >0, the algorithm outputs a number ˆZsuch that\nZ(G)exp(−ε)≤ˆZ≤Z(G)exp(ε) and runs in time poly(n,1/ε), where nis the\nsize of the graph G. This is called a fully polynomial-time approximationscheme\n(FPTAS). The randomized relaxation of FPTAS is called fully polynomial- time\nrandomized approximation scheme (FPRAS), which uses random bits and only\nrequires the final output be within the required accuracy with high p robability.\nThe spin systems ( β,γ,µ) are classified into two families with distinct phys-\nical and computational properties: ferromagnetic systems ( βγ >1) andanti-\nferromagnetic systems ( βγ <1). We shall denote the corresponding computa-\ntion problems respectively by Ferro(β,γ,µ) andAnti-Ferro (β,γ,µ), so as to\nemphasize which family these parameters belong to. Systems with βγ= 1 are\ndegenerate and trivial both physically and computationally. As a res ult, we only\nstudy systems with βγ/ne}a⊔ionslash= 1.\nGreat progress has been made recently for approximately comput ing the\npartition function for anti-ferromagnetic two-spin systems: it ad mits an FPTAS\nup to the uniqueness threshold [19,12,16,13], and is NP-hard to appro ximate in\nthe non-uniqueness range [17,6]. The uniqueness threshold is a phas e transition\nboundary in physics. It is widely conjectured that the computation al difficulty\nis related to the phase transition point in many problems; this is one of the very\nfew examples where a rigorous proof was obtained.\nFor ferromagnetic systems, the picture is quite different. The uniq ueness\ncondition does not coincide with the transition of computational diffic ulty and it\nis not clear whether it plays any role in the computational difficulty. In a seminal\npaper [10], Jerrum and Sinclair gave an FPRAS for ferromagnetic Isin g model\nβ=γ >1withanyexternalfield µ.Thus,thereisnotransitionofcomputational\ndifficulty for ferromagnetic Ising model, which contrasts the situat ion for anti-\nferromagnetic Ising model β=γ <1. For general ferromagnetic spin systems\nwith external field, the approximability is less clear. Since the Ising mo del (β=\nγ) is solved, we focus on the case β/ne}a⊔ionslash=γand always assume β < γby symmetry.\nBy transferring to Ising model, an FPRAS was known for the range o fµ≤/radicalbig\nγ/β[9].\nOn the other hand, a hardness result was obtained for Ising model with in-\nconsistent external fields [7]. This is a generalization of the spin syst em where\nthe external fields for different vertices can be different and take n from a setV.\nWe useSpin(β,γ,V) (Ferro(β,γ,V) orAnti-Ferro (β,γ,V) ) to denote this\ncomputation problem. It is proved that the Ising model with arbitra ry external\nfieldsFerro(β,β,(0,+∞)) is #BIS-hard,namelythe problemis atleast ashard\nas counting independent sets on bipartite graphs (# BIS). #BISis a problem ofintermediate hardness and has been conjectured to admit no FPRA S [5]. The\nreduction used here is called approximation-preserving reduction a s introduced\nin [4]: Let A,B:Σ∗→Rbe two functions. An approximation-preserving reduc-\ntionfromAtoBis a randomized algorithm that approximates Awhile using\nan oracle for B. We write A≤APBfor short if an approximation-preserving\nreduction exists from AtoB. To get that # BIS-hardness result, one need to use\n(or simulate) both arbitrarily small and large external fields. As β < γ, we can\nalways simulate some arbitrarily small external fields with gadgets. H owever,\nsimulating arbitrarily large external fields is only possible when βµ+1> µ+γ,\nin which case one gets a # BIS-hardness result similarly. If this is not the case,\nand in particular if β≤1< γ, no hardness result was known for any bounded\nexternal fields. These systems havecertain monotonicity proper ty, so all external\nfields that can be simulated by gadgets are inherently bounded by ab ove. It was\nnot even clear whether there is any hardness result or not. As a fir st result, we\nshow that the problem is already hard as long as we allow sufficiently larg e (yet\nstill bounded by above) and vertex-dependent external fields.\nTheorem 1. For anyβ < γwithβγ >1, there exsits a bounded set Vsuch that\nFerro(β,γ,V)is#BIS-hard.\nThe main difficulty is for the case of β≤1, for which we cannot simulate\nany external field larger than the upper bound of V. We overcome this difficulty\nby making use of a recent beautiful result in [3]. Instead of startin g with the\nindependent set problem on arbitrary bipartite graphs, we start w ith a soft\n(βγ >0)anti-ferromagnetictwo-spinsystemon boundeddegreebipart itegraphs\nfrom [3]. Then all the external fields needed for the reduction are b ounded.\nHowever, in the above reduction, we do need vertices to have differ ent ex-\nternal fields to make the reduction go through. This gives a hardne ss result for\nFerro(β,γ,V) but not Ferro(β,γ,µ) for a single µ. It is more interesting and\nintriguing (both physically and computationally) to understand the c omputa-\ntional complexity of a uniform spin system ( β,γ,µ) with the same external field\nµon all the vertices. As our main result of this paper, we also prove # BIS-\nhardness on this uniform case for sufficiently large single external fi eldµ. We\nprove that when µis sufficiently large, we can realize by sufficient precision of\nall the external fields which is smaller than µ∗(µ,β,γ), where µ∗(µ,β,γ) is a\nfunction of µ,βandγ, and approaches infinity as µgoes to infinity. Then by\nchoosinglargeenough µand makinguse ofTheorem1, weget ourmain theorem.\nTheorem 2. For anyβ < γwithβγ >1,there exist a µ0such that Ferro(β,γ,µ)\nis#BIS-hard for all µ≥µ0.\nOur main technical contribution is the construction of a family of gad gets\nto simulate a given target external field. We use a reverse idea of co rrelation\ndecay to do that. Correlation decay is proved to be a very powerfu l technique\nto design FPTAS for counting problems (see for examples [19,1,13,16,1 4,15]). In\ncorrelationdecaybasedFPTAS, one firstestablishesa treestruc ture andhope tocompute the marginal probability of the root. With a recursive relat ion, one can\nwrite the marginal probability of the root as a function of that of its sub-trees,\nthen truncate the computation tree at certain depth and do a rou gh guess at\nthe leaf nodes. A correlation decay property ensures that the er ror for the root\nis exponentially small although there are constant error for the lea ves. Here, we\nuse a similar idea to construct a tree gadget so that the marginal pr obability for\nthe root is very close to a target value. Using the same recursion, o ne translates\nthe target marginal probability for the root to that of its sub-tre es. In the leaf\nnodes, we simply use some basic gadgets to approximate the target marginal\nprobabilities. Again, although these approximation for leaves may ha ve constant\ngap, the error at the root is exponentially small thanks to the corr elation decay\nproperty. We believe that this idea of using an algorithm design techn ique to\nbuild gadgets and get hardness result is of independent interest an d may find\napplications in other problems.\nWe also make some improvements on the algorithm side showing that th ere\nis an FPRAS if µ≤γ/β. We remark that all the computational problem\nFerro(β,γ,µ) andFerro(β,γ,V) is no more difficult than # BIS, as we can\nuse the standard transformation to transform any ferromagne tic two-spin sys-\ntem to ferromagnetic Ising model with possibly different external fi elds and use\nthe #BIS-easiness result in [7]. Thus, the two # BIS-hardness theorems can also\nbe stated as # BIS-equivalent. We believe that the conjecture here is that for\nany fixed β < γ, there exists a critical µcsuch that it admits an FPRAS if the\nexternal field µ < µ c, and it is # BIS-equivalent if µ > µ c. The result of this\npaper is an important step towards this dichotomy.\n1.1 Related works\nThe approximationfor partition function has been studied extensiv ely with both\npositive and negative results [19,1,13,16,10,11,18,8,17,6]. For the algor ithm side\nof ferromagnetic two-spin systems, besides the FPRASes, there is also a recent\ndeterministic FPTAS for certain range of the parameters based on correlation\ndecay and holographic reduction [15].\n2 Bounded Local Fields\nIn the section, we show that bounded local fields are sufficient to es tablish a\nhardness result. The following theorem is a formal statement of Th eorem 1.\nTheorem 3. Letβ < γ,βγ >1,∆=⌊√βγ+1√βγ−1⌋+1andµ >/parenleftBig/radicalBig\nγ\nβ/parenrightBig∆\n. Then\nFerro(β,γ,[1,µ])is#BIS-hard.3\n3Technically, we should only define the problem by a finite set o f external fields. In\nthis paper and as in many others, we adopt the following conve ntion: when we say\na problem with an infinite set of external fields is hard, it mea ns that there exists a\nfinite subset of external fields to make the problem hard alrea dy.We first introduce our starting point from anti-ferromagnetic Isin g model on\nbipartite graphs, and show the reduction in the second subsection .\n2.1 Anti-ferromagnetic Spin Systems on Bipartite Graphs\n#BISis a special anti-ferromagnetic two-state spin system. Similar to # BIS,\none can also study other anti-ferromagnetic two-state spin syst ems on bipartite\ngraphs. We use a prefix Bi-to emphasize that input graphs are bipartite, and\na subscript ∆to indicate that maximum degree is ∆. For instance, the prob-\nlem ofAnti-Ferro (β,γ,µ) on bipartite graphs with maximum degree ∆, is\ndenoted shortly by Bi-Anti-Ferro ∆(β,γ,µ). The following theorem from [3]\nis the starting point of our reduction.\nTheorem 4 ([3]). Suppose a set of anti-ferromagnetic parameters (β,γ,µ)lies\nin the non-uniqueness region of the infinite ∆-regular tree T∆and that√βγ≥√∆−1−1√∆−1+1, andβ/ne}a⊔ionslash=γorµ/ne}a⊔ionslash= 1. ThenBi-Anti-Ferro ∆(β,γ,µ)is#BIS-hard.\nFor simplicity, we use the special anti-ferromagnetic Ising model β=γ <1\nin our reduction, where the non-uniqueness condition is easy to sta te.\nProposition 5. Ifβ <∆−1\n∆+1, then there exists a critical activity µc(β,∆)>1\nsuch that the Gibbes measure of Ising model (β,β,µ)on infinite ∆-regular tree\nT∆is unique if and only if |logµ|≥logµc(β,∆).\nProposition 5 is folklore, a proof can be found in, e.g. [16]. Combining t hese\ntwo results, we can get\nCorollary 6. For all0< β <1, there is an ε >0such that for any µ∈\n(1,1+ε),Bi-Anti-Ferro ∆(β,β,µ)is#BIS-hard, where ∆=⌊1+β\n1−β⌋+1.\nProof.As∆=⌊1+β\n1−β⌋+1, we know that β <∆−1\n∆+1. Then by Proposition 5, we\ncan choose ε=µc(β,∆)−1 and get that ( β,β,µ) is in the non-uniqueness region\nof the infinite ∆-regular tree T∆for allµ∈(1,1+ε). In order to make use of\nTheorem 4 and conclude our proof, we only need to verify that β≥√∆−1−1√∆−1+1.\nOur choice of ∆is the smallest integer to satisfy β <∆−1\n∆+1. As a result, we have\nβ≥∆−1−1\n∆−1+1≥√∆−1−1√∆−1+1. ⊓ ⊔\n2.2 The Reduction\nLemma 7. For any β < γwithβγ >1,µ >1and integer ∆ >1, we have\nBi-Anti-Ferro ∆/parenleftbigg1√βγ,1√βγ,µ/parenrightbigg\n≤APBi-Ferro ∆/parenleftBigg\nβ,γ,/bracketleftBigg\n1\nµ/radicalbiggγ\nβ,µ/parenleftbigg/radicalbiggγ\nβ/parenrightbigg∆/bracketrightBigg/parenrightBigg\n.Proof.Let bipartite graph G(L∪R,E) be an instanceofanti-ferromagneticIsing/parenleftBig\n1√βγ,1√βγ,µ/parenrightBig\nwith maximum degree ∆. We construct an instance of ferromag-\nnetic system with exactly the same graph. Each vertex u∈Lwith degree du\nhas weight µ/parenleftBig/radicalBig\nγ\nβ/parenrightBigdu\n, and each vertex v∈Rhas weight1\nµ/parenleftBig/radicalBig\nγ\nβ/parenrightBigdv\n. Then the\nmaximum possible external field is µ/parenleftBig/radicalBig\nγ\nβ/parenrightBig∆\nwhile the minimum one is1\nµ/radicalBig\nγ\nβ.\nTherefore, it is indeed an instance of Bi-Ferro ∆/parenleftbigg\nβ,γ,/bracketleftbigg\n1\nµ/radicalBig\nγ\nβ,µ/parenleftBig/radicalBig\nγ\nβ/parenrightBig∆/bracketrightbigg/parenrightbigg\n.\nLetZ1(G) be the partition function of the anti-ferromagnetic Ising instanc e,\nandZ2(G) be that for the ferromagnetic system. We shall prove that Z1(G) =\nγ−|F|µ|R|Z2(G). LetV/definesL∪R,A=/bracketleftBigg1√βγ1\n11√βγ/bracketrightBigg\n,A′=\n/radicalBig\nγ\nβγ\nγ/radicalBig\nγ\nβ\n,ˆA′=\n/bracketleftbigg\n1β\nγ1/bracketrightbigg\nandˆA=/bracketleftbigg\nβ1\n1γ/bracketrightbigg\n. Then\nZ2(G) =/summationdisplay\nσ∈{0,1}V/productdisplay\n(u,v)∈EˆAσu,σv/productdisplay\nu∈L/parenleftBigg\nµ/parenleftbigg/radicalbiggγ\nβ/parenrightbiggdu/parenrightBigg1−σu/productdisplay\nv∈R/parenleftBigg\n1\nµ/parenleftbigg/radicalbiggγ\nβ/parenrightbiggdv/parenrightBigg1−σv\n=/summationdisplay\nσ∈{0,1}V/productdisplay\n(u,v)∈EˆA′\nσu,σv/productdisplay\nu∈L/parenleftBigg\nµ/parenleftbigg/radicalbiggγ\nβ/parenrightbiggdu/parenrightBigg1−σu/productdisplay\nv∈R/parenleftBigg\n1\nµ/parenleftbigg/radicalbiggγ\nβ/parenrightbiggdv/parenrightBiggσv\n=/summationdisplay\nσ∈{0,1}V/productdisplay\n(u,v)∈EA′\nσu,σv/productdisplay\nu∈Lµ1−σu/productdisplay\nv∈R1\nµσv\n=µ−|R|γ|F|/summationdisplay\nσ∈{0,1}V/productdisplay\n(u,v)∈EAσu,σv/productdisplay\nu∈Lµ1−σu/productdisplay\nv∈Rµ1−σv\n=µ−|R|γ|F|Z1(G).\nThus we can get an approximation for the anti-ferromagnetic Ising model by an\noracle call to the ferromagnetic two-spin system. This concludes t he proof.⊓ ⊔\nNow,given the target µ >/parenleftBig/radicalBig\nγ\nβ/parenrightBig∆\nin Theorem3, we simply choosea µ′close\nenough to 1 in Lemma 7 and Corollary 6, such that/bracketleftbigg\n1\nµ′/radicalBig\nγ\nβ,µ′/parenleftBig/radicalBig\nγ\nβ/parenrightBig∆/bracketrightbigg\n⊆[1,µ]\nand #BIS≤APBi-Anti-Ferro ∆/parenleftBig\n1√βγ,1√βγ,µ′/parenrightBig\n. Then we can conclude that\n#BIS≤APBi-Ferro ∆(β,γ,[1,µ]) and finish the proof of Theorem 3.\n3 Uniform Local Field\nWe establish Theorem 2 in this section. If β >1, then one can use external field\nofµ >γ−1\nβ−1to simulate any external fields and get the # BIS-hardness result.This follows from a similar argument as that in [7]. To be self-contained , we also\ninclude a formal proof in the appendix. So we assume β≤1 in this section. We\nalso introduce a function h(x) =βx+1\nx+γwhich is used throughout this section.\nNote that since βγ >1,h(x) is monotonically increasing and1\nγ< h(x)< β≤1\nforx∈(0,+∞). We shall prove the following key reduction.\nLemma 8. Letβ≤1,βγ >1,dbe an integer such that β(βγ)d>1,µ∗be the\nlargest solution of xtox=µh(x)d, andµ >γd(βγ−1)\nβ/parenleftBig\n1+d+1\nln(β(βγ)d)/parenrightBig\n. Then\nFerro(β,γ,[1,µ∗])≤APFerro(β,γ,µ).\nAsµ∗=µh(µ∗)dand1\nγ< h(µ∗)< β, we have the following bound for µ∗.\nProposition 9.µ\nγd< µ∗< βdµ.\nWith this bound and Lemma 8, we can choose sufficiently large µso that this\nµ∗islargeenoughtoapplythehardnessresult(Theorem3)of Ferro(β,γ,[1,µ∗])\nto get the hardness result for Ferro(β,γ,µ). Formally, we have\nTheorem 10. Letβ≤1,βγ >1,dbe an integer such that β(βγ)d>1,\n∆=⌊√βγ+1√βγ−1⌋+ 1, andµ > γdmax/braceleftbigg/parenleftBig/radicalBig\nγ\nβ/parenrightBig∆\n,βγ−1\nβ/parenleftBig\n1+d+1\nln(β(βγ)d)/parenrightBig/bracerightbigg\n. Then\nFerro(β,γ,µ)is#BIS-hard.\nWe remark that there always exists such integer dsinceβ >0 andβγ >1.\nDifferent ds give different bounds for µand it is not necessarily monotone. For\na givenβ,γ, one can choose a suitable dto get the best bound4.\nIn the remaining of this section, we prove the key reduction stated in Lemma\n8. The main idea is to simulate any external field in [1 ,µ∗] by a vertex weight\ngadget.Inthefirstsubsection,westatethegeneralframewor kofsuchsimulation.\nThen in the second subsection, we present the detailed construct ion of a gadget.\n3.1 Vertex Weight Gadget\nDefinition 11 (Vertex weight gadget). LetG(V,E)be a graph with a special\noutput vertex v∗, defineµ(G) =ZG(v∗=0)\nZG(v∗=1)whereZG(v∗= 0)(resp.ZG(v∗= 1))\nis the partition function of G(V,E)in(β,γ,µ)-system conditioned on v∗= 0\n(resp.v∗= 1). We call Gavertex weight gadget that realizes µ(G).\nWe also use a family ofgraphsto approacha given externalfield. Let {Gi}i≥1\nbeafamilyofvertexweightgadgets.We say {Gi}realizesµiflimi→∞µ(Gi) =µ.\nVertex weight gadgets can be used to simulate external fields. For mally, we\nhave the following reductions.\n4We give one numerical example here to get some idea of this bou nd: ifβ= 1 and\nγ= 2, we can get ∆= 6 and choose d= 1; then the theorem tell us that the problem\nFerro(1,2,µ) is #BIS-hard ifµ >12.Lemma 12. LetGbe a vertex weight gadget of (β,γ,V). ThenSpin(β,γ,V∪\n{µ(G)})≤APSpin(β,γ,V).\nLet{Gi}be a sequence of vertex weight gadget of (β,γ,V)to realize µsuch\nthat for any ε >0there is a Giof sizepoly/parenleftbig\nε−1/parenrightbig\nwithexp(−ε)≤µ(Gi)\nµ≤exp(ε).\nThenSpin(β,γ,V∪{µ})≤APSpin(β,γ,V).\nProof.The proof of the first part is straightforward. For any instance Hof\nSpin(β,γ,V∪{µ(G)}) and a vertex of Hwith external field µ(G), we use one\ncopyofGandidentify the output vertexof Gwith that chosenvertexof H. After\nthe identification, the external field in that vertex is that of outpu t vertex of G.\nTherefore,afterthe modification,the new instanceisaninstance ofSpin(β,γ,V)\nand the partition function is equal to the partition function of Hscaled by a\npolynomial-time computable global factor/parenleftBig\nZ(G)\n1+µ(G)/parenrightBigj\n, wherejis the number of\nvertices with external field µ(G) inH.\nFor the second part, for an instance HofSpin(β,γ,V∪{µ}) and required\napproximation parameter ε, choose a gadget Giwhich is ε′=ε\n2nclose to realize\nµ; dothe samemodificationasaboveusingthis Giand callthe oracleforthe new\ninstance with approximation parameter ε′. This gives the desired approximation\nfor the original instance. ⊓ ⊔\n3.2 The Construction\nWe first define a gadget operation combas follows: for a given list of graphs\nG={G1,...,G k}, each with output v∗\nifori∈[k],comb(G) is a new graph\nG(V,E) thatcombinesthe graphsandjoinstheiroutputs.Fig.1isanillustr ation\nofcomb. Formally, we define V={u}∪/uniontext\ni∈[k]V(Gi) andE={(u,v∗\ni)|i∈\n[k]}∪/uniontext\ni∈[k]E(Gi), where uis the output of G. It is easy to verify that µ(G) =\nµ/producttext\ni∈[k]h(µ(Gi)).\n⇒\nFig.1.Result ofS5⇒comb({S5,S5}), the output vertex is marked as unfilled.\nWealsodefinetwobasicgadgets.Let Swbeaw-stargraph,withoutputbeing\nitscenter.Inparticular, S0isthesingletongraph.Notethat µ(Sw) =µh(µ)w.We\nalso defineTtbe ad-ary tree with depth t. For any external field ˆ µ∈(0,µ∗], we\nshallconstructalistofgadgetstosimulateit.Thetwoboundariesa reapproached\nbySwandTtrespectively.\nProposition 13. LetTtbe ad-ary tree with depth tandSwbe aw-star. Then(1){Sw}w≥1realizes0, or formally, µ(Sw) =µh(µ)w< µβw.\n(2){Tt}t≥0realizesµ∗, or formally, there exist two positive constants ιand\nc <1depending on µ,β,γanddsuch that 1<µ(Tt)\nµ∗≤exp(ctι).\nProof.(1) is obvious, we only prove (2).\nNote that µ(Tt) =µh(µ(Tt−1))d, we denote f(x) =µh(x)d. Recall that µ∗\nis the largest fixed point of f(x) andf(µ)< µ, we have 0 < f′(µ∗)<1. Define\ng(x) =xf′(x)\nf(x), theng(µ∗) =f′(µ∗). Sinceg(x) is a continuous function, we can\nchoose some η >0 such that 0 < g(x)≤c <1 for allx∈(µ∗−η,µ∗+η).\nWe now define a sequence {xi}i≥0such that x0=µandxi=f(xi−1)\nfor alli≥1. We claim that {xi}converges to µ∗asiapproaches infinity.\nTo see this, note that xi+1=f(xi)< xiandxi> µ∗for alli≥0. This\nimplies{xi}converges to some z≥µ∗. Moreover, since fis continuous, the\nsequence{f(xi)}i≥0also converges to z. These two facts together imply z=\nlimi→∞f(xi) =f(limi→∞xi) =f(z). In other word, zis a fixed point of fand\nthusz=µ∗. The claim implies that for some integer t0,xt0∈(µ∗,µ∗+η).\nWe define another sequence {yi}i≥0such that y0=µ(Tt0) andyi=f(yi−1)\nfor alli≥1. It holds that yi∈(µ∗,µ∗+η) and thus g(yi)≤c <1 for all i≥0.\nTherefore for all t≥1,\nlnyt−lnµ∗= lnf(yt−1)−lnf(µ∗)\n=˜yf′(˜y)\nf(˜y)·|lnyt−1−lnµ∗|for some y∈[µ∗,yt−1]\n=g(˜y)·|lnyt−1−lnµ∗|\n≤c·|lnyt−1−lnµ∗|\n≤ctη.\nWe denote ι= max{lnµ,ηc−t0}and conclude the proof. ⊓ ⊔\nOur main idea to realize a target external field ˆ µis to construct a list of\ngadgetsG={G1,...,G k}such that µ(comb(G))≈ˆµor more concretely ˆ µ≈\nµ/producttext\ni∈[k]h(µ(Gi)). All but one of these Giare basic gadgets of the following\nthree types: (1) isolate point S0withµ(S0) =µ; (2)Swwith large enough w\nsuch that µ(Sw)≈0; and (3)Ttwith large enough tsuch that µ(Tt)≈µ∗.\nThe remaining one Giis recursively constructed with a new target ˆ µ′so that\nideally ˆµ=µ/producttext\ni∈[k]h(µ(Gi)) holds. The combination of these basic gadgets\nare carefully chosen so that the new target ˆ µ′is also in the range (0 ,µ∗]. Then\nwe recursively construct this ˆ µ′by a subtree. We terminate the recursion after\nenough steps, and use a basic star gadget which is closest to the de sired value as\nan approximation in the leaf. With a correlation decay argument, we s how that\nthe error in the root can be exponentially small in terms of the depth , although\nthere may be a constant error in the leaf. A detailed construction w ith special\ntreatment for the boundary cases are formally given in Algorithm 1.Algorithm 1: Constructing Gℓ\nfunction construct (ℓ,^µ) :\ninput: Recursion depth ℓ; Target 0 <ˆµ≤µ∗to simulate;\noutput: Graph Gℓconstructed.\nbegin\nifℓ= 0then\nLetkbe the positive integer such that µh(µ)k+1<ˆµ≤µh(µ)k;\nreturn Sk;\nelse\nLetkbe the non-negative integer with µ∗h(µ)k+1<ˆµ≤µ∗h(µ)k;\nY′←k·S0;// a set of kcopies ofS0.\nµ1←ˆµ\nh(µ)k;\n// Invariant: µh(x′)d−i+1=µihas a solution 0< x′≤µ∗.\nfori←1tod−1do\nifµh(µ∗)h(0)d−i≥µithen\nyi←0;w←⌊ℓ·lnα−ln(dµ)\nlnβ⌋+1;Yi←Sw;\nelse\nyi←µ∗;t←⌊ℓ·lnα−lnd−lnι\nlnc⌋+1;Yi←Tt;\nµi+1←µi\nh(yi);\nLet ˆµ′be the solution of µh(x) =µdin (0,µ∗];\nY←Y′∪{Yi}d−1\ni≥1;\nδ←exp(−lnγlnα\nlnβℓ+lnγln(dµ)\nlnβ+lnµ\nγ);\nifˆµ′≤δthen\nChoose the largest integer wsuch that µ/parenleftBig\n1\nγ/parenrightBigw\n> δ;\nreturncomb(Y∪{S w});\nelse\nreturn comb(Y∪construct (ℓ−1,^µ′));\nBeforewe provethat the construction is correct,weobtain a few observations\nwhich is used in our proof. The condition on µin the key Lemma 8 is due to the\nfollowing property we need.\nProposition 14. Letµ >γd\nβ(βγ−1)/parenleftBig\n1+d+1\nln(β(βγ)d)/parenrightBig\n, for any µ1withµ∗h(µ)<\nµ1≤µ∗, the equation µh(x)d=µ1always has a solution with 0< x≤µ∗.\nProof.It suffices to show µ·h(0)d≤µ∗h(µ) andµ·h(µ∗)d≥µ∗. Sinceµ∗=\nµh(µ∗)d, the second part is trivial. As for the first part, it is sufficient to sho w/parenleftBig\nh(µ∗)\nh(0)/parenrightBigd\nh(µ)>1. Note that/parenleftBig\nh(µ∗)\nh(0)/parenrightBigd\nh(µ)> γdh(µ∗)d+1> γd/parenleftBig\nβ−βγ−1\nµ∗/parenrightBigd+1\n,\nγd/parenleftbigg\nβ−βγ−1\nµ∗/parenrightbiggd+1\n>1⇐⇒ln/parenleftbig\nβ(βγ)d/parenrightbig\n+(d+1)ln/parenleftbigg\n1−βγ−1\nβµ∗/parenrightbigg\n>0,(d+1)ln/parenleftbigg\n1−βγ−1\nβµ∗/parenrightbigg\n(♣)\n>−(d+1)βγ−1\nβµ∗\n1−βγ−1\nβµ∗(♠)\n>−ln/parenleftbig\nβ(βγ)d/parenrightbig\n,\nwhere (♣) is due to ln(1−x)>−x\n1−xforx∈(0,1), and (♠) is by the fact that\nβ(βγ)d>1 and the choice of µsuch that−βγ−1\nβµ∗>ln(β(βγ)d)+d+1\nβln(β(βγ)d).⊓ ⊔\nProposition 15. For every x,t≥0, it holds that h(x+t)≤(1 +t)h(x)and\nh((1+t)x)≤(1+t)h(x).\nProof.Note that x,t≥0,\nh(x+t)≤(1+t)h(x)⇐⇒/parenleftbiggβ(x+t)+1\nx+t+γ/parenrightbigg\n≤(1+t)/parenleftbiggβx+1\nx+γ/parenrightbigg\n⇐⇒t2(1+βx)+t/parenleftbig\n1+γ(1+β(x−1))+x+βx2/parenrightbig\n≥0.\nSince/parenleftbig\n1+γ(1+β(x−1))+x+βx2/parenrightbig\n>0, the inequality always holds.\nh((1+t)x)≤(1+t)h(x)⇐⇒x(1+t)β+1\nx(1+t)+γ≤(1+t)βx+1\nx+γ\n⇐⇒t2(x+βx2)+t(γ+2x+βx2)≥0\nAgain every term is non-negative, the last inequality is always true. ⊓ ⊔\nInthefollowing,westarttoverifythecorrectnessoftheconstr uction.Wefirst\nverify that the algorithm is well defined, namely µh(x) =µddoes havea solution\nˆµ′in (0,µ∗]. This can be done by verifying the loop invariant “ µh(x′)d−i+1=µi\nhas a solution 0 < x′≤µ∗” inductively.\nInitialization. Fori= 1, by Proposition 14, for some 0 <˜x≤µ∗it holds that\nµh(˜x)d−i+1=µi.\nMaintenance. Assuming µh(˜x)d−i+1=µihas solutions ˜ x∈(0,µ∗], we verify\nthatµh(x′)d−i=µi+1≡µi\nh(yi)has solutions x′∈(0,µ∗] fori∈[1,d−1].\nCaseµh(µ∗)h(0)d−i≥µi. By assumption we have µh(0)d−i+1< µi, also\nnote that µi≤µh(µ∗)h(0)d−i≤µh(0)h(µ∗)d−i, henceµh(0)d−i<µi\nh(0)≤\nµh(µ∗)d−i. Then by continuity, µh(x′)d−i=µi\nh(0)has solutions 0 < x′≤µ∗.\nCaseµh(µ∗)h(0)d−i< µi.Byassumption µh(µ∗)d−i+1≥µi,thusµh(0)d−i<\nµi\nh(µ∗)≤µh(µ∗)d−i, henceµh(x′)d−i=µi\nh(µ∗)has solutions 0 < x′≤µ∗.\nTermination. After the loop completes, µh(x′) =µdhas solutions0 < x′≤µ∗.\nNow we verify the vertex weight gadget returned by the construc tion satisfies\nour requirement by choosing ℓ=O(−logε).\nLemma 16. For0<ˆµ≤µ∗(β,γ,µ), and let G(V,E)be the graph returned\nbyconstruct (ℓ,ˆµ), we have the following: (1) exp(−(c+ℓ)·αℓ)≤µ(G)\nˆµ≤\nexp((c+ℓ)·αℓ), wherec= lnγandα=√βγ−1√βγ+1<1; (2)|G|= exp(O(ℓ)).Proof.We apply induction on ℓfor both statements. We prove for (1) first. For\nbase case ℓ= 0, we have\n|lnµ(G)−ln ˆµ|≤/vextendsingle/vextendsinglelnµh(µ)k−lnµh(µ)k+1/vextendsingle/vextendsingle=−lnh(µ)≤lnγ.\nAssume the statement holds for smaller ℓ. Letk,{yi}1≤i≤d−1and{Yi}1≤i≤d−1\nbe parameters chosen in the algorithm. Define\nF(z) = ln/parenleftBigg\nµh(µ)kd−1/productdisplay\ni=1h(yi)h(exp(z))/parenrightBigg\n,˜F(z) = ln/parenleftBigg\nµh(µ)kd−1/productdisplay\ni=1h(µ(Yi))h(exp(z))/parenrightBigg\n.\nWe note that F(z) is thecorrectrecursion to compute ln( µ(G)) and˜F(z) is our\napproximate recursion used in the algorithm.\nIn the following, we distinguish between ˆ µ′≤δand ˆµ′> δ.\n–If ˆµ′≤δ, then ln µ(G) =˜F(lnµ(Sw)) and ln ˆ µ=F(lnˆµ′). We have\nF(ln ˆµ′)≤˜F(lnµ(Sw)) = ln/parenleftBigg\nµh(µ)kd−1/productdisplay\ni=1h(µ(Yi))h(µ(Sw))/parenrightBigg\n(♥)\n≤αℓ+ln/parenleftBigg\nµh(µ)kd−1/productdisplay\ni=1h(yi)h(ˆµ′)/parenrightBigg\n=αℓ+F(ln ˆµ′)),\nwhere (♥) follows from following facts derived from Proposition 15:\n(i) Ifyi= 0, then 0≤µ(Yi)≤αℓ\nd, which implies h(µ(Yi))≥h(yi) and\nh(µ(Yi))≤/parenleftBig\n1+αℓ\nd/parenrightBig\nh(yi)≤exp/parenleftBig\nαℓ\nd/parenrightBig\nh(yi).\n(ii) Ifyi=µ∗, thenµ∗≤µ(Yi)≤exp/parenleftBig\nαℓ\nd/parenrightBig\nµ∗, which implies h(µ(Yi))≥\nh(yi) andh(µ(Yi))≤exp/parenleftBig\nαℓ\nd/parenrightBig\nh(yi).\n(iii) We claim that ˆ µ′< µ/parenleftBig\n1\nγ/parenrightBigw\n≤µ(Sw)≤µβw≤αℓ\nd. The only nontrivial\npart is to verify that µβw≤αℓ\nd. Sincewis the largest integer that\nˆµ′< µ/parenleftBig\n1\nγ/parenrightBigw\n, we have µ/parenleftBig\n1\nγ/parenrightBigw+1\n≤ˆµ′, which gives w≥lnµ−lnδ\nlnγ−1.\nPlug this into µβw≤αℓ\ndand letδ= exp(−lnγlnα\nlnβℓ+lnγln(dµ)\nlnβ+lnµ\nγ),\nthe inquality holds. Thus h(µ(Sw))≥h(ˆµ′) andh(µ(Sw))≤h(αℓ\nd)≤/parenleftBig\n1+αℓ\nd/parenrightBig\nh(ˆµ′)≤exp/parenleftBig\nαℓ\nd/parenrightBig\nh(ˆµ′).\n–If ˆµ′> δ, definex=µ(construct (ℓ−1,^µ′)), then by induction hypothesis,\nit holds that|lnx−ln ˆµ′|≤(c+(ℓ−1))·αℓ−1.ThensimilarlybyProposition15andthechoiceof wandt,wehave F(lnx)≤\n˜F(lnx)≤F(lnx)+αℓ. Thus by construction, we have\n|lnµ(G)−ln ˆµ|=/vextendsingle/vextendsingle/vextendsingle˜F(lnx)−F(ln ˆµ′)/vextendsingle/vextendsingle/vextendsingle\n≤αℓ+|F(lnx)−F(lnˆµ′)|\n≤αℓ+|F′(ln˜x)|·|lnx−ln ˆµ′|(for some ˜ x∈[ˆµ′,x].)\n≤αℓ+(ℓ−1)|F′(ln˜x)|αℓ−1+c|F′(ln˜x)|αℓ−1\nThus it is sufficient to show that |F′(ln˜x)|≤α.In fact,F′(lnx) =x·h′(x)\nh(x)=\n(βγ−1)x\n(x+γ)(βx+1)≤βγ−1\n(√βγ+1)2=α.\nNow we prove (2) of the Lemma. We denote s(ℓ) = max ˆµ|construct (ℓ,ˆµ)|\nand show that s(ℓ) =ℓexp(O(ℓ)) = exp( O(ℓ)).\nIfℓ= 0, since ˆ µis either the eventual external field (which is a constant\nbounded away from 0), or ˆ µ > δ, we have s(ℓ) =|Sk|=O(1).\nIfℓ >0, then|Yi|= exp(O(ℓ)) and thus|Y|= exp(O(ℓ)). By our choice of δ,\nit holds that w=O(ℓ) and thus|Sw|=O(ℓ). Therefore,\ns(ℓ) = exp(O(ℓ))+max{s(ℓ−1),O(ℓ)}=ℓexp(O(ℓ)) = exp( O(ℓ)).\nThis concludes the proof. ⊓ ⊔\nReferences\n1. Antar Bandyopadhyay and David Gamarnik. 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Dror Weitz. Counting independent sets up to the tree thre shold. In Proceedings\nof STOC , pages 140–149, 2006.\nA The β >1 case\nWe prove the hardness result for the case of β >1. This follows a similar argu-\nment as that in [7] and is known as a folklore. We include a formal proof here\nto be self-contain.\nTheorem 17. Letγ > β > 1andµ >γ−1\nβ−1. ThenFerro(β,γ,µ)is#BIS-hard.\nWe follow the same idea of simulating external field and making use of Th e-\norem 3 to conclude the proof. In this case, we can simulate all positiv e external\nfields.\nLemma 18. For every ˆµ >0, there is afamily of vertex weight gadgets {Gm}m≥1\nthat realizes ˆµ. Moreover, Gmis constructible in time mO(1)and\nexp(−1\nm)≤µ(Gm)\nˆµ≤exp(1\nm). (1)\nProof.For any m≥1, we add xself-loops and ybristles to a single vertex v,\nwherexandyare integers to be determined. Let vbe the output of Gm, then\nµ(Gm) =µ/parenleftBig\nβ\nγ/parenrightBigx/parenleftBig\nµβ+1\nµ+γ/parenrightBigy\n. Denote a= lnγ\nβ,b= lnµβ+1\nµ+γandc=ln ˆµ\nlnµ, then (1)\nis equivalent to\n|(y·b−x·a)−c|≤1\nm.We can use a procedure similar to extended Euclidean algoroithm to fin d such\nintegersx,yin timeO(lnm), such that it also guarantees x,y=mO(1).\nB Improved Tractable Result\nIn this section, we establish the following tractable result:\nTheorem 19. Letβ < γ,βγ >1andµ≤γ/β. Then there is an FPRAS for\nFerro(β,γ,µ).\nThe proof of this theorem follows by refining the proof in [9], where t hey\nestablish the tractable result for µ≤(γ/β)δ/2forδbeing the minimum degree\nof vertices in the graph. Specifically, we first contract all vertices with degree one\nand modify the external fields of their neighboring vertices, this on ly scales the\npartition function by a constant. Next, just as in [9], we shall reduc e a (β,γ,µ)\ninstance to a ferromagnetic Ising instance and apply the following ce lebrated\nresult, which is first introduced in [10] for uniform external fields an d refined for\nnon-uniform external fields in [9]:\nTheorem 20 ([10] and [9]). There is an FPRAS for Ising system (a,a,V)\nprovided that a >1and all external fields in Vare at most one.\nLetG(V,E) be an instance of ( β,γ,µ) system, we repeatedly apply the fol-\nlowing operations until no degree one vertices can be found:\n1. Pick a vertex uof degree one. Denote its incident edge by e= (u,v). Letµu\nandµvbe external fields on uandvrespectively.\n2. Remove uand edge ( u,v), update µv←µvh(µu).\nLetG′(V′,E′) be the remaining graph. G′either has no vertices of degree\none, or it only contains a single vertex. Moreover, for every v∈V′, the external\nfieldsµ′\nvsatisfiesµ′\nv≤µ. This can be easily verified given that µ≤γ/β. Let\nU={µ′\nv|v∈V′}, consider G′as an instance of ( β,γ,U) system, clearly\nZ(β,γ,µ)(G) =Z∗·Z(β,γ,U)(G′)whereZ∗isaneasilypolynomial-timecomputable\nfactor.\nLetV=/braceleftbigg\nµ′\nv/parenleftBig\nβ\nγ/parenrightBigdv/2/vextendsingle/vextendsingle/vextendsinglev∈V′/bracerightbigg\nwheredvis the degreeof vinG′. LetˆG(ˆV,ˆE)\nbe a copy of G′with ˆµv=µ′\nv/parenleftBig\nβ\nγ/parenrightBigdv/2\nfor every v∈ˆV. We are going to verify\nthatZ(β,γ,U)(G′) =/radicalBig\nγ\nβ|E′|\n·Z(a,a,V)(ˆG) fora=√βγ.\nDefineA=/bracketleftbiggβ1\n1γ/bracketrightbigg\n,A′=/bracketleftbiggγ/radicalbig\nγ/β/radicalbig\nγ/β γ/bracketrightbigg\nandˆA=/bracketleftbigg√βγ1\n1√βγ/bracketrightbigg\n. Then,Z(β,γ,U)(G′) =/summationdisplay\nσ∈{0,1}V′/productdisplay\n(u,v)∈E′Aσu,σv/productdisplay\nv∈V′µ′1−σv\nv\n=/summationdisplay\nσ∈{0,1}V′/productdisplay\n(u,v)∈E′A′\nσu,σv/productdisplay\nv∈V′\n/parenleftBigg/radicalBigg\nβ\nγ/parenrightBiggdv\nµ′\nv\n1−σv\n=/radicalbiggγ\nβ|E′|/summationdisplay\nσ∈{0,1}V′/productdisplay\n(u,v)∈E′ˆAσu,σv/productdisplay\nv∈V′\n/parenleftBigg/radicalBigg\nβ\nγ/parenrightBiggdv\nµ′\nv\n1−σv\n=/radicalbiggγ\nβ|E′|/summationdisplay\nσ∈{0,1}ˆV/productdisplay\n(u,v)∈ˆEˆAσu,σv/productdisplay\nv∈V′ˆµ1−σv\nv\n=/radicalbiggγ\nβ|E′|\n·Z(a,a,V)(ˆG)\nFinally, to apply Theorem 20, we only need ˆ µv≤1 for allv∈ˆV. Recall that\nˆGhasδ≥2, hence µ≤γ/βimplies ˆµv≤1. To sum up, this concludes the proof."    },    {        "title": "1405.1951v2.Peculiar_long_range_supercurrent_in_SFS_junction_containing_a_noncollinear_magnetic_domain_in_the_ferromagnetic_region.pdf",        "content": "arXiv:1405.1951v2  [cond-mat.supr-con]  19 May 2014Peculiar long-range supercurrent in SFS junction containi ng a noncollinear magnetic\ndomain in the ferromagnetic region\nHao Meng∗and Xiuqiang Wu\nNational Laboratory of Solid State Microstructures and Dep artment of Physics, Nanjing University, Nanjing 210093, Ch ina\n(Dated: August 27, 2018)\nWe study the supercurrent in a superconductor-ferromagnet -superconductor heterostructure con-\ntaining a noncollinear magnetic domain in the ferromagneti c region. It is demonstrated that the\nmagnetic domain can lead to a spin-flip process, which can rev erse the spin orientations of the\nsinglet Cooper pair propagating through the magnetic domai n region. If the ferromagnetic layers\non both sides of magnetic domain have the same features, the l ong-range proximity effect will take\nplace. That is because the singlet Cooper pair will create an exact phase-cancellation effect and\ngets an additional πphase shift as it passes through the entire ferromagnetic re gion. Then the\nequal spin triplet pair only exists in the magnetic domain re gion and can not diffuse into the other\ntwo ferromagnetic layers. So the supercurrent mostly arise s from the singlet Cooper pairs and the\nequal spin triplet pairs are not involved. This behavior is q uite distinct from the common knowl-\nedge that long-range supercurrent induced by inhomogeneou s ferromagnetism stems from the equal\nspin triplet pairs. The result we presented here provides a n ew way for generating the long-range\nsupercurrent.\nPACS numbers: 74.78.Fk, 73.40.-c, 74.50.+r, 73.63.-b\nThe interplay between superconductivity and ferro-\nmagnetism in mesoscopic structures has been extensively\nstudied because of the underlying rich physics and po-\ntential applications in spintronics and quantum informa-\ntion1–4. When a homogeneous ferromagnet ( F) is sand-\nwiched between two s-wave superconductors ( S) to form\na Josephson junction, the magnetic configuration of the\nFlayer may substantially modify the spatial properties\nof the superconducting order parameter. This behav-\nior is induced by the different action of the ferromag-\nnetic exchange field hon the spin-up and spin-down elec-\ntrons that form the Cooper pair. Then this pair inside\ntheFlayer acquire a relative phase shift Q·R, where\nQ≃2h//planckover2pi1vF,vFis the Fermi velocity and Ris the thick-\nness of the Flayer. This phase shift changes with Rand\nresults in an oscillation of critical current accompanied\nwith a rapid decay. The above oscillation will lead to the\ntransition from the so-called 0 state to the πstate1–4.\nTwo kinds of approaches have been proposed to pro-\nduce long-range supercurrent in SFSjunction. The first\napproachrequiresinhomogeneousmagnetism in Fregion\nso that the interested equal spin triplet pairs can be gen-\nerated2. One way to achieve this purpose is by arranging\nferromagnetictrilayerwith noncollinearmagnetizations5.\nInthisgeometry,thespin-flipprocessesattheinterface F\nlayers can convert the singlet Cooper pairs into the equal\nspin triplet pairs4,6. The triplet pairs can penetrate into\nthe central Flayer over a long distance unsuppressed by\nthe exchange interaction so that the proximity effect is\nenhanced. The second approach requires the Flayer to\nbe arrangedantiparallel. This situation was described by\nBlanteret al.for aSFFSjunction7. The physical origin\nof the enhanced proximity effect is described as a com-\npensation of the relative phase shift of a Cooper pair as\nit passes from the first Flayer into the second one. If the\ntwoFlayers have the same thickness, the net change inthe relative phase of the Cooper pair is zero in the clean\nlimit. This enhanced Josephson current has been proved\nby recent experiment8.\nIn this paper, we predict the third approach to gener-\nate the long-rangesupercurrent in SFSstructure, shown\nschematically in figure 1(a). The ferromagnetic region\nconsists of two ferromagnetic layers ( FLandFR) with\nmagnetizationsorientedin same directions. The FLlayer\nandFRlayer are separated by a clean magnetic domain\n(FM), whose magnetization is misaligned with direction\nof theFLlayer andFRlayer. The magnetic domain FM\ncaninduceaspin-flipprocess, whichreversesthe spinori-\nentations of the singlet Cooper pair propagating through\ntheFMregion. Thisprocesswill makethe singletCooper\npair create a phase-cancellation effect and obtain an ad-\nditionalπphase shift. If the FLlayer andFRlayer have\nthe same thickness and exchange field, the net change\nin the relative phase of a singlet Cooper pair is πwhen\nit passes through the entire Fregion. In this case, the\ncontribution to the long-rangesupercurrentmostly arises\nfrom the singlet Cooper pairs. This is because the equal\nspin triplet pairs only display in FMregion and can not\ndiffuse into the FLandFRlayers.\nIn our numerical calculation, the transport direction\nis along the yaxis, and the system is assumed to be\ninfinite in the x-zplane. The BCS mean-field effective\nHamiltonian1,9is\nHeff=/integraldisplay\nd/vector r{/summationdisplay\nα,βψ†\nα(/vector r)[He(ˆ1)αβ−(/vectorh·/vector σ)αβ]ψβ(/vector r)\n+1\n2[/summationdisplay\nα,β(iσy)αβ∆(/vector r)ψ†\nα(/vector r)ψ†\nβ(/vector r)+h.c.]},(1)\nwhereHe=−/planckover2pi12∇2/2m−EF,ψ†\nα(/vector r) andψα(/vector r) are cre-\nation and annihilation operators with spin α. ˆσand\nEFare Pauli matrices and the Fermi energy, respec-2\ndL dR(a)\ndMFMFL\nz\nxyθ s−wave Ss−wave SFR\n−2 024681012−4−3−2−101234\n(b)FM\nFLFRs−wave S s−wave S\nFIG. 1. (color online) (a) Schematic diagram of SFSstruc-\nture with two ferromagnetic layers FLandFRoriented along\nthezaxis and separated by a noncollinear magnetic domain\nFM. The lengths of FL,FRandFMare denoted by dL,dR\nanddM, respectively. The phase difference between the two\nSs isφ=φR−φL. (b) The transmission of electron and hole\nin above structure.\ntively. The superconducting gap is given by ∆( /vector r) =\n∆(T)[eiφLΘ(−y) +eiφRΘ(y−dF)] withdF=dL+\ndM+dR. Here, ∆( T) accounts for the temperature-\ndependent energy gap. It satisfies the BCS relation\n∆(T) = ∆0tanh(1.74/radicalbig\nTc/T−1) withTcthe supercon-\nducting critical temperature. Θ( y) is the unit step func-\ntion, andφL(R)is the phase of the left (right) S. The\nexchange field /vectorhdue to the ferromagnetic magnetizations\nin theFregion is described by\n/vectorh=\n\nhLˆz, 0<y<d L\nhM(sinθˆx+cosθˆz), dL<y<d L+dM\nhRˆz, d L+dM<y<d F,(2)\nwhereθis the misorientation angle of magnetization in\nthe magnetic domain FMregion. Based on the Bogoli-\nubov transformation ψα(/vector r) =/summationtext\nn[unα(/vector r)ˆγn+v∗\nnα(/vector r)ˆγ†\nn]\nand the anticommutation relations of the quasiparticle\nannihilation and creation operators ˆ γnand ˆγ†\nn, we have\nthe Bogoliubov-de Gennes (BdG) equation1,9\n/parenleftbiggˆH(y)ˆ∆(y)\n−ˆ∆∗(y)−ˆH∗(y)/parenrightbigg/parenleftbigg\nˆu(y)\nˆv(y)/parenrightbigg\n=E/parenleftbigg\nˆu(y)\nˆv(y)/parenrightbigg\n,(3)\nwhereˆH(y) =Heˆ1−hz(y)ˆσz−hx(y)ˆσxandˆ∆(y) =\niˆσy∆(y). Here ˆu(y) = (u↑(y),u↓(y))Tand ˆv(y) = (v↑(y),\nv↓(y))Tare two-component wave functions.\nSince the transversal momentum components are con-\nserved, we consider the configuration in the one dimen-\nsional regime for simplicity. The BdG equation can beeasilysolvedfor each superconductingelectrode and each\nFregion, respectively. The scattering problem can be\nsolved by considering the boundary conditions at the in-\nterfaces. Each interface gives a scattering matrix. The\ntotal scattering matrix of the system can be obtained\nby the combination of all these scattering matrices of\nthe interfaces. From the total scattering matrix, we can\nobtain the Andreev reflection amplitudes a1∼a4of the\njunction where a1(2)is forthe reflection from an electron-\nlike to a hole-like quasiparticle with spin up (down), and\na3(4)is for the reverse process with spin up (down)10,11.\nThe dc Josephson current can be expressed in terms of\nthe Andreev reflection amplitudes by using the finite-\ntemperature Green’s function formalism12–15\nIe(φ) =kBTe∆\n4/planckover2pi1/summationdisplay\nk/bardbl/summationdisplay\nωnke(ωn)+kh(ωn)\nΩn·\n[a1(ωn,φ)−a2(ωn,φ)\nke+a3(ωn,φ)−a4(ωn,φ)\nkh],(4)\nwhereωn=πkBT(2n+1)are the Matsubara frequencies\nwithn= 0,1,2,...and Ω n=/radicalbig\nω2n+∆2(T).ke(ωn),\nkh(ωn), andaj(ωn,φ) withj= 1,2,3,4 are obtained\nfromke,kh, andajby analytic continuation E→iωn.\nke(h)isthewavevectorforelectron(hole)inthe Ss. Then\nthe critical current is derived from Ic=maxφ|Ie(φ)|.\nIn principle, to obtain the spin singlet pair and the\nspin triplet pair amplitude functions as well as the lo-\ncal density of the states (LDOS), we need to solve the\nBdG equation (3) by Bogoliubov’s self-consistent field\nmethod9,16,17. We put the junction in a one dimen-\nsional square potential well with infinitely high walls.\nAccordingly, the solution in equation (3) can be ex-\npanded in terms of a set of basis vectors of the station-\nary states18,uα(/vector r)=/summationtext\nquα\nqζq(y) andvα(/vector r) =/summationtext\nqvα\nqζq(y)\nwithζq(y) =/radicalbig\n2/dsin(qπy/d). Hereqis a positive in-\nteger andd=dS1+dF+dS2.dS1anddS2are the\nlength of the left and right Ss, respectively. The BdG\nequation (3) is solved iteratively together with the self-\nconsistency condition ∆( y) =g(y)f3(y)9. Here the sin-\nglet pair amplitude is give by17\nf3(y) =1\n2/summationdisplay\n0≤E≤ωD[u↑(y)v∗\n↓(y)−u↓(y)v∗\n↑(y)]tanh(E\n2kBT),\n(5)\nwhereωDis the Debye cutoff energy. The effective at-\ntractivecoupling g(y) will be taken to be zerooutside the\nSand a constant within it. Iterations are performed un-\ntil self-consistency is reached, starting from the stepwise\napproximation for the pair potential. The spin triplet\npair with zero spin projection and the equal spin triplet\npair amplitude functions are defined as follows17\nf0(y,t) =1\n2/summationdisplay\nE>0[u↑(y)v∗\n↓(y)+u↓(y)v∗\n↑(y)]η(t),(6)\nf1(y,t) =1\n2/summationdisplay\nE>0[u↑(y)v∗\n↑(y)−u↓(y)v∗\n↓(y)]η(t),(7)3\n0 50 100 15000.20.40.60.8\nkFdL¯hIc/e∆\n00.25 0.50.75 100.10.20.3\nhL/EF¯hIc/e∆\n08162400.10.20.3\nkFdM¯hIc/e∆\n0 0.2 0.400.10.20.3\nhM/EF¯hIc/e∆(b) (a)\nFIG. 2. (Color online) (a) Critical current as a function of\nlengthkFdL(=kFdR) for exchange field hM/EF= 0 (black\nline) and hM/EF= 0.17 (magenta line) then kFdM= 10,\nand inset shows the critical current versus kFdMforkFdL=\nkFdR= 100. Parameters used in (a): hL/EF=hR/EF= 0.1\nandhM/EF= 0.17. (b) Critical current as a function of\nexchange field hL/EF(=hR/EF) forhM/EF= 0.17, and\ninset shows the critical current versus hM/EFforhL/EF=\nhR/EF= 0.1. Parameters used in (b): kFdL=kFdR= 100\nandkFdM= 10. In all plots θ=π/2.\nwhereη(t) = cos(Et)−isin(Et)tanh(E/2kBT). The\nabove singlet and triplet pair amplitudes are all normal-\nized to the value of the singlet pairing amplitude in a\nbulkSmaterial. The LDOS is given by17\nN(y,ǫ) =−/summationdisplay\nα[u2\nα(y)f′(ǫ−E)+v2\nα(y)f′(ǫ+E)],(8)\nwheref′(ǫ) =∂f/∂ǫis the derivative of the Fermi func-\ntion. The LDOS is normalized by its value at ǫ= 3∆0\nbeyond which the LDOS is almost constant.\nUnless otherwise stated, we use the superconducting\ngap ∆ 0as the unit of energy. The Fermi energy is\nEF= 1000∆ 0and temperature is T/Tc= 0.1. In self-\nconsistent field method, we consider the low tempera-\nture limit and take kFdS1=kFdS2=400,ωD/EF=0.1, the\nother parameters are the same as the ones mentioned\nabove.\nIn figure2(a), wepresent the dependence ofthe critical\ncurrentIcon the length kFdL(=kFdR) for two different\nexchange fields of FMregion. It is well known that, if\nhM/EF= 0 the spin-flip process does not occur in FM\nregion, the critical current Icexhibits oscillations with\na period 2πξFand simultaneously decays exponentially\non the length scale of ξF1. Here,ξFis the magnetic\ncoherence length. The main reason is described below:\na Cooper pair entering into the Flayer receives a finite\nmomentum Qfrom the spin splitting of up and down\nbands. So the spin singlet pairing state |↑↓∝an}b∇acket∇i}ht− |↓↑∝an}b∇acket∇i}ht in\nSwill be converted into the mixed state |↑↓∝an}b∇acket∇i}hteiQ·R− |↓↑\n∝an}b∇acket∇i}hte−iQ·RinFlayer, thus leading to a modulation of the\npair amplitude with the thickness RofFlayer. Then the\n0−πtransition will arise due to spatial oscillations of the\npair amplitude. In this case the phase shift induced by\ntheFlayer is additive and the relative phase is generally\nnonzero so that the supercurrents are suppressed.In contrast, when hM/EF= 0.17,Icwill slowly de-\ncrease with the thickness kFdL. The electron and hole\ntransport process is shown in figure 1(b). Because the\nmagnetization direction of the FMregion is along the x\naxis (θ=π/2), which is orthogonal to the magnetization\ninFLlayerandFRlayer,the spin-flip processwill appear\nin theFMregion. As a result, when a electron |↑∝an}b∇acket∇i}htetrans-\nmits fromFLlayer toFRlayer, the spin-flip can convert\n|↑∝an}b∇acket∇i}hteinto|↓∝an}b∇acket∇i}hte. Subsequently, the |↓∝an}b∇acket∇i}hteis Andreev reflected\natFRSinterface and will be further converted into hole\n|↑∝an}b∇acket∇i}hth. The|↑∝an}b∇acket∇i}hthmoving to left is consequently inverted to\n|↓∝an}b∇acket∇i}hth. Finally,this |↓∝an}b∇acket∇i}hthwillpropagatetothe SFLinterface\nandbereflectedbackastheoriginal |↑∝an}b∇acket∇i}hte. Hence, thespin-\nflip process occurring in the FMregion can reverse the\nspin orientations of the electron and Andreev-reflected\nhole transporting between FLlayer andFRlayer. While\nif the mixed state |↑↓∝an}b∇acket∇i}hteiQ·R− |↓↑∝an}b∇acket∇i}hte−iQ·Rpasses from FL\nlayer intoFRlayer, it will be converted into a new state:\n|↓↑∝an}b∇acket∇i}hteiQ′·R′− |↑↓∝an}b∇acket∇i}hte−iQ′·R′\n=−(|↑↓∝an}b∇acket∇i}hte−iQ′·R′− |↓↑∝an}b∇acket∇i}hteiQ′·R′)\n=|↑↓∝an}b∇acket∇i}htei(−Q′·R′+π)− |↓↑∝an}b∇acket∇i}hte−i(−Q′·R′+π).(9)\nTherefore, when the Cooper pair passes through the FL\nlayer, it will acquire a relative phase shift δχ1=QdL.\nSimilarly, traversing through the FRlayer, it gets the\nother phase shift δχ2=−Q′dR+π. As a result, this sit-\nuation can be described as a superposition of the phase\nshift of the Cooper pair as it travels through the entire\nFregion:χ=δχ1+δχ2. If theFLandFRlayer have\nthe same exchange field and thickness, the net change\nin the relative phase of the Cooper pair is χ=π, and\nit will not turn to 0 with increase of the length of FL\nlayer andFRlayer. Provided one does not take abso-\nlute value for Ie(φ) to define the critical current Ic,Ic\nis always negative and is correspond to the πstate. As\nthe magenta line shown in figure 2(a), Icalso displays\nan oscillatory behavior with increasing the length kFdL.\nFrom this behavior, it is easier to deduce that the spin\norientations of the part of Cooper pairs will be inverted\nby spin-flip in FMregion, so that these pairs can lead\nto a substantially enhanced critical current but do not\nprovide the oscillations for this current. However, the\nrest of Cooper pairs pass through the FMregion without\nspin-flipandcannotgeneratethecancellationoftherela-\ntive phase. Consequently, the transmission of these pairs\nmakes the critical current oscillate with the length of en-\ntireFregion. In addition, another interesting property\nis the nonmonotonic dependence of the critical current\nIcas a function of the length kFdM(see the inset in\nfigure 2(a)). We find that the maximum of Icis nearly\nlocated atkFdM= 10. It means that the spin-flip ratio\nreaches their maximal value in this condition.\nNext, we discuss the dependence of critical current\nIcon the exchange field hL/EF(=hR/EF). As plot-\nted in figure 2(b), Icalmost decreases monotonically to\n0 with the increase of hL/EF. This is easily under-\nstood, because normal Andreev reflection occurred at4\n−40−20 0204000.10.20.3\nkF(dL−dR)¯hIc/e∆\n−0.04−0.02 00.020.0400.10.20.3\n(hL−hR)/EF¯hIc/e∆\n00.25 0.50.75 1−0.3−0.2−0.100.10.2\nφ/π¯hIe(φ)/e∆\n  \n−2−1 0120.511.52\nǫ/∆LDOS\n  kF(dL−dR)=0 [A]\nkF(dL−dR)=10 [B]\nkF(dL−dR)=20 [C]A\nBC\n(c)(a) (b)\n(d)\nFIG. 3. (Color online) Critical current (a) as a function of\nthe length difference kF(dL−dR) forhL/EF= 0.1, and (b)\nas a function of the exchange field difference ( hL−hR)/EF\nforkFdL= 100. The current-phase relation Ie(φ) (c) and the\naveraged LDOS in all Fregion (d) corresponding to the point\nA, B and C in panel (a). The parameters in FMandFRlayers\nhave the fixed values kFdM= 10,kFdR=100,hM/EF= 0.17,\nhR/EF= 0.1 andθ=π/2. Here the LDOS is calculated at\nkBT= 0.001.\ntheSFLandFRSinterface will be suppressed by ex-\nchange splitting of FLlayer andFRlayer. Especially, if\nhL/EF=hR/EF= 1, theFLlayer andFRlayer are\nall converted into half metal and only one spin band can\nbe occupied, then the Andreev reflections at the inter-\nfaces will be completely prohibited. In this case, none of\nCooper pairs can transmit from the left Sto the right\none, so the Josephson current would be complete sup-\npressed. This feature further demonstrates that the cur-\nrent mostly arises from the contribution of the singlet\nCooper pairs but not the equal spin triplet pairs. That\nis because the equal spin triplet pairs can penetrate over\na long distances into the half metal and will scarcely be\naffected by exchange splitting6,19. In addition, the inset\nin figure 2(b) shows the hM/EFdependence of the criti-\ncal current Ic. It also displays a nonmonotonic behavior\nas thehM/EFis increased, and the maximum is nearly\nseated athM/EF= 0.17.\nTo further demonstrate the conclusion mentioned be-\nforehand, we now discuss intriguing influence of the\nlength and exchange field on the critical current Icwhen\ntheFLlayer andFRlayer have nonidentical physical fea-\ntures. As illustrated in figure 3(a) and (b), we present\nthe dependence of IconkF(dL−dR) and (hL−hR)/EF\nrespectively on condition that the exchange field and\nlength ofFRlayer are all fixed. Take first one for ex-\nample, we find that the dependence of Icon length dif-\nferencekF(dL−dR) look like a “Fraunhofer pattern”.\nWith the length difference close to 0, Icwill increase\nand also accompanies the transition between the 0 and π\nstates. Asmentionedabove,if FLlayerandFRlayerhavethe identical exchange fields ( hL/EF=hR/EF= 0.1)\nbut different lengths, the Cooper pair passing through\ntheFLlayer andFRlayer could acquire the phase shift\nχ=Q(dL−dR) +π. So the variation of length dif-\nference can lead to the oscillation of Ic. On the other\nhand, ifFLlayer andFRlayer have the same length\n(kFdL=kFdR= 100) but different exchange fields, the\nCooper pair can get the phase shift χ= (Q−Q′)dL+π.\nIcwill also oscillate with ( hL−hR)/EFbecause ofQ∝h\n(see figure 3(b)).\nIn addition, the current-phase relations Ie(φ) in par-\nticular points are illustrated in figure 3(c), and corre-\nsponding LDOSs are plotted in figure 3(d). If one take\nthe identical parameter in FLandFRlayer, such as\nkFLL=kFLR= 100 and hL/EF=hR/EF= 0.1,\nthe Cooper pairs will obtain a net phase shift χ=π.\nThen we could observe a negative Josephson current and\na shark zero energy conductance peak in LDOS, which\nindicate the junction is located in πstate. When the\nlength difference kF(dL−dR) = 10, it corresponds to\nthe transition point between the 0 and πstates of the\njunction. At this critical point, the harmonic I1sinφof\nthe current ( I(φ) =I1sinφ+I2sin(2φ) +...) vanishes,\nand theI2sin(2φ) will be fully revealed. Subsequently,\nthe sign of Icis changed with the increase of the length\ndifference. For kF(dL−dR) = 20, the junction is in the\n0 state, and the LDOS at ǫ= 0 will be converted from\nthe peak to a valley.\nFinally, we discuss the dependence of Icon the misori-\nentation angle θ, and the spatial distributions of the spin\nsinglet pair and the spin triplet pair for two different θ.\nOne can see from the inset in figure 4(a) that, when the\norientation of the magnetization in the FMregion is per-\npendicular to the direction of FLlayer andFRlayer, the\ncritical current reaches the maximum. However, it de-\ncreases to minimum on condition that the magnetization\nofFMregion is parallel or antiparallel to the one in FL\nlayer andFRlayer. For given thickness of the FMregion,\nit is possible to find the exchangefield at which switching\nbetween parallel and perpendicular orientations will lead\nto switching of Icfrom near-zero to a finite value. This\neffect may be used for engineering cryoelectronic devices\nmanipulating spin-polarized electrons. Furthermore, we\nfind the spin singlet pair amplitude f3oscillates in all F\nregion atθ= 0. But it will be coherent counteracted\natθ=π(see the main plot in figure 4(a)). In contrast,\nthe spin triplet pair amplitude f0are almost identical in\nabove two cases. For θ= 0, the equal spin triplet pair\namplitudef1is zero in entire Fregion. However, f1only\nsurvives in FMregion and can not exist in FLlayer and\nFRlayer under the condition of θ=π. From these con-\nsequences, we can derived that the long-range supercur-\nrent mostly arises from the coherent propagation of f3,\nand the contributions of f0andf1can be ruled out. To\nfurther uncover this contribution, we take into account\nf3induced by only one S. According to above theory,\nthe spin mixed state state in FLlayer is expressed as\n|↑↓∝an}b∇acket∇i}hteiQ·R− |↓↑∝an}b∇acket∇i}hte−iQ·R= (|↑↓∝an}b∇acket∇i}ht− |↓↑∝an}b∇acket∇i}ht )cos(Q·R) +i(|↑↓5\n050100150200−0.0500.050.10.15\nkFyf3\n0 100 200−0.0400.040.08\nkFyf0\n  \n0 100 200048x 10−3\nkFyf1\n050100150200−0.0400.040.08\nkFyf3\n  00.250.50.75100.10.20.3\nθ¯hIc/e∆θ=0\nθ=π/2\nkFdS1=400, kFdS2=0\nkFdS1=0, kFdS2=400(a) (b)\n(c) (d)\nFIG. 4. (color online) Spatial distributions of the spin sin glet\npair amplitude f3(a), the real parts of spin triplet pair am-\nplitudef0(b) andf1(c) for two angles θ= 0 and θ=π/2 at\nωDt= 12. The panels (a), (b) and (c) utilize the same legend.\nInset in (a) shows the critical current as a function of the an -\ngleθ. (d) The f3plotted as a function of kFyfor two cases\nkFdS1= 400,kFdS2= 0 and kFdS1= 0,kFdS2= 400 when\nθ=π/2. Parameters used in all figures: kFdL=kFdR= 100,\nkFdM= 10,hL/EF=hR/EF= 0.1,hM/EF= 0.17 and\nφ= 0.\n∝an}b∇acket∇i}ht+|↓↑∝an}b∇acket∇i}ht)sin(Q·R). This state in FRlayer can be\nconverted as |↑↓∝an}b∇acket∇i}htei(−Q′·R′+π)− |↓↑∝an}b∇acket∇i}hte−i(−Q′·R′+π)= (|↑↓\n∝an}b∇acket∇i}ht− |↓↑∝an}b∇acket∇i}ht)cos(Q′·R′+π) +i(|↑↓∝an}b∇acket∇i}ht+|↓↑∝an}b∇acket∇i}ht)sin(Q′·R′). We\ncan see the spin singlet pair has a phase shift πin theexpression of the FLlayer andFRlayer. But the spin\ntriplet pair with zero spin projection shows the same\ndescription in these two layers. These inferences are in\nagreement with our numerical results. As demonstrated\nin figure 4(d), f3shows a antisymmetric configuration\naround the middle FMregion. As a result, the superpo-\nsition of two f3stemming from the left and right Swill\nmake their amplitudes cancel each other.\nIn conclusion, we have studied numerically the long-\nrange supercurrent in a SFSstructure including a non-\ncollinear magnetic domain in the ferromagnetic region.\nWe find the magnetic domain could induce a spin-flip\nprocess, which can reverse the spin orientations of the\nsinglet Cooper pair when this pair propagate through\nthe magnetic domain region. This process will make the\nsinglet Cooper pair generate a phase-cancellation effect\nand acquire an additional πphase shift. If the ferromag-\nnetic layers on both sides of magnetic domain have the\nsame features (such as thickness and exchange field), the\nnet change in the relative phase of the singlet Cooper\npair isπwhen the pair passes through the entire Fre-\ngion. In this case, the long-range supercurrent mostly\nstems from the singlet Cooper pairs. The reason is that\nthe equal spin triplet pairs are only present in the mag-\nnetic domain region and can not spread to the other two\nferromagnetic layers. It is hoped that our results could\npropose a new way to generate the long-range Josephson\ncurrent.\nThis work is supported by the State Key Pro-\ngram for Basic Research of China under Grants No.\n2011CB922103andNo. 2010CB923400,andtheNational\nNatural Science Foundation of China under Grants No.\n11174125 and No. 11074109.\n∗menghao1982@shu.edu.cn\n1A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n2F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys. 77, 1321 (2005).\n3A. A. Golubov, M. Yu. Kupriyanov, and E. Ilichev, Rev.\nMod. Phys. 76, 411 (2004).\n4M. Eschrig, Phys. Today 64, No. 1, 43 (2011).\n5M. Houzet and A. I. Buzdin, Phys. Rev. B 76, 060504(R)\n(2007).\n6M. Eschrig, J. Kopu, J. C. Cuevas, and G. Schon, Phys.\nRev. Lett. 90, 137003 (2003); M. Eschrig and T. Lofwan-\nder, Nature Physics 4, 138 (2008).\n7Y. M. 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We raise the possibility that the rea son for the absence\nof spontaneous magnetization is suppression of itinerant f erromagnetism by electronic disorder in\nthe strong localization regime.\nPACS numbers: 72.15.Rn; 75.70.Ak; 73.61.At\nFerromagnetism in the transition metals (Fe, Co and\nNi) relies, at least partially, on the itinerancy of the con-\nduction band electrons. The balance between kinetic en-\nergy loss and exchange energy gain leads to polarization\nof the band electrons and to magnetic order. It is inter-\nesting to askwhat happens when the mobility ofthe elec-\ntrons is strongly suppressed by disorder. In the extreme\ncase electrons can be localized with localization length\nξ, that may be of atomic scale. Under these conditions\none may expect itinerant ferromagnetism to be entirely\nsuppressed since the electronic functions do not overlap\nand exchange energy is no longer relevant. This may be\nanalogous to the situation in uniform disordered super-\nconductors, where the superconductivity is suppressed\nby strong disorder as a result of disorder induced pair\nbreaking (see for example [1, 2, 3]).\nIn order to achieve high enough disorder for significant\nlocalizationinhomogeneousmetallicmaterialsitisneces-\nsary to use very thin amorphous films having thicknesses\nof a few mono-atomic layers in which the sheet conduc-\ntance can be of the order ofe2\nhor less. For this purpose\nthe samples studied in this work were Ni, Co and Fe thin\nfilms fabricated using ”quench condensation” (evapora-\ntiononacryo-cooledsubstrate). Thistechniqueallowsto\ndeposit sequential layers of ultrathin films and measure\ntransport without thermally cycling the sample orexpos-\ning it to atmosphere. If a thin underlayer of Ge or Sb is\npre-deposited before quench condensing a metal film, the\nunderlayerwetsthesubstratethusenablingthegrowthof\ncontinuousultrathinamorphouslayersevenatmonolayer\nthickness [4]. The underlayer being an insulator at low\ntemperatures is assumed to have negligible effect on the\nelectricpropertiesofthemetal[2, 4]. Theabilitytostudy\na single sample while driving it from strong localization\nto weak localization has been vastly used in the context\nof the superconductor-insulator transition. In the cur-\nrent work we used this technique to drive a ferromag-\nnetic amorphous film from strong to weak localization.\nWe studied Ni, Co and Fe films having resistance in the\nrange of 1 MΩ - 100 Ω and thicknesses of 0.2-8 nm. The\nfilmswereevaporatedona2nmthicklayerofSborGe, in0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 σ/σ 0\nd(mfp) 1 222k 23k 24k R4k =0.3k Ω\n  RS(Ω)\nlog(T) (a) (b) \n0.020 0.025 0.030 0.035 -2 -1 01R4k =150 k Ω\n  ln(R s)\n1/T 0.5  [1/K 0.5 ]\nFIG. 1: Relative conductivity versus the thickness in mean\nfree path units for a Fe sample . The solid line is a fit to eq.\n1. The insets show resistance as a function of temperature in\nthe weak (a) and strong (b) localization regime (for 3.5 and\n0.3 nm thick samples respectively).\na Hall bar geometry allowing 4 probe resistance and Hall\neffect measurements. Both sample resistance and thick-\nness were monitored during the growth and the process\nwas stopped at different evaporation stages. Room tem-\nperature AFM measurements of these films (performed\ninambientafterheatingthem up) revealthicknessrough-\nness smaller than 1nm.\nFigure 1 shows the normalized conductance versus\nthickness of a Fe sample. The data is fitted the following\nexpression extracted from the scaling theory of localiza-\ntion [5, 6]:\nσ/σ0= 1−3A/(2k2\nfdl)ln[τϕ/3τ], (1)\nwhereσ0is the Drude conductivity, kfis Fermi wave\nlength of the bulk, d is the thickness of the film and\nτandτϕare the transport and dephasing time respec-\ntively. Eq. 1. is well known in the theory of weak local-\nization. We have introduced a phenomenological factor\nA≈3 which enables us to extend the fit well beyond the\nexpected validity range into the strong disorder regime.\nThe discussion on justification of this procedure will be\npresented elsewhere [6]. The fitted relevant lengthscales:2\nl(nm) lϕ(nm) RC(Ω)dC(nm) µ(µB)\nFe 0.4 46 80k 0.5 2.2\nCo 0.29 21 10k 0.8 1.7\nNi 0.27 25 2k 1.8 0.6\nTABLE I: Mean free path, l, dephasing length, lϕ, critical\nresistance and thickness for the appearance of ferromagnet ic\nsignatures, RCanddC, and bulk magnetic moments, µB, for\nthe Ni,Co and Fe films use in this study.\nl-the mean free path and lϕ- the dephasing length are\nlisted in table 1. The insets depict the resistance versus\ntemperature curves for 0.3 nm and 3.5nm thick Fe film\nshowing Efros-Shklovskii like hopping behavior and log-\narithmic dependence typical of weak localization behav-\nior respectively. Extracting the localization lengths from\nsuch R(T) curves allows us to determine the crossover\nfrom strong localization ( ξ < L ϕ) to weak localization\n(ξ > Lϕ) at a sheet resistance of ∼10kΩ and thickness\nof∼0.8 nm.\nIn order to probe the magnetic state of the films we\nmeasured both Hall effect (HE) and magnetoresistance\n(MR) for each evaporation stage by applying a magnetic\nfield perpendicular to the films. All presented results\nwere obtained at T=4K. Hall resistivity in magnetic ma-\nterials is combined of the ordinary part, ρ0, observed\nin all normal metals due to the Lorentz force which is\nproportional to the magnetic field, H, and the extraor-\ndinary Hall effect (EHE), ρxy, which is proportional to\nthe magnetization of ferromagnetic films, M, so that\nρxy∝REHEµ0M,REHEbeing the extraordinary Hall\neffect coefficient. The two processes, which are believed\nto have the major contribution to the EHE, skew scat-\ntering and side jump, are both scattering processes, and\nthe EHE is expected to depend on the longitudinal re-\nsistivity (which is also due to scattering) in the following\nway:\nREHE=aρxx+bρ2\nxx (2)\nwhereais the skew scattering coefficient and bis the side\njump coefficient. Gerber et-al [7] showed that for very\nthin films the dominant factor is skew scattering and ρxx\nwas found to be proportional to ρxyfor Ni films between\n4 to 20 nm thick. For high magnetic fields the magneti-\nzation saturates and only the ordinary part contributes\nto the magneticfield dependence ofthe HE. Studying the\nEHE is an elegant way to measure magnetic properties\nof thin films using transport measurement, in particular,\nthe saturation magnetization can be obtained by extrap-\nolating the Hall resistance to zero field.\nFig 2a. shows the Hall effect of sequential quench con-\ndensed layers of Ni. Down to a sheet resistance of 2 kΩ\n(1.8 nm thick) only the ordinary Hall effect is observed.\nAs more material is added and the resistance further de-\ncreasedan EHE contribution develops and increaseswithH(T) (b) (a) \n-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 \n 6k Ω\n 4k Ω\n 2k Ω\n 1.5k Ω\n 1k Ω\n 0.7k Ω\n 0.3k ΩρH(µΩ cm) \n-4 -2 0 2 4-0.10 -0.05 0.00 0.05  ∆R/R( %)\nFIG. 2: Hall effect (a) and magnetoresistance (b) measure-\nments of sequential quench condensed Ni films. T=4.2K. The\ndash line in the negative MR of the 0.3 kΩ stage is the fit\nfor to the AMR phenomenological expression in eq. 3. The\nsolid lines in the positive MR curves show fits to the anti\nweak-localization theory of ref [9] with l=0.265 nm, lϕ= 25\nnm andlsoequal to 20, 21.6 and 22.7 nm for films with sheet\nresistance 6, 4, and 2 kΩ respectively.\nincreasing thickness. This defines a critical resistance,\nRC, and critical thickness, dC, for the appearance of the\nEHE. Fig 3 shows both the longitudinal resistivity, ρxx,\nandthe transverse(Hall) resistivity(determined fromex-\ntrapolating to H=0), ρxy, as a function of thickness for a\ntypical Ni sample. ρxxdecreasesmonotonously while ρxy\nis zero for the thinnest films. For films thicker than 2 nm\nρxyincreases with thickness until at d=4-5 nm it changes\nits trend. For thick enough samples (d >4 nm)ρxy∝ρxx\nin accordance with [7]. If the REHE∝ρxxexpression is\nextrapolated to smaller d, our results indicate that the\nmacroscopic magnetization, M, is zero for very resistive\nsamples in the strong localization regime, it grows in an\nintermediate regime until it reaches saturation for films\nhaving sheet resistance below 500 Ω.\nInterestingly, the magnetoresistance also undergoes a\nunique crossover at the same critical region. For high re-\nsistance (small thickness) a positive magnetoresistance is\nobserved, whereas for R < R C(d > dC) the MR changes\nsign and becomes negative, saturating at fields above a\nsaturationfield, HS. Thisisseeninfigure2bwhichshows\nthat the MR exhibits a change of MR sign for resistances\nbelow 2 kΩ ( R < R C).\nSimilar behavior is observedin Co and Fe as well, how-\neverRCanddCvary from material to material. Table\n1 summarizes the critical thicknesses and resistances for3\nwhich EHE appears and the MR changes sign for the\nthree transition metals.\nJust like the emergence of EHE, we interpret the MR\nsign change as a signature for the appearance of ferro-\nmagnetism in the film. Positive MR such as seen in our\nthin layers is also observed in quench condensed non-\nmagnetic layers such as Ag, where the MR remains pos-\nitive for films with resistance as low as 100 Ω. This be-\nhavior is understood as being due to weak localization\nin the presence of strong spin orbit. Since the films are\nultrathin, one can indeed expect strong Rashba spin or-\nbit scattering [8] on the surface leading to weak anti-\nlocalization and positive MR. The solid lines in fig 2b\nshows fits of our curves to the spin-orbit weak localiza-\ntion expression [9]. The spin orbit length, lSO, grows as\nthe film thickens as can be expected if Rashba spin orbit\nis the dominant factor.\nThe crossover to negative MR occurs only in ferro-\nmagnetic materials. Dugaev [10] showed that weak anti-\nlocalization due to spin orbit interaction does not oc-\ncur in ferromagnetic systems so that if ferromagnetism\nis present in a film, the MR should be always negative.\nHence, the MR sign change is also an indication for the\nappearance of magnetization in the film. We have tried\nto fit the negative MR curve to the Dugaev expression\n[10] but could obtain reasonable fits only for very thin\n(d <0.5nm) Fe layers [6]. For thicker samples we find\nthat the curves are well described by anisotropic magne-\ntoresistance (AMR) typical to ferromagnetic films. Since\nthe field is applied perpendicular to the current direc-\ntion one can expect a negative contribution that should\ndepend quadratically on the angle between the current\nand the magnetization [11]. We have fitted our curves to\nthe following phenomenological expression that assumes\nquadratic dependance of ∆ Ron H for low fields and sat-\nuration for fields larger than the saturation field, HS:\n∆R(H) = ∆R(∞)H2\nH2+H2\nS(3)\nA typical fit is seen in the 0.3 kΩ stage in fig. 2b. The\nextracted saturated fields from these fits match those ex-\ntracted from the Hall effect curves for all stages of evap-\noration.\nHence both HE and MR measurements imply that\nfor strong enough disorder, R > R C, the sample does\nnot show signs of spontaneous magnetization and that\nferromagnetism emerges only for films characterized by\nsmaller resistances. One may wonder whether this ef-\nfect can be due to granularity in the film. While the\nsamples are electrically continuous for films with thick-\nnesses of 0.2 nm we can not rule out the presence of\nsome film granularity. For comparison we have produced\ngranular films of Ni with grain sizes of about 20 nm in\ndiameter and 2 nm in height [12]. These are achieved by\nquench condensing Ni on a bare Si/SiO substrate with-\nout depositing a Sb or Ge wetting layer [13, 14]. For0 2 4 6 8 10 12 01000 2000 \n0.0 0.2 0.4 0.6 0.8 1.0 \nρxy (µΩ cm) ρxx (µΩ cm) \nd(nm) 10 210 310 410 510 624\nRXX (Ω)RXY (Ω)\n  \nFIG. 3: Longitudinal, ρxx, and saturated transverse (Hall)\nresistance, ρxyof a homogenous Ni sample versus the thick-\nness of the films. The inset shows the saturated transverse\n(Hall) resistance, Rxyversus longitudinal sheet resistance of\na Nigranular film.\nthe thinnest films of this type the grains are superpara-\nmagnetic showing no hysteresis in the magnetoresistance\ncurve [13]. In the granular case, EHE is observed even\nfor the highest-resistance measured samples R∼1MΩ.\nAs material is added the resistance drops considerably,\nhowever, the Hall effect resistance hardly changes until\nR <10kΩ, as can be seen in the inset of fig 3. Such\nbehavior has been also reported for granular Fe prepared\nby different methods [15] whereit wasunderstood that in\nthe granularcase the EHE is due to scattering within the\ngrains rather than being due to hopping between grains.\nForR <10kΩRxyreduces linearly with Rxx. This is\nthe regime in which the grains are expected to coalescing\nand the behavior approaches that of a continuous film.\nIn any case, a granular film behaves very differently than\nour uniform films.\nWe have also considered the possibility that appear-\nance of the magnetization is a function of the film thick-\nness. A number of experiments on epitaxially grown Ni,\nCoandFethinfilmsonCusubstrates[16, 17,18, 19]have\nshown that the Curie temperature, TCdrops sharply to\nzero for thicknesses approaching an atomic monolayer.\nThesesampleswereorderedcrystallinefilms andthe sub-\nstrates were metallic, hence, no effect of electron local-\nization could be expected. The suppression of TCwas\ninterpreted as the formation of magnetic dead layers due\nto electronic hybridization with the states in the Cu sub-\nstrate. This reduces the density of states and the Stoner\ncriterion is defied. A similar conclusion was drawn by\nBergman [20] who saw no signs for magnetization in the\nHall effect of quench condensed Ni evaporated on an\namorphousmetallic Pb75Bi25substratesforfilms thinner\nthan 2.5 monolayers. It seems that our results are differ-\nent from all the above. We note that our films are grown\non insulating substrates, hence electronic hybridization\nis not relevant. Consequently we are unable to detect\nmagnetization in Ni films that are 1 .8 nm thick ( ∼7\nmonolayers), which is much thicker than any of the pre-4\n-5 -4 -3 -2 -1 0246 1nm-Co \n 0.1nm-Ag \n 0.2nm-Ag \n 0.3nm-Ag \n 0.4nm-Ag \n 0.5nm-Ag RH(Ω)\nH(T) 0 2 4 6 80.04 0.06 0.08 0.10 \n  \n0 1 2 3 4 5 60.08 0.12 0.16 0.20 0.24 \n  \ndAg ( Å)RXy (Ω)(a) \n(b) RH(Ω)\nH(T) \nFIG. 4: Hall effect measurements of a series of a 1.1 nm Co\nfilm coated by Ag overlayers. The inset shows the saturated\ntransverse (Hall) resistance, Rxyof a 2nm thick Ni film (a)\nand a 1nm thick Co sample (b) versus the thickness of the Ag\noverlayer.\nvious experiments.\nIn order to test the notion that localization of the con-\nduction electrons and not the thickness of the films is\nthe cause for absence of magnetization we performed an\nexperiment designed to vary the degree of localization\nwithout changing the ferromagnetic layer thickness. For\nthis purpose we deposited thin layers of normal (non-\nmagnetic) Ag on top of a thin ferromagnetic layer in\nthe vicinity of RC. The normal metal is not expected\nto increase the amount of magnetic material but it can\nreduce electron localization and screen out electronic in-\nteractions. This is similar to an experiment performed\non ultrathin layers of superconductors in which the crit-\nical temperature is suppressed by disorder [21, 22]. Such\nfilms exhibit an ”inverse proximity effect”; Addition of\nultrathin normal metal layers acts to increase TCrather\nthan reduce it as expected from the proximity effect.\nIn our ferromagnetic films we find a similar effect. For\nthick films, adding any amount of an Ag overlayer re-\nduces the EHE signal. This is the case for Ni films for\nwhichdC∼1.8 nm. Indeed one expects that electronic\nhybridization with the Ag electrons should suppress fer-\nromagnetism. On the other hand, for 1 nm thick Co\nfilms, the first few Ag overlayers have an opposite effect.\nAs can be seen in fig. 4, adding ultrathin normal metal\nlayers plays the same role as adding ferromagnetic layers\nin that it increases the EHE signal. Thick enough Ag\noverlayers result in a change of trend and suppression of\nthe EHE. This is seen in the inset of fig 4 where we com-\nparethe outcomeofaddingultrathinlayersofAgto2nm\nthick and 1 nm thick ferromagnetic films. The results of\nthe 1nm Co film demonstrate that reducing the disorder\nwithout changing the ferromagnet thickness results in an\nincrease in the magnetization despite the fact that there\nis a competing effect of electron hybridization.\nIn summary, our results show that no ferromagnetism\ncan be detected by Hall effect or magnetoresistance mea-\nsurements for films having resistances larger than RC.\nWe note that RC, which is material dependent, seems tocorrelate with the material atomic magnetic moment µ,\nas can be seen in table 1. Hence it is possible that the\ncritical point at which magnetization appears in the film\nis related to the fraction of magnetization that is due to\nlocalized moments on the atoms. Clearly, more theoreti-\ncal work is required to clarify the interplay between itin-\nerant ferromagnetism and Anderson localization. Such\nresearch may prove to be very useful in shedding light on\nthe oldpuzzleofthe originofmagnetismin the transition\nmetals.\nWearegratefulforfruitfuldiscussionswithD.Golosov,\nA.M. Goldman and R. Berkovits. This research was sup-\nported by the Israeli academy of science (grant number\n249/05).\n[1] J.M. Valles Jr., R.C. Dynes and J.P. Garno, Phys. Rev.\nLett.69, 3567 (1992).\n[2] A. M.Goldman andN.Markovic, PhysicsToday, Novem-\nber, 39 (1998) and references therin.\n[3] A.M. Finkelstein, Physica B, 197, 636 (1994) and ref.\ntherin.\n[4] M. Strongin, R. Thompson, O. Kammerer and J. Crow,\nPhys. Rev. B1, 1078 (1970).\n[5] E. Abrahams, P. W. Anderson, D. C. Licciardello, and\nT. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).\n[6] N. Kurzweil, A. Frydman and E. Kogan, in preparation.\n[7] A. Gerber, A. Milner, A. Finkler, M. Karpovski, L. Gold-\nsmith, J. Tuaillon-Combes, O. Boisron, P. M´ elinon, and\nA. Perez, Phys. Rev. B, 69, 224403 (2004).\n[8] Yu. A. Bychkov and E. I. Rashba, J. Phys. C 17, 6093\n(1984); G. Dresselhaus, Phys. Rev. 100, 580 (1955).\n[9] S. Chakravarty and A. Schmid, Phys. Rep. 140, 195\n(1986).\n[10] V. K. Dugaev, P. Bruno, and J. Barna´ s, Phys. Rev. B\n64, 144423 (2001).\n[11] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11,\n1018 (1975).\n[12] A. Frydmanand R.C. Dynes, Phil. Mag. 81, 1153 (2001).\n[13] A. Frydman and R.C. Dynes, Sol. State Comm. 110, 485\n(1999).\n[14] A.Frydman,T.L.KirkandR.C.Dynes, SolidStateCom-\nmun.114, 481 (2000).\n[15] P. Mitra, R. Misra, A. F. Hebard, K. A. Muttalib and P.\nWolfle Phys. Rev. Lett., 99, 046804 (2007)\n[16] F. Huang, M. T. Kief, G. J. Mankey, and R. F. Willis,\nPhys. Rev. B 49, 3962 (1994).\n[17] F. Huang, G. J. Mankey, M. T. Kief, and R. F. Willis, J.\nAppl. Phys. 73, 6760 (1993).\n[18] X. Y. Lang, W. T. Zheng, and Q. Jiang, Phys. Rev. B\n73, 224444 (2006).\n[19] R. Zhang and R. F. Willis, Phys. Rev. Lett. 86, 2665\n(2001).\n[20] G. Bergmann, Phys. Rev. Lett. 41, 264 (1978).\n[21] O. Bourgeois, A Frydman and R. C. Dynes, Phys. Rev.\nLett.88, 186403 (2002).\n[22] O. Bourgeois, A Frydman and R. C. Dynes, Phys. Rev.\nB,68, 092509 (2003)."    },    {        "title": "2208.12799v3.Mechanics_of_a_ferromagnetic_domain_wall.pdf",        "content": "Mechanics of a ferromagnetic domain wall\nSe Kwon Kim1and Oleg Tchernyshyov2\n1Department of Physics, Korea Advanced Institute of Science and Technology,\nDaejeon 34141, Republic of Korea\n2William H Miller III Department of Physics and Astronomy and Institute for\nQuantum Matter, Johns Hopkins University, Baltimore, MD 21218, USA\nE-mail:1sekwonkim@kaist.ac.kr,2olegt@jhu.edu\nAbstract. This paper gives a pedagogical introduction to the mechanics of\nferromagnetic solitons. We start with the dynamics of a single spin and develop all\nthe tools required for the description of the dynamics of solitons in a ferromagnet.\nSubmitted to: J. Phys.: Condens. MatterarXiv:2208.12799v3  [cond-mat.mes-hall]  24 Dec 2022CONTENTS 2\nContents\n1 Introduction 3\n2 Single magnetic dipole 3\n2.1 Magnetic moment and angular momentum . . . . . . . . . . . . . . . . 3\n2.2 Spherical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.3 Precessional dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n2.4 Example: Larmor precession . . . . . . . . . . . . . . . . . . . . . . . . 5\n2.5 Conservative and gyroscopic forces . . . . . . . . . . . . . . . . . . . . 5\n2.6 Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.7 Example: precession near a potential minimum . . . . . . . . . . . . . 7\n2.8 Dissipative force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.9 Example: Larmor precession with dissipation . . . . . . . . . . . . . . 8\n2.10 Transformation to canonical variables . . . . . . . . . . . . . . . . . . 9\n2.11 Transformation to general variables . . . . . . . . . . . . . . . . . . . . 10\n2.12 Rayleigh's dissipation function . . . . . . . . . . . . . . . . . . . . . . 10\n2.13 Example: strong easy-plane anisotropy . . . . . . . . . . . . . . . . . . 11\n2.14 Recap: equations of motion . . . . . . . . . . . . . . . . . . . . . . . . 12\n3 Domain wall in a ferromagnetic wire 13\n3.1 Heisenberg spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . . 13\n3.2 Continuum theory of a ferromagnet . . . . . . . . . . . . . . . . . . . . 15\n3.3 Model with local interactions . . . . . . . . . . . . . . . . . . . . . . . 16\n3.4 Linear spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n3.5 Nonlinear spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n3.6 Static domain wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n3.7 Domain wall in a magnetic \feld . . . . . . . . . . . . . . . . . . . . . . 20\n3.8 Weakly perturbed domain wall . . . . . . . . . . . . . . . . . . . . . . 21\n3.9 Normal modes of a domain wall . . . . . . . . . . . . . . . . . . . . . . 22\n3.10 Angular momentum of a domain wall . . . . . . . . . . . . . . . . . . . 24\n3.11 Linear momentum of a domain wall . . . . . . . . . . . . . . . . . . . 25\n3.12 Counting states of a domain wall . . . . . . . . . . . . . . . . . . . . . 26\n3.13 Hamiltonian dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 26\n4 Walker's model of a domain wall 27\n4.1 Energy functional and domain-wall con\fguration . . . . . . . . . . . . 28\n4.2 Dynamics of collective coordinates . . . . . . . . . . . . . . . . . . . . 29\n4.3 Steady-state motion in a weak \feld . . . . . . . . . . . . . . . . . . . . 29\n4.4 Walker's breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n4.5 Oscillatory motion in a very strong \feld . . . . . . . . . . . . . . . . . 31\n4.6 D oring's mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\n5 Discussion 33\nAcknowledgements 33\nAppendix A 33CONTENTS 3\n1. Introduction\nThis article aims to provide a self-contained introduction to theory of ferromagnetic\nsolitons. The subject has been actively researched for more than 50 years and there\nare a number of excellent reviews and books [1, 2]. The last decades have seen new\ndevelopments in the \feld such as the use of collective coordinates to describe the\ndynamics of solitons [3, 4, 5, 6] as well as in technological applications such as soliton-\nbased racetrack memory [7, 8]\nA ferromagnet contains a large number of atomic magnetic dipoles that tend\nto line up with one another at su\u000eciently low temperatures. The interaction most\ncommonly responsible for the parallel alignment is Heisenberg's exchange, which\nrepresents a quantum-mechanical e\u000bect related to the fermionic statistics of electrons.\nOne of the most frequently used models of a ferromagnet is the Heisenberg model\nwith quantum spins of length S=~=2 on a lattice with nearest-neighbor exchange\ninteractions. The ground state has all spins pointing in the same direction. Elementary\nexcitations are magnons|quasiparticles carrying spin Sz=\u0000~along the direction of\nmagnetization. A magnon mode in a coherent state with a large amplitude represents\na classical wave of magnetization (a spin wave) [9].\nThis widespread approach, starting with a quantum model of spins on a lattice,\nis inconvenient for our purposes. Although it is suitable for obtaining weakly excited\nstates such as magnons, the treatment of topological solitons at the quantum level\nand on a lattice is too complicated. Instead, we shall start with continuum models of\na classical ferromagnet. This approach allows one to obtain both linear and nonlinear\nexcitations at the classical level and then quantize them.\nIn Section 2, we discuss the dynamics of a single spin and introduce various\nmathematical tools. The same tools, suitably generalized, are used for the description\nof soliton dynamics in a one-dimensional ferromagnet in Section 3. Sec. 4 deals with\na more complex model of a domain wall of Schryer and Walker [10].\n2. Single magnetic dipole\n2.1. Magnetic moment and angular momentum\nThe building block of a macroscopic magnet is an atom with angular momentum J\nand magnetic dipole moment \u0016related by the gyromagnetic ratio \r,\n\u0016=\rJ: (1)\nIf the angular momentum comes solely from the electron spins, the gyromagnetic ratio\nis\r=e=mec, wheree <0 is the electron charge and meits mass. More generally,\n\r=ge=2mec, where the Lande g-factor is determined by the lengths of spin, orbital,\nand total angular momenta S,L, andJ[11].\nWe shall treat both the angular momentum Jand the magnetic moment \u0016as\nclassical physical variables of \fxed lengths Jand\u0016. Often both are expressed in terms\nof a unit vector mparallel to J:\nJ=Jm;\u0016=\rJm;jmj= 1: (2)\nBecause of the linear proportionality between the three quantities, we will casually\nrefer to mas the magnetic moment or spin.CONTENTS 4\nem\ne✓e\u0000\nFigure 1. The local frame de\fned by the unit vectors (4).\n2.2. Spherical geometry\nTo resolve the constraint of the unit length jmj= 1, it is convenient to introduce two\nspherical angles: the polar angle \u0012(co-latitude) and the azimuthal angle \u001e(longitude):\nm= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012): (3)\nThey de\fne a local frame with three mutually orthogonal unit vectors\nem=m= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012);\ne\u0012=@m\n@\u0012= (cos\u0012cos\u001e;cos\u0012sin\u001e;\u0000sin\u0012); (4)\ne\u001e=1\nsin\u0012@m\n@\u001e= (\u0000sin\u001e;cos\u001e;0);\npointing up, south, and east, respectively (Fig. 1).\nSmall deviations of magnetization from a given direction mcan be written in the\nlocal frame as follows:\n\u000em=e\u0012\u000em\u0012+e\u001e\u000em\u001e; \u000em \u0012=\u000e\u0012; \u000em\u001e= sin\u0012\u000e\u001e: (5)\n2.3. Precessional dynamics\nThe dynamics of an atomic magnetic dipole is very di\u000berent from that of a Newtonian\nparticle. For a particle, an external force generates an acceleration proportional to the\nforce and inversely proportional to its mass. In contrast, a magnetic dipole does not\nhave inertia. Like a fast-spinning gyroscope, it precesses with a velocity proportional\nto the external torque.\nIt is helpful to picture a magnetic dipole whose south pole is pivoted and the\nnorth pole is allowed to move on a sphere of radius rso that its position is r=rm.\nThe force from the potential energy may then be calculated as F=\u0000dU\ndr=\u00001\nrdU\ndm\nand its torque as \u001c=r\u0002F=\u0000m\u0002dU\ndm. The torque determines the rate of change\nof angular momentum dJ=dt=Jdm=dt, hence the equation of motion\nJ_m=\u0000m\u0002dU\ndm; (6)CONTENTS 5\nwhere _mstands fordm=dt. Note that the geometrical size of the dipole rcanceled\nout.\nIt is worth keeping in mind that the derivative dU=dmis only taken in the\ndirections locally tangential to the sphere jmj= 1. In a formal sense, this derivative\nis de\fned as the coe\u000ecient in the Taylor expansion\nU(m+dm) =U(m) +dU\ndm\u0001dm+:::; (7)\nwheredmis an in\fnitesimal displacement tangential to the sphere and thus transverse\ntom. Even ifdU=dmhad a component parallel to m, it would not contribute to the\n\frst order in the expansion (7). For this reason, dU=dmshould be understood as the\ntransverse component of the energy derivative.\n2.4. Example: Larmor precession\nConsider the motion of a magnetic dipole in an external magnetic \feld h. Its potential\nenergy isU=\u0000\u0016\u0001h=\u0000\u0016m\u0001h. Upon substituting Uinto the equation of motion\n(6) we obtain\nJ_m=\u0000\u0016h\u0002m: (8)\nThe magnetic moment rotates uniformly about the direction of the \feld at the Larmor\nprecession frequency\n!L=\u0016h=J =\rh: (9)\nThe angle between the magnetic moment and the \feld remains constant, implying\nconservation of energy U=\u0000\u0016\u0001h.\nTo see this explicitly, rewrite the equation of motion in terms of spherical angles\n\u0012and\u001e(3):\n\u0000Jsin\u0012_\u001e\u0000@U\n@\u0012= 0; J _\u0012\u00001\nsin\u0012@U\n@\u001e= 0: (10)\nFor the \feld along the zaxis, h= (0;0;h), the potential energy U=\u0000\u0016hcos\u0012is\nindependent of the azimuthal angle \u001e. Thus the polar angle \u0012remains unchanged\nduring precession.\n2.5. Conservative and gyroscopic forces\nConservation of energy is a general property of the precession equation (6). After\nsome simple algebra, it can be transformed into the form\n\u0000J_m\u0002m\u0000dU\ndm= 0: (11)\nReturning to the picture of a dipole pivoted at the south pole, we may interpret this\nequation of motion as Newton's second law for a massless particle con\fned to move\non a unit sphere (hence no inertial term proportional to m). The second in Eq. (11)\nobviously represents a conservative force.\nWhat does the \frst term represent? It is similar to the Lorentz force acting on\na moving electric charge in a magnetic \feld as it is proportional to the velocity _m\nand is orthogonal to it. We thus may interpret the \frst term in Eq. (11) as a Lorentz\nforce acting on a particle with unit electric charge moving on the unit sphere jmj= 1\nin a radial \\magnetic \feld\" b(m) =\u0000Jm, as if there were a magnetic monopole ofCONTENTS 6\nstrengthJat the sphere's center. (We use the scare quotes to distinguish the \fctitious\n\\magnetic \feld\" bon the unit sphere jmj= 1 from the physical magnetic \feld h.)\nThe \\Lorentz force\" in Eq. (11) does no work and thus cannot be described in\nterms of potential energy. It belongs to a class of gyroscopic forces \frst discussed by\nKelvin [12]. Precession of a spin is indeed reminiscent of the motion of a fast-spinning\ngyroscope.\nEq. (11) can be used to establish conservation of potential energy during motion:\ndU\ndt=_m\u0001dU\ndm=_m\u0001(\u0000J_m\u0002m) = 0: (12)\n2.6. Lagrangian\nThe equation of motion (11) can be obtained by minimization of action S=R\nLdt.\nThe Lagrangian\nL=a(m)\u0001_m\u0000U(m) (13)\nhas no kinetic energy [9]. The \frst term gives rise to the gyroscopic (\\Lorentz\") force\nand contains a gauge potential a(m) whose curl gives the \\magnetic \feld\" of the\nmonopole,\nrm\u0002a(m) =b(m) =\u0000Jm: (14)\nThe \frst term in the Lagrangian (13) is peculiar for a number of reasons.\n(i) Its action\nSg=Z\na(m)\u0001_mdt=Z\na(m)\u0001dm (15)\nis independent of how fast the vector mmoves on the unit sphere and depends\njust on the geometry of its path. Hence the name geometric action forSgand\ngeometric (or Berry) phase for Sg=~[13].\n(ii) This term is not uniquely de\fned. A gauge transformation\na(m)7!a(m) +rm\u001f(m); (16)\nwhere\u001f(m) is an arbitrary scalar function on the unit sphere, changes the vector\npotential a(m) but leaves the \\magnetic \feld\" b(m) the same.\n(iii) Strictly speaking, the magnetic \feld of a monopole b(m) cannot be represented\nby a gauge potential because this \feld con\fguration is not divergence-free: it\nhas a source with magnetic charge \u0000Jat the origin m= 0 [9, 14]. Although\nthe magnetic monopole is not accessible, we can still observe that there is a net\n\\magnetic \rux\"\u00004\u0019Jby integrating the \\\feld\" over the unit sphere.\n(iv) As a result of this problem, any choice of the gauge potential a(m) contains a\nsingularity. For example, the gauge potential [14]\na(m) =Jms\u0002m\n1\u0000ms\u0001m; (17)\nhas a singularity at m=ms. It is associated with a localized \\magnetic \rux\"\nof strength +4 \u0019J, which compensates the uniform \rux \u00004\u0019Jspread over the\nsphere. If we were allowed to explore the entire 3-dimensional space of m, we\nwould discover that there is a Dirac string carrying \\magnetic \rux\" +4 \u0019Jfrom\nthe origin to in\fnity in the direction ms. With the Dirac string attached, the\n\\magnetic \feld\" becomes solenoidal, which makes it possible to describe it in\nterms of a vector potential.CONTENTS 7\n(v) Vector potential (17) describes the \\magnetic \feld\" of a monopole b(m) =\u0000Jm\nalmost everywhere on the unit sphere m, with the exception of point ms, where\nthe \\magnetic \rux\" of the Dirac string pierces the sphere. As long as the\ntrajectory m(t) stays away from the Dirac string, we can use the gauge potential.\nTo complete this section, we write down the gauge term in spherical angles for\nthe azimuthally symmetric gauge choices with the Dirac string attached to the north\nand south poles, ms= (0;0;\u00061):\na(m)\u0001_m=J(cos\u0012\u00061)_\u001e: (18)\nThese are the two most commonly used gauge choices [9].\n2.7. Example: precession near a potential minimum\nChoose the axes so that the energy has a minimum at the north pole, m0= (0;0;1).\nIn the vicinity of the energy minimum,\nm= (mx; my;q\n1\u0000m2x\u0000m2y); (19)\nwe may expand the potential energy in powers of the transverse deviations mx\u001c1\nandmy\u001c1. To the second order,\nU(m) =U(m0) +Kxm2\nx\n2+Kym2\ny\n2+::: (20)\nHereKxandKyare positive constants and the omitted terms are of higher orders in\nthe transverse components of magnetization. Choose the gauge potential (17) with the\nsingularity at the south pole, ms= (0;0;\u00001), so that a\u0019(Jmy=2;\u0000Jmx=2;0) +:::\nThe resulting Lagrangian\nL\u0019\u0002\nJ( _mxmy\u0000_mymx)\u0000Kxm2\nx\u0000Kym2\ny\u0003\n=2 +::: (21)\nyields the equations of motion for mxandmy:\n\u0000J_my\u0000Kxmx= 0; J _mx\u0000Kymy= 0: (22)\nThe spin follows an elliptical trajectory Kxm2\nx+Kym2\ny= const precessing at the\nfrequency!=\u0000p\nKxKy=J.\n2.8. Dissipative force\nIn the real world, a magnetic dipole will not precess about the magnetic \feld forever\nand will eventually line up with it, settling into a minimum of potential energy. Energy\ndissipation can be caused by various processes, e.g., radiation of electromagnetic waves\nor friction. We shall add a simple phenomenological term to the equations of motion\nthat represents|in the analogy with a particle on a sphere|a dissipative force \u0000\u000bJ_m\nproportional to the velocity _mand directed against it. Eq. (11) now reads\n\u0000J_m\u0002m\u0000dU\ndm\u0000\u000bJ_m= 0: (23)\nIt expresses the balance of gyroscopic, conservative, and dissipative forces acting on\nthe dipole. The dimensionless constant \u000b>0 introduced by Gilbert [15] characterizes\nthe strength of energy dissipation. It is usually small, \u000b\u001c1, with typical values\nranging from 10\u00004in insulating magnets to 10\u00002in metallic ones.CONTENTS 8\nFigure 2. Larmor precession of a spin in an external magnetic \feld in the\npresence of dissipation. The energy minimum is at the north pole (black point).\nThe spin trajectory is shown for 1 =(2\u000b) Larmor periods with the starting point\non the equator. The Gilbert damping \u000b= 0:05; the gyromagnetic ratio \r >0.\nThe equation of motion, expressed in spherical angles, reads\n\u0000Jsin\u0012_\u001e\u0000@U\n@\u0012\u0000\u000bJ_\u0012= 0;\nJ_\u0012\u00001\nsin\u0012@U\n@\u001e\u0000\u000bJsin\u0012_\u001e= 0: (24)\nIn the presence of the dissipative force, potential energy is dissipated at the rate\ndU\ndt=_m\u0001dU\ndm=_m\u0001(\u0000J_m\u0002m\u0000\u000bJ_m) =\u0000\u000bJj_mj2\u00140: (25)\n2.9. Example: Larmor precession with dissipation\nWe return to the problem of a dipole in a magnetic \feld, this time with dissipation.\nWith energy U=\u0000\u0016hcos\u0012, Eq. (24) reads\n\u0000Jsin\u0012_\u001e\u0000\u0016hsin\u0012\u0000\u000bJ_\u0012= 0;\nJ_\u0012\u0000\u000bJsin\u0012_\u001e= 0: (26)\nThe introduction of dissipation leads to a slight slowdown of the Larmor precession\nand to a gradual approach to the energy minimum:\n\u001e(t) =\u001e(0)\u0000~!Lt;~!L=!L\n1 +\u000b2;\ncos\u0012(t) = tanh (\u000b~!L(t\u0000t0)): (27)\nHere!L=\rhis the bare Larmor frequency (9) and t0is the time when the spin lies\nin the equatorial plane, cos \u0012= 0. We see that the azimuthal motion is precessional\nand fast, with the frequency of order \u000b0, whereas the polar motion is relaxational and\nslow, with the relaxation rate of order \u000b. Fig. 2 shows the trajectory of a spin for\n1=(2\u000b) Larmor periods starting at the equator and approaching the energy minimum\nat the north pole.CONTENTS 9\n2.10. Transformation to canonical variables\nAlthough our physical quantities|magnetization and angular momentum|are\nrepresented by a three-dimensional vector m, its three components are not\nindependent because of the \fxed length jmj= 1. It may sometimes be convenient to\nwork with two independent variables, which can be introduced in a number of ways.\nOne possible route is to use a pair of canonical variables ( q;p), one of which is a\ncoordinate and the other its conjugate momentum. To obtain them, we may use the\nLagrangian in one of the standard gauges (18),\nL=J(cos\u0012\u00061)_\u001e\u0000U(\u0012;\u001e): (28)\nWe choose the azimuthal angle \u001eas the coordinate and obtain the corresponding\ncanonical momentum in the usual way:\nq=\u001e; p =@L\n@_\u001e=J(cos\u0012\u00061) =Jz\u0006J: (29)\nAs one might expect, the momentum conjugate to \u001eis (up to an additive constant)\nthe angular momentum Jz=Jcos\u0012. It is somewhat alarming that the polar angle \u0012\ndoes not have a conjugate momentum (no _\u0012term in the Lagrangian). We will address\nthis problem in a later section. For now, we shall dismiss this concern by noting that\n\u0012is present in the canonical momentum so that both angles are represented in the\ncanonical pair ( q;p).\nWe now realize that our two-dimensional space is a phase space . The number of\nstates in a phase space is proportional to phase-space volume:\nd\u0000 =dqdp\n2\u0019~=J\n2\u0019~sin\u0012d\u0012d\u001e: (30)\nIntegrating over the sphere, we obtain the total number of states\n\u0000 =J\n2\u0019~Z\u0019\n0sin\u0012d\u0012Z2\u0019\n0d\u001e=2J\n~= 2j: (31)\nwherej=J=~is the length of angular momentum in the units of ~. The exact answer,\nwhich we have learned in a quantum mechanics course, is \u0000 = 2 j+1. Our approximate\nanswer, obtained within the scope of classical mechanics, is correct in the limit of a\nlarge length of angular momentum, j\u001d1.\nWith canonical variables identi\fed, we have the Poisson bracket. In particular,\nf\u0012;\u001eg=\u0000f\u001e;\u0012g=@\u0012\n@q@\u001e\n@p\u0000@\u0012\n@p@\u001e\n@q=1\nJsin\u0012: (32)\nThe Hamiltonian is obtained in a standard way,\nH=p_q\u0000L=U(\u0012;\u001e): (33)\nIt contains potential energy only. The _\u001eterm in the Lagrangian (28) is notkinetic\nenergy. Hamilton's equations of motion,\n_q=fq;Hg=@H=@p; _p=fp;Hg=\u0000@H=@q; (34)\nreproduce the equations of motion for the spherical angles (10).CONTENTS 10\n2.11. Transformation to general variables\nWe may not always want to use pairs of canonically conjugate variables and instead\nperform a more general transformation from mto a pair of coordinates ( q1;q2), e.g.,\nq1=\u0012andq2=\u001e. Equations of motion for the new variables can be obtained by\nstarting from the equation (23) expressing force balance.\nTo obtain the equations of motion for the new variables fqig, wherei= 1;2, we\n\frst note that the time evolution of m(q1;q2) comes solely through the time evolution\nof the coordinates: _m= _qj@m=@qj, where summation is implied over the doubly\nrepeated index j. We substitute this expression into Eq. (23) and multiply the result\nby@m=@qito obtain an equation of motion for the new coordinates:\nGij_qj\u0000@U=@qi\u0000Dij_qj= 0: (35)\nIt expresses the balance of the gyroscopic, conservative, and dissipative forces for\neach coordinate qi. The coe\u000ecients GijandDijof the antisymmetric gyroscopic and\nsymmetric dissipative tensors are de\fned as follows:\nGij=\u0000Jm\u0001\u0012@m\n@qi\u0002@m\n@qj\u0013\n; Dij=\u000bJ@m\n@qi\u0001@m\n@qj: (36)\nThe gyroscopic tensor Gijis the inverse of the of Poisson tensor \u0005ij=fqi;qjg, i.e.,\nGij\u0005jk=\u000ek\ni[16]. It can be obtained as the curl of a gauge potential Ai:\nGij=@Aj\n@qi\u0000@Ai\n@qj; Ai=a(m)\u0001@m\n@qi: (37)\nThe Lagrangian, expressed in terms of the new coordinates, reads\nL=Ai(q) _qi\u0000U(q): (38)\nThe \frst term of the Lagrangian gives rise to the geometrical action\nSg=Z\nAi_qidt=Z\nAidqi: (39)\nThe number of states in an in\fnitesimal rectangle with sides ( dq1;dq2) is\nd\u0000 =1\n2\u0019~jG12jdq1dq2=p\ndetG2Y\ni=1dqi\np\n2\u0019~: (40)\nFor the choice of coordinates q1=\u0012andq2=\u001e, the nonzero components of the\ntensors are\nG\u0012\u001e=\u0000G\u001e\u0012=\u0000Jsin\u0012; D\u0012\u0012=\u000bJ; D\u001e\u001e=\u000bJsin2\u0012: (41)\nUpon substituting these tensor components into Eq. (35), we obtain the equations of\nmotion for the spherical angles (24).\n2.12. Rayleigh's dissipation function\nDissipative forces cannot be described as a term in a Lagrangian. It is an emergent\nforce, arising from interaction of a macroscopic object with numerous microscopic\ndegrees of freedom. In thermodynamics, it plays a major role in the relaxation of the\nphysical system toward thermal equilibrium. In that context, the dissipative force can\nbe obtained from the Rayleigh dissipation function Rquadratic in velocities [17],\nR=1\n2\u000bJj_mj2=1\n2Dij_qi_qj: (42)CONTENTS 11\nThe modi\fed Euler-Lagrange equations are\n@L\n@qi\u0000d\ndt@L\n@_qi\u0000@R\n@_qi= 0: (43)\nThe last term is the dissipative force.\n2.13. Example: strong easy-plane anisotropy\nLet us discuss the dynamics of a spin whose potential energy\nU(\u0012;\u001e) =\u0000K\u0012\n2sin2\u0012+V(\u0012;\u001e) (44)\nis dominated by the easy-plane anisotropy term with K\u0012>0. Whatever the starting\nstate, after a period of initial relaxation the spin will \fnd itself near the equatorial\ncircle\u0012=\u0019=2. Its long-term dynamics will be in\ruenced by the azimuthal potential\nlandscape de\fned by V(\u0019=2;\u001e).\nTo get a \frst look at the long-term dynamics, consider the motion in the vicinity\nof the potential energy minimum at \u0012=\u0019=2 and some \u001e=\u001e0, where the potential\nenergy is approximately quadratic in the deviations \u000e\u0012and\u000e\u001efrom the minimum:\nU(\u0019=2 +\u000e\u0012;\u001e 0+\u000e\u001e)\u0019K\u0012\u000e\u00122\n2+K\u001e\u000e\u001e2\n2; (45)\nwhereK\u001e\u001cK\u0012. Equations of motion (24), linearized in the deviations from\nequilibrium, read\n\u0000J\u000e_\u001e\u0000K\u0012\u000e\u0012\u0000\u000bJ\u000e _\u0012= 0; J\u000e _\u0012\u0000K\u001e\u000e\u001e\u0000\u000bJ\u000e_\u001e= 0: (46)\nIn the absence of dissipation, \u000b= 0, we obtain elliptical precession near\nthe equatorial plane with the frequency !=p\nK\u0012K\u001e=Jand the amplitude ratio\n\u000e\u00120=\u000e\u001e0=p\nK\u001e=K\u0012\u001c1. Turning on dissipation at \frst makes the precession\nunderdamped, with the relaxation rate \u0000 \u0019\u000bK\u0012=2Jin the limit \u000b\u001c2p\nK\u001e=K\u0012.\nWhen the damping constant exceeds the critical value \u000bc= 2p\nK\u0012K\u001e=(K\u0012\u0000K\u001e)\u001c1,\nthe spin dynamics becomes overdamped, with two distinct relaxation rates, \u0000 1\u0019\n\u000bK\u0012=Jand \u0000 2\u0019K\u001e=\u000bJ\u001c\u00001in the limit 2p\nK\u001e=K\u0012\u001c\u000b\u001c1.\nOur simple analysis of the \fnal approach to equilibrium demonstrates that, even\nwhen dissipation is nominally weak, \u000b\u001c1, spin dynamics does not necessarily\nretain its precessional character and can be purely relaxational. An apparently\nsmall damping constant \u000b\u001c1 may represent relatively strong dissipation when it\nis compared to another small parameter, the ratio of azimuthal and polar anisotropiesp\nK\u001e=K\u0012. Put another way, increasing the strength of the easy-plane anisotropy K\u0012\nat a \fxed\u000b\u001c1 will eventually bring us into a regime where the spin motion separates\ninto two independent components, fast relaxation, during which the spin approaches\nthe equatorial plane, and slow relaxation within the equatorial plane. (The fast mode\ninclude azimuthal as well as polar motion.)\nNext we shall derive the long-term dynamics of spin motion in a more general\nsetting, when the polar angle is already near equilibrium, \u0012=\u0019=2 +\u000e\u0012with a small\n\u000e\u001e, but\u001emay not be. The Lagrangian can then be approximated as follows:\nL\u0019\u0000J\u000e\u0012 _\u001e\u0000K\u0012\u000e\u00122=2\u0000V(\u0019=2;\u001e): (47)\nIn the Rayleigh dissipation function,\nR\u0019\u000bJ(\u000e_\u00122+_\u001e2)=2\u0019\u000bJ_\u001e2=2; (48)CONTENTS 12\nwe have neglected the contribution of polar motion to damping because it is restricted\nby the strong easy-plane anisotropy, so \u000e_\u0012\u001c_\u001e. This is an important general point,\nso let us emphasize it: dissipation is dominated by soft modes; dissipation from hard\nmodes can often be neglected .\nFrom the Lagrangian (47) and Rayleigh function (48) we obtain the equations of\nmotion:\n\u0000J_\u001e\u0000K\u0012\u000e\u0012= 0; J\u000e _\u0012\u0000@V=@\u001e\u0000\u000bJ_\u001e= 0: (49)\nThe \frst equation lets us express the polar angle (a hard mode of no interest to us)\nin terms of the azimuthal angle (the soft mode we are interested in). Then the second\nequation will be for the azimuthal angle alone:\nI\u001e\u0000\u000bJ_\u001e\u0000@V=@\u001e = 0; (50)\nwhere we introduced a \\moment of inertia\" I=J2=K\u0012.\nBy integrating out the hard mode \u000e\u0012, we have generated inertia (mass) for the\nsoft mode\u001e. This is a su\u000eciently frequent occurrence in ferromagnetic dynamics to\nmerit a proper name, the D oring mass [18]. The Lagrangian of the azimuthal angle\nalone (with the polar angle integrated out) acquires kinetic energy:\nL(\u001e) =I_\u001e2=2\u0000V(\u0019=2;\u001e): (51)\nThis elegant result hinges on neglect of dissipation for the hard mode \u000e\u0012. Had we\nnot neglected it, the equation relating \u0012to\u001ewould be more involved|non-local in\ntime|and this simple picture would not have emerged. The non-locality in time is\nnegligible on time scales longer than \u001c1= 1=\u00001.\nReturning to the equation of motion for the azimuthal angle (50), we see that it\ndescribes a massive rotator subject to an external potential and dissipative friction.\nOn long time scales, we may neglect inertia and obtain a simpler equation of dissipative\ndynamics,\n\u0000\u000bJ_\u001e\u0000@V=@\u001e = 0: (52)\nThe azimuthal velocity _\u001eis proportional to the potential torque \u0000@V=@\u001e . Doing\nso is permissible on time scales longer than the characteristic acceleration time\n\u001c1=I=\u000bJ =J=\u000bK\u0012= 1=\u00001.\nWe should check that our long-term dynamics reproduces the previously obtained\napproach to equilibrium, when \u001e=\u001e0+\u000e\u001e. For small \u000e\u001e, the equatorial torque\n\u0000@V=@\u001e\u0019\u0000k\u000e\u001eand we recover exponential relaxation of \u000e\u001eat the rate \u0000 2=K\u001e=\u000bJ,\nin agreement with our preliminary analysis.\nOn a \fnal note, we observe that the long-term dynamics of the soft mode \u001egiven\nby Eq. (52) can be obtained directly from the Lagrangian (47) and Rayleigh dissipation\nfunction (48) by merely ignoring the hard mode \u000e\u0012. This is a general principle that we\nwill apply in many contexts in what follows: the response of a system to weak external\nperturbations is dominated by its soft modes .\n2.14. Recap: equations of motion\nWe end the discussion of a single magnetic dipole by gathering in one place di\u000berent\nforms of its equations of motion, all of which are equivalent to one another.\n\u000fThis form of the equation expresses the balance of the gyroscopic, conservative,\nand dissipative forces acting on a magnetic dipole:\n\u0000J_m\u0002m\u0000dU\ndm\u0000\u000bJ_m= 0: (53)CONTENTS 13\n\u000fThis equation expresses the rate of change of angular momentum in terms of\nconservative and dissipative torques acting on the dipole:\nJ_m=\u0000m\u0002dU\ndm\u0000\u000bJm\u0002_m: (54)\n\u000fThe previous equation is often written in terms of an \\e\u000bective \feld\" he\u000b=\n\u0000dU=d\u0016as follows:\n_m=\u0000\rhe\u000b\u0002m\u0000\u000bm\u0002_m;he\u000b=\u00001\n\u0016dU\ndm: (55)\nThis form of the equation of motion is known as the Landau-Lifshitz-Gilbert\n(LLG) equation.\n\u000fThe LLG equation (55) contains two terms with a time derivative _m, which\nis inconvenient when the equation is integrated numerically. Through simple\nalgebra, it can be converted into the form used originally by Landau and Lifshitz,\n(1 +\u000b2)_m=\u0000\rhe\u000b\u0002m\u0000\u000bj\rjhe\u000b: (56)\nThis form of the equation contains a single term with the time derivative _m\nand is used in numerical micromagnetic solvers such as OOMMF and MuMax.\nOne cautionary remark is on order. The e\u000bective \feld he\u000bde\fned in Eq. (55)\nis understood as a derivative dU(m)=dmwith respect to transverse variations of\nm, see Eq. (7). A sloppy evaluation may produce a longitudinal component of\nhe\u000bparallel to mand create a problem for the Landau-Lifshitz equation (56),\nchanging the length of the unit vector m. Just to be on the safe side, one may\nreplace he\u000bin the second term of Eq. (56) with its transverse part m\u0002(he\u000b\u0002m).\nThe same applies to Eq. (53).\n\u000fWritten in terms of spherical angles, the equations of motion are\n\u0000Jsin\u0012_\u001e\u0000@U\n@\u0012\u0000\u000bJ_\u0012= 0;\nJ_\u0012\u00001\nsin\u0012@U\n@\u001e\u0000\u000bJsin\u0012_\u001e= 0: (57)\n\u000fFor arbitrary coordinates q1andq2parametrizing the unit vector m, the equations\nof motion for these coordinates again expresses the balance of the gyroscopic,\nconservative, and dissipative forces associated with each coordinate:\nGij_qj\u0000@U=@qi\u0000Dij_qj= 0: (58)\nThe antisymmetric gyroscopic and symmetric dissipative tensors are\nGij=\u0000Jm\u0001\u0012@m\n@qi\u0002@m\n@qj\u0013\n; Dij=\u000bJ@m\n@qi\u0001@m\n@qj: (59)\n3. Domain wall in a ferromagnetic wire\n3.1. Heisenberg spin chain\nTheory of a single magnetic dipole can be readily extended to that of many dipoles.\nA convenient way to accomplish that is to start with the Lagrangian of a single spin\n(13) and to generalize it to many spins fmng:\nL=X\nna(mn)\u0001_mn\u0000U(fmng); (60)CONTENTS 14\nwhere once again the vector potential a(m) describes the \feld of a magnetic monopole,\nrm\u0002a(m) =\u0000Jm, andU(fmng) is the potential energy of the system. The Rayleigh\nfunction is generalized in the same way,\nR=1\n2X\nn\u000bJj_mnj2: (61)\nThen the Euler-Lagrange equations of motion for every spin n,\n@L\n@mn\u0000d\ndt@L\n@_mn\u0000@R\n@_mn= 0; (62)\nyields an obvious generalization of single-spin dynamics (54),\nJ_mn=\u0000mn\u0002@U\n@mn\u0000\u000bJmn\u0002_mn; (63)\nor its variants, Eqs. (53) and (55). The e\u000bective \feld acting on spin mnishe\u000b\nn=\n\u0000\u0016\u00001@U=@ mn.\nHere we examine a Heisenberg chain with magnetic dipoles arranged periodically\nwith lattice constant a. Adjacent dipoles mnandmn+1interact with energy\n\u0000Amn\u0001mn+1, whereAis the exchange constant:\nU=\u0000AX\nnmn\u0001mn+1: (64)\nThe equation of motion for dipole mnis\nJ_mn=Amn\u0002(mn\u00001+mn+1)\u0000\u000bJmn\u0002_mn: (65)\nAn equilibrium state, _mn= 0, is reached when all dipoles line up along the same\ndirection, mn=m. Such a state clearly minimizes the potential energy (64) for A>0.\nThe common direction of magnetization mis arbitrary.\nIt is instructive to investigate the dynamics of magnetization close to equilibrium,\nmn=m+\u000emn, where in\fnitesimal deviations from equilibrium \u000emnare transverse\ntomin order to preserve the unit length of mn.\nIt is convenient to introduce a reference frame with three unit vectors ex,ey, and\nezforming a right triple, ei\u0002ej=\u000fijkek. We align one of them with the equilibrium\ndirection, ez=m. Then\u000emn=mn;xex+\u000emn;yey. After expanding Eq. (65) to the\n\frst order in \u000emn, we obtain the dynamical equation for the deviations:\nJ_mn;x =\u0000A(mn\u00001;y\u00002mn;y+mn+1;y) +\u000bJ_mn;y;\n\u0000J_mn;y=\u0000A(mn\u00001;x\u00002mn;x+mn+1;x) +\u000bJ_mn;x: (66)\nThe two transverse components can be conveniently combined into a complex number\n n=mn;x+imn;y. These complex variables have the following equation of motion:\ni(J+i\u000bJ)_ n=\u0000A( n\u00001\u00002 n+ n+1): (67)\nSolutions of this equation have the form of waves \u000emn(t) =Ceikna\u0000i!t, whereais the\nlattice period and kis the wavenumber; the frequency is\n!=2A(1\u0000coska)\nJ+i\u000bJ: (68)\nTwo things are worth noting, First, the frequency of a spin wave !vanishes when\nthe wavenumber k!0. This is a consequence of spontaneous breaking of a continuous\nsymmetry, in this case of global spin rotations, by the ground state with spontaneousCONTENTS 15\nmagnetization m. Second, in the presence of dissipation ( \u000b > 0), the frequency has\nan imaginary part,\nIm!=\u00002\u000bA(1\u0000coska)\nJ(1 +\u000b2): (69)\nBecause Im !<0, a spin wave  n(t) =Ceikna\u0000i!tdecays exponentially in time.\nThe mathematics can be simpli\fed for waves whose wavelength is long compared\nto the lattice spacing, ka\u001c1. For such waves, magnetizations of adjacent dipoles\nare almost the same, mn+1\u0019mn. In this limit, we may treat discrete variables mn\nas a slowly varying and continuous function m(z), wherez=nais the coordinate\nalong the chain. Then \fnite di\u000berences can be transformed into spatial derivatives via\nTaylor expansion:\nmn+1\u0000mn=m(z+a)\u0000m(z)\u0019@m(z)\n@za;\nmn+1\u00002mn+mn\u00001\u0019@2m(z)\n@z2a2; (70)\nand so on.\nIn the continuum approximation, the equation of motion for transverse\n\ructuations  (z;t) =mx(z;t) +imy(z;t) (67) reads\niJ@ \n@t=\u0000Aa2@2 \n@z2: (71)\nFor simplicity, we switched o\u000b dissipation, \u000b!0. The resulting wave equation\nresembles the Schr odinger equation for a free particle with a nonrelativistic energy-\nmomentum relation E=p2=2mand the mass m=J~=(2a2A). These particles were\n\frst introduced by Felix Bloch [19] and are now known as magnons, the quanta of\nspin waves.\n3.2. Continuum theory of a ferromagnet\nTo develop a continuum description of a ferromagnetic medium, on a formal level,\nwe promote the unit vector mencoding the magnetic moment of a single dipole to\na unit-vector \feld m(r) varying smoothly in space r. The mathematical apparatus\ndeveloped in the previous section is generalized accordingly.\nFor example, the equation of motion (53) undergoes two very minor changes:\n\u0000J_m\u0002m\u0000\u000eU\n\u000em\u0000\u000bJ_m= 0: (72)\nFirst, the \\length\" of angular momentum Jhas been replaced with the corresponding\nintensive quantity, the angular momentum Jper unit volume. Second, the potential\nenergy function U(m) has become the potential energy functional U[m(r)] and thus\nthe ordinary derivative dU=dm(7) has turned into the functional derivative \u000eU=\u000em(r).\nThe Landau-Lifshitz equations (55) and (56) retain their forms, albeit with an\ne\u000bective \feld\nhe\u000b(r) =\u00001\nM\u000eU\n\u000em(r)(73)\ncontaining the functional derivative.CONTENTS 16\nEquations of motion for the spherical-angle \felds \u0012(r) and\u001e(r) are\n\u0000Jsin\u0012_\u001e\u0000\u000eU\n\u000e\u0012\u0000\u000bJ_\u0012= 0;\nJ_\u0012\u00001\nsin\u0012\u000eU\n\u000e\u001e\u0000\u000bJsin\u0012_\u001e= 0: (74)\nA \feld m(r) has in\fnitely many degrees of freedom, so if we attempt to describe\nit in terms of some coordinates qi, the number of these coordinates will be in\fnite.\nNonetheless, the equations of motion for these coordinates remain unchanged,\nGij_qj\u0000@U=@qi\u0000Dij_qj= 0; (75)\nand only the gyroscopic and dissipative tensors are promoted to functionals of the\nmagnetization \feld:\nGij=\u0000JZ\ndVm\u0001\u0012@m\n@qi\u0002@m\n@qj\u0013\n; Dij=\u000bJZ\ndV@m\n@qi\u0001@m\n@qj:(76)\nThe gyroscopic tensor can be obtained from a gauge potential Ai:\nGij=@Aj\n@qi\u0000@Ai\n@qj; Ai=Z\ndVa(m)\u0001@m\n@qi: (77)\nThis is entirely analogous to Eq. (37) for a single spin. Equations of motion (75) can\nbe obtained from the Lagrangian\nL=Ai(q) _qi\u0000U(q) (78)\nsupplemented by the Rayleigh dissipation function [17]\nR=1\n2Dij_qi_qj: (79)\nOur model is a ferromagnet in one spatial dimension with coordinate z. It\ncan describe a ferromagnetic wire whose transverse dimensions are smaller than the\ncharacteristic length \u00150de\fned in Eq. (82). Exchange interaction tends to align spins,\nso a ground state should have a uniform magnetization \feld m(z) = const. However,\nsome of these uniform states will have lower energies than others. In particular, long-\nrange dipolar interactions prefer that the dipoles line up along the direction of the\nwire, leaving just two ground states,\nm(z) =\u0006ez= (0;0;\u00061): (80)\n3.3. Model with local interactions\nWe will use a simple phenomenenological form of the potential energy functional\nminimized by these two ground states:\nU[m(z)] =Z\ndz\u0000\nAjm0j2+Kjm\u0002ezj2\u0001\n=2: (81)\nHere m0\u0011dm=dz. The \frst term models the exchange interaction and penalizes\nspatially non-uniform states. It can be derived as a continuum approximation to the\nHeisenberg chain. The second term penalizes states with magnetization deviating\nfrom the direction of the wire ez.\nBoth the exchange constant Aand anisotropy strength Kare positive. Together\nwith the density of angular momentum J, they de\fne the characteristic scales of\nlength\u00150, time\u001c0, and energy \u000f0:\n\u00150=p\nA=K; \u001c 0=J=K; \u000f 0=p\nAK: (82)CONTENTS 17\nWe stress that this model is not very realistic as it relies on a local anisotropy term\nto select the two ground states (80). A more realistic model with dipolar interactions\nwould require a non-local energy functional as spins would interact over large distances.\nThe advantage of our toy model is in its relative simplicity: with not too much e\u000bort,\nwe can study its linear and nonlinear excitations.\n3.4. Linear spin waves\nTo obtain spin waves near one of the uniform ground states, m0=ez, we consider a\nweakly excited state with m(z) =m0+\u000em(z), where\u000em=exmx+eymyrepresents\nsmall transverse deviations of magnetization. Expand the Lagrangian density to the\nsecond order in \u000emas in Sec. 2.7:\nL=1\n2Z\ndz\u0010\n_mxmy\u0000_mymx\u0000m0\nx2\u0000m0\ny2\u0000m2\nx\u0000m2\ny\u0011\n+::: (83)\nHere we have switched to natural units (82), in which J=A=K= 1. The\nLagrangian (83) is quadratic in the transverse spin components, The resulting\nequations are linear in them, hence the term linear spin waves .\nIn the absence of dissipation, \u000b= 0, the equations of motion for the transverse\nmagnetization components are\n\u0000_my+m00\nx\u0000mx= 0;_mx+m00\ny\u0000my= 0: (84)\nIt is convenient to combine the two transverse components into a complex \feld\n =mx+imy, whose equation of motion,\ni_ =\u0000 00+ ; (85)\nresembles the Schr odinger equation i_ =H with the Hamiltonian\nH=\u0000d2\ndz2+ 1; (86)\nwhose energy spectrum is \u000fk= 1+k2. A traveling spin wave  (t;z) = (0;0)e\u0000i!t+ikz\nhas the frequency\n!=K+Ak2\nJ; (87)\nwhere we have restored the units. The frequency spectrum has a gap with a minimal\nexcitation frequency lim k!0!=K=J= 1=\u001c0.\n3.5. Nonlinear spin waves\nTo describe spin waves with a large amplitude, we introduce an Ansatz\nm(z;t) = (sin \u0002 cos ( kz+ \b);sin \u0002 sin (kz+ \b);cos \u0002): (88)\nThe polar angle \u0002 between mand the easy axis ezis the wave's amplitude; the\nazimuthal \b is its phase, Fig. 3. We will treat the amplitude \u0002( t) and the phase \b( t)\nas two collective coordinates of the nonlinear wave (88). Eq. (75) determines their\ndynamics.\nThe energy of the wave (88) scales linearly with the wire length and diverges in an\nin\fnite wire. It is therefore convenient to assume a \fnite wire length `and to imposeCONTENTS 18\nzxy\nFigure 3. Snapshot of a large-amplitude spin wave. The spins have a \fxed polar\nangle \u0002. The azimuthal angle varies in space as \u001e=kz+ \b.\nperiodic boundary conditions to eliminate edge e\u000bects. The wire length `must then\nbe commensurate with the wavelength \u0015= 2\u0019=k. The energy of the wave (88) is then\nU(\u0002;\b) =1\n2(Ak2+K)`sin2\u0002: (89)\nThe lack of a \b dependence is related to a symmetry, in this case the symmetry of\nglobal spin rotations about the zaxis. We say that \b is a zero mode .\nThe gyroscopic and dissipative tensors are obtained with the aid of Eq. (76).\nTheir nonvanishing coe\u000ecients are\nG\u0002\b=\u0000G\b\u0002=\u0000J`sin \u0002; D \u0002\u0002=\u000bJ`; D \b\b=\u000bJ`sin2\u0002:(90)\nThe following equations of motion result for the collective coordinates:\n\u0002 :\u0000J`sin \u0002 _\b\u00001\n2(K+Ak2)`sin 2\u0002\u0000\u000bJ`_\u0002 = 0;\n\b : +J`sin \u0002 _\u0002\u0000\u000bJ`sin2\u0002_\b = 0: (91)\nThe equation for \b shows that \u0002 is the slower variable of the two as _\u0002 =\u000bsin \u0002 _\b\nand\u000b\u001c1. It would be conserved in the absence of dissipation ( \u000b= 0). After some\nsimpli\fcation, we \fnd that the spin rotation frequency depends on the amplitude \u0002:\n_\b =\u0000K+Ak2\nJ(1 +\u000b2)cos \u0002: (92)\nEq. (92) reproduces the spectrum of linear spin waves (87) in the limit of a small\namplitude (\u0002!0) and weak dissipation ( \u000b!0).\n3.6. Static domain wall\nThe uniform ground states m(z) =\u0006ezare global minima of the potential energy\n(81). There are also local minima, in which magnetization interpolates between \u0000ez\non one end of the wire and + ezon the other. In such a state, magnetization m(z)\nforms two (nearly) uniform domains, in which the energy density is (nearly) zero. The\nregion in between has non-uniform magnetization deviating from the easy axis has a\npositive energy density and is known as a domain wall.\nTo obtain a formal solution for a domain wall, it is convenient to write the\npotential energy (81) in terms of spherical angles,\nU[\u0012(z);\u001e(z)] =Z\ndzh\nA(\u001202+ sin2\u0012\u001e02) +Ksin2\u0012i\n=2: (93)CONTENTS 19\nzxyσ=+1σ=−1\nFigure 4. Domain walls in a ferromagnetic wire with Z2topological charges\n\u001b=\u00061.\nMinimization of the energy \u000eU= 0 yields \feld equations\n\u0000A\u001200+1\n2(A\u001e02+K) sin 2\u0012= 0; A (sin2\u0012\u001e0)0= 0: (94)\nSolutions with a uniform azimuthal angle, \u001e0= 0 automatically satisfy the second of\nthese equations. The \frst one can be integrated once to obtain \u0000A\u001202+Ksin2\u0012=C.\nThe constant of integration can be obtained by noting that at z!\u00061 we recover\nuniform states with \u00120= 0 and sin \u0012= 0. Therefore C= 0. Integrating this equation\nyields domain-wall solutions (Fig. 4),\ncos\u0012(z) =\u001btanhz\u0000Z\n\u00150; \u001e(z) = \b: (95)\nSolutions (95) have three parameters, one discrete and two continuous.\n\u000fTopological charge \u001b=\u00061 determines the ground states on the two sides of\nthe domain wall: m=\u0000\u001bezatz=\u00001 andm=\u001bezatz= +1. The\nword \\topological\" indicates that this characteristic cannot be changed by small\ncontinuous deformations of the soliton.\n\u000fPositionZdetermines the location of the domain wall. The change from\nm=\u0000\u001bezto\u001bezhappens around z=Z, in a region of width \u00150(82).\n\u000fAngle \b de\fnes an azimuthal plane in which the spins interpolate between\nm=\u0000\u001bezand\u001bez.\nNote that the domain-wall solution (95) contains two parameters, Zand\n\b, de\fning the position of the domain wall and its azimuthal plane. These\ncollective coordinates of a domain wall quantify its rigid translations and rotations,\ntransformations that leave the energy (81) invariant and thus known as zero modes .\nIt is helpful to picture the con\fguration space\n\u00001<Z < +1;0\u0014\b\u00142\u0019 (96)\nas the surface of a cylinder of the unit radius. The state of a domain wall is then\ndepicted by a point (cos \b ;sin \b;Z) on that surface. We will use this representation\nto illustrate the dynamics of a domain wall in Walker's model in Sec. 4.CONTENTS 20\n3.7. Domain wall in a magnetic \feld\nA magnetic \feld applied along the easy axis, h=hez, exerts a force on a domain wall,\nas can be seen from the following consideration. The \feld couples to magnetization\nthrough the Zeeman term with energy density \u0000Mh\u0001m, where the magnetization\ndensity isM=\rJ. By moving the domain wall, we can reduce its Zeeman energy by\nincreasing the length of the domain where mpoints along h. Displacing the domain\nwall (95) by \u0001 Zchanges the Zeeman energy by 2 \u001bMh\u0001Z, which indicates that the\napplied \feld exerts force\nFZ=\u00002\u001bMh (97)\non the wall.\nIt is natural to expect that the domain wall will move in the direction of the\napplied force. However, we shall see shortly, that the primary e\u000bect of the force will be\nprecession of the domain wall, rather than translational motion. In fact, in the absence\nof dissipation the domain wall will not move at all, _Z= 0, because conservation of\nenergy precludes that. The presence of dissipation will allow translational motion with\na velocity proportional to the damping constant \u000b.\nTo determine the dynamics of the domain wall in a magnetic \feld, we use the\nequations of motion for the \felds \u0012(z) and\u001e(z) (74) with the energy functional\nU=Z\ndz\u001aA\n2(\u001202+ sin2\u0012\u001e02) +K\n2sin2\u0012\u0000Mhcos\u0012\u001b\n: (98)\nThe equations of motion are\n\u0000Jsin\u0012_\u001e+A\u001200\u00001\n2(A\u001e02+K) sin 2\u0012\u0000Mhsin\u0012\u0000\u000bJ_\u0012= 0;\nJ_\u0012\u0000A(sin2\u0012\u001e0)0\u0000\u000bJsin\u0012_\u001e= 0: (99)\nAlthough this looks like a big mess, it turns out that our previous solution can still\nbe made to work, with one minor modi\fcation. We simply assume that the collective\ncoordinates Zand \b in Eq. (95) become dependent on time. Eqs. (99) are then\nsatis\fed if the linear and angular velocities obey the relations\n\u0000J_\b\u0000Mh\u0000\u001b\u000bJ_Z=\u0015 0= 0; \u001bJ_Z\u0000\u000bJ\u00150_\b = 0: (100)\nWe thus obtain the velocities\n_\b =\u0000~!L;_Z=\u0000\u001b\u000b\u0015 0~!L: (101)\nHere ~!Lis the Larmor precession frequency renormalized by dissipation de\fned in\nEq. (27).\nTwo things immediately stand out. First, as we have already mentioned, the\nprimary e\u000bect of the applied force (97) is precession: the angular velocity _\b is of order\n\u000b0, whereas the linear velocity _Zis of order \u000b1. Second, the precession frequency\nof a domain wall is the same as that of a single spin (27). The complexity brought\nby interactions of spins has no e\u000bect on the precession frequency. Simple results call\nfor simple explanations, but our purely mathematical approach to the problem|\fnd\na solution to coupled partial di\u000berential equations|obscures the physical picture.\nFurthermore, closed-form solutions are available only in a very restricted set of\nproblems. It worked for a magnetic \feld along the easy axis but it would fail for\na \feld orthogonal to it. We need to \fnd another approach.CONTENTS 21\n3.8. Weakly perturbed domain wall\nFortunately, there is a general approach that will provide a path to solution for all\nkinds of perturbations and will be provide more physical intuition. Of course, there\nis a price that we will pay for the universality of the approach: it is only guaranteed\nto work for weak perturbations.\nThe method is based on rigidity of a domain wall (and solitons in general). It\ntakes a \fnite amount of energy to alter the width of a domain wall from its equilibrium\nvalue\u00150. On the other hand, no energy is required to translate a domain wall or to\nrotate its azimuthal plane. One can make an analogy with a tennis ball that is hard\nto squeeze but easy to move. Thus we may expect that a weak enough perturbation\nwill produce substantial response only in the zero modes Zand \b of a domain wall\nbut not in other modes (such as its width \u00150).\nOn a formal level, although a domain wall can be deformed in an in\fnite number\nof ways and is therefore described by in\fnitely many coordinates qi, we only need to\ntake into account two of them|position Zand azimuthal angle \b|when we consider\nthe motion of a domain wall under a weak external perturbation.\nThus the recipe is to use the equations of motion (75) with qi=Zand \b only.\nTo that end, we need the gyrotropic and dissipative tensors GijandDij(76). The\ngyroscopic tensor Gijis antisymmetric, so with 2 coordinates we just need to compute\none of its components, say, GZ\b. The dissipative tensor Dijis symmetric and, for this\nchoice of collective coordinates, diagonal; so it only has 2 nonzero components.\nWe compute in detail only the gyroscopic coe\u000ecient\nGZ\b=\u0000JZ1\n\u00001dzm\u0001\u0012@m\n@Z\u0002@m\n@\b\u0013\n: (102)\nAsmis a function of z\u0000Z, we may replace @m=@Z with\u0000@m=@z. On the other\nhand,\u001e= \b everywhere, so\n@m\n@\b=@m\n@\u001e=e\u001esin\u0012; (103)\nwhere we used the local spin frame de\fned in Eq. (4) and Fig. 1. Next, because only\n\u0012varies in space and \u001edoes not, application of the chain rule yields\n@m\n@z=d\u0012\ndz@m\n@\u0012=d\u0012\ndze\u0012: (104)\nWith the aid of the identity m\u0001(e\u0012\u0002e\u001e) = 1, we obtain\nGZ\b=\u0000JZ1\n\u00001dzdcos\u0012\ndz=\u0000Jcos\u0012(z)\f\f+1\n\u00001=\u00002\u001bJ: (105)\nIt is remarkable that the gyroscopic coe\u000ecient GZ\bdepends just on the density of\nangular momentum Jand the topological charge \u001bbut is completely insensitive to\nthe precise shape \u0012(z\u0000Z) of the domain wall!\nThe dissipative coe\u000ecients are computed along similar lines:\nDZZ= 2\u000bJ=\u00150; D \b\b= 2\u000bJ\u00150; D \bZ=DZ\b= 0 (106)\nThey give the following the equations of motion for our collective coordinates:\nZ:\u00002\u001bJ_\b +FZ\u00002\u000bJ_Z=\u0015 0= 0;\n\b : + 2\u001bJ_Z+F\b\u00002\u000bJ\u00150_\b = 0; (107)\nwhereFZ=\u0000@U=@Z is conservative force and F\b=\u0000@U=@ \b conservative torque.CONTENTS 22\nReturning to the previously solved problem of a domain wall in an applied \feld\nparallel to the easy axis, we have a force FZ=\u00002\u001bMhand torque F\b= 0. With\nthese, the new approximate equations of motion (107) reproduce the exact ones (100).\nThe new approach also provides an insight into the vexing question: why is the\nprecession frequency of a domain wall the same as that of a single spin in the same\nmagnetic \feld? In the absence of friction, \u000b= 0, the precession frequency can be\nobtained from the \frst equation (107):\n_\b =\u0000FZ\nGZ\b=\u00002\u001bMh\n2\u001bJ=\u0000\rh: (108)\nBoth the Zeeman force on the domain wall FZ=\u00002\u001bMhand the gyroscopic\ncoe\u000ecientGZ\b=\u00002\u001bJare insensitive to the detailed structure of the domain wall.\nTheir ration determines the precession frequency.\nEquations of motion (107) also paint a simple physical picture of the mechanics of\na domain wall in a ferromagnetic wire. With two degrees of freedom, position Zand\nangle \b, it resembles a bead on a string, which can move along the string and rotate.\nHowever, the dynamics of the two objects are very di\u000berent. For a bead, a force\ncreates linear acceleration and a torque creates angular acceleration. For a domain\nwall, a force generates angular velocity, whereas a torque produces linear velocity.\nWhen pushed, a domain wall rotates, and when twisted, it moves!\nWe can also understand now how the addition of dissipation enables a domain\nwall to move under an external force. When it rotates (its primary response to a force),\ndissipation creates a dissipative torque directed against the direction of rotation. This\ntorque causes linear motion. The proportionality of the dissipative torque to the\ndamping constant \u000bexplains why the velocity of the soliton is also of order \u000b.\n3.9. Normal modes of a domain wall\nSo far we have approximated the dynamics of a domain wall by its two most prominent\nmodes|position Zand global azimuthal angle \b. We have justi\fed the neglect of all\nother modes (such as the width of the domain wall \u0015) by their hard nature. In this\nsubsection, we shall make a quantitative analysis of the hard modes and deduce time\nscales on which their dynamics can be neglected. These results were \frst obtained by\nYan et al. [20].\nIn natural units of length, time, and energy (82), the Lagrangian density is\nL(\u0012;\u001e) = (cos\u0012\u00001)_\u001e\u0000(\u001202+ sin2\u0012\u001e02+ sin2\u0012)=2: (109)\nA static domain wall with \u001b= +1,Z= 0 and \b = 0 is\ncos\u00120(z) = tanhz; \u001e 0(z) = 0: (110)\nExpand the Lagrangian density in small deviations from this state \u000e\u0012(z) and\u000e\u001e(z)\nto the second order:\nL(\u00120+\u000e\u0012;\u001e 0+\u000e\u001e) =L0+L1+L2+:::; (111)\nwhereL0=L(\u00120;\u001e0). The \frst-order term L1= (cos\u00120\u00001)\u000e_\u001eis a time derivative;\nit thus does not contribute to the equations of motion and can be neglected. The\nsecond-order term is\nL2=\u0000sin\u00120\u000e\u0012\u000e_\u001e\u0000(\u000e\u001202+ sin2\u00120\u000e\u001e02+ cos 2\u00120\u000e\u00122)=2 (112)CONTENTS 23\nRewriting it in terms of transverse increments \u000em\u0012and\u000em\u001e(5) reveals a hidden O(2)\nsymmetry:\nL2=\u0000\u000em\u0012\u000e_m\u001e\u0000(\u000em\u0012H\u000em\u0012+\u000em\u001eH\u000em\u001e)=2; (113)\nwhere we have introduced the operator\nH=\u0000d2\ndz2+ 1\u00002 sech2z: (114)\nTo utilize the symmetry, we switch to complex \felds\n =\u000em\u0012+i\u000em\u001ep\n2;  \u0003=\u000em\u0012\u0000i\u000em\u001ep\n2: (115)\nTheir Lagrangian density\nL2=i \u0003_ \u0000 \u0003H (116)\nyields the equations of motion\ni_ =H ;\u0000i_ \u0003=H \u0003: (117)\nThe spin-wave \\Hamiltonian\" H(114) is similar to its counterpart for the ground\nstate (86). The main di\u000berence is the presence of an attractive P oschl{Teller potential\nwellU(z) =\u00002 sech2z[21] in Eq. (114) caused by the presence of the domain wall.\nThis Hamiltonian has remarkable properties. (See Appendix A for a brief review and\nRef. [22] for much more.) Its energy spectrum has a single bound state with exactly\nzero energy,H\t0= 0, and a localized wavefunction\n\t0(z) = 2\u00001=2sechz: (118)\nIts continuum states are re\rectionless and are in a one-to-one correspondence with\nthe plane waves eikzof the free spin-wave Hamiltonian (86):\nH k=\u000fk k; \u000fk= 1 +k2;  k(z) =tanhz\u0000ikp\n1 +k2eikz: (119)\nEigenstates with di\u000berent kare orthogonal to the zero mode \t 0(z) and to one another,\nZ\ndz \u0003\nk0(z) k(z) = 2\u0019\u000e(k\u0000k0); (120)\nThe zero mode \t 0(z) corresponds to the in\fnitesimal translations and rotations of\nthe domain wall, whereas the free states  k(z) describe spin waves in the presence of\nthe domain wall.\nAn arbitrary in\fnitesimal deformation of the domain wall can be expanded in\nterms of the eigenmodes,\n (z;t) =Q0(t)\t0(z) +Zdk\n2\u0019qk(t) k(z); (121)\nComplex amplitudes Q0(t) andqk(t) form a complete set of collective coordinates for\na domain wall. The amplitude of the zero mode is related to position and azimuthal\nangle,\nQ0=Z+i\b; (122)\nwhereasqkare complex amplitudes of the spin waves (119).\nThe Lagrangian of these collective coordinates is\nL2=Z\ndzL2=iQ\u0003\n0_Q0+Zdk\n2\u0019\u0000\niq\u0003\nk_qk\u0000\u000fkjqkj2\u0001\n: (123)CONTENTS 24\nThe Rayleigh function is\nR=\u000bj_Q0j2+\u000bZdk\n2\u0019j_qkj2: (124)\nFrom these we obtain the equations of motion:\ni_Q0\u0000\u000b_Q0= 0; i_qk\u0000\u000fkqk\u0000\u000b_qk= 0; (125)\nwhich yields the eigenfrequencies\n\n0= 0; !k=\u000fk\n1 +i\u000b=1 +k2\n1 +i\u000b: (126)\nUpon restoring the units, we obtain the spectrum\n!k=K+Ak2\nJ(1 +i\u000bJ): (127)\nThe relaxation rate \rk=\u0000Im!k\u0019\u000b\u000fkfor\u000b\u001c1 is slowest for modes with k!0,\nlim\nk!0\rk=\u000bK=J=\u000b=\u001c0: (128)\nThus the in\ruence of spin waves can be neglected on time scales longer than \u001c0=\u000b.\nThis criterion can be further relaxed. If external perturbations and the response\nof the domain wall occur on time scales longer than \u001c0then spin waves, whose spectrum\nhas a gap!k\u00151=\u001c0, are not excited.\n3.10. Angular momentum of a domain wall\nSymmetry with respect to rotations about the zaxis implies conservation of angular\nmomentum Jz. In our one-dimensional model, there is no orbital contribution to this\nquantity. Therefore angular momentum comes solely from spin. This gives\nJz(Z) =Z\ndzJm(z\u0000Z)\u0001ez: (129)\nThis quantity is not well de\fned in an in\fnite wire, where the integration region\nextends from z=\u00001 to +1because the integrand approaches the constant values\n\u0006Jaway from the domain wall, jz\u0000Zj\u001d\u0015. It is convenient to consider a \fnite wire\nof \fnite length `\u001d\u0015so that the integration range is \u0000`=2<z <`= 2. If the domain\nwall is located exactly in the middle of this wire, Z= 0, then the contributions of spins\nwith positive and negative projections onto the zaxis cancel each other and we \fnd\nJz= 0. Shifting the domain wall to a position Z6= 0 shifts the balance by elongating\none domain and shortening the other, creating a net angular momentum proportional\nto the displacement of the domain wall, Jz(Z)/Z. To compute the proportionality\nconstant, we may take the derivative\n@Jz\n@Z=JZ+`=2\n\u0000`=2dz@m(z\u0000Z)\n@Z\u0001ez=\u0000JZ+`=2\n\u0000`=2dz@m(z\u0000Z)\n@z\u0001ez\n=\u0000Jm(z\u0000Z)\u0001ezj+`=2\n\u0000`=2\u0019\u00002\u001bJ: (130)\nIn the last step, we assumed that the magnetization at the ends of the wire returns\nto the ground-state values \u0006\u001bez, which is justi\fed if the domain wall is not too close\nto the ends.\nWe thus obtain an important result:\nJz(Z) =\u00002\u001bJZ: (131)CONTENTS 25\nThe angular momentum of a domain wall in a ferromagnetic wire is proportional\nto its position Zrelative to the center of the wire. The proportionality coe\u000ecient\nis insensitive to the detailed structure of the domain wall and depends only on the\ndensity of angular momentum J(per unit length) and the topological charge of the\ndomain wall \u001bde\fned in Eq. (95).\nThis result has interesting consequences. Angle \b and angular momentum Jz\nform a canonical pair with the commutator [\b ;Jz] =i~. With the aid of Eq. (131),\nwe \fnd that coordinate Zand azimuthal angle \b do not commute,\n[\b;Z] =\u0000i~\n2\u001bJ: (132)\nIt follows that the position and azimuthal orientation of a domain wall cannot be\nsimultaneously measured. The uncertainty product is \u0001 Z\u0001\b\u0015~=4J. In an atomic\nchain with the lattice constant aand atomic spins of length ~S(with 2San integer),\nJ=~S=a and the uncertainty product \u0001 Z\u0001\b\u0015a=4Sis of the order of the\ninteratomic distance.\n3.11. Linear momentum of a domain wall\nIn a uniform wire, translational symmetry implies conservation of linear momentum\nPz. We cannot derive it along the lines leading to Eq. (131). Instead, we will reverse-\nengineer it from the commutator of Zand \b (132).\nTo that end, we recall that coordinate Zand its momentum Pzhave the\ncommutator [ Pz;Z] =\u0000i~. Comparison with Eq. (132) shows that a plausible answer\nwould be\nPz= 2\u001bJ\b: (133)\nThe linear momentum of a domain wall in a ferromagnetic wire is thus proportional\nto its azimuthal angle.\nAlthough this result is hard to internalize, we can check that it is compatible\nwith the dynamics of a domain wall derived earlier. As we have seen previously, an\nexternal magnetic \feld hparallel to the easy axis ezexerts on the domain wall the\nforceFz=\u00002\u001bMh. This external force determines the rate of change for the linear\nmomentum, _Pz=\u00002\u001bMh. Our conjectured result for Pz(133) tells us that the\ndomain wall will precess at the rate _\b =\u0000\rh, in accordance with Eq. (108).\nEqs. (131) and (133) reveal a pleasing symmetry. The angular momentum is\nproportional to the linear coordinate of the domain wall and vice versa, with the\nsame proportionality coe\u000ecient, up to a sign.\nThere is, however, a subtle di\u000berence between the linear and angular coordinates:\nwhereas the former lives on a straight line \u00001< Z < +1and is unambiguously\nde\fned, the latter lives on a circle, so that values of \b di\u000bering by 2 \u0019describe the\nsame physical state and must be identi\fed. This leads to an ambiguity in linear\nmomentum (133): it is de\fned modulo 4 \u0019J. There is no such problem with the\nangular momentum.\nThe paradox of linear momentum was resolved by Haldane [14]. It turns out\nthat the continuum and classical theory of the ferromagnet used throughout this\nsection \\remembers\" its discrete and quantum origins. In a periodic atomic chain\nwith the lattice constant a, linear momentum is de\fned modulo 2 \u0019~=athanks to\nBragg scattering. If the atomic spins have length ~S, the spin density is J=~S=a\nand the momentum ambiguity is 4 \u0019~S=a. This is indeed an integer multiple of 2 \u0019~=a\nbecause 2Sis an integer.CONTENTS 26\n3.12. Counting states of a domain wall\nAt su\u000eciently low energies (de\fned below), position Zand angle \b of a domain wall\nfully describe its dynamics. Since the two variables are, up to a multiplicative constant,\nmomenta of one another, the low-energy con\fguration space ( Z;\b) can be viewed as\na phase space, in which the number of states is proportional to the volume:\nd\u0000 =dZdPz\n2\u0019~=d\bdJz\n2\u0019~=JdZd\b\n\u0019~: (134)\nThe same result can be obtained if we extrapolate Eq. (40) from a single spin to\na domain wall by using q1=Z,q2= \b, andG12=GZ\b=\u00002\u001bJ.\nIn our count, we included the two zero modes of a domain wall and discarded all\nof the other modes, which (at low amplitudes) are spin waves with \fnite frequencies\n!\u0015!0=K=J(127). The spin waves are frozen out at energies below ~!0.\n3.13. Hamiltonian dynamics\nIn Sec. 2.10, we expressed the dynamics of a single spin in terms of canonical variables.\nThere we chose the azimuthal angle \u001eas coordinate qand thezcomponent of spin\nJz=Jcos\u0012as canonical momentum p. This can also be done for a magnetic soliton.\nFor a domain wall in a ferromagnetic wire, we also select the azimuthal angle \b and\nangular momentum Jzas a canonical pair:\nq= \b; p =\u00002\u001bJZ; (135)\nsee Eq. (131).\nLet us consider a domain wall perturbed by a weak external magnetic \feld h\nparallel to the easy axis and small anisotropy breaking the axial symmetry. Then the\nenergy of the domain wall will have the following form:\nU(Z;\b) = 2\u001bMhZ+V(\b): (136)\nThe \frst term expresses the Zeeman coupling to the magnetic \feld (Sec. 3.7), the\nsecond is a phenomenological energy of the additional anisotropy. By expressing the\nenergy in terms of the canonical variables (135), we obtain the Hamiltonian\nH(p;q) =\u00002!Lp+V(q): (137)\nHere!Lis the Larmor precession frequency (9). Hamilton's equations (34) read\n_q=\u0000!L;_p=\u0000dV=dq: (138)\nIt is straightforward to check that these equations reproduce the equations of motion\n(107) in the absence of dissipative forces, \u000b= 0.\nThe formulation of dynamics in terms of canonical variables underpins the\nstandard procedure of quantization. With physical variables promoted to operators,\nthe Poisson bracket turns into the commutator. Quantization of vortex and skyrmion\ndynamics along these lines was considered in Refs. [23, 24].\nIt is worth noting that our experience with classical mechanics of massive particles\nand objects made from them has conditioned us to expect that canonical momenta\nare proportional to velocities: the linear momentum of a nonrelativistic particle is\npx=m_xand the angular momentum of a planar rotator is L=I_\b. In contrast, the\nlinear momentum of a domain wall in a wire is proportional to its azimuthal angle\nand the angular momentum to its coordinate, see Eqs. (131) and (133). Roughly\nspeaking, the two coordinates Zand \b of a domain wall also serve as its canonicalCONTENTS 27\nx(a)yz\n(b)xyz\nΦX\nΦX\nFigure 5. Domain wall in Walker's model: (a) \b = 0. (b) \b = \u0019=2. The shaded\nplane is the plane of the domain wall, where m(x) is orthogonal to the easy axis\nez. The dotted blue lines are minima of the anisotropy energy at \b = \u0006\u0019=2. The\nblack dot on the surface of the cylinder depicts the con\fguration ( X;\b) of the\ndomain wall.\nmomenta! Instead of two pairs of canonical variables in this case we have only one,\nso one coordinate has to serve as a canonical coordinate and the other as canonical\nmomentum.\nFor a ferromagnetic soliton, the separation of collective coordinates into canonical\ncoordinates and momenta is arbitrary and therefore somewhat unsatisfactory. It\ncan be avoided if we do not insist on a canonical structure and treat all collective\ncoordinates on the same footing. In this treatment, the Poisson bracket can be\nobtained from its inverse known as the Lagrange bracket [25]. For ferromagnetic\nsolitons, the Lagrange bracket is the gyroscopic tensor. See Ref. [16] for further\ndetails.\n4. Walker's model of a domain wall\nSchryer and Walker [10] investigated the dynamics of a realistic domain wall in a 3-\ndimensional ferromagnet with intrinsic easy-axis anisotropy. In their model, the easy\ndirection is zand the domain wall forms a sheet parallel to the yzplane, Fig. 5, so\nthat the spin \feld depends on the xcoordinate only. The axial symmetry of global\nrotations about the zaxis encountered in the previous model of a ferromagnetic wire\nis broken by magnetostatic (dipolar) interactions and the energy of the domain wall\ndepends on its azimuthal orientation.CONTENTS 28\n4.1. Energy functional and domain-wall con\fguration\nThe magnetization \feld m(x) has the following energy per unit area in Walker's model:\nU0=Z\ndx\u0014A\n2(\u001202+ sin2\u0012\u001e02) +K\n2sin2\u0012+M2\n8\u0019sin2\u0012cos2\u001e\u0015\n:(139)\nHereMis the magnetization related to the spin density Jby the gyroscopic ratio,\nM=\rJ. The \frst two terms in Eq. (139) are similar to those of our simpli\fed\n1-dimensional model (93); Ais the strength of Heisenberg exchange and Kis the easy-\naxis anisotropy. The last, magnetostatic term represents the energy of the magnetic\n\feldh= (\u0000M sin\u0012cos\u001e;0;0) generated by the inhomogeneous magnetization on the\ndomain wall and can be viewed as additional anisotropy breaking the symmetry of\nrotations about the easy z-axis. Thexdirection is made harder by the magnetostatic\nenergy. Therefore, in equilibrium the magnetization will stay in the yzplane.\nThe characteristic length, time, and energy scales are the same as in the model\nof the axially symmetric wire (Sec. 3):\n\u00150=p\nA=K; \u001c 0=J=K; \u000f 0=p\nAK; \u00162=M2=4\u0019K: (140)\nThe dimensionless ratio \u00162quanti\fes the relative strengths of the magnetostatic and\nintrinsic anisotropy terms.\nMinimizing the energy functional (139) at a \fxed and uniform azimuthal angle,\n\u001e(x) = \b, yields a pro\fle shown in Fig. 5,\ncos\u0012(x) =\u001btanhx\u0000X\n\u0015(\b); \u0015(\b) =\u00150p\n1 +\u00162cos2\b: (141)\nThe energy density (per unit area) of a domain wall,\nU0(\b) =\u000f0p\n1 +\u00162cos2\b; (142)\nis the lowest for \b = \u0006\u0019=2, i.e., when the domain wall interpolates the ground states\nm=\u0000ezandm= +ezin theyzplane.\nAs in the model of a ferromagnetic wire, we will add an external perturbation\nin the form of the Zeeman coupling to a magnetic \feld h= (0;0;h) aligned with the\neasy axis,\nU1(X) =\u0000Z\ndxMhcos\u0012= 2\u001bMhX: (143)\nPositionXand angle \b are the collective coordinates of Walker's domain wall.\nIt will be useful to picture the state of the domain wall in con\fguration space\n\u00001<X < +1;0\u0014\b\u00142\u0019; (144)\nthe surface of a cylinder, Fig. 5. The width of the domain wall \u0015(\b) is a slave of the\nangle \b, not an independent variable.\nThe gyrotropic and dissipative tensors have the following nonvanishing\ncoe\u000ecients:\nGX\b=\u0000G\bX=\u00002\u001bJ;\nDXX=2\u000bJ\n\u0015(\b); D \b\b= 2\u000bJ\u0015(\b)\"\n1 +\u0012\u00150(\b)\n\u0015(\b)\u00132#\n: (145)\nObserve that the dissipative coe\u000ecients depend on the width of the domain wall \u0015\n(which in turn depends on the azimuthal angle \b), indicating that the viscous force\ndepends on the precise geometry of the domain wall (in this case, its width). In\ncontrast, the gyroscopic coe\u000ecients are not sensitive to the geometry and depend only\non the topology of the domain wall (through the Z2topological charge \u001b).CONTENTS 29\n4.2. Dynamics of collective coordinates\nThe motion of a domain wall in Walker's model is more complex than in the simpler\nmodel of a magnetic wire considered above. The increased complexity can be traced\nto a more elaborate energy landscape of the domain wall: its energy now depends on\nboth collective coordinates,\nU(X;\b) = 2\u001bMhX+U0(\b); (146)\nso there are both conservative force and torque (per unit area):\nFX=\u0000@U\n@X=\u00002\u001bMh;\nF\b=\u0000@U\n@\b=\u000f0\u00162sin 2\b\n2p\n1 +\u00162cos2\b: (147)\nThe force grows linearly with the applied \feld h. The torque attains a peak value\nFmax\n\bat a critical angle \b c, where\nFmax\n\b=\u000f0\u0010p\n1 +\u00162\u00001\u0011\n;\bc= arctan4p\n1 +\u00162: (148)\nThe equations of motion for the collective coordinates read\nX:\u00002\u001bJ_\b +FX\u0000DXX_X= 0;\n\b : + 2\u001bJ_X+F\b\u0000D\b\b_\b = 0: (149)\nIn the absence of the external \feld, H= 0, the domain wall is at rest, _X= 0, at\none of its equilibrium azimuthal orientations, \b = \u0006\u0019=2. Turning on the \feld sets\nthe domain wall in motion. Its dynamics has two distinct regimes depending on the\nstrength of the applied \feld h.\n4.3. Steady-state motion in a weak \feld\nBelow a critical \feld strength hc, the domain wall moves at a constant velocity,\n_X= const, without changing its shape, _\b = 0. Its velocity grows with an increasing\n\feld,\n_X=FX\nDXX=\u0000\u001b\rh\u0015 (\b)\n\u000b: (150)\nAs the velocity grows, the azimuthal angle \b deviates more from the equilibrium\nvalue \b =\u0006\u0019=2. The azimuthal angle in the state of dynamical equilibrium is found\nfrom the balance of torques, given by the second of Eqs. (149),\n2\u001b_X+\rM\u0015(\b)\n4\u0019sin 2\b = 0: (151)\nFig. 6 shows the motion of a domain wall through its con\fguration space ( X;\b)\nfor weak \felds h < hc. The domain wall starts from a minimum of the anisotropy\nenergyU0(\b) at \b = \u0019=2 and reaches a new dynamic equilibrium at an angle \b <\bc.\nAll diagrams show the motion for the same length of time 2 Tc, whereTc= 2\u0019=(\rhc)\nis the Larmor period for the critical \feld hc. Thus the axial (vertical) displacement\nserves as a proxy for the linear velocity _X. As can be seen from Fig. 6, the steady-state\nvelocity increases with an increasing \feld.CONTENTS 30\nh=00.6hc0.8hc0.95hc0.2hc0.4hc\nFigure 6. Motion of Walker's domain wall in weak \felds, h < h c. Numerical\nsolutions of Eqs. (149) for the anisotropy ratio \u00162= 1 and Gilbert damping\n\u000b= 0:05 Solid black line: trajectory of the domain wall in con\fguration space\n(X;\b) depicted as the surface of a cylinder, as in Fig. 5. Dashed blue lines:\nminima of the anisotropy energy U0(\b). Dashed red lines: critical angles\n\bc. Trajectories for all \felds hare shown for the same length of time 2 Tc,\nwhereTc= 2\u0019=(\rhc) is the Larmor period for the critical \feld. The vertical\ndisplacement is an indicator of the domain wall velocity.\nA steady state can be maintained as long as the gyroscopic torque 2 \u001bJ_Xis\nbalanced by the conservative torque F\b. The peak torque (148) determines the critical\nspeed\nj_Xjc=Fmax\n\b\n2J; (152)\nabove which the steady-state motion cannot be sustained (Walker's breakdown). The\ncritical \feld is\nhc=\u000bM\n16\u0019sin 2\bc: (153)\nNote that the critical velocity (152) is independent of Gilbert's damping \u000b,\nwhereas the critical \feld (153) is of order \u000b.\n4.4. Walker's breakdown\nAbove the critical \feld, the steady-state velocity (150) becomes so large that the\ngyroscopic torque G\bX_Xexceeds the peak conservative torque (148) and the steady-\nstate equilibrium is no longer possible. The motion of the domain wall becomes\ncomplex, combining non-steady precession of the azimuthal angle with oscillation and\ndrift of position X.\nFig. 7 shows trajectories of a domain wall in con\fguration space ( X;\b) for several\nvalues of the \feld h>hc. As in the steady-state regime, the trajectories start at t= 0\nat the minimum of the anisotropy energy \b(0) = \u0019=2 and end at t= 2Tc, where\nTc= 2\u0019=(\rhc) is the Larmor period for the critical \feld. In contrast to the steady-\nstate regime, increasing the \feld beyond hcleads to a reduction in the average velocity\nof the domain wall.CONTENTS 31\nh=1.05hc2hc3hc5hc1.25hc1.5hc\nFigure 7. Motion of Walker's domain wall in strong \felds, h>h c. See Fig. 6 for\ndetails. The shrinking axial displacement over the same time period 2 Tcindicates\na decreasing average velocity of the domain wall in higher \felds.\n4.5. Oscillatory motion in a very strong \feld\nIn a very strong applied \feld, h\u001dhc, the domain wall moves in an oscillatory manner.\nTo understand the nature of this motion, we may at \frst neglect Gilbert's damping\nand set\u000b= 0. The energy (146) is then conserved,\n2\u001bMhX+U0(\b) = const : (154)\nThis limits the range of motion for the domain wall's coordinate to\n\u0001X=U0(0)\u0000U0(\u0019=2)\n2Mh: (155)\nThe azimuthal angle decreases steadily at the Larmor frequency (9),\n\b(t) = \b(0)\u0000!Lt: (156)\nThe linear velocity can then be found from the balance of the gyroscopic and\nconservative torques, Eq. (151), which yields\n_X=\u0000\u001b\rM\u00150\n8\u0019sin 2\b(t): (157)\nIn the absence of damping, the velocity oscillates around zero average.\nAdding weak damping, \u000b\u001c1, breaks the conservation of energy. The fast\noscillatory motion will lead to a gradual reduction of the energy with time. Averaged\nover an oscillation period, the domain-wall energy U0(\b) will remain constant over\ntime, so the reduction must come from the Zeeman energy 2 \u001bMhX. That creates\na slow drift on top of the oscillatory motion with an average velocity proportional\nto Gilbert's damping \u000b. In contrast, the steady-state velocity (150) is proportional\nto\u000b\u00001. The drift velocity above Walker's breakdown is thus much lower than the\nsteady-state velocity below it. See Schryer and Walker [10] for further details and\nBeach et al. [26] for experimental examples.CONTENTS 32\n4.6. D oring's mass\nLet us return to the regime of a weak magnetic \feld. We will now investigate the\ntransient dynamics of the wall before it reaches the steady state considered in Sec. 4.3.\nThe azimuthal angle of a weakly perturbed wall remains close to one of its\nequilibrium values, \b = \u0006\u0019=2 +\u000e\b with\u000e\b\u001c1. The azimuthal angle is now a\nhard mode . We will integrate it out to obtain the dynamics of the remaining soft\nmodeX. The elimination of the hard mode is also justi\fed on practical grounds: it is\neasier for experimentalists to measure the position Xof a domain wall than to observe\nits azimuthal orientation \b.\nNear equilibrium, the conservative torque (147) is approximately linear in the\ndeviation\u000e\b from equilibrium,\nF\b\u0018\u0000\u0014\u000e\b; (158)\nwith the sti\u000bness constant \u0014=\u00162\u000f0. We rewrite the equations of motion (149) as\nfollows:\nX:GX\b_\b +FX\u0000DXX_X= 0;\n\b :G\bX_X\u0000\u0014\u000e\b = 0: (159)\nWe have dropped the dissipative force \u0000D\b\b_\b for the hard mode \b because its\nmotion is restricted. We will see later that this is a reasonable approximation.\nThis simpli\fcation enables us to express the hard mode in terms of the soft mode,\n\u000e\b =G\bX_X=\u0014 and to eliminate it from the equation of motion for the soft mode,\nwhich acquires the familiar form of Newton's second law:\n\u0000MX+FX\u0000DXX_X= 0: (160)\nIntegrating out the hard mode \b has endowed the soft mode Xwith inertial mass\nM=G2\nX\b=\u0014 (161)\nand kinetic energy M_X2=2.\nEquation (160) has a characteristic relaxation time\n\u001c=M\nDXX=\u001c0\n\u000b\u00162=4\u0019\n\u000b\rM; (162)\nduring which the velocity of the domain wall reaches its terminal value _X=FX=DXX.\nWe derived the e\u000bective inertial dynamics of the soft mode X(160) while neglecting\nthe dissipative force \u0000D\b\b_\b for the hard mode. To check the accuracy of that\napproximation, we extract the relaxation time from the full equations of motion (149):\n\u001c=G2\nX\b+DXXD\b\b\n\u0014DXX= (1 +\u000b2)M\nDXX: (163)\nWe can see now that dissipation of the hard mode gives a small correction O(\u000b2) to\nthe relaxation time, which justi\fes its neglect.\nAs dissipative forces are not essential for the emergence of inertia, we may derive\nit in the Lagrangian framework. The low-energy Lagrangian of the position Xand\nazimuthal angle \b = \u0006\u0019=2 +\u000e\b can be written as\nL(X;\u000e\b) =\u0000GX\b\u000e\b_X\u0000\u0014\u000e\b2\n2\u0000U(X): (164)\nThe \frst term represents the geometric (Berry-phase) part of the Lagrangian (78)\nwith the gauge choice AX=\u0000GX\b\u000e\b andA\b= 0. Minimization of the actionCONTENTS 33\nS=R\nL(X;\u000e\b)dtwith respect to the hard variable \u000e\b yields the optimal \u000e\b =\n\u0000GX\b_X=\u0014 for a given trajectory X(t). Substituting this optimal \u000e\b(t) into the\noriginal Largangian (164) yields a new Lagrangian for the soft mode Xalone,\nL(X) =M_X2\n2\u0000U(X); (165)\nwith the D oring mass (161). It yields Eq. (160) without the dissipative force, which\ncan be added via Rayleigh's dissipation function.\nThe emergence of inertia in ferromagnetic solitons through the elimination of a\nhard mode was \frst noted by D oring [27]. Inertia has been observed in the dynamics\nof domain walls [28, 29] and skyrmions [30, 31].\n5. Discussion\nThe method of collective coordinates can be applied to more complex domain walls,\ne.g., composite [32] and extended [33] ones, as well as to other types of solitons such\nas vortices [34] and skyrmions [30]. An extension to antiferromagnets [35] shows that\nantiferromagnetic solitons generally possess kinetic energy and inertia.\nThe method has been successful in describing the dynamics of magnetic solitons\nin classical spin systems. One potentially promising direction for further development\nis to generalize the collective-coordinate approach to quantum spin systems where\nquantum e\u000bects are prominent in the dynamics of solitons.\nThe expert reader will note a glaring omission in our review: we chose not to\ncover the spin-transfer torque exerted by a spin-polarized electric current in a metallic\nferromagnet [36]. Spin-transfer torque has been treated in the framework of collective\ncoordinates with the electrical current included as a background \feld [37]. It has been\nrecognized for some time that this approach is \rawed and that a more sound treatment\nshould include the elect ric charge as one of the physical degrees of freedom, on the\nsame footing as the collective coordinates of a magnetic soliton [38]. A separate article\nin this issue by one of us addresses that problem [39].\nAcknowledgements\nS.K.K. was supported by Brain Pool Plus Program through the National\nResearch Foundation of Korea funded by the Ministry of Science and ICT (NRF-\n2020H1D3A2A03099291) and by the National Research Foundation of Korea funded\nby the Korea Government via the SRC Center for Quantum Coherence in Condensed\nMatter (NRF-2016R1A5A1008184). O.T. was supported by the U.S. Department of\nEnergy, O\u000ece of Science, Basic Energy Sciences under Award No. DE-SC0019331.\nAppendix A.\nThe Schr odinger equation H1 =\u000f with a P oschl{Teller potential [21],\nH1=\u0000d2\ndx2+ 1\u00002 sech2x; (A.1)\ncan be solved with the aid of supersymmetric quantum mechanics [22]. The\nHamiltonian is factorized, H1=AyA, in terms of non-Hermitian operators,\nA=d\ndx+ tanhx; Ay=\u0000d\ndx+ tanhx; (A.2)CONTENTS 34\nwith the commutator [ A;Ay] = 2 sech2x.\nThe supersymmetric partner of H1is\nH2=AAy=AyA+ [A;Ay] =H1+ 2 sech2x=\u0000d2\ndx2+ 1; (A.3)\nthe Hamiltonian of a free non-relativistic particle. Its eigenstates are plane waves,\n 2(x) =eikx, with energy eigenvalues \u000fk= 1 +k2.\nFrom each eigenstate of H2we can construct an eigenstate of H1with the same\neigenvalue as follows. Suppose AAyj 2i=\u000fj 2i. Make a new state j i=Ayj 2i. It\nis an eigenstate of H1:\nAyAj i=AyA(Ayj 2i) =Ay(AAyj 2i)\n=Ay(\u000fj 2i) =\u000fAyj 2i=\u000fj i: (A.4)\nThe norm of the new state is\nh j i=h 2jAAyj 2i=h 2j\u000fj 2i=\u000fh 2j 2i: (A.5)\nTherefore, a properly normalized eigenstate of H1would be\nj 1i=Ay\np\u000fj 2i: (A.6)\nWe thus obtain eigenstates of H1with eigenvalues \u000fk= 1 +k2:\n k(x) =1p\n1 +k2\u0012\n\u0000d\ndx+ tanhx\u0013\neikx=tanhx\u0000ikp\n1 +k2eikx: (A.7)\nNote that these states become plane waves far away from the potential well,\n k(x)\u0018\u00061\u0000ikp\n1 +k2eikx=eikx+i\u001e\u0006asx!\u00061; (A.8)\nwhere\u001e\u0006=\u0000\u0019=2\u0006arctan (1=k). It is remarkable that the P oschl{Teller potential\nis re\rectionless: a plane wave with wavenumber kis fully transmitted and merely\nexperiences a phase shift \u001e+\u0000\u001e\u0000= 2 arctan (1 =k).\nEigenstates ofH1have the same inner product as their supersymmetric partners,\nthe plane waves:Z\ndx \u0003\nk0(x) k(x) =Z\ndxe\u0000ik0xeikx= 2\u0019\u000e(k\u0000k0): (A.9)\nIn addition to free states (A.7), H1has a bound state with exactly zero energy,\nAyA\t0= 0. This state is annihilated by the operator A, which yields a \frst-order\ndi\u000berential equation\n\t0\n0(x) + tanhx\t0(x) = 0: (A.10)\nIts solution, properly normalized, is\n\t0(x) = 2\u00001=2sechx: (A.11)CONTENTS 35\n[1] Kosevich A M, Ivanov B A and Kovalev A S 1990 Phys. 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B 98104411\n[34] Huber D L 1982 Phys. Rev. B 263758{3765\n[35] Tveten E G, Qaiumzadeh A, Tretiakov O A and Brataas A 2013 Phys. Rev. Lett. 110127208\n[36] Ralph D C and Stiles M D 2008 J. Magn. Magn. Mater. 3201190{1216\n[37] Tserkovnyak Y, Brataas A and Bauer G E W 2008 J. Magn. Magn. Mater. 3201282{1292\n[38] Dasgupta S and Tchernyshyov O 2018 Phys. Rev. B 98224401\n[39] Tchernyshyov O 2023 J. Phys.: Condens. Matter 51014001"    },    {        "title": "2310.01721v2.Stray_magnetic_fields_from_elliptical_shaped_and_stadium_shaped_ferromagnets.pdf",        "content": "Stray magnetic fields from elliptical-shaped and stadium-shaped ferromagnets\nTomohiro Taniguchi∗\nNational Institute of Advanced Industrial Science and Technology (AIST),\nResearch Center for Emerging Computing Technologies, Tsukuba 305-8568, Japan\n(Dated: October 27, 2023)\nAn artificial spin ice consisting of numerous ferromagnets has attracted attention because of its\napplicability to practical devices. The ferromagnets interact through their stray magnetic field and\nshow various functionality. The ferromagnetic element in the spin ice was recently made in elliptical-\nshape or stadium-shape. The former has a narrow edge, expecting to generate a large stray magnetic\nfield. The latter has a large volume and is also expected to generate a large stray magnetic field.\nHere, we estimate the stray magnetic field by numerically integrating the solution of the Poisson\nequation. When magnetization is parallel to an easy axis, the elliptical-shaped ferromagnet generates\na larger stray magnetic field than the stadium-shaped ferromagnet. The stray magnetic fields from\nboth ferromagnets for arbitrary magnetization directions are also investigated.\nI. INTRODUCTION\nComputation of magnetic field from a ferromagnetic\nbody, or more generally solving a Possion equation, has\nbeen an attractive problem in both mathematics and\nphysics [1–12], and still prompts publications recently\n[13–18]. Even after great development of numerical\nsolvers [19–24], deriving a solution of potential, or field,\nin terms of analytical functions or in forms of some inte-\ngrals with simplified approximations, such as macrospin\nor rigid-vortex assumption, is highly demanded, particu-\nlarly when many-body problems are of interest [25–29].\nThis is because it enables us to estimate various param-\neters, such as coercive and stray fields, with adequate\ncalculation cost. The past works have mainly focused\non the internal magnetic field and derived the solutions\nof the demagnetization coefficients for various shapes of\nferromagnets such as ellipsoid, cylinder, and cuboid [1–\n6, 8–10], while the stray magnetic field, originated from\nmagnetostatic interaction, for vortex, cylinder, and so on\nhas also been investigated [7, 11–18].\nAn interesting target nowadays related to this is\nto compare stray magnetic fields from two kinds of\nuniformly magnetized ferromagnets, namely elliptical-\nshaped and stadium-shaped ferromagnets [30, 31], which\nare schematically shown in Figs. 1(a) and 1(b), respec-\ntively. They are recently used as a fundamental element\nof artificial spin ice, and their stray magnetic fields are\nthe origin of the frustration in it through magnetostatic\ninteraction [32–39]. The frustrated states of the ferro-\nmagnets are expected to be used in several applications\nsuch as multibit memory and neuromorphic computing\n[40–42]. Therefore, the evaluation and comparison from\nthese two kinds of ferromagnets are of great interest.\nLet us assume that the length and width of these two\nferromagnets, which are 2 aand 2 bin Fig. 1, are the\nsame. Accordingly, the stadium-shaped ferromagnet has\na larger volume compared with the elliptic-shaped fer-\nromagnet. From this perspective, the stadium-shaped\n∗tomohiro-taniguchi@aist.go.jpferromagnet is expected to generate a larger stray mag-\nnetic field. However, we should note that the magnetic\npole of the elliptic-shaped ferromagnet is concentrated in\na narrower region (around x≃ ±ain Fig. 1) compared\nwith that in the stadium-shaped ferromagnet, when the\nmagnetization points parallel to the easy (long) axis. In\nthis sense, the elliptic-shaped ferromagnet is expected to\ngenerate a larger stray field. Then, a question arises as to\nwhich shape of the ferromagnet generates a larger stray\nfield. Here, we compute their stray magnetic fields with\nmacrospin assumption and study this question. Starting\nfrom Maxwell equations in a steady state, ∇·B= 0 and\n∇×H=0, where BandHare the magnetic flux density\nand the magnetic field, respectively, the magnetic poten-\ntialV, satisfying H=−∇V, obeys∇2V= 4π∇·M,\nwhere Mis the magnetization. The solution of this Pos-\nsion equation is\nV(r) =I\ndS′n·M\n|r−r′|, (1)\nwhere dSis an infinitesimal cross-section area of the inte-\ngral region, while nis a unit vector orthogonal to the sur-\nfacedS. The stray magnetic field is estimated from Eq.\n(1) as H=−∇V. It is found that the elliptical-shaped\nferromagnet generates a larger stray magnetic field along\nthe direction of the easy axis while the stadium-shaped\nferromagnet generates a larger stray magnetic field along\nthe in-plane hard-axis direction.\nII. STRAY FIELD FROM AN\nELLIPTICAL-SHAPED FERROMAGNET\nHere, we derive a formula of the stray magnetic field\nfrom an elliptical-shaped ferromagnet. The geometry\nis schematically shown in Fig. 2(a). In the follow-\ning, we denote that magnetization direction in terms\nof the zenith and azimuth angles, θandφ, asM=\nM(sinθcosφ,sinθsinφ,cosθ). Since the Poisson equa-\ntion is a linear equation, its solution is a superposition\nof the solutions, where the magnetization points to in-\nplane ( xy-plane) and perpendicular ( z-axis) directions.arXiv:2310.01721v2  [cond-mat.mes-hall]  25 Oct 20232\n(a)\n+a -a -a+b\n-bxy(b)\n+a\n+a+b\n-bx\nxy\n-a+h/2\n-h/2z\n+a x -a+h/2\n-h/2z\nFIG. 1. (Color online) Top and side views of (a) elliptical-\nshaped and (b) stadium-shaped ferromagnets. The length in\nthe long-axis ( x-axis) direction and the width in the short-\naxis ( y-axis) direction are 2 aand 2 b, respectively, while the\nheight in the zdirection is h.\nr=(X,Y,Z)\ny\nx+a-a\n+b-b\n-h/20θ\nφ(a) (b)\n-ayz\nx0 x0y0 yt\nζβ +a+b\n-b+h/2ynM\nFIG. 2. (Color online) (a) Bird’s eye view of the elliptical-\nshaped ferromagnet. (b) Top view of the elliptical-shaped\nferromagnet, where the tanget ytand normal ynwith at a\npoint ( x0, y0), as well as the angles βandζ, are indicated.\nTherefore, we denote the solution of the magnetic po-\ntential Vfor the in-plane and perpendicularly magne-\ntized ferromagnets as ViandVp, respectively. The mea-\nsurement point of the potential is r= (X, Y, Z ). The\nmagnetic field at this point is H(r) =−∇V|x=r=\n(−∂V/∂x, −∂V/∂y, −∂V/∂z )x=r.A. Magnetic potential\nTo derive the solution of Vfor the elliptical-shaped\nferromagnet, it is useful to introduce an elliptic coordi-\nnate, where the positions xandyin the Cartesian co-\nordinate are related to new variables, αandβ, asx=\nccoshαcosβandy=csinhαsinβ, where −∞ ≤ α≤ ∞\nand−π≤β≤π, while c=√\na2−b2. We also introduce\nαe=1\n2ln\u0012a+b\na−b\u0013\n. (2)\nLet us denote a point on the circumference of the ellipse\nas (x0, y0); see also Fig. 2(b). The tangent and normal\nwith respect to the point are, respectively, given by\nyt=b2\ny0\u0010\n1−x0\na2x\u0011\n, (3)\nyn=a2y0\nb2x0x−y0\u0012a2\nb2−1\u0013\n. (4)\nThe angle ζbetween ynand the x-axis satisfies\ncosζ=b2coshαecosβp\nb4cosh2αecos2β+a4sinh2αesin2β,(5)\nsinζ=a2sinhαesinβp\nb4cosh2αecos2β+a4sinh2αesin2β.(6)\nTherefore, the projection of the magnetiza-\ntion to the normal vector, n·M, becomes\nMn=Msinθ(cosφcosζ+ sin φsinζ). An infinitesimal\ncross-section area around the point ( x0, y0) is a product\nof infinitesimal lines in the xy-plane,p\nx2\n0+y2\n0dβ, and\nalong the z-direction, dz, as dS=p\nx2\n0+y2\n0dβdz =\ncp\ncosh2αecos2β+ sinh2αesin2βdβdz . Accordingly,\nthe magnetic potential Vigenerated from the in-plane\nmagnetized elliptical-shaped ferromagnet becomes\nVi=MsinθZh/2\n−h/2dz′Zπ\n−πdβcp\ncosh2αecos2β+ sinh2αesin2βp\n(x−ccoshαecosβ)2+ (y−csinhαesinβ)2+ (z−z′)2\n× \nb2coshαecosβcosφ+a2sinhαesinβsinφp\nb4cosh2αecos2β+a4sinh2αesin2β!\n.(7)\nWhen the magnetization points to the perpendicu-\nlar (z) direction, the infinitesimal cross-section area is\ndxdy =Jdαdβ , where the Jacobian Jis (c2/2)(cosh 2 α−\ncos 2β). Note that there are two magnetic poles, n·M,appearing on the xy-planes at z=h/2 and z=−h/2,\nwith the opposite sign. Thus, the magnetic potential Vp\ngenerated from the perpendicularly magnetized elliptical-\nshaped ferromagnet is3\nVp=McosθZαe\n0dαZπ\n−πdβc2(cosh 2 α−cos 2β)\n2\"\n1p\n(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z−h/2)2\n−1p\n(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z+h/2)2#\n.(8)\nB. Magnetic field\nThex,y, and zcomponents of the magnetic field are\ngiven by\nHx=MsinθZh/2\n−h/2dz′Zπ\n−πdβcp\ncosh2αecos2β+ sinh2αesin2β(x−ccoshαecosβ)\n[(x−ccoshαecosβ)2+ (y−csinhαesinβ)2+ (z−z′)2]3/2\n× \nb2coshαecosβcosφ+a2sinhαesinβsinφp\nb4cosh2αecos2β+a4sinh2αesin2β!\n+McosθZαe\n0dαZπ\n−πdβc2(cosh 2 α−cos 2β)\n2(x−ccoshαcosβ)\n×\u00141\n[(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z−h/2)2]3/2\n−1\n[(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z+h/2)2]3/2\u0015\n,(9)\nHy=MsinθZh/2\n−h/2dz′Zπ\n−πdβcp\ncosh2αecos2β+ sinh2αesin2β(y−csinhαesinβ)\n[(x−ccoshαecosβ)2+ (y−csinhαesinβ)2+ (z−z′)2]3/2\n× \nb2coshαecosβcosφ+a2sinhαesinβsinφp\nb4cosh2αecos2β+a4sinh2αesin2β!\n+McosθZαe\n0dαZπ\n−πdβc2(cosh 2 α−cos 2β)\n2(y−csinhαsinβ)\n×\u00141\n[(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z−h/2)2]3/2\n−1\n[(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z+h/2)2]3/2\u0015\n,(10)\nHz=MsinθZh/2\n−h/2dz′Zπ\n−πdβcp\ncosh2αecos2β+ sinh2αesin2β(z−z′)\n[(x−ccoshαecosβ)2+ (y−csinhαesinβ)2+ (z−z′)2]3/2\n× \nb2coshαecosβcosφ+a2sinhαesinβsinφp\nb4cosh2αecos2β+a4sinh2αesin2β!\n+McosθZαe\n0dαZπ\n−πdβc2(cosh 2 α−cos 2β)\n2\n×\u0014z−h/2\n[(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z−h/2)2]3/2\n−z+h/2\n[(x−ccoshαcosβ)2+ (y−csinhαsinβ)2+ (z+h/2)2]3/2\u0015\n.(11)4\n+a -a+b\n-bxy\na-b0φ\n(a-b)+bcos φ\n-(a-b)+bcos φφ b\nFIG. 3. (Color online) Top view of the stadium-shaped fer-\nromagnet, which can be regarded as a cylinder with a radius\nbseparated by a cuboid at the center. Definition of ϕof the\ncylinder coordinate is also shown. The value of the xcoordi-\nnate of a point on the circumference is ±(a−b) +bcosϕ.\nIII. STRAY FIELD FROM A\nSTADIUM-SHAPED FERROMAGNET\nHere, we derive a formula of the stray magnetic field\nfrom a stadium-shaped ferromagnet. We note that the\nstadium-shaped ferromagnet can be regarded as a cylin-\nder ferromagnet with a radius band height h, which isseparated by a cuboid with a long side 2( a−b), short\nside 2 b, and height h; see Fig. 3. Accordingly, the stray\nmagnetic field of this ferromagnet can be obtained as a\nsuperposition of the stray magnetic fields from cylinder-\nshaped and cuboid-shaped ferromagnets. The magnetic\nfields generated by these ferromagnets can be expressed\nin terms of analytical and special functions, as shown in\nSupplementary Material of Ref. [13]. We keep, however,\nintegral forms of the magnetic field because the expres-\nsion of the field by these functions will be greatly com-\nplex.\nA. Stray magnetic field generated by\nmagnetization pointing in xdirection\nLet us first consider a case where the magnetization M\npoints to the xdirection. The magnetic pole, n·M, in\nthis case appears only in the side of the cylinder parts.\nThus, it is convenient to introduce a cylinder coordi-\nnate ( ρ, ϕ, z ), which relates to the Cartesian coordinate\nasx=ρcosϕandy=ρsinϕ. An infinitesimal cross-\nsection area on the side of the cylinder is bdϕdz , while\nthe magnetic pole n·Mon this area is Mcosϕ. There-\nfore, the magnetic potential is given by (see also Fig. 3)\nVi,x=MZπ/2\n−π/2dϕZh/2\n−h/2dz′ bcosϕp\n[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2\n+MZ3π/2\nπ/2dϕZh/2\n−h/2dz′ bcosϕp\n[x+ (a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2,(12)\nor by redefining ϕin the second integral as ϕ−π,\nVi,x=MZπ/2\n−π/2dϕZh/2\n−h/2dz′(\nbcosϕp\n[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2\n−bcosϕp\n[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2)\n.(13)\nThe stray magnetic field H=−∇Vi,xis given by\nHxx=MZπ/2\n−π/2dϕZh/2\n−h/2dz′\u0012[x−(a−b)−bcosϕ]bcosϕ\n{[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2}3/2\n−[x+ (a−b) +bcosϕ]bcosϕ\n{[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2}3/2\u0013\n,(14)5\nHyx=MZπ/2\n−π/2dϕZh/2\n−h/2dz′\u0012(y−bsinϕ)bcosϕ\n{[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2}3/2\n−(y+bcosϕ)bcosϕ\n{[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2}3/2\u0013\n.(15)\nHzx=MZπ/2\n−π/2dϕZh/2\n−h/2dz′\u0012(z−z′)bcosϕ\n{[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2}3/2\n−(z−z′)bcosϕ\n{[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2}3/2\u0013\n.(16)\nB. Stray magnetic field generated by\nmagnetization pointing in ydirection\nNext, we calculate the stray magnetic field when the\nmagnetization points to the ydirection. The magneticpotential in this case is the sum of those generated by\nthe magnetic poles appear on the sides of the cylinder\nand cuboid. The former contribution, similar to Vi,xis\nVi,y,cy=MZπ/2\n−π/2dϕZh/2\n−h/2dz′ bsinϕp\n[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2\n+MZ3π/2\nπ/2dϕZh/2\n−h/2dz′ bsinϕp\n[x+ (a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2,(17)\nor equivalently,\nVi,y,cy=MZπ/2\n−π/2dϕZh/2\n−h/2dz′(\nbsinϕp\n[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2\n−bsinϕp\n[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2)\n,(18)\nwhile the latter contribution is\nVi,y,cu=MZ(a−b)\n−(a−b)dx′Zh/2\n−h/2dz′\"\n1p\n(x−x′)2+ (y−b)2+ (z−z′)2−1p\n(x−x′)2+ (y+b)2+ (z−z′)2#\n.(19)\nThe stray magnetic field H=−∇(Vi,y,cy+Vi,y,cu) is given by\nHxy=MZπ/2\n−π/2dϕZh/2\n−h/2dz′\u0012[x−(a−b)−bcosϕ]bsinϕ\n{[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2}3/2\n−[x+ (a−b) +bcosϕ]bsinϕ\n{[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2}3/2\u0013\n+MZ(a−b)\n−(a−b)dx′Zh/2\n−h/2dz′\u001ax−x′\n[(x−x′)2+ (y−b)2+ (z−z′)2]3/2−x−x′\n[(x−x′)2+ (y+b)2+ (z−z′)2]3/2\u001b\n,\n(20)6\nHyy=MZπ/2\n−π/2dϕZh/2\n−h/2dz′\u0012(y−bsinϕ)bsinϕ\n{[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2}3/2\n−(y+bsinϕ)bsinϕ\n{[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2}3/2\u0013\n+MZ(a−b)\n−(a−b)dx′Zh/2\n−h/2dz′\u001ay−b\n[(x−x′)2+ (y−b)2+ (z−z′)2]3/2−y+b\n[(x−x′)2+ (y+b)2+ (z−z′)2]3/2\u001b\n,\n(21)\nHzy=MZπ/2\n−π/2dϕZh/2\n−h/2dz′\u0012(z−z′)bsinϕ\n{[x−(a−b)−bcosϕ]2+ (y−bsinϕ)2+ (z−z′)2}3/2\n−(z−z′)bsinϕ\n{[x+ (a−b) +bcosϕ]2+ (y+bsinϕ)2+ (z−z′)2}3/2\u0013\n+MZ(a−b)\n−(a−b)dx′Zh/2\n−h/2dz′\u001az−z′\n[(x−x′)2+ (y−b)2+ (z−z′)2]3/2−z−z′\n[(x−x′)2+ (y+b)2+ (z−z′)2]3/2\u001b\n.\n(22)\nC. Stray magnetic field generated by magnetization pointing in zdirection\nNext, we calculate the stray magnetic field when the magnetization points to the zdirection. The magnetic pole,\nn·M, in this case appears on the circular and rectangular surfaces on z=±h/2 with the opposite sign. Thus, the\nmagnetic potential is\nVp=MZπ/2\n−π/2dϕZb\n0ρdρ(\n1p\n[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z−h/2)2\n−1p\n[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z+h/2)2\n+1p\n[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z−h/2)2\n−1p\n[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z+h/2)2)\n+MZ(a−b)\n−(a−b)dx′Zb\n−bdy′\"\n1p\n(x−x′)2+ (y−y′)2+ (z−h/2)2\n−1p\n(x−x′)2+ (y−y′)2+ (z+h/2)2#\n.(23)\nTherefore, the stray magnetic field H=−∇Vpbecomes\nHxz=MZπ/2\n−π/2dϕZb\n0ρdρ\u0012x−(a−b)−ρcosϕ\n{[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z−h/2)2}3/2\n−x−(a−b)−ρcosϕ\n{[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z+h/2)2}3/2\n+x+ (a−b) +ρcosϕ\n{[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z−h/2)2}3/2\n−x+ (a−b) +ρcosϕ\n{[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z+h/2)2}3/2\u0013\n+MZ(a−b)\n−(a−b)dx′Zb\n−bdy′\u0014x−x′\n[(x−x′)2+ (y−y′)2+ (z−h/2)2]3/2\n−x−x′\n[(x−x′)2+ (y−y′)2+ (z+h/2)2]3/2\u0015\n,(24)7\nHyz=MZπ/2\n−π/2dϕZb\n0ρdρ\u0012y−ρsinϕ\n{[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z−h/2)2}3/2\n−y−ρsinϕ\n{[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z+h/2)2}3/2\n+y+ρsinϕ\n{[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z−h/2)2}3/2\n−y+ρsinϕ\n{[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z+h/2)2}3/2\u0013\n+MZ(a−b)\n−(a−b)dx′Zb\n−bdy′\u0014y−y′\n[(x−x′)2+ (y−y′)2+ (z−h/2)2]3/2\n−y−y′\n[(x−x′)2+ (y−y′)2+ (z+h/2)2]3/2\u0015\n,(25)\nHzz=MZπ/2\n−π/2dϕZb\n0ρdρ\u0012z−h/2\n{[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z−h/2)2}3/2\n−z+h/2\n{[x−(a−b)−ρcosϕ]2+ (y−ρsinϕ)2+ (z+h/2)2}3/2\n+z−h/2\n{[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z−h/2)2}3/2\n−z+h/2\n{[x+ (a−b) +ρcosϕ]2+ (y+ρsinϕ)2+ (z+h/2)2}3/2\u0013\n+MZ(a−b)\n−(a−b)dx′Zb\n−bdy′\u0014z−h/2\n[(x−x′)2+ (y−y′)2+ (z−h/2)2]3/2\n−z+h/2\n[(x−x′)2+ (y−y′)2+ (z+h/2)2]3/2\u0015\n.(26)\nD. Total stray magnetic field generated from a\nstadium-shaped ferromagnet\nSummarizing the results above, the stray magnetic\nfield from the stadium-shaped ferromagnet, when the\nmagnetization points to an arbitrary direction as M=\nM(sinθcosφ,sinθsinφ,cosθ), is\nH=\nHxxHxyHxz\nHyxHyyHyz\nHzxHzyHzz\n\nsinθcosφ\nsinθsinφ\ncosθ\n (27)\nIV. COMPUTATION OF MAGNETIC FIELD\nHere, we examine the stray magnetic fields estimated\nby using the above formulas. Before that, however, let\nus mention the validity of the above calculation briefly.\nWe used the macrospin assumption in the derivation of\nthe stray magnetic field. Let us also assume there is\nno bulk nor interfacial magnetic anisotropy. Thus, the\nmagnetic anisotropy is solely determined by the shape\nmagnetic anisotropy. Then, the xaxis is the easy axis,\nwhile the yaxis is the in-plane hard axis. We performedmicromagnetic simulation by MuMax3 [22] and found\nthat, in the absence of an external magnetic field, the\nmacrospin assumption works well when the magnetiza-\ntion points to the xdirection, while it does not work for\nthe magnetization pointing in the ydirection; see Ap-\npendix. We, however, evaluate the stray magnetic field\nwith the macrospin assumption even when the magneti-\nzation direction is not parallel to the xdirection. This is\nbecause, in some practical applications such as physical\nreservoir computing by an artificial spin ice, a large ex-\nternal magnetic field is applied to the whole system [43].\nIn such a case, the macrospin assumption works even\nwhen the magnetization direction is deviated from the x\n(easy) axis; see also Appendix. Therefore, we assume in\nthe following that an external magnetic field is applied to\nthex,y, orzdirection to make the macrospin assump-\ntion used in the above calculations applicable. A further\ncomparison with micromagnetic simulation is kept as a\nfuture work because it strongly depends on the system\nsize, and its comprehensive study is beyond the scope of\nthis work.8\nA. Magnetic fields when magnetization points to x,\ny, orzdirections\nHere, we show the stray magnetic fields from\nthe elliptical-shaped and stadium-shaped ferromagents,\nwhere the magnetization is parallel to the x,y, orzaxis.\nThe values of the parameters are M= 1500 emu/cm3,\na= 200 nm, b= 75 nm, h= 20 nm [30, 31]. We com-\npute the stray magnetic fields in a plane parallel to the\nxy-plane at z= 15 nm, which is 5 nm above the upper\nferromagnetic surface at z= +h/2 = 10 nm.\nLet us first consider the case where the magnetization\nis parallel to the easy ( x) axis. Figures 4(a)-4(c) are\nthex,y, and zcomponents of the stray magnetic fields\ngenerated from the elliptical-shaped ferromagnet, while\nFigs. 4(d)-4(f) show those generated from the stadium-\nshaped ferromagnet. Since the magnetization points to\nthe + xdirection, the xcomponent of the stray magnetic\nfield points to the + xdirection for |x|> a, while it points\nto the −xdirection in |x|< abecause the magnetic field\nemitted near x= +areturns near to x=−a; see Figs.\n4(a) and 4(d). The yandzcomponents of the stray\nmagnetic fields in Figs. 4(b), 4(c), 4(e), and 4(f) can\nalso be explained in a similar way.\nNow let us remind our motivation to this study here.\nThe stadium-shaped ferromagnet has a larger volume\nthan the elliptical-shaped ferromagnet. In fact, their vol-\numes are Vs= [πb2+4b(a−b)]handVe=πabh , and thus,\ntheir difference satisfies Vs− V e= (4−π)b(a−b)h >0.\nSince the stray magnetic field is, roughly speaking, total\nnumber of the magnetic moments, which is the product of\nthe magnetization and the volume, the stadium-shaped\nferromagnet is expected to produce a larger stray field.\nThe elliptical-shaped ferromagnet, however, has a nar-\nrower area near |x| ≃a, and thus, the density of the\nmagnetic pole is larger than that of the stadium-shaped\nferromagnet. Therefore, it is unclear which type of the\nferromagnet can generate a larger stray field. We thus\nevaluate the stray magnetic field along the xdirection\nand notice that the stray magnetic fields generated from\nthe elliptical-shaped ferromagnets are mainly larger than\nthose generated from the stadium-shaped ferromagnet.\nFor example, the stray magnetic field generated from the\nelliptical-shaped ferromagnet at x=−300 nm and y= 0\nnm, i.e., 100 nm external to the ferromagnetic edge at\nx=−a=−200 nm, is 441 Oe, while that generated\nfrom the stadium-shaped ferromagnet is 281 Oe.\nThe situation is changed when the magnetization\npoints to the ydirection. Figures 5(a)-5(c) show the\nx,y, and zcomponents of the stray magnetic field gen-\nerated from the elliptical-shaped ferromagnet, where the\nmagnetization points to the ydirection. Those generated\nfrom the stadium-shaped ferromagnet is shown in Figs.\n5(d)-5(f). The field distribution is, roughly speaking, ro-\ntated 90◦from those shown in Fig. 4. In particular, let\nus focus on Figs. 5(b) and 5(e). We notice that the y\ncomponent of the stray field is larger for the stadium-\nshaped ferromagnet that that for the elliptical-shapedferromagnet. For example, the stray magnetic field at\nx= 0 nm and y= 100 nm, i.e., 25 nm external to the\nferromagnetic edge at y= +b= 75 nm, is 1498 Oe for the\nstadium-shaped ferromagnet, while that generated from\nthe elliptical-shaped ferromagnet is 597 Oe. The large\ndifference may arise from the difference of the ferromag-\nnetic boundary. The magnetic field becomes large when\nthe surface density of the magnetic pole, given by M·ndS,\nis large. For the stadium-shaped ferromagnet, the bound-\nary of the ferromagnet in −(a−b)≤x≤+(a−b) is paral-\nlel to the xaxis, i.e., the vector n, which is normal to the\nareadS, is parallel to the magnetization. Thus, the den-\nsity of the magnetic pole, as well as the stray magnetic\nfield, is large. On the other hand, for the elliptical-shaped\nferromagnet, the boundary is bent, and thus the vector\nnis not parallel expect at x= 0. Thus, the magnetic\npole density, M·ndS, becomes relatively small, resulting\nin a small stray magnetic field in the ydirection.\nWhen the magnetization points to the zdirection, as\nshown in Fig. 6, the stray magnetic fields from the\nelliptical-shaped and stadium-shaped ferromagnets have\nsimilar strength, although their distributions are differ-\nent depending on the sample shape.\nIn Fig. 7(a), the dependence of the xcomponents of\nthe stray magnetic fields generated from the elliptical-\nshaped and the stadium-shaped ferromagnets on the dis-\ntance from the sample edge at x= +a= 200 nm are\nshown by the red-solid and blue-dotted lines, respec-\ntively. The yandzcoordinates are 0 nm, i.e., Fig.\n7(a) shows the stray magnetic field along the xaxis,\nwhen the magnetization points to + xdirection. It is\nshown that the stray magnetic field generated from the\nelliptical-shaped ferromagnet is larger than that gener-\nated from the stadium-shaped ferromagnet, as implied\nfrom Figs. 4(a) and 4(d). Similarly, Fig. 7(b) shows the\ndependence of the ycomponent of the stray magnetic\nfield from both ferromagnets on the distance from the\nsample edge at y= +b= 75 nm, where x=z= 0 nm.\nThe magnetization in this case is aligned in + ydirection.\nIt can be confirmed that the stray magnetic field gener-\nated from the stadium-shaped ferromagnet is larger than\nthat generated from the elliptical-shaped ferromagnet, as\nmentioned above.\nIn previous studies [32–38] on the artificial spin ice con-\nsisting of ferromagnets, the magnetization alignments in\nequilibrium were on interest, where the magnetization\nin each ferromagnet approximately points to the easy-\naxis direction, i.e., + xor−xdirection. The frustra-\ntion caused by the stray magnetic field results in a large\nnumber of degenerated ground state. It provides sev-\neral interesting physical phenomena, such as residual en-\ntropy and spin liquid. The presence of a large number\nof the degenerated ground state is also of interest from a\nviewpoint of practical applications such as multibit mem-\nory and logic device [40]. In this sense, the elliptical-\nshaped ferromagnet will be preferable, compared with\nthe stadium-shaped ferromagnet, because it can generate\na larger stray field in the xdirection and drives a strong9\n\"x1.dat\" u 1:2:3\n\"x2.dat\" u 1:2:3x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)6\n4\n2\n0\n-2\n-4Field (kOe)6\n4\n2\n0\n-2\n-4(a)\n(d)x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)4\n2\n0\n-2\n-4Field (kOe)4\n2\n0\n-2\n-4(b)\n(e)\"x1.dat\" u 1:2:4\n\"x2.dat\" u 1:2:4x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)15\n10\n5\n0\n-5\n-10\n-15Field (kOe)15\n10\n5\n0\n-5\n-10\n-15(c)\n(f)\"x1.dat\" u 1:2:5\n\"x2.dat\" u 1:2:5\nFIG. 4. (Color online) Stray fields in (a) x, (b) y, and (c) zdirections generated from an elliptical-shaped ferromagnet. The\nmagnetization points to the + xdirection. Those fields generated from a stadium-shaped ferromagent are shown in (d), (e),\nand (f).\nx (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)2\n1\n0\n-1\n-2Field (kOe)2\n1\n0\n-1\n-2(a)\n(d)x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)2\n0\n-2\n-4Field (kOe)2\n0\n-2\n-4(b)\n(e)x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)6\n4\n2\n0\n-2\n-4\n-6Field (kOe)6\n4\n2\n0\n-2\n-4\n-6(c)\n(f)\"y1.dat\" u 1:2:3\n\"y2.dat\" u 1:2:3\"y1.dat\" u 1:2:4\n\"y2.dat\" u 1:2:4\"y1.dat\" u 1:2:5\n\"y2.dat\" u 1:2:5\nFIG. 5. (Color online) Stray fields in (a) x, (b) y, and (c) zdirections generated from an elliptical-shaped ferromagnets. The\nmagnetization points to the + ydirection. Those fields generated from a stadium-shaped ferromagents are shown in (d), (e),\nand (f).\nfrustration. In addition, the stadium-shaped ferromag-\nnet sometimes includes vortexes at the edges, depend-\ning on the size, and breaks the macrospin assumption\n[30, 31]. It results in a disappearance of magnetic pole,\nwhich is a topological charge in a spin ice [40] and plays\na central role in frustration. This is another reason pre-\nferring the elliptical-shaped ferromanget, rather than the\nstadium-shaped ferromagnet, applied to an artificial spin\nice. The stadium-shaped ferromagnet might be, however,\nof use to different purposes, such as reservoir computing\nutilizing a ferromagnetic ensemble [41–43], where an ex-\nternal magnetic field is applied to the direction deviated\nfrom the easy ( x) axis direction. In such a case, the stray\nmagnetic field, pointing in a direction different from thex-axis, may be generated and determines the magneti-\nzation alignment of the other ferromagnets. Then, the\nhistory of the applied field can be recognized from the\nmagnetization alignment. The shape of the ferromag-\nnet should be designed, depending on the purpose, and\nlarge-scale simulations, as well as experiments, will be\nnecessary in future.\nV. CONCLUSION\nIn summary, the stray magnetic fields generated from\nan elliptical-shaped and stadium-shaped ferromagnets\nwere evaluated from the integral forms of the solutions10\nx (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)4\n2\n0\n-2\n-4Field (kOe)4\n2\n0\n-2\n-4Field (kOe)4\n2\n0\n-2\n-4\nField (kOe)4\n2\n0\n-2\n-4(a)\n(d)x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)(b)\n(e)x (nm)0100 200 300 -200 -100 -300\nx (nm)0100 200 300 -200 -100 -300-1000100\n50\n-50y (nm)\n-1000100\n50\n-50y (nm)Field (kOe)4\n2\n0\n-2Field (kOe)4\n2\n0\n-2(c)\n(f)\"z1.dat\" u 1:2:3\n\"z2.dat\" u 1:2:3\"z1.dat\" u 1:2:4\n\"z2.dat\" u 1:2:4\"z1.dat\" u 1:2:5\n\"z2.dat\" u 1:2:5\nFIG. 6. (Color online) Stray fields in (a) x, (b) y, and (c) zdirections generated from an elliptical-shaped ferromagnets. The\nmagnetization points to the + zdirection. Those fields generated from a stadium-shaped ferromagents are shown in (d), (e),\nand (f).\nx (nm)x componet of \nstray magnetic field (kOe)250 300 350 400 450 5001.5\n1.0\n0.5\n0(a)\ny (nm)y componet of \nstray magnetic field (kOe)100 120 140 160 180 2001.52.02.5\n1.0\n0.5\n0(b)\n: elliptical-shaped\n: stadium-shaped: elliptical-shaped\n: stadium-shaped\nFIG. 7. (Color online) (a) The xcomponents of the stray mag-\nnetic fields generated from the elliptical-shaped (red-solid)\nand the stadium-shaped (blue-dotted) ferromagnets, where\nthe magnetization points to the + xdirection. The measure-\nment points locate at y=z= 0. (a) The ycomponents of\nthe stray magnetic fields generated from the elliptical-shaped\n(solid red) and the stadium-shaped (dotted blue) ferromag-\nnets, where the magnetization points to the + ydirection. The\nmeasurement points locate at x=z= 0.\nof the Poisson equation. The results here will be use-\nful for the estimation of the stray magnetic field with\nlow calculation cost, as well as relatively high reliability\nthan a point-particle approximation [43]. The elliptical-\nshaped ferromagnet generates a larger stray field than\nthe stadium-shaped ferromagnet, when the magnetiza-\ntion points to the easy (long) axis direction. This is due\nto a larger density of the magnetic pole concentrated near\nthe sample edge. The stadium-shaped ferromagnet, on\nthe other hand, generates a larger stray magnetic field\nthan the elliptical-shaped ferromagnet along the in-plane\nhard (short) axis direction because of the large number\nof the magnetic pole generated on the boundary. Re-\ngarding these results, the elliptical-shaped ferromagnet\nwill be suitable when used in, for example, an artificial\nspin ice utilizing a large number of degenerated ground\nstate. This is because, in such a system, the stray mag-\nnetic field generated by the magnetization which is ap-proximately parallel to the easy axis causes a frustration\nand provides functionalities for multibit memory and so\non. The stadium-shaped ferromagnet might be, on the\nother hand, of interest for reservoir computing utilizing\nferromagnetic ensemble. This is because the magnetiza-\ntion direction is forcibly deviated from the easy axis by\nan external magnetic field, and thus, the stray magnetic\nfield pointing in an arbitrary direction affects the mag-\nnetization alignment of the other ferromagnets.\nACKNOWLEDGEMENT\nThe author is grateful to Takehiko Yorozu for his enor-\nmous contribution to this work. The author is also thank-\nful to Hitoshi Kubota for discussion. This work was sup-\nported by a JSPS KAKENHI Grant, Number 20H05655.\nAppendix A: Verification of macrospin assumption\nThe applicability of the macrospin assumption was\nstudied by performing micromagnetic simulation with\nMuMax3 [22]. The elliptical-shaped ferromagnet with\na= 200 nm, b= 75 nm, and h= 20 nm was used for this\npurpose. The mesh number is 200, 75, and 10 in the x,y,\nandzdirection, respectively, where we remind that the\nsample length (width) in the x(y) direction is 2 a(2b),\nand thus, the mesh scale is 2 nm in every direction. The\nexchange stiffness is assumed to be 1 .3×10−6erg/cm.\nFigure 8(a) shows the magnetization alignment after\nrelaxation, where the initial state of the magnetization\nwas set to be uniform in the xdirection. An exter-\nnal magnetic field is absence in this example. In this\ncase, even after the relaxation, the magnetization ap-11\n(a) (b) (c)\nFIG. 8. (Color online) Magnetization alignment of an\nelliptical-shaped ferromagnet, in the absence of an external\nmagnetic field, when the initial state is parallel to (a) xand\n(b)yaxis. (c) Magnetization alignment in the presence of an\nexternal magnetic field of 1 .0 kOe in the ydirection, where\nthe initial state is parallel to the yaxis.\nproximately points to the xdirection uniformly. When\nthe initial state was set to be uniform in the ydirection,\non the other hand, the magnetization alignment after re-\nlaxation becomes nonuniform, as schematically shown in\nFig. 8(b). In this case, the magnetization state is en-\nergetically unstable, and the magnetic moments start tochange their direction to minimize the magnetic energy.\nThe magnetization moments near the edge try to change\ntheir direction along the tangent of the ferromagnet, and\nas a result, magnetic vortexes appear. In this case, the\nmacrospin assumption does not work well. The magnetic\nvortexes will also appear near the edges of the stadium-\nshaped ferromagnet due to its circular shape [30, 31].\nWhen an external magnetic field is applied in the ydi-\nrection, however, a uniform alignment of the magnetiza-\ntion is approximately recovered, as shown in Fig. 8(c),\nwhere the initial state was uniform to the ydirection and\nthe magnitude of the external magnetic field is 1 .0 kOe.\nNote that, for a= 200 nm, b= 75 nm, and h= 20 nm,\nthe demagnetization coefficients [9] are Nx= 0.0406···,\nNy= 0.1549···, and Nz= 0.8045···, and thus, the\nin-plane magnetic anisotropy field with M= 1500 is\n4πM(Ny−Nx) = 2 .153 kOe.\n[1] J. A. Osborn, Demagnetizing Factors of the General El-\nlipsoid, Phys. Rev. 67, 351 (1945).\n[2] R. I. Joseph and E. Schl¨ omann, Demagnetizing Field in\nNonellipsoidal Bodies, J. Appl. Phys. 36, 1579 (1965).\n[3] R. I. Joseph, Ballistic Demagnetizing Factor in Uniformly\nMagnetized Cylinders, J. Appl. Phys. 37, 4639 (1966).\n[4] W. F. 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Kampmeier, 3D. Grützmacher,  \n4E. Schmoranzerova  , 5K. Olejn ík, 5H. Reichlová , 6C. Heiliger,  1C. Denker, 1J. Walowski, 2T. \nKampfrath , 1M. Münzenberg  \n \n1. Institut für Physik, Universität Greifswald, Felix -Hausdorff -Straße 6, 17489 \nGreifswald, Germany  \n2.  Department of Physical C hemistry, Fritz Haber Institute, Faradayweg 4 -6, 14195 \nBerlin, Germany  \n3.  Peter Grünberg Institut (PGI -9) and Jülich -Aachen Research Alliance (JARA -FIT), \nForschungszentrum Jülich, 52425 Jülich, Germany  \n4. Faculty of Mathematics and Physics, Charles Univer sity, Ke Karlovu 3, 121 16  Prague, \nCzech Republic  \n5.  Institute of Physics ASCR v.v.i., Cukrovarnická 10, 162 53 Prague, Czech Republic  \n6. Justus L iebig Univers ität, Gießen, Germany  \n \n \nAbstract  \n \nWe study laser driven spin -current effects at ferromagnet /topological -insulator interfaces  \nby two complementary experimental approaches.  The DC photocurrent is studied in \nferromagnet/ topological -insulator bilayers  with high spatial resolution . Dynamic interface \ncurrents are explored via the emission of terahertz  radiation emitted by  these currents with \nhigh temporal resolution.  From our experiments , we reveal  a lateral and dynamic \ninteraction of the ferromagnet and the topological insulator  inter face. \n \nI. Introduction   \n \nA topological insulator is a bulk electrical insulator with a two -dimensional metallic surface. \nThese films are o f large interest because of spin -momentum locking of the surface states \nwhich allows one to drive spin dynamics at ferromagnet/ topological -insulator  (FM/TI)  \ninterfaces  [1, 2] . The combination of a ferromagnet and a topological insulator enables to \nmanipulate the magnetization of the ferromagnet by spin -orbit torques.  \nRecent results have shown that a surprisingly large  conversion factor of a  charge current into  a \nspin currents seem possible  [3]. The conversion  efficiency is given by the spin Hall (SH) angle  \nwhich is defined  by 𝛩SH=𝑗⃗𝑠𝑝𝑖𝑛 𝑗⃗𝑐ℎ𝑎𝑟𝑔𝑒 ⁄  where J spin and J charg e represent the corresponding \nspin- and charge current densities . While normal metals show efficiencies up to a maximum of \n0.3, the record value for FM/TI  interfaces suggest s that 𝛩SH can amount up to 100  and even \nmore , making these interfaces very interesting for charge -to-spin conversion  in spintronic \ndevices . Current theoretical investigations of these interfaces suggest that the efficiency can be \nonly explained when the interface channel play s a significant role, and the bulk transport channels are suppressed  [4]. Therefore,  granular film s of a topological insulator could  be more \nefficient for spin-orbit -torque -based  switching of the ferromagnet’s magnetization . Second , a \nferromagnet can also be used to confine and control the surface states  [5]. In FM/TI  films , the \nedge s show quantized conductance, and their conduction can be c ontrolled by the magnetic \nstate. The direction of the surface state is, owing to spin-mom entum locking , spin polarized \nand unidirectional  [6] [7] [8] [9]. It has been suggested to use domain walls  to shape complex \ndevices of a topological insulator based spintronics  [10, 11] . Third, the use of a ll-optical \ncontrol of the transient magnetization of topological insulators has  been demonstrated  [12]. \nExploiting the selection rules and the helicity of the light , spin currents can be driven a ll-\noptically by the circular photogalvanic effect. In the following , we aim to combine all -optical \nexcitation of the surface and interface currents  in ferromagnet/ topological -insulator  double \nlayers (Bi 1-xSbx)2Te3/CoFeB and focus on  the composition of around  Sb=50% . We excite the \nferromagnet/ topological -insulator sandwich structure optically and probe  the photo current \nresponse of the electrons for different helicit ies of the exciting laser beam . The photogalvanic \neffects are detected  in two ways. We measure them with  standard  contacts in lateral devices  \nand then , we will detect the transient interface current by the terahertz ( THz) emission \nspectroscopy in a  contact -free manner . The absence of contacts  allows us to also measure the \ntemporal evolution of the currents  and at buried interfaces . \n \nII. Experiment   \n \nThe topological -insulator films were prepared  by molecular -beam epitaxy  [13] and \ncharacterized  by structural characterization: x -ray diffraction measurements , Raman \nscattering, angle -resolved photoemission spectroscopy  (ARPES)  and transport experiments to \nextract their Fermi level and carrier  density  for an Sb content ranging from x = 0 to 1  [14]. \nStrain free films are realized  by van der Waals  epitaxy on  a 70 nm thick  Si-(111) surface  with \nan insul ating layer of oxidized silicon  of 300 nm thickness  top layer  on the  Si wafer . The \nchosen  composition of (Bi 57Sb43)2Te3 films is intrinsic or has very little n -doping . It is found that \nit is as close as possible to the Dirac point . The carrier density is below 3 x 1013 cm-2.  \nTo prepare the FM/TI bilayer s, we used 15 to 160  nm thick  films that  undergo  surface cleaning \nand in-situ Ar -ion etching . The surface layer  composition  is checked  by its XPS spectra during \nthe whole  process . The few-nanometer thick oxidized surface is cleaned  until the pure Te peak \nappears  in the XPS spectra and the oxide peak is removed . The spectra for  Te, as well as for TeO 2 \nis shown  in Fig. 1 a. A careful  5 minute long in -situ annealing at 250°C heal s the  surface defect s \ndue to  etching , but does not change the surface stoichiometry  signi ficantly , as confirmed  by \nARPES . A successful  reconstruct ion of  the Dirac cones and surface state s was demonstrated [15] \nand is presented in Fig. 1c . Subsequently,  a 3 nm thick Co 0.2Fe0.6B0.2 layer  was deposited by \nplasma sputtering in  the same UHV  system  at a rate of 0. 4 Å/s at a base pressure of 10-9 mbar . \nSince the in-situ Ar-ion etching beam is focused and scanned on the substrate, it allows to prepare wedges of the (Bi 0.57Sb0.43)2Te3 film. Finally, a 2.5 nm MgO capping was added for \nprotection .  \n \nA.  Photocurrent microscop y \n \nThe device layout  is presented  in Fig. 2.  It consists of a large  square  (area 200200 m2) of plain  \ntopological insulator film . A smaller square ( 4040 m2) containing the ferromagnetic  layer is \npatterned in the middle. The whole device is contacted by four contact s along the sides and four \nat the  edges , thereby  allow ing for  a measurement of the photo - and thermo voltages along all \ndirections and . In addition , by using the contacts, a current can be driven through the structure . \nThe device was prepared by three -step electron -beam lithography. In the first and the second \nstep, the squares were defined by writing with the electron beam, and a hard etching mask o f Al \n(50 nm) was deposited using lift -off technique. The individual structures were defined by Ar+ ion \nmilling, and the mask was removed after each step. Finally, large Au (90 nm) contacts were \ndefined by optical lithography and lift -off.     \n \nThe experime nt schematic  for the DC photocurrent measurements  is presented in Fig. 2a, \nwhile Fig.  2b shows a large -scale optical reflectivity map of our device . A Gaussian laser  \nbeam (wavelength 785 nm , 1mW ) was focused down to a waist of 3 µm scanned over the \nsample . For e ach point of the map , the photovoltage 𝑉photo  is measured for different \npolarization of the light. By rotating the quarter -wave -plate angle 𝜃 before the sample by 360° , \nthe polarization is change d from linear ly to right -handed  circular ly to linear ly and left -handed  \ncircularly polarized within one circle  of 360° . A modulation frequency of 77 Hz was applied \nby square -type intensity modulation  of the laser beam . The resulting si gnal was analyzed using \nthe relationship  \n𝑉photo =𝐶sin(2𝜃)+𝐿1sin(4𝜃)+𝐿2cos(4𝜃)+𝐷, (1) \ntaken from Ganichev et al. [16] [17]. By fitting, we obtain the parameters D, L 1 , L2 and C  at \neach point  of the laser spot . The coefficient D  correspond s to the  thermovoltage , while C \nquantifies the strength of the  circular photogalvanic voltage  driven by right and left circularly \nphotons in the sample. L1 and L2 correspond to the real part and imaginary part of the linear \nphotogalvanic voltage , in phase and  out of phase by /2 with respect to 𝜃=0. \n \nThe result ing map of D is shown along with an optical reflectivity micrograph in Fig.  3a. To \ninterpret the signals, we emphasize that the laser scanning generates a thermal gradient and a \n(helicity) dependent photocurrent. At each sample position, the laser generates a different \nthermal profile and thermovoltage. For laser excitation s close to the Au contact pad  we find \nlarge values of D, which indicate  an inhomogeneous  thermovoltage driven at the \nAu/SiO/ Si(111) interface from the semiconductor/metal  contact.  A detailed close -up of the 2D map and averaged line profiles above  and at the side  are shown  (top and right side panel) . \nWhen moving the laser spot perpendicular to the contacts no slope indicate constant \nthermovoltages  (right panel) . When moving the laser spot illumination parallel to the contacts  \n(top panel) , we observe a sign change  with different slopes  on the topological insulator \nslightly steeper as on the ferromagnetic element with steps at the edges.   \nThe reflectivity signal is determined by the reflected laser light at the same time. This is \nhelpful  to quantify the local absorption . The reflectivity map shows that the ferromagnetic \nsquare  element  has a higher reflectance , cause d by the ferromagnetic layer and residual \nremaining Al etch-layer on top . Therefore , we expect a smaller  thermoelectric signal  here. \nThis is reflected in the behavior of D,  the thermovoltage levels  off on the ferromagnet element \nand shows  a plateau. On the topological insulator t he thermovoltages  show a larger  slope  \nreverse the sign , while the laser spots move from one contact to  the other , demonstrating that \nwe can successfully create controlled temperature gradients by the laser heating in between the \ncontacts in the topological insulator film . \n \nNext,  we want to study the area at around  the topological insulator and the interface to the \nstructure d FM/TI  element. The resulting map of C, sensitive to the circular chirality of the \nphotogalvanic voltages  extracted is shown in Fig.  4. A schematic representation how this spin \naccumulation is detected by the spin -galvanic is given in Fig. 4. The optical excitation in the \narea of the accumulated spins results in a small , detectable circular photo -galvanic current, \nresulting in a voltage in  the V range. While the size of the detected voltages is  small , the \nabsolute change in signal is quite large , which points indirectly to a high degree of spin \npolarization.  Again,  a detailed close -up of the 2D map and averaged line profiles above and at \nthe side are shown (top and right side panel). The C parameter, shows a  completely different \nbehavior.  This shows that it allows to discriminate between thermal effects , and effects driven \nby the circular photogalvanic effect . Both effects can be well separated .  \n \nIn the area of the free topological insulator film, the val ue of C stays constant. In  the \nproximity of  the edge s of the ferromagnetic structure within a short distance below a few  \nmicrometer s, significant changes are found. The circular photo -galvanic effect  for ri ght and \nleft shows  two peaks  for the vertical average over a line section . One positive and one \nnegat ive peak  is found. The negative an d positive peak  occur s perpendicu lar to the \nthermocurrent flow , e.g. perpendicular to the thermal gradient , next to the FM/TI element \nedges . They have  the signature of a spin-Hall effect , arising from a spin -to-charge conversion \nby spin -orbit scattering  [18], driven by the thermal gradient  (spin-Nernst effect ). In addition,  \nwe used a  +31mT or -31mT magnetic field applied in -plane and along the temperature \ngradient  as a control  of the magnetization direction . While the right panel remains unchanged, \nwe observe now a splitting of the circular photogalvanic voltage  parallel to the thermocurrent \nflow , e.g. along  the thermal gradient . This splitting can be controlled by the field di rection . \nThe voltage signal switches with the +31mT or -31mT magnetic field  rotating the magnetization of the  TI/FM element. The signature of the ferromagne t top ological insulator \ninteraction can only be understood by a significant modification of the surface states . On the \nright side of Fig. 4, a schematic representation is given of the band structure of the topological \ninsulators surface state (TSS) is given as inset. Schematically we show magnetic interfaces or \ndopants interact with the Tis surface b ands opening a gap, and shifting the ban ds along the k -\naxis. The spin -currents arising from the spin -Nernst effect appear to be blocked at the edges of \nthe ferromagnetic element on top of the topological insulator film . Our spatial mapping allow s \nto determine  the length scale of the accumu lation at the interface of two to three  micrometer. \nWe conclude that in this case the ferromagnetic structure acts in a similar way as if th e \ntopological insulator would have  finite edges . This results in a spin -accu mulation  at the edges \nof the ferromagnetic element . Two main effects have been discussed at these interfaces, by \ncalculation of the interface band structure in presence of the ferromagnetic layer on top of the \nTSS by density functional theory and by developing model Hamiltonians of the effect of the \nexchange  interaction . A spin Hamiltonian developed by Kim et al.  [27] is used in the \nfollowing ,    \n \n𝐻𝑇𝑆𝑆=𝑣𝐹(−𝑖ℏ𝛁×𝒛)∙𝝈− 𝐽(𝑚𝑥𝜎𝑥)+𝑚𝑦𝜎𝑦)− 𝐽⊥(𝑚𝑧𝜎𝑧),                     (2) \n \nwhere 𝑣𝐹 is the Fermi velocity , 𝝈 is the spin operator ( Pauli spin matrices ) of the conduction \nelectrons , 𝑚𝑖 denotes the magnetic moment  of the magnet, and J is a coupling parameter. This \nresults in the following modification of the band structure:  \n \n𝐸±=±√(𝑝𝑥−𝐽𝑚𝑦)2+(𝑝𝑦−𝐽𝑚𝑥)2+ 𝐽 (𝑝𝑥−𝐽𝑚𝑦)2   ,𝒑=ℏ𝑣𝐹𝒌          (3)    \n \nIn this equation , we observe  two effects: (i) a band splitting at the Dirac crossing point, \nproportional to J  and (ii) a lateral shift of momentum  depending on the direction of M. We \nhave in our case a magnetization that is in -plane along ±x direction. W e observe two different \neffect s along the x and y direction of the edge of the ferromagnet . A band splitting could result \nin a channel that blocks the spin -transport and correspond s to the finding of a spin \naccumulation that we observed located at the edge of topological insulator films  with finite \nedges along the y direction perpendicular to M. This effect will not be affected by the \nmagnetization’s sign but only by its  absolute va lue. In the case of the momentum shift of k, \nwe expect a direction magnetization M dependent component . In the configuration that the \neffect of the momentum shift dominates, this will result in and increase and decrease for the \ntwo spin directions respectively and thus a spin accumulation that will change its sigh as we \nobserve in the cross section  of the photogalvanic signal along the x direction indicating a \nsplitting spin o ccupation before and after the ferromagnetic structure that changes its sign. It \ndemonstrates the capability to switch the spin -occupation in the topological insulator via the magnetization of a micron or nanostuc tured ferromagnetic element in proximity  to a \ntopological insulator . \n \nB. THz emission spectroscopy  \n \nThe question remains , if below the ferromagnet , in proximity to the topological insulator to the \nferromagnetic film , currents are  flowing, or if the spin -currents below the ferromagnet are \nblocked completely as one might suspect at first glance . To detect an optically driven transfer \nof spin polarization from the ferromagnet to the topological insulator, we performed \nphotocurrent experiments with ultrashort optical pulses (duration ~10  fs, wavelength 800  nm) \nas schematically shown in Fig.  5. Any optically induced spin current flowing from the \nferromagnet to the topological insulator [28,29] will be converted into an in -plane charge \ncurrent by spin -to-charge -current  conversion  (S2C) . As the resulting current burst is ultrashort, \nit will emit electromagnetic radiation with frequencies extending into the terahertz (THz) \nrange.  The transient THz field can be measured by ele ctro-optic sampling . \n \nTypical electro -optic waveforms of emitted THz pulses  are shown in  Fig. 5 a. When we reverse \nthe direction of in -plane magnetization 𝑴 of the FM layer by an external magnetic field, we \nobserve an almost complete reversal of the THz signal. By taking the difference of the two \nwaveforms, we extract the THz signal component linear (odd) in the magnetization.  Note that \nthe THz emission signal can also  arise from the magnetic -dipole radiation that is emitted by \nthe optically quenched magn etization of the FM layer [ 33]. For this purpose, we repeated the \nexperiment with a single FM layer and found that the odd-in-𝑴 signal component was one \norder of magnitude smaller than from the TI/ FN stack  (not shown) . This observation strongly \nsuggests that the THz signal from the TI/ FN stack indeed arises from optically induced spin \ntransport [28,29] across the FM -TI interface  in conju nction with spin-to-charge -current \nconversion (see Fig.  5).  \nOur results indicate  that the interface  between the FM and the TI is “active” . A spin curr ent \ncan be injected and converted into a charge current . A comparison to the emission from FM|Pt \nlayers shows, h owever , that  at THz frequencies , our FM/ TI stacks are not yet as efficient THz \nemitter s as heav y-metal based stacks such as FM/ Pt. \nIt is not yet clear where and by which microscopic mechanism spin-to-charge -current \nconversion  occurs. Important conversion  mechanisms include the inverse spin Hall effect \n(ISHE) in the bulk of the TI and/or FM and at the interface through the inverse Rashba \nEdelstein effect (IREE). In an attempt to address this question, we varied the thickness of the \nTI by using a wedge -like TI film. The resulting THz emission amplitude as a function of the \nTI thickness is displayed in Fig.5 . We observe that the THz signal amplitude saturates for a TI \nfilm thickness above 10  nm. This phenomena is also known for CoFeB /Pt layers and can arise \nfrom (i)  a decreased absorptance of the FM film with decreasing TI thickness and (ii)  the back  \nreflection of the spin current at the TI-substrate interface [ 34].  If scenario  (ii) were dominant, the signal drop below a TI thickness of 10  nm would imply a \nspin relaxation length of 10  nm in the TI. This value is much larger than the values seen for \nheavy metals (~1  nm), Cu (4  nm) [ 35] and ferromagnets (~1  nm) [36]. We , therefore, consider \nscenario  (ii) rather unlikely. To tes t scenario  (i), we conducted c alculations of the pump -pulse \nenergy deposited in the FM layer by using a transfer -matrix approach and the optical constants \nof the TI and FM at the pump wavelength . The calculations (not shown) indeed confirm  that \nthe drop of  the THz signal below a TI thickness of 10  nm can be well explained by the TI -\nthickness -dependent pump ab sorptance.  Therefore, our thickness -dependent measurements \nstill leave the possibility open that spin-to-charge -current conversion  can occur in the bul k of \nboth the FM and the TI layer as well as at the interface in -between.   \nIt will be interesting for the future, in how far concepts of using the FM magnetization \ndirection to confine edge currents inside the topological insulator . As suggest ed by the Tokura \ngroup [ 7], this apporach  would provide access to the shaping of complex spin -current devices \nin ferromagnet/topological insulator hybrid systems.  Indeed, recently, 2D materials have \nsuccessfully been exploited as spin -filter efficient materials  [30-32].  \n \nIII. Summary   \n \nWe presented investigations of the circular photo -galvanic currents for intrinsic doping (Bi 1-\nxSbx)2Te3 with x = 43% CoFeB bilayers  patterns of micrometer dimens ions that are investigate \nby scanning optical excitation using circular polarized light and detecting the thermo - and \ncircular photo -galvanic contributions. We show that a detailed separation of contacts and films i s \nnecessary. Close to the ferromagnet/ topological insulator structure , we observed a spin  \naccumulation that can be compared to the spin  accumulation effects observed at a topological \ninsulator boar der. The spin accumulation occurs on a length scale of below 2 to 3 µm. We \ninterpret this signal as a modification of the circular photocurrent spin current by the \nthermo current in the topological insulator st ructure. In addition , by using THz emission \nspectroscopy, we demonstrated that the interface is “active” and a significant spin -current flow \nacross the TI -FM interface can be driven by laser excitation . Additional  investigations have to \nfollow to clarify whether the spin-to-charge -current conversion  predominantly  proceeds at the  \ninterface or in the bulk . Our observation may provide new methods to structure and control the \nconfinement of the spin -flow in the topological insulator . The spin  accumulation  can be \nmodified on micrometer length scales. This shows that by engineering the ferromagn et, the \nlateral spin  polarization in the topological insulator can be modified  [19] and that we have \nactive interfaces in our devices that allow a control of the spin -current via the magnetizati on of \nthe top ferromagnetic layer.   \n  Figures:  \n \n \nFigure 1: Etching , surface  cleaning and  characterization of a (Bi 0.57Sb0.43)2Te3 film. a) Lateral \n(in)homogeneity , b) X-ray photoemission spectroscopy (XPS) during surface preparation \nusing Ar+ -ion beams  and c) ARPES data ( APRES data from [ 15]). \n \n  \n \n \n \nFigure 2: a) Schematic of a f erromagnetic CoFeB element (40 m  40 m) on to p of the  200 \nm  200 m topological insulator (Bi 0.57Sb0.43)2Te3 film along with driving laser beam and \ncontacts for photocurrent analysis . By changing the helicity of the laser beam polarization, we \ncan separate various contributions to the photo current at each illuminated point, namely the \nthermoelectric and light -helicity -dependent photo -galvanic contributions. b) S canning \nreflectivity micrograph of the device.  \n  \n \nFigure 3 : Details of the ferromagnetic (FM) film structure on top of the topological insulator \nfilm. a) Optical r eflectivity map recorded simultaneously to the b) thermovoltage map. In \naddition to the color map , the averaged cross sections are shown perpendicular and parallel to \nthe contact pads.  \n  \n \n \n \nFigure 4: a) Value of the circular photogalvanic effect C. Cross sections are plotted for + -30 \nmT applied field.  b) Schematics of a control of surface curre nt in the topological insulator  (FM: \nferromagnet, TI: topological insulator) . The Dirac cone can be opened, depending on the \nmagnetizations orientation or the shifted. The shades area shows the difference in chirality \ndependent circular galvanic current for + - B field and the Nernst eff ect at the edges of the \nstructure respectively.  \n \n  \n  \n \n  \n \nFigure 5: THz emission spectroscopy of ultrafast interfacial spin transport. a) Concept schematic. \nAn ultrashort optical laser pulse induces spin transport between a ferromagnet ic (CoFeB) and \ntopological insulator (TI , (Bi 0.57Sb0.43)2Te3) thin film . Due to spin -to-charge current conversion, \nan ultrafast in -plane charge current results that emits a detectable THz electromagnetic pulse. \nb) Typical waveforms of THz electro -optic signals. They reverse almost completely when the in -\nplane magnetization 𝑴 of the FM layer is reversed. c) Amplitude of the THz emission signal as a \nfunction of the thickness of the TI layer.  \n  \n \n \n[1]  H. Wang, J. Kally, J. S. Lee, T. Liu, H. Chang, D. R. Hickey, K. A. Mkhoyan, M. Wu, A. \nRichardella and N. 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Meanwhile, for a Ni 80 Fe 20  film, the EMF due to the inverse spin8Hall effect in the \nNi 80 Fe 20  film itself under the FMR is mainly generated. Thi s tendency is qualitatively explained  \n 2with differences of the spin polarization, the spin  Hall conductivity, the anomalous Hall \nconductivity, the magnetization saturation, and the  resistivity of the FM films.  \n \na) E8mail: shikoh@elec.eng.osaka8cu.ac.jp \n  \n 3     In spintronics, recently, the spin8pumping wit h the ferromagnetic resonance (FMR) and \nthe inverse spin8Hall effect (ISHE) have become pow erful techniques to generate a spin current \nand to detect the spin current, respectively. 1812  First, those techniques were used for spin \ninjection from a ferromagnetic metal Ni 81 Fe 19  as a spin injector into a nonmagnetic Pt layer \nworking as a spin detector, by using the Ni 81 Fe 19 /Pt bi8layer structure samples.1 In the spin \ninjection by using the spin8pumping with the FMR, t he conductance mismatch problem between \nthe ferromagnetic material and the target8material,  which causes lowering the spin injection \nefficiency in the case of an electrical spin inject ion method,13,14  is negligible. 4,5,7,8,10,12  Therefore, \nwhile those techniques are very fundamental, those apply to investigate the spin transport \nproperty of various materials, for example, by usin g the ferromagnetic metal (FM) \n/target8material/non8magnetic metal (NM) junction s tructure samples.587,12  Meanwhile, it was \ndiscovered that the electromotive force (EMF) was g enerated in a single layer Ni 80 Fe 20  film \nitself under the FMR.15  The suggested EMF generation mechanism in the sing le8layer Ni 80 Fe 20  \nfilm itself under the FMR is as follows; a spin cur rent is generated due to the magnetic \ninhomogeneity of the Ni 80 Fe 20  film under the FMR condition and converted to a ch arge current \nwith the spin8orbit interaction of Ni 80 Fe 20  film, that is, the ISHE in the Ni 80 Fe 20  film.15  \nPreviously, by using the Y 3Fe 5O12 (YIG)/ferromagnet “bi8layer” structure samples, the  ISHE \ngenerated in 3d8transition ferromagnetic metal film s of Fe, Co, Ni 80 Fe 20  and Ni under the FMR  \n 4of the YIG was observed,9,11  while the EMF generated in a single8layer ferromag netic metal film \nitself under the FMR is not investigated except for  the Ni 80 Fe 20  film.15  If those EMF generation \nphenomena by using the spin8pump with the FMR of a FM material were applied for practical \nuse, the single8layer structure is better than any other multi8layer structures, in terms of the \nmaterial saving and easiness of manufacturing devic es. In this study, the EMF properties \ngenerated in single8layer 3d8transition FM films th emselves under the respective FMR were \ninvestigated. \n \n     Our sample structure and experimental set up a re illustrated in Figure 1. On a \nthermally8oxidized silicon substrate, a ferromagnet ic metal (FM = Fe, Co, and Ni 80 Fe 20 ) was \ndeposited to a thickness of 25 nm by using a conven tional DC magnetron sputtering system. No \nprotection layer was formed on the FM films, as sim ilar to the previous study.15  After forming \nthe FM films, the sample substrates were cut as a r ectangular shape of 4.0 ×1.5 mm 2, to \nmeasure the physical properties. \n     A sample substrate was set into the microwave TE 011 8mode cavity of an electron spin \nresonance (ESR) system (JEOL, JES8TE300) to excite the FMR of the sample. The microwave \nfrequency f to excite the FMR was 9.45 GHz. The EMF property o f the FM sample was \nmeasured by using a nano8voltmeter (Keithley Instru ments, 2182A). Leading wires to detect the  \n 5output voltage properties from a sample were direct ly attached with silver paste at both ends of \nthe film sample. All of the measurements were perfo rmed at RT. \n \n     Figures 2 (a)8(c) show the FMR spectra of samp les at an external magnetic field \norientation angle θ of 0°, with the microwave power PmW  of 200 mW to excite the respective \nFMR. Figs. 2 (a), (b) and (c) are for an Fe sample,  for a Co sample, and for a Ni 80 Fe 20  sample, \nrespectively. As expected, the FMR was observed in all FM films at the respective FMR field \nHFMR  of 1061 Oe for the Fe, 1094 Oe for the Co and 472. 8 Oe for the Ni 80 Fe 20 . The saturation \nmagnetization MS was estimated to be 1061 emu/cc for Fe, 1094 emu/c c for Co and 472.8 \nemu/cc for Ni 80 Fe 20  by using the following equation: 2,3  \n)π4 (S FMR FMR 0 M H H + =γω ,   (1) \nwhere ω0 (= 2 πf) and γ are respectively, the angular frequency of the mic rowave and the \ngyromagnetic ratio of the respective FMs. \n     Figs. 2 (d)8(f) show the output voltage proper ties at the θ of 0° and 180°, with the PmW  of \n200 mW for the excitation of the respective FMR. Fi gs. 2 (d), (e) and (f) are for an Fe sample, \nfor a Co sample, and for a Ni 80 Fe 20  sample, respectively. For the experimental data (o pen \ncircles), components which do not relate to the mag netic field orientation angles are removed by \nusing the eq.(2):   \n 62180 0V VV−=,   (2) \nwhere the V0 and V180  correspond to the EMFs at the θ of 0° and 180°, respectively. Output \nvoltages were observed in all FM films themselves u nder the respective FMR. The polarity of \noutput voltages is inverted in all FM films against  the magnetization reversal of the respective \nFM films. The output voltages increased with the in crease of PmW . These polarity inversion to \nthe magnetization reversal and PmW  dependence of output voltages are similar to previ ous \nstudies using the spin8pump and ISHE.388,10,12,15  To analyze those output voltage properties, the \ndata in Figs. 2 (d)8(f) were fitted by the followin g equation: \nBG 2 2\nFMRFMR\nAHE 2 2\nFMR2\nISHE) () (2\n) ()( VHHHHVHHV HV +Γ+−−Γ−+Γ+−Γ= , (3) \nwhere the first and second terms of the eq. (3) cor respond to the EMF due to the ISHE and the \nEMF due to the extra ordinary Hall effect in FMs, t hat is, the anomalous Hall effect (AHE) in \nFMs, respectively.1,488,10,12,15  The VISHE and VAHE  indicate the magnitude of the EMF due to the \nISHE and that due to the AHE, respectively. Γ is a damping constant in these fittings. The ISHE \nterm is a Lorenz function, in other words, symmetry  to the HFMR , while the AHE term is \nderivative of a Lorenz function, which is anti8symm etry to the HFMR . VBG is background signals \non experiments, which are independent of the extern al magnetic field. The fitting results are \ndrawn with the solid lines in Figs. 2 (d)8(f). The above analysis indicated that the electromotive \nforces generated in the 3d8transition FM metal film s under their FMR were successfully  \n 7observed. The ISHE properties of the FM films were similar to the studies by using “bi8layer” \nstructure samples. 9,11  \n     Table 1 shows the summary of the analysis, whe re the VISHE , VAHE,  the absolute value of \nthe ratio of | VISHE /VAHE |, MS, the ratio to the VISHE  of Co, and the ratio to the VAHE  of Ni 80 Fe 20  are \ndescribed. From the values of the | VISHE /VAHE |, we can say that the AHE is dominant for the Fe \nand Co samples, while the ISHE is dominant for the Ni 80 Fe 20  sample. This tendency has the \nreproducibility and may be due to the difference of  the spin polarization ( PS), spin8Hall \nconductivity ( σSHE ), anomalous Hall conductivity ( σAHE ), MS, and resistivity ( ρ) of the FM films. \nTable 2 shows the data of the PS,16,17  the σSHE ,15,18  the σAHE ,15,18  and the ρ, where only the ρ \nvalues were experimentally obtained in this study. \nIn the ISHE regime, 1 the spin current density 0\nSj is converted to a charge current density \njC, as follows:  3,5  \nσθ×∝0\nS SHE C j j ,   (4) \nwhere the σ and θSHE  are the spin polarization vector of the spin curre nt and the spin8Hall \nangle, which is a kind of a conversion efficiency f rom the 0\nSj to the jC,. Thus, the absolute \nvalue of the jC can be expressed as follows: \n0\nSHE CSj jθ∝ .  (5) \nThe θSHE  is equal to \nCSHE\nσσ, where the σC is the electrical conductivity of a FM film and  \n 8corresponds to 1/ ρ. Thus, the eq. (5) is rewritten as follows; \n0\nSHE CSj jρσ∝ .  (6) \nIn this study, the ISHE is not observed as a charge  current but as an EMF via the sample \nresistance. Under an assumption that the 0\nSj is approximately proportional to the PS in a FM \nfilm, the absolute value of VISHE  is expressed as follows;  \nSPlVSHES2\nISHEσρ∝ ,  (7) \nwhere the l is the length of an FM sample (4 mm in this study) , and the S is the sectional area of \nthe FM sample (1.5 mm × 25 nm). Using the values in  the Table 2, the VISHE  values of Fe, Co, \nand Ni 80 Fe 20  samples to the VISHE  for a Co sample ratio is estimated to be Fe : Co :  Ni 80 Fe 20  = 8 : \n1 : 6, while the experimentally obtained data were Fe : Co : Ni 80 Fe 20  = 36 : 1 : 13 as shown in \nthe Table 1. Those were qualitatively consistent, a lthough the quantitative consistency lacks. \n     The VAHE  in FM films is simply described as follows:19 \nCS AHE IM V∝ ,  (8) \nwhere the IC is a charge current in an FM film. In this study, the IC is generated due to the ISHE \nin FMs. Therefore, the eq. (8) can be rewritten as follows; \nρISHE\nAHEVMVS∝ .    (9) \nSimilarly to the VISHE , using the values in the Table 2 for parameters in  the eq.(9), the VAHE  \nvalues to the VAHE  for a Ni 80 Fe 20  sample ratio is estimated to be Fe : Co : Ni 80 Fe 20  = 5: 1.1 : 1,  \n 9while the values ratio on the experiments is Fe : C o : Ni 80 Fe 20  = 134 : 11 : 1 as shown in the \nTable 1. The tendency of the AHE among the Fe, Co, and Ni 80 Fe 20  samples was also \nqualitatively consistent between the above consider ation and our experiments. The \ndiscrepancies between the above estimation and expe riments for both the VISHE  and VAHE  may \ncome from the difference of the magneto8crystalline  anisotropy of the FM films. In general, it is \nnegligible for Ni 80 Fe 20 , while it strongly affects to the magnetic propert ies for Fe and Co films. \nThe MS values were estimated by using the eq. (1), which is established for a uniform \nmagnetization rotation mode and not considered the strong correlation between the neighbor \nspins. The way how to take the magneto8crystalline anisotropy of FM films into account must \nbe found. Also, the polarities of the respective EM Fs have not considered in this study, yet. \nHowever, the polarities of the output voltages are different among the researches.9,11  Because of \nthe lack of amount of the related studies, the disc ussion about the polarity of output voltages is a \nnext issue. For further investigation, other 3d 8transition FM films with different 3d 8electron \nnumbers are tested. \nFinally, we compared the VISHE  value of the Ni 80 Fe 20  sample in this study with the VISHE  of \nsome Ni8Fe alloy/NM multi8layer samples in previous  studies.183 In this study, the VISHE  was \nestimated to be 14.7 µV for a single layer Ni 80 Fe 20 film sample, while the VISHE  was estimated to \nbe 10 µV ~ 30 µV for “bi8layer” Ni 80 Fe 20 /Pt or Ni 80 Fe 20 /Pd samples.183 Thus, it was indicated  \n 10 that the single FM layer structure under the FMR ge nerates high enough EMF compared to the \nFM/NM multi8layer structure. That is, we successful ly demonstrated a better method to obtain \nthe EMF under the FMR than using any other multi8la yer structures, which has a merit of the \nmaterial saving, and easiness of manufacturing devi ces.  \n \nThe EMF properties generated in Fe, Co, and Ni 80 Fe 20  films themselves under their FMR \nwere investigated. For Fe and Co films, the EMF due  to the AHE was dominantly generated \nunder their FMR, while for Ni 80 Fe 20  films, the EMF due to the ISHE in the Ni 80 Fe 20  film itself \nunder the FMR was mainly generated. This tendency w as qualitatively explained with the PS, \nthe σSHE , the σAHE , the MS, and the ρ of the FM films. \n \n[Acknowledgement] \n     This research was partly supported by a Grant8 in8Aid from the Japan Society for the \nPromotion of Science (JSPS) for Scientific Research  (B) (26286039). \n  \n 11 References \n1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, App l. Phys. Lett. 88 , 182509 (2006). \n2K. Ando, Y . Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, \nPhys. Rev. B 78 , 014413 (2008). \n3K. Ando, and E. Saitoh, J. Appl. Phys. 108 , 113925 (2010). \n4K. Ando, and E. Saitoh, Nat. Commun. 3, 629 (2012). \n5E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi, Phys. Rev. Lett. 110 , \n127201 (2013). \n6Y . Kitamura, E. Shikoh, Y . Ando, T. Shinjo, and M. Shiraishi, Sci. Rep. 3, 1739 (2013) \n7Z. Tang, E. Shikoh, H. Ago, K. Kawahara, Y . Ando, T . Shinjo, and M. Shiraishi, Phys. Rev. B \n87 , 140101 (R) (2013). \n8J.C. Rojas8Sánchez, M. Cubukcu, A. Jain, C. Vergnau d, C. Portemont, C. Ducruet, A. Marty, L. \nVila, J.8P. Attané, E. Augendre, G. Desfonds, S. Ga mbarelli, G. Jaffrès, J.8M. George, and M. \nJamet, Phys. Rev. B 88 , 064403 (2013). \n9B.F. Miao, S.Y . Huang, D. Qu, and C.L. Chien, Phys.  Rev. Lett. 111 , 066602 (2013). \n10 Y . Ando, K. Ichiba, S. Yamada, E. Shikoh, T. Shinjo , K. Hamaya, and M. Shiraishi, Phys. Rev. \nB 88 , 140406 (R) (2013). \n11 H. Wang, C. Du, P.C. Hammel, and F. Yang, Appl. Phy s. Lett. 104 , 202405 (2014).  \n 12 12 Y . Tani, Y . Teki, and E. Shikoh, Appl. Phys. Lett. 107 , 242406 (2015). \n13 G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip,  and B.J. van Wees, Phys. Rev. B 62 , \nR4790 (2000). \n14 A. Fert, and H. Jaffrès, Phys. Rev. B 64 , 184420 (2001). \n15 A. Tsukahara, Y . Ando, Y . Kitamura, H. Emoto, E. Sh ikoh, M. P. Delmo, T. Shinjo, and M. \nShiraishi, Phys. Rev. B 89 , 235317 (2014). \n16P.M. Tedrow, and R. Meservey, Phys. Rev. B 7, 318 (1973). \n17E. Villamor, M. Isasa, L.E. Hueso, and F. Casanova,  Phys. Rev. B 88 , 184411 (2013). \n18T. Naito, D.S. Hirashima, and H. Kontani, Phys. Rev . B 78 , 014413 (2008). \n19R. Karplus, and J. M. Luttinger, Phys. Rev. 95 , 1154 (1954). \n \n  \n 13 Figure captions: \nFig. 1. (Color online) A schematic illustration of our single ferromagnetic metal (FM) layer \nsample and experimental set up. The dimensions of t he FM (Fe, Co and Ni 80 Fe 20 ) layer are 1.5 \nmm × 4.0 mm and the thickness is 25 nm. Two electro des are attached on both ends of the FM \nfilm using silver paste.  \n \nFig. 2. (Color online) (a)8(c) FMR spectra of (a) a n Fe sample, (b) a Co sample, and (c) a \nNi 80 Fe 20  sample under the microwave power of 200 mW. The I is the microwave absorption \nintensity. (d)8(f) Static magnetic field H dependence of the electromotive force (EMF), V, for θ \n= 0° (red line and circles) and 180° (blue line and c ircles). (d), (e) and (f) are for an Fe sample, \nfor a Co sample, and for a Ni 80 Fe 20  sample, respectively.  \n \nTable 1. The analysis results for our Fe, Co, and N i 80 Fe 20  samples. \n \nTable 2. The parameters of Fe, Co, and Ni 80 Fe 20  samples. Only ρ values are obtained in this \nstudy. \n  \n 14 \n \nK. Kanagawa, et al.: FIG. 1.  \n 15 \n \n \nK. Kanagawa, et al.: FIG. 2. \n \n \n \n \n \n \n \n \n  \n 16 K. Kanagawa, et al.: Table 1. \n  Fe Co Ni 80 Fe 20  \nVISHE  (µV)  34.0  80.939  12.4  \nVAHE  (µV) 873.1  85.85  0.544  \nThe ratio to the VISHE  of Co 36  1 13  \nThe ratio to the VAHE  of Ni 80 Fe 20  134  11  1 \n|VISHE /VAHE | 0.466  0.161  22.8  \nMS (emu/cc)  1061  1094  472.8   \n 17  \nK. Kanagawa, et al.: Table 2. \n  Fe  Co  Ni80 Fe 20  \nPS 16,17 0.44  0.34  0.38  \nσSHE  (Ω 81cm 81) 15,18  400  200  133  \nρ (Ωcm) (experimental data) 7.2×10 85 4.2×10 85 12×10 85 \nσAHE  (Ω 81cm 81) 15,18 806  341  73  "    },    {        "title": "0903.4945v1.Magnetic_structure_and_charge_ordering_in_Fe3BO5_ludwigite.pdf",        "content": "arXiv:0903.4945v1  [cond-mat.str-el]  28 Mar 2009Magnetic structure and charge ordering in Fe 3BO5ludwigite\nP. Bordet1,∗and E. Suard2\n1Institut NEEL\nCNRS & UJF, BP 166, 38042 Grenoble Cedex 9, France\n2Institut Laue Langevin\nBP 156, 38042 Grenoble Cedex 9, France\nThe crystal and magnetic structures of the three-leg ladder compound Fe 3BO5have been inves-\ntigated by single crystal x-ray diffraction and neutron powd er diffraction. Fe3BO5contains two\ntypes of three-leg spin ladders. It shows a charge ordering t ransition at 283 K, an antiferromagnetic\ntransition at 112 K, ferromagnetism below 70 K and a weak ferr omagnetic behavior below 40K. The\nx-ray data reveal a smooth charge ordering and an incomplete charge localization down to 110K.\nBelow the first magnetic transition, the first type of ladders orders as ferromagnetically coupled an-\ntiferromagnetic chains, while below 70K the second type of l adders orders as antiferromagnetically\ncoupled ferromagnetic chains.\nPACS numbers: 75.25.+z, 71.27.+a\nKeywords: charge ordering, spin ladder, magnetic structur e, neutron diffraction\nI. INTRODUCTION\nThe so-called homometallic ludwigite Fe 3BO5is a\nmixed valence low dimensional semiconductor which\nshows a number of exciting electronic and magnetic\nproperties1,2. The general chemical formula for ludwig-\nites isM2+\n2M3+O2BO3whereM2+orM3+are 3d tran-\nsition metal ions. The M2+andM3+ions are located\nat the centers of edge sharing oxygen octahedra forming\nzig-zag walls. The crystal structure of the Fe 3BO5high\ntemperature phase is depicted in Figure 1, where the\natom labeling is shown. The symmetry is orthorhom-\nbic, space group Pbam, with a=9.462 ˚A, b=12.308 ˚A,\nc=3.075˚A. TheFe2+andFe3+cations occupy four dis-\ntinct metal sites. The quasi-two dimensional structure\ncan be viewed as resulting from the presence of two types\nof three-leg spin ladders sub-units of Fecations. Lad-\nders I are built upon the Fe2 andFe4Fe3+cations\nhaving localized high-spins S=5/2 with one additional\nitinerant electron per rung. Ladders II involve the Fe1\nandFe3Fe2+cationswith S=2localizedspinsaccording\nto M¨ossbauer spectroscopy2,3. Ladders I show a struc-\ntural and charge ordering transition at Tc = 283K such\nthat long and short bonds alternate along the ladder c-\naxis4. This transition is accompanied by an anomaly in\nthe magnetizationand a changeofslope ofthe resistivity.\nThe origin of this transition has been discussed in terms\nof excitonic instability6.Fe3BO5shows an antiferro-\nmagnetic transition at 112K,weakferromagnetismbelow\n70Kand another magnetic transition at 40K where ferro-\nmagnetism disappears1,2. It is of primary importance to\ndetermine the evolution of charge and magnetic ordering\nin the whole temperature range between 300K and 10K\nin order to gain insight into the complex physical behav-\nior of this material. Previous x-ray diffraction studies4,5\nonly provide structural information at 300K, 144K and\n15K.Magneticstructureinvestigationat5Kwasreported\nby Attfield et al.9, but nothing is known about mag-\nnetic orderings at intermediate temperatures. Here wepresent the results of a x-ray single crystal diffraction ex-\nperiment and of a neutron powder diffraction experiment\nboth as function of temperature, aiming at providing de-\ntailed information about the charge and spin behavior of\nthis complex and originalcompound. Preliminaryresults\nwere given in refs.10,11.\nII. EXPERIMENTAL\nSingle crystal and powder sample used for this work\nwere synthesized and characterized according to ref.2.\nA. Single crystal x-ray diffraction\nIn order to investigate in close detail the charge order-\ning occurring at 283K, we performed a single crystal x-\nraydiffractionexperimentasafunctionoftemperatureto\nfollow the evolution of the structural parameters across\nthe transition and in a large temperature range. The\nexperiment was carried out with a Nonius KappaCCD\ndiffractometer equipped with graphite monochromatized\nAgKαradiation( λ=0.5608˚A).The temperature wascon-\ntrolled with an Oxford Cryosystem nitrogen gas blower.\nA single crystal sample was ground to a sphere of 0.1\nmm radius. 180˚omega scans with 2˚frame size were\nused to collect the data up to sin θ/λ=0.9, with a 35 mm\nsample to detector distance, every 20K between 320K\nand 110K on cooling. CCD data were processed, aver-\nagedandcorrectedforabsorptionwith the Denzo-SMN12\nand Maxus14packages, refinements and structure ana-\nlyzes were performed using the Jana2000 software15. Al-\nthough the transition was reported to appear at 283K16,\nwe could detect superstructure spots characteristic of a\ndoubling of the c axis parameter already on the 290K\ndata. These data and those at lower temperature were\ntherefore collected for the refinement on the basis of this\ndoublecell, while the 300K,310Kand320Kdatawerere-2\nfined with the high temperature phase unit cell. For the\nlow temperature phase, each experiment leads to about\n2450 unique reflections of which approximately one half\nwere superstructure reflections (l odd), and ≈1750 were\nconsidered as observed, i.e. had I >3σ(I) (where Iand\nσ(I) denote a Bragg reflection intensity and its standard\ndeviation, respectively). Forthe high temperaturephase,\nabout 1260 unique reflections were obtained, of which\nabout 1050 were observed. The structure of ludwigite\nhas space group Pbamwith cell parameters a=9.465 ˚A,\nb=12.310 ˚A, c=3.077 ˚Aat 320K. As stated above, all su-\nperstructure spots appearing below 290K could be in-\ndexed by doubling the c axis parameter. The systematic\nextinction condition lead to space group Pbnm, as re-\nported by Mir et al.16. The refined atomic coordinates\nat 320K and 110K are shown in Table I. It can be\nT=320K\nAtomPos. x y z\nFe12a 0 0 0\nFe22d1/2 0 1/2\nFe34g0.00029(3) 0.27433(3) 0\nFe44h0.74436(3) 0.38746(3) 1/2\nB4h0.2687(3) 0.3617(2) 1/2\nO14h0.8431(2) 0.0427(1) 1/2\nO24g0.3874(2) 0.0787(1) 0\nO34h0.6229(2) 0.1382(1) 1/2\nO44g0.1130(2) 0.1408(1) 0\nO54h0.8409(2) 0.2360(1) 1/2\nT=110K\nAtomPos. x y z\nFe14a 0 0 0\nFe24c0.51174(3) -0.00266(2) 1/4\nFe38d0.50017(2) 0.22596(2) 0.00257(3)\nFe4a4c0.74924(3) 0.39071(3) 1/4\nFe4b4c0.25988(3) 0.61513(3) 1/4\nBa4c0.2652(3) 0.3629(2) 1/4\nBb4c0.7292(3) 0.6400(2) 1/4\nO1a4c0.8451(2) 0.0437(1) 1/4\nO1b4c0.1585(2) 0.9583(1) 1/4\nO28d0.8880(1) 0.4218(1) -0.0096(2)\nO3a4c0.6255(2) 0.1387(1) 1/4\nO3b4c0.3809(2) 0.8628(1) 1/4\nO48d0.6133(2) 0.3591(1) -0.0025(2)\nO5a4c0.8435(2) 0.2376(1) 1/4\nO5b4c0.1620(2) 0.7650(1) 1/4\nTABLE I: Positional parameters for ludwigite at 320K and\n110K from single crystal x-ray diffraction\nseen that the main consequence of the phase transition\non the structure is that some atomic positions split into\ntwo positions which are no longer equivalent by symme-\ntry in the low temperature phase. It is also noteworthyFIG. 1: Structure of the high temperature phase of Fe 3BO5\nabove 290K projected along the c axis. Light gray octahe-\ndra contain Fe cations with a +2 valence, dark gray ones Fe\ncations with an average valence of +2.66.\nFIG. 2: Structural arrangement of the Fe4-Fe2-Fe4 ladders\nabove and below the structural phase transition in Fe 3BO5.\nThe vertical direction corresponds to the c axis.\nthat among the Fe cations, only the Fe4 cations undergo\nsuch a splitting, which suggests that the transition can\nbe driven by charge ordering in the Fe4-Fe2-Fe4 ladders\nalong which one electron is delocalized at high temper-\nature. This is clearly visible in Figure 2 : above the\ntransition, the Fe2 cations present a strongly anisotropic\natomic displacement parameter (a.d.p.) along the rung,\nwhile below it, this a.d.p. drastically decreases and the\nFe2 position is displaced toward Fe4a or Fe4b in an alter-\nnatewayalongthe c axis. In ordertocharacterizethis ef-\nfect, the ionicvalenceswerecalculatedusingthebond va-\nlence sum (b.v.s.) method17. In Figure 3 are represented\nas function of temperature the evolution of the Fe-Fe dis-\ntances along the rungs of the Fe4-Fe2-Fe4 ladders, the Fe3\nFIG. 3: Charge ordering and Fe cation pair formation in\nFe3BO5at the structural phase transition. Top : Fe-Fe dis-\ntances along rungs of the Fe4-Fe2-Fe4 ladders. Middle : Fe\ncation valences calculated with the b.v.s. method. Bottom\n: Atomic displacement parameters for the Fe2 cation. Error\nbars are smaller than the marker sizes\ncation valences and the Fe2 a.d.p.’s. These parameters\nwere found to display the most prominent modifications\nacrossthetransition. Athighertemperature,theFe2-Fe4\ndistances is ≈2.79˚A; well below the transition, Fe2-Fe4a\n≈2.61˚A, an extremely short distance (the Fe-Fe distance\nin metallic ironis ≈2.48˚A) while Fe2-Fe4bincreasesto ≈\n2.95˚A. At the same time, the valences of Fe4a and Fe4b\nbecome different. Above the transition, the Fe4 cation\nvalence is found to be +2.67, which is the expected value\nif one extra electron is shared in a disordered way among\nthe three Fe3+cations (2 Fe4 and 1 Fe2) which consti-tute the ladder rung. However, the Fe2 cation valence\nis found to be close to +2.5. This low value, together\nwith the large and anisotropic a.d.p. for Fe2 strongly in-\ndicate that this cation is in a disordered position above\nthe transition, very probably dynamic, i.e. the cation\nvibrates between to positions symmetric about the rung\ncenter, coming alternately close to the two Fe4 cations at\nthe rung extremities.\nBelow the transition, ordering occurs and the Fe2 dis-\nplacementstowardtherungextremities(Fe4acations)al-\nternateinazigzagwayalongthecaxis. Atthesametime\nthe Fe2 a.d.p.’s become isotropic and similar to those of\nthe other atoms. The Fe4a valence decreases to ≈+2.53,\nclose to the Fe2 cation valence, while the Fe4b valence\nincreases to ≈+2.86. Clearly, this structural phase tran-\nsition is related to the ordering of charges between Fe4a\nand Fe4b. A schematic picture can be proposed where\nthe Fe2 and Fe4a cations form a pair by sharing the com-\nmon electron which was delocalized over the whole rung\nin the high temperature phase, while the Fe4b valence\nbecomes +3. However, as indicated by the anomalous\nbehavior of the Fe2 cations, the Fe2-Fe4 pairs probably\nalready exist in the high temperature phase but are dy-\nnamically disordered between the two sides of the rung.\nTherefore, the transition can also be viewed as an order-\ning of the pairs or of the electrons which are responsible\nfor their existence. On cooling towards the transition,\nthe decrease of the available thermal energy could be the\ndriving force to induce the localization of the Fe2 cations\n(or Fe2-Fe4 pairs). The zigzag order along the c axis of\nthe Fe2-Fe4 pairs would be a way to minimize the struc-\ntural distortions effects induced by this localization.\nAs can be seen in Figure 3, the phase transition takes\nplace over a quite wide temperature range of more than\n50K before all parameter values are stabilized. This in-\ndicates that a large amount of disorder is still present\ndown to ≈200K. Furthermore, the values at which the\nevolution of the Fe cation valences saturate do not corre-\nspond to complete charge ordering even down to 110K.\nThis will have important consequences for the magnetic\nbehavior of this compound, since the magnetic ordering\nprocess will take place on a context of dynamical charge\ndisorder, at least within the Fe4-Fe2-Fe4 ladders.\nB. Neutron powder diffraction experiment\nInordertoinvestigatetheevolutionofthemagneticor-\ndering for Fe 3BO5, a neutron powder diffraction (NPD)\nexperiment was performed on the D20 instrument of the\nI.L.L. The 5g sample was put in a cylindrical vanadium\ncan inside an orange cryostat. A wavelength of 1.3 ˚Afrom\nthe Cu(200)monochromator reflection was used, which\ncorresponds to the optimized flux configuration of the\ninstrument. The sample was first heated to 320K, that\nis above the structural phase transition, and then cooled\nat an approximately constant rate of 1K/9minutes down\nto 10K while diffractograms were continuously measured4\nFIG. 4: Low angle part of the neutron diffractograms for\nFe3BO5at 125K, 75 and 14K. The two different sets of mag-\nnetic peaks corresponding to the two ordering transitions a re\nclearly visible, as well as magnetic diffuse scattering at th e\nhigher temperatures.\nFIG. 5: Evolutionwith temperature of thediffracted intensi ty\nfor Fe 3BO5at three positions corresponding to background\n(crosses), higher temperature magnetic phase (circles) lo wer\ntemperature magnetic phase (squares).\nevery 10 minutes. Each diffractogram covers a tempera-\nture range of ≈1.1K. The data were analyzed using the\nprogram Fullprof18. The structural phase transition is\nalmost undetectable with these data. The only observ-\nable though very weak superstructure reflections are the\n(1 3 1/2) and (2 2 1/2) at about 23.4˚.\nThe thermal evolution of the low angle part ofthe neu-\ntron diffractograms is shown in Figure 4. The evolution\nof the diffracted intensities at three selected angles are\nshown in Figure 5. The chosen angles correspond to thepositions of two magnetic peaks and of a background\npoint in between them. At high temperatures, a broad\nhump centered at about 14˚is clearly visible in the back-\nground in Figure 4. Its intensity increases steadily with\ndecreasing temperature, as shown in Figure 5. At 115\nK, a first set of magnetic peaks appears, accompanied by\na small decrease of background intensity. At 70K new\nmagnetic peaks set in, and the background decreases to\nits high angle value. In the whole temperature range,\nthe magnetic peaks can be indexed on the basis of the\ncrystallographic unit cell. These observations indicate\nthat the compound undergoes two successive magnetic\nordering transitions on cooling. Since a strong diffuse\nscattering of magnetic origin persists between the two\ntransitions, the higher temperature magnetic order must\nconcern only part of the spin system, the rest remaining\ndisordered down to the second transition. The two mag-\nneticstructuresweresolvedandrefinedusingdataat82K\nand 10K, and their evolutions were followed by sequen-\ntial refinement using Fullprof. The treatments included\na full structural refinement of the crystal structure, with\na single isotropic a.d.p. for each atom type. The peak\nshapes were described with a pseudo-Voigt function with\nreflection widths following the Cagliotti law. The peak\nasymmetry evidenced in figure 4 is essentially due to ax-\nial divergence of the experimental setup. It was taken\ninto account in the Rietveld refinement using the Berar\nempirical model13.\nMagnetic Structure at T=82K\nAtomMxMy Mz\nFe202.3(1) 0\nFe4a 02.3(2) 0\nFe4b 02.4(2) 0\nMagnetic Structure at T=10K\nAtomMxMy Mz\nFe13.3(2) 0 0\nFe203.9(1) 0\nFe34.0(1) 0 0\nFe4a 02.74(7) 0\nFe4b 02.74 0\nTABLE II: Magnetic Structure parameters for ludwig-\nite at 82K (Rp=6.57, Rwp=6.61, Rexp=1.44, Chi2=21.2,\nRBragg=1.8, Rmag=6.6) and 10K (Rp=7.86, Rwp=8.18,\nRexp=1.31, Chi2=39.1, RBragg=2.6, Rmag=5.3).\nThe results of the magnetic structure refinements at\nthese two temperatures are given in Table II. The Ri-\netveld plot of the 10K refinement is shown in Figure 6.\nThe high temperature magnetic structure (HTMS) be-\ntween 70K and 115K was refined to Rmag=6.6%. It in-\nvolves only the Fe2, Fe4a and Fe4b cations from ladder\nI. As shown in Figure 7, it consists in ferromagnetically\ncoupledantiferromagneticchainsrunningalongthecaxis\nwithin each ladder. All moments are aligned along the b5\n 7 14 21 28 35 42 49 56 63 70 77 -40000 -10000 20000 50000 80000 110000 140000 170000\n 2 θ  (ë) Intensity (arb. units) \nFIG. 6: Rietveld plot of the NPD data refinement at 10K for\nFe3BO5. The tick marks correspond to the nuclear (top) and\nmagnetic (bottom) reflection positions.\nFIG. 7: Magnetic structure at 82K (left) and 10K (right) for\nFe3BO5. Only the Fe atoms are shown. At 82K : Fe1 and\nFe3 : white (no moment); Fe2 : gray, Fe4a, Fe4b : black. At\n10K : Fe1 and Fe3 : light gray; Fe2 : dark gray, Fe4a, Fe4b :\nblack\naxis. The refined valuesofthe momentson Fe2, Fe4a and\nFe4b were equal at 2.3(1) µB. During sequential refine-\nment as function of temperature, a more stable behavior\nwas obtained by constraining the moments on Fe4a and\nFe4b to be equal. The highest moment value obtained\njust above the second transition were 2.7(1) µB for Fe2\nand 2.51(7) µB for Fe4a and Fe4b. The low temperature\nmagnetic structure (LTMS)is induced by the ordering of\nladders II. As shown in Figure 7, ladders II areformed by\nantiferromagnetically coupled ferromagnetic chains run-\nning along c, all moments being aligned along the a axis.\nThe evolution of the ordered magnetic moments is de-\npicted in Figure 8. At 10K, the Fe1 and Fe3 moment\nvalues are 3.3(2) µB and 4.0(1) µB, respectively. The spin\narrangement on ladders I remains similar to the HTMS\none, but the transition is accompanied by an increase\nof the Fe2 moments to 3.9(1) µB at 10K, while the Fe4a\nand Fe4b moments stay practically constant at 2.7(1) µBFIG. 8: Evolution with temperature of the magnetic moments\nfor Fe 3BO5from Rietveld refinement of NPD data. The mod-\nuli of Fe4a and Fe4b are constrained to be equal at all tem-\nperatures.\ndown to the lowest temperatures. This arrangement is\nclose to the one reported by Attfield et al., but in their\ncase all moments were considered as equal. If we try\nto refine our data with the same constraint, the refine-\nment becomes clearly worse, with Rmag ≈12% compared\nto 5.3%. We can also obtain slightly better refinements\nby allowing some of the moments to rotate in the (a,b)\nplane. Only Fe4b and Fe3 rotations out of their direction\nalong the b, resp. a axis yield an improvement of Rmag\nby≈1%. However, these positions were quite unstable as\nfunction of temperature. We consider that they are not\nwell demonstrated by the refinements and that the most\nprobable structure is the one described in Figure 7. In\norder to check the consistency of the magnetic structures\ndetermined experimentally with the space group sym-\nmetry, we performed a group theory analysis based on\nthe method of Bertaut19, using the Basireps software20.\nFor thePbnmspace group, there are 8 one-dimensional\nreal irreducible representations for the little group Gk\nassociated to the observed magnetic propagation vector\n(0,0,0). For sites Fe2, Fe4a and Fe4b, the magnetic ar-\nrangement is given by the basis vectors of the represen-\ntation 4 : (u,v,0), (-u,-v,0), (-u,v,0), (u,-v,0), where only\nthe component along the b axis is found non-zero exper-\nimentally. For sites Fe1 and Fe3, the magnetic arrange-\nment is given by the basis vectors of the representation 5\n: (u,v,w), (u,-v,w), (u,-v,-w), (u,v,-w),[(u,v,w), (u,-v,w),\n(u,-v,-w), (u,v,-w)]. This corresponds to a ferromagnetic\ncomponent along the a axis. It could also imply an an-\ntiferromagnetic component in the (b,c) plane which is\nnot observed experimentally. In the case of a magnetic\nstructure with several magnetic sites (here 5 sites cor-\nresponding to the 5 non-equivalent Fe cations), the on-\nset of the magnetic transitions depends on the relative\nstrengths of the intra-site and inter-site interactions. In6\nthe present case, the Fe2, Fe4a, and Fe4b cations order\nfirst with basis vectors belonging to the same represen-\ntation. This means that the inter-site interactions are\ndominant between these sites forming ladder I. The sit-\nuation is similar for the Fe1 and Fe3 cations on ladder\nII which order at lower temperature with the basis vec-\ntor of the same representation (but different from the\none for ladder I). These different transition temperatures\nand representations for to the two ladders indicate that\nthe magnetic interactions between them are weaker than\nthose prevailing within each of them.\nIII. DISCUSSION\nThe single crystal x-ray diffraction analysis down to\n110K indicates in ladder I an electron localization and\nFe2-Fe4a pair formation with a very short Fe-Fe bond\nlength. In view of the calculated bond valences, it seems\nhowever that the electron remains delocalized between\nthe Fe2 and Fe4a cations. The 15K data reported by\nMir et al.5are consistent with this picture. However\nthe presence of a single data point at 15K prevents from\ndetecting a possible further electronic localization corre-\nlated to the magnetic ordering transitions. Our neutron\npowder diffraction data, aimed at solving the magnetic\nstructures, do not have sufficient resolution to tackle this\nquestion and are hindered by the presence of magnetic\npeaks. The present NPD experiment shows that a con-\nsiderable dynamic magnetic scattering is already present\na high temperature down to the 70K transition. This in-\ndicates the existence of strong magnetic correlations at\ntemperatures 4 times higher than the ordering temper-\nature. The magnetic ordering transition at 112K con-\ncerns only the Fe cations in ladders I, while ladders II\nremain disordered and still display very strong magnetic\ncorrelations. This is a direct proof of the very weak mag-\nnetic coupling between the two types of ladders in this\nsystem. In ladders I, the coupling along a rung is ferro-\nmagnetic, with negligible difference of moments between\nthe 3 Fe cations. The rungs are antiferromagnetically\ncoupled along the ladder (and c axis) direction. This\nis in contrast to the theoretical predictions of Whangbo\net al.7based on the Strongly Interacting Spin Units ap-\nproach, who proposed antiferromagnetic coupling in the\nladders rungs. Recently, Vallejo et al.8have proposed an\ninterpretation taking into account the prominent role of\nitinerant electrons in this system, which was able to pre-\ndict the right magnetic structure. This is also supported\nby the relatively moderate values of the resistivity ( ≈10\nΩ.cm at room temperature4) which is only slightly modi-\nfiedbythechargeorderingtransition. Thevalueof2.3 µB\nfor the Fe cation moments is much smaller than what is\nexpected for Fe cations (5 and 4 µB forFe3+andFe2+,\nrespectively). This may be another indication that mag-\nnetic and electronic ordering is not complete even in lad-\nders I in the 70K-112Ktemperature range. Below 70K, a\nnew magnetic order sets in, where the Fe cations of lad-ders II form ferromagnetic chains along the c axis which\nare coupled antiferromagnetically. The magnetic order\non ladders I is not modified, except for an increase of the\nFe2 moments. Examining Figure 7, one can notice that\nthe Fe2 antiferromagnetic chains of ladders I are weakly\nconnected to ladders II through two Fe3 ferromagnetic\nchains. Therefore, this Fe2 moment increase is probably\nnot a consequence of the magnetic ordering of ladders\nII. It might be related to a further ordering of charges\nbetween the Fe2 and Fe4a cations, but as stated above,\nwe do not have any direct proof of this effect yet. The\ndifference of magnetic coupling along the rung of ladders\nI and ladders II probably originates in the strong struc-\ntural difference between the two types of ladders. The\nFe2, Fe4a and Fe4b cations of ladders I are connected\nalong a rung via edge-sharing at distances between 2.6\nand 2.95 ˚A. The Fe1 and Fe3 cations of ladders II are\nconnected via corner sharing at distances above 3.35 ˚A.\nAt 10K, the Fe1 and Fe3 moment values (3.3 and 3.9 µB)\nare reasonable for a Fe2+cation, but those for Fe2, Fe4a\nand Fe4b cations of ladders I are still small compared to\nexpected values. Since there are twice as many Fe3 than\nFe1 cations per cell, the magnetic arrangement leads to\na net moment along the a axis of 18.9 µB per cell (i.e.\n0.79µB per Fe cation). Indeed, the presence of ferromag-\nnetism was demonstrated in this compound below 70K2\nby magnetization measurements. However, the same au-\nthors observed its disappearance below 40K. We do not\ndetect any anomalous change in the NPD diffractograms\naround this temperature. If the magnetic structure is\npreserved, the disappearance of the net magnetic mo-\nment along the a axis would be possible only if the mag-\nnitude of the Fe1 moments becomes twice that of the Fe3\nmoments, which would certainly be detectable with our\ndata. The origin of this magnetization change at 40K as\nto be searched elsewhere.\nIV. SUMMARY\nIn this study, we have used single crystal x-ray diffrac-\ntion and neutron powder diffraction as function of tem-\nperature to investigate the charge and magnetic ordering\nin Fe3BO5ludwigite. Achargeorderingtransitionoccurs\nclose to room temperature. X-ray diffraction indicates\nthatittakesplaceoverawidetemperaturerangebetween\n200Kand300K.Itconsistsinanorder-disordertransition\nwithin one of the two three-leg ladders contained in the\nstructure and is related to the localization of an electron\nto form a pair of iron cations within the ladder’s rung.\nAbove the transition this electron is dynamically delo-\ncalized over the three iron cations forming the rung. To\nminimize the effects of the structural distortion created\nbythiselectronlocalization,the ironcationpairsorderin\na zig-zag way along the ladder’s direction. At lower tem-\nperatures, ludwigite has been reported to undergo two\nsuccessive antiferromagnetic and ferromagnetic ordering\ntransitions at 112K and 70K, respectively. Refinements7\nof neutron powder diffraction data indicate that the first\ntransition is due to magnetic ordering of the ladders\nwhich charge-order around room temperature, forming\nferromagnetically coupled antiferromagnetic chains run-\nning along the ladder axis. At 70K, magnetic order of\nthe second type of ladderssets in. It consists in antiferro-\nmagnetically coupled ferromagnetic chains running along\nthe ladder axis. This ferrimagnetic arrangement remains\nunchanged down to 10K and leads to a net magnetic mo-\nment of≈19µB per cell at low temperature. We observe\nno modification of the NPD diffractograms around 40K,where the ferromagneticcomponent has been reported to\ndisappear.\nAcknowledgments\nTheauthorswouldliketothankV.Simonet, J.Dumas,\nM. Avignon and M. Continentino for fruitful discussions,\nJ. Fernandes for providing us with the samples and T.\nHansen for his help during the D20 experiment.\n∗Electronic address: pierre.bordet@grenoble.cnrs.fr\n1M. A. Continentino, J.C. Fernandes, R.B. Guimaraes, B.\nBoechat, and A. Saguia, in ”Frontiers in Magnetic Mate-\nrials”, Vol. 24, p.385, ed. A.V. Narlikar, Springer Publ.\n(2005),\n2R.B. Guimaraes, M. Mir, J.C. Fernandes, M.A. Con-\ntinentino, H.A. Borges, G. Cernicchiaro, M.B. Fontes,\nD.R.S. Candela, E. Baggio-Saitovitch, Phys. Rev. B60,\n6617 (1999).\n3J. Larrea, D.R. Sanchez, F.J. Litterst, E. Baggio-\nSaitovitch, J.C. Fernandes, R.B. Guimaraes, M.A. Con-\ntinentino, Phys. Rev. B70, 174452 (2004); J. Larrea,\nD.R. Sanchez, F.J. Litterst, E. Baggio-Saitovitch, J. Phys .\nCond. Mat. 13, L949 (2001); J. Larrea, D.R. Sanchez, E.\nBaggio-Saitovitch, J.C. Fernandes, R.B. Guimaraes, M.A.\nContinentino, F.J. Litterst, J. Magn. Mag. Mat. 226-230 ,\n1079 (2001).\n4M. Mir, R.B. Guimaraes, J.C. Fernandes et al., Phys. Rev.\nLett.87, 147201 (2001).\n5M. Mir, J. Janczak, Y.P. Mascarenhas, J. of Appl. Cryst.\n39, 42 (2006)\n6A. Latge, M.A. Continentino, Phys. Rev. B66, 094113\n(2002).\n7M.H. Whangbo, H.J. Koo, J. Dumas, M.A. Continentino,\nInorg. Chem. 41, 2193 (2002).\n8E. Vallejo, M. Avignon, Phys. Rev. Lett. 97, 217203\n(2006); E. Vallejo, M. Avignon, J. Magn. Magn. Mat. 310,\n1130 (2007).9J. Paul Attfield, John F. Clarke and David A. Perkins,\nPhysica B 180 & 181 (1992) 581-584\n10P. Bordet , E. Suard, Acta Cryst. A61, C57 (2005).\n11P. Bordet, B. Anterion, M. Mir, R.B. Guimaraes, J.C. Fer-\nnandes, M.A. Continentino, Acta Cryst. A58 (Suppl) ,\nC363, (2002).\n12Z. Otwinowski and W. Minor In Methods in Enzymology,\n276,edited by C.W. Carter, Jr. & R.M. Sweet pp. 307-326,\n(1997)New York:Academic Press..\n13J.-F. Brar & G. Baldinozzi, J. Appl. Cryst. 26, 128 (1993).\n14S. Mackay, C.J. Gilmore, C. Edwards, N. Stewart & K.\nShankland,(1999). maXusComputerProgram fortheSolu-\ntion and Refinement of Crystal Structures. Bruker Nonius,\nThe Netherlands, MacScience, Japan & The University of\nGlasgow.\n15V. Petricek, M. Dusek & L. Palatinus,(2000). Jana2000.\nThe crystallographic computing system. Institute of\nPhysics, Praha, Czech Republic.\n16M. Mir, R.B. Guimaraes, J. C. Fernandes, M. A. Conti-\nnentino, A. C. Doriguetto, Y. P. Mascarenhas, J. Ellena,\nand E. E. Castellano, R. S. Freitas and L. Ghivelder, Phys.\nRev. Lett. 87, 147201 (2001).\n17N.E. Brese & M. O’Keeffe, Acta Cryst. B47, 192 (1991).\n18J. Rodriguez-Carvajal, Physica B 192, 55 (1993).\n19E. F. Bertaut, Acta Cryst. A24, A24, 217 (1968).\n20J. Rodriguez-Carvajal, program available with the fullpro f\ndistribution"    },    {        "title": "1302.3062v2.Theory_of_ferromagnetism_in_vanadium_oxide_based_perovskites.pdf",        "content": "Theory of ferromagnetism in vanadium-oxide based perovskites\nHung T. Dang1and Andrew J. Millis1\n1Department of Physics, Columbia University, 538 West 120th Street, New York, New York 10027, USA\n(Dated: June 16, 2021)\nThe conditions under which ferromagnetism may occur in transition metal oxides with partially\n\flledt2gshells such as vanadium-based perovskites are studied using a combination of density\nfunctional and single-site dynamical mean \feld methods. For reasonable values of the correla-\ntion strength, rotations of the VO 6octahedra play an important role in enabling ferromagnetism,\nwith ferromagnetism typically occurring for rotations larger than a nonzero critical value. Fer-\nromagnetism is suppressed near the Mott-insulating phase but the phase boundary is otherwise\nonly weakly dependent on carrier concentration. Design rules are suggested for new oxide systems\nexhibiting ferromagnetism.\nPACS numbers: 71.28.+d,75.10.Lp\nI. INTRODUCTION\nTransition metal oxides1are of great interest in con-\ndensed matter physics because they exhibit a rich va-\nriety of exotic phenomena which remain incompletely\nunderstood. While \\late\" transition metal oxides (in-\nvolving Cu or Ni) have been very extensively stud-\nied due to their connection to high- Tcsuperconductiv-\nity and Mn-based compounds have attracted attention\nfor their colossal magnetoresistance, the \\early\" transi-\ntion metal oxides such as vanadium oxides have, despite\nsome important studies,2received less attention in re-\ncent years. However, following the pioneering work of\nOhtomo and Hwang,3early transition metal oxides are\nincreasingly used as components of atomic-scale oxide-\nbased heterostructures.4,5One goal of heterostructure\nresearch is to design materials exhibiting phases not ob-\nserved in bulk.6,7An essential step towards realizing this\ngoal is obtaining a clear understanding of the relation-\nship between physical structure and observed electronic\nphenomena.\nIn this paper we investigate the relationship between\nlattice structure, correlation strength and electronic\nproperties in the context of ferromagnetism in early tran-\nsition metal oxides. Ferromagnetism is a correlated elec-\ntronic property which is both technologically important\nand (because only a spin symmetry and not translation\nor gauge symmetry is broken) more straightforward from\nthe theoretical and computational points of view than\nother phases such as antiferromagnetism or supercon-\nductivity. We choose the early transition metal oxides\nin part because of the intriguing recent report5of ferro-\nmagnetism in superlattice systems involving LaVO 3and\nSrVO 3. The report is of interest because ferromagnetism\nis reported for the superlattice even though no ferromag-\nnetism is observed in bulk solid solutions of the form\nLa1\u0000xSrxVO3. One possible explanation is that the su-\nperlattice enables a di\u000berent crystal structure, more fa-\nvorable to ferromagnetism than that found in the ob-\nserved bulk structures. Understanding whether this ex-\nplanation is viable, and more generally being able to de-sign superlattices with desired magnetic properties, re-\nquires deeper insight into the conditions\nThe conditions under which ferromagnetism may oc-\ncur is a question of long-standing theoretical interest.8{11\nThe development of dynamical mean \feld theory12has\nopened a new avenue of research, but apart from some\npioneering investigations of the Curie temperatures of Fe\nand Ni13the studies have mainly been based on model\nsystems. Vollhardt, Ulmke, and collaborators have stud-\nied the single-band Hubbard model, \fnding that in this\nmodel ferromagnetism occurs at generic carrier concen-\ntrations only when there is a large density of states peak\nat or near the lower band edge.14,15For a \fxed value\nof the Hubbard U, the Curie temperature Tcwas found\nto depend sensitively on the peak position, becoming un-\nmeasurably small as the density of states peak was moved\na small distance away from the lower band edge.16How-\never, many materials of physical and technological inter-\nest involve transition metals with partially \flled degen-\nerate (or nearly degenerate) dlevels, where the Hund's\ninteraction may play an important role. While the im-\nportance of the Hund's interaction in partially \flled d\nlevels has been appreciated for decades, the issue has\nbeen systematically studied only in the case of the Bethe\nlattice17{19in which the density of states has a simple\nsemicircular structure. In this situation large values of\nthe Hund's coupling and correlation strength (the Hub-\nbardU) are required for ferromagnetism.\nIn this paper, we study the conditions under\nwhich ferromagnetism may occur in systems related to\nLa1\u0000xSrxVO3(LSVO). In this family of materials the key\norbitals are vanadium dlevels. These are partly \flled (in\nLa1\u0000xSrxVO3the mean number of delectrons is 2\u0000x)\nand the electrons are subject to a strong Hund's coupling,\nfavoring magnetism. The end-member LaVO 3is classi-\n\fed as a Mott insulator.1It has structural and antiferro-\nmagnetic transitions at around 140K but is insulating to\nhigh temperatures.20,21Our investigations are based on\ndensity functional theory (DFT) calculations and include\nin particular the e\u000bects of GdFeO 3-type rotations of the\nBO6structural motif of the ideal perovskite structure.\nWe treat the many body physics using the single-site dy-arXiv:1302.3062v2  [cond-mat.str-el]  2 May 20132\nFM/metal AFM\nPM/metal(a)\nFM/metal IAFM\nPM/metal(b)tilt angle\n1.5density 2-x2.0FM/metal IAFM\nPM/metal(c)tilt angle tilt angle\nFIG. 1: Three possible magnetic phase diagrams for bulk\nsolid solution La 1\u0000xSrxVO 3as a function of carrier density\n(xaxis) and tilt angle ( yaxis). The phase boundaries are the\nblack curves. (a) Weak correlations: Slater antiferromagnet\natx= 0. (b) Intermediate correlations. (c) Strong correla-\ntions: Mott insulator at x= 0. The notations are: FM for\nferromagnetic, PM for paramagnetic, AFM for antiferromag-\nnetic order, and I for insulating state.\nnamical mean \feld approximation,12which is widely used\nin real-materials many-body physics calculations22,23and\nin particular in studies of ferromagnetism.14,24\nOur main result is an understanding of the dependence\nof ferromagnetism on carrier concentration, octahedral\nrotation and correlation strength. At non-integer elec-\ntron concentration we \fnd that increasing the octahe-\ndral rotation amplitude increases the tendency to fer-\nromagnetism. The physics underlying this result is a\ndensity of states e\u000bect related to that previously noted\nin the one-band Hubbard model.14,24We also \fnd that\nat \fxed rotation amplitude the carrier concentration de-\npendence of ferromagnetism depends on the correlation\nstrength. For weak to moderate correlations (so that at\nthe carrier concentration n= 2 corresponding to LaVO 3\nthe paramagnetic phase is metallic in contradiction to\nexperiment1,20,21) decreasing the carrier concentration\nbelown= 2 weakly decreases the tendency to ferromag-\nnetism. For stronger correlations (so that at the carrier\nconcentration n= 2 corresponding to LaVO 3, the in-\nsulating phase is a Mott insulator), doping away fromLaVO 3strongly increases the tendency towards ferro-\nmagnetism. An intermediate situation may also occur.\nThe three panels of Fig. 1 present schematic representa-\ntions of these three cases.\nWe remark that in the literature, the many-body\nphysics properties of transition metal oxides have been\nmodeled in two ways. One, which we refer to as the \\ d\nonly model,\" treats the frontier orbitals (in the case of the\nvanadium-based materials, antibonding bands derived\nmainly from V t2g-symmetry dorbitals admixed with\noxygenp\u0019states) as a multiorbital Hubbard model. This\napproach is generally accepted1as the relevant descrip-\ntion of the early transition metal perovskites such as the\nsystems we study here. A second approach is based on\natomic-like dorbitals which are subject to on-site corre-\nlations and are hybridized with the full pmanifold. This\napproach is used in the DFT+U and DFT+DMFT22,23\napproaches. It is generally believed to be essential to\na correct description of the late (e.g. Cu-based or Ni-\nbased) transition metal oxides25but has been less widely\nused in the study of early transition metal oxides. We\nconsider both approaches in this paper, \fnding similar\nqualitative conclusions but signi\fcant quantitative dif-\nferences which arise from the richer physics of the \\ pd\nmodel\" situation.\nThe paper has following structure. The model and\nmethods are described in Sec. II. Section III derives the\nmagnetic phase diagram of LaVO 3as a function of carrier\nconcentration, tilt angle and correlation strength using\nthe widely used donly model. In this analysis, most at-\ntention is given to the strongly correlated case presented\nin Fig. 1c. Section IV analyzes the pdmodel. The \f-\nnal Sec. V is the summary and conclusion. Appendices\npresent details of some of the results.\nII. MODEL AND METHODS\nA. Model\nWe study systems derived from SrVO 3and LaVO 3.\nBulk SrVO 3crystallizes in the ideal cubic ABO 3per-\novskite structure.26The crystal structure of LaVO 3may\nbe thought of as a cubic perovskite with an additional\nGdFeO 3-type rotational distortion leading to a tilted\nstructure with symmetry group Pnma20Fig. 2 shows\nthe lattice structure of LaVO 3. The basic structural mo-\ntif of the perovskite structure is the oxygen octahedron.\nIn the GdFeO 3rotated structure there are four inequiva-\nlent octahedra characterized by di\u000berent directions of the\nprincipal axes and by slight ( \u00185%) distortions of the V-\nO bond length. We have found (not shown) that varying\nthe relative magnitudes of the V-O bond lengths over\nthe physical range causes only small changes to the non-\ninteracting DOS or to the DMFT solution. We therefore\nset all V-O distances to d= 1:95\u0017A and focus on the\ne\u000bect of rotation by studying a range of \u0012and\u001e.\nThe octahedral rotations in the GdFeO 3-distorted3\nFIG. 2: (Color online) Representation of ABO3perovskite\nstructure with GdFeO 3-type (octahedral rotation) distortion\ngenerated using the VESTA program.28Large spheres (green\nonline): A atoms (here La or Sr), small spheres (red online):\nO atoms, intermediate size spheres at the centers of the octa-\nhedra (blue online) : B-site atoms (here, V).\nPnma systems can be characterized by two angles, \u0012\nand\u001e,27with corresponding rotation axes ^ n\u0012, ^n\u001eand\nwavevectors ~Q\u0012,~Q\u001echaracterizing the changes in rota-\ntion axis from cell to cell. We choose coordinates such\nthat the rotation axis ^ n\u0012is [110] while ^ n\u001eis [001]. The\ncorresponding wavevectors are ~Q\u0012= (\u0019;\u0019;\u0019 ) and~Q\u001e=\n(\u0019;\u0019; 0). For bulk LaVO 3,\u0012= 11:5\u000eand\u001e= 8:8\u000e.20As\nLa is progressively replaced by Sr in bulk solid solution\nLSVO,\u0012and\u001ego to 0. Appendix A provides an esti-\nmate of the x-dependence of \u0012and\u001ein the actual solid\nsolution.\nWe now turn to the electronic structure. The large\nenergy splitting between the transition metal dbands and\noxygenpbands characteristic of early transition metal\noxides such as La/SrVO 3is generally believed1to justify\na downfolding to a model containing only the frontier\nbands, which in the present case are composed mainly\nof V-derived t2gsymmetrydstates with only a small\nadmixture of oxygen pstates. We refer to this as the\ndonly model. In a subsequent section we present an\nexamination of a more general \\charge transfer\" model\nin which the full p dcomplex is considered.\nThe Hamiltonian of the donly model is\nH=Hkin+Honsite; (1)\nwhereHkindescribes the electron hopping between sites\nandHonsite describes the ddinteractions, which we take\nto be on-site.The kinetic Hamiltonian has the quadratic form\nHkin=X\nk;\u000b;\f;\u001bH\u000b\f\nband(k)cy\nk\u000b\u001bck\f\u001b; (2)\nwherecy\nk\u000b\u001bandck\f\u001bare electron creation and annihila-\ntion operators in reciprocal space with wavevector k.\u000b\nand\fare orbital indices, and \u001bis the spin index.\nFor the cubic structure, the ^Hband(k) describing the\nt2g-derived antibonding bands is easily constructed using\nphenomenological tight binding methods.29For example,\nfor calculations with only antibonding bands composed\nmainly oft2gorbitals, ^Hband(k) is almost diagonal and\nwell approximated by a two-dimensional energy disper-\nsion with nearest neighbor hopping tand next-nearest\nneighbor hopping t0. The explicit form for the band aris-\ning from the xy-symmetry orbital is\n\u000f(k)xy=\u00002t(coskx+ cosky)\u00004t0coskxcosky:(3)\nand for the xz=yz -derived bands one relabels the momen-\ntum indices appropriately.\nFor the tilted structure, the form of ^Hband(k) is more\ndi\u000ecult to express in the simple tight binding language\nused for the cubic perovskite structure. First, the natural\nbasis for de\fning local dorbitals is aligned with the axes\nof the local VO 6octahedron and so varies from site to\nsite making the Slater-Kanamori procedure much more\ninvolved. Second, the V-O-V bond is buckled, decreas-\ning the hopping tboth in absolute terms and relative to\nt0, as well as opening many other channels of hopping\nfrom one V-site to another. For tilted structures we con-\nstruct ^Hband(k) numerically from density functional the-\nory (DFT) band structure calculations using maximally-\nlocalized Wannier function (MLWF) techniques.30We\nhave veri\fed that when applied to the cubic structure\nthe method reproduces the phenomenological tight bind-\ning model discussed above.\nWe found that in the resulting MLWF band structures\n(not shown), the bands derived from t2gand oxygen p\norbitals agree very well with the DFT results, but the\negantibonding bands do not because they are entangled\nwith many other bands. However, the egstates derived\nfrom these bands are well above the Fermi level and, in an\nappropriate local basis aligned with the symmetry axes\nof a VO 6octahedron, do not mix with t2gbands. They\ntherefore do not a\u000bect the \fnal results signi\fcantly.\nThe on-site interaction term in the Hamiltonian,\nHonsite , is assumed to take the standard Slater-Kanamori\nform31{33\nHonsite =UX\ni\u000bni\u000b\"ni\u000b#+U1X\ni\u000b6=\fni\u000b\"nj\f#+\n+U2X\ni;\u000b>\f;\u001bni\u000b\u001bni\f\u001b+\n+JexX\ni\u000b6=\f y\ni\u000b\" i\f\" y\ni\f# i\u000b#+\n+JpairX\ni\u000b6=\f y\ni\u000b\" i\f\" y\ni\u000b# i\f#;(4)4\nwhereUis the on-site intra-orbital interaction, U2and\nU1are on-site inter-orbital interactions for electrons with\nthe same and di\u000berent spin respectively and JexandJpair\nare the exchange and pair hopping parts of the interac-\ntion. For a transition metal ion in free space the rela-\ntions between these coe\u000ecients are: Jex=Jpair=J,\nU1=U\u00002J,U2=U1\u0000J=U\u00003J.34It is generally\nbelieved that the relations also hold for the t2gshell1and\nwe make this assumption here. We consider several val-\nues ofUandJbut focus most attention on the values\nU= 6eV\u001822tandJ= 1eV. These values of UandJ\nare similar to but slightly larger than those used in re-\ncent papers35{38and are chosen to reproduce the crucial\nfeature of the phase diagram, which is that LaVO 3is a\nMott insulator.\nB. Dynamical mean-\feld theory\nTo treat the on-site interaction terms we use single-\nsite dynamical mean \feld theory (DMFT),12which ob-\ntains the electron self-energy ^\u0006(!) (assumed to be site-\nlocal but dependent on spin and orbital indices) from\nthe solution of a quantum impurity model. We use the\nhybridization-expansion CTQMC solver39to solve the\nquantum impurity model. In the present case, the im-\npurity model is a three-orbital model speci\fed by the U\nandJinteractions and by a hybridization function \fxed\nby the self-consistency condition\n^Gi\nimp(!) =\u001ah\n(!+\u0016)1\u0000^Hband\u0000^\u0006(!)i\u00001\u001b\nii:(5)\nwherefgiidenotes the on-site Green's function at vana-\ndium sitei.\nIn applying the procedure to the GdFeO 3-distorted\nstructure, it is important to formulate the impurity prob-\nlem corresponding to a given V site in a local basis\naligned to the V-O bond directions of the VO 6octahe-\ndron at this site. The self energy derived from solving\nthe impurity problem for each octahedron is then ro-\ntated back to the global basis to de\fne the lattice Green's\nfunction used in the self-consistency condition. There\nare three advantages. First, the hybridization function\nis nearly diagonal in the local basis, which greatly re-\nduces the sign problem in the quantum Monte Carlo so-\nlution of the impurity model. Second, the t2gandeg\nbands are clearly distinguished and well separated in the\nlocal basis, enabling removal of the egbands from the\nmany-body problem (they can be neglected or treated\nin mean \feld) so the dynamical impurity model involves\nthree rather than \fve bands. Third, the self-consistency\nconditions corresponding to the four inequivalent V-sites\nofPnma LaVO 3are related by rotations so only one im-\npurity model needs to be solved.C. Determining the magnetic phase boundary\nIn order to detect magnetic order, one can allow for\nbroken symmetry states in the DMFT procedure and\nlower the temperature until an ordered state is reached.\nHowever, ordering temperatures are typically so low rel-\native to the basic energy scales in the problem that our\nsimulations become prohibitively expensive. We there-\nfore compute the susceptibility which is found to have a\nCurie-Weiss form \u001f\u00001\u0018T\u0000Tc. Extrapolation of \u001f\u00001\nto 0 yields an estimate of the Curie temperature Tc. We\ninterpret positive Tcas evidence of magnetism. To com-\npute\u001f\u00001, we add a uniform \feld ~H\u0001~ \u001bto our Hamiltonian,\ncompute the magnetization m(H), verify that mis linear\ninHand then de\fne \u001f\u00001=H\nm.\nT\nC\n=0.234m2 (μ B 2 )\n0\n0.5\n1\n1.5\n2\n2.5χ-1\n0\n0.05\n0.1\nT/t\n0.1\n0.2\n0.3\n0.4\nm\n2\nH = 0.01\nH = 0.005\nFitting lineInverse susceptibility χ -1  (arb. unit)\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\nTemperature T/t\n0\n0.25\n0.5\n0.75\n1\n1.25\nFIG. 3: (Color online) Inverse magnetic susceptibility \u001f\u00001=\nH\nm(H)plotted vs. temperature Tobtained from single-site\nDMFT solution to Eqs. (1 and 3) with Hkintaken to have\nthe form appropriate to a simple cubic lattice with nearest\nneighbor hopping tand second neighbor hopping t0=\u00000:3t.\nCarrier density is \fxed to n= 1:5 and applied magnetic \feld\nH= 0:005 (circles) and H= 0:01 (diamonds). The Curie\ntemperature Tc= 0:234tis estimated by linear extrapolation\nof\u001f\u00001in the region where mis linear inH. Inset: Expanded\nview of the near- Tcregion, together with the magnetization\nsquared (m2) (leftyaxis) calculated by allowing for broken\nsymmetry DMFT solution at H= 0 which also shows that\nTc\u00190:222t. The parameters are t= 0:5;t0=\u00000:3t;U=\n16 = 32tandJ=U\n6.\nFig. 3 demonstrates the method on a model with a high\nCurie temperature, where the magnetic state can easily\nbe constructed. This model is de\fned by Eqs. (1, 3)\nwithHkintaking the same form as in the simple cubic\nlattice but with t0chosen ast0\nt=\u00000:3, so that the sign5\noft0=tis opposite to the sign implied by band structure\ncalculations on the actual materials. For this unphysical\nsign oft0=tthe model has a ferromagnetic ground state\nwith a high Curie temperature. We set carrier density\nn= 1:5 and calculate Tcin two ways: by lowering the\ntemperature until ferromagnetic order is observed, which\nis atTc\u00190:222t; or by measuring \u001f\u00001(T) for several\nvalues ofTaboveTcand linearly extrapolating to \u001f\u00001=\n0. The extrapolation shows Tc= 0:234t. The two values\nare very close. We conclude that extrapolating Tcfrom\n\u001f\u00001(T) is a reliable way to determine whether the model\nexhibits ferromagnetic order.\nIII. MAGNETIC PHASE DIAGRAM\nIn this section, we study the magnetic phase dia-\ngram of model systems derived from the calculated band\nstructure of LaVO 3but with variable amplitude for the\nGdFeO 3distortion and di\u000berent values for the carrier\nconcentration.\n020ϕ=ϴ=0\n020ϕ=ϴ=4\n020ϕ=ϴ=8\n020total density of statesϕ=8.8, ϴ=11.5\n020ϕ=ϴ=11.5\n020ϕ=ϴ=14\n020ϕ=ϴ=16\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5\nenergy (eV)020ϕ=ϴ=18\nFIG. 4: Density of states for bulk LaVO 3(dcarrier density\nn= 2) calculated from DFT with di\u000berent values of tilt angle\n\u0012and rotation angle \u001ein which\u0012=\u001etogether with the\nDOS of realistic LaVO 3structure (\u0012= 11:5\u000e;\u001e= 8:8\u000e). A\nvan Hove peak at the Fermi level develops as \u0012and\u001eincrease,\nand is well-formed at \u0012=\u001e= 14\u000eand above. The bandwidth\nWdecreases as the \u0012and\u001eincreases. The Fermi energy is at\n0.\nFig. 4 shows the evolution of the non-interacting den-\nsity of states with increasing amplitude of the GdFeO 3\ndistortion. If the structure is cubic, the energy disper-\nsion in the t2gmanifold is almost two dimensional. The\nsign and magnitude of t0place the van Hove peaks of\nthe non-interacting DOS at high energy (see top panel\n(\u0012=\u001e= 0) of Fig. 4). As the distortion ampli-\ntude is increased the lowest DOS peak shifts to lower\nenergy and the bandwidth decreases. Previous litera-\nture suggests a connection between peak position and\nferromagnetism;14,24while standard Stoner theory8sug-\ngests that bandwidth may also be important.\nχ\n-1\n for W=2.4eV\nχ\n-1\n for W=1.6eV\nR\nW\n for W=2.4eV\nR\nW\n for W=1.6eVWilson ratio R W\n2\n2.5\n3\n3.5\n4\n4.5\n5Inverse susceptibility χ -1\n0\n0.05\n0.1\n0.15\n0.2\n0.25\ntemperature (eV)\n0\n0.05\n0.1\n0.15\n0.2\n0.25FIG. 5: (Color online) Inverse magnetic susceptibility \u001f\u00001\n(closed symbols, black on-line; right axis) and Wilson ra-\ntioRW(open symbols, red on-line, left axis) calculated for\nLa1\u0000xSrxVO 3cubic structure as functions of temperature at\ncarrier density n= 1:5 and on-site interactions U= 6eV,\nJ= 1eV using two bandwidths, W= 2:4eV (circles and solid\nlines) andW= 1:6eV (triangles and dashed lines).\nWe begin our study of the connection by examining\nthe cubic structure. The solid curve with \flled circles\n(black online) in Fig. 5 is the inverse susceptibility calcu-\nlated for temperatures down to T= 0:03eV for the case\nU= 6eV,J= 1eV and carrier density n= 1:5 using\nthe DOS shown in top panel of Fig. 4, which has the\nbandwidth W\u00192:4eV. The inverse susceptibility curve\nis seen to deviate from the Curie form at low tempera-\nture. The extrapolation to zero of the high temperature\nlinear regime implies a Tc<0.\nWe attribute the \rattening out of the susceptibility\ncurve to the onset of Fermi-liquid coherence. To ver-\nify this, and to gain additional insight into the na-\nture of spin correlations in this model, we computed\nthe Wilson ratio RW=\u00192\n3\u001f\n\r,40(in our conventions\nthe dimensional factorskB\n\u0016B= 1). Here \ris the co-\ne\u000ecient of the linear speci\fc heat \r=dCV\ndT\f\f\f\f\nT=0=\n\u00192\n3Tr[\u0017FZ\u00001].41We estimated the linear coe\u000ecient of\nthe speci\fc heat from the density of states \u0017Fat the\nFermi level and calculated renormalization factor Z,\nwhich are obtained from the measured imaginary time\nGreen's function via \u0017F\u0019 \u0000\f\n\u0019G(\u001c=\f=2) and from\nthe Matsubara frequency self energy via Z\u00001\u00191\u00006\nIm\u0006(!1= 3\u0019=\f)\u0000Im\u0006(!0=\u0019=\f)\n!1\u0000!0(not shown).\nThe solid curve with open circles (red online) in Fig. 5\nshows that, for the DOS in the top panel of Fig. 4 with\nU= 6eV and J= 1eV, the Wilson ratio extrapolates\nto the value RW= 2 as temperature T!0. The\nvalueRW= 2 is expected for a Kondo lattice with a\nlow quasiparticle coherence scale but no intersite corre-\nlations, while a system with strong ferromagnetic cor-\nrelations would be characterized by an RW\u001d2. We\ntherefore conclude that there is no evidence for ferro-\nmagnetism in the cubic structure at U= 6;J= 1eV and\nW= 2:4eV and that the \rattening of the \u001f\u00001(T) curve\nindicates the onset of the Fermi liquid coherence.\nWe now turn to the e\u000bects of the GdFeO 3distortion.\nFor simplicity of presentation, we focus mainly on the\ncase\u0012=\u001e. Fig. 4 shows the evolution of DOS with\ntilting angle. We see that, as the tilt angle is increased,\nthe position in energy of the lowest density of states peak\nshifts down in energy.\ntilt angle 8\no\n0\n0.05\n0.1\n0.15\n0.2\ntilt angle 8\no\nzoom in\ntilt angle 12\no\n0\n0.05\n0.1\n0.15\ntilt angle 12\no\nzoom in\nN\nd\n = 1.55\nN\nd\n = 1.95\ntilt angle 14\nosusceptibility χ -1  ( μ B 2 /eV)\n0\n0.05\n0.1\n0.15\ntilt angle 14\no\nzoom in\ntilt angle 16\no\n0\n0.05\n0.1\n0.15\ntilt angle 16\no\nzoom in\ntilt angle 18\no\n0\n0.05\n0.1\n0.15\ntemperature T (eV)\n−0.1\n0\n0.1\n0.2\n0.3\n0.4\ntilt angle 18\no\nzoom in\ntemperature T (eV)\n−0.02\n0\n0.02\n0.04\nFIG. 6: (Color online) Temperature dependence of inverse\nsusceptibility computed for model La 1\u0000xSrxVO 3computed\nfor tilt angles \u0012=\u001efrom 8\u000eto 18\u000ecomputed at densities\nn= 1:55 (black circle lines) and n= 1:95 (diamond dashed\nlines) and interaction parameters U= 6eV,J= 1eV from\nEq. (1) with Hkinderived from MLWF \fts to band struc-\nture. The diamonds and circles are data points measured by\nDMFT. The lines are \ftted from data points. Left column:\nplot over wide temperature range; right column: expanded\nview of small \u001f\u00001region. The vertical dashed line marks zero\ntemperature.\nFig. 6 shows the evolution of the inverse susceptibility\nwith increasing amplitude of GdFeO 3distortion at U=\n6eV andJ= 1eV for two carrier density values n= 1:55\nandn= 1:95. Forn= 1:55, ferromagnetic order canbe observed starting at \u0012=\u001e= 14\u000e. In contrast, for\nn= 1:95, the Curie temperature is nonzero within errors\nonly for\u0012=\u001e>16\u000e.\n1.5 1.6 1.7 1.8 1.9 2.0\nfilling81012141618 tilt angle (Degree)\n(La/Sr)VO3 solid solutionPMFM\nphase boundary U=4eV\nphase boundary U=5eV\n0.0040.0080.0120.0160.0200.0240.0280.032\nFIG. 7: (Color online) The magnetic phase diagram in the\nspace of carrier density n(x-axis) and tilt and rotation an-\ngles\u0012=\u001e(y-axis) for the solid solution La 1\u0000xSrxVO 3with\non-site interactions U= 6eV,J= 1eV. The white region indi-\ncates paramagnetic order ( Tc<0:004eV), the colored regions\ndenote ferromagnetic order with the transition temperature\nTcgiven by the scale bar (color on-line). The solid curve is\nthe\u0012=\u0012(n) curve estimated for La 1\u0000xSrxVO 3as described\nin Appendix A. The estimated phase boundaries for U= 4\n(circle and dashed line, blue on-line) and 5eV (squares and\nsolid line, green on-line) with J= 1eV are also plotted.\nThe shaded areas (color online) in Fig. 7 show the\nphase diagram resulting from a detailed study of the de-\npendence of Curie temperature on carrier density and\ntilt (rotation) angle at U= 6eV and J= 1eV obtained\nfrom curves such as those shown in Fig. 6. The expense\nof the computation and the uncertainties inherent in our\nextrapolation means that the phase boundary is not pre-\ncisely determined. We regard Tc<0:004eV as consistent\nwithTc= 0 within uncertainties. The width of the strip\nseparating Tc= 0:004 and 0:008 gives a measure of the\nerror bars on the locations of the phase boundary. The\ndivergence in tilt angle required to obtain a nonzero tran-\nsition temperature as n!2 may be understood from the\nfact that for the correlation strength considered here the\nn= 2 compound is a Mott insulator. Ferromagnetism\nis favored by metallic motion of the carriers and is sup-\npressed in proximity to the Mott insulating phase which\nfor the parameters we consider is antiferromagnetically\nordered. To summarize, we see that for these values of U\nandJobtaining a ferromagnetic state requires two con-\nditions: (1) large hole doping away from LaVO 3and (2)\nlarge GeFeO 3-type distortion.\nWe now consider the physics behind the dependence of\ntransition temperature on tilt angle. Inspection of Fig. 4\nshows that increasing the magnitude of the GdFeO 3both\nreduces the bandwidth (from 2 :4eV at\u0012=\u001e= 0 to\n1:8eV at the critical angle) and moves the position of7\nthe lowest density of states peak to lower energy. To in-\nvestigate the relative importance of the two e\u000bects we\nhave performed computations for the cubic structure us-\ningU= 6eV,J= 1eV but with reduced bandwidth\nW= 1:6 eV. The dashed curves in Fig. 5 show the Wil-\nson ratioRW(open triangles, red on-line) and the inverse\nsusceptibility \u001f\u00001(\flled triangles, black on-line) for the\nsmaller bandwidth W= 1:6eV. The susceptibility curve\nindicates that the intercept is increased relative to the\nlarger bandwidth case, but remains negative. The sharp\ndownturn in the Wilson ratio at the lowest temperatures\nsuggests that the decrease in intercept is a consequence\nof a lowered Fermi liquid coherence scale and does not\nindicate stronger ferromagnetic correlations. We there-\nfore conclude that for reasonable values of the bandwidth\nand correlation strength the crucial factor for ferromag-\nnetism is the position of the density of states peak. For\nvery large values of the Hund's coupling J(>\u00183eV), fer-\nromagnetism may occur over a much wider parameter\nrange, as seen in previous Bethe lattice studies.17{19\nCurie temperature T c  (eV)\n−0.1\n−0.05\n0\n0.05\nDOS maximum relative to the Fermi level (eV)\n−0.15\n−0.1\n−0.05\n0\n0.05\n0.1\n0.15\nFIG. 8: (Color online) The dependence of Curie temperature\nTcon van Hove peak position with respect to the Fermi level\nof fully spin-polarized state calculated from Eqn. (3) with\nt= 0:264eV,t0=tchanges from\u00000:3!0:3. On-site interac-\ntionsU= 6eV and J= 1eV. The dashed line marks the zero\ntemperature.\nThe importance of the position of the density of states\npeak was previously noticed in the context of the one-\nband Hubbard model.14{16However, in that case ferro-\nmagnetism was only found if the density of states peak\nis far below the Fermi level of the paramagnetic state.\nIn the present case inspection of Fig. 4 shows that it is\nonly necessary for the density of states peak to be not\ntoo far above the Fermi level. We believe that the di\u000ber-\nence arises from the \\double exchange\" physics of Hund's\ncoupling in partially \flled dshells. The Hund's coupling\nfavors high spin states, which means that hopping be-\ntween two sites is optimal if the spins are parallel and is\nsuppressed if they are not parallel. This strongly favorsferromagnetism. A natural question is how far above the\nFermi level can the density of states peak be and still\nsupport ferromagnetism. For a reasonable range of J\n(\u00181eV, see Fig. 8) we \fnd that a good rule of thumb is\nthat ferromagnetism occurs if the density of states peak\nlies at or below the Fermi level of the fully polarized fer-\nromagnetic state . This answer is clearly not universal\nsince model system studies17{19indicate that increasing\nJto very large values favor ferromagnetism even if the\ndensity of states peak lies very high in energy.\nWe have also studied selected densities at the smaller\ncorrelation strengths U= 4eV and U= 5eV. Estimates\nof the resulting phase boundaries are shown as dashed\nand solid lines in Fig. 7. We see that for intermediate n\nthe phase boundary is only weakly dependent on Uand\nn. For carrier concentrations near n= 2 weakening the\ncorrelations to move the system out of the Mott phase\ngreatly increases the range in which ferromagnetism is\nobserved. These calculations are the basis for the three\nschematic phase diagrams shown in Fig. 1.\nIV. THE EFFECT OF OXYGEN BANDS\nIn early transition metal oxides, the bands that cross\nthe Fermi level are well separated from other bands and\nare of mainly transition metal dcharacter, arising from\nt2gsymmetry orbitals. In many body calculations, it is\ncommon to focus only on these bands, which are treated\nas a multiband Hubbard model, while all other bands\nare neglected.1,2However, it is generally believed that\nthe fundamentally correct model of the transition metal\noxides should be based on atomic-like dorbitals coupled\nto weakly correlated oxygen pstates.25In this section\nwe examine the magnetic phase diagram of the vanadate\nsystem by applying the methods described in previous\nsections to the \\ pd\" model which describes this situation\nand comparing the \\ pd\" model results to those from the\n\\donly\" model.\nThepdmodel has two important energy scales: the\non-site interaction Uwhich as before is the energy cost\nfor changing doccupancy and the charge transfer energy,\nin other words, the energy cost for an electron to hop\nfrom a ligand to a transition metal atom. The charge\ntransfer energy may be de\fned in di\u000berent ways but the\ncorrect value is not well established, either from ab ini-\ntioconsiderations or from experiment. The important\npoints for our subsequent discussion are that the physics\ndepends sensitively on the charge transfer energy as well\nas onUand that for reasonable Ua metal-insulator tran-\nsition may be driven at integer band \flling by varying the\ncharge-transfer energy.25We will see that, as was found\nin thedonly model, the ferromagnetic phase boundary\ndepends on whether the parameters are such as to place\nLaVO 3on the metallic or insulating sides of the metal-\ninsulator phase diagram.\nThe Hamiltonian describing the panddstates of a8\n(a) cubic, \nΔ\n largeχ-1\n0\n0.05\n0.15\n0.2\nθ\n=\nφ\n=16\n๐\n(b) tilted, \nΔ\n large\n(c) cubic, \nΔ\n smallχ-1\n0\n0.05\n0.15\nθ\n=\nφ\n=16\n๐\n(d) tilted, \nΔ\n small\n(e) cubic, d-only model\nfilling 1.55\nfilling 1.90\nfilling 2.00χ-1\n0\n0.05\n0.15\ntemperature T (eV)\n−0.1\n0\n0.1\n0.2\n0.3\nθ\n=\nφ\n=16\n๐\n(f) tilted, d-only model\ntemperature T (eV)\n−0.1\n0\n0.1\n0.2\n0.3\n0.4\nFIG. 9: (Color online) Inverse susceptibility vs. tempera-\nture for three di\u000berent \flling values 1 :55;1:90 and 2:00 for\ncubic (left column) and GdFeO 3-distorted LaVO 3with tilt\nangle\u0012=\u001e= 16\u000e(right column). (a,b) pdmodel with\nlarge \u0001 (approximately that predicted by standard42dou-\nble counting correction); (c,d) pdmodel with small \u0001; (e,f)\ndonly model. On-site interactions U= 6eV;J= 1eV. For\ncubic case, \u0001 large = 10:97eV;\u0001small = 2eV. For tilted case,\n\u0001large = 10eV;\u0001small = 0eV. The vertical dashed lines mark\nzero temperature.\nperovskite transition metal oxide has the form43\nH=Hd+Hp+Hhyb; (6)\nwhereHd=Hd\nkin+Honsite withHd\nkindescribing bare\non-site energies and electron hoppings between dorbitals\nandHonsite as in Eq. (4); Hpdescribes the dispersion\nof the oxygen pbands in the absence of p dhybridiza-\ntion;Hhybis the hybridization between dandporbitals.\nWe takeHonsite to have the usual rotationally invariant\nSlater-Kanamori form with U= 6eV and J= 1eV. The\nband termHkin=Hd\nkin+Hp+Hhybhas the same generic\nform as Eq. (2) but now the space of orbital indices \u000b;\f\nis expanded to label both vanadium dand oxygen por-\nbitals. The corresponding H\u000b\f\nband(k) is generated using\nMLWF methods with a wide energy window including\nbothdandpbands (see the MLWF procedure in Sec. II\nand the Appendix B).\nHowever,22,42,44the Hartree shift arising from Honsite\nmeans that the dlevel energy \u000f0\ndobtained from the\nMLWF procedure must be renormalized by a \\double\ncounting correction\" which we denote by \u0001: \u000fd=\u000f0\nd\u0000\u0001.\nWe consider two values for \u0001: one, which we refer to\nas \u0001large\u001810eV, is essentially the value obtained by\napplying the standard42double counting correction to\nbasic band theory and is such that LaVO 3is predicted\nto be metallic, in contradiction to experiment. The other\nvalue, which we refer to as \u0001 small\u00180\u00002eV, is such thatthe material is insulating at n= 2 in agreement with ex-\nperiment.\nThe resulting model is solved using single-site DMFT\nas described in Sec. II, but with one important addition.\nThe fullpdmanifold includes V egorbitals. While the eg\nenergy lies above the Fermi level, so that the eg-derived\nantibonding bands are empty, some of the \flled bands\nareeg-oxygen bonding states which have a small but non-\nzeroegcontent. One must therefore solve a 5-orbital im-\npurity model. Performing this calculation in full general-\nity would be prohibitively expensive. We therefore follow\nstandard procedure and treat the egorbital contribution\nto the impurity model in a Hartree approximation.\nFig. 9 shows representative computations of the inverse\nsusceptibility for cubic and tilted systems at several car-\nrier concentrations. The top row (panels a and b) dis-\nplayspdmodel results obtained for the standard dou-\nble counting correction (so that LaVO 3is wrongly pre-\ndicted to be a metal) while the middle rows (panels c\nand d) show results obtained if the double counting cor-\nrection is tuned so that the calculation places LaVO 3in\nthe Mott/charge-transfer insulating regime of the phase\ndiagram. The bottom two panels (e and f) present donly\nmodel results for comparison. In the donly model the U\nvalue is such as to place the n= 2 (LaVO 3) material in\nthe Mott insulating region of the phase diagram). The\ntransition temperature estimates obtained by linearly ex-\ntrapolating the \u001f\u00001curves to 0 are given in Table I.\nn= 1:55n= 1:90n= 2:00\ncubic, \u0001 large = 10:97eV -0.1230 -0.0139 -0.0040\ntilted, \u0001 large = 10eV 0.0963 0.1127 0.1167\ncubic, \u0001 small = 2eV -0.0161 0.0025 0.0007\ntilted, \u0001 small = 0eV 0.0452 0.0303 0.0178\ncubic,donly model -0.0602 -0.0200 -0.0349\ntilted,donly model 0.0185 0.0042 -0.0025\ncubic,donly model, U= 4eV -0.2347 -0.1368 -0.1250\ntilted,donly model, U= 4eV 0.0267 0.0345 0.0366\nTABLE I: Values for Curie temperature Tc(in eV) for each\ncase considered in Fig. 9 together with results for U= 4eV\nfor three di\u000berent \fllings n= 1:55;1:90 and 2:00. All compu-\ntations are for J= 1eV; except where indicated, U= 6eV is\nused.\nExamination of the results in Table. I shows that the\nqualitative trends are the same in the pdanddonly model\ncalculations. In particular, in both models increasing the\ntilt angle increases the tendency towards ferromagnetism.\nHowever, signi\fcant di\u000berences are visible; in particular\nthepdmodel has a signi\fcantly greater tendency to fer-\nromagnetism than does the donly model and (especially\nin the small-\u0001 case) the di\u000berences are more pronounced\nfor the cubic than for the tilted structure.\nWe believe that there are two main origins for the dif-\nferences. First, in the small \u0001 cubic system case, the\nchange in the charge transfer energy relative to band the-9\nory a\u000bects the density of states, moving the peaks closer\nto the Fermi level while for the tilted structure the shift\nin charge transfer energy does not change the peak po-\nsitions as much (see Appendix C). Second, and perhaps\nmore important, the egstate occupancy arising from the\np\u0000dbonding bands (omitted in the donly model) in-\ncreases the e\u000bective moment on the dsite, thereby en-\nhancing the tendency towards magnetism. This e\u000bect is\nmore pronounced in the larger \u0001 (smaller charge transfer\nenergy) case, because the p dmixing is larger.\npd cubic with e\ng\npd cubic w/o e\ng\npd tilted with e\ng\npd tilted w/o e\ng\n(a)χ-1\n0\n0.05\n0.15\n0.2\ntemperature T(eV)\n−0.1\n0\n0.1\n0.2\n0.3\n0.4\nd-only cubic\npd cubic w/o e\ng\nd-only tilted\npd tilted w/o e\ng\n(b)\ntemperature T(eV)\n−0.1\n0\n0.1\n0.2\n0.3\n0.4\nFIG. 10: (Color online) Inverse susceptibility vs. temperature\nfor cubic (blue color) and 16\u000e-tilted (red color) structures at\n\fllingn= 2:0. (a) Comparison for pdmodel with (solid lines)\nand without (dashed lines) egband spin polarization. (b)\nComparison between donly model (solid lines) and pdmodel\nwithoutegband spin polarization (dashed lines). Parameters\nfor calculations with pdmodel are the same as in Fig. 9a,b.\nCalculations with donly model use U= 4eV< U c;J= 1eV\nfor both structures.\nTo demonstrate this point we present in Fig. 10a,b cal-\nculations of the inverse susceptibility under di\u000berent con-\nditions. The curves in the left panel compare calculations\nin which the egoccupancy is frozen at the spin unpolar-\nized values (dashed lines) and calculations in which the eg\norbitals are treated within the Hartree-Fock approxima-\ntion as described above. We see that the feedback from\nthe polarization of the egorbitals makes a signi\fcant con-\ntribution to the transition temperature. The right pan-\nels show that calculations performed in the \\frozen eg\"\nmodel agree reasonably well with the corresponding cal-\nculations in the donly model for U= 4eV <Uc. Further\nanalysis is given in Appendix D.\nV. CONCLUSIONS\nIn this paper we have investigated the conditions under\nwhich ferromagnetism might be observed in bulk solid\nsolutions derived from LaVO 3. Our theoretical studies\nused realistic band structures derived from maximally-\nlocalized Wannier function \fts to density functional (typ-\nically PWscf/GGA) band structure calculations, along\nwith the single-site DMFT approximation which contains\nthe local physics of partially \flled orbitally degenerate\ndorbitals with strong Hund's coupling. Our main the-oretical \fnding is that for physically reasonable values\nof the Hund's coupling ferromagnetism is driven mainly\nby density of states e\u000bects, being favored by a density of\nstates peak lying not too high in energy. The dependence\non correlation strength and carrier concentration is rela-\ntively weak except that if the correlation strength is large\nenough to drive a Mott transition to an antiferromag-\nnetic Mott/charge transfer insulator, ferromagnetism is\nsuppressed in the vicinity of the insulating state.\nAn association between ferromagnetism and the den-\nsity of states was previously noticed in studies of the\none band Hubbard model.14,19,24However, there is a sig-\nni\fcant di\u000berence between ferromagnetism in the model\nstudied here and that found in the single-band Hubbard\nmodel. In the single-band Hubbard model, ferromag-\nnetism only occurs when the DOS peaks are located far\nbelow the chemical potential, and indeed very close to the\nlower band edge.24In the orbitally degenerate, Hund's\ncoupled systems studied here, the requirements on the\nposition of the density of states peak are substantially re-\nlaxed. We \fnd that for physically reasonable correlation\nstrengths, ferromagnetism can occur as long as the low-\nest density of states peak is at or below the Fermi level of\nthe fully polarized ferromagnetic state. The importance\nof orbital degeneracy and Hund's coupling has been pre-\nviously noted in studies of models with a semicircular\n(Bethe-lattice) density of states17{19and in the periodic\nAnderson model45but the important role played (for rea-\nsonable values of J) by the near-Fermi-surface density of\nstates peaks seems not to have previously been noticed.\nIn the vanadate systems, the density of states is con-\ntrolled by the amplitude of the GdFeO 3distortion away\nfrom the ideal cubic perovskite structure. We \fnd\nthat increasing the distortion amplitude favors ferromag-\nnetism. Indeed, if the Uinteraction were set to the un-\nphysically small value U= 4eV so that LaVO 3were not\na Mott insulator, then the observed GdFeO 3distortion\nwould be large enough to put the material at or very near\nto the ferromagnetic phase boundary. One may then ask\nwhy doping does not induce ferromagnetism in bulk solid\nsolutions such as La 1\u0000xSrxVO3. We believe that the an-\nswer is that in the physical system, the distortion ampli-\ntude and the carrier concentrations are both determined\nby the La/Sr ratio in such a way that, as xis varied, the\ndistortion amplitude decreases so that the system follows\na path in parameter space which remains outside the fer-\nromagnetic region (solid line in Fig. 7).\nWe remark that while the general experience has been\nthat the single-site DMFT approximation provides a\ngood qualitative representation of the physics, in partic-\nular of trends as material parameters are varied,2,22,23,46\nits quantitative accuracy in producing magnetic phase\nboundaries has not been established. It is possible\nthat single-site DMFT underestimates magnetic order.\nThe reports of ferromagnetism in vanadate superlattices5\nwhere both the doping and the tilt angle may be locally\nlarge, suggests that the true ferromagnetic phase bound-\nary may be shifted to lower tilt angle but still above the10\nbulk LaVO 3\u0012(n) curve. Increases in computer power\nand improvements in algorithms may soon make cluster\ndynamical mean \feld studies of realistic systems with or-\nbital degeneracy and Hund's coupling feasible, and it will\nbe important to use these methods to assess the reliabil-\nity of single-site DMFT predictions.\nThe second point of materials theory is the issue\nof what is the relevant theoretical model. Consistent\nwith most theoretical literature on early transition metal\noxides,1,2,46we focused mainly on a donly \\multiband\nHubbard model\" approach where the electrons in the\nnear Fermi surface dderived bands were correlated by lo-\ncalU;Jinteractions. The results are in reasonable accord\nwith experiment. It has been argued22,23,25that a more\ngenerally valid description may be obtained from a model\nin which correlations are applied to atomic-like dorbitals\nwhich are hybridized to the other orbitals, in particular\nto the oxygen porbitals. In this \\ pd\" model approach, a\nkey parameter is the renormalized dlevel energy, which is\nshifted from the band theory value by a \\double counting\ncorrection\" for which there is no generally agreed value.\nIf the standard expression47for the double counting cor-\nrection is used, in the single-site DMFT approximation\nLaVO 3is found to be a ferromagnetic metal rather than\nan antiferromagnetic insulator and indeed the solid so-\nlution La 1\u0000xSrxVO3is predicted to be ferromagnetic for\na wide range of x, in reasonable correspondence to the\nsmallUcalculation in the donly model. If the double\ncounting contribution is shifted such that the model for\nLaVO 3is in its Mott insulating regime, then the result-\ning phase diagram is very similar to that shown in Fig. 7\nfor thedonly model. Quantitative di\u000berences arise from\nthe changes in density of states arising from the large\nrenormalization of \u000fdrequired to make the n= 2 case a\nMott insulator and from the contribution of the (small\nbut non-negligible) egoccupancy arising from the the p-d\nbonding states. Determining the appropriate theoretical\napproach for the early transition metal oxide is an im-\nportant open question.\nOur \fndings suggest several routes to creating ferro-\nmagnetism in arti\fcially fabricated systems. The gen-\neral aim should be to manipulate the band structure so\nas to move density of states peaks close to the Fermi\nlevel. Inducing octahedral tilts by compressive strain is\nan important route. Further, in early transition metal\noxides, increasing the p dhybridization clearly increases\nthe e\u000bective dmoment and this provides a self-consistent\nampli\fcation of the Curie temperature. Therefore ma-\nnipulation of the p denergy di\u000berence can be used to\ncontrolTc.\nWe also observe that some transition metal oxides such\nas the Ru-based perovskites and Ruddlesden Popper ma-\nterials Srn+1RunO3n+1involve holes in the t2gbands and\nin a qualitative sense may be thought of as the particle-\nhole transforms of the models studied here. In this case\nfor the physical sign of t0the van Hove peaks are on the\nother side of the Fermi level, suggesting that the theoret-\nical models will be more likely to exhibit ferromagnetism.However, increasing the tilt angles shifts the peaks in the\nwrong direction, reducing the tendency to magnetism. A\nmore detailed investigation of this physics is in progress.\nACKNOWLEDGEMENTS\nWe thank U. L uders, J. Okamoto and C. Marianetti\nfor helpful conversations. We acknowledge support from\nDOE-ER046169. HTD acknowledges partial support\nfrom the Vietnam Education Foundation (VEF). A por-\ntion of this research was conducted at the Center for\nNanophase Materials Sciences, which is sponsored at Oak\nRidge National Laboratory by the Scienti\fc User Fa-\ncilities Division, O\u000ece of Basic Energy Sciences, U.S.\nDepartment of Energy. We used the code for CT-HYB\nsolver39written by P. Werner and E. Gull, based on the\nALPS library.48\nAppendix A: \u0012(x)Curve for La 1\u0000xSrxVO 3\nThe relation between tilt angle \u0012and the doping level\nxis obtained based on the fact that the (pseudo)cubic\nlattice constant apof La 1\u0000xSrxVO3is linearly dependent\nonx.49Bulk SrVO 3hasaSVO\np = 3:84\u0017A,26bulk LaVO 3\nhasaLVO\np = 3:92\u0017A,20hence for doped material, ap(x) =\naSVO\npx+aLVO\np(1\u0000x).\nWe further assume that the e\u000bective V-O distance d\nalso has similar relation, d(x) =dSVOx+dLVO(1\u0000x).\nBulk SrVO 3is cubic, thus dSVO =aSVO\np\n2= 1:92\u0017A; while\ndLVO = 2:00\u0017A.20The angle\u0012is calculated from ap=\n2dcos\u0012, and with n= 2\u0000xwe obtain the \u0012(n) curve\nshown in Fig. 7.\nAppendix B: Band Calculations and Wannier Fitting\nOur procedure for obtaining the band theoretic contri-\nbution to the local Green's functions is as follows. First,\ngiven a rotation angle \u001eand tilt angle \u0012we construct\nthe 4-unit-cell structure corresponding to a GdFeO 3-type\ndistortion of the cubic perovskite. The PWscf code of\nQUANTUM ESPRESSO50,51is then used to calculate\nthe band structure for that system. Finally, we use\nthe Wannier90 implementation52of the MLWF proce-\ndure with an appropriate energy window to generate our\n^Hband(k).\nFor the band structure calculations with PWscf, we\nused a cuto\u000b energy Ecutoff = 30Ry\u0019408eV and a\n10\u000210\u000210 Monkhorst-Pack k-mesh. The MLWF proce-\ndure involves the choice of both an overall energy window\nand a \\frozen\" window within which the MLWF bands\nare forced to coincide with the DFT bands. For MLWF\n\ftting with Wannier90, we set the overall energy win-\ndow for the dbands to run from \u00001:5eV to 6eV with11\nrespect to the chemical potential. For larger octahedral\nrotations, the egandt2gbands are well separated and\nchoosing the frozen energy window to run from \u00001:5eV\nto 1:5eV represents the t2gbands well. For smaller ro-\ntations, band overlap becomes important and the upper\ncuto\u000b of the frozen energy window must be reduced to\navoid overlap with the egbands; this means that the\nhigh-lying (unoccupied) part of the t2gbands is not per-\nfectly represented, but these issues do not a\u000bect our main\nresults, which concern the location of the ferromagnetic\nphase. If oxygen bands are included in the calculation,\nthe energy window is enlarged from \u000010 to 6eV, while the\nrange\u000010!1:5eV is set as the frozen energy window.\nAppendix C: Charge Transfer Model\nIn the charge transfer ( pd) model, the value of the\ncharge transfer energy \u0001 a\u000bects the ferromagnetic phase\nboundary and also changes the band structure. In Fig. 11\nwe demonstrate the e\u000bect of varying \u0001 on the band struc-\nture of cubic LaVO 3.\nWe use the same parameters as in Sec. IV for the\ncubic structure and employ a Hartree-Fock approxima-\ntion to derive the spin/orbital unpolarized DOS. Fig. 11\nshows such DOS for \u0001 = 2eV and \u0001 = 10 :97eV. Within\nHartree-Fock calculation, oxygen pbands are located at\nthe same positions as in corresponding DMFT results,\nso thep dhybridization may be expected to be simi-\nlar in the two calculations. We therefore consider the\nHartree-Fock DOS as the \\non-interacting\" DOS for the\npdmodel, including charge transfer e\u000bects.\n2.0\n 1.5\n 1.0\n 0.5\n 0.0 0.5 1.0 1.5\nenergy (eV)0.00.20.40.60.81.01.21.4 t2g-band density of statesDelta = 2eV\nDelta = 10.97eV\nFIG. 11: (Color online) Hartree-Fock orbital/spin unpolarized\nt2gband density of states with U= 6eV and J= 1eV for two\ncases of cubic structure used in Sec. IV: \u0001 small = 2eV and\n\u0001large = 10:97eV. The dashed vertical line marks the Fermi\nlevel.\nThe two curves shown in Fig. 11 demonstrate that\nchanging \u0001 changes both the position of the van Hove\npeaks relative to the Fermi level and the bandwidth.\nFor \u0001large = 10:97eV, the bandwidth is W= 2:58eVand the ratio between next nearest neighbor and nearest\nneighbor hoppings is t0=t= 0:23. For \u0001 small = 2eV,\nthe bandwidth Wof the antibonding band is smaller\n(W= 2:03eV) and the van Hove peak moves closer to\nthe Fermi level ( t0=t= 0:18). These two factors act to\nmake the model more ferromagnetic.\nAppendix D: Contribution of egBands\nIn present-day applications of the DFT+DMFT\nmethod to transition metal oxides, atomic-like dorbitals\nare de\fned via a projector or Wannier construction and\nare coupled to other bands (mainly oxygen p, in practice).\nThe resulting theory di\u000bers from the multiband Hubbard\nmodels often used2in two ways. First, the explicit inclu-\nsion of oxygen orbitals means that charge transfer physics\nin the sense of Ref. 25 may play a role. Second, the full d\nmanifold is included, in contrast to the multiband Hub-\nbard approach where only the frontier orbitals (in the\npresent case, the t2gstates) are treated. This latter di\u000ber-\nence is important for the vanadates because even though\ntheegorbitals are notionally empty, hybridization with\nthe oxygen bands leads to a non-vanishing density of eg\nelectrons.\nA fully interacting treatment of the entire dmanifold is\ntoo expensive to be computationally practicable. In this\npaper we take the virtual occupation of the egorbitals\ninto account via a Hartree approximation. In each iter-\nation of the DMFT self consistent calculation, egoccu-\npancies obtained from the previous iteration are used to\ncalculate the egfeedback to the t2gbands via the Hartree\napproximation to the inter-orbital self energy:\nSH\n\u001b=X\niinegbands[(U\u00002J)ni\u0016\u001b+ (U\u00003J)ni\u001b]:(D1)\nThe two panels of Fig. 10 show that it plays an impor-\ntant role in the magnetic phase diagram. In what follows\nwe give analytic Stoner-style arguments explaining these\nresults.\nTo estimate the contribution of egbands to the polar-\nization, we \frst observe that although the egorbitals are\nnotionally empty, hybridization with the oxygen states\nmeans that the egoccupancy is non-vanishing; roughly\nif thepegenergy di\u000berence is \u0001 eand thep\u0000dhopping\nistpdthen\nneg\u0018tpd\n\u0001e: (D2)\nA perturbation to the egenergy of magnitude hthus\nleads to a change in occupancy\n\u000eneg\u0018tpdh\n\u00012e; (D3)\nso that we expect a susceptibility given by\n\u001feg\u0018tpd\n\u00012e: (D4)12\nThe absence of egcharacter at the Fermi level and the\nfact that the eg\u0000penergy di\u000berence (\u0001 e) is larger than\nthat of the t2g\u0000penergy di\u000berence suggests that \u001feg\nwill be smaller than the band susceptibility, but not by\norders of magnitude.\nWe will assume that the intra- t2gcontributions lead to\na\u001ft2g=A=(T\u0000T0) which would diverge at some tem-\nperatureT0. 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Commun. 178, 685\n(2008)."    },    {        "title": "0905.1762v1.Trends_in_Ferromagnetism_in_Mn_doped_dilute_III_V_alloys_from_a_density_functional_perspective.pdf",        "content": "arXiv:0905.1762v1  [cond-mat.mtrl-sci]  12 May 2009Trends in Ferromagnetism in Mn doped dilute III-V alloys fro m a\ndensity functional perspective\nRoby Cherian and Priya Mahadevan\nS.N. Bose National Centre for Basic Sciences,\nJD-Block, Sector III, Salt Lake, Kolkata-700098, India\nClas Persson\nDepartment of Materials Science and Engineering,\nRoyal Institute of Technology, SE-10044, Stockholm, Swede n\n(Dated: November 12, 2018)\nAbstract\nMn doping in dilute III-V alloys has been examined as a route t o enhance ferromagnetic stability.\nStrong valence band bowing is expected at the dilute limit, i mplying a strong modification of the\nferromagnetic stability upon alloying, with even an increa se in some cases. Using first principle\nelectronic structure calculations we show that while codop ing with a group V anion enhances the\nferromagnetic stability in some cases when the effects of rela xation of the lattice are not consid-\nered, strong impurity scattering in the relaxed structure r esult in a reduction of the ferromagnetic\nstability.\nPACS numbers:\n1I. Introduction\nDilute magnetic semiconductors have been intensively studied in rece nt times with the\nintention of replacing conventional electronic devices with those ba sed on these materials\n[1]. No real devices have been realized so far and the interest in this fi eld has been based on\nconcept devices. An essential aspect of dilute magnetic semicondu ctors based devices is the\nmanipulation of the spin of the electron which one would like to do at roo m temperature.\nHence the search is on for a room temperature ferromagnet. One material that has been\nintensively studied for this purpose is Mn doped GaAs [2]. The ferrom agnetic Curie tem-\nperature ( Tc) are still far from desirable reaching a maximum of 250 K in specially des igned\nsuperlattices [3]. Alternate materials such as transition metal atom s doped in II-VIsemicon-\nductors [4], perovskite oxides [5], other oxides [6], III-V semiconduc tors and chalcopyrites\n[7] have been synthesized and studied and the search is on for the m ost suitable material.\nIn this work we confine our attention to Mn doping in III-V semicondu ctors. Theoretical\ncalculations predict a higher ferromagnetic stabilization energy for Mn in GaNthan in GaAs\n[8]. However, the main conclusion from experiments is that the effect ive interaction between\nthe Mn atoms is antiferromagnetic [9] or could be weakly ferromagne tic [10]. Strong ferro-\nmagnetism has been observed though the origin seems to be the pre sence of ferromagnetic\nclusters of Mn present at large doping concentrations [11].\nIn this work we examine an alternate strategy to obtaining high Curie temperatures by\nconsidering dilute alloys of III-V semiconductors. Strong valence b and bowing is expected\nin the dilute limit in some cases, and we want to use that to modify the fe rromagnetic\nstability. These ideas are not new to the current work and have bee n proposed earlier in\nthe literature [12]. However there is no detailed theoretical work wh ich has examined the\nelectronic structure, modified interaction strengths and the con sequent implications on the\nTc. The basic idea that one aims to exploit here is to use semiconductors with band edges\nenergetically closer to the Mn 3 dlevels. This increases the hydridization between the host\nanionpstates and the Mn dstates. This however results in a deep acceptor level in the\nband gap. Thus the ferromagnetism-mediating holes are more localiz ed. A system that\nhas been proposed as a strong candidate is a dilute alloy of GaAs with G aP. The hope is\nthat the itinerancy of the carriers is retained while the p-dexchange is enhanced because of\ntheshorterMn-anionbondlengths. LDA+U/coherentpotentiala pproximation(CPA)based\ncalculations [13] have looked at 25%, 50%, 75% alloys of GaAs and GaP a nd found a modest\n2increase in Tc. Experiments [14, 15] have looked at the dilute limit of Ga1−xMnxAs1−yPy\nandGa1−xMnxAs1−yNywith y∼0.01-0.04. There is a strong decrease in Tcwith increasing\ny even at this dilute limit. As the dilute limit has not been studied theoret ically we examine\nwhether the experimental trends may be captured within our calcu lations and if not, then\nthe cause is due to extraneous factors not included in the present approach. We are able to\ncapture the reduction of Tcin GaAs alloyed with GaN as well as the case of GaAs alloyed\nwith GaP.\nII. Methodology\nIn order to address these issues we have considered 216 atom sup ercells of GaAs, GaP\nand GaN. One Mn was placed at the origin while the second was placed at the fourth FCC\nneighbor position as earlier work has shown that the ferromagnetic stability is strongest for\nMn atoms at these positions [16]. We consider dilute III-V alloys as the host semiconductor\ninto which Mn is doped. In order to form the dilute III-V alloys a group V anion impurity is\ncodoped at different distances on the line perpendicular to the line jo ining the two Mn sites.\nFor GaAs alloyed with GaP we included higher dopant concentrations a lso. The optimized\nlattice constants for the GaAs, GaP and GaN are 5.72, 5.49 and 4.52 ˚Arespectively. The\nlattice constant of the supercell is set according to Vegard’s law fo r the III-V alloy. Full\noptimization of the internal positions is carried out within the first pr inciple electronic\nstructure calculations using a plane wave pseudopotential implemen tation of the density\nfunctional theory [17]. A plane wave cutoff of 400 eV was used for th e basis set. The\nelectronic structure was solved considering the generalised gradie nt approximation [18] for\nthe exchange at gamma point alone using projected augmented wav e (PAW) potentials [19].\nIII. Results and Discussion\nWe first consider the case of the As doped into GaN. GaNwe know has a large band offset\nwith GaAs with the former having a deeper valence band maximum. Mn w hen doped into\nGaN has been shown to introduce states into the band gap with signifi cant Mn character.\nHowever, in contrast, Mn when doped into GaAs is found to introduc e states with weak Mn\ncharacter in the band gap. Although in both cases the formal oxida tion state of the Mn is\n3+, in the former case the configuration is d4while in the latter case is {d5+ hole}[20]. Since\nat this dilute limit, alloys show significant band bowing effects, we invest igate Mn doping\nin this limit to probe modifications in the ferromagnetic stability. In Fig.1 we have plotted\nthe Mndpartial density of states. The N ppartial density of states of a N atom which is\n3nearest neighbor of the Mn as well the As ppartial density of states have been shown. The\nvalence band maximum seems to comprise of primarily As pstates with some N padmixture.\nHowever there is hardly any As pcharacter at the Fermi level. We can understand this\neffect within a simple model that was proposed earlier to explain the ele ctronic structure of\ntransition-metal (TM) impurities in semiconductors [20]. The domina nt interaction seems\nto be between the transition-metal impurity and its nearest neighb ors. Atoms farther away\nfrom the TM impurity interact to a much lesser extent.\nThe next question we ask was how is the ferromagnetic stability modifi ed. Should the\npresence of the As impurity affect the ferromagnetic stability? As t he As levels are between\nthe Mndand the N plevels one would expect a modification in the ferromagnetic stability,\npossibly a value between the two end limits of Mn in GaN and GaAs. In this dilute alloy\nlimit the valence band maximum is intermediate between that for GaAs a nd GaN. The\nferromagnetic stability as well as the corresponding mean field estim ate ofTcare given in\nTable. I for various distances of the As impurity from either Mn atom . In the absence of As\nimpurity, the ferromagnetic stability is ∼84 meV and at the other limit of Mn in GaAs it is\n∼164 meV. This drastically drops to ∼21 meV in the presence of an As impurity at 3.76\n˚A. As the As impurity is moved farther away the ferromagnetic stabilit y is partly regained\nthough it still remains less than the value in the absence of As impurity . This is contrary to\nour expectations and indicates that one must consider other fact ors such as alloy scattering\nin addition to the modified energy denominator for the interaction.\nExamining the opposite limit of GaAs into which the N is doped we find a simila r trend\nin the ferromagnetic stabilization energy. In GaAs, Mn doping gives r ise to a ferromagnetic\nstabilizationenergy of164meV(Table. II). Thisdrastically dropsto 100meV(Table. II) for\na N impurity introduced at 4.73 ˚A. Ferromagnetic stability corresponding to the unalloyed\nlimit is partially regained for the N atoms farther away from Mn as show n in Table. II.\nRecent experimental work [14] on dilute magnetic semiconductors h ave focussed on tran-\nsition metal impurities in dilute alloys with an aim of understanding the me chanism of the\nferromagnetism better. If the hole introduced by Mn doping is a vale nce band hole the\nbelief is that it should be weakly perturbed by alloying effects at the dilu te limit. However\na hole which resides in an impurity band is expected to be strongly affec ted by alloying\neffects. We artificially tune the hole introduced by the Mn doping with a introduction of\nUon the Mn dstates in ( Ga,Mn)As0.99N0.01. With a Uof 4 eV on the Mn dstates, the\n4hole moves towards a valence band hole. Hence alloy scattering effec ts are expected to be\nweaker. Indeed our calculated ferromagnetic stability results effe ct this. The ferromagnetic\n(FM) stability results reflect this. The FM stability changes by about 64 meV in the absence\nofUto 15 meV in the presence of U. Stone et al. [14] have examined P doping in GaMnAs.\nAn insulator to metal transition is observed as a function of doping. Usually high effective\nmass have to be assumed for the carriers to explain the experiment al observations. This\nreinforces the idea of the carriers residing in an impurity band.\nWhile Mn in GaN has usually been accepted as a system in which the hole int roduced\nby Mn doping resides in an impurity band, the case of Mn in GaAs is heavily debated with\nsupporters on both sides. In order to quantify our observations further we examined Mn\ndoping in dilute alloys formed by P introduction in GaAs. If our earlier re sults of the N\ndoping in GaAs established the validity of the impurity band model, then we should see\nstrong effects on the ferromagnetic stability here also. The result s as a function of the\nimpurity distance are given in Table. III. The perturbation seems to be very weak and\nconsequently the variations in the ferromagnetic stabilization ener gy from the unperturbed\ncase are small. Hence modifications of ferromagnetic stability in alloye d systems is not proof\nenough for the impurity band picture. The deviation between our re sults and experimental\nresults of GaMnAs 1−yPycould be due to various reasons. One cause could be that the\nalloying concentration are large. Indeed when we introduced two P im purities, such that\neach P atom is the nearest neighbor to one of the Mn atom and also th ese P atoms connects\nthe Mn atoms via a Ga atom, we found a reduction in the ferromagent ic stability from\n164 meV in the unalloyed limit to 110 meV. Thus impurity scattering is res ponsible for the\nreduction in ferromagnetic stability.\nCloser analysis revealed that there are two parts which need to be c onsidered when an\nimpurity atom is introduced. The first part is that associated with th e modified electronic\ninteraction strengths associated with the impurity and the second associated with the strain\nin the host lattice. Indeed we cannot decouple the two effects comp letely but the effects of\nthese perturbations may be discussed within calculations performe d under some constraints.\nIn Table. IV we consider the case of N codoped into GaAs at a distanc e of 4.73 ˚A. The\nferromagnetic stability is evaluated in the unrelaxed case where the calculation is performed\nassuming that N merely replaces an As atom. This calculation would cap ture the effects\nof modified interaction strengths as a result of N doping. Ferromag netic stability of the\n5Mn pairs is increased from 164 meV in the absence of N impurity to 203 m eV. Examining\nthe density of states corresponding to the unrelaxed limit, we find t hat Npstates are\nenergetically closer to the Mn dstates and hence the increased interaction between the two\ncould explain the increased ferromagnetic stability. This is reflected in the inset of Fig.\n2(c) which shows the N pcharacter near Fermi energy. Ga-N bonds are much smaller than\nthe Ga-As bonds. Allowing the atoms to optimize their internal positio ns by total energy\nminimization Wefind that the Ga-Nbonds are ∼2.08˚Along while the Ga-Asbonds close to\ntheNatomare2.53 ˚Along. Those far away fromtheNimpurity are2.48 ˚Along. Examining\nthe ferromagnetic stability in such a configuration, we find that it is d rastically reduced to\n100 meV. Examining the density of states (Fig. 2) we find that there is hardly any change\nin the As pdensity of states plotted for the nearest neighbor As atom as well as the Mn d\npartial density of states with and without relaxation. N pdensity of states show significant\nchanges with relaxation with the N pstates moving from -1 eV to -3.5 eV below the Fermi\nlevel. In the inset of Fig. 2 (c) we magnify the near Fermi energy reg ion of the N pdensity\nof states and see that there is significant N pcharacter at the Fermi energy in the unrelaxed\ncase which is reduced upon relaxation. Since movement of N plevels should imply reduced\ninteraction with Mn, the maineffect of ferromagneticstability reduc tion hasto be the lattice\nstrain. Considering the case of P in GaAs doped with Mn (Fig. 3), we fin d that the effects\nof relaxation are very weak as a result of which ferromagnetic stab ility is hardly affected\n(Table. III). Thus we are able to elucidate the microscopic origin of t he reduction of Tc\nwhen we alloy III-V semiconductors with another group V anion. Alth ough our calculations\nprovide us with qualitative trends, an exact numerical expression f or theTcas a function\nof concentration in the dilute limit is difficult. This is because the ferrom agnetic stability\nis a strong function of the disorder position as well as of concentra tion. Strong disorder\npotentials caused by clustering of the alloying atoms results in a stro nger decrease in Tc\nthan when one has a random distribution.\nIV. Conclusion\nWe have examined Mn doping in the dilute alloy limit where strong valence b and bowing\nis observed, as a possible route to enhance the ferromagnetic tra nsition temperature. This\nhas been quantified in our calculation as the ferromagnetic stabilizat ion energy for a pair of\nMn atoms occupying lattice sites for which ferromagnetic stability ha s been observed to be\nstrong. Contrary to expectations of enhanced ferromagnetic s tability on alloying we find a\n6reduction in the case of Ga1−xMnxAs1−yNyandGa1−xMnxAs1−yPywhere y ∼0.01. Mn\ndoping in Ga1−xMnxAs1−yPyshows very small changes in the ferromagnetic stability for\n1% anion doping but for larger percentage of doping shows a reduct ion inTc. The origin\nof reduced ferromagnetic stability is traced to the strong strain e ffects that accompany the\nintroductionoftheanionimpurity. Thisstronglyscatterstheelect roncomprisingthevalence\nband maximum and therefore modifies the ferromagnetic stability.\nPM thanks the Indo French centre for financial support. CP than ks Swedish Energy\nAgency and Swedish Foundation for International Cooperation in R esearch and Higher Ed-\nucation (STINT).\n[1] T. Jungwirth, J. Sinova, J. Masek, J. Kucera, and A. H. Mac Donald, Rev. Mod. Phys. 78,\n809 (2006).\n[2] K. M. Yu, W. Walukiewicz, T. Wojtowicz, I. Kuryliszyn, X. Liu, Y. Sasaki, andJ. K. Furdyna,\nPhys. Rev. B 65, 201303 (2002). M. J. Seong, S. H. Chun, H. M. Cheong, N. Samar th, and\nA. Mascarenhas, Phys. Rev. B 66, 033202 (2002).\n[3] A. M. Nazmul, T. Amemiya, Y. Shuto, S. 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R.C. et. al\nFIG. 1: (a) The up (solid line) and down (dashed line) spin pro jected density of states for Mn d,\n(b) Np, which is the nearest neigbor of the Mn, and (c) for the As pcalculated for (Ga,Mn)N\ncodoped with As. The zero of energy corresponds to the Fermi e nergy.\n9-4 -2 0 2 402468up_spin-no_relax\nup_spin-relax\n-4 -3 -2 -1 0 1 2 3 400.20.40.60.8DOS (/eV/Cell)\n-4 -3 -2 -1 0 1 2 3 4\nEnergy (eV)01234\n-1 -0.5 0012a) Mn-d\nb) As-p\nc) N-p\nFig.2. R.C. et. al\nFIG. 2: (a) The spin up projected density of states for Mn d, (b) As p, which is the nearest\nneigbor of the Mn, and (c) for the N pcalculated for (Ga,Mn)As codoped with N. Inset of (c)\nshows a magnified view of the N pdensity of states near the Fermi energy region. The solid lin es\ncorresponds to the unrelaxed case while the dashed lines cor responds to the relaxed case. The zero\nof energy corresponds to the Fermi energy.\n10-4 -3 -2 -1 0 1 2 3 402468 up_spin-no_relax\nup_spin-relax\n-4 -3 -2 -1 0 1 2 3 400.20.40.60.8DOS (/eV/Cell)\n-4 -3 -2 -1 0 1 2 3 4\nEnergy (eV)00.511.52a) Mn-d\nb) As-p\nc) P-p\nFig.3. R.C. et. al\nFIG. 3: (a) The spin up projected density of states for Mn d, (b) As p, which is the nearest\nneigbor of the Mn, and (c) for the P pcalculated for (Ga,Mn)As codoped with P. The solid lines\ncorresponds to the unrelaxed case while the dashed lines cor responds to the relaxed case. The zero\nof energy corresponds to the Fermi energy.\n11TABLE II: Effect on the stability by N codoping in (Ga,Mn)As hos t. The distance of the N\natom from either Mn atoms in both the ferromagnetic (FM) and a ntiferromagnetic (AFM) relaxed\nconfiguration is also given.\nDistance ( ˚A)Ferromagnetic stability (meV) Tc(K)\nFMAFM (EFM−EAFM)\n-- -163.8 633.6\n4.734.75 -100.1 387.2\n7.417.42 -154.3 596.9\n8.448.44 -161.7 625.5\n10.1810.19 -147.5 570.6\nTABLE III: Effect on the stability by P codoping in (Ga,Mn)As ho st. The distance of the P\natom from either Mn atoms in both the ferromagnetic (FM) and a ntiferromagnetic (AFM) relaxed\nconfiguration is also given.\nDistance ( ˚A)Ferromagnetic stability (meV) Tc(K)\nFMAFM (EFM−EAFM)\n-- -163.8 633.6\n4.734.75 -154.3 596.9\n7.427.44 -162.1 627.0\n8.458.45 -169.1 654.1\n10.2110.21 -172.7 668.0\nTABLE IV: Effects of relaxation on the stability in the case of ( Ga,Mn)As host codoped with N.\nEnergy (eV) Ferromagnetic stability (meV) Tc(K)\nEFMEAFM (EFM−EAFM)\nUn-Relaxed -907.192 -906.989 -203.2 786.0\nRelaxed -909.264 909.164 -100.1 387.2\n12"    },    {        "title": "1910.01959v1.Intrinsic_2D_Ferromagnetism_in_V5Se8_Epitaxial_Thin_Films.pdf",        "content": " 1Intrinsic 2D Ferromagnetism in V 5Se8 Epitaxial Thin \nFilms \nMasaki Nakano,*,†,‡,¶ Yue Wang,†,¶ Satoshi Yoshida,†,¶ Hideki Matsuoka,† Yuki Majima,† Keisuke \nIkeda,§ Yasuyuki Hirata,§,‖ Yukiharu Takeda,⊥ Hiroki Wadati,§,‖,★ Yoshimitsu Kohama,‖ Yuta \nOhigashi,† Masato Sakano,† Kyoko Ishizaka,†,‡ and Yoshihiro Iwasa†,‡ \n†Quantum-Phase Electronics Center and Department of Applied Phys ics, the University of \nTokyo, Tokyo 113-8656, Japan.  \n‡RIKEN Center for Emergent Matter  Science (CEMS), Wako 351-0198,  Japan.  \n§Department of Physics, the Uni versity of Tokyo, Tokyo 113-0033,  Japan.  \n‖The Institute for Solid State Phys ics, the University of Tokyo,  Kashiwa 227-8581, Japan.  \n⊥Materials Sciences Research Cent er, Japan Atomic Energy Agency,  Hyogo 679-5148, Japan.  \n \n   2ABSTRACT : The discoveries of intrinsic ferromagnetism in atomically-thi n van der Waals \ncrystals have opened up a new re search field enabling fundament al studies on magnetism at two-\ndimensional (2D) limit as well as development of magnetic van d er Waals heterostructures. To \ndate, a variety of 2D ferromagne t i s m  h a s  b e e n  e x p l o r e d  m a i n l y  b y mechanically exfoliating \n‘originally ferromagnetic (FM)’ va n der Waals crystals, while b ottom-up approach by thin film \ngrowth technique has demonstrated emergent 2D ferromagnetism in  a variety of ‘originally non-\nFM’ van der Waals materials. Here we demonstrate that V 5Se8 epitaxial thin films grown by \nmolecular-beam epitaxy (MBE) exh ibit emergent 2D ferromagnetism  with intrinsic spin \npolarization of the V 3 d electrons despite that the bulk counterpart is ‘originally ant iferromagnetic \n(AFM)’. Moreover, thickness-dependence measurements reveal that  this newly-developed 2D \nferromagnet could be classified a s an itinerant 2D Heisenberg f erromagnet with weak magnetic \nanisotropy, broadening a lineup of 2D magnets to those potentia lly beneficial for future spintronics \napplications.  \n \nKEYWORDS : 2D magnet, vanadium selenide, molecular-beam epitaxy, anomalo us Hall effect, \nx-ray magnetic circular dichroism   \n   3Exploring novel 2D magnets that exhibit long-range magnetic ord er at the 2D limit is one of \ncentral topics in modern condense d-matter researches based on 2 D materials and their \nheterostructures.1-5 There, the most-frequently-used  approach is to mechanically ex foliate FM van \nder Waals crystals down to the ultrathin regime, providing a si mple but versatile route to \nexploration of intrinsic 2D ferromagnetism as initially demonst rated on insulating ferromagnets, \nCr2Ge2Te6 ( C G T )  a n d  C r I 3.1,2 Those pioneering works as well as the recent study on metallic  \nferromagnet, Fe 3GeTe 2 ( F G T ) ,3 have experimentally verified the fundamental aspect of 2D \nferromagnets at two extreme condi tions; long-range FM order cou ld be stabilized down to the \nmonolayer limit in Ising-type ferromagnets with strong uniaxial  magnetic anisotropy ( i.e., CrI 3 and \nFGT), whereas it is strongly suppressed at the 2D limit in Heis enberg-type ferromagnets with weak \nanisotropy ( i.e., CGT) due to thermal fluctuations in accord with the Mermin-Wa gner theorem.6 \nHowever, very intriguingly, it turned out that 2D Heisenberg fe rromagnets give rise to an emergent \nfunctionality enabling control of the FM transition temperature  by external magnetic fields,1 which \nis unique to this class of 2D ferr omagnets with weak magnetic a nisotropy.  \nApart from mechanical exfoliation, recent development of thin f ilm growth technique has \nenabled us to investig ate magnetic properties of a wide variety  of van der Waals materials down \nto the monolayer limit even including hardly-cleavable and ther modynamically-metastable \ncompounds, providing a unique and powerful path to exploration of novel 2D magnets. In fact, \nthose atomically-thin samples grown by bottom-up approach have often provided emergent room-\ntemperature 2D ferromagnetism in originally non-FM compounds as  demonstrated in monolayer \nVSe 2 and MnSe x epitaxial thin films grown by MBE as well as in VTe 2 a n d  N b T e 2 ultrathin \nnanoplates grown by chemical-vapor deposition.7-9. However, in those studies, magnetic properties \nwere mainly characterized by magnetization measurements using a  superconducting quantum  4interference device magnetometer, which could not distinguish i ntrinsic FM signals from extrinsic \nones in particular when the total magnetic moments were very sm all in the ultrathin regime. Recent \nspectroscopic studies on MBE-grown monolayer VSe 2 suggest that the ground state of this \ncompound should be the charge-density wave state rather than th e FM state,10-14 indicating the \nessential importance of  evaluation of magne tic properties by di fferent approaches.  \nI n  t h i s  s t u d y ,  w e  e x a m i n e d  m a g n e t i c  p r o p e r t i e s  o f  v a n a d i u m  s e l e nide epitaxial thin films \ngrown by our MBE process15-17 through anomalous Hall effect (AHE) and x-ray magnetic circula r \ndichroism (XMCD) measurements, which should enable us to probe only intrinsic FM signals. \nImportantly, detailed structural c haracterization verified that  our films evolve from VSe 2 to V 5Se8 \nsoon after the formation of the first layer. Figure 1a,b illust rates a schematic structure of V 5Se8 \ncrystal,18 which is characterized by the 2D VSe 2 sheets separated by the layers of the periodically-\naligned V atoms (labelled as VI). Figure 1c shows the cross-sectional scanning transmission \nelectron microscope image of our typical film, indicating the e xistence of the well-ordered VI \natoms in between the VSe 2 layers. We confirmed that those V interlayers were formed just  after \nthe formation of the first VSe 2 layer. Figure 1d shows the reflection high energy electron \ndiffraction (RHEED) pattern taken at the end of the growth, and  Figure 1e exhibits the evolution \nof the in-plane lattice paramete r during the growth evaluated f rom the spacing between the RHEED \nstreaks. The lattice parameter initially matched the a-axis length of VSe 2,19 while it largely \ndeviated from that of VSe 2 during the formation of the second VSe 2 layer and stayed almost \nunchanged until the end of the g rowth. The saturated value tune d out to match the a*-axis length \nof V 5Se8,18 indicating the formation of V 5Se8 after the second layer. The formation of V 5Se8 lattice \ncould be also confirmed by the out-of-plane x-ray diffraction p attern shown in Figure 1f, indicating  5that the out-of-plane lattice parameter was close to the c*-axis length of V 5Se8 rather than the c-\naxis length of VSe 2.18,19  \nBulk V 5Se8 is known to be an itinerant antiferromagnet with the Néel temp erature TN ~ 30 \nK,20,21 while our V 5Se8 epitaxial thin films turned out to exhibit the spontaneous FM order that is \nmissing in bulk. Figure 2a shows the Hall-effect data of our 30  ML-thick film, demonstrating \nnegative AHE signals with clear magnetic hysteresis loop at T = 2 K. This hysteresis disappeared \nwith increasing temperature at around T ~ 10 K, while the AHE signals survived up to T ~ 20 K. \nThese results verified that our film entered the FM phase below  T ~ 20 K while the spontaneous \nFM order developed below T ~ 10 K. On the other hand, the magnetoresistance (MR) \nmeasurements proved the existence of the AFM order together wit h the FM order. The previous \nstudies on a sister sulfide compound V 5S8 demonstrated unique anisotropy in MR in the AFM \nphase.22,23 Figure 2b shows the MR curves of our 30 ML-thick film with the  out-of-plane and in-\nplane magnetic fields taken at d ifferent temperatures. Negative  MR was observed in both \nconfigurations, while the magnitude was different depending on the direction of the magnetic \nfields, which should be associated with the existence of the AF M order.22,23 The onset temperature \nwas determined to be around T = 30 K, which is consistent to TN of bulk V 5Se8 and slightly above \nthe onset temperature of the FM order. These results therefore suggest that our film hosts the AFM \nphase with TN ~ 30 K together with the FM phase below T ~ 20 K.  \nThickness dependence measurements  revealed that the observed sp ontaneous FM order was \nstrongly suppressed at the monolayer limit. Figure 3a shows the  Hall-effect data at T = 2 K for the \ndifferent-thickness samples, the 30 ML-, 8 ML-, and 3 ML-thick films. All the films demonstrated \nclear AHE signals, while the hysteresis loop associated with th e spontaneous FM order nearly \ndisappeared for the 3 ML-thick film. Considering that the 3 ML- thick film should have the same  6structure and composition as those of the thicker films (see Fi gure 1e and Supporting Information), \nthe observed suppression of the spontaneous FM order at the 2D limit should be attributed to the \ndimensionality effect. Figure 3b shows the results of the detai led temperature-dependence \nmeasurements, where the anomalous Hall resistivity ( AH) at the magnetic field  0H = -3 T (AH, \nsat, corresponding to the saturation magnetization) and at 0H = 0 T (AH, rem, corresponding to the \nremanent magnetization) are both plotted as a function of tempe rature (see Supporting Information \nfor the corresponding AHE data). We here define TC* and TC as the temperatures at which AH, sat \nand AH, rem start to develop, respectively,  which were determined to be 17  K and 7 K for the 30 \nML-thick film. With reducing thickness, on the other hand, TC turned out to be significantly \nsuppressed down below 2 K for the 3 ML-thick film, while TC* remained almost unchanged. Those \nresults are essentially the same  as those observed in CGT nano- thick crystals,1 where the \nspontaneous FM order was suppre ssed at the monolayer limit, whi ch is a characteristic feature of \nHeisenberg-type ferroma gnets with weak magnetic anisotropy. Tak en together, we conclude that \nthis newly-developed ferromagnet should be classified as an iti nerant 2D Heisenberg ferromagnet, \na new type of 2D magnets that ha ve not been developed so far in cluding exfoliated flakes. Figure \n3c summarizes the evolutions of TC* and TC as a function of the layer number together with that of \nthe AH/xx value calculated with AH, sat at T = 2 K.  \nThe observed AHE signals verified the existence of the spin-pol arized carriers in our V 5Se8 \nepitaxial thin films, while XMCD measurements elucidated that t he emergent FM phase in fact \narises from intrinsic spin polarization of the V 3 d electrons. Owing to the element-specific nature \nas well as the very high sensitivity, XMCD has been widely used  to investigate the origin of the \nmagnetism even in diluted systems as well as interface systems.24,25 Figure 4a shows the x-ray \nabsorption spectra (XAS) of the 30 ML-thick film near the V L2,3-edges taken at the base  7temperature T =  5  K  w i t h  r i g h t - h a n d e d  c i r c u l a r l y - p o l a r i z e d  ( R C P )  a n d  l e f t - h a nded circularly-\npolarized (LCP) incident light under the out-of-plane magnetic fields. The helicity switching of \ncircularly-polarized light was operated at 1 Hz using the twin helical undulator26. There was clear \ndifference in the XAS spectra between RCP and LCP setups, which  changed its sign under reversal \nof the magnetic field direction, indicating the existence of th e intrinsic XMCD signals. The \nresultant XMCD spectra are shown in Figure 4b, demonstrating th a t  t h e  f i n i t e  X M C D  s i g n a l s  \nappeared only near the absorption peaks. This indicates that th ose signals are indeed originating \nfrom intrinsic spin polarization of the V 3 d electrons. Figure 4c shows the magnetic field \ndependence of the XMCD intensity taken at the photon energy of h =  5 1 7 . 8  e V ,  c l e a r l y  \ndemonstrating non-linear behavior, which should be arising from  the FM order. This non-linear \nbehavior turned out to disappear at T = 25 K, suggesting that the onset temperature of the FM order \nshould be located between T = 5 K and T = 25 K, which is consistent to the transport results with \nTC* = 17 K for the 30 ML-thick film . The linear response observed at T = 25 K should arise either \nfrom the AFM phase or from the enhanced paramagnetic response n ear TC*, although further \nstudies are needed to address its origin. Nevertheless, the obt ained results strongly support that the \nemergent FM phase in our V 5Se8 epitaxial thin films should hav e intrinsic origin arising from  spin \npolarization of the V 3 d electrons.  \nAs a driving force that could stabilize the FM phase in V 5Se8 epitaxial thin films, we here \npropose three possible origins. Considering that bulk V 5Se8 hosts the AFM order and that the FM \norder in our V 5Se8 films appears just below TN, it is natural to think that the observed \nferromagnetism could be categorized as the weak ferromagnetism (or the canted \nantiferromagnetism in other words), which should originate from  the Dzyaloshinskii-Moriya (DM) \ninteraction induced by the broken space inversion symmetry. Giv en that V 5Se8 i s  a   8centrosymmetric material,18 the DM interaction should appear as a consequence of the local  \nsymmetry breaking at the substrate-film interface. A second pos sibility could be the effect of the \nepitaxial strain. The in-plane lattice parameters of our V 5Se8 films evaluated by the RHEED \nanalysis were somewhat larger than that of bulk V 5Se8 (see Figure 1e), implying the existence of \nthe tensile strain in our V 5Se8 films. This suggests that the distance between the interlayer VI atoms \nis a bit increased, which might influence the magnetic exchange  interaction in V 5Se8. Another \npossibility could be the dimensionality effect. The recent stud y on a sister sulfide compound V 5S8 \nreported a signature of the evol ution of the magnetic ground st ate from the AFM phase to the FM \nphase with reducing thickness, which was attributed to the dime nsionality effect.27 Although the \nmaterial is different, a similar effect might be relevant to ou r V 5Se8 films. And also, this implies \nthat reducing dimensionality not only leads to the suppression of the spontaneous FM order at the \n2D limit, but also triggers the evolution of the magnetic groun d state at the relatively thicker regime \nin this material family. Those three mechanisms are just exampl es, and further studies are required \nto address the underlying microscopic mechanism both from theor y and experiment viewpoints. \nHowever, we note that the AFM pha se and the FM phase should be almost energetically degenerate \nin this material family due to its unique magnetic structure, a nd therefore, a small modification in \ntheir structural and electronic properties could trigger the ev olution of the magnetic ground state. \nDetailed discussions on the magne tic structure are provided in Supporting Information.  \nThe present study demonstrates t he emergence of intrinsic ferro magnetism in V 5Se8 epitaxial \nthin films grown by MBE. The thi ckness dependent AHE verified t hat this material system forms \na new class of 2D magnets categorized as an itinerant 2D Heisen berg ferromagnet, which has not \nbeen developed so far including exfoliated flakes. These Heisen berg-type ferromagnets are \ncharacterized with weak magnetic anisotropy, which should be ad vantageous for controlling  9magnetic properties by external means including magnetic and el ectric fields as well as by the \nproximity effect in heterostructures, potentially providing a n ovel functionality unachievable with \nIsing-type ferromagnets with strong uniaxial magnetic anisotrop y. The current work also \ndemonstrates the fundamental imp ortance of employing AHE and XM CD for evaluation of \nintrinsic ferromagnetism in 2D ma terials in the ultrathin regim e.  \n ASSOCIATED CONTENT  Supporting Information  \nThe Supporting Information is ava ilable free of charge on the A CS Publications website at DOI:  \nExperimental details, magnetic structure of V\n5S8, further characterization of the AFM phase, and \nthe detailed transport data including the temperature dependenc e of the resistivities and the Hall \ncoefficients as well as the detailed AHE data of the different- thickness samples. (PDF)  \n \nAUTHOR INFORMATION  \nCorresponding Author  \n*E-mail: nakano@ap.t.u-tokyo.ac.jp  \nPresent Addresses  \n★Graduate School of M aterial Science, University of Hyogo, Hyogo  678-1297, Japan.  \nAuthor Contributions   10¶M.N., Y.W., and S.Y. contributed equally to this work. Y.W., S. Y., H.M., and Y.M. grew and \ncharacterized the thin films. M.N ., Y.W., H.M., and Y.M. perfor med transport measurements and \nanalyzed the data. M.N., S.Y., K.Ikeda, Y.H., Y.T., and H.W. pe rformed x-ray magneto-optical \nmeasurements. M.N., H.M., and Y. K. performed magneto-transport measurements at high fields. \nY.O. and M.S. supported the expe riments. M.N., K.Ishizaka, and Y.I. supervised the study. M.N. \nwrote the manuscript. All the authors discussed the results and  commented on the manuscript.  \nNotes  \nThe authors declare no compe ting financial interest.  \n \nACKNOWLEDGMENT  \nWe are grateful to S. M. Bahramy for valuable discussions and t o Y. Kashiwabara, T. Shitaokoshi, \nK. Matsui, M. Kawasaki, M. Uchid a, and T. Fujita for experiment al help. This work was supported \nby Grants-in-Aid for Scientific R esearch (Grant Nos. 19H05602, 19H02593, and 19H00653) and \nA3 Foresight Program from the Japan Society for the Promotion o f Science (JSPS). M.N. was \npartly supported by The Murata Science Foundation. H.M. was sup ported by JSPS through \nProgram for Leading Graduate Schools (MERIT). The synchrotron x -ray magneto-optical \nmeasurements were performed at B L23SU in SPring-8 (Proposal No.  2018A3843).  \n \nREFERENCES  \n(1) Gong, C.; Li, L.; Li, Z.; Ji, H.; Stern, A.; Xia, Y.; Cao, T.; Bao, W.; Wang, C.; Wang, Y.; \nQiu, Z. Q.; Cava, R. J.; Louie, S. G.; Xia, J.; Zhang X. Nature  2017 , 546, 265.   11(2) Huang, B.; Clark, G.; Navarro-Moratalla, E. ; Klein, D. R.; Chen g, R.; Seyler, K. L.; Zhong, \nD.; Schmidgall, E.; McGuire, M. A.; Cobden, D. H.; Yao, W.; Xia o, D.; Jarillo-Herrero, P.; \nXu, X. Nature  2017 , 546, 270.  \n(3) Fei, Z.; Huang, B.; Malinowski, P.; Wang, W.; Song, T.; Sanchez , J.; Yao, W.; Xiao, D.; \nZhu, X.; May, A. F.; Wu, W.; Cobden, D. H.; Chu, J.-H.; Xu, X. Nature Mater.  2018 , 17, \n778.  \n(4) Burch, K. S.; Mandrus, D.; Park, J.-G. Nature  2018 , 563, 47.  \n(5) Gong, C.; Zhang, X. Science  2019 , 363, eaav4450.  \n(6) Mermin, N. D.; Wagner, H. Phys. Rev. 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Y.; Kao, C.-C. Nature \nMater.  2013 , 12, 703.  \n(26) Saitoh, Y.; Fukuda, Y.; Takeda, Y. ; Yamagami, H.; Takahashi, S. ;  A s a n o ,  Y . ;  H a r a ,  T . ;  \nShirasawa, K.; Takeuchi, M.; Tanaka, T.; Kitamura, H. J. Synchrotron Rad.  2012 , 19, 388.  \n(27) Niu, J.; Yan, B.; Ji, Q.; Liu, Z.; Li, M.; Gao, P.; Zhang, Y.; Yu, D.; Wu, X. Phys. Rev. B  \n2017 , 96, 075402.  \n   14 \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \nFigure 1.  Basic structural properties of V\n5Se8 epitaxial thin films grown by MBE. (a) Schematic \ntop view and (b) side view of V 5Se8 crystal. (c) The cross-sectional scanning transmission electro n \nmicroscope image of the 30 ML-thick film taken along [1―\n10] azimuth of the substrate (the direction \nindicated by an arrow in a). The yellow dots indicate the posit ions of the interlayer V atoms. The \ninset shows the corresponding schematic crystal structure. (d) The RHEED pattern of a typical \nfilm taken along [110] azimuth of the substrate after the whole  growth process. The Miller indices \nof the film are converted from original ones characterized with  a-, b-, and c-parameters defined in \na and b to the setting of VSe 2 lattice with a*-, b*-, and c*-parameters, which were calculated using \nthe relations, a* = b/2, b* = (c/2)sin × 2/√3, and c* = a/2. (e) The evolution of the in-plane lattice \nparameter during the film growth calculated from the spacing be tween the RHEED streaks. The \nlayer number is defined as t he number of the host VSe 2 layer. The a-axis length of VSe 2 and the \na*-axis length of V 5Se8 are taken from the literatures.18,19 (f) The out-of-plane x-ray diffraction \npattern of the 30 ML-thick film. The expected peak positions of  VSe 2 and V 5Se8 are calculated \nfrom the c-axis length of VSe 2 and the c*-axis length of V 5Se8 taken from the literatures.18,19  \n \n  \n 15 \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \nFigure 2.  The magneto-transport prope rties of the 30 ML-thick V\n5Se8 epitaxial thin film. (a) AH \nof the 30 ML-thick film calculated with the anti-symmetrized tr ansverse resistivity as a function \nof the magnetic fields at different temperatures. The data at T = 30 K, 20 K, 10 K, 5 K, and 3 K \nare vertically shifted by 0.30, 0.24, 0.18, 0.12, and 0.06 cm, respectively. (b) The symmetrized \nMR curves of the 30 ML-thick film with the out-of-plane (red) a nd in-plane (blue) magnetic fields \nat different temperatures. T he MR value is defined as [ (0H)-0]/0. The data at T = 20 K, 10 K, \n5 K, 3 K, and 2 K are vertically shifted by -0.25, -0.50, -0.75 , -1.00, and -1.25 %, respectively. \nThe inset shows the difference in MR defined as the MR with the  out-of-plane fields minus that \nwith the in-plane fields a t different temperatures.  \n \n  \n 16 \n \n \n \n \n \n \n \n  \n \nFigure 3.  Dimensionality effect on ferromagnetism of V\n5Se8 epitaxial thin films down to the \nmonolayer limit. (a) The AH versus 0H curves of the 30 ML- (red), 8 ML- (green), and 3 ML- \n(blue) thick films at T = 2 K. The data of the 30 ML- and 3 ML-thick films are vertica lly shifted \nby 0.1 and -0.1 cm, respectively. The data of the 30 ML-thick film is the same as that shown \nin Figure 2a. (b) The temperature dependences of AH of the same samples. AH, sat (filled symbols) \ncorresponds to AH at 0H = -3 T while AH, rem (open symbols) corresponds to AH at 0H = 0 T. \nThe data of the 30 ML- and 8 ML-thick films are vertically shif ted by 0.20 and 0.10 cm, \nrespectively. TC* and TC are defined as the temperatures at which AH, sat and AH, rem start to develop, \nrespectively. (c) The resultant phase diagram, where TC* (filled symbols) and TC (open symbols) \nare plotted as a function of the layer number. The monolayer li mit corresponds to the 2 ML-thick \nfilm, which consists of two VSe 2 sheets separated by one layer of  the periodically-aligned V at oms. \nTN (black cross) of the 30 ML-thick film is also plotted. The ins et shows the corresponding AH/xx \nvalues of those films evaluated with AH, sat at T = 2 K.  \n \n  \n 17 \n \n \n \n \n \n \n \n  \n \nFigure 4.  The x-ray magneto-optical properties of the 30 ML-thick V\n5Se8 epitaxial thin film. (a) \nThe XAS of the 30 ML-thick film near the V L2,3-edges taken under illuminations of RCP and \nLCP with the out-of-plane magnetic fields 0H = ±9 T at the base temperature T = 5 K. (b) The \naveraged XMCD spectra at the different magnetic fields. Each da ta was calculated by subtracting \nthe IRCP-ILCP data taken at the positive magnetic fields from those at the n egative fields. (c) The \nmagnetic field dependences of th e anti-symmetrized XMCD intensi ty taken at the photon energy \nof h = 517.8 eV indicated by an arrow in b measured at T = 5 K (red  and blue symbols fo r \ndecreasing- and increasing-field sweeps, respectively) and at T =  2 5  K  ( b l a c k  s y m b o l s  f o r  \ndecreasing-field sweep).  \n \n  \n 18TOC FIGURE:  \n \n \n \n \n \n \n \n \n \n1  \nSupporting Info rmation for \nIntrinsic 2D Ferromagnetism in V 5Se8 Epitaxial Thin \nFilms \nMasaki Nakano,*,†,‡,¶ Yue Wang,†,¶ Satoshi Yoshida,†,¶ Hideki Matsuoka,† Yuki Majima,† Keisuke \nIkeda,§ Yasuyuki Hirata,§,‖ Yukiharu Takeda,⊥ Hiroki Wadati,§,‖,★ Yoshimitsu Kohama,‖ Yuta \nOhigashi,† Masato Sakano,† Kyoko Ishizaka,†,‡ and Yoshihiro Iwasa†,‡ \n†Quantum-Phase Electronics Cent er and Department of Applied Phys ics, the University of Tokyo, \nTokyo 113-8656, Japan. \n‡RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198,  Japan. \n§Department of Physics, the  University of Tokyo, Tokyo 113-0033,  Japan. \n‖The Institute for Solid State Physics, the University of Tokyo,  Kashiwa 227-8581, Japan. \n⊥Materials Sciences Research Cen ter, Japan Atomic Energy Agency,  Hyogo 679-5148, Japan. \n \n \n \nA. Experimental details  \nThin film growth and struc tural characterizations  \nAll the films were grown on insulating Al 2O3 (sapphire) (001) substrates by molecular-beam \nepitaxy (EIKO Engineering) by following our recently-establishe d growth process.15-17 The growth \ntemperature was set to 450 oC. During the growth, V was supplied by an electron beam evapor ator \nwith the evaporation rate of ~ 0.01 Å/s, while Se was supplied by a standard Knudsen cell \nthroughout the growth process with  the rate of ~ 2.0 Å/s. The o btained films were annealed in-situ  \nat the same temperature for 30-60 minutes to improve the crysta llinity. All the films were covered \nby thick insulating Se capping lay ers to protect their surfaces  from oxidization. The crystallinities of \nthe obtained films were characterized in-situ  by reflection high energy electron diffraction and \nex-situ  by a four-circle x-ray diffract ometer (PANalytical, X’Pert MRD ). The local struc ture w as  \ncharacterized ex-situ  by scanning transmission electr on microscope (JEOL, JEM-ARM200 F).  \n 2 Transport measurements  \nAll the films were cut into Hall-bar shape by mechanical scratc hing over the Se capping layers in \norder to define the channel regi o n s  ( t y p i c a l l y  a  f e w  h u n d r e d s  o f micrometers). The electrical \ntransport properties were characterized by Physical Property Me asurement System (Quantum \nDesign, PPMS). The magnetoresist ance curve measured up to 0H ~ 60 T was taken by the \npulsed-field system at the Inst itute for Solid State Physics, t he University of Tokyo.  \n \nX-ray magnetic circular dichroism measurements  \nThe synchrotron x-ray magneto- optical measurements including XA S and XMCD measurements \nwere performed at BL23SU in SPring-8. The helicity switching of  circularly-polarized light was \noperated at 1 Hz using t he twin helical undulator.26 All the data were taken in t he total electron yield \nmode at different temperatures under different magnetic fields.  The magnetic field direction was \nperpendicular to the sample a nd parallel to the incident beam d irection.  \n \nB. Magnetic structure of bulk V 5X8 (X = S, Se)  \nA family of V 5X8 (X = S, Se) has been studied for a long time since ’70. Figure S1 a shows a \nschematic structure of V 5Se8.18 There are three inequivalent V a toms drawn by three different colors. \nVI (red) represents the interlayer V atoms while VII (blue) and VIII (green) correspond to the V \natoms constituting the VSe 2 layers. The difference between VII and VIII is whether the V atoms are \nbonded to the VI atoms or not. Magnetic propertie s of this material system have  been well studied \nmainly on its sister sulfide compound V 5S8 (TN ~ 25-35 K),21-23,28-32 showing that magnetism is \nhosted by the VI atoms while itinerancy is provided by the 2D VS 2 sheets containing the VII and VIII \natoms. Figure S1b illustrates the magnetic structure of V 5S8 in the AFM phase determined by the \nneutron diffraction measurements.32 The magnetic moments are though t to be arising mainly from \nthe VI atoms, whereas those from the VII and VIII atoms are believed to be negligible. The local \nspins at the VI atoms are ferromagnetically a ligned along the out-of-plane dir ection, forming a 2D \n‘FM sheet’ within the a-b plane. The neighboring two FM sheets form one ‘FM block’, whic h is \nantiferromagneticall y coupled to the next FM block, forming a m agnetic superstructure. This unique \nmagnetic structure suggests that the FM coupling between two FM  sheets and the AFM coupling \nbetween two FM blocks are almost energetically degenerate. The direction of each spi n is a bit tilted \nfrom the a-axis direction toward the c-axis direction within the a-c plane by approximately 10 \ndegree. Although the magnetic structure of V 5Se8 has not yet been determined by neutron \ndiffraction measurements, we assume it should be similar to tha t of V 5S8 because of many \nsimilarities in terms of the cryst al structure as well as the e lectronic and magne tic properties.  \n \nC. Further characterizations of the AFM phase  \nFigure S2a shows the MR curve of another 30 ML-thick V 5Se8 epitaxial thin film with the \nout-of-plane magnetic fields taken at T = 2 K. There was small but finite magnetic hysteresis loop \nobserved near 0H = ±5 T, most likely originating from the spin-flop transition i n  t h e  A F M  3 phase.22,23. Figure S2b presents the MR curve measured up to 0H ~ 60 T with the out-of-plane \nmagnetic fields at T = 0.7 K by the pulsed-field system. The negative MR saturated at around 0H ~ \n50 T, which is consistent to the saturation of the magnetizatio n measured with bulk V 5Se8 \npolycrystalline samples,21 indicating that the negative MR originates from the AFM order.  The \nZeeman energy at 0H = 50 T corresponds to the thermal energy at T ~ 30 K, which roughly \nmatches TN of our 30 ML-thick film charact erized by the MR anisotropy mea surements (see the \nmain text for details). Figure S3 a-f shows the MR curves of the  30 ML-, 8 ML-, and 3 ML-thick \nfilms with the out-of-plane and in-plane magnetic fields taken at different temperatures. The \nanisotropy in MR could be clearly seen in the thinner films as well, which was pronounced with \ndecreasing temperature while the onset temperature of this anis otropy in MR was determined to be \naround T ~ 30 K for all the samp les irrespective of  the film thickness.   \n \nD. The basic transport propertie s of the different-thickness sa mples  \nFigure S4a shows the l ongitudinal resistivity ( xx) versus temperature curves of the \ndifferent-thickness samples used in this study, the 30 ML-, 8 M L-, and 3 ML-thick films, \nrespectively. The 30 ML- and 8 ML-thick films showed metallic b ehavior, whereas the 3 ML-thick \nfilm exhibited weakly insulating behavior. Similar evolution fr om metallic to insulating behavior \nwith reducing thickness was also observed in another itinerant 2D ferromagnet, Fe 3GeTe 2,33 \npresumably originating from carri er localization effect at the 2D limit. Figure S4b shows the Hall \ncoefficient ( RH) versus temperature curves of the same films together with tha t of bulk V 5S8.30 The \ncharacteristic temperature depe ndence with the sign change in RH were commonly observed, \nindicating that the structures a nd compositions are essentially  the same among those samples.  \n \nE. The detailed AHE data of the different-thickness samples  \nFigure S5a-c shows the AH versus 0H curves of the different-thickne ss samples taken at different \ntemperatures. The normal Hall co mponents were subtracted for al l the data.  \n \nAdditional references  \n(28) De Vries, A. B.; Haas, C. J. Phys. Chem. Solids  1973 , 34, 651.  \n(29) Oka, Y.; Kosuge, K.; Kachi, S. Phys. Lett.  1974 , 50A, 311.  \n(30) Nozaki, H.; Ishizawa, Y.;  Saeki, M.; Nakahira, M. Phys. Lett.  1975 , 54A, 29.  \n(31) Kitaoka, Y.; Yasuoka, H. J. Phys. Soc. Jpn.  1980 , 48, 1949.  \n(32) Funahashi, S.; Nozaki, H.; Kawada, I. J. Phys. Chem. Solids  1981 , 42, 1009.  \n(33) Deng, Y.; Yu, Y.; Song, Y.; Zhang, J.; Wang, N. Z.; Sun, Z.; Yi , Y.; Wu, Y. Z.; Wu, S.; Zhu, \nJ.; Wang, J.; Chen, X. H.; Zhang, Y. Nature  2018 , 563, 94.  \n 4  \n  \n \n  \n \n  \n \n  \n \n  \nFigure S1.  (a) The crystal structure of V\n5Se8.18 VI, VII and VIII represent the inequivalent V atoms. \n(b) The magnetic structure of V 5S8 in the AFM phase determined by the neutron diffraction \nmeasurements.32  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \nFigure S2.  (a) The symmetrized MR curve of  another 30 ML-thick film measu red with the \nout-of-plane magnetic fields at T = 2 K. The magnetic hysteresis loop is most likely originating  \nfrom the spin-flop transition in the AFM phase. (b) The symmetr ized MR curve measured up to \n0H \n~ 60 T with the out-of-pla ne magnetic fields at T = 0.7 K by the pulsed-field system.  \n \n \n5  \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \nFigure S3.  The symmetrized MR curves of the (a) 30 ML-, (c) 8 ML-, and (e ) 3 ML-thick films, \nrespectively, taken with the out-of-plane and in-plane magnetic  fields at different temperatures. The \nMR value is defined as [\n(0H)-0]/0. The corresponding difference in MR are shown for the (b) \n30 ML-, (d) 8 ML-, and (f) 3 ML-th ick films, respectively.  \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \nFigure S4.  (a) The \nxx-T and (b) the RH-T curves of the 30 ML-, 8 ML-, and 3 ML-thick films. The \nRH-T curve of bulk V 5S8 is also shown.30  \n \n6  \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \nFigure S5.  The anti-symmetrized \nAH-0H curves of the (a) 30 ML-, (b) 8 ML-, and (c) 3 ML-thick \nfilms, respectively, taken a t different temperatures.  \n \n"    },    {        "title": "1904.01458v3.Quantum_storage_in_quantum_ferromagnets.pdf",        "content": "Quantum storage in quantum ferromagnets\nYingkai Ouyang1, 2, 3,\u0003\n1Department of Physics and Astronomy, University of Sheffield, Sheffield, UK\n2Singapore University of Technology and Design, 8 Somapah Road, Singapore\n3Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore\nWe must protect inherently fragile quantum data to unlock the potential of quantum technologies. A perti-\nnent concern in schemes for quantum storage is their potential for near-term implementation. Since Heisenberg\nferromagnets are readily available, we investigate their potential for robust quantum storage. We propose to use\npermutation-invariant quantum codes to store quantum data in Heisenberg ferromagnets, because the ground\nspace of any Heisenberg ferromagnet must be symmetric under any permutation of the underlying qubits. By\nexploiting an area law on the expected energy of Pauli errors, we show that increasing the effective dimension\nof Heisenberg ferromagnets can improve the storage lifetime. When the effective dimension of Heisenberg fer-\nromagnets is maximal, we also obtain an upper bound for the storage error. This result relies on perturbation\ntheory, where we use Davis’ divided difference representation for Fr ´echet derivatives along with the recursive\nstructure of these divided differences. Our numerical bounds allow us to better understand how quantum mem-\nory lifetimes can be enhanced in Heisenberg ferromagnets.\nIntroduction.— Decoherence quickly renders unprotected\nquantum data unreliable. To combat this, it becomes necessary\nto encode quantum data into quantum error correction codes.\nThe challenge in designing robust quantum memories arises\nfrom the difficulty of simultaneously (i) utilizing an easily ac-\ncessible physical system, (ii) having a quantum code that lies\nwithin the ground space of the system’s Hamiltonian H, and\n(iii) having an increased storage lifetime twith an increasing\nnumber of qubits Nin the physical system. Self-correcting\nquantum memories [1, 2] should satisfy (ii) and (iii), but are\nchallenging to implement in a multitude of desirable settings\n[3–14]. Indeed, constraint (i) does easily not hold, which frus-\ntrates the design of reliable quantum storage. For instance,\nquantum memories based on stabilizer codes which correct at\nleast one error and also satisfy (ii) unfortunately reside in un-\nphysical systems with many-body interactions, and can only\nbe approximately constructed [4, 15–17]. Of these constraints,\nit is most pertinent to satisfy (i), because physically unrealistic\nquantum memories will be difficult to engineer.\nThere are two reasons to store quantum data within the\nground space and thereby satisfy constraint (ii). First, a grow-\ning energy gap can suppress excitations from the ground space\n[18]. Second, storing quantum data within the ground space\navoids unnecessary errors that can occur even in the complete\nabsence of noise. Any state within the ground space is an\neigenstate of the physical system, and for such states, they are\nleft unchanged by a unitary operation Utthat the system’s nat-\nural dynamics induces, after a storage time of telapses. By\navoiding the need to uncompute Ut, we would not suffer from\nan imperfect reversal of Utcaused by our imprecise knowl-\nedge of t.\nStorage within the ground space, while satisfying constraint\n(ii), is not enough to result in self-correcting quantum memo-\nries and thereby satisfy constraint (iii). Moreover, many phys-\nically realistic systems satisfying constraint (i) comprising of\ntwo-local terms are surprisingly incompatible with constraint\n(iii) [12]. However, this no-go result does not preclude phys-\nical systems comprising of non-commuting two-body interac-\n\u0003y.ouyang@sheffield.ac.uktions from satisfying constraint (iii). Consequently, determin-\ning whether such physical systems can satisfy constraint (iii)\nis especially pertinent. In this paper, we study Heisenberg fer-\nromagnets as media for quantum storage because they com-\nprise of non-commuting two-body interactions and therefore\nsidestep the no-go result of [12]. We also study to what extent\nHeisenberg ferromagnets satisfy constraint (iii).\nThe Heisenberg ferromagnet (HF) is a model of quantum\nmagnetism, and is prevalent in many naturally occurring phys-\nical systems, and thereby satisfies constraint (i). For instance,\nthe HF is found in various cuprates [19, 20], in solid Helium-3\n[21], and more generally in systems with interacting electrons\n[22]. Even in many physical systems that cannot be naturally\ninterpreted as ferromagnets, effective HFs can nonetheless be\nengineered, for instance by symmetrizing systems dominated\nby dipole interactions using dynamic pulse sequences [23].\nEffective HFs have also been engineered in ultracold atomic\ngases [24] and quantum dots [25]. Specifically, we study spin-\nhalf HFs in the absence of an external magnetic field, with\nHamiltonian of the form\nH=\u0000å\nfi;jg2EJ(1\u0000pi;j): (1)\nHere, 1denotes an N-qubit identity operator, pi;jdenotes a\nswap operator on the ith and jth qubits, Jdenotes the exchange\nconstants, and Edenotes the set of interactions. Such HFs\nhave J>0 and ground state energy set to zero.\nBy storing quantum data using permutation-invariant ( PI)\ncodes in HFs, we automatically satisfy constraint (ii). This\nis because symmetric states lie within the ground space of\nany HF, and quantum data in PIcodes, by being invari-\nant under any permutation of their underlying qubits, are\nsymmetric states. The possibility of quantum error correc-\ntion using a PIcode was first affirmed by Ruskai, via con-\nstruction of an explicit PI-J9;1;3Kcode that corrects one\nerror [26]1. Ruskai’s result [26] was subsequently im-\nproved to yield PI-J7;1;3Kcodes [27]. In [28], Ouyang\n1Here, JN;K;DKdenotes a quantum code that encodes Klogical qubits into\nNphysical qubits and with distance D. When one needs to correct terrors\non a quantum code, it suffices to have D>2t.arXiv:1904.01458v3  [quant-ph]  3 Feb 2020constructs PI-J(2t+1)2;1;2t+1Kcodes that correct tarbi-\ntrary errors, thereby generalizing Ruskai’s nine-qubit code.\nFurther generalizing these codes, Ouyang introduces PI-\nJ(2t+1)2(d\u00001);log2d;2t+1Kcodes [29]. Approximate\nquantum error correcting PIcodes were also investigated in\nqubit [28, 30, 31] and bosonic settings [32]. Research on PI\ncodes shows that quantum correction is possible within the\nground space of HFs, which is only suggestive that constraint\n(iii) can be compatible with PIcodes. This is because the cod-\ning parameters of PIcodes alone, being independent of the\nparameters in HFs, are not enough to determine what happens\nwhen physical noise applies to PIcodes stored in HFs. To\nbetter understand the extent in which HFs with PIcodes can\nsatisfy constraint (iii), we study bounds on the storage error of\nPIcodes under the action of two different noise models, where\nboth bounds depend on properties of the underlying HF.\nOur first noise model applies to HFs of any geometry, and\nintroduces Pauli errors. These Pauli errors occur with proba-\nbilities that are thermodynamically related to their expected\nenergies on the codespace of a specific family of PIcodes\n[28]. To derive an upper bound on the storage error, we find\nan area law on the expected energy of a Pauli error, which\ndemonstrates that a quantum memory based in a HF can have\na macroscopic energy barrier for Pauli errors. This allows us\nto show that the storage error decreases with increasing dimen-\nsionality of the HF.\nOur second noise model introduces unitary errors proba-\nbilistically, where each unitary arises from a local perturba-\ntion of the underlying Hamiltonian. Such a noise model can\ndescribe the effects of unwanted physical interactions, such\nas spurious local fields afflicting each particle independently.\nWe use perturbation theory to bound the storage error by us-\ning Davis’ divided difference representation of these taking\nFr´echet derivatives. Because we require complete knowledge\nof the Hamiltonian’s spectrum, we restrict our analysis to ex-\nactly solvable mean-field HFs. In such HFs every pair of spins\ninteracts with equal strength. Since such HFs have an infinite\neffective dimension, analyzing them is indicative of the ulti-\nmate limits of robust quantum storage in HFs.\nWith respect to both noise models, we provide upper bounds\nfor the storage error of quantum memories in HFs that are nu-\nmerically tractable. In both cases, we find that quantum mem-\nories in HFs are partially self-correcting in the sense that is an\noptimal system size for fixed noise parameters that minimizes\nour upper bounds on the storage error.\nEnergy of Pauli errors and their geometry.— We use GNU\ncodes [28] to elucidate the dependence of a HF’s dimension\nwith respect to the storage error of quantum data. GNU codes\ndepend on three parameters g,n, and u, and encode a single\nqubit into N=gnuqubits. Here gandnquantify the distance\nof the GNU code with respect to bit-flip and phase-flip errors\nrespectively, while uis a scaling parameter where u\u00151. When\ng=n=2t+1, the GNUcode corrects terrors. A GNUcode has\nlogical codewords\njrLi=å\n0\u0014j\u0014n\nmod(j;2)=rs\u0000n\nj\u0001\n2n\u00001jDgnu\ng ji; (2)where r=0;1, andjDN\nwiare Dicke states of weight w2[33,\n34].\nWe quantify a HF’s dimension using properties of its un-\nderlying graph of interactions [35]. This graph Ghas vertices\nlabeled from 1 to N, and edges Ethat correspond to the in-\nteractions in the HF’s Hamiltonian H. Given a subset Sof\nf1;:::; Ng, let¶ESdenote its edge-boundary with respect to\nthe edge set E, which is the set of edges in Ewith exactly\none vertex in S. When every subset Ssatisfies the inequality\nj¶ESj\u0015cmin(jSj;N\u0000jSj)1\u00001=d, the graph and HF have di-\nmension dwith isoperimetric constant c.\nGiven a set PofN-qubit Pauli errors that afflict at most\nN=2 qubits, let\nhPi=min\njyi2ChyjPHPjyi (3)\ndenote the minimum expected energy of P2Pon the code\nC. When Cis aGNUcode, we derive a lower bound on hPiin\nterms of the edge boundary V(P), where V(P)denotes the set\nof vertices on which Pacts non-trivially. In particular, Theo-\nrem 1 below gives an area law on the minimum size of hPi.\nTheorem 1. LetCbe an N-qubit GNU code with parameters\ng=n=2t+1and u =2, where N =2(2t+1)2and t\u00151. Then\nwith respect to the Hamiltonian H given by (1)with exchange\nconstants J and set of interactions E, for every N-qubit Pauli\nP inP, we have\nhPi\u0015cJj¶E(V(P))j;\nwhere c=minf2m;1\u00004mgand\nm= (1+5t+6t2)=(4+32t+32t2):\nThe significance of Theorem 1 lies in the geometric inter-\npretation it imparts to hPi. Namely, when the graph Ghas\ndimension dand isoperimetric number c, we have the isoperi-\nmetric inequality\nhPi\u0015JccjPj1\u00001=d; (4)\nwherejPj=jV(P)jdenotes the weight of P. For a HF on a\n1D spin-chain,hPi\u0015Jc:For a HF on a square lattice, [36]\nwith our result implies that hPi\u0015Jcp\njPj:Whenever d>1,\nthe expected energy of Pgrows with its weight, and we have\na macroscopic energy barrier [37]. This suggests that when\nd>1, HFs can be good quantum memories. To see this, con-\nsider a noise model that introduces Pauli errors P2Pwith\nprobability proportional to e\u0000bhPiand with effective inverse\ntemperature b. The corresponding channel is T, which ran-\ndomly introduces Pauli errors, and has the form\nT(r) =å\nP2P(e\u0000bhPi=Z)PrP; (5)\n2Dicke states are uniform superpositions of computational basis states la-\nbeled by binary vectors of Hamming weight w.\n22 4 6 8\ndimension \n1016\n1014\n1012\n1010\n108\n106\n104\n102\nstorage error ut\nt=1,N=18\nt=2,N=50\nt=3,N=98\nt=4,N=162\nt=5,N=242\nt=6,N=338FIG. 1: When a HF stores an encoded qubit within a GNU code on\nN=2(2t+1)2physical qubits that corrects terrors, we use (7) to\nobtain upper bounds on the storage error utwith respect to the HF’s\ndimension dandt. Here bJ=13 and c=1.\nwhereZ=åP2Pe\u0000bhPi. The corresponding probability of\nobtaining an uncorrectable error, which is also the storage er-\nror after perfectly performing quantum error correction, is then\nut=1\nZå\nP2P\njPj\u0015t+1e\u0000bhPi: (6)\nFrom the isoperimetric inequality (4) and the bound\nj¶EV(P)j\u0014DjPj, where Dis the maximum vertex degree of\nG, we obtain\nut\u0014 N=2\nå\nw=t+1\u0012N\nw\u0013\n3we\u0000bJccw1\u00001=d! N=2\nå\nw=0\u0012N\nw\u0013\n3we\u0000bJDw!\u00001\n:\n(7)\nWe illustrate (7) in Figure 1 with D=4;c=1;bJ=13, and\nvary the dimension dand number of correctible errors t. Fig-\nure 1 shows that long-range interactions can help to suppress\nutbecause increasing ddecreases ut. From Figure 1, when\nd<4, the optimal code corrects between 1 error to 4 errors.\nThis shows that for a system with sufficient few dimensions,\nour scheme exhibits a partial self-correction property, where\nincreasing the system size cannot indefinitely reduce the stor-\nage error.\nRandom coherent noise and storage error.— A good quan-\ntum memory preserves entanglement. Given a quantum code\nCwith logical codewords j0Li;:::;j(M\u00001)Li, consider the\nentangled statejYCi=åM\u00001\nj=0jji\nj jLi=p\nM:Thestorage er-\nrorofCwith respect to a noisy channel Nis\ne(N;C) =min\nR1\n2\r\rjYCihYCj\u0000R(N(jYCihYCj))\r\r\n1;\nwhereR=I\nR,N=I\nN,Iis an identity chan-\nnel, the minimization is over all recovery maps R, andk\u0001k 1\ndenotes the trace norm. To keep notation simple, when the\ncodeCand noise model Nare already specified, we use the\nshorthand e=e(N;C).\n0 50 100 150 200\nstorage lifetime  (ns)\n106\n105\n104\n103\n102\nstorage error \nJ=10 GHz, t=1,N=7\nJ=103 GHz, t=1,N=7\nJ=103 GHz, t=2,N=25\nJ=104 GHz, t=2,N=25\nJ=105 GHz, t=2,N=25\nJ=106 GHz, t=3,N=49FIG. 2: When a mean-field HF stores an encoded qubit within an\nN-qubit PIthat corrects terrors, we use Theorem 3 to obtain upper\nbounds for the corresponding storage error eafter a target storage\nlifetime of t. The baseline lifetime and storage error for an unpro-\ntected qubit are 12ns and 0 :00048 respectively. The shaded region\nindicates where eis smaller than the baseline.\nNow, let perturbations A1;:::; Aato the Hamiltonian Hoc-\ncur with probabilities p1;:::; parespectively. These perturba-\ntions model the effect of qubits coupling to spurious classical\nfields. Each perturbation Ajis a linear combination of oper-\nators that acts non-trivially on a single qubit, and induces a\nunitary evolution Uj=g(H+Aj), where g(x) =e\u0000ixtdenotes\nan exponential function. The corresponding noisy channel is\nNt, where for any initial state r, we have\nNt(r) =a\nå\nj=1pjUjrU†\nj: (8)\nIn what follows, we focus our attention solely Nt, which we\ncall a random coherent noise channel, which is parameterized\nby its noise strength a=max jkAjk=N.\nFor any perturbation Aj, the Taylor series of the unitary\ng(H+Aj)gives\ng(H+Aj) =g(H)+¥\nå\nk=1D[k]\ng(H;Aj)=k!; (9)\nwhere\nD[k]\ng(H;Aj) =dk\ndxkg(H+xAj)jx=0 (10)\nare Fr ´echet derivatives of g(H)in the matrix direction Aj\n[38, 39]. Now the correctible component of g(H+Aj)with\nrespect to a code that corrects terrors comprises of Fr ´echet\nderivatives of order at most t, because these Fr ´echet derivatives\nare polynomials in Ajof order no more than t. Therefore, we\nstudy only the high order Fr ´echet derivatives. These Fr ´echet\nderivatives allow us to bound the storage error.\nLemma 2. Given a quantum code Cthat corrects t errors, let\nRj=å¥\nk=t+1D[k]\ng(H;Aj)and definekRkC=max jfkRjjyik:\njyi2Cg. Then e\u0014kRkC+kRk2\nC:\nFrom the integral representation of Rj[40], we exploit the\nfact that g(H+Aj)is unitary for Hermitian HandAto get\nkRkC\u0014max\njmaxfkD[t+1]\ng(H;Aj)jyik:jyi2Cg\n(t+1)!;(11)\n3which depends only on a single Fr ´echet derivative instead of\ninfinitely many. Given (11), one can clearly bound kRkCin\nterms of justkHkandaN. However, such a bound increases\nwith increasingkHk, and exhibits a behavior contrary to the\nnumerical evidence in Figure 1. Increasing the number of\nlong-range interactions increases both kHkand dimensional-\nity, but since increasing dimensionality should decrease the\nstorage error, this suggests that the storage error should instead\ndecrease with increasing kHk. We solve this conundrum by\nusing Davis’ representation [41] of Fr ´echet derivatives, which\nreveals the intricate dependence of Fr ´echet derivatives on the\nspectral decomposition H=åj\u00150ljPj. Here, ljstrictly in-\ncrease with jandPjare eigenprojectors. Namely, we can\nwrite D[k]\ng(H;Aj)=k! as\nå\nn0;:::;nkg(ln0;:::;lnk)(PnkAj):::(Pn1Aj)Pn0: (12)\nHere, g(ln0;:::;lnk)are divided differences that arise nat-\nurally from the theory of Lagrange interpolation. To un-\nravel (12), we leverage on the remarkable properties of di-\nvided differences. First, divided differences are invariant un-\nder any permutation of their arguments. Hence, we can al-\nways arrange the arguments of a divided difference in non-\ndecreasing order. Second, divided differences generalize\nscalar derivatives, because the divided difference of a vec-\ntor with kidentical arguments is proportional to the (k\u0000\n1)th derivative of the underlying function. For the exponen-\ntial function, we have jg(y1;:::; yk)j=tk\u00001=(k\u00001)! when\ny1=\u0001\u0001\u0001=yk. For instance,jg(2;2)j\u0014t. Third, a di-\nvided difference when not evaluated on identical arguments\ncan be recursively defined; whenever yiandyjare distinct,\ng(y) = ( g(y[noti])\u0000g(y[notj]))=(yi\u0000yj);where y[noti]de-\nnotes a vector obtained from yby deleting its ith component.\nFrom (11) and (12), we find that\nkRkC\u0014(aN)t+1\u0012\nht+1+tt+1\n(t+1)!\u0013\n: (13)\nHere tt+1=(t+1)! arises from the divided difference with all\narguments equal to zero. The term ht+1is the sum of all\njg(0;ln1;:::;lnt+1)jwhere n1+\u0001\u0001\u0001+nt+1>0.\nEvaluating a bound on ht+1requires knowing the eigenval-\nues of H. Since finding the eigenvalues of Hfor arbitrary\nEis in general a difficult problem [35], we study an exactly\nsolvable HF where every pair of spins interacts equally with\nJi;j=J. We call such a HF a mean-field HF, and its ground\nstate energy is l0=0 and its higher energy eigenvalues are\nl1=JN;l2=2J(N\u00001),l3=3J(N\u00002), and in general,\nlj=J j(N+1\u0000j)[35]. We proceed to bound dj, which is the\nminimum energy needed to transition away from lj. For in-\nstance, d0=l1\u0000l0=JN,d1=l2\u0000l1=J(N\u00002), and d2=\nl3\u0000l2=J(N\u00004). In general, dbN=2c=2+(N\u00002bN=2c)and\ndj=J(N\u00002j)for all j=0;:::;bN=2c\u00001. Importantly, djis\nnon-increasing in jand is maximal when j=0. Exploiting the\nrecursive structure of divided differences, one can show that\njg(0;ln1;:::;lnt)j\u00142t+1d\u00001\n0(dn1:::dnt+1)\u00001: (14)\nWhen there are repeated arguments for the divided differ-\nence, the bound is more complicated than (14), because itinvolves maximization. To avoid maximizing, we overesti-\nmate the size of ht+1by overcounting the contributions from\ng(0;ln1;:::;lnt+1)when there are repeated arguments in g.\nFirst we use (14) for divided differences even when there are\nrrepeated arguments. Second, for divided arguments with r\nrepeated entries, we count the contributions from leaves that\nterminate with all possible divided differences with repeated\nidentical arguments.\nIf the contribution to the divided differences is domi-\nnated by leaves with no repeating indices, the total con-\ntribution of such leaves to ht+1is at most St2t+1=d0,\nwhere S=d\u00001\n0+\u0001\u0001\u0001+d\u00001\nt+1. The contribution to ht+1\nby leaves that terminate with rrepeated arguments is\n(tr\u00001=(r\u00001)!)St+1\u0000r=d0. From this we get ht+1\u0014qwhere\nq=St+1\nd0+n=2+1\nd0t+1\nå\nr=2\u0000t+1\nr\u00001\u0001\ntr\u00001\n(r\u00001)!St+2\u0000r: (15)\nThis leads us to our final result, which follows directly from\nLemma 2, (13) and (15).\nTheorem 3. Let H be a mean-field HF where every pair of\nqubits interacts with exchange constant J, and Cbe any PI\ncode that corrects t errors. Let Ntbe the random coher-\nent noise channel as given in (8). Then when we have per-\nfect quantum error correction, e(Nt;C)\u0014Q+Q2, where\nQ= (aN)t+1(q+tt+1=(t+1)!), and qis given in (15).\nFrom Theorem 3, we can see that the quantum memory is\npartially self-correcting, because the bound on the storage er-\nror contains a term (Nt)t=t! which diverges for large N, since\nNis quadratic in t. Hence for every noise parameter aand\nexchange constant J, our scheme for a quantum memory has\nan optimal system size. Figure 2 illustrates only results for\noptimal system sizes.\nRecently, a superconducting qubit was stored between 12ns\nto 20ns with a fidelity of 0.9995 [42]. Using our noise model,\nthese experimental parameters can be recast into a baseline\nstorage error of 5 (10\u00004)with a memory lifetime of 12ns and\na noise strength of a=0:04MHz. Given this noise model, we\nuse Theorem 3 to obtain upper bounds on the storage error eof\nan encoded qubit within a PIcode in a mean-field HF, and we\ndepict these numerical results in Figure 2. Here, the number of\nqubits for t=1 is seven [27], and when t\u00152,N= (2t+1)2.\nFrom Figure 2, if one uses a seven-qubit PIcode with J=\n103GHz, the qubit’s storage lifetime can be improved to over\n100ns. In addition, if one uses a 25 qubit PIcode that corrects\n2 errors [28], the qubit’s storage lifetime can be enhanced to\nover 120ns when J=104GHZ. Similarly, if J=106GHZ, the\nqubit’s storage lifetime can be enhanced to over 150ns using a\n49 qubit PIcode that corrects 3 errors. From this, we can see\nhow increasing Jand the number of qubits in HFs can enhance\nthe storage lifetime.\nDiscussions.— Here, we study quantum storage in a phys-\nically abundant physical system, the HF. Our scheme, being\nbased on a physical model that is simple to realize, will be\neasier to implement than those built upon many-body interac-\ntions. Because Pauli errors on PIcodes can exhibit a macro-\nscopic energy barrier, we see evidence that a quantum memory\nbased in a HF can become increasingly robust with increasing\ndimensionality of the HF. In addition, we study the impact of\n4increasing the coupling strengths within HFs on the targeted\nstorage error. For this, we analyze an infinite-dimensional HF,\nnamely the mean-field HF, and find numerically tractable up-\nper bounds on the storage error. Our derivation of the bounds\nrelies on a novel approach based on the connection between\nmatrix perturbation theory and divided differences. We find\nthat increasing the size of the coupling strengths can be bene-\nficial in extending the storage lifetime of a HF-based quantum\nmemory when used in concert with a PIcode.\nSince our analysis technique extends to any physical system\nwith a completely understood spectral structure, it applies also\nto other code-inspired Hamiltonians, and lays the foundations\nfor analyzing using quantum memories using our new method-\nology. While PIcodes can be robustly prepared in supercon-\nducting charge qubits, by inducing Rabi-like oscillations be-\ntween Dicke states of adjacent weights [43], it remains to see\nhow the initialization Hamiltonian can be integrated with the\nHF. Furthermore, constructing explicit protocols for the de-\ncoding of PIcodes can bring quantum memories in HFs closerto implementation.\nI. ACKNOWLEDGMENTS\nAcknowledgements.— YO thanks Joseph F. Fitzsimons for\ncoming up with the idea for studying Heisenberg ferromagnets\nfor quantum storage, and also for helping to prove Theorem 1.\nYO also thanks Jon Tyson for bringing to his attention the di-\nvided difference representation of matrix Fr ´echet derivatives.\nYO acknowledges support from the Singapore National Re-\nsearch Foundation under NRF Award NRF-NRFF2013-01, the\nU.S. Air Force Office of Scientific Research under AOARD\ngrant FA2386-18-1-4003. YO acknowledges support from the\nEPSRC (Grant No. EP/M024261/1) and the QCDA project\n(Grant No. EP/R043825/1) which has received funding from\nthe QuantERA ERANET Cofund in Quantum Technologies\nimplemented within the European Unions Horizon 2020 Pro-\ngramme.\n[1] B. M. Terhal, “Quantum error correction for quantum memo-\nries,” Rev. Mod. Phys. , vol. 87, pp. 307–346, Apr 2015.\n[2] B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. 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By definition of hPi, we have\nhPi=\u0000å\nfi;jg2EJ(Tr(PrPpi;j)\u0000Tr(rpi;j)):Since ris permutation-invariant, both pi;jrandrpi;jare equal\ntor, and hence\nhPi=Jå\nfi;jg2ETi;j; (A1)\nwhere\nTi;j=1\u0000Tr(r(pi;jPpi;j)P): (A2)\nIf swapping the ith and jth qubits leaves Pinvariant, we have\n(pi;jPpi;j)P=I\nN, and the corresponding term Ti;jis zero.\nNow denote Xi,YiandZias Pauli operators that apply a single\nX,YandZPauli on the ith qubit and leave the remaining qubits\nuntouched.\nConversely, if swapping the ith and jth spins changes P, the\noperator (pi;jPpi;j)Pis either Xi;j,Yi;j, orZi;j. Because ris\npermutation-invariant, it suffices to evaluate\nTr(rX1;2);Tr(rY1;2);Tr(rZ1;2): (A3)\nSince ris supported on a PI-code that corrects at least a single\nerror, the Knill-Laflamme conditions [44] imply that for all\nQ2Q=fX1;2;Y1;2;Z1;2g, we haveh0LjQj1Li=h1LjQj0Li=\n0, andh0LjQj0Li=h1LjQj1Li[28]. Hence for all Q2Qand\nrsupported on our PI-codespace,\nTr(rQ) =h0Ljrj0Lih0LjQj0Li+h1Ljrj1Lih1LjQj1Li\n=h0LjQj0Li=h0LjQj0Li+h1LjQj1Li\n2\n=å\n0\u0014`\u0014n2\u0000n\u0012n\n`\u0013\nhDm\ng`jQjDm\ng`i:\nUsing the Vandermonde binomial identity\u0000N\ng`\u0001\n=\u0000N\u00002\ng`\u0001\n+\n2\u0000N\u00002\ng`\u00001\u0001\n+\u0000N\u00002\ng`\u00002\u0001\n, it follows that\nTr(rQ) =\u001a\n1\u00004m;Q=Z\n2m; Q2fX;Yg; (A4)\nwhere for our GNUcode with parameters g;nandu, we have\nm:=å\n0\u0014`\u0014n2\u0000n\u0012n\n`\u0013\u0000N\u00002\ng`\u00001\u0001\n\u0000N\ng`\u0001:\nIn particular, for all u>1, we have 0 <m<1=2, which im-\nplies that either Ti;j=0 or Ti;jis strictly positive. The lower\nbound forhPithen depends crucially on the number of edges\nfor which Ti;jis strictly positive. This implies that\nhPi\u0015Jj¶E(V(P))jminf1\u00004m;2mg: (A5)\nSubstituting u=2;g=n=2t+1 then gives the result.\nAppendix B: Proof of Lemma 2\nConsider N(r) =åjpjUjrU†\njfor any density matrix r,\nwhere pjare probabilities and Ujare unitary matrices. Let\nCbe a quantum code with logical codewords jrLiforr=\n0;:::; d\u00001. Let Uj=Gj+Bj, where GjandBjdenote the\n6correctible and uncorrectable parts of Ujrespectively with re-\nspect to C. It suffices to show that\ne(N;C)\u0014å\njpj\ndd\u00001\nå\nr=0\u0012q\nhrLjB†BjrLi+hrLjB†BjrLi\u0013\n:\n(B1)\nWe now proceed with the proof. From the linearity of the\nchannels NandR, it is clear that\ne(N;C) =min\nR1\n2\r\rjYCihYCj\u0000R(N(jYCihYCj))\r\r\n1\n=min\nR1\n2\r\r\r\r\r1\ndd\u00001\nå\nr;s=0(jrihsj\njrLihsLj\n\u0000jrihsj\nå\njpjR\u0010\nUjjrLihsLjU†\nj\u0011!\r\r\r\r\r\n1:\nUsing the triangle inequality for the trace norm and the mono-\ntonicity of the trace distance under the partial trace, we get\ne(N;C)\u0014å\njpj\n2dd\u00001\nå\nr=0\r\rMr;Uj\r\r\n1; (B2)\nwhere\nMr;U=jrLihrLj\u0000R(UjrLihrLjU†):\nIt remains to obtain an upper bound on kMr;Uk1.\nLemma 4.\nkMr;Uk1\u00142q\nhrLjB†BjrLi+2hrLjB†BjrLi: (B3)\nWe defer the proof of Lemma 4, which essentially involves\ndecomposing Mr;Uinto a block matrix on the correctible and\nuncorrectable subspaces of the quantum code C. Thus, it read-\nily follows that\ne(N;C)\u0014å\njpj\ndd\u00001\nå\nr=0\u0012q\nhrLjB†BjrLi+hrLjB†BjrLi\u0013\n:\nSince probabilities sum to one, we can use H ¨older’s inequality\nand find that the maximization over rgives the upper bound\nrequired in the lemma.\nAppendix C: Proof of Lemma 4\nLetPgandPbdenote projectors onto the correctible and\nuncorrectable subspaces of the quantum code Crespectively.\nIt follows that PgjrLi=jrLiandPbjrLi=0. Hence, by writing\nMr;Uinto a size 2 block matrix induced by the projectors Pg\nandPbwe get\nMr;U=\u0012\nmg;gmg;b\nmb;gmb;b\u0013\n; (C1)where\nmg;g=jrLihrLj\u0000PgR(UjrLihrLjU†)Pg;\nmg;b=PgR(UjrLihrLjU†)Pb;\nmb;g=PbR(UjrLihrLjU†)Pg;\nmb;b=PbR(UjrLihrLjU†)Pb: (C2)\nLetU=G+Bwhere Gdenotes the correctible part of Uand\nBdenotes the uncorrectable part of Uwith respect to the code\nC. Second, since Rcorrects errors on the correctible subspace\nand its Kraus operators do not induce interactions between the\ncorrectible and uncorrectable subspaces, it follows that\nPgR(UjrLihrLjU†)Pg\n=PgR(GjrLihrLjG†)Pg+PgR(BjrLihrLjG†)Pg\n+PgR(GjrLihrLjB†)Pg+PgR(BjrLihrLjB†)Pg\n=jrLihrLjhrLjG†GjrLi: (C3)\nSincehrLjU†UjrLi=1, it follows from the definition of Gand\nBthathrLjG†GjrLi+hrLjB†BjrLi=1. Hence\nMr;U=\u0012\nhrLjB†BjrLijrLihrLjPgR(UjrLihrLjB†)Pb\nPbR(BjrLihrLjU†)PgPbR(BjrLihrLjB†)Pb\u0013\n:\n(C4)\nWe now require a block-matrix generalization of the Ger ˘sgorin\ncircle theorem [45] given by the following lemma.\nLemma 5. Let M be a Hermitian matrix with the block de-\ncomposition\nM=\u0012A C\nC†B\u0013\n;\nwhere A and B are Hermitian matrices of the same size. Then\nkMk1\u0014kAk1+kBk1+2kCk1: (C5)\nProof. Since Mis Hermitian, there is a unitary matrix Vsuch\nthatkMk1=Tr(V M). Such a unitary matrix Vadmits a block\ndecomposition\nV=\u0012\nV11V12\nV21V22\u0013\n: (C6)\nThen we can see that\nkMk1=Tr(V11A+V12C+V21C†+V22B): (C7)\nSince the trace of the product of two matrices is an inner prod-\nuct, we can use H ¨older’s inequality to obtain an upper bound\nonkMk1:Such a bound requires knowledge on the operator\nnorm (the maximum singular value) of the submatrices of V.\nFor this, we use the parallelogram law. Note that\nV\u0012\nx\ny\u0013\n=\u0012\nV11x+V12y\nV21x+V22y\u0013\nV\u0012\nx\n\u0000y\u0013\n=\u0012\nV11x\u0000V12y\nV21x\u0000V22y\u0013\n: (C8)\n7Whenever (x;y)is a vector of unit norm, it follows that the\nvectors on the right side of the above equation are also vectors\nof unit norms. Now let us apply the parallelogram law, which\ngives us\n2kV11xk2+2kV12yk2=kV11x+V12yk2+kV11x\u0000V12yk2:\n(C9)\nSince Vis unitary and (x;y)is a vector of unit norm, we al-\nways have\n2kV11xk2+2kV12yk2\u00142: (C10)\nWhenever y=0, we must havekxk=1, and in this scenario\nwe find thatkV11xk2\u00141. This implies that kV11k\u00141. Re-\npeating this argument, we find that the operator norm of all\nsubmatrices of Vis at most 1. Hence the result follows from\napplying H ¨older’s inequality on (C7).\nUsing Lemma 5, we get\nkMr;Uk1\u0014hrLjB†BjrLi+2kPgR(UjrLihrLjB†)Pbk1\n+kPbR(BjrLihrLjB†)Pbk1: (C11)\nNow recall that for matrices SandT,kSTk1\u0014kSkkTk1and\nkSTk1\u0014kTkkSk1. Using this fact repeatedly, we find that\nkPgR(UjrLihrLjB†)Pbk1\u0014kR(UjrLihrLjB†)k1;(C12)\nkPbR(BjrLihrLjB†)Pbk1\u0014kR(BjrLihrLjB†)k1:(C13)\nFurthermore, because Ris a trace-preserving map, we have\nkPgR(UjrLihrLjB†)Pbk1\u0014kjrLihrLjB†k1; (C14)\nkPbR(BjrLihrLjB†)Pbk1\u0014kBjrLihrLjB†k1: (C15)\nUsing that fact that for a matrix S, we havekSk1=Trp\nS†S, it\nfollows that\nkjrLihrLjB†k1=Trq\nBjrLihrLjB†\n=q\nhrLjB†BjrLi; (C16)\nand\nkBjrLihrLjB†k1=Trq\nBjrLihrLjB†BjrLihrLjB†\n=hrLjB†BjrLi: (C17)Hence\nkMr;Uk1\u00142q\nhrLjB†BjrLi+2hrLjB†BjrLi; (C18)\nand the result follows.\nAppendix D: Proof of Eq. (14)\nIn Figure 3, we depict g(l0;l2;l2;l3)as a binary tree\nwith root labeled by g(l0;l2;l2;l3). Children of the root\nare obtained by removing the largest and the smallest argu-\nments from g(l0;l2;l2;l3), which are l0andl3respectively.\ng(λ0,λ2,λ2,λ3)\ng(λ0,λ2,λ2)\ng(λ0,λ2) g(λ2,λ2)\ng(λ2) g(λ0)g(λ2,λ2,λ3)\ng(λ2,λ3) g(λ2,λ2)\ng(λ2) g(λ3)3\n0\n2\n0 23\n02 3\n2\nFIG. 3: A tree illustrates the recursive structure of the divided differ-\nence of g(l0;l2;l2;l3), where gis the exponential function.\nHence, the root’s children inherit a factor of (l3\u0000l0)\u00001from\nthe root. This procedure iterates until no distinct arguments\nin the divided difference remain. We can see that the left-\nmost vertex in Figure 3 is a leaf labeled by g(l3)that con-\ntributes at most (l3\u0000l0)\u00001(l3\u0000l2)\u00002to the bound. The\nblue vertex labeled by g(l2;l2)contributes at most (l3\u0000\nl0)\u00001(l3\u0000l2)\u00001tto the bound. Using the triangle inequal-\nity and the monotonicity of di, we find that g(l0;l2;l2;l3)\u0014\nmaxf23d\u00001\n0d\u00001\n2d\u00001\n3;22d\u00001\n0d\u00001\n3tg:Here, the factor of d\u00001\n0\narises as a coarse upper bound to (l3\u0000l0)\u00001, while d2and\nd3correspond to the smallest energy gaps obtainable when de-\nscending the tree from the root’s children onwards. General-\nizing this argument for distinct non-zero ln1;:::;lnt+1where\nn1>\u0001\u0001\u0001>nt+1gives the result.\n8"    },    {        "title": "0804.3546v1.Pressure_temperature_phase_diagram_of_ferromagnetic_superconductors.pdf",        "content": "arXiv:0804.3546v1  [cond-mat.supr-con]  22 Apr 2008Pressure-temperature phase diagram of ferromagnetic supe rconductors\nV. P. Mineev\nCommissariat ` a l’Energie Atomique, INAC / SPSMS, 38054 Gre noble, France\n(Dated: October 26, 2018)\nThe symmetry approach to the description of the ( P,T) phase diagram of ferromagnet supercon-\nductors with triplet pairing is developed. Taking into acco unt the recent experimental observations\nmade on UCoGe it is considered the case of a crystal with orthorhombic stru cture and strong\nspin-orbital coupling. It is shown that formation of ferrom agnet superconducting state from a\nsuperconducting state is inevitably accompanied by the firs t order type transition.\nPACS numbers: 74.25.Dw, 74.20.Rp, 74.20.De, 74.70.Tx\nThere are several metallic compounds demonstrating\nthe coexistence of superconductivity and itinerant fer-\nromagnetism. These are UGe2,1URhGe,2and recently\nrevealed UCoGe.3The large band splitting and the high\nlow temperature value of upper critical field4,5,6in ura-\nnium ferromagnetic superconductors indicate that here\nwe deal with Cooper pairing in the triplet state. The su-\nperconductivity in UGe2andURhGearises at tempera-\ntures far below the correspondingCurie temperature and\ncoexists with ferromagnetism in some pressure interval\nsuch that at the ( P,T) phase diagram the whole region\noccupied by the superconducting state is situated inside\nof more vast ferromagnetic region.7,8The phase diagram\nof new ferromagnetic superconductor UCoGeis found to\nbe qualitatively different.9At ambient pressure the ferro-\nmagnetism ( TCurie= 2.8K) coexists with superconduct-\ning state ( Tsc= 0.8K) arising close to ferromagnetic in-\nstability. ThenatappliedpressuretheCurietemperature\nis diminished such that no indication of ferromagnetic\norder has been observed above P∗≈1GPa. The resis-\ntive superconducting transition is, however, quite stable\nin temperature and persists up to the highest measured\npressure of about 2 .4GPa.\nThe apparent intersection of pressure dependent tran-\nsition lines TCurie(P) andTsc(P) raises the problem\nabout the general properties of ( P,T) phase diagram in-\ncluding the region of coexistence of ferromagnetic and\ntriplet pairing superconducting states. It has its own\npeculiarity other than that of singlet superconductivity\ncoexisting with ferromagnetism in form is known as the\nAnderson-Suhl or cryptoferromagnetic superconducting\nstate (for review see10) characterized by formation of a\ntransverse domain-like magnetic structure. The struc-\nture period or the domain size is larger than interatomic\ndistance and smaller than the superconducting coherence\nlength that decreasesthe depairing effect of the exchange\nfield. The latter is irrelevant in the case of triplet super-\nconductivity, hence there is no reason for formation of a\ncryptomagnetic state.\nThe regionof coexistence ofsuperconductivity and fer-\nromagnetism is separated from the normal state by the\nregion of ferromagnet normal state at P < P∗and by\nthe region of superconducting state at P > P∗. Let us\nlook first at P < P∗region.\nAll the mentioned ferromagnetic superconductors arethe metals with orthorhombic symmetry and magnetic\nmoment directed along of one of the crystallographicaxis\nwhich we choose as ˆ zaxis. The symmetry of the nor-\nmal paramagnetic state is determined by the elements of\ngroup\nG=D2×U(1)×R, (1)\nwhereD2= (E,Cz\n2,Cx\n2,Cy\n2) is the point symmetry\ngroup,U(1) is the group of gauge transformations and\nRis the time reversal operation. At the transition to the\nnormal ferromagnet state the symmetry reduces to the\nGM=DM\n2(Cz\n2)×U(1), (2)\nwhereDM\n2(Cz\n2) = (E,Cz\n2,RCx\n2,RCy\n2) is so called mag-\nnetic class11. The symmetries and the order parameters\nof unconventional superconducting states arising from\nthe normal state with ferromagnetic order in orthorhom-\nbic crystals with strong spin-orbital coupling have been\npointed out in the paper.12They belong to two different\ncorepresentations AandBsuch that two ferromagnetic\nsuperconducting classes of different symmetry denoted\nA1,A2andB1,B2arerelatedtoeachofthemcorrespond-\ningly. Then it was pointed out13,14that superconducting\nstates in triplet ferromagnet superconductors represent a\nspecial type of two band superconducting states, namely,\nconsisting of pairing states occupied either by spin up or\nspin down electrons. The two-component order parame-\nters have a form\nd(k) =1\n2[−(ˆx+iˆy)∆↑(k)+(ˆx−iˆy)∆↓(k)].(3)\nHere ˆx, ˆyare the unit vectors of the spin coordinate sys-\ntem pinned to the crystal axes.\n∆↑(k) =−η↑f−(k),∆↓(k) =η↓f+(k).(4)\nFunctions f±(k) are odd functions of momentum direc-\ntions of pairing particles on the Fermi surface. The\ngeneral forms of these functions for the different co-\nrepresentations in ferromagnetic superconductors with\northorhombic symmetry are listed in the papers.12,14\nThe complex order parameter amplitudes η↑andη↓\nare not completely independent:\nη↑=|η↑|eiϕ, η↓=±|η↓|eiϕ. (5)2\nThus, being different by their modulos they have the\nsame phase with an accuracy ±π. At the phase tran-\nsition to the normal ferromagnet state the amplitudes of\nthe both components of the superconducting order pa-\nrameter turn to zero.\nThe superconducting orderparameterform (3) is quite\nnatural in ferromagnetic state with spin up - spin down\nband splitting. We shall use another form of the order\nparameterwhich is more appropriatewhen the ferromag-\nnetism (but not superconductivity !) is suppressed by\npressure, and the band splitting plays no role. So, we\nchoose the order parameter consisting of sum of real and\nimaginary parts. For the states SA1andSA2they are\ndA1(k) =η1ϕ1(k)+iη2ϕ2(k), (6)\ndA2k) =iη1ϕ1(k)+η2ϕ2(k), (7)\nwhere\nϕ1(k) =u1ˆxkx+u3ˆyky,ϕ2(k) =u2ˆxky+u4ˆykx,(8)\nandu1,....are real functions of k2\nx,k2\ny,k2\nz. Due to the\nproperty (5) the amplitudes η1andη2can be chosen\nhaving equal phase factors. Both states with the order\nparameters (6) and (7) belong to the same corepresenta-\ntionAand have common critical temperature of transi-\ntion from the normal ferromagnet state. But they obey\nthe different symmetry(belong to different ferromagnetic\nsuperconducting classes)\nGA1=D2(Cz\n2) = (E,Cz\n2,RCx\n2,RCy\n2),(9)\nand\nGA2=˜D2(Cz\n2) = (E,Cz\n2,RCx\n2eiπ,RCy\n2eiπ).(10)\nThey also differ each otherby the direction ofthe Cooper\npair spin momentum\nSA1=i/angbracketleftd∗\nA1×dA1/angbracketright\n/angbracketleftd∗\nA1dA1/angbracketright(11)\nSA2=−SA1. (12)\nHere the angular brackets mean the averaging over the\nFermi surface.\nAt the second order type transition from the nor-\nmal ferromagnet to the superconducting ferromagnet the\nstatesSA1andSA2appear in the ferromagnet domains\nwith the opposite direction of magnetization12. Thus the\norder parameter of superconducting state proves to be\ncoordinate dependent and can be written as\nd(r,k) =eiα(r)η1(r)ϕ1(k)+e−iα(r)+iπ/2η2(r)ϕ2(k),\n(13)\nwhere the phase α= 0 in the magnetization up domains\nwithSA1superconducting state and α=π/2 in the\nmagnetization down domains with SA2superconductingstate. The phase is changed near atomic thickness fer-\nromagnet domain walls in the layer of the order of su-\npeconducting coherence length. At the phase transition\nto the normal ferromagnet state the amplitudes of the\nboth components of the superconducting order parame-\nter (13) turn to zero.\nLet us consider now what is going on at P > P∗. Here,\nat temperature decrease, the system can pass from the\nnormal to the superconducting state. It is quite natural\nto think that it happens by means of the second order\ntype transition to conventional superconducting state S1\nwith the order parameter\ndS1(k) =η1ϕ1(k) (14)\nhaving the full point symmetry of the crystal lattice and\nrelating to the trivial superconducting class\nGs1=D2×R. (15)\nAs any one component superconducting state formed di-\nrectly from a nonmagnetic normal state by means of the\nsecond order type transition this state is nonmagnetic.15\nThe spontaneous magnetism appears at a transition to\nthe ferromagnet state where the superconducting order\nparameter acquires the second component. By means of\nthe second order phase transition the state S1can turn\nonly to the homogeneous superconducting ferromagnet\nstateSA1with symmetry group GA1which is the sub-\ngroup of Gs1. So, we come to the ferromagnet state with\nonly one type of superconducting domains. The second\ntypeofsuperconductingdomainsformedbythestate SA2\ncan not be smoothly created from the state S1simply be-\ncause its symmetry group Gs1does not contain GA2as\nits subgroup.\nSimilar situation occurs if instead of conventional su-\nperconducting state the system pass first to nonconven-\ntional superconducting state with the order parameter\ndS2(k) =η2ϕ2(k), (16)\nhaving the symmetry group or superconducting class\nGs2=Ds\n2(Cz\n2)×R, Ds\n2= (E,Cz\n2,eiπCx\n2,eiπCy\n2).(17)\nThis state by the second order phase transition can be\ntransformed only to the homogeneous superconducting\nferromagnet state SA2with symmetry group GA2which\nis the subgroup of Gs2. The superconducting domains\nformed by the state SA1can not be smoothly created\nfrom the state S2because its symmetry group Gs2does\nnot contain GA1as its subgroup. Thus, the formation\nof multi domain superconducting ferromagnet state (13)\nfrom homogeneous superconducting state either conven-\ntional type with an order parameter (14), or nonconven-\ntional type with an order parameter (16) by means the\nsecond order type transitions is impossible.\nWe come to the conclusion that formation of multi do-\nmain ferromagnetsuperconducting state at P > P∗is in-\nevitably related with first order type transition. Either,3\n \n P T \nP*  N \nF \nS1 F+SA1 +SA2  \nFig. 1a  \nP T \nP*  N \nF \n F+SA1 +SA2  \nFig. 1 b \nFIG. 1: Schematic pressure - temperature phase diagrams:\n(a) and (b) (see the text). Here, Nis the normal paramag-\nnetic phase, Fis the ferromagnet phase, S1is the homoge-\nneous superconducting phase, F+SA1+SA2is multi domain\nferromagnet superconducting phase. The thin lines are the\nlines of the second order transithions, the thick lines are t he\nlines of the first order transitions.\nat temperature decrease first we have second order type\ntransition to the homogeneous superconducting state fol-\nlowed at lower temperature by the first order type tran-sition to the multi domain superconducting ferromagnet\nstate (see Fig.1a). Or, there is only one first order type\ntransitionfrom the normalparamagnetstate tothe multi\ndomain superconducting ferromagnet state (see Fig.1b).\nBoth of these possibilities are compatible with an impor-\ntant observation made by resistivity measurements on\npolycrystalline specimens of UCoGe9that the supercon-\nducting phase transition in the paramagnetic state be-\ncomes much broader than in the magnetically ordered\nphase. Toestablishcompletepressure-temperaturephase\ndiagram of single crystal UCoGethe application of the\nother experimental methods is desirable.\nThe external field of about 100 Oe directed along the\neasymagnetizationaxisof UCoGetransformsit from the\nmulti domain to the single domain ferromagnet6. The\ntransformation process should accompanied by weaken-\ning of the first order phase transition depicted at Fig.1\nand smooth conversion of it to the second order type\ntransition.\nIn conclusion, we have found the pressure-temperature\nphase diagram of a ferromagnetic triplet superconductor\nwith strong spin-orbital coupling and orthorhombic crys-\ntal structure. Its shape is dictated by the symmetry of\ntheorderparameter. Theexperimentalrevelationofsuch\ntype phase diagram in UCoGeor similar compounds can\nserve as the direct indication to the type of supercon-\nducting state we deal with.\nACKNOWLEDGMENTS\nThe author is indebted to M. Houzet and A. Buzdin\nforvaluablediscussions,toD.AokiandJ.Flouquetforthe\ninterest to the work and to A. Muratov and S. Blundell\nfor the kind assistance in the manuscript preparation.\n1S.S.Saxena, P.Agarval, K.Ahilan, F.M.Grosche,\nR.K.W.Hasselwimmer, M.J.Steiner, E.Pugh, I.R.Walker,\nS.R.Julian, P.Monthoux, G.G.Lonzarich, A.Huxley,\nI.Sheikin, D.Braithwaite and J.Flouquet, Nature 406, 587\n(2000).\n2D.Aoki, A. Huxley, E.Ressouche, D.Braithwaite,\nJ.Flouquet, J.-P.Brison, E.Lhotel and C.Paulsen, Nature\n413, 613 (2001).\n3N.T.Huy, A.Gasparini, D.E. de Nijs, Y.Huang,\nJ.C.P.Klaasse, T.Gortenmulder, A. de Visser, A.Hamann,\nT.Gorlach, and H.v. Lohneysen, Phys. Rev. Lett. 99,\n067006 (2007).\n4A.Huxley, I. Sheikin, E.Ressouche, N.Kernavanois,\nD.Braithwaite, R.Calemzuk and J.Flouquet, Phys. Rev. B\n63, 144519 (2001).\n5F.Hardy and A.D.Huxley, Phys. Rev. Lett. 94, 247006\n(2005)\n6N.T.Huy, D.E. de Nijs, Y.Huang, and A. de Visser,\nPhys.Rev.Lett. 100, 077002 (2008)\n7A.Huxley, E.Ressouche, B.Grenier, D.Aoki, J.Flouquet\nand C.Pfleiderer, J. Phys.: Condens. Matter 15, 1945(2003).\n8F.Hardy, A.Huxley, J.Flouquet, B.Salce, G.Knebel,\nD.Braithwaite, D.Aoki, M.Uhlarz, C.Pfleiderer, Physica B\n359-361 , 1111 (2005).\n9E.Hassinger, D.Aoki, G.Knebel, and J.Flouquet, cond-\nmat/0804.0500 and Journ. Phys. Soc. Jap. to be published\n(2008).\n10L.N.Bulaevskii, A.I.Buzdin, M.L.Kulic, andS.V.Panjukov ,\nAdv. Phys. 34, 175 (1985).\n11L.D.LandauandE.M.Lifshitz ”Electrodynamics ofContin-\nuous Media”, Pergamon Press, Oxford, 1984.\n12V.P.Mineev, Phys. Rev. B 66, 134504 (2002).\n13K.V.Samokhin and M.B.Walker, Phys. Rev. B 66, 174501\n(2002).\n14V.P.Mineev and T.Champel, Phys. Rev.B 69, 144521\n(2004).\n15V.P.Mineev, K.V.Samokhin ”Introduction to unconven-\ntional Superconductivity”, Gordon and Breach Science\nPublishers, 1999."    },    {        "title": "1906.10974v2.Room_temperature_insulating_ferromagnetic__Ni_Co_1_2xTi1_xO3_thin_films.pdf",        "content": "1 \n  ROOM -TEMPERATURE INSULATING \nFERROMAGNETIC   \n(NI,CO)1+2 XTI1-XO3 THIN FILMS  \nYukari Fujioka1, Johannes Frantti*1, Christopher Rouleau2, Alexander Puretzky2, Zheng Gai2, Nickolay \nLavrik2, Andreas Herklotz3,4, Ilia N. Ivanov2 and Harry M. Meyer2 \n1Finnish Research and Engineering, Helsinki 00180, Finland  \n2Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, \nUSA  \n3Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tenne ssee \n37831, USA  \n4Institute of Physics, Martin -Luther -University Halle -Wittenberg, Halle 06099, Germany  \nKeywords  ilmenite, c orundum, nickel cobalt titanate , oxide , vacancy , thin film,  insulating ferromagnet , \nuniaxial  \n*Corresponding author: e -mail johannes.frantti@fre.fi  \nABSTRACT  \nInsulating uniaxial room -temperature ferromagnets are a prerequisite for commonplace  spin wave -\nbased devices , the obstacle in contemporary ferromagnets being the coupling of  ferromagnetism with  \nlarge conductivity . We show that the uniaxial A1+2 xTi4+1-xO3 (ATO), A=Ni2+,Co2+ and 0 .6<x≤1, thin films \nare electrically insulating ferromagnets  already at room -temperature . The octahedra network of the  \nATO  and the corundum and ilmenite structures are the same yet different octahedra -filling prove d to be \na route to switch from the antiferromagnetic to ferromagnetic regime . Octahedra can continuously be \nfilled up to x=1, or vacated (-0.24 <x<0) in the ATO structure . TiO-layers , which separat e the \nferromagnetic (Ni,Co) O-layers and intermediate the antiferromagnetic coupling between the \nferromagnetic layers in the NiTiO 3 and CoTiO 3 ilmenites , can continuously be replaced by (Ni,Co )O-\nlayers  to convert the ATO -films  to ferromagnetic insulator  with abundant direct cation inte ractions .  \n  2 \n  INTRODUCTION  \nMagnetic , notably conducting ferro - and ferr imagnetic , materials  are commonplace in modern \nmicroelectronics  devices . Ferromagnetic insulators  – still being sparse - have  apparent  adva ntages over \nthe conducting magnetic materials. Oxides of magnetic cations are a natural starting point to look for \nferromagnetic insulators, though they commonly  order antiferromagnetically , as dictated by the oxygen \nmediated superexchange between the cations  [1]. Ferromagnetism  due to  double exchange interaction  \nwas treated in ref. 2, which correlates the ferromagnetism and electric conductivity in (La 1-xAx)MnO 3, A \nis Ca, Sr, or Ba  [3,4]. The double exchange  interaction  explains  the correspondence between the largest \nmagnetization and largest electrical conductivity.  Cobalt modified TiO 2 anatase has been under keen \ninterest due its ferromagnetic characteristics, though the observed ferromagnetism and electric \nconductivity seem to depend on the oxygen content and the origin of the ferromagnetism is not yet fully \nclear [ 5]. The subject of the present report is to show how  ferromagnetic insulators can be formed by \naltering the strength  and sign of the i ndirect and direct interactions , doable in the corundum  and \nilmenite -derived  crystal structures . We also show how ma gnetization can be adjusted  by adjusting \ncompositio n, and briefly address  potential thin film and interface applications.  \nThough  insulating  ferromagnets are rare, electrically insulating oxides with net magnetization  due to \nuncompensated magnetization from the antiparallel alignment of magnetic sublattices  are known. Also \nin these materials the ordering results in from antiferromagnetic interactions. Prime example s are \nferrites, with either the spinel, garnet or hexaferrite structure , commonly applied in the RF -devices as \nfilms with thickne ss often exceeding 100 microns . Magnetic thin films find applications in miniaturized \nRF-devices  utilizing  spin waves as information carriers [ 6,7]. Miniaturization is not solely based on the \nsmaller dimensions of individual devices  but also on the possibilit y to diminish the number of devices  \nvia tunability . By forming layer structures from functionally different materials  devices for versatile  \napplication s can be prepared. Examples are  magnetic sensor , memory , and spintronic devices  [8,9] and \ntunable phase shifter s [10]. Insulating oxides are often  transparent to visible and ultraviolet light. In the \nrecent study t ransparency of  the Co 3O4 nanoparticles  covered by L- and D -cys amin o acids , to circularly \npolarized light in the ultraviolet r ange was reversi bly modulated by magnetic field [11]. Bulk Co 3O4 is \nnot optically active since it has the cubic spinel  structure . The study demonstrated the chirality transfer \nfrom amino acids  to the crystal line core of the  Co 3O4 nanoparticles , creating chiromagnetic \nnanoparticles. Light beam modulation by external magnetic field can be applied  in magnetooptic s. \nSpintronics is an area in which devices based on insulating ferromagnetic thin films are intensively \nstudied. A spin filter tunnel junction was envisaged in ref. 12, and a tunnel junction spin filtering device, \nutilizing insulating ferromag netic EuS layers decoupled by  an Al2O3 layer was  demonstrated  in ref.  13. \nSpin filtering characteristics were measured at 1 K , the Curie temperature of EuS being  16 K. Garnet \nstructured ferrites, mainly Y3Fe5O12 (YIG) and its derivatives , are commonl y utilized in room -\ntemperature applications  based on spin -wave propagation  [8].  Since i ron is a major metallic impurity \nin semic onductor ind ustry (see, for instance ref. 1 4), there is a demand for non-ferrous materials . Due \nto the cubic symmetry  the magne tocrystalline anisotropy of YIG is modes t, requiring an external \nmagnetic field for practical devices. An external magnetic field is impractical for miniaturized devices. \nInstead , ferromagnetic materials with uniaxial anisotropy can be applied in devices requir ing uniform  \nmagnetization. In magnetic tunnel junctions CoFeB is frequently applied , though it requires buffer \nlayers  (commonly heavy metals Mo or Ta ) to preserve the correct structure and composition. Due to the \nelectric conductivity, s pin-waves quickly attenuate  in metallic ferromagnets .  3 \n  Exchange bias - a phenomenon in which an antiferromagnet unidirectionally pins the adjacent \nferromagnetic layer - plays a crucial role  in magnetic  devices . The material choice and layer growth \ntechnique affects the magnitude and the possible reduction  of the exchange bias  upon subsequent field \ncycling . The reduction is called the training effect, which appears both in in -plane and out -of-plane \nmagnetized ferromagnetic films. The latter is often referred to as perpendicular exchange bias and is \nsuggested to have a crucial role for applications [15]. A common  device  consist s of a ferromagnetic \nlayer whose magnetization direction is pinned  by the antiferromagnet , and a ferromagnetic layer whose \nmag netization direction  can be realign ed. The device  forms  a basis for memory cells and spin filters. \nGood interfaces are critical for  proper device functionality [1 6]. Also the oxidation level  affects magnetic \nproperties [1 7]. Attach ment of structurally simi lar layers diminish es defect , such as misfit dislocations \nor strain , densiti es. \nWe report  on an insulating uniaxial  A1+2 xTi1-xO3 (ATO) thin films [1 8], deposited by alternating ta rget \npulsed laser deposition [1 9]. As show n below, in the ATO film s the largest magnetization coincides with \nthe largest resistivity , in contrast to the ferromagnetism due to the double exchange interactions. The \nATO structure  can be visualized as the ilmenite structure in which the octahedra are vacated ( x<0) or \nfilled ( x>0). Oxygen atoms are arranged similarly in the ATO, ilmenite and corundum structures. Cations \ninstead are arranged differe ntly , described in the Supplemental  Material. Origin of the e lectrical and \nmagneti c properties  in NiTiO 3 and CoTiO 3 ilmenites [20-23] (structurally similar to ATO with  x=0) and \nTi2O3 (bears similarity to ATO w ith x≈-1), V2O3 and -Fe2O3 corundum compounds are first summarized  \nto understand the predominance of antiferromagnetic ordering and the influence of 3 d-orbital \noccupancy . Instead of the corundum structure, Co 2O3 [24] and Ni 2O3 [25] adopt the bixbyite structure at \nambient conditions  and thus do not suit for a direct comparison in the present case . V2O3 and Ti 2O3 \ndemons trate the impact  of direct orbital overlap interactions and disordered distribution of cations at \nthe octahedra sites to electrical resistivity , which play s an importan t role in the present material .  \nMAGNETISM AND CONDUCTIVITY IN THE ILMENITE AND CORUNDUM COMPOUNDS  \nThe m agneti c structure of the transition metal oxide s depends on the direct orbital overlap of cations , \nbesides oxygen mediated interactions.  According to ref. 20, magnetic interactions between 3 d-cations \nare divided to oxygen intermediated and direct orbital overlap interactions. Due to the short distance \nand the orientation of t2g-orbital lobes, direct orbital overlap interactions are relevant when octahed ra \nshare an edge or a face, summarized in Supplemental Material.  The sign of the interactions depends on \ngeometrical factors and the occupation of d-orbitals.  \nILMENITE  COMPOUND S \nAntiferromagnetic ordering  in NiTiO 3 and CoTiO 3 \nThe valence band edge  of the ATiO3 ilmenite compounds, where  the A-cation is Ni, Co or Fe  is derived \nfrom oxygen 2 s and 2 p states and the conduction band edge from cation 4 s and 4 p states  [21]. The \ncation 3 d states are localized  within the energy gap , and thus the magnetism due to the localized 3 d-\norbital states is described in terms of direct and indirect interactions .  The compounds are electrically \ninsulating, though the localized states can result in  polaron conductivity at elevated temperatures.   \nNiTiO 3 and CoTiO 3 ilmenites undergo a phase transition from paramagnetic to antiferromagnetic phase \nat 23 and 38 K, respectively. The low -temperature antiferromagnetic structure of NiTiO 3 and CoTiO 3 \nilmenites is explained by oxygen mediated 90° interaction s in the Ni2+-O (Co2+-O) hexagonal basal ab-\nplane , and the interactions between the Ni2+-O (Co2+-O) planes which  separated by diamagnetic Ti4+-O \nlayers. The 90 ° ab-plane interaction is dominated by the coupling of the t2g and eg orbitals  by oxygen . In 4 \n  the case of the more than half filled t2g orbitals and half -filled eg orbitals the interaction is \nferromagnetic, which is the case of Ni2+ (t2g6eg2) and Co2+ (t2g5eg2). The interactions between the layers \nperpendicular to the c-axis are cation -oxygen -oxygen -cation  (cooc)  type, and are antiferromagnetic \n[21]. The interactions are rather weak, as shown by the low Néel temperatures. Magnetic anisotropy \ndictates the spin direction in bulk crystals [ 21], which  is in the ab-basal plane in NiTiO 3 and CoTiO 3.  \nCORUNDUM COMPO UNDS  \nPara - and antiferromagneti c ordering  in Ti2O3 and V2O3 \nThe electronic energy band structure of V 2O3 [26] and Ti 2O3 [26,27] corundum structures bear \nsimilarities  and both exhibit a metal -insulator transition  (MIT)  related to the anomaly in the c/a ratio \nwhen temperature  is varied through the transition . The covalent bonding between cations occurs both \nparallel and perpendicular to the hexagonal c-axis [2 8]. Below the transition temperature the symmetry \nof V 2O3 is lowered to monoclinic I2/a [29]. In literature  electronic -energy -band -structure computations \nrefer to the trigon al symmetry. To explain the MIT,  a band gap should open at the transition  \ntemperature . In trigonal  crystals the three -fold degenerate t2g states are split to nondegenerate a1g and \ntwo-fold degenerate eg states. The simplest explanation offered for Ti 2O3 is that a gap is opened \nbetween a full a1s and empty 𝑒𝑔𝜋 states with decreasing temperature due to the anomalously decreasing \nc/a ratio.  Band structure computations [ 26-28,30 -32] indicate that the Fermi level is located within a \nband formed from Ti/V 3 d t2g-orbital derived  from the overlapping  a1g and 𝑒𝑔𝜋 states  (see particularly \nrefs. 26-28), whereas the antibonding a1g and 𝑒𝑔𝜋 states do not overlap  [26]. O2p-orbital derived states \nlay deeper in the valence band, and the Ti/V 3d 𝑒𝑔𝜎-orbital derived states are few eV above the \nconduction band edge. Due to the overlapping of partially filled a1g and 𝑒𝑔𝜋 states both Ti 2O3 and V 2O3 \nshould be metallic [2 7]. To find an explanation for the discrepancy between experimentally observed \ninsulating state and the band structure prediction valid for an ideal structure , the cases in which the V -\nsite is vacant and the ideally vacant octahedron site is occupied by V were considered in ref. 32. While \nthe density -of-states per V atom at Fermi level for the V 11O18 (one V -site vacant) is increased by 40% \nover the reference V 12O18, the corresponding vacancy -interstitial V 12O18 bands is diminished by 30%  \n[32]. Thus, cation dis order has dramatic effec t on the transport properties . The band gap can also be \nintroduced by  taking electron correlation into account [ 33,34].  \nIn the corundum structure the ≈135 ° cation -oxygen -cation (coc) inte ractions , shown in Fig. 1(a),  \nbetween  the adjacent ab-planes competes wi th the cooc -interactions .  \n \nFig. 1. Panel (a) shows the direct and oxygen intermediated interactions in the corundum structu re. \nPanels (b) and (c) indicate two possible spin orderings in the corundum structure, respectively. The \nordering +--+ shown in (b) is found in -Fe2O3 at low -temperatures and the ordering +-+- shown in (c) \nis found in Cr 2O3. Panels (d) and (e) illustrate the ATO -structure for x=1, and magnetic ordering .  \ncc\nOOc\nc≈135 (a) (b) (c) (d)\n (e)\n+--+\n+-+-\n-+\n+-5 \n  The latter interactions connect next neighbor layers. The coc -angle coupling the layers above and below \nis equivalent only if the octahedra are symmetric and the cation occupies the center. In the corundum \nstructure the angles typically deviate by 10 °. The angles depend on the lattice parameters and the \nfractional coordinates of the oxygen and cations within the unit cell. In compounds in which the eg \norbitals are half filled the 135° coc-interaction is strong and antiferromagnetic. A prediction of the  role \nof direct interactions in Ti2O3 and V 2O3 is given in refs. 22 and 2 3. Based on the 3 d shell occupancies \n(Ti3 d1 and V3 d2), Ti2O3 and V 2O3 were predicted to exhibit one and t wo transitions, respectively [2 2]. In \nV2O3, the high -temperature transition is due to the bond formation through the octahedra face along the \nc-axis, and the lower transition due to the bond formation through the octahedra edge in the ab-plane is \ndiscussed in  refs. 22 and 2 3. Ti2O3 has only one broad transition in which Ti -cations form bonds parallel \nto the c-axis.  Experiments are consistent with the bonding idea  [23]. Straightforward indication  of \nelectron  pairing  is a reduced susceptibility  and conductivity. With decreasing temperature the  \nsusceptibility  of V 2O3 sharply drops at around 180 K . A decrease in susceptibility is also seen in Ti 2O3. \nThe low -temperature p hase of V 2O3 is antiferromagnetic [ 35], though the details of the magnetic \nstructure may be more complicated [ 36]. In contrast to V 2O3, Ti 2O3 exhibits no antiferromagnetic \nordering  [37] as the 3 d1 electrons are paired at low -temperatures .  \n-Fe 2O3 – a weak ferromagnet  \nFe2O3 stabilizes to the  and  phases (the corundum and the bixbyite phases, respectively) at ambient \nconditions. The valence and conduction band edges of the -Fe2O3 are essentially derived from the O 2 p \nstates  and Fe 3 d states, respectively, which predicts the compound to be of charge transfer type  [38]. \nThe low -conductivity of the -Fe2O3 is due to the very heavy carrier masses in the conduction band \nminimum so that the conduction is polaronic. In -Fe2O3 the 135 ° coc-interaction is stronger than the \ndirect interaction and explains the magnetic order [21-23]. The layers perpendicular to the c-axis are \nferromagnetically ordered . Basically, the direct cation -cation interactions favor antiferromagnetic \nordering as do the 90 ° coc-interactions , analog ousl y to Mn2+ in MnTiO 3 [21]. However, t he b asal planes \nare ferromagnetic, since the strong 135 ° coc-interactions couple the ferromagnetic layer \nantiferromagnetically to the layer  below and above. Thus, strong 135 ° coc-interactions  explain  the + --+ \nspin arrangement [39] along the c-axis , shown in Fig. 1(b) . The magnetic ordering in -Fe2O3 occurs \nwithout a change of the unit cell [ 40]. Due to the spin canting -Fe2O3 exhibits weak ferromagnetism.  In \ncontrast, Cr 2O3 exhib its + -+- spin arrangement [4 1] and the direct cation -cation interactions through \nshared  octahedra face have dominant  role , since  the eg orbitals are empty.  \nATO  THIN FILMS  – CHANGE IN MAGNETIC O RDERING VIA INCREASED DIRECT INT ERACTIONS  \nIn contrast to t he ilmenite structure , the  ATO structure  of (Ni,Co) 1+2 xTi1-xO3 is no longer formed from \nalternating (Ni,Co) -O- and Ti -O-layers  (x = 0 being a possible exception ), but the layers  possess three \ntypes of cations.  In the x = 1 limit , all Ti -O-layers of the ilmenite struct ure are replaced by (Ni,Co) -O-\nlayers and the number of cations in  each layers is increased by 50 %. The increase of the cation density \nis due to the complete filling of the octahedra, which affects the direct  and oxygen mediated  interactions \nbetween cation s. In the ilmenite and corundum structures each  octahedr on forms a pair with the \noctahedron above (or below) along the c-axis direction  via face sharing , and the pair has a vacant \nocta hedron above and below. This favors direct interaction, an d ultimately bonding, through the shared \noctahedron  face. Filling of octahedra (x>0) corresponds to  longer filled  octahedra  chains , which in turn \naffects the bonding , cation displacements and magnetis m. For instance, the number of cations sharing \noctahedra  faces  both below and above increases with increasing x, which tends to displace the cation \ntowards octahedron center. When x = 0 or 1, the  ATO structure possess a well -characterized space 6 \n  group  symmetry , 𝑅3̅ and 𝑃63/𝑚𝑚𝑐 , respectively . For other values of x, notably in the Ti -rich regime, \nthe formation of local bonds expectably results in  a non-cooperative phase transition spread over a \nbroad temperature range.  Figs. 1(d) and (e) illustrate  the subject material  of the present paper . By \ngradually filling the vacant octahedra sites by cations  a ferromagnetic insulating phas e is introduced. \nFerromagnetism combined with electrical insulation is not obvious, as will be discussed.  \nEXPERIMENTAL  \nTable 1 summarizes  the samples  addressed in this study . The compositions are given in terms of \nformula A1+2 xTi1-xO3. Sample naming convention was introduced in ref. 1 8, except for the sample \nM(0.03) , and for consistency the same convention is applied here except that from now on we give (x) \nafter the sam ple name. The deposition procedure and results of the structural char acterization are \ngiven in Table 1 in ref. 1 8. Briefly, NiTiO 3, CoTiO 3 and NiCoO 4 targets were repeatedly ablated, different \nsubstrate in -situ heating temperature s between 650 and 900K  were tested , and the oxygen partial \npressure was kept low, at 5mTorr in the case of sample A (-0.24)  and at 10mTorr  for the other samples \nreported in this study . Laser beam fluence was 3.0Jcm-2, except the sample A (-0.24)  which was grown \nby 4.8 Jcm-2 fluence.  Film thickness estimations are based on x -ray diffraction diffraction (XRD) and X -\nray photoelectron spectroscopy (XPS ) measurements . Homogeneous cation distribution in the films \nwas confirmed by repeating a cycle in which a thin layer of film was  sputtered  and XPS spectra were \nsubsequently collected until substrate was reached.  This also yielded a thickness estimate of the films.  \nTable 1. The composition , substrate,  thickness  and fraction of the filled octahedra zv. Except  for the \nsamples F(1) and I(0.61), t he c-axis is perpendicular to the substrate plane . Thickness estimates in \nbraces were obtained from XPS measurements.  Ni, Co and Ti cation valences were +2, +2 and +4, \nrespectively.   \nSample  Composition  Substrate  Thickness (nm)  zv \nA(-0.24)  (Ni 0.57Co0.43)0.51Ti1.24O3 Al2O3 (112̅0) 53 (40)  0.58  \nC(0.24)  (Ni 0.39Co0.61)1.49Ti0.76O3 Al2O3 (0001) 35.5 (34.6)  0.75  \nD(0.37)  (Ni 0.35Co0.65)1.75Ti0.63O3 Al2O3 (0001) 39.0 (60)  0.78  \nE(1) (Ni 0.39Co0.61)3O3 Al2O3 (0001) 44.3  1.00  \nF(1) (Ni 0.39Co0.61)3O3 Al2O3 (101̅0) - 1.00  \nH(0.61)  (Ni 0.42Co0.58)2.22Ti0.39O3 Al2O3 (0001) 53.7  0.87  \nI(0.61)  (Ni 0.42Co0.58)2.22Ti0.39O3 Al2O3 (101̅0) - 0.87  \nJ(0.03)  (Ni 0.46Co0.54)1.05Ti0.97O3 Al2O3 (0001) 39.4  0.67  \nM(0.03)  (Ni 0.46Co0.54)1.05Ti0.97O3 Al2O3 (101̅0) - 0.67  \nMagnetization measurements were performed by a superconducting quantum interference device \n(SQUID) magnetometer (Quantum D esign XL7). F ield-cooled (FC) DC magnetization runs were carried \nout over the temperature range down to 5 K un der 0.1 T  magnetic field. Field -dependent magnetization \ndata as a function of applied field were collected at 5 and 300 K. Data presented below were corrected \nfor the substrate contribution  by subtracting data measured from pure substrate under similar \ncondition as the samples . Two measurement geometries were used, the applied field Ha was either \nparallel (referred to as in -plane) or perpendicular (referred to as out -of-plane)  to the substra te plane . \nDemagnetization field correction was applied for the out -of-plane data according to the equation  \nHi=Ha-4M, where Hi is the  field inside of the film  and M is the magnetization in units of emucm-3.   \nFor electrical measurements interdigitated gold electrodes were patterned by photolithography and a \nmask.  Electrodes were aligned differently in order to see possible orientation dependent conductivity.   7 \n  RESULTS  \nFC-MAGNETIZATION  AND HYSTERESIS  \nFigure 2 shows the magnetization measured with constant magnetic field of 1000  Oe (FC-\nmagnetization) as a function of temperature. Low -temperature anomalies are observed in all samples \nexcept to sample H(0.61) . Sample  J(0.03)  has two-low temperature magnetic trans itions. The humps \nseen in samples A(-0.24) , C(0.24) , D(0.37)  and E(1) are weak and broad. None of the samples exhibits a \ncurve typical to a pure antiferromagnetic order . Sample J (0.03) , as shown below, exhibits co -existing \nantiferromagnetism and ferromagnetism a t low -temperatures , and a clear connection between the \nhysteresis and FC -magnetization  measurements . This is not inconsistent with the single phase nature of \nthe films as discussed below . Weak antiferromagnetic feature is also seen in the s ample E (1) at 5K.   \n \nFig. 2. FC magnetization as a function of temperature. Data were collected with a constant magnetic \nfield 0.1 T with in -plane geometry. Room -temperature magnetization M(300K) was subtracted to each \ncurve, except for the curve shown in inset. The inset shows the H/M curve of the sample J(0.03). Arrows \nindicate two transitions, one occurring at 78 K and the other one at around 25 K. The lower \ntemperature transition was rev ealed by a slight change in the slope. Fits to two data ranges are given in \nthe inset.  \nIn sample E (1), the covalent bonding related to the t2g orbital overlap should first  occur through the \noctahedra face along the c-axis, as it is the shortest direct distance between cations, see Table S1 in \nSupplemental Material. However, t he anomaly observed in sample E (1) between 30 and 50 K does not \ncorrespond to a noticeable  change in the hysteresis loop: M increases slightly with decr easing \ntemperature  and t hus, th e anomaly is not related to a bond formation. Assuming that not all magnetic \nmoments are ordered in high temperature region, the anomaly can be related to an ordering of the \nremaining moments. Another noticeable feature of th e E(1) sample is that the magnetization is weaker \nthan in the samples possessing less Ni and Co, such as the sample J (0.03) . Also the difference between \nthe in -plane and out -of-plane magnetization is small, though the magnetic moments lay in the ab-plane.  \nAn indication of the role of the direct orbital overlap interactions is seen in Ti-rich sample A (-0.24)  \nwhich becomes ferromagnetic only at low -temperatures. Within the direct interaction scheme this \nsuggests that the hybrid state of the Co t2g and Ti t2g is half -filled by an electron originating from the \nhalf-filled Co t2g orbital. The appearance of ferromagnetism is consistent with the idea that the anomaly \nis not related to a covalent bond formation.  Structural distortions, present already at room \n8 \n  temper ature, are observed also in the bulk Ni 1−xCo xTiO 3 Solid Solution [ 42,43]. In thin films [1 8], similar \ndistortions appear in the ilmenite -like sample J (0.03) . \nFig. 3 shows the room -temperature and 5 K magnetization as a function of field, M-H, loo ps for in -plane \ngeometry. Fig. 4 plots the out -of-plane M-H loops . Extracted values and the interpretations of the \nmagnetic data are summarized in Table 2.  \n \nFig. 3. In-plane M-H loops measured at 300 K [panels (a) and (b)] and 5 K [panels (c) and (d)]. P anels \n(b) and (d) give the linear part subtracted M-H loops.  \n \nFig. 4. Out -of-plane M-H loops measured at 5 K. Panel (a) shows the data including paramagnetic part, \nand panel (b) gives the linear part subtracted M-H loop.  \nSample A (-0.24)  is paramagnetic at room temperature. Except for the sample C (0.24) , the linear part  \ncorresponding paramagnetic contribution  is weak  in the samples . The paramagnetic contribution of the  \n(a) (b)\n(c) (d)\n(a) (b)\n9 \n  sample C (0.24)  is larger in in -plane geometry, compare Figs. 3 (c) and 4(a).  The curves from which the \nlinear part is subtracted, Fig. 3 (b) and (d), show that, besides sample A (-0.24) , all samples possess a \nremnant magnetization at room -temperature.  In the case of sample E (1), we note that M slightly \ndecreases with increasin g H at 300K, Fig. 3(b). This behavior is absent at 5 K. We assign it to an \n“overcorrected” substrate contribution. Similar behavior is seen in sample C (0.24)  at 5K, Fig. 3(d).  \nTable 2. Magnetic properties as a function of composition and temperature. Abbre viations are: AFM – \nAntiferromagnetic, FM – Ferromagnetic, FiM – Ferrimagnetic, PM – Paramagnetic, Hc – coercive field (in \nOe), Mr – remanence magnetization (emucm-3), Ms – saturation magnetization (emucm-3), M(T) – \nmagnetization as a function of temperature. Labelings ∥ and ⏊ refer to the easy axis directions parallel \nand perpendicular to the c-axis direction, respectively, and coc(90 °), cooc, cc(e) and cc(f) refer to 90° \nbasal plane cation -oxygen -cation, cation -oxygen -oxygen -cation interactions, and interactions through \nthe shared edge and f ace of octahedra, respectively.   \nSample  In-plane M(T) In-plane \nmagnetic \nstate at 300K, \nHc, Mr, Ms In-plane \nmagnetic state \nat 5K, Hc, Mr, \nMs Out-of-plane \nmagnetic state \nat 5K, Hc, Mr, Ms Easy axis \ndirection  Dominating \ninteractions  \nA(-0.24)  Weak and \nbroad peak \nwith a \nmaximum at \naround 30K  PM PM FM, 170,  2, 6 ∥ cc(f)  \nC(0.24)  Weak and \nbroad peak \nwith a \nmaximum at \n≈40K PM+FM,  131,  \n2, 19 PM+FM,  250,  \n5, 23 PM+FM,  175,  5, \n25 ∥ cc(f), \ncoc( 90°), \ncc(e)  \nD(0.37)  Broad peak \nonset at 70K \nwith a \nmaximum at \n≈40K FM, 153,  1, 5 FM, 210,  1, 14 FM, 490,  2, 8 ⏊ cc(f), \ncoc( 90°), \ncc(e)  \nE(1) Broad peak(s) \nat around 35 \nand 52 K FM, 99, 4, 28 FM, 230,  8, 33 FM, 291,  5, 32 ⏊ cc(f), cc(e), \ncoc( 90°) \nH(0.61)  No anomalies  FM, 93, 5, 43 FiM,  262,  12, \n50 FiM,  386,  12, 56 ⏊ cc(f), cc(e), \ncoc( 90°),cooc  \nJ(0.03)  Slope change \nat 78K and \n25K FiM,  66, 5, 47  FiM,  667,  25, \n64 FiM,  1867,  38, \n72 ⏊ cc(f), cc(e), \ncoc( 90°) \ncooc  \n \nEASY AXIS  AND COERCIVE FIELD  \nAll samples have the c-axis perpendicular to the substrate which provides information about the easy \naxis direction.  As is seen by comparing Figs. 3(d) and 4 (b), magnetization reaches saturation quicker in \nin-plane geome try. Clearest example is exhibited by the susceptibility curve of the sample J (0.03) , which \nfollows a curve characteristic to magnetic moments parallel to the substrate  plane, shown in the inset of \nFig. 2 . Sample J (0.03)  exhibits a double loop at 5 K, showing that the film is  ferrimagnetic. FC curve, Fig. \n2, reveals that this sample undergoes two phase transitions at 78 and 25 K.  The composition and the 10 \n  structure of the sample J (0.03)  is close to the prototype ilmenite , and its properties can be contrasted to \nthe properties found in bulk materials.  In the bulk (Ni 0.60Co 0.40)TiO 3 there are two transi tions, at around \n25 and 69 K [ 42], the lower transition corresponding to a transition to fe rrimagnetic phase with FC \ncurves ch aracteristic to an antiferromagnetic cusp. M-H measurements showed that the \n(Ni 0.60Co 0.40)TiO 3 sample possesses a remnant magnetizatio n at temperatures below 69 K [ 42]. The \ncomposition of the sample J (0.03)  is similar to the bulk sample, and the FC curve is no longer of simple \nantiferr omagnetic type . In contrast to the bulk sample, s ample  J(0.03)  is ferromagnetic  at room \ntemperature. The magnetic moments are parallel to ferromagnetic layers and substrate  plane , as can be \nconcluded from the data shown in Fi g. 2. The layers  perpendicular to the c-axis  are ferromagnetically \ncoupled at room -temperature . A clear onset of antiferromagnetic coupling initiates at around 78 K. In \nthis model the inverse susceptibility below 78 K is 1/ ∥. The model is given in simples t terms and does \nnot describe finer details, such as the change in the inverse susceptibility slope at around 25 K , seen \nalso in the bulk sample . The difference with the phenomenological model given for uniaxial \nantiferromagnet in which the transition take s place between paramagnetic and antiferromagnetic \nphases (see, e.g. ref. 40, p. 124) is that the present transition occurs between phases in which the \ncoupling between the ferromagnetic layers changes as a function of temperature. A clear double loop is \nseen at 5 K, Fig. 3(d) and 4(b) , indicative of a sig nificantly large  fraction of antiferromagnetically \ncoupled layers.   \nCoercive fields are smaller in in -plane case (see Table 2), the exception being sample C (0.24) . This \nbehavior can be explained by assuming that the easy axis is perpendicular to the c-axis in all samples \nexcept sample s A(-0.24)  and C(0.24) . The easy axis of the sample s A(-0.24)  and C(0.24)  is assigned to be \nthe c-axis . Sample A (-0.24)  is the most Ti -rich sample, which sh ows a weak but evident out -of-plane \nremnant magnetization at 5 K. No in -plane magnetization was found for sample A (-0.24)  at 5 and 300 K, \nsee Figs. 3(b) and (d). It is worth to note that though the sample A (-0.24)  was grown on ( 11̅20) \nsubstrate, the film exhibited  preferred 00 l orientation [1 8]. Shape anisotropy  favors magnetization M in \nthe plane of the film and often  has a la rge effect on the direction  of M. The s hape anisotropy due to the \ndemagnetization energy affects the spin orientation and favors in -plane M: the energy cost re lated to \nthe orientation of M is proportional to cos2 , where  is the angle between the film normal and M. \nDespite shape anisotropy , the magnetization of the samples A (-0.24)  and C (0.24)  prefers to stay \nperpendicular to the substrate plane.  \nSATURATION AND REMNANT MAGNETIZATION  \nThe saturation magnetization values extracted from the in -plane and out -of-plane geometries at 5 K are \nsimilar, except for sample D(0.37) , see Table 2. The  magnetization of the samples C(0.24) , H(0.61)  and \nE(1) saturated in both geometries . In sample D (0.37)  magnetization parallel to the c-axis is not \nsaturated, as is seen by comparing the  Ms values measured at 5 K for in -plane and out -of-plane \ngeometries given in Table 2 , consistently with the idea tha t the magnetic moments prefer to be in  the \nab-plane.  The largest remnant and saturation magnetization values are found in sample J (0.03) . Sample \nH(0.61)  exhibits a weak double loop at 5 K, see Figs. 3(d) and 3(b). At room -temperature the sample \nexhibits re mnant magnetization with a magnitude between the values measured from samples E(1) and \nJ(0.03) . Even though the density of Ni/Co cations is largest in the sample E (1), it had a saturation \nmagnetization roughly half of the value found in sample J (0.03) .  \nELECTRICAL RESISTANCE  \nTable 3  lists specific resistance  measured from thin films grown on Al2O3 (101̅0) substrates .  11 \n  Table 3 . Specific resistance , in MΩcm, measured at 1Hz frequency.  \nSample   \nF(1)  2.2 \nI(0.61)  135  \nM(0.03)  170  \nIn terms of the hexagonal axes, only 220 reflection in the sample F (1), and 110 and 220 reflections in \nthe sample I (0.61) , were observed and thus the conductivity values correspond to the conductivity in \nthe 110 -plane (112̅0-plane  in hexagonal indices ).  Despite the substrate orientation, sample M (0.03)  \nhad c-axis perpendicular to the substrate plane , and thus t he orbital overlap interactions through \noctahedra edges are relevant for understanding the conductivity .  \nMeasurements on interdigitated electrodes, oriented r andomly  on the sample surface, indicate d no  \ndependence on the crystal orientation on the plane. Among the three samples, increa ses with \ndecreasing x, i.e., with decreasing octahedra filling fraction . Though all films are insulating, the nearly \ntwo orders of magnitude smaller , when compared with the similarly oriented film I (0.61) , found in \nfilm F (1) suggests  that there is a dependency between the extent of direct cation overlap and increased \nconduc tivity. In sample F (1), every octahedra is filled by Ni or Co. Cation -cation interactions due to the \noverlapping t2g orbitals can result in formation of a fully occupied band.  \nFERROMAGNETIC AND ANT IFERROMAGNETIC COUPL ING \nIn the ATO films the m agnetic moments are ferromagnetically  arranged  in the ab-planes , as in NiTiO 3 \nand CoTiO 3 [21]. The 90° coc-interaction between  Ni2+ and Co2+ cations remains  ferromagnetic  when \nthe filling fraction is increased . In NiTiO 3 and CoTiO 3 the ferromagnetic ab-plane coupling cooperates \nwith the antiferromagnetic cooc-interaction . The ferromagnetic order , stable at room -temperature in \nthin films , is related to the compressive ab-plane stress which enhances the 90 ° coc-interactions as the \ndistances become shorter. Due to the difference between Al 2O3 substrate lattice parameters and the film \nlattice parameters the ab-plane is under compressive stress, whereas the c-axis is stretched [1 8], thus \nthe c ooc-interaction becomes weaker. This, however, does not expl ain the ferromagnetic coupling \nbetween the ab-planes . Instead, we assign the coupling by cation -cation interaction s.  \nIn strained piezomagnetic films , magnetization has a term linearly proportional to the strain . Whether \nthe film exhibits piezomagnetism depends on the magnetic point group symmetry . As an example, the \nmagnetic point groups of the structures shown in Fig. 1(b) and (c) do not exhibit piezomagnetic effect, \nwhereas the ++ -- and ++++ arrangement s corresp ond to the piezomagnetic symmetries  [40]. The \nsymmetry depends on the direction of the magnetic moments . Ferromagnetic phase s (phases with \ndifferent magnetic moment directions  counted distinct ) of the x=1 film s are piezomagnetic  when the  \nmagnetic moments are directed along the x- or y-axis  [44]. The m agnetic point group for both  case s is \n2𝑚⁄2𝑚′⁄2𝑚′⁄ . In this case, the magnetization  depends linearly on stress.  In contrast, if the magnetic \nmoments are parallel to the z-axis, the magnetic po int group is 6𝑚⁄2′𝑚′⁄2′𝑚′⁄ , which does not allow \npiezomagnetic effect.  Table 2 shows that magnetization does not monotonically increase with \ndecreasing x, which could be due to different strain in films . \nTo understand the ferromagnetic coupling along the c-axis, we consider the role of direct cation -cation \ninteractions . Degenera cy of the  t2g states is a valid assumption for Ni and Co, which have 8 and 7 d-\nelectrons, respectively.  The Co -Co direct orbital interactions through shared octahedra face and edge \nare ferromagnetic  [23], consistently with decreasing antiferromagnetic coupling with decreasing Ti -\ncontent . Also the correlation superexchange mechanism for 90 ° coc-interaction is ferromagnetic for \nintera cting Ni/Co cations [ 20]. As illustrated in Suppleme ntal Material, the layers adjacent along the c-12 \n  axis are coupled via 7 1° coc-interactions.  In the case the t2g orbitals are more than half -filled, two p \nelectrons from different p orbitals ( p for each cation) of like spin are simultaneously excited from the \noxygen ion to the eg orbitals of the coupled cations  [20]. In contrast to the corundum structure, in \nsamples E (1) and F (1) each cation has equivalent direct interaction with the cations above and below. \nNo susceptibility decrease, indicative of a covalent bond formation, is seen in Fig. 2. As samples E (1) \nand J (0.03)  show  (Figs. 3 and 4) , the competing 135 ° coc-interaction is still present and is sufficiently \nstrong to couple significant fraction of the ab-planes antife rromagnetically at low -temperature . Further \nincrease in the Ti -content decreases  the magnetic cation content and magnetization.  Though all \noctahedra sites can be filled by Co and Ni, pure c obalt oxide  thin films , grown under similar conditions \nas (Ni,Co) -rich samples, stabilized to spinel structure [1 8]. \nA further contrast to the antiferromagnetic corundum and ilmenites compounds is that the \nferromagneti c and antiferromagnetic interactions  are not cooperative  in the thin films . This is seen \nfrom  the behavior of the samples  H(0.61)  and J(0.03) : both are ferromagneti c at room -temperature, \nwhereas the cooperative antiferromagnetic transition takes place at low -temperature in sample J(0.03) . \nAs Figs. 3 and 4  show , sample H (0.61)  is ferri magnetic at low -temperatures yet exhibits no anomaly in \nmagnetization as a fun ction of temperature, see Fig. 2 . Thus, the antiferromagnetic ordering occurs \ngradually. The anti ferromagnetic coupling does not correspond to a chan ge in the ferromagnetic \ncompo nent . The ferromagnetic coupling occurs at higher temperature than the antiferromagnetic \ncoupling also in the bulk ilmenite (Ni 0.60Co 0.40)TiO 3 [42], and the ferromagnetic magnetization  is \npreserved at lowest temperatures.  The magnetic phase of a single phase sample can be simultaneously \nantiferromagnetic and ferromagnetic, if the symmetry groups of the ferromagnetic and \nantiferromagnetic vectors are the same [ 40]. This is demonstrated in -Fe2O3, in which the magnetic \nmoments are canted and res ult in the weak ferromagnetic phase via Dzyaloshinskii -Moriya interaction . \nIn (Ni+Co) -rich samples, the lack of antiferromagnetic ordering implies that the net magnetization is \nnot due to the Dzyaloshinskii -Moriya interaction , and the 135° coc-interactions no longer dominate.   \nCONCLU DING REMARKS  \nMagnetic properties of (Ni,Co) 1+2 xTi1-xO3, -0.24< x<1, thin films with ATO  structure  were addressed. \nIncreased octahedra -filling increases the direct orbital interactions between the cations, which str ongly \nalters the electrical and magnetic properties. ATO films possess uniaxial magnetocrystalline anisotropy  \nand the octahedra filling and magnetization can be controlled during film growth  by adjusting x. Films \nwith magnetization parallel and perpendicular to the substrate plane were grown. 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Brokchouse. Antiferromagnetic structure in Cr2O3. J. Chem. Phys. 21, 961 - 962 (1953).  \n42. Y. Fujioka, J.  Frantti, A. Llobet, G. M. King, S. N. Ehrlich. Structure and magnetic properties of triclinic \nNi06Co0.4TiO3 ilmenite oxide . Materials Today: Proceedings.  3, 265 -276  (2016) . \n43. Y. Fujioka, J. Frantti, A. Puretzky, and G. King. Raman Study of the Structura l Distortion in the \nNi1−xCo xTiO 3 Solid Solution. Inorganic Chemistry. 55, 9436 -9444 (2016).  \n44. The z-axis coincides with the hexagonal c-axis, the x-axis with the a-axis, and y-axis is chosen so that \nx, y and z form a right -hand ed system. This follow s the convention [38] that the z, x and y axes are \nparallel to the symmetry axes given in the first, second and third entry in the full Hermann -Mauguin \nsymbol , which is here the space  group  symbol  𝑃63𝑚⁄2𝑚⁄2𝑐⁄.   16 \n  \nSUPPLEMENTARY  MATERIAL S \nA2+1+2 XTI4+1-XO3 (ATO)  STRUCTURE  \nThe ATO structure possess es the same oxygen octahedra network as the ilmenite structure. The \nessential difference between the  ilmenite and ATO  structures is the filling fraction of the octahedra and \nthe cation arrangement followed by considerable changes in the crystal symmetry and magnetism. The \nprototype ABO3 ilmenite structure is layered with AO6 and BO6 octahedra alternating along the \nhexagonal c axis. Each layer consists of two, slightly separated, two -dimensional triangular cation nets. \nWithin the layers the AO6 and BO6 octahedra share edges, and out of the layers (along the hexagonal c \naxis ) the octah edra are connected via  shar ed octahedral faces and corners. Each octahedron shares \nedges with three filled and three empty octahedra and a face with one filled octahedron and one empty \noctahedron. This is the origin of the cation displacements towards the vacant octahedron. Two thirds of \nthe octahedra are occupied by cations and the remaining one third of the octahedral -site vacancies are \nordered so as to minimize the cation−sublattice electrostatic repulsive forces.   \nFigure S1(a) illustrates the ATO struc ture, obtained from the ilmenite structure by either filling (0< x1) \nor vacating ( -½<x<0) the octahedra. The Ti/(Ni+Co) ratio is adjustable in the range -0.25 < x < 1, and \ncan be continuously changed during the growth cycle while preserving the crystal stru cture the same. \nThe symmetry changes from rhombohedral to hexagonal (space group P63/mmc ) at around x=0.61. The \nfraction of the filled octahedra, zv, is a function of x, and the ilmenite and corundum str uctures are \nspecial case s of the ATO structure at which x = 0. The dark -grey  (TiO 6) and light -grey  (Ni/CoO 6) \noctahedra indicate octahedra filled in the ilmenite structure. In a general case there is no particular \nordering of the cations. The octahedra vacant i n the ilmenite and the corundu m structures are indicated \nin Fig. S1(a) . The corundum structure is obtained by placing (often in an average sense) the same type \nof cations into the dark - and light -grey octahedra . When -½<x<0 each excess Ti4+ (1) replaces a Ni/Co2+ \ncation and (2) creates  a vacant octahedron.  \n \nFig. S1. (a) The ATO structure , given in terms of the hexagonal axes . Both the ilmenite and corundum \nstructures correspond to the case x=0, zv=2/3, the Ti -rich ATO structure to the case x<0, zv<2/3 and the \n(Ni,Co) -rich  ATO structure to the case x>0, 2/3≤ zv≤1. (b) The ATO structure when all octahedra (dark \ngrey) are filled. The light -grey polyhedra are formed from two face -to-face tetrahedra and are \npredominantly empty.  \nIn practice the Ti valence ch anges at high Ti concen trations - titanium is in a valence state +3 in the \nTi2O3 corundum structure. If 0< x<1, two excess Ni/Co2+ cations are required to replace one Ti4+: the first 17 \n  excess Ni/Co2+ replaces a Ti4+ cation and the second one occupies a vacant octahedron. When x<0 less \nthan 2/3 of the octahedra are filled. The octahedra filling mechanisms are consistent with the charge \nneutrality requirement and experimentally verified compositions and cation valences [1 7]. The ATO \nstructure allows a large variety of cation arrangem ents.  \nAs shown in Fig. S1(b), the ATO structure also contains chains of tetrahedra , connected via alternating \nvertex and face -sharing . No indication of cations in the tetrahedral sites was found  [18], though it is \npossible  that some of the sites are occup ied. When occupied the distance between the  nearest neighbor \noctahedron  and tetrahedron cations through a shared face is particularly short , 1.7Å . If the tetrahedral \nsites would be occupied, there would be 180 ° cation -oxygen -cation interactions along the c-axis.  \nOCTAHEDRA SHARING  \nMagnetic ordering  is controlled by the type of cations and the octahedra filling  configuration . \nMagnetization and electric dipole moment depend on octahedra filling fraction zv. The statistics of the \ncation -cation interaction via s hared octahedra face and edge changes as a function of zv. In the \nprototype ilmenite and corundum structures each octahedron shares one face with one octahedron and \nfor each filled octahedron the number of octahedra sharing just one face is decreased by tw o. When x = \n1, every octahedron shares two faces, which are perpendicular to the c-axis.  \nIn the ATO structure cations are displaced towards the octahedra centers with increasing  zv. When  x = 1, \neach oxygen is shared by six octahedra and each octahedr on is connected to 20 octahedra . Connection \nis via one (shared corner) , two  (shared edge)  or three oxygen  (shared face), as illustrated in Fig. S2. \nWithin the basal plane each octahedron has six edge sharing neighbor ing oct ahedra  (e.g., octahedron 1 \nshares an  edge with octahedra 2 and 3), and above (similarly below) it has six corner sharing octahedra \n(e.g., octahedron 1 shares a corner with octahedra 5 and 6) and two face sharing neighbor s (e.g., \noctahedron  1 shares a face with octahedron 4). Table S1 lists th e corresponding cation -oxygen -cation \nangles and bond lengths. The bond lengths and angles depend on structural parameters, which change \nas a function of zv, and strain due to the lattice mismatch between the substrate and thin film.  \n \nFig. S2. (a) The six octahedra referred to in Table S1. Numbers refer to i and j used in Table S1. Left -\nhand picture illustrates the face, edge and corner sharing octahedra. (b) Right -hand picture shows that \nin the ab-basal plane each octahedron shares each edge with a neighbo ring octahedron.   \nTable S1. Cation -oxygen -cation angles and cation -cation distances in the x = 1 ATO structure. C, E and F \nrefer to Corner, Edge and Face sharing of octahedra pairs i-j specified in Fig. S2, respectively . Ai and Aj \n18 \n  refer to cations in octahedra i and j. Angles and distances are computed for space group 𝑃63/𝑚𝑚𝑐  with \na = 2.978 0 Å and c= 4.8703 Å. cc and coc refer to direct cation -cation overlap and cation -oxygen -cation \ninteractions , respectively . Interactions expec ted to be dominant are given for each pair. The expectation \nis based on the known ordering found in the 3 d-transition metal ilmenite and corundum oxides.  \ni-j Sharing  Ai -O- Aj angle ( °) Ai -Ai distance (Å)  Interactions  \n3-6 F 70.6  2.44  cc \n1-4 F 70.6  2.44  cc \n2-5 F 70.6  2.44  cc \n1-3 E 89.9  2.98 cc, coc  \n4-6 E 89.9  2.98  cc, coc  \n1-2 E 89.9  2.98  cc, coc  \n2-3 E 89.9  2.98  cc, coc  \n4-5 E 89.9  2.98  cc, coc  \n5-6 E 89.9  2.98  cc, coc  \n1-5 C 131.8  3.85 coc \n1-6 C 131.8  3.85 coc \n2-4 C 131.8  3.85 coc \n2-6 C 131.8  3.85 coc \n3-4 C 131.8  3.85 coc \n3-5 C 131.8  3.85 coc \n "    },    {        "title": "0706.4208v1.Penetration_of_Andreev_bound_states_into_the_ferromagnet_in_a_SrRuO___3____110_YBa__2_Cu__3_O___7_δ___bilayer__a_scanning_tunneling_spectroscopy_study.pdf",        "content": "arXiv:0706.4208v1  [cond-mat.supr-con]  28 Jun 2007Penetration of Andreev bound states into the ferromagnet in a\nSrRuO 3/(110)YBa 2Cu3O7−δbilayer: a scanning tunneling spectroscopy study\nItay Asulin,1Ofer Yuli,1Israel Felner,1Gad Koren,2and Oded Millo1,∗\n1Racah Institute of Physics, The Hebrew University, Jerusal em 91904, Israel\n2Department of Physics, Technion - Israel Institute of Techn ology, Haifa 32000, Israel\nScanning tunneling spectroscopy of thin epitaxial SrRuO 3/(110)YBa2Cu3O7−δferromag-\nnet/superconductor bilayers, reveal a clear penetration o f the Andreev bound states into the fer-\nromagnetic layer. The penetration is manifested in the dens ity of states of the ferromagnet as a\nsplit zero bias conductance peak with an imbalance between p eak heights. Our data indicate that\nthe splitting occurs at the superconductor side as a consequ ence of induced magnetization, con-\nfirming recent theoretical predictions. The imbalance is at tributed to the spin polarization in the\nferromagnet.\nPACS numbers: 74.45.+c, 74.50.+r, 74.78.Fk, 74.81.-g\nI. INTRODUCTION\nThe study of superconductor (S)/ ferromagnet (F)\nproximity systems allows a direct investigation of the in-\nterplay between the two competing orders of supercon-\nductivity and ferromagnetism. In a highly transparent\nS/normal-metal (N) junction, superconducting correla-\ntions are induced in N while they are weakened in the\nS side. The mechanism underlying the proximity effect\n(PE) is the Andreev reflection (AR): a hole-like quasi-\nparticle impinging on the interface from the N side is\nretro-reflected as an electron-like quasiparticle with op-\nposite momentum and spin while destroying a Cooper\npair in the S. Consequently, in S/F junctions, the prop-\nerties of the PE are significantly modified due to the\nspin polarization and the presence of an exchange field\nin F. Theoretical works predict a rapid oscillatory decay\nof the induced superconducting order parameter inside\nF.1,2,3These predictions, that were confirmed by vari-\nous experiments4,5,6, stem from the singlet pairing in the\nS, and hold for both s-wave and d-wave (along antinodal\ndirections)superconductors.7,8However,theunderstand-\ning of how the anisotropyof the d-wavesymmetry within\nthe a-b plane manifests itself in the PE in the presence of\nan exchange field, is still rudimentary. In particular, it is\nstill unclear what happens to the Andreev bound states\n(ABS) that reside at the nodal surfaces of d-wave S and\nhow, in turn, they affect the density of states (DOS) of\nF.\nSplitting the current carrying ABS requires the re-\nmoval of either their directional or spin degeneracy. The\nformer may result from the admixture of a subdominant\norder parameter9while the latter may be due to a small\nmagnetization at the surface. A possible origin of such\na surface magnetization is the background antiferromag-\nnetic correlations in the underdoped regime of the high-\nTcsuperconductors.10Alternatively, a magnetization in-\nside the S may also result from a proximity to a mag-\nnetic layer. Recent theoretical works that considered the\ninverse PE in S/F bilayers predict an induced magneti-\nzation in the S side over a length-scale of the supercon-ducting coherence length, ξS.11,12,13,14The direction of\nthe induced magnetic moment near the interface may be\nparallelorantiparalleltothemagnetizationinF,depend-\ning on the interface transparency, the thickness of F and\nthe strength of the exchange field. Experimentally, there\nis little evidence for such an induced magnetization.15,16\nThe induced magnetization is expected to remove the\nspin degeneracy of the ABS at the S/F interface and\nshift them to finite energies. However, this effect was not\nobserved so far in the DOS of F and is the focus of this\npaper.\nWe have previously shown in N/nodal-oriented d-wave\nS bilayers that as long as the interface is highly transpar-\nent, the N layer is ballistic and phase coherence is main-\ntained, ABS can penetrate into the N layer and give rise\nto a zero-bias conductance peak (ZBCP) in its DOS.17\nThe question that arises now is what happens when an\nexchange field is introduced in the normal layer. The\npossible penetration of the (spin degenerate) ABS into a\nspin polarized F layer was not observed experimentally\nand remains unclear theoretically.\nIn this work we employed scanning tunneling spec-\ntroscopy on thin epitaxial SrRuO 3/(110)YBa2Cu3O7−δ\n(SRO/(110)YBCO) S/F bilayers below full coverage of\nthe YBCO by the SRO layer. Our spectroscopy mea-\nsurements clearly reveal an intricate penetration of the\nABS into the F layer to distances of at least 9 nm. The\npenetration is reflected in the DOS of the F layer as a\nsplit ZBCP with a pronounced imbalance between split\npeak heights. The split is attributed to the removal of\nthe spin degeneracy of the ABS due to induced magneti-\nzation in the S. The imbalance between the peak heights\nis attributed to the spin polarization in the F layer.\nII. EXPERIMENT\nSRO is an itinerant ferromagnet ( TCurie= 150Kin\nthin films) with lattice parameters that are similar to\nthose of YBCO18and therefore they can form highly\ntransparent interfaces, essential for the existence of AR2\n20 40 60 80 100 120 140 160 180 -2.0x10 -3 -1.0x10 -3 1.6x10 -5 2.0x10 -5 2.4x10 -5 \n130 K \n  \n150 K 88 K Moment (emu) \nTemperature (K) b) a) \nFIG. 1: (a) SEM image (0.8X0.6 µm2) of a 3 nm thick SRO\nlayer on a (110)YBCO film demonstrating the SRO island\n(bright areas) topography. Inset: a zoom on one SRO crys-\ntallite. (b) Magnetization measurement (at 1000 Oe after\nzero-field cooling) depicting the ferromagnetic transitio n on-\nset at 150 K and the S transition at 88 K.\nand the PE. A total of 16 bilayers of SRO (2-17 nm nom-\ninal thickness) on (110) YBCO were prepared by laser\nablationdepositionon (110) SrTiO 3substrates. In order\nto achieveYBCO layerswith the nodal (110) orientation,\nfirsta10nmthicktemplatelayerofYBCOwasdeposited\nat 660◦C, and then another YBCO layer of 50 nm thick-\nness was deposited at 750◦C. The (110) orientation was\nverified by x-ray diffraction. Finally, the SRO layer was\ndeposited at 785◦C. Annealing for obtaining optimally\ndoped YBCO was done in-situunder 50 Torr oxygen gas\npressureand a dwell ofone hour at 450◦C (the SRO layer\nis insensitive to the oxygen annealing process). The bi-\nlayers showed S transition temperatures at around 88 K\nwith a transition width of about 3 K, implying nearly\noptimally doped films. The tunneling spectra (dI/dV vs.\nV characteristics, proportional to the local quasi-particle\nDOS) were obtained at 4.2 K, much lower than both the\nsuperconducting and ferromagnetic transitions.III. RESULTS AND DISCUSSION\nThe bare (110)YBCO films feature ∼40×100 nm2\nelongated crystallite with uniform directionality over ar-\neasofa few µm2(see Ref. 17). Figure 1apresentsa scan-\nning electron microscope (SEM) image of a (110)YBCO\nfilm covered with a 3 nm thick (nominal) SRO layer. At\nlow thicknesses the SRO layer exhibits island-film topog-\nraphy where the underlying elongated crystallite struc-\nture of the (110)YBCO is clearly apparent in between\nthe islands. This island topography enabled us to mea-\nsure, on the same sample, the evolution of the DOS from\nthat of a bare (110)YBCO surface to the DOS of a fer-\nromagnet in proximity to a (110) surface. Full coverage\nby the SRO layer was obtained at average thicknesses\nof above 10 nm. Below full coverage, the islands fea-\ntured varying thickness even within each island. This is\napparent in the inset of Fig. 1b which focuses on one\nSRO crystallite. It is important to note that in thin\nSRO films of thicknesses lower than 5 unit cells, TCurie\ndecreases substantially.19Therefore it is expected that\nthe exchange field and local magnetization in films that\nfeature thickness variations at these ranges will not be\nuniform. Fig. 1b depicts a SQUID magnetometer mag-\nnetization measurement performed on a 3 nm thick SRO\nlayer over-coating a (110)YBCO layer. The onset of the\nferromagnetic transition at ∼150 K is clearly observed,\nindicating that our SRO layers at these thicknesses are\nferromagnetic. However, it is also apparent that there is\na distribution of the value of TCurie: the transition is not\nsharp and takes place over a range of temperatures start-\ning from 150 K down to 130 K (marked by arrows in Fig.\n1b), indicating that the SRO layer is magnetically inho-\nmogeneous. This inhomogeneity stems from two main\nfactors. First is the island topography that yields differ-\nences between the mean exchange field at the crystallite\nedge and its interior. Second is the thickness variations,\nas discussed above.19The diamagnetism of the YBCO\nsets in at about 88 K.\nThe tunneling spectra of the bare (110)YBCO films\nfeatured a pronounced ZBCP with gap-like features all\nover the YBCO crystallites (see Ref. 17 for data on sim-\nilar films). Interestingly, ABS were also detected in the\ntunneling spectra obtained on the SRO crystallites. Fig-\nure 2a presents an STM image of a 2 nm thick SRO layer\novercoating a (110)YBCO film where an SRO crystallite\nis apparent. The tunneling spectra presented in Fig. 2b\nwere taken at the points marked in the topographic im-\nage. Far enough from the crystallite (point 1), on a bare\nYBCO region, a pronounced centered ZBCP was mea-\nsured (upper black curve), similar to those measured on\nbare (110) films.17The middle (red) dI/dV curve was\nobtained at the vicinity of the crystallite edge where the\nthickness of the SRO is lower compared to the center\nof the crystallite (point 2). The amplitude of the peak\nis largely suppressed and it is shifted by ∼0.8 meV to\npositive energies. The lower (blue) dI/dV curve was ob-\ntained on the SRO crystallite (point 3). Here, the ZBCP3\n-20 -10 0 10 20 0.2 0.4 0.6 0.8 1.0 1.2 1.4 \n21\n3\n3 meV \n  dI/dV (arb. units) \nBias (mV) 0.8 meV b) \n123a) \nFIG. 2: (color online) (a) 80 nm2STM image of a 2 nm thick\nSRO layer on a (110)YBCO film showing an SRO crystal-\nlite (yellow brighter area). (b) Tunneling spectra (vertic ally\nshifted for clarity) obtained at the points marked in (a)\nis shifted to +3 meV and the peak appears to be broad-\nened. In fact, our overall accumulated data (see below)\nimplies that this shifted and broadened peak reflects a\nsplit ZBCP in which the negative energy peak (in this\ncase) is largely suppressed and smeared (due to noise\nand life-time broadening effects).\nThe typical tunneling spectra that were obtained on\nthe SRO crystallites featured a pronounced split of the\nZBCPs with a varying degree of imbalance between the\npositive and negative peak heights. This imbalance\nranges from a nearly symmetrical, double-peak structure\nto an asymmetricalsplit in which one ofthe peaks is fully\nsuppressed. Interestingly, the suppression of both the\npositive and negative peaks were observed with roughly\nthe same occurrence (even on the same sample). More-\nover, the amplitude of the split (energy shift of ABS)\nalso featured pronounced changes ranging between less\nthan 1 meV and a maximum energy shift of ∼5 meV\n(much larger than the known split amplitude due to the\nadmixture of a sub-dominant OP at optimal doping).9\nWe note that a wide range of both the degree of imbal-\nance and the energy shift were observed on all samples\nwith average SRO thickness up to 9 nm. Figure 3 shows\nthe range of peak imbalance and the suppression of neg-\native as well as positive peaks. Here, the dI/dV curves\nwere all obtained on the same sample (but in different\nareas), with an SRO thickness of 3 nm. The peaks in\nFig. 3a (with shifts of up to ∼3.5 meV) exhibit a small,\nbut pronounced, imbalance between peak heights where\nthe positive peak is suppressed relative to the negative\npeak. The imbalance is larger in Fig. 3b in which the\npositive energy peak seems to be fully suppressed and\nthe negative peak is found at -3.1 meV. The degree of\nimbalance in Fig. 3c is roughly the same as in 3a but\nhere the negative energy peak is suppressed. In Fig. 3d\nthe suppression is, again, at the negative side and only\na trace of the negative peak is still detectable, yet more\nthan in 3b.\nThe range of the energy shift is demonstrated in Fig.\n4. Here, the dI/dV curves were acquired sequentially at\nequal steps along a 200 nm long line taken on a 2 nm\nthick SRO layer. The ZBCP evolves from a centered-20 -10 0 10 20 0.6 0.8 1.0 1.2 \n3.5 meV \n  \n-20 -10 0 10 20 0.4 0.6 0.8 1.0 \n  \n3.1 meV \n-20 -10 0 10 20 0.4 0.6 0.8 1.0 1.2 \n3.2 meV \n  \n-20 -10 0 10 20 0.4 0.6 0.8 1.0 \n2.7 meV \n  dI/dV (arb. units) \nBias (mV) a) b) \nc) d) \nFIG. 3: (color online) Tunneling spectra obtained on a 3 nm\nthickSROlayerovercoatinga(110)YBCO filmdemonstrating\nthe range of imbalance between split ZBCP peak heights and\nthe suppression of both positive and negative peaks.\n-20 -10 0 10 20 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 \n4.5 meV \n3.5 meV \n  dI/dV (arb. units) \nBias (mV) 2.8 meV \nFIG. 4: Tunnelingspectra obtained sequentially at fixedste ps\nalong a 200 nm line on a 2 nm thick SRO layer overcoating\na (110)YBCO film demonstrating the range of energy shift.\nThe curves are shifted vertically for clarity.\nZBCP (bottom curve) into a shifted ZBCP where the\nenergy shift increases up to a value of 4.5 meV (upper\ncurve). This variation of the energy shift occurs on a\nsingle SRO island (as demonstrated also in Fig. 2). It is\nimportant to note that there is no clear correlation be-\ntween the magnitude of the split (energy shift) and the\ndegree of imbalance between peak heights (e. g., com-\npare figure 3a and 3b), suggesting that these two effects\nhave a different origin.\nWe shall first address the mechanism by which ABS\ncan penetrate the F layer. The formation of ABS at\nthe bare nodal (110) surface in YBCO can be under-\nstood within the framework of a quasiclassical insula-\ntor (I)/N/S quantum well model20, where an effective N4\nlayer of thickness ξSresults from the pair breaking na-\nture of nodal surfaces. Here, the current carrying ABS\ncorrespond to closed quasiparticle trajectories consisting\nof ARs at the S/N interface and normal reflections at\nthe free interface. A metallic N layer, that is in good\nelectrical contact with the S, serves as an extension of\nthe above-mentioned inherent N region at the nodal sur-\nface and the INS model is still valid.17The picture is\nmore intricate if the N layer is replaced with a spin po-\nlarized F layer. The AR is suppressed due to the im-\nbalance between up and down spin populations. More-\nover, the wave-vector mismatch between quasiparticles\nwith up and down spins in F leads to breaking of the\nretro-reflectivityoftheARprocess.21Thismeansthataf-\nter two consecutive Andreev and normal reflections the\nquasiparticle will reach the S/F interface at some dis-\ntance, L, from the originally impinging quasiparticle and\nthus a closed trajectory will not be realized. However, as\nlong asL < ξS, ABS can still be formed. The above con-\nstraints, and the virtual Andreev reflection process21,22,\nwillresultinadecreasedspectralweightoftheABS.Nev-\nertheless, the penetration of ABS into an F layer is still\npossible and is observed here by the pronounced ZBCP\nwe measured in the DOS of the F layer. The reduced\nspectral weight is also demonstrated in the above data\n(figures 2 and 4). The ABS will survive in the F layer as\nlong as its thickness is smaller then the mean free path\nand phase coherence is maintained between the electron-\nlike and hole-like quasiparticles along their trajectories.\nOur results show a clear penetration of the ABS up to\nthicknesses of 9 nm. The mean free path in our SRO is-\nlands couldbe significantlysmallerthan 50nm, the value\nmeasuredinsingle-crystalsat1.8K.23Phasecoherencein\nFismaintainedonthelength-scaleofthetheF-coherence\nlength, whichis estimatedto be 3nm inthe cleanlimit.24\nTherefore,oneshouldnotexpectpenetrationtodistances\nlarger than ∼9 nm. We emphasize that the measured\nZBCP was continuous all over the SRO crystallites and\nwas not localized along narrow and elongated strips. For\nthat reason, the observed penetration of ABS into the\nSRO layer can not be accounted for by effects related\nto magnetic domain walls, as was the case in our previ-\nous work on (110)YBCO/SRO (antinodal-YBCO/SRO)\nbilayers.24In that study, the OP penetrated the F layer\nto distances above 26 nm, an order of magnitude larger\nthan the F-coherence length. This penetration, however,\ntook place only along narrow and elongated strips, sep-\narated by at least 200 nm, consistent with the known\nmagnetic domain wall structure in SrRuO 3. This behav-\nior was attributed to crossed Andreev reflections, taking\nplace in the vicinity of the magnetic domain walls.24\nThe observedimbalancebetween peak heights is a con-\nsequence of the spin polarization inside the F layer: the\npeak that corresponds to the minority spins in F is sup-\npressed with respect to that corresponding to the ma-\njority spins. However, the exchange field in F is not ex-\npected to split the penetrating ABS, as it does not split\nthe coherence peaks of the observed2,4induced gaps inS. We therefore suggest that the splitting of the ABS\ntakes place at the interface or inside S. Our observa-\ntion that the negative and positive energy peaks are sup-\npressed with the same occurrence supports this conjec-\nture. When the spin degeneracy of the ABS is lifted\nand the ZBCP is split, the peak that corresponds to the\nminority (majority) spins is shifted to positive (nega-\ntive) energies. Consequently, if the splitting occurs in-\nside F, then the spin polarization should always suppress\nthe peak that corresponds to the minority spins, i. e.,\nthe suppression is always expected to occur in the same\nside regardless of the direction of magnetization in F.\nWhich of the sides (positive or negative) will be sup-\npressed in F is determined by the sign of its spin polar-\nization, which is known to be negative in SRO (i.e., the\nmajority spin at the Fermi surface is antiparallel to the\nbulk magnetization).25Therefore, only the negativepeak\nis expected to be suppressed in that case, in contrast to\nour observation. This strongly indicates that the split-\ntingcannotoccurinsidetheFislands. Moreover,thelack\nof correlation between the degree of splitting and imbal-\nance providesfurther evidence that the splitting does not\noccur inside F. We believe that the observed splitting is\na consequence of an induced magnetization inside S that\npenetrates it to a distance of ξS. As noted above, this\ninduced magnetization can be parallel or antiparallel to\nthat in F, depending on local changes in the thickness\nof the F layer and the interface transparency.11,12,13,14\nTherefore, both negative and positive sides can be sup-\npressed (on the same sample), depending on the relative\ndirection of magnetization with respect to that of F.\nWe now turn to discuss the magnitude of the split.\nThe relevant energy scale is the exchange energy, Eex∼\nKBTCurie, which sets the upper limit for the value of the\nenergy shift. In SRO, Eex∼13meVand would corre-\nspond to a maximum energy shift of ∼6.5 meV to each\nside. The induced magnetization should be weaker and\nhence the lower values we typically observe. As demon-\nstrated in figures 2 and 4, the observed energy shift can\nchange at different lateral locations on the same F layer,\neven on the same crystallite. This can be explained by\nlocal changes in thickness of the F layer and/or the in-\nterface transparency that cause a non-uniformity in the\ninduced magnetization. In general, the energy shift (or\nsplit amplitude) become larger as we move towards the\ncenter of the island, where the thickness is usually larger\ncompared to the rims. This is consistent with stronger\nmagnetization and consequently stronger induced mag-\nnetizationinS,yieldinglargersplits(shifts)attheYBCO\nside of the interface. The magnetic non-uniformity of the\nSRO layer, manifested in the distribution of the SRO’s\nTCuriein Fig. 1b, supports this assumption. The junc-\ntion transparency, on the other hand, is not expected\nto change within an island over such distances. So, it\nseems that the local magnetization affects our spectra,\nalthough the interface transparency and its variations\n(mainly from island to island) may also play an impor-\ntant role.5\nA theoretical model that is seemingly related\nto our experiment was introduced by Kashiwaya\net al., who studied the conductance properties\nof ferromagnet/ferromagnet-insulator/d-wave supercon-\nductor junctions.21Their calculated I-V characteristics\nexhibit (in the case of the nodal S surface) spectral fea-\ntures similar to some of those observed in our measure-\nments, namely, an asymmetric split of the ZBCP, where\nthe peak at positive energy is suppressed. As in our\ncase, the polarization of the F electrode suppresses the\npeak that correspondsto the minority spins. However, in\ntheir model the splitting of the ZBCP is a consequence of\n’spin filtering’, namely, different effective barrier heights\nthat are felt by quasi-particles with opposite spins that\ntunnel through the ferromagnetic insulator.21In spite of\nthe fact that we could achieve good fits to our spectra\nusing this model, we do not believe that it applies to\nour experiment, as explained below. Very low F/S in-\nterface transparencies are needed in order to reproduce\nour spectral features by this model, even lower than the\ntypical values corresponding to the tunnel junction be-\ntween the STM tip and the sample ( Z∼5 within the\nBTK formalism26). Such low transparencies are not con-\nsistent, to say the least, with our previous STS studies\nof SRO/YBCO bilayers24and magnetoresistance inves-\ntigations in YBCO/SRO/YBCO trilayers27where prox-\nimity effects were observed, implying highly transparent\n(lowZof less than 1) interfaces (that also allow for in-\nduced magnetization in the YBCO layer). Consequently,\nin our measurements transport is dominated by the tip-\nsample tunnel junction and therefore our spectra reflectthe DOS at the SRO surface and thus monitor the pen-\netration of the ABS into the SRO layer. In contrast, the\nI-V spectra calculated by the Kashiwaya et al. model\nreflect the properties of the S/F interface, in particular\nthe effect of spin filtering. We also like to point out that\nthe spin filtering scenario cannot naturally account, in\nour bilayers, for those spectra in which the negative peak\nis suppressed. For this to occur, one has to impose the\nexistence of an insulating ferromagnetic layer (between\nthe YBCO and SRO layers) having spin polarization op-\nposite to that of the bulk SRO, which is unrealistic.\nIV. SUMMARY\nIn summary, our scanning tunneling spectroscopy\nmeasurements of SrRuO 3/(110)YBa2Cu3O7−δbilayers\nclearly reveal a penetration of the ABS into the F layer\nand provide evidence for the predicted splitting of ABS\nat the S/F interfaces due to an effect of induced magne-\ntization at the S side.\nV. ACKNOWLEDGMENTS\nWe thank G. Deutscher, D. Orgad and L. Klein for\nhelpful discussions. This work was supported by the Is-\nrael Science Foundation, Center of Excellence program,\n(grant # 1565/04). O. M. acknowledges the Harry deJur\nchair of applied sciences at the Hebrew University.\n∗Electronic address: milode@vms.huji.ac.il\n1E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys.\nRev. B55, 15174 (1997).\n2A. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n3M. Zareyan, W. 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Koren, Phys.Rev.B. 72, 184515 (2005)."    },    {        "title": "1111.2421v3.Static_ferromagnetic_materials__from_the_microscopic_to_the_mesoscopic_scale.pdf",        "content": "arXiv:1111.2421v3  [math.AP]  2 May 2013Static ferromagnetic materials:\nfrom the microscopic to the mesoscopic scale\nBrigitte Bid´ egaray-Fesquet, Quentin Jouet, and St´ ephane Lab b´ e\nLaboratoire Jean Kuntzmann and Universit´ e de Grenoble,\nBP 53, 38041 Grenoble Cedex 1, France\nNovember 3, 2018\nAbstract\nThanks to averaging processes and Γ-convergence technique s, we\nare able to link a microscopic description of ferromagnetic materials\nbased on spin lattices and their mesoscopic description in t he static\nframework for thethree fundamental contributions: exchan ge, magne-\ntostatics and external.The results are in accordance with t he classical\ncontinuous description of ferromagnetic phenomena and jus tifies it.\nThis work is a seed towards a dynamic description of ferromag netic\nmaterials.\n1 Introduction\nThe continuous description of ferromagnetic materials has been int roduced\nsince the 1960 viathe micromagnetism model developed by Brown [5]. This\nmodel, based on a thermodynamical description of ferromagnetic p henom-\nena, has proved its efficiency in numerous works viarelevant simulations\n([3, 7, 9, 14]). Nevertheless, several problems persist in the desc ription, from\nthe thermic effects to the magnetostrictive behaviors. These pro blems are\nvery sharp and, in order to understand their modeling, one need to under-\nstand the microscopic behavior of magnetization (atomic scale) and the link\nbetween this scale and the mesoscopic scale (continuous magnetic m atter\nscale of ferromagnetic effects). In this paper, we focus on the be ginning of\nthis program: the link between a microscopic description of ferroma gnetic\nmaterials and their mesoscopic description in the static framework f or the\nthree fundamental contributions: exchange, magnetostatic an d external field\n1[1, 10, 12]. Here, the microscopic scale designates the atomic scale w here\natom nuclei are assumed to be pointwise electric charges bearing on e mag-\nnetic moment induced by the atom electronic cloud; the mesoscopic s cale\ndesignates the continuous description of matter for which the fer romagnetic\neffects are significative. One of the first formal approach of the lin k be-\ntween the mesoscopic behavior of ferromagnetic materials and a mic roscopic\nthermodynamic description of this behavior can be found in [6] for a model\nfocused on the magnetoelastic phenomenon. In this book, the aut hors gives\na formal approach of the asymptotics.\nWe can cite also several mathematical studies connected to the pr oblem\nwe study in this paper. First, in [11], authors study the convergen ce of ener-\ngies for given sequences of magnetizations and various oscillatory b ehaviors\nof the magnetization. This approach does not consider the conver gence of\nminimizers but of energies and shows the importance of controlling loc al\noscillations of the magnetization in order to obtain a micromagnetic mo del\nwhich ensures that the modulus of the limit magnetization is constant on the\nferromagnetic domain. We can also mention the study performed in [1 3] who\nemphasizes the importance of controlling local oscillations of the mag netiza-\ntion in order to compute the limit magnetostatic interactions. This st udy is\na derivation of the interaction forces between two magnetic bodies for given\nmagneticconfigurationsbuiltfromthemicroscopicinteractionsbet weenmag-\nnetic moments. It gives interesting informationabout the magneto staticfield\nof a body depending on the local regularity of the magnetization. Th e study\nof the limit mechanical stress is also studied in [8] where the authors build\na theory for the description of deformable magnetic bodies. A micro –meso\napproach is adopted in order to justify, from microscopic well inter actions of\ndipoles, the interaction at the meso-scale, but this does not tackle the prob-\nlem of the limit of minimizers. Nevertheless, in [2], a study of the minimizer\nconvergence is performed in the case of Ginzburg–Landau system s. Authors\nfocus on a Γ-convergence with a Dirichlet energy where the limit cons traint\nis the divergence-free condition and not the constraint on the loca l modulus\nof the obtained field. In this case, the constraint on the modulus de rives\nfrom a penalization term and not from the modeling hypothesis.\nIn our paper, we are interested in the convergence of minimizer seq uences\nof the microscopic description of ferromagnetic materials toward a contin-\nuous micromagnetic description which respects the constraint of u niformity\nof the local modulus of the magnetization. This constraint is obtaine d by a\nhypothesis on the local variations of the magnetization. In particu lar, this\ncontrol of the local variation induced by the Heisenberg interactio n model\nhas to be tempered with a hypothesis on allowed defects. Forgettin g defects\nwould give riseto aconstrained system andinparticular limits in HspΩqfors\n2strictly greater than1. Such aregularitywould notallowkey microst ructures\nof the micromagnetism theory: the vortices.\nAt the microscopic scale, we describe the material as a regular perio dic\nspinlatticeintersected withthemagneticdomain(aboundedopense tofR3).\nSection 2 is dedicated to the mathematical description of the micros copic\nmodel (the spin lattice) and to an averaging process towards a mes oscopic\nmodel which leads to a constant norm magnetic field, in accordance w ith\nusual models of micromagnetism. The microscopic energies are intro duced\nand several modeling hypotheses are set. The main hypothesis is ind uced by\ntheadiabaticbehavior oftheHeisenberg energy comparedto theg lobalferro-\nmagnetic energy. This hypothesis gives a constraint on neighboring magnetic\nmoments. In fact, this constraint is verified by a set of minimizers fo r a given\nlattice.\nThe energy induced by the Heisenberg interaction is more difficult to\ntreat. Section 3 addresses the study of this contribution and, in p articu-\nlar, its asymptotic behavior for sequences of lattices verifying the modeling\nhypothesis. This hypothesis ensures compactness which allows to u se Γ-\nconvergence tools in H1. The limiting energy constructed from the discrete\nmagnetization is the exchange energy (Theorem 1).\nInSection4, weintroducethedemagnetizationandexternal ener giesboth\nfor discrete lattices and the continuous model and finally obtain a co nver-\ngence result for the sum of the Heisenberg, demagnetization, and external\ncontributions (Theorem 2).\n2 Mathematical descriptions of a spin lattice\n2.1 Atomic lattice description\nWe consider a collection of spins which are located on the nodes of a pe riodic\nlatticeLin theRdspace (d“1,2,3) with mesh size aą0. In the scope\nof this paper we will restrict to the case of 1D, square or cubic lattic es,Lis\nsimplyaZd, but we can think of more complex lattices. Here all the nodes\nplay the same role to ensure a unique definition of neighbors. In the o pposite\ncase a multi-species model should be used.\nThe nodes are indexed by iPNand we denote by xitheith spin location\nandµxithe corresponding spin value (magnetic moment). The norm of thes e\nmagnetic moments are scaled to the unit value and therefore for all iPN,\nµxiPS2, whereS2is the unit sphere of R3.\nInstead of describing a collection of magnetic moments, we can gath er all\n3the values in one single vector field µdefined by\n@xPRd, µpxq “ÿ\niPNµxiδxipxq,\nwhereδxiis the Dirac delta function centered at xi.\n2.2 Scaling\nWe want to obtain an homogenized model of the spin lattice, i.e. give a\ndescription when this lattice is seen from far. Instead of really doing this,\nwe will perform some dual transformation, i.e. consider only nodes t hat are\nincluded in some fixed bounded domain Ω, and shrink the lattice (as sho wn\non Figure 1 for d“2). More precisely, we suppose that 0 P˚Ω and for all\nnPN˚, using the homothety hnpxq:“x{n,@xPRd, we define\n• Ln“hnpLq, the shrunk lattice;\n• Ln,Ω“LnXΩ, the nodes of the shrunk lattice that belong to Ω;\n•µnP pS2qLn,Ω, the shrunk vector field.\nWe notice that for all yPRd,\nµnpyq “ÿ\nxPLn,Ωµn,xδxpyq,\nwhereµn,x“µh´1\nnpxq. We assume that Ω has a sufficiently regular boundary\nin order that the number of nodes belonging to Ln,Ωis\n#Ln,Ω“Cndˆ\n1`Oˆ1\nn˙˙\n.\nwhereCisa constant which onlydepends on L,aandΩ(which areconstants\nof our problem).\n4(a) (b) (c)\nFigure 1: Scaling of a 2D spin lattice: each sub-plot represents the d omain\nΩ and (a) the square lattice L; (b) the shrunk square lattice L2; (c) the\nhomogeneized lattice in Ω as nÑ 8.\n2.3 Regularity assumptions\nIn order to pass to the limit as nÑ 8, we have to assume that the magnetic\nmoments are locally almost aligned. The definition of locality is given by an\ninteger multiple kPN˚of the shrunk mesh size a{n.\nWedefine a first regularity assumption that onlydepends onthe dist ance.\nFor allxPRdandrą0, we denote by Bpx,rqthe ball of center xand radius\nrinRd.\nHypothesis 1. For allnPN˚, there exists ζną0such that\n@xPΩ,@y,zPLn,ΩXBˆ\nx,ka\nn˙\n,1´ζnďµn,y¨µn,zď1,\nwhereζn“Op1{n2q.\nWeareindeedinterested inthelimitwhenwehaveamoreandmoredens e\nlattice of more and more aligned spins. We therefore perform a diago nal\nprocess and correlate nand the spin alignment.\nTo define the averaging process we will also need to assume the Ω has the\nuniform cone property .\nHypothesis 2. There exists an angle αand a radius r, such that for all yPΩ\nthere exists a cone Cyof angleαwith vertex at ysuch thatBpy,rq XCyĂΩ.\n52.4 Partitions adapted to the lattices\nLet us first work on the initial lattice. To this aim, we define a partition of\nunity pρxqxPkLadapted to the dilated lattice kL. Since our lattice is uniform\nand all the nodes are equivalent, all the ρxare equal up to a translation (see\nFigure 2), i.e. there exists ρ‹PC8\n0pRd;Rqsuch that\n@xPkL,@yPRd, ρxpyq “ρ‹py´xq.\na ka\nFigure 2: Partition pρxqxPkLin dimension 1.\nWithout lack of generality, we assume that ρ‹ą0 and supp ρ‹PBpkaq.\nBy definition of a partition\n@yPRd,ÿ\nxPkLρxpyq “ÿ\nxPkLρ‹py´xq “1.\nLetnkbe the number of nodes in Bp0,kaq, which is e.g. nkď p2k`1qdfor a\ncubic lattice (number of nodes contained in the cube around the sph ere). If\nwe now sum over all the xPL, we have the same translation property and\n@yPRd,ÿ\nxPLρ‹py´xq “nk.\nDefiningρ“n´1\nkρ‹, we have\n@yPRd,ÿ\nxPLρpy´xq “1,\nand therefore a continuous partition of unity on Rd.\nNow, we adapt this partition to the shrunk lattice Ln. Defining\n@nPN˚, ρnpxq “ρph´1\nnxq,\nwe have a continuous partition adapted to the shrunk lattice (see F igure 3)\n@yPRd,ÿ\nxPLnρnpy´xq “1 and supp ρnĂBˆ\n0,ka\nn˙\n.\n6a{n ka{n\nFigure 3: Partition pρxqxPLnin dimension d“1.\nSince∇ρ‹is uniformly bounded (i.e. Op1q), then∇ρnis uniformly Opnq.\nTo define an averaging process we will have to use a truncated part ition\nof unity, namely\nΦnpyq “ÿ\nxPLn,Ωρnpy´xq.\nWe clearly have 0 ďΦnpyq ď1 and as a finite sum of C8functions, Φ nP\nC8pRd;Rq. We also have a stronger result, namely Φ npyqis bounded from\nbelow uniformly in nandyPΩ: there exists bą0 andn0PN˚such that\n@yPΩ,@něn0, b ďΦnpyq ď1.\nThis result stems from the ”cone property” (Hypothesis 2). Besid es∇Φnis\nuniformlyOpnq.\n2.5 Averaging process\nFrom the sequence pµnqnPN˚of vector fields, we now define a new sequence\npmnqněn0of regular vector fields on Ω: for all něn0, we define\nmn“1\nΦnµn‹ρn,\nRemark 1. SinceρnandΦnPC8pRd;Rq, andΦnis bounded from below by\nbą0, it immediately follows that mnPC8pΩ;R3q. In particular, since Ωis\nbounded,mnPH1pΩ;R3q.\nLemma 1. The sequence pmnqněn0is bounded in L8pΩ;R3qandL2pΩ;R3q.\nProof.Recall\nµn“ÿ\nxPLn,Ωµn,xδx,\n7hence, for all yPΩ,\nmnpyq “1\nΦnpyqÿ\nxPLn,Ωµn,xρnpy´xq.\nWe clearly have\nˇˇˇˇˇˇ1\nΦnpyqÿ\nxPLn,Ωµn,xρnpy´xqˇˇˇˇˇˇď1\nΦnpyqÿ\nxPLn,Ω|µn,x|ρnpy´xq “Φnpyq\nΦnpyq“1,\n}mn}L8pΩ;R3qď1,\n}mn}L2pΩ;R3qďa\n|Ω|.\nProposition 1. Under Hypothesis 1, pmnqněn0is a bounded sequence in\nH1pΩ;R3q.\nProof.For allyPΩ, we write\n∇mnpyq “∇ˆ1\nΦnpyq˙ÿ\nxPLn,Ωµn,xρnpy´xq `1\nΦnpyq∇ÿ\nxPLn,Ωµn,xρnpy´xq,\n“ ´∇Φnpyq\nΦnpyqbmnpyq `1\nΦnpyqÿ\nxPLn,Ω∇ρnpy´xq bµn,x.\nLetxyPLn,ΩXsuppρnpy´ ¨q, we can write\nÿ\nxPLn,Ω∇ρnpy´xq bµn,x “ÿ\nxPLn,Ω∇ρnpy´xq b pµn,xy` pµn,x´µn,xyqq\n“ÿ\nxPLn,Ω∇ρnpy´xq bµn,xy\n`ÿ\nxPLn,Ω∇ρnpy´xq b pµn,x´µn,xyq\n“∇Φnpyq bµn,xy\n`ÿ\nxPLn,Ω∇ρnpy´xq b pµn,x´µn,xyq.\nHence\n∇mnpyq “ ´∇Φnpyq\nΦnpyqbmnpyq `∇Φnpyq\nΦnpyqbµn,xy\n`ˆ1\nΦnpyq˙ÿ\nxPLn,Ω∇ρnpy´xq b pµn,x´µn,xyq\n8“ ´∇Φnpyq\nΦnpyqb¨\n˝µn,xy`1\nΦnpyqÿ\nxPLn,Ωpµn,x´µn,xyqρnpy´xq˛\n‚\n`∇Φnpyq\nΦnpyqbµn,xy`ˆ1\nΦnpyq˙ÿ\nxPLn,Ω∇ρnpy´xq b pµn,x´µn,xyq\n“∇Φnpyq\nΦnpyq2b¨\n˝ÿ\nxPLn,Ωpµn,xy´µn,xqρnpy´xq˛\n‚\n`ˆ1\nΦnpyq˙ÿ\nxPLn,Ω∇ρnpy´xq b pµn,x´µn,xyq.\nIn both sums there is only an Op1qnumber of terms for which ρnor∇ρn\nis non-zero. In the first sum, by Hypothesis 1, |µn,x´µn,xy| “Opζ1{2\nnq “\nOp1{nq. Besides |∇Φnpyq| “Opnqand Φnis bounded from below. The\nfirst term is therefore Op1q. In the second sum |µn,x´µn,xy| “Op1{nqand\n∇ρnpy´xq “Opnq. Hence there exists C1ą0 such that, for all yPΩ and\nněn0,|∇mnpyq| ďC1.\nFinally, Ω being bounded, hence p}∇mn}L2pΩ;R3dqqněn0is a bounded se-\nquence.\n2.6 Asymptotics towards a mesoscopic model\nSinceH1pΩ;R3qis weakly compact, Proposition 1 implies that there exists\nmPH1pΩ;R3qsuch that\nmnámweakly inH1pΩ;R3q.\nFrom now on, we have to assume that Ω is compact (closed) and has a\npiecewise C1boundary, to ensure that this implies that this convergence is\nstrong inL2pΩ;R3q.\nProposition 2. Under Hypothesis 1, mhas a constant norm equal to 1\nalmost everywhere on Ω.\nRemark 2. We recover here a property of the magnetization field in Brown ’s\nmodel of micromagnetism [5], where the constant norm is assu med.\nProof.We have seen that for all něn0andyPΩ,|mnpyq| ď1. We want to\nshow that it is also bounded from below and pass to the limit. For all yPΩ,\n9since |µn,x|2“1,\n|mnpyq|2“ˇˇˇˇˇˇ1\nΦnpyqÿ\nxPLn,Ωµn,xρnpy´xqˇˇˇˇˇˇ2\n“1\nΦnpyq2ÿ\nx,x1PLn,Ωµn,x¨µn,x1ρnpy´xqρnpy´x1q.\nSince suppρnĂBp0,ka{nq, the sum runs indeed on Ln,ΩXBpy,ka {nq:\n|mnpyq|2“1\nΦnpyq2ÿ\nx,x1PLn,ΩXBpy,ka\nnqµn,x¨µn,x1ρnpy´xqρnpy´x1q.\nHypothesis 1 implies that\n@x,x1PLn,ΩXBˆ\ny,ka\nn˙\n,1´ζnďµn,x¨µn,x1ď1.\nand we also have that\n@yPΩ,ÿ\nx,x1PLn,Ωρnpy´xqρnpy´x1q “Φnpyq2,\ntherefore\n1\nΦnpyq2p1´ζnqΦnpyq2ď |mnpyq|2ď1\nΦnpyq2Φnpyq2,\n1´ζnď |mnpyq|2ď1.\nIn theL2pΩ;R3qlimit, we therefore have |m| “χΩalmost everywhere.\n3 Towards the exchange energy\n3.1 Heisenberg interaction\nThe interaction of the spins is described by the Heisenberg energy, which can\nbe written as follows:\nex,y“ ´1\n2Ax,ypµx¨µy´1q,\nwhereAx,yě0 only depends on the distance between xandy. We make the\nassumption that each node xonly interacts with its neighbors Nx.\n10LetN0ĂLbe the set of neighboring nodes in interaction with the node\np0,0,0qviathe Heisenberg energy. In the shrunk lattice, we will restrict the\ncomputation of this energy to elements xandyinLn,Ω. Since the lattice is\nhomogeneous, the neighbors of any given node xPLn,Ωcan be deduced from\nthedefinitionof N0:Nn,Ω,x“ ptxu`h´1\nnpN0qqXΩ. Wealsoassumethat(a) N0\n(andhence Nn,Ω,x)isafiniteset; (b) N0ĂBp0,kaq(i.e.Nn,Ω,xĂBpx,ka {nq).\nWe therefore define the node energy by\nex“ÿ\nyPNxex,y“ ´1\n2ÿ\nyPNxAx,ypµx¨µy´1q.\nInwhat follows we will restrict thestudy to 3Dcubic lattices. Dimensio ns\n1 and 2 are also possible to treat in the same way. The fact that the la ttice\nis cubic is used in the explicit computations, but our proof may be exte nded\nto other classes of regular lattices. We also consider as neighbors o nly the 6\nclosest ones on the cubic lattice, which are at the same distance. Sin ceAx,y\nonly depends on the distance, which is now equal for all the neighbor s, we\ncan set\nAx,y“#\nAą0,ifyPNx,\n0,else.\nFor allnPN˚, we define the exchange energy of the domain Ω associated\nto the spin distribution µnsumming up the node energies\nEn,expµnq “ÿ\nxPLn,Ωex“a\nnÿ\nxPLn,Ωÿ\nyPNn,Ω,xA|µn,y´µn,x|2.\n3.2 Spaces and convergence\nWe define the space sequence pWnqnPN˚byWn“ pR3qLn,Ω. In the sequel, we\nsetW“ś\nnPN˚Wn.\nHypothesis 3. Letµ“ pµnqnPN˚Pś\nnPN˚Wn. There exists cą0such that\n@nPN˚,@xPLn,Ω,@yPNn,Ω,x, |µn,x´µn,y|2ďc\nn2,\nexcept possibly for a subset ln,ΩĂLn,Ωsuch that #pln,Ωq “Opnq, and there\nexists a sequence pcnqně1PRN˚such thatcnÝ Ý Ý Ñ\nnÑ80and\n@nPN˚,@xPln,Ω,@yPNn,Ω,x,|µn,x´µn,y|2ďcn.\n11More precisely, given the constant c, we can state that xPln,Ωif there\nexistsyPNn,Ω,xsuch that |µn,x´µn,y|2ąc{n2. Hypothesis 3 is a little\nweaker than Hypothesis 1.\nWe denote by WhĂWthe set of µ“ pµnqnPN˚for which Hypothesis 3\nholds andWh\n1ĂWhthe set of µPWhfor which |µn,x| “1,@nPN˚,@xP\nLn,Ω.\nWe want to define the convergence of elements of Wtowards elements of\nthe limit space H1pΩ;R3q. To this aim we define a partition of the domain\nΩĂR3in tetrahedra, by groups of 5 tetrahedra (see Figure 4 and Sectio n\n3.3.1).\nFigure 4: Partition of a single 3D lattice cell. These five tetrahedra ar e ele-\nmentsTn, andTnforthecorner tetrahedra, and T˚\nnforthecenter tetrahedron\n(see Section 3.3.1).\nDefinition 1. We define the projection Pn:WnÝ ÑH1pΩ;R3qsuch that\nPnpµnqis equal toµnon the lattice nodes and linear on each element of the\npartition.\nDefinition 2. We say that µPWconverges to mPH1pΩ;R3qifPnpµnq\nconverges weakly in H1pΩ;R3qtomasnÑ 8. We denote this µnÝ Ý Ý Ñ\nnÑ8m.\nDefinition 3. We define the projection pn:CpΩ;R3q Ý ÑWn, such that for\nallmPCpΩ;R3q,pnpmq PWnand for all xPLn,Ω,ppnpmqqx“mpxq.\n3.3 Main result\nThe exchange energy En,exis a functional defined on Wn. The main result of\nthis paper is the following.\nTheorem 1. LetpµnqnPN˚PWh. In the sense of the topology defined by\nDefinition 2,\nEn,exΓÝ Ý Ý Ñ\nnÑ8E8,ex,\n12where\nE8,ex:H1pΩ;R3q Ý ÑR`\nmÞÝ Ñ2Aż\nΩ|∇mpxq|2dx.\nMoreover, this convergence is compatible with the unit norm constraint.\nRemark 3. TheΓ-convergence result is two-fold [4]:\nconstruction: for allmPH1pΩ;R3q, there exists µPWsuch thatµnÑm\nandlimsupnÑ8En,expµnq ďE8,expmq.\nlower semi-continuity: for allµPW, such that }µn} ďB,µnÑm8P\nH1pΩ;R3qandliminfnÑ8En,expµnq ěE8,expm8q.\nProof.The proof of Theorem 1 splits into many steps to which various lem-\nmas are devoted, the technical proofs of which are postponed.\nLemma 2. LetpµnqnPN˚PWh. Then there exists m8PH1pΩ;R3qsuch that\nµnÝ Ý Ý Ñ\nnÑ8m8.\nNow this function m8is only inH1pΩ;R3q, and not continuous, and\nwe need to use pointwise values of this function. Therefore, for all ε, we\napproximate m8byaC1pΩ;R3qwhichcoincideswith m8onasmallerdomain\nΩε. We denote by λthe Lebesgue measure on Rd.\nLemma 3. LetuPH1pΩ;R3q, for allεą0, there exists an open set ωu,ε\nand a function uεPC1pΩ;R3qsuch that for all xPΩzωu,ε, uεpxq “upxq,\nλpωu,εq ăε, and\nlim\nnÑ8#pLn,ΩXωu,εq\n#pLn,Ωqăε.\nCorollary 1. LetuPH1pΩ,R3q; for allεą0, there exists an open set\nΩu,εĂΩwith piecewise C1boundary such that λpΩzΩu,εq ďεand\nu|Ωu,εPC1pΩu,ε;R3q.\nLemma 4. LetuPC1pΩ;R3q. Then ppnpuqqnPN˚PWh,pnpuq Ý Ý Ý Ñ\nnÑ8uand\nlimnÑ8En,exppnpuqq “E8,expuq. Moreover if uhas a constant unit norm,\nthen ppnpuqqnPN˚PWh\n1.\nFor allεą0, we can therefore associate to m8an open set Ω εon which\nit isC1and define the approximate energies:\nEε\nn,expµnq “a\nnÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xA|µn,y´µn,x|2for allnPN˚,\nEε\n8,expm8q “2Aż\nΩε|∇m8pxq|2dx.\n13Comparing Eε\nn,expµnqandEε\nn,exppnpm8qq, we first show the following.\nLemma 5. For allεą0,liminfnÑ8Eε\nn,expµnq ěEε\n8,expm8q.\nEstimating the remainder of the sums and integrals on ωε, we can then\nprove the following.\nLemma 6. liminfnÑ8En,expµnq ěE8,expm8q.\n3.3.1 Proof of Lemma 2: Limit of a lattice of spins\nLemma 2 is proved using an explicit computation of the projection Pnpµnq.\nTo this aim, we define, for all nPN˚,\n• Tn, the set of tetrahedra which form the partition of Ln,Ω;\n•Tn, the set of corner tetrahedra (4 for each mesh of the lattice, se e\nFigure 4);\n•T˚\nn, the set of center tetrahedra (1 for each mesh of the lattice, se e\nFigure 4);\n•En, the set of edges of the mesh of Ln,Ω(given by couples of the indices\nof the lattice nodes);\n•Cn, the set of edges of elements of T˚\nn;\n•Sn, the set of outer surfaces of Tn(triplets pi,j,k qwhere pi,jq PCn,\npi,kqand pj,kq PEn).\nsPSnePEn\nePCn\nFigure 5: Different sets of tetraedra and edges.\nRemark 4. Generically Ωn”Ť\nτPTnτŁΩ. We suppose that the geometry of\nΩis such that λpΩzΩnq “Op1{nq, i.e. there are order n2tetrahedra covering\nthe difference set. We set Pnpµnqto zero on ΩzΩn.\n14Step 1. Estimate ofş\nΩ|Pnpµnqpxq|2dx.\nLetxPΩn, then there exist τPTnandxPτ. SincePnpµnqis linear\nonτ, thenPnpµnqpxqis a pondered mean of the µn,xτ\ni,i“1,...,4, where\npxτ\n1,xτ\n2,xτ\n3,xτ\n4q P pLn,Ωq4are the vertices of τ. Therefore |Pnpµnqpxq| ď1.\nHence\nż\nΩ|Pnpµnqpxq|2dx“ż\nΩn|Pnpµnqpxq|2dx“ÿ\nτPTnż\nτ|Pnpµnqpxq|2dx\nďÿ\nτPTnλpτq “λpΩnq.\nStep 2. Explicit computation ofş\nΩ|∇Pnpµnqpxq|2dx.\nWe computeş\nΩ|∇Pnpµnqpxq|2dxexplicitly using the tetrahedron decom-\nposition of the lattice. On each tetrahedron Pnpµnqis linear, and therefore\nits gradient is constant.\nIfτPTn, we can construct an orthogonal system using the lattice nodes\npxτ\n1,xτ\n2,xτ\n3,xτ\n4q P pLn,Ωq4, for example pµn,xτ\ni´µn,xτ\n1qi“2,...,4. Each direction\nyields one component of the gradient. Since the length of the edges area{n,\nthe components of thegradient arethe pµn,xτ\ni´µn,xτ\n1qn{a. Besides the volume\nofτis1\n6pa{nq3, therefore\nż\nτ|∇Pnpµnqpxq|2dx“a\n6n4ÿ\ni“2|µn,xτ\ni´µn,xτ\n1|2.\nFor a center tetrahedron τ˚ PT˚\nn, it is a bit more tricky since the edges are\nnot orthogonal. The volume is of course the complement of the volum es of\nthe corner tetrahedra, namely1\n3pa{nq3. The computation yields\n|∇Pnpµnqpxq|2“n2\n4a2ÿ\n1ďi,jď4|µn,xτ˚\ni´µn,xτ˚\nj|2\nand hence\nż\nτ˚|∇Pnpµnqpxq|2dx“a\n12nÿ\n1ďi,jď4|µn,xτ˚\ni´µn,xτ˚\nj|2.\n15Gathering all the contributions\nż\nΩ|∇Pnpµnqpxq|2dx\n“ÿ\nτPTnż\nτ|∇Pnpµnqpxq|2dx\n“ÿ\nτPTnż\nτ|∇Pnpµnqpxq|2dx`ÿ\nτ˚PT˚nż\nτ˚|∇Pnpµnqpxq|2dx\n“4ÿ\npi,jqPEna\n6n|µn,xi´µn,xj|2`2ÿ\npi,jqPCna\n12n|µn,xi´µn,xj|2´Sn,Ω.\nThe coefficient 4 in front of the sum on Enoccurs because each element of\nEnis an element of four τPTn. Similarly each element of Cnbelongs to two\nτ˚PT˚\nn. The positive error Sn,Ωis due to an over-estimation because some\nof the edges ePEnare on the outer surface of Ω nand have been counted\ntoo many times. Following Remark 4, the contribution of Sn,Ωwill always\nbeOp1{nqless than that of the other terms, and therefore will tend to zero\nasnÑ 8.\nWe rewrite the first sum\n4ÿ\npi,jqPEna\n6n|µn,xi´µn,xj|2“ÿ\nxPLn,Ωÿ\nyPNn,Ω,xa\n3n|µn,x´µn,y|2\n(here each edge is counted twice through the couples px,yqand py,xq). For\nxPln,Ω, we can only estimate pa{3nq|µn,x´µn,y|2ď4a{3n, but there are\nonlyOpnqsuch terms. For xPLn,Ωzln,Ω,pa{3nq|µn,x´µn,y|2ďac{3n3and\nthere areOpn3qsuch terms. Therefore\n4ÿ\npi,jqPEna\n6n|µn,xi´µn,xj|2“Op1q.\nFor the second sum\nÿ\npi,jqPCna\n6n|µn,xi´µn,xj|2“1\n2ÿ\npi,j,k qPSna\n6n|µn,xi´µn,xk`µn,xk´µn,xj|2\n“ÿ\npi,j,k qPSna\n12n´\n|µn,xi´µn,xk|2` |µn,xj´µn,xk|2\n`2pµn,xi´µn,xkq ¨ pµn,xk´µn,xjq¯\n“ÿ\npi,jqPEna\n3n|µn,xi´µn,xj|2`ÿ\npi,j,k qPSna\n6npµn,xi´µn,xkq ¨ pµn,xk´µn,xjq\n“ÿ\nxPLn,Ωÿ\nyPNn,Ω,xa\n6n|µn,x´µn,y|2`ÿ\npi,j,k qPSna\n6npµn,xi´µn,xkq ¨ pµn,xk´µn,xjq.\n16As for the previous sum, we can decompose these sums into two Op1qcon-\ntributions. There are at most Opnqterms contributing to Sn,Ωand stem-\nming from an xPln,Ω, therefore Sn,Ω“Op1q. Therefore }Pnpµnq}H1pΩ;R3q\nis uniformly bounded and Pnpµnqis weakly convergent in H1pΩ;R3q. This,\ntogether with Definition 2, leads to Lemma 2.\nA by-product of this proof is the fact that we can write\n}∇Pnpµnq}2\nL2pΩ;LpR3;R3qq“ż\nΩ|∇Pnpµnqpxq|2dx“1\n2AEn,expµnq `αnpµnq\nwhere\nαnpµnq “a\n6nÿ\npi,j,k qPSnpµn,xi´µn,xkq ¨ pµn,xk´µn,xjq ´Sn,Ω.\n3.3.2 Proof of Lemma 3: C1approximation of a H1function\nFollowing Ziemer theorem (see [15], Theorem 3.11.6), we know that for\nany function uPH1pΩ;R3qand for all εą0, there exists a function\nuεPC1pΩ;R3qsuch thatλpωu,εq ďε, where\nωu,ε:“ txPΩ such that upxq ‰uεpxqu.\nWe want to extend this result and be able to localize the irregularities o f\nuwith respect to a shrinking lattice.\nLetuPH1pΩ;R3q, andXube the set of points where uis notC1. Since\nthe gradient of uPH1pΩ;R3qhas to be defined almost everywhere, there\ncannot be an open ball in Xuand therefore˝\nXu“ H.\nWe now fix εą0. The Lebesgue and H3Hausdorff measures coincide in\nR3and therefore we both have λpXuq ďεandH3pXuq ďε.\nHence there exists a sequence of open balls pBiqiPNsuch that\nXuĂď\niPNBiand8ÿ\ni“1diam pBiq3ăε.\nSinceXuis closed and bounded, it is compact and we can extract from\nthis open cover a finite subcover.\nHence there exists NPNsuch that\nXuĂNď\ni“1BiandNÿ\ni“1diam pBiq3ăε.\n17ObviouslyŤN\ni“0Biis bounded with a piecewise C1boundary, hence\nlim\nnÑ8#pLn,ΩŞ ŤN\ni“1Biq\n#pLn,Ωq“λ˜Nď\ni“1Bič\nΩ¸\năε.\nBesidesu|ŤN\ni“1BiPH1pŤN\ni“1Bi;R3q, and we therefore can choose uεsuch\nthatωu,εĂŤN\ni“1Bi. Finally\nlim\nnÑ8#pLn,ΩŞωu,εq\n#pLn,Ωqďε.\nIn our proof, we set Ω ε“8ŔΩzωm8,εand begin to work on the restricted\nshrunk lattice Ln,Ωε“Ln,ΩŞΩε. We also denote Dn,Ωεthe subset of ele-\nmentsxPLn,Ωεsuch that # pNn,Ω,xŞΩεq ‰6, that is the set of nodes which\nare too close to BΩεto have their 6 nearest neighbors in Ω ε.\nSince BΩεis piecewise C1for allεą0, we know that # Dn,Ωε“Opn2q.\n3.3.3 Proof of Lemma 4: Construction\nLetxPLn,ΩandyPNn,Ω,x. SinceuPC1pΩ;R3q,∇uis bounded by some\nconstantCon Ω and\n|pnpuqx´pnpuqy|2ďC2|x´y|2ďC2´a\nn¯2\n“C2a2\nn2.\nThereforepnpuq PWh. Clearly if |u| “1 on Ω, for all xPLn,Ω,|pnpuqx| “1\nandpnpuq PWh\n1.\nLemma 2 implies that there exists u8PH1pΩ;R3qsuch thatpnpuq Ý Ý Ý Ñ\nnÑ8\nu8. SinceuPC1pΩ;R3q,Pnppnpuqqconverges towards u. This convergence\nis pointwise and even uniform on Ω. Hence u8“u.\nLast\nEn,exppnpuqq “a\nnÿ\nxPLn,Ωÿ\nyPNn,Ω,xA|pnpuqy´pnpuqx|2\n“a\nnÿ\nxPLn,Ωÿ\nyPNn,Ω,xA|upyq ´upxq|2.\nSinceuPC1pΩ;R3q, for allxPLn,Ωand allyPNn,Ω,x,\n|upyq ´upxq|2\n|y´x|2“ˇˇˇˇ∇upxq ¨y´x\n|y´x|ˇˇˇˇ2\n`en,x,y,\n18and therefore for all xPLn,ΩzDn,Ω\n´n\na¯2ÿ\nyPNn,Ω,x|upyq ´upxq|2“2|∇upxq|2`en,x,\nand the errors en,xare uniformly op1qasnÑ 8. Hence\nEn,exppnpuqq “2Aÿ\nxPLn,Ω´a\nn¯3\np|∇upxq|2`en,xq `Oˆ1\nn˙\n,\nwhere theOp1{nqstems from xPDn,Ω. Since lim nÑ8ř\nxPLn,Ωpa{nq3en,x“0,\nand the border of Ω is piecewise C1\nlim\nnÑ8ÿ\nxPLn,Ω´a\nn¯3\n|∇upxq|2“ż\nΩ|∇upxq|2dx\nand\nlim\nnÑ8En,exppnpuqq “E8,expuq.\n3.3.4 Proof of Lemma 5: Lower semi-continuity\nWith Definition 2 for the convergence, Pnpµnq ám8inH1pΩ;R3q, and\nliminf\nnÑ8}Pnpµnq}H1pΩ;R3qě }m8}H1pΩ;R3q\n(see [4], Proposition 2.3).\nSince we have assumed that Ω is compact and has a piecewise C1bound-\nary, we have already seen that the convergence is strong in L2pΩ;R3qand\ntherefore\nliminf\nnÑ8}∇Pnpµnq}2\nL2pΩ;LpR3;Rdqqě }∇m8}2\nL2pΩ;LpR3;Rdqq.\nLet us first fix εand work in Ω ε. Thanks to Proposition 4, we know that\nm8is continuous and\n}∇Pnppnpm8qq}L2pΩε;LpR3;RdqqÝ Ý Ý Ñ\nnÑ8}∇m8}L2pΩε;LpR3;Rdqq.\nThis implies that\nliminf\nnÑ8´\n}∇Pnpµnq}2\nL2pΩε;LpR3;Rdqq´ }∇Pnppnpm8qq}2\nL2pΩε;LpR3;Rdqq¯\ně0.\n19According to Lemma 2, we can write\n}∇Pnpµnq}2\nL2pΩε;LpR3;Rdqq“1\n2ApEε\nn,expµnq `αε\nnpµnqq.\nWe therefore know that\nliminf\nnÑ8ˆ1\n2AEε\nn,expµnq ´1\n2AEε\nn,exppnpm8qq `αε\nnpµnq ´αε\nnppnpm8qq˙\ně0.\nWe split this into 5 parts, namely\n1\n2AEε\nn,expµnq ´1\n2AEε\nn,exppnpm8qq “an`bn,\nαε\nnpµnq ´αε\nnppnpm8qq “cn`dn`en`fn,\nwhich can be expressed using ψn,x:“µn,x´m8pxq, defined for all nPN˚\nandxPLn,Ωε:\nan“a\n2nÿ\nxPLn,Ωεÿ\nyPNn,Ωε,x|ψn,x´ψn,y|2,\nbn“a\n2nÿ\nxPLn,Ωεÿ\nyPNn,Ωε,x2pm8pxq ´m8pyqq ¨ pψn,x´ψn,yq,\ncn“a\n6nÿ\npi,j,k qPSnpm8pxiq ´m8pxkqq ¨ pψn,xk´ψn,xjq,\ndn“a\n6nÿ\npi,j,k qPSnpψn,xi´ψn,xkq ¨ pm8pxkq ´m8pxjqq,\nen“a\n6nÿ\npi,j,k qPSnpψn,xi´ψn,xkq ¨ pψn,xk´ψn,xjq,\nfn“Sn,Ωεppnpm8qq ´Sn,Ωεpµnq.\nWe show below that enďan, andbn,cn,dnandfntend to zero. This implies\nthat\nliminf\nnÑ8pαnpµnq ´αnppnpm8qqq ď1\n2Aliminf\nnÑ8pEε\nn,expµnq ´Eε\nn,exppnpm8qqq\nand therefore\nliminf\nnÑ8pEε\nn,expµnq ´Eε\nn,exppnpm8qqq ě0.\nLemma 4 implies that lim nÑ8Eε\nn,exppnpm8qq “Eε\n8,expm8q, hence\nliminf\nnÑ8Eε\nn,expµnq ěEε\n8,expm8q.\nwhich ends the proof.\n20Proof ofenďan.For allxi,xj,xkPLn,Ωε,\npψn,xi´ψn,xkq ¨ pψn,xk´ψn,xjq ď1\n2p|ψn,xi´ψn,xk|2` |ψn,xk´ψn,xj|2q.\nSince each couple pi,jq PEnis an element of 4 triples in Sn, we have\nÿ\npi,j,k qPSnpψn,xi´ψn,xkq ¨ pψn,xk´ψn,xjq ď2ÿ\npi,jqPEn|ψn,xi´ψn,xj|2\nďÿ\nxPLn,Ωεÿ\nyPNn,Ωε,x|ψn,x´ψn,y|2,\nwhich is a much stronger result than enďan.\nProof ofcnanddnÑ0.Letpψn,x´ψn,yq ¨ pm8pyq ´m8pzqqbe one term\nof the sum in cnand setv“z´y. Then\npψn,x´ψn,yq ¨ pm8pyq ´m8pzqq “ pψn,x´ψn,yq ¨ pm8pyq ´m8py`vqq\nand in the same sum there is also a term pψn,x´ψn,yq ¨ pm8pyq ´m8py´vqq,\nexcept foryPDn,Ωε.\nNow sinceµnPWh, there exists a constant Cψě0 such that\n|ψn,x´ψn,y| ďCψa\nn,\nexcept foryPln,Ωε, but since there are Opnqsuch nodes, their contribution\nincntends to 0. We also have\nn\nappm8pyq ´m8py´vqq ` pm8pyq ´m8py`vqqq\nÝ Ý Ý Ñ\nnÑ8p∇m8pyq ´∇m8pyqq ¨v\n|v|“0,\nand therefore\na\n6nÿ\npi,j,k qPSnpψn,xi´ψn,xkq ¨ pm8pxkq ´m8pxjqq “Oˆ1\nn˙\nOpn3qOˆ1\nn˙\noˆ1\nn˙\n“op1q.\nRemark 5. WhenyPDn,Ωε, we can only say that |ψn,x´ψn,y| ďCψa{n\n(except onln,Ωε) and |m8pyq ´m8pzq| ďCma{n, and since #Dn,Ωε“Opn2q,\nthe contribution of these nodes in cnis a\nOpn2qOˆ1\nn˙\nOˆ1\nn˙\nOˆ1\nn˙\n`OpnqOˆ1\nn˙\nOˆ1\nn˙\n“Oˆ1\nn˙\n.\nThe sumdnis treated in the same way.\n21Proof offnÑ0.The quantity Sn,Ωεpµnqis a sum of Opn2qterms reading\nlikepa{6nq|µn,xi´µn,xj|2forpi,jq PEn, orpa{12nq|µn,xi´µn,xj|2forpi,jq P\nCn. By Hypothesis 3, only Opnqamong these terms can be only op1{nqand\nthe others are Op1{n3q. HenceSn,Ωεpµnq “op1q.\nOn the other hand, the fact that u8PC1pΩε,R3qensures that all the terms\ninSn,Ωεppnpm8qqareOp1{n3q, and therefore Sn,Ωεppnpm8qq “Op1{nq.\nProof ofbnÑ0.We use the fact that C8pΩε;R3qis dense in H1pΩε;R3q\nfor the } ¨ }H1pΩε;R3qnorm. Let ηą0, there exists mηPC8pΩε;R3qsuch\nthat }m8´mη}H1pΩε;R3qďη. Thenbncan be split into two contributions\nbn“bη\nn`b1η\nnwhere\nbη\nn“a\nnÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xpmηpxq ´mηpyqq ¨ pψn,x´ψn,yq,\nb1η\nn“a\nnÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xppm8´mηqpxq ´ pm8´mηqpyqq ¨ pψn,x´ψn,yq.\nFor the first term, we notice that\nbη\nn“a\nnÿ\nxPLn,Ωεÿ\nyPNn,Ω,xpmηpxq ´mηpyqq ¨ψn,x\n´a\nnÿ\nyPLn,Ωεÿ\nxPNn,Ω,ypmηpxq ´mηpyqq ¨ψn,y.\nAs in the previous proof, we write yasx`vand\nbη\nn“a\nnÿ\nxPLn,Ωεÿ\nvPVn,0ppmηpxq ´mηpx`vqq ´ pmηpx´vq ´mηpxqqq ¨ψn,x.\nWe estimate\npmηpxq ´mηpx`vqq ´ pmηpx´vq ´mηpxqq\n“ż0\nt“1∇mηpx`tvq ¨vdt´ż´1\nt“0∇mηpx`tvq ¨vdt\n“ ´∇mηpxq ¨v\n|v|`∇mηpxq ¨v\n|v|`Op|v|2q ďca2\nn2,\nwhere the constant conly depends on the second derivative of mη, which is\nbounded on Ω ε. Thus\n|bη\nn| ďca3\nn3ÿ\nxPLn,Ωεÿ\nyPNn,Ω,x|ψn,x|.\n22Thanks to the compact injection of H1pΩ;R3qinL1pΩ;R3q, we deduce\nthat this term vanishes. We obviously have\n|b1η\nn| ďa2Cψ\nn2ÿ\nxPLn,Ωεÿ\nyPNn,Ωε,x|pm8´mηqpxq ´ pm8´mηqpyq|.\nFor any function uPC1pΩ;R3q, andy“x`v, ifrx,ys ĂΩ, then\nupyq ´upxq “ż0\nt“1∇upx`tvq ¨vdt\n“∇upxq ¨v`ż0\nt“1p∇upx`tvq ´∇upxqq ¨vdt.\nApplying this to m8andmη,\npm8´mηqpyq ´ pm8´mηqpxq “ p∇m8pxq ´∇mηpxqq ¨v\n`ż0\nt“1p∇m8px`tvq ´∇m8pxqq ¨vdt\n´ż0\nt“1p∇mηpx`tvq ´∇mηpxqq ¨vdt.\n‹By definition of mη,}m8´mη}H1pΩ;R3qďη, which implies that }∇m8pxq´\n∇mηpxq}L2pΩε;R3qďη. Hence\na2Cψ\nn2ˇˇˇˇˇˇÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xp∇m8pxq ´∇mηpxqq.py´xqˇˇˇˇˇˇ\nď2Cψÿ\nxPLn,Ωε´a\nn¯3\n|∇m8pxq ´∇mηpxq|\nď2Cψ¨\n˝ÿ\nxPLn,Ωε´a\nn¯3˛\n‚1{2¨\n˝ÿ\nxPLn,Ωε´a\nn¯3\n|∇m8pxq ´∇mηpxq|2˛\n‚1{2\n.\nSince\nlim\nnÑ8ÿ\nxPLn,Ωε´a\nn¯3\n|∇m8pxq ´∇mηpxq|2“ż\nΩε|∇m8pxq ´∇mηpxq|2dx,\nlim\nnÑ8a2Cψ\nn2ÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xp∇m8pxq ´∇mηpxqq.py´xq|\nďC}∇m8pxq ´∇mηpxq}L2pΩε;R3qďCη.\n23‹Let us treat the second contribution involving the C8functionmη. Since\nż0\nt“1p∇mηpx`tvq ´∇mηpxqq ¨vdt“ż0\nt“1żt\nt1“0∇2mηpx`t1vq ¨vbvdt1dt,\nand∇2mηis uniformly bounded on Ω ε, we can estimate\na2Cψ\nn2ÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xż0\nt“1p∇mηpx`tpy´xqq ´∇mηpxqq ¨ py´xqdt“Op1\nnq.\n‹In the last contribution, function m8is onlyC1:\na2Cψ\nn2ÿ\nxPLn,Ωεÿ\nyPNn,Ωε,xˇˇˇˇż0\nt“1p∇m8px`tpy´xqq ´∇m8pxqq ¨ py´xqdtˇˇˇˇ\nďa2Cψ\nn2ÿ\nxPLn,Ωεÿ\nvPNn,0ż1\nt“0|p∇m8px`tvq ´∇m8pxqq ¨v|dt\nďa3Cψ\nn3ż1\nt“0ÿ\nxPLn,Ωεÿ\nvPNn,0|p∇m8px`tvq ´∇m8pxqq ¨v\n|v||dt.\nSincem8PC1pΩε;R3q, the elements of the sum converge uniformly towards\n0 asnÑ 8, and therefore the integral is opn3q.\nHence lim nÑ8|b1η\nn| ďηand therefore lim nÑ8|bn| ďη, for allηą0. This\nleads to lim nÑ8|bn| “0, and the lemma is proved.\n3.3.5 Proof of Lemma 6\nThe terms that occur in En,expµnqbut not in Eε\nn,expµnqare those involing\ncouples px,yqwhere one at least of the nodes belong to Ω zΩε. There are\nεOpn3qsuch nodes. Hence, by Hypothesis 3, the contribution of ln,Ωin\nEn,expµnqtendstozero, andthereexists cą0suchthatforall tµnunPN˚PWh,\nlim\nnÑ8|Eε\nn,expµnq ´En,expµnq| ďcε.\nWe have already estimated Pnpµqon Ω zΩn,ε, where Ω n,εis a polyhedral\nsubset of Ω which, like ωε, has anOpεqLebesgue measure. We already know\nthatPnpµnq|ΩzΩn,εPH1pΩzΩn,ε;R3qandPnpunq|ωεám8|ΩzΩn,εin the sense\nofH1pΩzΩn,ε;R3q, therefore\nż\nΩzΩn,ε|∇m8pxq|2dxďliminf\nnÑ8ż\nΩzΩn,ε|∇Pnpµnqpxq|2dxďCε.\n24liminf\nnÑ8En,expµnq “liminf\nnÑ8pEε\nn,expµnq ` pEn,expµnq ´Eε\nn,expµnqqq\něliminf\nnÑ8pEε\nn,expµnqq `liminf\nnÑ8pEn,expµnq ´Eε\nn,expµnqq\něEε\n8,expm8q `liminf\nnÑ8pEn,expµnq ´Eε\nn,expµnqq\něE8,expm8q ` pEε\n8,expm8q ´E8,expm8qq\n`liminf\nnÑ8pEn,expµnq ´Eε\nn,expµnqq\něE8,expm8q ´ pc`c1qε.\nSince this holds for all εą0, we finally deduce that\nliminf\nnÑ8En,expµnq ěE8,expm8q.\n4 Other energy contributions\n4.1 Magnetostatics: demagnetizing energy\nWe can define a mapping hd:L2pR3;R3q ÞÑL2pR3;R3qby: for all uP\nL2pR3;R3q,hdpuqis solution in the sense of distributions to\n\"\nrothdpuq “0,\ndivhdpuq “ ´divu.\nWhenuis the magnetization, hdpuqis the demagnetizing field. Its energy is\nEdpuq “µ0\n2}hdpuq}L2pR3;R3q.\nForuPL2pΩ;R3q, we denote by ˜ utheL2pR3;R3qfunction which equals u\ninside Ω, and 0 outside Ω. Hence for a spin distribution µnP pS2qLn,Ω, the\ndemagnetizing energy is defined by\nEn,dpµnq “µ0\n2›››hd´ČPnpµnq¯›››\nL2pR3;R3q.\n4.2 External energy\nThe external energy, or Zeeman contribution, models the influenc e of an\nexternal magnetic field on the magnetization. Given such a field hZin\nC0pR3;R3q, for alluPL2pR3;R3qsupported in Ω, we set\nEZpuq “ ´ż\nΩhZ¨udx,\n25this energy in maximized when uis almost everywhere in Ω in the direction\nofhZ.\nAt the micro-scale, for a spin distribution µnP pS2qLn,Ω, we set\nEn,Zpµnq “ ´´a\nn¯3ÿ\nxPLn,ΩhZpxq ¨µn,x.\n4.3 Total energy\nWe define the total energy summing up the exchange, demagnetizin g, and\nexternal energies both in the lattice context:\n@µnP pS2qLn,Ω,Enpµnq “En,expµnq `En,dpµnq `En,Zpµnq,\nand the limit continuous one:\n@uPH1pΩ;R3q,E8puq “E8,expuq `E8,dpuq `EZpuq.\nTheorem 2. LetpµnqnPN˚PWh. In the sense of the topology defined by\nDefinition 2\nEnΓÝ Ý Ý Ñ\nnÑ8E8.\nMoreover, this convergence is compatible with the unit norm constraint.\nProof.We have already shown that\nEn,exΓÝ Ý Ý Ñ\nnÑ8E8,ex.\nMoreoverhd:L2pR3;R3q ÞÑL2pR3;R3qis linear and continuous. We\ntherefore choose pµnqnPN˚PWhsuch thatµnÝ Ý Ý Ñ\nnÑ8µPH1pΩ;R3q. This\nmeans that the sequence pPnpµnqqis weakly convergent in H1pΩ;R3qtoµ.\nHence\nPnpµnqL2pΩ;R3qÝ Ý Ý Ý Ý Ñ\nnÑ8µandČPnpµnqL2pR3;R3qÝ Ý Ý Ý Ý Ý Ñ\nnÑ8˜µ.\nIn particular En,dpµnq Ý Ý Ý Ñ\nnÑ8E8,dpµq.\nBesides ifµPH1pΩ;R3q, we know that pnpµq Ý Ý Ý Ñ\nnÑ8µinH1pΩ;R3q, which\nends the proof of\nEn,dΓÝ Ý Ý Ñ\nnÑ8E8,d.\nLast, we notice that\nEn,Zpµnq “ ´´a\nn¯3ÿ\nxPLn,ΩhZpxq ¨µn,x“ ´´a\nn¯3ÿ\nxPLn,ΩhZpxq ¨Pnpµnqpxq,\n26which can be re-written as the approximation of ´ş\nΩhZpxq ¨Pnpµnqpxqdx\nthanks to piecewise approximations. Then, using the regularity of hZand\nthe convergence of µninL2pΩ;R3qtowardµ, we prove that\nEn,Zpµnq Ý Ý Ý Ñ\nnÑ8EZpµq.\n5 Conclusion\nIn this paper, we prove a Γ-convergence result from a discrete de scription\nof ferromagnetic materials at the microscopic scale to the continuo us one.\nThis result has been shown thanks to a rigidity hypothesis on the latt ice of\nmagnetic moments. This modeling hypothesis is based on the Heisenbe rg\ninteraction phenomenon and could be justified by a time multi-scale st udy.\nThe new hypothesis would take into account the speed of the Heisen berg\nrelaxation compared to the Larmor precession process.\nThe results in this paper are the seed in order to address the micro–\nmesoscopic limit for dynamic processes to be able to better underst and the\ndissipation phenomena involved in the mesoscopic Landau–Lifchitz sy stem.\nReferences\n[1] Amikam Aharoni. Introduction to the theory of ferromagnetism . Oxford\nScience Publications, 1996.\n[2] Roberto Alicandro and Marco Cicalese Variational analysis of the\nasymptotics of the XYmodel.Archive for Rational Mechanics and\nAnalysis, 192(3):501–536, 2009.\n[3] Fabrice Boust, Nicolas Vukadinovic, and St´ ephane Labb´ e. 3D d ynamic\nmicromagneticsimulations ofsusceptibility spectra insoft ferromag netic\nparticles. ESAIM Proceedings , 22:127–131, February 2008.\n[4] Andrea Braides. Gamma-Convergence for Beginners , Oxford University\nPress, 2002.\n[5] William Fuller Brown, Jr. Micromagnetics , volume 18 of Interscience\nTracts on Physics and Astronomy . John Wiley and Sons, 1963.\n[6] William Fuller Brown, Jr. Magnetoelastic Interactions , volume 9 of\nTracts in Natural Philosophy . Springer-Verlag, 1966.\n27[7] William Fuller Brown, Jr and Anton E. LaBonte. Structure and ener gy\nof one-dimensional domain walls in ferromagnetic thin films. Journal of\nApplied Physics , 36(4):1380–1386, 1965.\n[8] Antonio DeSimone and Paolo Podio-Guidugli. On the continuum theor y\nof deformable ferromagnetic solids. Archive for Rational Mechanics and\nAnalysis, 136(3):201–233, 1996 .\n[9] R. Fisher, Thomas Shrefl, Helmut Kronm¨ uller, and Josef Fidler. G rain-\nsize dependence of remanence and coercive field of isotropic nanoc rys-\ntalline composite permanent magnets. Journal of Magnetism and Mag-\nnetic Materials , 153:35–49, 1996.\n[10] Laurence Halpern and St´ ephane Labb´ e. La th´ eorie du mi-\ncromagn´ etisme. Mod´ elisation et simulation du comportement des\nmat´ eriaux magn´ etiques. Matapli, 66:77–92, 2001.\n[11] Richard D. James and Stefan M¨ uller. Internal variables and fin e-scale\noscillations in micromagnetics. Continuum Mechanics and Thermody-\nnamics, 6(4):291–336, 1994.\n[12] St´ ephane Labb´ e and Pierre-Yves Bertin. Microwave polariza bility of\nferrite particles with non-uniform magnetization. Journal of Magnetism\nand Magnetic Materials , 206:93–105, 1999.\n[13] Anja Schlmerkemper. Mathematical derivation of the continuu m limit\nof the magnetic force between two parts of a rigid crystalline mater ial.\nArchive for Rational Mechanics and Analysis , 176(2):227–269, 2005.\n[14] Nicolas Vukadinovic, Jamal Ben Youssef, and Henri Le Gall. Influ ence\nof magnetic parameters on microwave absorption of domain mode fe r-\nromagnetic resonance. Journal of Magnetism and Magnetic Materials ,\n150:213–222, 1995.\n[15] William P. Ziemer. Weakly differentiable functions: Sobolev Spaces and\nFunctions of Bounded Variation , volume 120of Graduate Texts in Math-\nematics. Springer, 1989.\n28"    },    {        "title": "1008.0299v1.A_Search_for_Defect_Related_Ferromagnetism_in_SrTiO__3_.pdf",        "content": "arXiv:1008.0299v1  [cond-mat.mtrl-sci]  2 Aug 2010A Search for Defect Related Ferromagnetism in SrTiO 3\nD.A. Crandles, B. DesRoches, F. S. Razavi\nDepartment of Physics, Brock University,\nSt. Catharines, Ontario, Canada, L2S 3A1∗\n(Dated: September 18, 2018)\nAbstract\nRoom temperature ferromagnetic hysteresis is observed in c ommercial SrTiO 3substrates pur-\nchased from a variety of suppliers. It is shown that the ferro magnetic signal comes from the\nunpolished surfaces. Surface impurity phases cannot be det ected using either x-ray diffraction or\nenergy dispersivex-ray spectraon the unpolishedsurfaces . However, a possiblecorrelation between\nsurface disorder (xray diffraction peak linewidth) and ferro magnetism is observed. Ar ion bom-\nbardment (10keV-90 keV) can be used to produce surface layer disorder but is not found to induce\nferromagnetism. Annealing of the substrates at temperatur es ranging from 600 to 1100◦C is found\nto alter the hysteresis curves differently depending on wheth er the annealing is performed in air\nor in vacuum. Identical annealing behaviour is observed if t he substrates are artificially spiked\nwith iron. This suggests that the ferromagnetic hysteresis of as purchased SrTiO 3could be due to\nFe contamination of the unpolished surfaces. In addition, i t is shown that no ferromagnetism is\nobserved in samples that contain 1019-1021cm−3of oxygen vacancies if all the faces are polished.\n1I. INTRODUCTION\nClaims of room temperature ferromagnetism (FM) have been made f or several semicon-\nducting oxides that do not contain any magnetic ions. The phenomen on has been observed\nin a variety of binary oxides prepared in nanoparticle1and thin-film form2–8. There has\nbeen considerable interest in these otherwise non-magnetic oxides as a part of the effort to\ndevelop suitable materials for spintronic devices.\nThis work isconcerned withthe originofroomtemperature magnetic hysteres isobserved\nin single crystal SrTiO 3(001) substrates (STO) purchased from a variety of suppliers. T he\nhysteresis, which cannot be explained by non-interacting paramag netic impurities, is inter-\nesting in light of recent work which highlights the considerable potent ial of STO surfaces for\nelectronic and spintronic devices. Consider, for example, the two d imensional electron gas\nformed at the interfaces of LaAlO 3/STO heterostructures9as well as other phenomena such\nas blue luminescence10and high conductivity11produced by Ar+irradiation.\nThe observation of FM hysteresis in commercial STO raises several questions, which\ninclude whether the hysteresis is a bulk or surface effect, whether or not it can be associated\nwith an impurity phase, or whether or not it can be associated with a p articular kind\nof point or extended defect that can be controlled through mater ials processing. It has\nbeen established that the surfaces of STO can be quite complex. Fo r example, slicing a\nVerneuil grown STO crystal into substrates suitable for thin film gr owth produces a large\nconcentration of dislocations and voids12near the cut surface while annealing in oxidizing\nor reducing conditions can promote segregation into Ti-rich or Sr-r ich regions in the surface\nskin layer13. In the experiments described below, particular attention is paid to the effect\nof annealing in reducing and oxidizing atmospheres since it has been fo und in other oxides\nwhere FM has been observed, for example TiO 26and HfO 27, that the magnitude of the\nremanent moment can be altered by vacuum annealing.\nIt is interesting to note that in STO oxygen vacancies donate free e lectrons to the hy-\nbridized Ti(3d)-O(2p) conduction band and can produce supercon ductivity with maximum\nTc≈0.3 K14. The possibility of the coexistence of ferromagnetism and superco nductivity\nis a compelling reason to study the magnetic properties of reduced S TO. Finally, STO is\na common substrate for metal oxide thin film growth. Since certain t hin film materials\nwith interesting magnetic and charge ordering properties such as L aVO315can only be pro-\n2duced in reducing atmospheres16, it is important to be aware of how reduction at elevated\ntemperatures affects the magnetic properties of the substrate .\nII. RESULTS AND DISCUSSION\nA. Bulk or Surface Effect?\nSTO (100) single crystal substrates were purchased from a varie ty of suppliers: Alfa\nAesar, Crystec , MTI corporation and Semiconductor Wafer Inc. Substrates were purchased\nin both one side polished (1sp) and two side polished (2sp) form. Extr eme care was taken\n- for example always using teflon tweezers that had never been in co ntact with stainless\nsteel - to avoid contamination of the samples with magnetic elements17. Magnetic moment\nmeasurements were performed using a Quantum Design SQUID magn etometer. Some small\ndegree of magnetic hysteresis at 300 K was measured in every single one of the over 50\nsubstrates tested.\nTo establish whether the hysteresis was a bulk or a surface effect, two sorts of tests were\ndone. HCl and HNO 3acids were both found to reduce the size of the remanent moment\nwhile polishing the unpolished surfaces in stages down to mirror-like sm oothness was found\nto completely remove all traces of FM hysteresis. Fig. 1(b) compar es measurements of\nmoment versus field on STO substrates with one side polished (1sp) - the as purchased state\n- with substrates with both 5 ×5 mm surfaces polished (2sp) as well as substrates with all\nsix sides of the substrate polished (asp). Each round of polishing re moved a surface layer\napproximately 10 to 15 µmthick. The hysteresis is clearly associated with the unpolished\nsurfaces of the substrate. The saturation magnetization can be extracted by subtracting out\nthe diamagnetic contribution. Fig. 1(a) summarizes the whole set of measurements of the\nsaturation moment versus unpolished surface area.\nEnergy Dispersive X-ray Spectroscopy (EDX) and X-ray Diffractio n Spectra (XRD) were\nmeasured on both the unpolished and polished surfaces of the STO s ubstrates. These\nmeasurementsrevealednosignificantdifferencebetweenthepolish edandunpolishedsurfaces\nexceptthattheXRDlineswereconsiderablywiderforthestressed anddisorderedunpolished\nside of the substrates, as expected18. Data for the unpolished surfaces are shown in Fig.\n2 which reveal no impurity phases or magnetic elements to the level o f sensitivity of these\n3techniques. The peak near 3.7 keV indicates the presence of a small amount of calcium in\nthese substrates.\nB. Defect Ferromagnetism\nThere are at least two possibilities for the FM hysteresis of the unpo lished surfaces.\nFirstly, it could be due to particles of FM material left from either the diamond saw or\nwire saw cuts used to produce substrates from the crystal boule , or from handling by the\nsuppliers oftheunpolishededgesusing stainless steel tweezers. A tlowenoughconcentration,\nthese particles would not be detected by the EDX and XRD measurem ents illustrated in\nFig. 2. Secondly, the hysteresis could be associated with extended or point defects such as\noxygen vacancies in the vicinity of the cut surface. Two different ap proaches to this question\nwere taken: (i) A systematic study was undertaken of the effect o f annealing atmosphere\nand temperature on both the room temperature magnetic hyster esis as well as the width\nof the XRD lines. The reasoning was that annealing should heal some o f the disorder\nsimultaneously reducing the width of the XRD lines and remanent mome nt. In addition,\nannealing in vacuum allows one to introduce a controlled density of oxy gen vacancies via the\nannealing temperature. (ii) Artificial disorder was introduced by Ar+or Ar2+bombardment\nof polished STO surfaces to see if a magnetic moment could be induced which would clearly\nbe associated with disorder. The goal was to find a way of introducin g controlled amounts\nof disorder associated with surface ferromagnetism.\nAnnealing of the STO substrates was performed at different tempe ratures in either a vac-\nuumofapproximately5 ×10−6Torr(reducing environment) orinair(oxidizing environment)\nby placing the substrate in an Al 2O3crucible inside a quartz evacuation chamber in the fur-\nnace. A pattern emerged which is illustrated in Fig. 3. Annealing in air, f orT >600◦C,\ndestroyed or significantly decreased the room temperature rema nent magnetization while\nannealing in vacuum for T <900◦C did not destroy the remanent magnetization.\nThe vacuum annealing experiments produced samples with a wide rang e of oxygen va-\ncancy density. Increasing the annealing temperature from 800◦C to 1100◦C systematically\ndecreased the dc resistivity (measured using the van der Pauw tec hnique) and increased the\ndark colouration of the STO substrates. It is possible to produce d arker, even less resistive\nsamplesbyeitherannealingthesubstratealongwithpowderedTiina sealed, evacuatedtube\n4or by intentionally growing grossly oxygen deficient crystals21. Sample magnetization data\nforoneofthereduced substrates aswell asthatofaSrTiO 2.72crystal areshown inFig. 4 (a).\nNote that data of Fig. 4 (a) are for samples that have no unpolished surfaces. The M vs. H\nrelationsofalltheoxygendeficient samples werelinear, whileonlytwo such curves areshown\nin Fig. 4 (a) for simplicity. This demonstrates clearly that oxgyen vac ancies do not produce\nFM in STO. The vacancy densities were not directly measured but the re is a well known\ncorrelation between the room temperature dc resisitivity and free carrier concentration19\nwhere, in the simplest picture, each oxygen vacancy donates two f ree electrons to the con-\nduction band. In Fig. 4(b), the susceptibility of pure STO, which is du e to the diamagnetic\nresponse of the closed shell ions plus a Van Vleck term22, isχ=−9.2±0.5×10−8emu/g-G\nand becomes less negative with the positive Pauli contribution of an in creasing number of\nfree carriers. This was the explanation given by Frederikse and Can dela22and the data illus-\ntrated in Fig. 4(b) agrees to within 10% with their earlier measuremen ts22. Almost all of the\nsamples included in Fig. 4(b) have carrier concentrations compatible with low temperature\nsuperconductivity14. Because the samples which have carrier density ( ≈1019−1021cm−3)\ncompatible with superconductivity exhibit linear magnetic susceptibilit ies, it appears that\nSTO is not a system where FM and superconductivity coexist.\nHaving concluded that oxgyen vacancies do not produce the FM hys teresis, one might\nhypothesize that other kinds of disorder such as dislocations and in terstitials produced by\nthe diamond saw cuts may be associated with the FM hysteresis. X-r ay diffraction reveals\nthat annealing does indeed relieve the strain on the unpolished surfa ces as shown in Fig.\n5(a) which illustrates how the linewidth of the (003) diffraction peak f rom the unpolished\nside decreases with annealing temperature. The line on the polished s ide is also illustrated,\nfor comparison. The decrease in linewidth with annealing temperatur e is seen in all diffrac-\ntion peaks for both vacuum and air annealing. However, Fig. 5(b) wh ich is a plot of of\nthe remanent moment versus linewidth of the x-ray diffraction peak s, suggests that the cor-\nrelation between remanent moment and surface strain and disorde r is not strong not only\nbecause of the scatter in the data but because air annealing and va cuum annealing have\nsuch different effects on magnetization but produce similar decreas es in the linewidth of the\nx-ray diffraction peaks\nAs mentioned earlier, Ar ion bombardment was utilized to produce sur face disorder in\nan attempt to mimic the disorder observed in the x-ray spectra of t he unpolished surfaces.\n5This was done by taking samples with polished surfaces exhibiting linear magnetization\nversus applied field relations and mounting them with non-ferromagn etic adhesive onto 5-\ncm diameter Si wafers and placing them in an Ar ion beam to receive a do se of 1016ions.\nTwo beam energies were attempted: either 10 keV (range 92 ˚A straggle 42 ˚A estimated using\nSRIM software (Stopping Rangeof Ions in Matter20)) or 90 keV (range 570 ˚Astraggle 250 ˚A).\nA dose of 101690 keV ions produces a huge amount of damage, milling away part of th e\nsurface and producing an amorphous layer than can be observed u sing x-ray diffraction. Fig.\n6(b) contrasts the (002) diffraction peaks from both irradiated a nd non-irradiated surfaces\nof the same substrate. Note the creation of a broad peak at lower diffraction angle which is\nassociated with the amorphous layer of thickness roughly 60 nm. Fo rcomparison, diffraction\npeaks in amorphous Si can have widths well over one degree23. Note that the main (002)\npeakof theAr bombardedside ofthe crystal isnarrower thantha t ofthenon-irradiatedside.\nOn the non-irradiated but polished surface, x-rays are scattere d fromboth a surface strained\nlayer plus a less strained region deeper in the crystal. On the irradiat ed side, the narrow\npeak represents diffraction from deeper in the crystal than the s urface amorphous layer.\nMore importantly, a lack of magnetic hysteresis was measured afte r ion bombardment as\nshown in Fig. 6 (a) for both incident Ar ion kinetic energies. It has bee n conclusively shown\nthat FM is not associated with oxgyen vacancies, nor with an amorph ous layer produced by\nAr ion bombardment while Fig. 5(b) offers only a weak correlation betw een surface strain\nand disorder and the observed saturation moment.\nIt is difficult to prove the second hypothesis, in which the FM hystere sis of the unpolished\nsurfaces is due to contamination by FM particles, because of the low particle density. Recall\nthat neither XRD nor EDX spectra illustrated in Fig. 2 showed any evid ence of impurity\nelements or phases. The approach taken here was to see whether artificially contaminated\nsamples produced the same annealing behaviour as the raw substra tes. For the final set of\nexperiments performed samples were prepared with no moment by s oaking a 1sp substrate\nin nitric acid for 10 minutes in an ultrasonic bath as shown in Fig. 7(a). S ubsequently\nsuch samples were artificially contaminated with iron by rubbing the un polished surface on\nstainless steel. Comparison of Figs. 7(a),(b) and (c) demonstrate s that artificial contami-\nnation creates a moment of roughly the same magnitude as observe d in the as purchased\n1sp substrates. Finally the iron contaminated samples were subjec t to the same annealing\ntreatments as untreated 1sp substrates. Comparison of Fig. 3 a nd Fig. 7(b) and (c) illus-\n6trates that the iron contaminated samples demonstrate the same annealing behaviour as the\nuntreated 1sp substrates.\nA natural explanation of Fig. 3 and Fig. 7 comes from a comparison of the magnetic\nproperties of iron and the iron oxides which are listed in Table I. Iron p articles on the\nrough surfaces of the substrates coming from either diamond saw cuts or handling with\nstainless steel tweezers, are the source of the FM hysteresis ob served on the unpolished\nsurfaces. Below 900◦C, vacuum annealing would either maintain these particles in the Fe\nstate, or at 1100◦C there may oxidation to antiferromagnetic wustite FeO whose pres ence\nis indicative of a highly reducing environment25. On the other hand air annealing would\noxidize the iron particles to antiferromagnetic α−Fe2O3at all temperatures. α−Fe2O3is\nmore thermodynamically stable than γ−Fe2O331. Definite confirmation of the presence of\niron on unpolished surfaces awaits detection by more sensitive surf ace elemental probes like\nPIXE (particle induced x-ray emission). It is of interest to compare the present work with\nthe only other systematic study of the ferromagnetic properties of commercial substrates by\nSalzeret al.who considered sapphire (Al 2O3) substrates purchased from Crystec32. The\nauthors studied the magnetic hysteresis curves of both 1sp and 2 sp samples of a variety of\norientations and sizes. Significantly they also tried to measure surf ace magnetic impurity\nconcentration using PIXE (proton irradiated x-ray emission) meas urements. The PIXE\nmeasurements revealed that Fewas themain impurity and that thea verage Feconcentration\non unpolished c-faces was 230 ng/cm2while on polished c-faces the concentration was 10\nng/cm2. It is very interesting to note that a linear fit of Fig. 1(a) yields Fe de nsity of\n190±40 ng/cm2if one assumes that the moment of the substrates is produced by p ure Fe\nwith a room temperature magnetization of 220 emu/g.\nIII. DISCUSSION\nThere has been an on-going discussion surrounding ferromagnetis m in magnetic ion free\noxides. On one side are those who argue that ferromagnetism is pro duced by defects in\nthe thin flims2–8, while on the other side are those who argue that it is a spurious effec t\nproduced by iron contamination17,33,34. Since some of the strongest claims linking oxygen\nvacancies and ferromagnetism have been made for TiO 2−δthin films, it is useful to compare\nthe TiO 2−δdata with the present measurements on single crystal SrTiO 3.\n7A summary of the properties of TiO 2−δthin films grown by a number of techniques\non a variety of substrates appears in Table II. The most notable fe ature of the data is\nthat several groups have found that the saturation magnetizat ion depends on the oxygen\npartial pressure during film deposition and have also observed that air annealing destroys\nthe ferromagnetism. All these authors have concluded that this is evidence that oxygen\nvacanciesareproducingtheferromagnetichysteresis ofthefilms andonehascitedtheoretical\nsupport35for oxygen-vacancy-induced-ferromagnetism in TiO 2. However, no measurements\nof oxygen vacancy were actually made in these works.\nOn the other hand, in the present work, reasonable estimates of o xygen vacancy density\ncan be inferred from the free carrier density measurements made and Fig. 4 shows that sam-\nple magnetization is a linear function of applied field even in samples with a large density\nof bulk oxygen vacancies. It seems very clear that oxygen vacanc ies do not produce ferro-\nmagnetism in STO. The effect of vacuum annealing on STO and TiO 2is quite similar where\noxygen vacancies produce an increase in the electrical conductivit y. Hence, the differences in\nferromagnetism associated with oxygen vacancies in these two mat erials must be associated\nwith differences in the Ti-O network where STO contains a network of corner shared TiO 6\noctahedra, while there is a combination of edge and corner sharing in rutile.\nThis is not the first work to have shown that commercial substrate s without films them-\nselves exhibit ferromagnetic hysteresis. Saturation moments for various substrates as mea-\nsured by a number of groups are listed in Table III. It is interesting t o note that aside from\nthe anomalously large moment measured for LaAlO 3by Hong et al.4, there appears to be a\nrough correlation between substrate moment and the hardness o f the substrate which are all\nharder than stainless steel. It is difficult to compare the moments of the TiO 2−δfilms with\nthe substrates because usually it is the magnetization of the sample s, rather than the raw\nmoments that are listed in the papers. However, it is curious that bo th Yoon et al.2,3and\nSudakar et al.6find that the magnetization to be a decreasing function of film thickn ess.\nThis data could be partially explained by iron contamination on the unpo lished substrate\nsurfaces. It is also puzzling that Sudakar et al.measure magnetization of films produced by\nreactive RF sputtering to be ten times less than films produced by pu lsed laser deposition.\nThey suggest that it is because the polycrystalline sputtered films o n fused quartz may have\nless defects than the epitaxial PLD films on LaAlO 3, but this does not seem convincing.\nA simpler explanation could be that different substrates have differe nt amounts of surface\n8contamination. It is also interesting that Yoon et al., found the Curie temperature of TiO 2−δ\nthin film samples to be 880 K which is close to the transition temperatur e ofγ−Fe2O3as\nshown in Table I. The present annealing experiments illustrated in Fig. 3, which can be ex-\nplained by the presence of iron on the unpolished surfaces of subst rates, raise the possibility\nthat the supposed dependence of saturation moment on oxygen p artial pressure in the TiO 2\nfilms might be the response of the substrates themselves.\nIV. CONCLUSIONS\nFerromagnetic hysteresis with a saturation moment of the order o f 5×10−5emu/cm2is\nregularly observed in STO (001) substrates with 1 or 2 sides polished as purchased from a\nvariety of suppliers. Oxygen vacancies do not appear to be associa ted with ferromagnetism\nin STO nor can a magnetic moment be created by Ar ion bombardment. Reduced STO is\nnot a system where FM and superconductivity coexist. In the as pu rchased samples, the\nhysteresis is associated with the unpolished surfaces and is likely due to Fe contamination\nfrom handling with stainless steel tweezers or the diamond or wire sa w cuts rather than\noxygen vacancies or other lattice defects. It is possible that all sin gle crystal substrates used\nfor thin film growth contain ferromagnetic iron or iron oxide on the un polished surfaces.\nPolishing and simple acid washes can remove the weak ferromagnetic s ignal. Extreme care\nmust be taken in handling samples to avoid contamination and to interp ret magnetic data\nmeasured on thin films grown on STO substrates with unpolished surf aces. One possible\napplication of the results of this paper is for thin film growers using ST O substrates. By\nannealingathightemperaturebeforefilmgrowth, alltracesofthe ferromagneticsignalofthe\nsubstrate can be removed giving film growers confidence that any s ubsequent ferromagnetic\nsignals are due to the films themselves. It may also be of interest to f urther study iron\nimplantation as a way to produce ferromagnetic surface layers in a c ontrolled manner.\n9∗Electronic address: dcrandles@brocku.ca\n1A. Sundaresan, R. Bhargavi, N. Rangarajan, U. Siddesh and C. N. R. Rao, Phys. Rev. B.\n74, 161306(R) (2006).\n2Soack Dae Yoon, Yajie Chen, AriaYang, Tevor L.Goodrich, XuZ uo, DarioA Arena, Katherine\nZiemer, Carmine Vittoria, Vincent G. Harris, J. Phys. Cond. Matt.18, L355 (2006).\n3Soack Dae Yoon, Yajie Chen, Aria Yang, Tevor L. Goodrich, Xu Z uo, Katherine Ziemer,\nCarmine Vittoria, Vincent G. Harris, J. Mag. Mag. Mat. 309, 171 (2007).\n4N. H. Hong, J. Sakai, N. Poirot, V. Briz´ e, Phys. Rev. B, 73, 132404 (2006).\n5A.K. Rumaiz, B. Ali, A. Ceylan, M. Boggs, T. Beebe, S. I. Shah, Sol. St. Comm. 144, 334\n(2007).\n6C. Sudakar, P. Kharel, R. Suryanarayanan, J. S. Thakur, V. M. Naik, R. Naik, G. Lawes, J.\nMag. Mag. Mat. 320, L31 (2008).\n7J.M.D. Coey, M. Venkatesan, P. Stameov, C.B. Fitzgerald and L.S. 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RRL, 3148 (2009).\n11TABLE I: Properties of iron and iron oxides.aFeO is typically formed in reducing atmospheres.\nbThe magnetization of α-Fe2O3does not saturate. The value listed is for T=295 K and H=1T ,\nwell above any fields used in this study.\nMaterial Magnetic Order Tcor TN(K)Msat(300K) emu/g References\nFeFerromagnetic 1043 220 24\nFeOAntiferromagnetic 200a25\nFe3O4Ferrimagnetic 858 30-90 24,26–28\nγ-Fe2O3Ferrimagnetic 878 50-75 28\nα-Fe2O3Antiferromagnetic 955 0.5b29,30\nTABLE II: Summary of data collected on TiO 2−δthin films. PLD signifies pulsed laser deposition.\nMOD signifies metal-organic decomposition. RRFS signifies r eactive radio frequency sputtering.\naAnnealing in O2 for 8h at 650 C destroyed the FM.bThe substrates were held at 27◦C for RF\nsputtering in an oxygen partial pressure of 1.5 mTorr using A r as the sputtering gas. These films\nwere then annealed at 900C in vacuum to obtain rutile structu re.\nMethod Substrate Deposition T (C) P(O2)Thickness (nm) Msat(300K)References\n(C) (mTorr) (nm) emu/cm3\nPLD LaAlO 3 700 0.3 250 33 2,3\n25 250 12\nPLD SiO2 800 0.02 na 0.32 5\n0.2 na 0.03\nPLD LaAlO 3 600-700 0.001 200 20 4\n0.001 200 0a\nRRFSfused quartz 27b1.5 25 40 6\n50 10\n100 5\nMOD Al2O3 500 air 500-900 0.8 6\n2sp - 25 mm2\n12TABLE III: Magnetic moments measured in 5mm ×5mm×0.5 mm (1sp) commercial substrates.\nYSZ is yttria stabilized zirconia . It is of interest that the moh’s hardness of stainless steel is listed\nbetween 5 and 6.\nMaterial 300 K Saturation Moment (emu) Moh’s Hardness References\nMgO 1×10−65.8 Ref. 4\nLaAlO 3 1×10−55-6 Ref. 4\nLaAlO 3 1×10−65-6 Ref. 33\nSrTiO3 1.6±0.6×10−56-6.5 this work\nYSZ 2.5×10−59 Ref. 4\nAl2O3 3±1×10−59 Ref. 34\nAl2O3 3±2.5×10−59 Ref. 32\n13-400x10-6-2000200400Magnetization (emu/g)\n-5000 0 5000\nApplied Field(G)25x10-6\n20\n15\n10\n5\n0saturation moment 0.4 0.3 0.2 0.1 0.0\nunpolished surface area (cm2 )(emu)(a)\n(b)\n 1sp\n 2sp\n asp\nFIG. 1: (a) The average saturation moment as a function of unp olished surface area. The uncer-\ntainty bars represent the sample to sample standard deviati on. For example, for the point where\nthe unpolished surface area is 0.35 cm2, which is for 1sp substrates 0.5 mm thick, the data are\nfor 19 different samples from two different suppliers. For the po int where the unpolished surface\narea is approximately 0.2 cm2which is for 2sp substrates 1 mm thick, the data are for 15 differ ent\nsamples from four different suppliers. (b) Magnetization ver sus applied field at 300 K for a 5 ×5\nmm STO substrate polished on one side (open squares marked 1s p), hand polished on both 5 ×5\nmm faces ( open triangles marked 2sp), polished on all six fac es of the parallelpiped ( closed circles\nmarked asp).\n141000\n800\n600\n400\n200\n0Counts\n8 6 4 2 0\nEnergy (keV)\n1024100241000 Normalized Intensity\n80 60 40 20\n2θ (degrees)001002 003(a)\n(b)Ti KβTi Kα\nSr LαCa\nFIG. 2: Upper Panel (a): Energy Dispersive Xray Spectrum of t he unpolished side of STO (001)\nsubstrate. Lower Panel (b): Xray Diffraction pattern of the un polishedside of STO (001) substrate\n15/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s120/s49/s48/s45/s54/s32\n/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s120/s49/s48/s45/s54/s32\n/s45/s51/s48/s48/s120/s49/s48/s45/s54/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s101/s109/s117/s47/s103/s41/s54/s48/s48/s111/s67\n/s86/s97/s99/s117/s117/s109\n/s57/s48/s48/s111/s67\n/s86/s97/s99/s117/s117/s109/s54/s48/s48/s111/s67\n/s65/s105/s114\n/s57/s48/s48/s111/s67\n/s65/s105/s114\n/s49/s49/s48/s48/s111/s67\n/s86/s97/s99/s117/s117/s109/s49/s48/s53/s48/s111/s67\n/s65/s105/s114\n/s45/s52 /s45/s52\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s107/s71/s41/s48/s52 /s48/s52\nFIG. 3: Magnetization versus applied field at 295 K for 1sp STO substrates before (open circles)\nand after annealing (closed circles) at various temperatur es and in different atmospheres. Vacuum\nimplies a pressure of approximately 5 ×10−6Torr.\n16-300x10-9-200-1000100200300χ  (emu/g-G)\n1014  1017  1020  \nCarrier Concentration (cm-3 )-1.0x10-3-0.50.00.51.0M (emu/g)\n-5000 0 5000\nH (G)\nTiSrTiO2.72\n1100oC 800oCSTO(a)\n(b)SrTiO2.72\n1100oC\nFIG. 4: (a) Magnetization (M) versus applied field (H) at 295 K for a sample of SrTiO 2.72and\nfor a STO substrate that has been annealed at 1100C for 10 h in a vacuum of 5 ×10−6Torr.\nThe M vs. H curves of substrates with no unpolished surfaces a nnealed between 800 and 1100\n◦C in vacuum are linear, but only one representative curve (11 00◦C) is shown for clarity. (b)\nRoom temperature magnetic susceptibility of SrTiO 3−xas a function of free carrier concentration\n- measured via the room temperature dc resistivity as discus sed in Ref. 19 - which in a first\napproximation is proportional to oxygen vacancy density. P oints are included for a series of STO\nsubstrates annealed for 10 h in vacuum at 800,900,1000, and 1 100C. The point marked ‘Ti’ is for\na substrate annealed in a vacuum sealed quartz tube containi ng powdered Ti. The point marked\nSrTiO2.72is for a grossly non-stoichiometric single crystal. The sol id horizontal line indicates the\nmeasured room temperature susceptibility of pure SrTiO 3.\n171.0\n0.8\n0.6\n0.4\n0.2\n0.0normalized counts\n72.5 72.4 72.3 72.2\n2θ (degrees)\n1.2x10-6\n0.8\n0.4\n0.0Moment (emu/mm2)\n0.12 0.08 0.04 0.00\nFWHM (degrees)STO 003unpolished\n 600 oC\n 900 oC\n 1100 oC\n polished\n(a)\n(b) as grown\n vacuum\n air \nFIG. 5: Upper Panel (a): (003) line of STO XRD spectrum for sub strates annealed in vacuum of\n5×10−6Torr at various temperatures or as purchased. Except for the long dash curve which is for\na polished side, all the spectra are for unpolished surfaces . The data are similar for air-annealed\nsamples. Lower Panel (b): Remanent moment at 295K versus the full width at half maximum of\nthe (003) line of the XRD spectrum taken on unpolished surfac es unnannealed and annealed (air\nor vacuum) at a variety of temperatures. The uncertainty in m oment is for a single measurement\nof the hysteresis curve on a single sample.\n18101102103104105106counts\n48 46 44 42\n2θ  (degrees)-400x10-6-2000200400M (emu/g)\n-5000 0 5000\nH (G)\n as grown\n Ar++(002)\n peakAr Irradiated STO\n(a)\n(b)\nFIG. 6: (a) Linear magnetization versus applied field at 295 K for an STO substrate that had\nbeen bombarded by a dose of 101690 keV Ar2+ions. (b) Portion of 2 θxray scan comparing the\ndiffraction peak before and after irradiation by 90 keV Ar2+ions\n19/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s120/s49/s48/s45/s54/s32\n/s45/s52/s48/s120/s49/s48/s45/s54/s45/s50/s48/s48/s50/s48/s52/s48/s77/s111/s109/s101/s110/s116/s32/s40/s101/s109/s117/s41\n/s45/s53/s48/s48/s48 /s48 /s53/s48/s48/s48\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s71/s41/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s120/s49/s48/s45/s54/s32\n/s86/s97/s99/s117/s117/s109\n/s32/s54/s48/s48/s111/s67\n/s65/s105/s114\n/s49/s49/s48/s48/s111/s67/s40/s97/s41\n/s40/s98/s41\n/s40/s99/s41/s65/s99/s105/s100\nFIG. 7: All panels show moment versus applied field at 295 K for 5×5×0.5 mm STO substrates.\nThe unfilled symbols show data before the acid or annealing tr eatment while the filled symbols\nshow the data after the treatment. Panel (a): Effect of Nitric A cid Treatment on as purchased\nsubstrates. Panel (b): Effect of annealing at 600◦C in a vacuum of 5 ×10−6Torr on STO substrate\nartificially spiked with Fe. Panel (c): Effect of high temperat ure air annealing (1100◦C) on STO\nartificially spiked with Fe.\n20"    },    {        "title": "1910.14338v1.Terahertz_emission_from_anomalous_Hall_effect_in_a_single_layer_ferromagnet.pdf",        "content": "1  Terahertz  emission  from  anomalous Hall effect in a single -layer  ferromagnet  \n \nQi Zhang ,1,2 Ziyan Luo ,1 Hong Li , 2,3 Yumeng Yang ,1 Xinhai Zhang ,2,* and Yihong \nWu1,†    \n \n1 Department of Electrical and Computer Engineering, National University of \nSingapore, 4 Engineering Drive 3, Singapore 117583,  Singapore  \n2 Department of Electrical & Electronic Engineering, Southern University of \nScience and Technology, Xueyuan Rd 1088, Shenzhen 518055, China  \n3 Institute of Applied Physics and Material Engineering, University of Macau, \nAvenida da Universidade, Macau 999078, C hina  \n \nWe report on terahertz  emission from a single  layer  ferromagnet which  involves the generation of \nbackflow nonthermal charge cur rent from the ferromagnet/dielectric interface  by femtosecond laser \nexcitation and subsequent conversion of the charge current  to a transverse transient charge current \nvia the anomalous Hall effect , thereby generating the THz radiation. The THz emission can be either \nenhanced or suppressed, or even the polarity can be reversed, by in troducing a magnetization gradient \nin the thickness direction of the ferromagnet. Unlike spintronic THz emitters reported previously , it \ndoes not require a dditional  non-magnetic layer or Rashba interface .  \n \n \n \n \n \n \n \nCorresponding authors: Email:  *zhangxh@sustc.edu.cn , †elewuyh@nus.edu.sg  \n  2  I. INTRODUCTION  \nRecently spin-to-charge conversion in femtosecond (fs)  laser excited magnetic heterostructures  \nhas attracted attention as a promising mechanism for producing terahertz (THz) wave  with \nmagnetically controllable polarization state  [1]. The key to the THz emission  is the generation of \nspin-polarized super -diffusive charge current from a ferromagnetic layer by fs laser excitation and \nsubsequent conversion of the spin polarized current to a transverse charge current by either inverse \nspin Hall effect (ISHE)  [1-10] or inverse Rashba-Edelstein effect  (IREE)  [11-13]. The former \ninvolves a ferromagnet (FM) / non -magnet (NM) heterostructure wherein when a n fs laser is \nirradiated on the FM/NM heterostructure , non -equilibrium electrons are excited to the states above \nFermi level, generating a super -diffusive spin polarized current normal to the interface  [14,15] . When  \nthe spin polarized current enters the NM layer with large spin -orbit coupling (S OC), it is converted \nto a charge current in the transverse direction by ISHE, thereby generating the THz wave  [1,2] . On \nthe other hand, the THz emitter based on IREE typically consists of a FM with an adjacent  Rashba \ninterface, e.g., FM/ Ag/Bi  [16]; in this case, the super -diffusi ve spin polarized current launched by \nthe FM layer is converted to transverse charge current at the Ag/Bi interface  via the IREE  [11-13], \nwhich in the same way as the ISHE, generat es the THz emission. In this letter , we examine the \npossibility of using anomalous Hall effect (AHE)  [17] to generat e THz emission  in a single layer FM \nwith a large SOC  and demonstrate efficient THz emission in samples  with a magnetization  gradient .  \nAs shown schematically in Fig.  1, when a thin metallic film is irradiated by an fs laser, the \nelectrons are excited to states above the Fermi level (Fig.  1(a)). Immediately after the excitation, \nequilibration takes place through two dominant mechanisms, i.e., electron -electron and electron -\nphonon interactions [18]. Due to the much smaller heat capacity, the electron subsystem quickly \nreaches a high temperature  (\neT) Fermi -Dirac distribution within 0.1 ps (Fig.  1(b)), whereas the lattice \nstays close to the ambient temperature (\npT ) [19]. The electrons subsequently cool and thermalize 3  with its own lattice within a few picoseconds (Fig.  1(c)). In addition to phonons, the magnon also \nplays a role i n the equilibration process of hot electrons in ferromagnets. The magnon temperature \n(\nmT) is typically higher than that of phonons before all the three subsystems reach the thermal \nequilibrium [20,21] . \nBefore the electron subsystem reaches the equilibrium (i.e., \nt < 0.1 ps), the nonthermal \nelectrons move at a fast speed (~106 m/s) in a super -diffusive manner  (Fig.  1(d))  [14,15] . For electrons \nat a distance of at least a mean -free-path (\ne) away from the top  (e.g., with MgO  capping layer ) and \nbottom ( e.g., with quartz substrate ) interfaces , they will collide with other electrons  to reach the \nthermal equilibrium  within the electron subsystem . However, for electrons with a distance shorter \nthan \ne  from the interface, they will be reflected back to form a backflow current  due to reflect ion at \nthe FM/dielectric interface  [15]. The amplitude of the backflow current depends strongly on the \nmaterial properties and roughness of the interface. For metallic films deposited on a smoo th substrate, \ntypically the bottom interface will be smoother than the top interface. In this  case, the backflow \ncurrents from the bottom  (\nl1j) and (\nl2j ) interfaces  will not be completely canceled out, leading to a \nnet backflow current  (\nl1 l2j - j ). We argue that the backflow current will be converted to a transverse \ntransient current w hen the FM has a large AHE. Recently, we have demonstrated that (Fe xMn 1-x)yPt1-\ny thin films with optimum compositions exhibit a sizable AHE, and the backflow of spin \naccumulation induced by the AHE gives the anomalous Hall magnetoresistance (AHMR) [22]. Here, \nwe show that (Fe xMn 1-x)yPt1-y is also a promising material  for AHE -based THz emitter. We \nsubstantiate our arguments by examine both the thickness and pumping fluence -dependence of the \nemission efficiency in (Fe xMn 1-x)yPt1-y emitters. We sh ow that t he use of Pt composition gradient can \neither enhance or suppress t he THz emission, depending on the gradient direction relative to normal \ndirection of the two interfaces.  4  II. EXPERIMENTAL RESULTS AND DISCUSSION  \nA. AHE origin of THz emission  \nThe sample preparation and THz measurement methods are given in the Supplemental Material  \n[23,24] . Fig. 2(a) illustrates the THz generation from fs laser pumped single layer  FM with either  the \nquartz (up) or MgO  (down)  side pumping . We first measured the THz emission from a FeMnPt (3) \nthin film capped by MgO (4) (hereafter the number inside the brackets indicate film thickness in nm) , \nand the results  are shown in Fig. 2(b), where the upper and lower pair of waveforms are obtained \nfrom quartz side and MgO  side pumping  at a fluence of 555 μJ/cm2, respectively  (hereafter, FeMnPt \nrefers  to (Fe 0.8Mn 0.2)0.67Pt0.33). The solid and dashed lines correspond to the applied field in + y and –\ny direction, respectively. It is apparent that the  THz polarity is reversed when either the applied \nmagnetic field or pumping direction is reversed; the former indicates tha t the THz emission is of \nmagnetic origin  [25-27] whereas the  latter implies that there is a reversal of the sign of the driving \nforce that is responsible for the generation of the THz wave. Similar sign dependence on pumping \ndirection has been observed  in FM/NM bilayer emitters  [2,6,8] . But, in those cases, it is understood \nthat the sign reversal of THz wave polarization is  caused by the reversal of the direction of spin \npolarized current  due to sample flipping  [3,10] . In the present case, however, there is only  a thin layer \nof FM ; the flipping of the sample will not break the symmetry as long as the sample is homogenous \nin the thickness direction, which is presumably the present  case. Therefore, the most likely cause for \nthe sign reversal is the presence of a net longitudinal current  in the film thickness direction as \ndiscussed above. With presence of  AHE, the longitudinal current is converted to a transient transverse \ncurrent  \nAHE=tlj m j  with \nm the magnetization direction,  \nlj the net longitudinal current, and \nAHE\n the AHE  angle.  Since  \nlj is physically related to two interfaces, the reverse of pumping \ndirection, i.e., sample flipping, will lead to the sign reversal of \nlj . This explains why the polarity of \nTHz reverses when either the pumping direction or magnetization direction is reversed . In addition  5  to FeMnPt, we also conducted similar experiments by replacing FeMnPt with Fe0.8Mn 0.2 (3), \nCo0.2Fe0.6B0.2 (3), Ni 0.8Fe0.2 (3) and the rest remains the same. Indeed, we observed THz radiations \nfrom all the three  samples, though  their amplitudes are much  smaller compared to that of FeMnPt . \nInterestingly, the polarity of THz wave also reverses when either the pumping direction or the \nmagnetic field directio n is reversed for all the samples. We plot the peak values of THz waveforms \nof the four emitters in F ig. 2(c) with  a positive magnetic field. The positive (negative ) values \ncorrespond  to quartz (MgO ) side pumping. The NiFe shows a larger amplitude compared to FeMn \nand CoFeB, which is in agreement with the observation of a  large r AHE  in NiFe  [28-30], though the \nroughness of the interface may also play a role. The above  results provide strong evidences that the \nTHz emission  from FeMnPt is originated from the AHE .  \n \nB. Thickness dependence of THz emission  \nTo further test the relevance of the proposed AHE origin , we studied the thickness dependence \nof THz emission from single layer FeMnPt  from  3 nm to 9 nm . Figs. 3(a) and 3(b) show  the emission \nwaveforms (shifted a long the time axis for clarity ) and the corresponding peak amplitude (symbol), \nrespectively . As can be seen, the THz amplitude  increases from 3 nm to 4 nm and then decreases \nmonotonically as the thickness increases further, with a broad maximum at around 4 nm. The \nthickness dependence can be modeled by involving the following sub -processes: i) generation of  \nnon-equilibrium  electr ons by laser excitation , ii) generation of backflow current at the two interfaces , \niii) creat ion of  transverse transient charge current via  AHE , and i v) conversion of the charge current \nto THz emission.  Without los ing generality, we assume that the electron reflection coefficients at the \nquartz/FM and FM/ MgO interface s are \n1r and \n2r, respectively  (Fig. 1(d)) . We also assume that the \nresulted super -diffusive current decays exponential ly from the two interfaces . Based on this model, \nthe electric field of the THz emission may be written as: 6   \n/2 /2\n120\n01( ) ( ) ( ) ( )ee TTx x d d x d x\nd\nTHz s e AHE\nair subE M d FA d v d re e r e e dxn n Z d−−−−−−\n   − ++ \n \n                \n12\n1 1 2 2\n01( ) ( ) ( ) [ (1 ) (1 )]Td d d\ns e AHE\nair subM d FA d v d r e e r en n Z d    − − −\n=   − − − ++          (1) \n \nwhere \nd is the FM thickness , \nF  the fluence,  \n()Ad  the absor ptance of the FM layer , \n1r , \n2r  the \nreflection coefficient at two interfaces,  \ne the non-equilibrium electron  mean -free-path, \nT the \ndecay length of THz emission inside the FM layer , \n()evd  the average electron speed , \n()sMd  the \nsaturation magnetization , \nairn(\nsubn ) the reflective index of air (substrate),  \n0Z the impedance of free \nspace , and \n1/2\n/2eT\nTe=− , \n2/2\n/2eT\nTe=+ . Here, we have included the explicit thickness \ndependence of \nA, \nev and \nsM in the equation . \n()Ad  is obtained from fitting to the experimental data , \nwhich yields  \n( ) 0.0314 0.1767A d d=+  (with d in nm) . \n()evd  is estimated from the electron \ntemperature (\neT), which itself is related to the absorbed laser energy per unit volume  (\naE) [31,32] : \n22\n0 ()\n2e\naTTE−=\n, where \n  is the electronic specific heat and \n0T  the initial electron temperature ( i.e., \nroom temperature). Substitut e \n( ) /aE A d F d=  into previous equation , one obtain s \n2 1/4\n02 ( )()eeA d Fv T Td = +\n, where F is the laser fluence . In addition to  \nA and \nev, the saturation \nmagnetization also depends on  \nd, which drops quickly with reducing  \nd when \nd  < 5 nm, as revealed  \nin our previous study  [22]. A polynomial fitting to the experimental data is used to calculate  \nTHzE  in \nEq.(1)  (see Supplemental Material  [23]). The first  and second  term in the integrand of Eq. (1) \nrepresents the longitudinal to transverse current conversion efficiency as well as small absorption of 7  the THz wave by the metal layer  near the two interfaces . The last term of Eq. (1) denotes the \ntransverse current to THz conversion efficiency.  As shown by the solid -line in Fig. 3(b), the thickness \ndependence of THz peak amplitude can be fitted well using Eq. (1)  with the following parameters : \ne\n = 1.2 nm, \nT = 14 nm, \n1r = 0.6, \n2r = 0.1 (see Supplemental Material for more details [23]), F = \n555 μJ/cm2 (experimental value) , \n = 0.7 mJ/cm3K2, \nairn  = 1, \nsubn  = 1.453,  \n0Z = 377 Ω  and \n = \n5.32×106 S· m-1. As the electron specific heat of  FeMnPt  is not available , we use the reported value \nfor Fe [32] instead . Assuming  every single photon excite s an electron -hole pair, at an absorptance of \n0.4, a laser bea m with a fluence of \nF  = 555 μJ/cm2 will lead to  a nonthermal electron density of \nabout 0.88 nm-3, which is equivalent to an average electron spacing of 1.05 nm. Therefore, the fitted \nvalue of  \ne = 1.2 nm is reasonable . The fitted THz decay length of 14 nm is compar able to the \nreported  value in FM/NM bilayers [8]. The conductivity of 5.3 2\n106 S· m-1 is obtained \nexperimentally  for the 9 nm FeMnPt  [22]. The good agreement between fitted and measured \nthickness dependence provides further support to the proposed AHE origin of THz emission . \n \nC. Pump fluence dependence of THz emission  \nNext , we discuss  the pumping fluence dependence of THz radiation from the FeMnPt  (d) \nemitter  as shown  in Fig. 4. The symbols denote the measured values and solid lines are fitting curves  \naccording to Eq. (1) . At a fluence of 555 μJ/cm2, \n112e  = 0.6 nm.  Therefore, the fluence \ndependence o f \nTHzE  is mainly  determined by  \ns e eM Fv  and the last term of Eq. (1)  with \n1/4 1/4( ( ) / ) ( )ev A d d F  +\n where  \n2 ( )d\nAd=   (note: \n0 eTT  ) and \n1/3[ ( ) ]eA d F− . The laser \nheating induced magnetization reduction may be approximated by  \n00 ( ) ( )[1 ( )]s s eM d M d T T  = − − , \nwhere  \n0()sMd  is the saturation magnetization at  \n0T and \n a thickness -dependent constant. It is 8  thickness -dependent because the temperature -dependence of magnetization is different at different \nthickness when \nd  is small.  As shown in Fig. 4, the experimental results can be fitted very well using \nEq. (1)  by treating  \n as a thickness dependent fitting parameter . The inset shows the thickness \ndependence of  \n used in the fitting . As expected,  \n decreases with increasing \nd  as the heating \neffect becomes weak er at large r thickness  (see more discussion in the Supplemental Material  [23])  \n[33-35].  \n \nIII. D ISCUSSION ON  ENHANCEMENT OF TERAHERTZ EMISSION  \n With  the confirmation  of AHE origin , now  we discuss  how we can enhance the THz emiss ion. \nAn obvious approach is to optimize the Pt composition so as to increase the difference between AHE  \nnear the two interfaces, thereby increasing the THz efficiency . As discussed in the Supplemental \nMaterial  [23], at a thickness of 3 nm, the THz peak -to-peak amplitude ratio of FeMnPt emitter s \ndeposited at a Pt power of 15, 35 and 50 W is 14.1:1.7:1, with the sample deposited at 15 W showing \nthe strongest THz emission. This prompted us to  introduce a Pt grad ient in the thickness  direction . \nWhen the film is irradiated by a laser, i mmediately after the laser excitation, the spin polarization of \nthe non -equilibrium electrons is assumed to be the same as those in the equilibrium state at Fermi \nlevel  [14]. Therefore, the Pt composition gradient will be converted to a spin chemical po tential \ngradient  \nsμ  in the thickness direction  [20,36] .  With the presence of  \nsμ , the total transverse \ncurrent may be written as  \n( / 2 )AHE SHE e   =  + t bf sj m j s μ , where  \nSHE  is the spin Hall angle,  \ns\n the net spin polarization direction of non -equilibrium electrons and  \nsμ the spin chemical potential \n[37,38] . Ther efore, the introduction of Pt composition gradient may either enhance or reduce the THz \nemission, depending on the sign of  \nsμ  and relative contributions from the two terms .   \nFig. 5(a) shows the THz waveform from FeMnPt (9) with uniform Pt composition (15 W), 9  which is almost the same as that of FeMnPt (3) shown in Fig. 2(b). Fig. 5(b) shows THz emission \nfrom FeMnPt (9) with positive Pt gradient ( i.e., Pt power increases from 15  W to 50 W during \ndeposition). Compared to the waveform shown in Fig. 5(a), the polarity remains the same but the \npeak -to-peak amplitude is increased by a factor of 5.4 (212 versus 39 in arbitrary unit). This means \nthat a positive Pt composition gradient h elps to enhance the THz emission. On the other hand, as \nshown in Fig. 5(c), the sample with a negative Pt composition gradient ( i.e., Pt power decreases from \n50 W to 15 W during deposition) gives a much smaller THz signal with a peak -to-peak amplitude of \n– 84. The negative sign indicates that the polarity is reversed compared to the previous two cases. \nThe ratio of the peak -to-peak amplitudes of the three samples is 1:5.4: -2.15.  If we simply estimate \nthe peak -to-peak amplitude ratio of the three samples by  using \n1r  = 0.6, \n2r  = 0.1 and the amplitude \nratio from samples with uniform Pt composition as discussed above, the ratio of these samples turned \nout to be 1:1.18: -0.11. Obviously the two sets of values don’t tally, suggesting that the magnetization \ngradient plays an importan t role. The discrepancy can be resolved by taking into account the \ncontribution from \nsμ , as discussed in Supplemental Material [23]. These results demonstrate that \nthe spin chemical potential gradient plays an important role in TH z emission from the single layer \nFM. With further optimization of the materials and composition gradient, the current approach can \npotentially lead to more efficient THz emission.  \n \nIV. CONCLUSIONS  \n In summary, we have demonstrated an alternative mechanism for generating THz emission \nfrom ultrathin FM layer via the AHE, though the presence of AHE in the THz regime has been \nreported before  [39-41]. The process involves generation of backflow super -diffusive  current at  the \nFM/dielectric interfaces  and subsequent conversion of the charge current to transverse current via \nAHE , and thereby generating the THz radiation.  The THz generation is mainly caused by the non -10  thermal super -diffusive current near the two interfaces; the contribution from laser excitation induced \nhot carrier gradient inside the sample, if any, should be very small considering the large effective  \npenetration depth  (see Supplemental Material [23]) compared to the film thickness  [42]. Further \ninves tigations can be carried out in future using multiple wavelength excitations when the laser \nsources are available [43,44] . There is tremend ous room for improvement  of emission efficiency \nonce materials with large AHE are found. In the meantime, as we demonstrated in this paper, the \nemission efficiency can also be enhanced by introducing a gradient in the FM magnetization.  \n \nACKNOWLEDGEMENTS  \nY.H.W. would like to acknowledge support by Ministry of Education, Singapore un der its Tier 2 \nGrants (grant no. MOE2017 -T2-2-011 and MOE2018 -T2-1-076). 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Papaioannou, G. Torosyan, S. Keller, L. Scheuer, M. Battiato, V. K. Mag -Usara, J. \nL’huillier, M. Tani, and  R. Beigang, Efficient Terahertz Generation Using Fe/Pt Spintronic \nEmitters Pumped at Different Wavelengths, IEEE Trans. Magn. 54, 9100205 (2018).  \n[44] R. I. Herapath, S. M. Hornett, T. S. Seifert, G. Jakob, M. Klä ui, J. Bertolotti, T. Kampfrath, \nand E. He ndry, Impact of pump wavelength on terahertz emission of a cavity -enhanced 16  spintronic trilayer, Appl. Phys. Lett. 114, 041107 (2019).  \n  17  FIGURE CAPTIONS  \nFIG. 1. (a)-(c) Sub -processes at different time scale after ultrafast laser excitation: (a) Excitation of \nnon-equilibrium electrons. (b) High -temperature Fermi -Dirac distribution of electron subsystem. (c) \nNearly thermal equilibrium state among electron, phonon and magnon. (d) Schematic of nonthermal \nelectron reflection at quartz/FM and FM/MgO in terfaces and the formation of backflow current.  \nFIG. 2. (a) Schematics of THz emission setup with pumping from the quartz (up) and MgO (down) \nside, respectively. (b) THz wave from FeMnPt. The upper pair of waveforms is from quartz side \npumping and the lower pair is from MgO side pumping. Solid and dashed -lines correspond to positive \nand negative fields, respectively. (c) THz peak values from different emitters: FeMnPt, FeMn, CoFeB \nand NiFe, with positive applied H field (p ositive amplitude: quartz s ide pumping ; negative amplitude: \nMgO side pumping ). \nFIG. 3. Thickness dependence of THz waveform from FeMnPt single -layer  FM emitter: (a) THz \nwaveform at different FeMnPt thickness (shifted in time axis for clarity) and (b) Peak -to-peak THz \namplitude as a function of thickness (symbol: measured data; solid -line: fitting using Eq. (1)).  \nFIG. 4. Pumping fluence dependence of T Hz peak amplitude from the FeMnPt single -layer  FM \nemitter: Peak amplitude as a function of fluence for emitters with different FeMnPt thickness. Inset: \ncoefficient (\n ) of laser heating effect on saturation magnetization.  \nFIG. 5. (a) THz emission from FeMnPt with uniform Pt composition. (b) THz emission from FeMnPt \nwith positive Pt composition gradient. (c) THz emission from FeMnPt with negative Pt composition \ngradient. In all the figures, the upper pair of waveforms is from quart z side pumping and lower pair \nis from MgO side pumping. Solid and dashed -lines correspond to positive and negative fields, \nrespectively. Darker color indicates the reg ion with higher Pt composition.  \n  18   \nFIG. 1 . \n  \n \n  \n -\n+-\n+-\n+  \n  \n  \n  \n \n                Excitation(a) (b) (c)\n(d)        ps    ps t = 0-phonon/\nmagnon\nFM Quartz MgO\nElectron\nReflectionElectron\nReflection\n    \n        \n 19   \nFIG. 2.  \n  \n-60-40-200204060\nFeMnPt FeMn CoFeB NiFeTHz peak value (arb. units)Quartz side pumping\nMgO side pumping(c) (b) (a)\nH+\n-140-105-70-3503570\n0 1 2 3 4 5Magnitude ( arb. u nits)\nTime ( ps)H+\nH-FeMnPt 3nm\nx yz\nHex\nTHzfs\nlaser\nHex\nTHzfs\nlaser\n20   \nFIG. 3.  \n  \n-200204060\n0 3 6 9 12 15Magnitude ( arb. u nits)\nTime ( ps)3 nm\n4 nm\n5 nm\n6 nm\n7 nm\n8 nm\n9 nm\n304050607080\n2 4 6 8 10Magnitude  ( arb. u nits)\nThickness ( nm)Fitting\nPeak to peak value(a) (b)21   \nFIG. 4.  \n  \n0102030405060\n0 200 400 600 800Peak value (arb. units)\nFluence (μJ/cm2)3nm\n4nm\n5nm\n6nm\n7nm\n8nm\n9nm\n0.00020.00040.0006\n246810\nThickness (nm)γ22   \nFIG. 5.  \n"    },    {        "title": "1903.12022v1.High_spin_polarization_in_the_low_Curie_temperature_complex_itinerant_ferromagnet_EuTi___1_x__Nb__x_O__3_.pdf",        "content": "High spin-polarization in the low Curie temperature complex itinerant ferromagnet\nEuTi 1\u0000xNb xO3\nSuman Kamboj,1Deepak K. Roy,2Susmita Roy,3Rajeswari Roy\nChowdhury,1Prabhat Mandal,3Mukul Kabir,2, 4and Goutam Sheet1,\u0003\n1Department of Physical Sciences, Indian Institute of Science Education and Research Mohali,\nSector 81, S. A. S. Nagar, Manauli, PO: 140306, India\n2Department of Physics, Indian Institute of Science Education and Research, Pune 411008, India\n3Condensed Matter Physics Division, Saha Institute of Nuclear Physics, Kolkata, PO: 700064, India\n4Centre for Energy Science, Indian Institute of Science Education and Research, Pune 411008, India\n(Dated: March 29, 2019)\nThe physical systems with ferromagnetism and \\bad\" metallicity hosting unusual transport prop-\nerties are playgrounds of novel quantum phenomena. Recently EuTi 1\u0000xNbxO3emerged as a fer-\nromagnetic system where non-trivial temperature dependent transport properties are observed due\nto coexistence and competition of various magnetic and non-magnetic scattering processes. In the\nferromagnetic state, the resistivity shows a T2temperature dependence possibly due to electron-\nmagnon scattering and above the Curie temperature Tc, the dependence changes to T3=2behaviour\nindicating a correlation between transport and magnetic properties. In this paper, we show that\nthe transport spin-polarization in EuTi 1\u0000xNbxO3, a low Curie temperature ferromagnet, is as high\n(\u001840%) as that in some of the metallic ferromagnets with high Curie temperatures. In addition,\nowing to the low Curie temperature of EuTi 1\u0000xNbxO3, the temperature ( T) dependence of Ptcould\nbe measured systematically up to Tcwhich revealed a proportionate relationship with magnetiza-\ntionMsvs.T. This indicates that such proportionality is far more universally valid than the\nferromagnets with ideal parabolic bands. Furthermore, our band structure calculations not only\nhelped understand the origin of such high spin polarization in EuTi 1\u0000xNbxO3but also provided\na route to estimate the Hubbard Uparameter in complex metallic ferromagnets in general using\nexperimental inputs.\nSpintronic devices involve manipulation of the spin of\nelectrons. Spin polarized electrons can, in principle, be\nsourced from itinerant ferromagnets with high degree of\nspin polarization[1{3]. In this regard, search, determi-\nnation and understanding of spin polarization in novel\nmaterials is of utmost importance. Since device appli-\ncations are desirable at ambient temperatures and most\nof the traditional \\transport\" spin polarization measure-\nment techniques rely on low-temperature experiments,\nit is most important to understand temperature evolu-\ntion of the magnitude of spin polarization in materials.\nSuch measurements are particularly interesting in mate-\nrials that cannot be characterized by conventional metal-\nlicity. On the other hand, for a complete understanding\nof such temperature evolution up to the Curie tempera-\nture (Tc), it is important to have a ferromagnet where Tc\nfalls within the low-temperature limit where such mea-\nsurements are possible. In this context, EuTi 1\u0000xNbxO3\nis found to be the ideal candidate because it is a \\bad\"\nmetal and at the same time, a ferromagnet with low Tc\u0018\n9.5 K forx= 0.15[41].\nEuTi 1\u0000xNbxO3is an important member of the per-\novskite titanate family RTiO 3(R = rare earth ion),where\nTi usually has 3d1electronic con\fguration. This family\nhas attracted researchers since last two decades owing to\ntheir several fascinating properties [4{7]. There has been\na considerable amount of theoretical and experimental\n\u0003goutam@iisermohali.ac.inwork done for understanding the nature of the ground\nstate in this family. Among the perovskites, EuTiO 3\nwith unique divalent rare-earth ion Eu and tetravalent\nTi has generated substantial interest since the discovery\nof magnetoelectric coupling in this compound below the\nantiferromagnetic (AFM) transition temperature TN=\n5.5 K [8{13]. EuTiO 3has a positive Curie- Weiss con-\nstant and exhibits strong coupling between magnetic or-\ndering and phonon modes. E\u000borts have been made to\ndrive this system from antiferromagnetic phase to other\nmagnetic phases upon modi\fcations due to partial sub-\nstitution at the Ti site. Small amount of substitution at\nTi site by Nb drives the system from antiferromagnetic\nto ferromagnetic [40]. Substitution by Nb4+ion at the\nTi4+results in the introduction of electrons in EuTiO 3\nwithout disturbing the Eu2+moments chain. Previous\nstudies in EuTi 1xNbxO3revealed notable increase in elec-\ntrical conductivity and large values of magnetocaloric re-\nsponse of about 23 :8 J kg\u00001K\u00001even at low magnetic\n\feld 0-2T[11]. Therefore, introduction of Nb in antifer-\nromagnetic EuTiO 3results in ferromagnetic behaviour in\nEuTi 1\u0000xNbxO3(x= 0.15) with T C\u00189.5 K and satura-\ntion magnetization at 2 K is estimated to be \u00187\u0016B/Eu\n[11].\nIn this paper, we have employed spin-resolved Andreev\nre\rection (PCAR) spectroscopy using conventional su-\nperconducting tip of Nb to measure the transport spin-\npolarization in EuTi 1\u0000xNbxO3(x= 0.15) [14{22]. Spin\npolarization ( P) in a metallic ferromagnet is generally\nde\fned asP= (N\"(EF) -N#(EF)/(N\"(EF) +N#(EF)),arXiv:1903.12022v1  [cond-mat.mtrl-sci]  28 Mar 20192\nwhereN\"(EF) andN#(EF) are the density of states\n(DOS) of the up and down spin channels respectively\nat the Fermi level. However, in a transport experi-\nment like PCAR, the relevant quantity is not the ab-\nsolute spin polarization but the so called \\transport spin\npolarization\" which is de\fned as: Pt= (hN\"vF\"iFS-\nhN#vF#iFS)/(hN\"vF\"iFS+hN#vF#iFS) in the ballistic\nregime, where N\"andN#are the density of states (DOS)\nof the up and down spin channels respectively at the\nFermi level and vF\"andvF#are the respective Fermi\nvelocities [23].\n-6 -4 -2 0 2 4 60.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 (dI/dV) N\nVdc (mV) T = 1.98 K \nP = 40% \nZ = 0.388 \nD = 1.55 meV \n-6 -4 -2 0 2 4 60.90 0.95 1.00 1.05 1.10 1.15 (dI/dV) N\nVdc (mV) T = 1.98 K \nP = 41.5% \nZ = 0.333 \nD = 1.45 meV \n-6 -4 -2 0 2 4 61.00 1.05 1.10  (dI/dV) N\nVdc (mV) T = 1.98 K \nP = 41 % \nZ = 0.265 \nD = 1.252 meV \n0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 020 40 60 80 100  P (%) \nZ Experimental data \n Linear fit (a) (b) \n(c) (d) \nFigure 1. (a),(b),(c) Normalized dI/dV spectra for point-\ncontacts on EuTi 1\u0000xNbxO3with x=0.15 using a Nb tip. The\nblack lines show BTK \fts with spin-polarization included. (d)\nSpin-polarization (P t) vs. barrier strength (Z) plot. The solid\nlines show extrapolation to Z = 0 where the spin-polarization\napproaches 45 %\nPCAR spectroscopy measurements were performed on\nsingle crystal EuTi 0:85Nb0:15O3using Nb tips. The bal-\nlistic point-contacts between the sample and the tips\nwere fabricated and controlled by moving the tip up\nand down by rotating a di\u000berential screw-based head as-\nsembly manually. The measurements involved obtaining\nthe point contact spectra (i.e. di\u000berential conductance\ndI=dV vs.Vcurves) for di\u000berent contacts with di\u000berent\nvalues ofZ, the interface transparency as in the Blonder-\nTinkham-Klapwijk (BTK) theory. The spectra thus ob-\ntained were analyzed using BTK theory modi\fed to in-\ncorporate the e\u000bect of spin polarized bands [24, 25]\nIn Figure 1(a,b,c) we show three representative PCAR\nspectra (red points). The overall spectral features indi-\ncate that the point contacts belong to the ballistic or dif-\nfusive regime of transport where the two peaks symmetric\naboutV= 0 appear due to Andreev re\rection across a\nsuperconducting interface with \fnite transparency ( Z).\nThe black lines show the \fts to the experimentally ob-\ntained spectra using spin-polarized BTK theory. The su-\nperconducting gap \u0001 ranges between 1.2-1.5 meV for dif-\nFigure 2. (a) Normalized dI/dV spectra with varying tem-\nperature over a temperature range of 1.98 K to 9.33 K for\npoint-contacts on EuTi 0:85Nb0:15O3using a Nb tip. The col-\nored lines show experimental data points and the black lines\nshow BTK \fts with spin-polarization included. (b) Tempera-\nture dependence of transport spin-polarization and the broad-\nening parameter \u0000. The solid lines are guide to the eye.\nferent contacts. The extracted values of spin-polarization\nPtand \u0000 are also shown. For di\u000berent values of Zbe-\ntween 0.265 - 0.388 the transport spin polarization is\nmeasured to be around 41%. The extracted values of Pt\nis also seen to slightly depend (linearly) on Zas shown\nin Figure 1(d). The solid lines in Figure 1(d) show linear\nextrapolation of the Z-dependence of PttoZ= 0 which\ngives the expected intrinsic value of the spin-polarization\n(forZ= 0). The intrinsic spin-polarization extracted in\nthis case is found to be approximately 45% which is com-\nparable to elemental ferromagnetic metals like Fe ( P=\n40%), Co (P= 42%) and Nickel ( P= 39%)[26].\nFigure 2(a) shows temperature dependence of\nnormalized dI/dV spectra for point-contacts on\nEuTi 0:85Nb0:15O3. The dip at V= 0 in the spectra\nis maximum at the lowest observed temperature of3\n2 3 4 5 6 7 80.750.800.850.900.951.00MS(T)/MS(0)\nT (K)1618202224262830323436384042\nPt (%)\n20222426283032343638400.750.800.850.900.951.00\nMS(T)/MS(0)\nPt(%)\n  \nFigure 3. Temperature dependence of transport spin-\npolarization ( P) and spontaneous magnetization M S(T) nor-\nmalized by that at T=0 K,in the temperature range 2 - 9\nK. Inset shows variation of normalized magnetization with\ntransport spin-polarization. The solid lines are guide to the\neye.\n1.98 K. With increase in temperature, the di\u000beren-\ntial conductance spectra gradually undergo thermal\nsmearing. At temperatures greater than 8.34 K the\nmost dominant feature of the spectra, namely, the two\npeaks in the conductance spectra associated with the\nsuperconducting energy gap gets \rattened.\nThe BTK \fts of the temperature dependence of nor-\nmalizeddI=dV spectra (solid line in Figure 2(a)) give\nan estimate of temperature dependence of the mea-\nsured transport spin-polarization Ptand also the broad-\nening parameter \u0000 as shown in Figure 2(b). Trans-\nport spin-polarization decreases with increase in temper-\nature whereas the parameter \u0000 is observed to increase\nwith temperature. \u0000 gradually increases and reaches a\nmaximum value of 0.4meV at 8 K close to the T Cof\nEuTi 0:85Nb0:15O3. The parameter \u0000 is associated with\nlifetime (\u001c) of superconducting quasiparticle in the way\n\u0000 = ~/\u001c. The increase in \u0000 with temperature signi\fes\na corresponding decrease in quasiparticle lifetime at the\nsame temperature which might be due to enhanced spin\n\ructuations at higher temperatures. Figure 3 shows tem-\nperature dependence of transport spin-polarization and\nspontaneous magnetization in the same panel for a direct\ncomparison between the two. Ptis observed to decrease\nwith temperature following the same behaviour as Ms.\nThis is clearly seen in the inset of Figure 3 where Msand\nPtare seen to linearly dependent on each other. This is\nremarkable because such proportional dependence of Pt\nandMswith temperature is expected only for materi-\nals with strictly parabolic bands and EuTi 0:85Nb0:15O3\nis known to be a system with large correlation e\u000bects[15].\nTo gain insight into the experimental results, we have\nperformed \frst-principles calculations within the density\nfunctional theory as implemented in the VASP. [27{30]\nThe spin-polarized wavefunctions are described within\nthe projector augmented wave formalism with 550eV cuto\u000b for the kinetic energy. [31] The exchange-\ncorrelation energy is described with the Perdew-Burke-\nErnzerhof functional. [32] The strong electron correlation\nin Eu-4f, Ti-3d, and Nb-4delectrons is accounted by an\non-site Hubbard-type Coulomb interaction Uwithin the\nrotationally invariant Dudarev's approach. [33] While ap\n2\u0002p\n2\u00022 tetragonal supercell is used to determine\nthe correct magnetic ground state of pure EuTiO 3, a\n2p\n2\u0002p\n2\u00022 orthorhombic supercell is used to determine\nthe electronic structure and spin transport polarization\nin EuTi 1\u0000xNbxO3. The Brillouin zone is sampled with\n4\u00028\u00026 k-point mesh according to the Monkhorst-Pack\nscheme, [34] while the atomic positions, volume and the\nshape of the supercell are optimized until all the force\ncomponents are less than 0.01 eV/ \u0017A threshold. The cal-\nculated lattice parameter of 3.97 \u0017A for EuTiO 3compares\nwell with the experimental value of 3.91 \u0017A. [9]\nIn agreement with the previous experimental and the-\noretical results, the present DFT+ U(UEu= 3.5 andUTi\n= 2 eV) calculations predict an antiferromagnetic (AFM)\ninsulating ground state for EuTiO 3. [35, 36] The AFM\nsuperexchange between the half-\flled 4 fstates of Eu2+\n(S=7/2) via the Ti-3 dstates wins over the ferromag-\nnetic (FM) indirect exchange via Eu-5 dstates. [37] The\ncompeting FM state lies only 1.2 meV/f.u. higher in en-\nergy. Thus, within the DFT+ Uapproach, the tuneable\nHubbardUparameter dictates the long-range magnetic\nordering. [38, 39] In the absence of an on-site Coulomb\ninteraction for the Ti-3 delectrons, the UEuabove 5 eV\nfavours the competing FM state. In contrast, while UTi\n= 2 eV is used, a lower UEu<4 eV describes the AFM\nground state correctly. Such decrease in UEuwith an\nonset ofUfor the Ti-3 delectrons is suggestive, since, if\nthe Ti electrons are assumed to be more localized then\nthe Eu electrons need to be more delocalized in order to\nmaintain the superexchange interaction described above.\nThe long-range AFM order can be tuned to FM through\nstrain, pressure, and chemical doping. [36, 40] It is known\nthat upon doping EuTiO 3with more than 5% Nb in place\nof Ti in EuTi 1\u0000xNbxO3, the magnetic structure changes\nfrom AFM to FM. [40] The Nb atoms in the lattice intro-\nduce one itinerant electron per Ti-atom replaced, and as\na result, EuTi 1\u0000xNbxO3becomes metallic. The itinerant\nNb-4delectrons mediate the Ruderman-Kittel-Kasuya-\nYoshida exchange interaction between the Eu2+ions, and\nthe FM ground state emerges. [41]\nThe EuTi 1\u0000xNbxO3withx= 0.15 (UEu= 3.5 and\nUTi=Nb= 2 eV) shown in Figure 4(a) becomes FM metal\n[Figure 4(b)], and we estimate the transport spin polar-\nization in the ballistic limit. The magnetic moment of\n6.85\u0016Bat the Eu-site is in agreement with an earlier\nexperiment, [35] which arises from the localized Eu-4 f\norbitals [Figure 4(b)], which are around 0.5 eV below\nthe Fermi level. A deeper investigation into the density\nof states (DOS) indicates that the states at the Fermi\nlevelEFis composed of Ti-3 dstates with small contri-\nbutions from the Eu-4 fand Nb-4dstates [Figure 4(b)].\nThe spin-polarized DOS at the EFis calculated to be4\nc ab\nEu Ti Nb O\n-4.0-3.0-2.0-1.00.01.0\nΓ Y S X Γ Z TR U Z Energy E-EF (eV)\n 2 2\nDOS (states/eV/unit-cell)Total ↑\nTotal ↓\nEu-4 f\nNb-4 d\nTi-3d\nO-2p\n-0.4 0 0.4\nΓ Y S X Γ Z T R U ZE-EF (eV)\nSpin ↑\nSpin ↓\nUTi = 1 eV\n2 eV\n3 eV-0.4 0 0.4E-EF (eV)\nSpin ↑\nSpin ↓\nUEu = 2 eV\n3.5 eV\n5 eV(a)(b) (c)\n(d)\nFigure 4. (a) Crystal structure of EuTi 1\u0000xNbxO3indicates the oxygen octahedra tilting. (b) Band structure and the corre-\nsponding density of states for x=0.125. Majority (minority) spin channel is drawn with black (red) color. The localized Eu-4 f\nstates lie about 0.5 eV below the Fermi level. The Fermi level is composed of Ti-3 d, Eu-4 fand Nb-4 delectrons. (c)-(d) Energy\nevolution of the low-energy bands with Hubbard Uparameter. While UEuis varied with \fxed UTi=2 eV in (c), the evolution\nof bands with di\u000berent UTi(UEu= 3.5 eV) is shown in (d).\nN\"(EF) = 0.574 states/eV/unit-cell which is 68% higher\nthan theN#(EF). The average Fermi velocity hvF\"i=\n3.47\u0002105m/s is also found to be higher for the spin-up\nchannel when compared to the spin-down channel, hvF#i\n= 1.99\u0002105m/s. These result in a very high transport\nspin polarization of 49% in the ballistic regime, which\nis in excellent agreement with the present experimental\nresults with x= 0.15.\nIt is important to discuss that the electronic structure\nand the concurrent Ptcrucially depends on the on-site\nCoulomb interaction UEu=Ti=Nb, which is treated as a pa-\nrameter within the DFT+ Uformalism. [42] We \fnd that\nwith increasing UEu[Figure 4(c)] both the channels are\npushed down in energy and the corresponding Ptmono-\ntonically decreases to reach 25% for UEu= 5 eV, while\ntheUTiis kept \fxed at 2 eV. In contrast, with increasing\nUTi[Figure 4(d)], the minority channel is pushed higher\nin energy and the corresponding Ptincreases monotoni-\ncally. Ultimately a half-metallic solution emerges for UTi\n= 3 eV with 100% spin polarization. Thus, in the ab-\nsence of a \frst-principles estimation of Hubbard U, one\ncan use the experimental knowledge of transport spin po-\nlarization to better estimate the theoretical Uparameter\nin ferromagnetic metals.\nIn conclusion, we report point contact An-\ndreev re\rection spectroscopy on polycrystalline of\nEuTi 0:85Nb0:15O3. EuTi 0:85Nb0:15O3is ferromagneticwith Curie temperature T C= 9.5 K. Transport spin\npolarization ( Pt) estimated from di\u000berential conductance\nspectra is about 45% which is close to elemental fer-\nromagnets. Temperature dependence of transport spin\npolarization closely follows the temperature dependence\nof spontaneous magnetization ( MS). Furthermore,\ntemperature dependence of the broadening parameter\n(\u0000) estimated from PCAR spectra exhibits increase with\ntemperature. This implies with increase in tempera-\nture superconducting quasiparticle lifetime decreases.\nThis decrease can be attributed to the e\u000bect of spin\n\ructuations close to the critical temperature of the\nferromagnetic system. Thus, our study can provide\nan alternative way to probe spin \ructuations in ferro-\nmagnetic materials. Based on a comparative analysis\nbetween the experimental and theoretical investigations\nwe have also discussed a possible route to directly\nestimate the Hubbard Uparameter in complex metallic\nferromagnets in general.\nGS acknowledges \fnancial support from the research\ngrants of (a) Swarnajayanti fellowship awarded by the\nDepartment of Science and Technology (DST), Govt.\nof India under the grant number DST/SJF/PSA-\n01/2015-16 and (b) from SERB under the grant num-\nberEMR/2015/001650 . M. K. acknowledges fund-\ning from the Science and Engineering Research Board\nthrough Nano Mission project SR/NM/TP-13/2016 .\n[1] I. Zutic, J. Fabian, and S. 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B 44, 943 (1991) ."    },    {        "title": "1709.05898v1.Magnetic_field_driven_enhanced_ferroelectric_switching_in_self_grown_ferroelectric_ferromagnetic_composite_in_the_BiFeO3_BaTiO3_multiferroic_alloy_system.pdf",        "content": "1 \n \n1 \n Magnetic field driven enhanced ferroelectric switching in self -grown \nferroelectric -ferromagnetic composite in the BiFeO 3-BaTiO 3 multiferroic \nalloy system  \nAmit Kumar, Bastola Narayan, Rohit P, and Rajeev Ranjan*  \nMaterials Engineering Department, Indian Inst itute of Science, Bangalore -560012.  \n \nOver the years attempts have been made to compensate  for the inherent weaknesses in the bulk \nstate of the multiferroic BiFeO 3, such as high leakage current and the absence of ferromagnetic \ncorrelation, and exploit its m agnetoelectric potential by forming solid solutions with other \nperovskites. Studies in the recent few years have shown that alloying of BiFeO 3 with BaTiO 3, \nboth with and without additives , can induce both ferroelectric and ferromagnetic switching . \nWhile the coexistence of both the ferroic orders  is encouraging from the view point of \ntechnological applications, the origin of ferromagneti sm in this system remains elusive . Here, we \nsynthesize d such compositions and carried out a detailed structural  analysis employing magnetic \nseparation of the powder particles . We found that the origin of ferromagnetism lies in the \nspontaneous precipitation  of a minor (  ~ 1 wt % ) barium hexaferrite (BaFe 9O19) phase, often \nundetected in routine x -ray diffraction studies  of powde rs sampled from the entire specimen . We \nalso demonstrate that inspite of the insignificant fraction the ferrimagnetic phase , this self -grown \ncomposite exhibit noticeably enhanced ferroelectric switching in the presence of external \nmagnetic field. We obtain ed a dc magnetoelectric coupling of ~ 9 x 10-8 s/m, a value which is \ncomparable  to what has been reported for layered ferroelectric/ferromagnetic laminates and \nbilayer thin film ferroelectric -ferromagnetic hetrostructure s. Our study suggests that reasonabl y \nlarge magnetoelectric coupling is realizable in simple 0 -3 ferroelectric -ferromagnetic bulk \ncomposites provided synthesis strategies are developed which induce s spontaneous precipitation \nof the ferromagnetic phase in small volume fraction  to ensure good insulating behaviour of the \ncomposite thus developed . \n \n* ranjanrajeeb@gmail.com  2 \n \n2 \n I. Introduction   \nMagnetoelectric multiferroic materials exhibiting coexistence of ferromagnetism and \nferroelectricity  are of great interest both from the scientific an d technologi cal viewpoints  [1-3]. \nMost s ingle phase multiferroic s however exhibit  antiferromagnetic ordering  [4, 5] . As a \nconsequence, e ven though the se systems show intrinsic coupling between the antiferromagnetic \norder and polarization via the structural degrees of freedom  [6, 7], they are not suitable for \nmagnetoelectric devices as the applied magnetic field has no influenc e on the system. BiFeO 3 \n(BFO) is one of the most investigated room temperature multiferroic material [7-10]. The \nferroelectricity in BFO arises d ue to off -centered displacement of the Bi+3 along the [111] of the \nperovskite cell by virtue of the stereo chemical activity of its 6s2 lone pair electron  [7]. The \nmagnetic order, on the other hand, is associated with moments on the Fe-sublattice . In bulk BFO \nthese moments form  a long wavelength ( ~ 62 nm ) cycloidal configuration superposed on a  G-\ntype antiferromagnetic order  [11], and preclude a linear magnetoel ectric coupling . Attempts have \nbeen made to suppress th is long period modulation and induce a ferromagnetic correlation by \nsize reduction [ 12], strain engineering in epitaxially grown films [ 8,9], and chemical \nmodification [ 3, 6] . While, onset of ferromagnetism and electric -field induced changed in the \nmagnetization has been demonstrated in epitaxial BFO films [9], the challenge persist s in bulk \nspecimens.  \nA popular strategy among experimentalists is to alloy BFO with other stable perovskites  \nsuch as such  BaTiO 3 [13-17], PbTiO 3 [18-21], rare-earth  elements  [22- 26], non -magnetic ions at \nthe Fe -site [2 7], and look for improvements in the desired properties . Among others, (1-\nz)BiFeO 3-(z)BaTiO 3 (BF-BT) has attracted significant attention in the recent past [13-17]. The \nsystem exhibits complete solid solubility with rhombohedral structure in the composition  range \n0z<0.33, cubic -like for 0.33 z0.92, and tetragonal for 0.92< z1 [28]. Of particular interest in \nthe context of magnetoelectric behaviour  of this alloy system are  the reports of ferromagnetism \n[6, 15, 29 -31]. Although the coexistence of ferroelectr icity and ferromagnetism in BF -BT is very \ninteresting, exploitation of its magnetoelectric potential rests on our understanding of the origin \nof ferromagneti sm in this system . While Park et al [15] argued that the ferromagnetism is \nintrinsic in nature and arise due to release of the latent magnetization locked in the long ranged 3 \n \n3 \n cycloidal spin structure of the  parent compound  BiFeO 3 [15], other groups who reported \nferromagnetism  in this system [29-32] have not discussed /speculated its origin. Here, we \nexami ne the origin of ferromagnetism in the BiFeO 3-BaTiO 3 system in detail by carrying out a \ncombined magnetization and structural study  involving magnetic separation of the powder \nparticles . We prove t hat the ferromagnetism is extrinsic in nature and owes its origin to \nspontaneous precipitation of a minor (~ 1 wt. %) barium hexaferrite phase, often undetected in \nroutine x -ray diffraction of powder sampled from the whole specimen. We also demonstrate that \ninspite of the insignificant fraction of the  ferromagnet ic phase, the bulk specimen e xhibit a very \nhigh dc magnetoelectric coupling  ~ 9 x 10-8  s/m, a value which is comparable to what has \nbeen reported for ferroelectric -ferromagnetic laminates [3 3] and thin film s [34].       \n \nII. Experimental  \nFollowing liter ature, we focused on synthesizing dense ceramics of BF -BT compositions \nexhibiting good ferroelectric and ferromagnetic properties  using CuO and MnO 2 as additives \n[30-32].  For this study, w e chose the composition 0.675 BiFeO 3-0.325  BaTiO 3 and added x wt %  \nCuO + y wt %  MnO 2 as grain growth additives . The relative fractions of the two additives were \nvaried as (i) x=0, y=0; (ii) x=0.1, y=0.15; (iii) x=0.2, y=0.30; and (iv) x=0.4, y=0.6. In the \nremaining of this paper, we will label t hese samples as Sample -1, Sa mple -2, Sample -3 and \nSample -4, respectively . The ceramics were synthesized  by conventional solid state reaction \nmethod using BaCO 3, Fe 2O3, Bi 2O3, TiO 2, CuO, and MnO 2 as raw materials  of high purity grade \n(~99.9%, Alfa Aesar). The powders were mixed accordi ng to the stoichiometric formula and ball \nmilled in acetone medium for 12 h to get a homogeneous mixture. The powders were calcin ed at \n820 0C for 2 h. The calcined powders were  pressed into disks of 20 mm diameter and ~1 mm \nthickness using 2 wt % polyvinyl  alcohol (PVA) as a binder. The pellets were sintered at 990 0C \nfor 2  h in closed alumina crucible with calcined powder of the same composition used as spacer . \nThis s acrificial medium was used to reduce the evaporation of Bi during sintering. For electrica l \nmeasurements, the opposite faces of the sintered pellets  were painted with silver paste.  X-ray \npowder diffraction study was carried out using a rotating anode diffractometer ( Rigaku , \nSmartLab ) equipped with a Johansson monochoromater  in the  incident beam  to remove the Cu –4 \n \n4 \n Kα2 radiation.  Magnetization study was carried  out using the vibrating sample  magnetometer \n(VSM - LakeShore ). Microstructural characterization was done with a  scanning electron \nmicroscopy ( ESEM Quanta 200 ). Polarization electric –field ( P-E) loops were measured with a  \nPrecision Premier II loop tracer.   \n \nIII. Results   \nA.  Structur al, microstructural , ferroelectric  and magnetic  properties  \nFigure 1 shows the XRD patterns of Sample -1, Sample -2, Sample -3, and Sample -4. In \nconformity with the earlier reports [30 -32], the XRD patterns suggest formation of perovskite \nphase  with n o visible impurity peaks . The singlet nature of the Bragg profiles, confirms the \nstructure to be cubic -like, in agreement with the reported structure for this composition [ 28]. The \nscanning electron micrograph  (SEM) images revealed the average grain size to increase with \nincreasing  the additive content (from submicron grains - 0.6 µm for x=0, y=0  (Sample 1)  to ~ 4 \nµm for x=0.4, y=0.6 (Sample 4) , Fig. 2 , confirming that these ad ditives acts as grain growth \nagents . The enhancement in the ferroelectric switching  with increasing additive contents , shown \nin Fig. 3(a), proves a direct correspondence between increased grain size and increased domain \nswitching.  Interestingly, and a s was  intended, the ferromagnetic behaviour of the specimen also \nimproves dramatically with increasing the additive content with sample 4 showing a nice M -H \nhysteresis with a  magnetization of ~ 0.45 emu/g  at a 20 kOe, Fig. 3b. Sample 4 is therefore \nequivalent to the BF -BT specimens reported in the past by different groups exhibiting both \nferroelectric ity and ferromagneti sm [30-32]. We chose this Samp le for a detailed investigation . \nB. Ferromagnetism : intrinsic or extrinsic ?  \nWe examined the two scenarios: (i) ferro magnetism is intrinsic to the perovskite phase  as \nproposed by Park et al [15] , or (ii) ferromagnetism is extrinsic , and arise from another phase the \nvolume fraction of which is so small that it is not revealed in XRD patterns of powder sampled \nfrom the ent ire specimen. To answer this puzzle, w e adopted the strategy of magnetic separation \nof the powder  particles. If the perovskite phase is ferromagnetic, then all the  powder particles of \nthe sample should be attracted towards a magnet , if placed in its proxim ity.  On the other hand, if \nthe ferromagnetism is due to another phase, then only those few particles would be attracted. 5 \n \n5 \n With this in view, w e ground the sintered pellet of Sample 4 to fine powder and brought a \nmagnet bar in its vicinity. We found that on ly a very small fraction of the ground powder got \nattached to the bar magnet. These powder particles were collected separately for structural \nanalysis. Fig. 4a shows the XRD pattern of the particles collected on the bar magnet . The \ndiffraction pattern clea rly reveals additional Bragg peaks corresponding to a non -perovskite \nphase.  Indexing of these non -perovskite peaks revealed the phase to be barium hexaferrite \nBaFe 12O19. The high intensity peaks in this pattern correspond to the perovskite phase. This \nconfirms that the particles attached to the bar magnet comprise of bonded grains of the \nperovskite and barium hexaferrite. For the sake of direct comparison, we also synthesized \nBaFe 12O19, the XRD pattern of which is also shown in Fig. 4a.  Figs. 4 ( b) and (c ) show the \nmagnetic hysteresis data of the powders collected on the bar  magnet , and BaFe 12O19 respectively. \nWe used the magnetization data of BaFe 12O19 as a reference to estimate the weight fraction of the \nBaFe 12O19 phase in Sample  4. At 20 kOe the magnetiza tion of pure BaFe 12O19 is 52 emu/gm  \n(Fig. 4e) , and that of the Sample 4 (before magnetic separation) is 0.45 emu/g (Fig. 3b). This \nsuggests that the weight fraction of BaFe 12O19 in Sample 4 is ~ 0.9 %. It is therefore not \nsurprising that the Bragg peaks co rresponding to the BaFe 12O19 are not detectable in the XRD \npattern  shown in Fig. 1 .  \nHaving proved  that the origin of ferromagnetism in our sample is extrinsic , in the next \nstep we investigated the origin ferromagnetism in a lower BaTiO 3 content alloy, i.e . 0.9BiFeO 3-\n0.1BaTiO 3 reported in the past [ 6].  To be in conformity with the earlier studies, we did not add \nthe grain growth additive s CuO and MnO 2 during the synthesis  of this composition. As \nanticipated for this composition [6], t he splitting of the ps eudocubic {111} pc  Bragg profile  into \ntwo (shown in the inset of Fig. 5a ), confirms the rhombohedral structure. We also noticed a weak \nreflection (marked with * in Fig. 5a) correspond ing to a non -ferromagnetic impurity phase \nBi2Fe4O9 often reported in the BiFeO 3 rich end of the BFO -based alloys. As shown in Fig. 5c, \nthis specimen show s ferromagnetism with magnetization  0.12 emu/g at 20 kOe . As with Sample \n4, we carried out magnetic separation of the powder particles of 0.9BiFeO 3-0.1BaTiO 3. The \nXRD pattern o f the magnetic  rich particles, shown in Fig. 5b, confirms the additional Bragg \npeaks corresponding t o the barium hexaferrite phase. In contrast to sample 4 , however, the pellet \nof 0.9BiFeO 3-0.1BaTiO 3 did not show  polarization -electric field hysteresis loop  typical of a 6 \n \n6 \n normal ferroelectric material due to significantly enhanced conductivity  in conformity with the \nearlier studies [6].   \n \nC. Magnetoelectric coupling  \nHaving proved that the observed f erromagnetism in this system is extrinsic,  i.e., they \narise du e to the precipitation of ~ 1 wt % barium hexaferrite grains in between ferroelectric \ngrains, we investigated the coupling between the two phases  and its response on macroscopic \nproperties . We carried out polarization switching experiments under magnetic f ield, and also \nmagnetization switching study before and after subjecting the specimen to strong external \nelectric field. The experiments were carried out on sample 4 as it exhibited the best ferroelectric \nand ferromagnetic properties in our series. Fig. 6 a shows the P-E loops of sample 4 at different \nvalues of dc magnetic field.  The saturation polarization  (Ps) increases from 20.83 C/cm2 at H = \n0 kOe  to 27.85C/cm2 at 10 kOe i.e. by ~ 34 % , Fig. 6b.  We also  noted a slight decrease in the \ncoercive field ( Ec) from 22.5 kV/cm to 21.5 kV/cm when the field was increased to 10 kOe . The \ndc magnetoelectric response calculated using ∆Ps/∆H is ~ 8.81×10-8 s/m. This value is \ncomparable with the reported magnetoelectric coupling α of a Pb(Zr,Ti)O 3/terfenod -D laminate \ncomposite (~10-8) [33] and bilayer epitaxial thin film of BaTiO 3/(La, Sr)MnO 3 [34] (~ 2 x 10-7 \ns/m).  \nWe also carried out P -E measurements in the subcoercive field region . In this regime, the \npolarization can be approximately considered to be a linear func tion of field. Often P-E \nmeasurements in the subcoercive regime are performed to understand domain switching \nbehaviour in the framework of Rayleigh analysis  [35-38]. Rayleigh analysis assumes movement \nof domain walls in a medium comprising of pinning cente rs (defects) or varying strengths [ 35]. \nFor small fields the domain wall motion is assumed to be reversible. The extent of the \nreversibility decreases as the coercive field is approached. The magnitude of the property \n(dielectric or piezoelectric) in the s ub-coercive regime is proportional to the amplitude of the \nfield [35].   \n                                           (3) 7 \n \n7 \n where Eo is the amplitude of the cyclic sub -coercive electric field. \n  and  Eo represent the \nreversible and the  irreversible contributions, respectively to the measured dielectric response.  \nThe dielectric coefficient is calculated from the P -E loops using the expression  \nr = P p-p /2E o                                    (4) \nWhere Pp-p is peak -to-peak polarization measured for applied electric field amplitude  Eo.  For \nRayleigh analysis P -E measurements are carried out at different electric field amplitude s to \ngenerate the  - E0 data. Here, o ur intention is not to carry out Rayleigh analysis on our \nspecimen , but to e xamine the effect of magnetic field on the polarization of the specimen in the \nquasi -linear P -E regime. Accordingly, we fix the electric field amplitude of the measuring signal  \nin the subcoercive region and varied the strength of the dc magnetic field appl ied parallel to the \nelectric -field direction . The resultant plots are shown in Fig. 6c. The increase in polarization with \nincreasing magnetic field is consistent with the trend observed even in the high field regime (Fig. \n6a).  The magnetic field dependenc e of the dielectric coefficient determined usi ng equation 4 is \nshown in Fig. 6d.  \n \nIV. Discussion  \nA. Sensitivity of the phase formed to synthesis conditions  \nA perusal of literature suggests that there is no unanimity on the magnetic state in the BF -\nBT system . While ferromagnetic M -H loops has been reported by Park et al [15],  Kim et al [17], \nand Wei et al  [30]. No such loop is evident in the M -H data reported by Yang et al [14] and  \nBuscagalia et al [39].  Park et al have claimed th eir specimen s, prepared u sing molten salt \nsynthesis method  to be single phase and attributed the observed ferromagnetism to the release of \nthe latent magnetization due to dissol ution of the toroidal spin structure by finite size effect [ 15].  \nThe authors reported maximum magnetiza tion of 1.88 emu/g at 5 Tesla and a remanent \nmagnetization of 0.75 emu/g in  a composition z =0.2. For the same composition prepared by \nconventional ceramic synthesis method, Kim et al. also reported a ferromagnetic loop  but with \nan order of magnitude lower  remanent magnetization [17]. In contrast, s imilar compo sitions \nprepared by Yang et al., did not show ferromagnetic loop [ 14]. Whil e Park et al have attributed 8 \n \n8 \n the ferromagnetism in their specimen to the relatively small size and cube shaped particles, a  \nperusal of the XRD data of (1 -z)BiFeO 3-(z)BaTiO 3 presented in Fig. 2c of  their paper  reveals \nweak peaks on the higher 2  side of the pseudocubic {110} pc doublet  [15]. These peaks are not \nlikely to be associ ated with the perovskite phase. The barium hexafe rrite phase also shows a peak \non the right side of the {110} pc in our s pecimen , Fig.4a. From  this analogy we suspect that the \nadditional peak in the XRD patterns of BF -BT specimens prepared by Park et al is also likely to \nbe barium hexaferrite phase. That this peak can be seen in their pattern suggests that the volume \nfraction of the hexaferrite phase in their specimen is noticeably larger than in ours. We suspect \nthat t he ferromagnetism in their specimen is not likely to be intrinsic bu t due to the precipi tation \nof the hexaferrite phase.  As in our sample where in the ferromagnetic barium hexaferrite phase is \nprecipitated by the CuO and MnO 2 additives, the used of salts in the molten salt synthesis \nmethod adopted by Park et al appears to assist the formation  of the separate magnetic phase, not \nsuspected by the authors.  That this is also dependent on the concentration of the BaTiO 3 is \nevident from the fact that the hexaferrite phase is formed even without the additives in \n0.9BiFeO 3-0.1BaTiO 3. This is in confo rmity with the M-H data reported by Singh et al for the \nsame composition [6]. The  M-H data in ref. [6] however did not show saturation, presumably \nbecause the data was colle cted up to a maximum field of 6 kOe only.  In view of our results, and \nthe varied m agnetic states reported by different research groups , it becomes evident that the \nprecipitation of the barium hexaferrite phase in the BiFeO 3-BaTiO 3 system is very sensitive to \nthe details of the synthesis conditions . A better understanding of the correlat ion between the \nsynthesis conditions and the formation of the hexaferrite phase wo uld enable better control of the \nvolume fraction of the barium hexaferrite phase and the overall magnetoelectric behaviour of this \nmaterial system.  \nB. Significance of spontaneo us precipitation of the magnetic phase  \nThe significant increase in the dielectric response with increasing magnetic field (shown \nin Figs. 6c and 6d) suggests that domain wall mobility is significantly enhanced by the external \nmagnetic field. In convention al Rayleigh analysis the dielectric coefficient increases with \nincreasing amplitude of the measuring field (within the sub -coercive region). This is due to the \nincreased ability of the domain walls to disentangle with the pinning centers. Since in our \napproach the electric field amplitude is fixed, the enhancement of the dielectric coefficient by the 9 \n \n9 \n external magnetic field proves that magnetic field enhances ferroelectric -ferroelastic domain \nswitching in the majority ferroelectric grains of our self -grown composite.  The enhancement in \nthe ferroelectric -ferroelastic switching with magnetic field is most likely to be associated with \nthe transfer of the magnetostrictive strain from the barium hexaferrite grains to the surrounding \nferroelectric grains. Our resu lts suggest that the magnetostrictive stress increasingly depins the \nferroelectric -ferroelastic domain walls in the ferroelectric grains. Our investigations have proved \nthat inspite of extrinsic origin of the ferromagnetism in this system, electric field c an \nsignificantly change the magnetization behaviour (the magnetic coercive field reduced by ~ 25 % \nafter electric poling), and magnetic field could increase the polarization by ~ 33 % (when the \nmagnetic field was increased to 10 kOe). These large changes s uggest a very good coupling \nbetween the ferroelectric and the ferromagnetic grains. At this point w e may caution that the \nelliptical loops observed in the P -E measurements in the subcoercive regime may as well be \nmimicked by leaky dielectrics. Such ellipti cal loops need not necessarily be associated with \ndomain switching phenomena unless the materials exhibit saturat ion polarization, characteristic \nof a normal ferroelectric material , well above the coercive field.   \nWhile unlike single phase multiferroic sys tems, wherein the  magnetoelectric coupling is \nfundamentally limited by the crystal and magnetic structure of the system, the magnetoelectric \ncoupling in ferroelectric -ferromagnetic composites is determined by how efficiently the str ain in \none phase is tran sferred to the other. In this context, t he quality of the contact between the \nferroelectric and the ferromagnetic grains is an important factor. When the ferromagnetic phase \nspontaneously precipitates out in the piezoelectric matrix system it is likely to form a smooth and \ncoherent interface with the neighboring piezoelectric grains, making stress /strain  transfer very \nefficient.  Moreover, in process involving spontaneous precipitat ion, the precipitated grain s are \nlikely to be homogeneously distributed over  the entire specimen instead of getting clustered as is \nlikely to be the case when the powders of the ferromagnetic phase is physically mixed with the \npiezoelectric powder to fabricate the composite . The fact that we could see a significant increase \nin the  polarization even with ~ 1 wt % of hexaferrite phase confirms the efficient coupling \nbetween the ferroelectric and the ferromagnetic grains in our self -grown composite . We also \nperformed magnetic  hysteresis experiment before and after electric poling of S ample 4, and \nfound the magnetic coercive field ( Hc) to decrease substantially from 2700 Oe to 2000 Oe , i.e by 10 \n \n10 \n ~ 26 %, Fig. 7. The value of the saturation magnetization was however not noticeably affected \nby electric poling.  This experiment proves that ele ctric poling has considerable influence on the \nmagnetization behaviour of the embedded barium hexaferrite grains.  \n \nC. Comparison with previous studies on in -situ grown ferroelectric -ferromagnetic \ncomposites  \n We may note that the idea of developing self -grow ing ferroelectric -ferromagnetic \ncomposites is not new. Run et al have reported in-situ grown BaTiO 3-CoFe 2O4 ferroelectric -\nferromagnetic eutectic composit e [40]. They reported the ferromagnetic component to be ~ 40 \nvolume percent and obtained dynamic magnet oelectric coupling coefficient ~ 50 mV/cm -Oe. \nMajumder and Bhattacharyya attempted to grow in situ  BaTiO 3-CoFe 2O4 composite but could \nachieve dynamic magnetoelectric coupling coefficient of ~ 5 mV/cm -Oe [41]. The low value is \nattributed to the formation of  other unwanted phases during high temperature synthesis. A \nsimilar value was reported by Duong et al who developed core -shell microstructure of BaTiO 3-\nCoFe 2O4 composite [42 ]. Since t he oxide ferrimagnets are also semiconducting in nature , their \npresence i n large quantity is likely to increase the leakage current of the composite specimen. It \nis therefore not surprising that most studies on bulk ferroelectric -ferromagnetic magnetoelectric \ncomposites do not show polarization -electri c field loop characteristi c of ferroelectric materials.  \nAccurate estimation of quasi dc -magnetoelectric coupling as determined in our study is therefore \nnot possible in such specimens.  Although , at present, we do not understand the mechanism \nassociated with the formation of barium hexaferrite phase in our system, the fact that additives \nsuch as CuO and MnO 2 in our case are able to do so, is an encouraging development. Our results  \noffer a hope that it might be possible to develop self-grown ferroelectric ferromagnetic \ncomposites in o ther systems as well by slight tweaking with the synthesis conditions , including \nadditives  which helps in controlled precipitation of a ferromagnetic phase . Such naturally grown \nferroelectric -ferromagnetic composites are expected to perform better than the  artificial \nfabricated composite due to reasons discussed above.   \n \n \n 11 \n \n11 \n V. Conclusion s \n In summary, we demonstrate that the ferromagnetic behaviour often reported in the \nBiFeO 3-BaTiO 3 multiferroic system arise due to precipitation of minor barium hexaferrite  phase \n(~ 1 wt %) often unnoticed in conventional characterization of the specimens.  The compositions \nexhibiting good ferroelectric and ferromagnetic switching are spontaneously formed ferroelectric \n–ferromagnetic 0 -3 particulate composites. We found that  inspite of the very insignificant \nfraction of the magnetic phase, this composite exhibit a significantly large change in polarization \non application of magnetic field. The dc magnetoelectric coupling was found to be 9 x 10-8 s/m, a \nvalue comparable to tha t reported in bi -layer ferroelectric -ferromagnetic thin films. We argue \nthat this extraordinary property arise due to the efficient transfer of the magnetostrictive strain \nfrom the embedded barium hexaferrite grains to the surrounding piezoelectric grains by virtue of \nthe smooth interphase boundaries, and preservation of the insulating nature of composite due to \nsmall fraction of the semiconducting phase. Our study provides a guideline to design natural 0 -3 \nferroelectric ferromagnetic with remarkably improv ed magnetoelectric coupling.  \n \nAcknowledgement:  \nOne of the authors (Amit Kumar)  would like to thank  UGC (New D elhi) for providing Dr D S \nKothari Post Doc Fellowship for this work.  R. Ranjan gratefully acknowledges the Nano Mission  \nProgramme of Department  of Science and Technology, Govt. of India for financial support \n(Grant No. SR/NM/NS -1010/2015 (G) ). Science and Engineering Research Board (SERB) of the \nMinistry of Science and Technology is also acknowledged for financial assistance (Grant no.  \nEMR/2016/0 01457 ).  \n \nReferences  \n[1]. W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759 (2006).  \n[2]. M. Fiebig, J. Phys. D. Appl. Phys. 38 R123 (2005)  \n[3]. J. Ma, J. Hu, Z. Li and C. -W. Nan, Adv. Mater. 23, 1062 (2011).  \n[4]. P. Mandal, M. J. Pitcher, J. A laria, H. Nui, P. Borisov, P. Stamenov, J. B. Claridge and M. J. \nRosseinsky, Nature 525, 363 (2015).  \n[5]. J. F. Scott, NPG Asia Materials, 5 e72 (2013); doi: 10.1038/am.2013.58  12 \n \n12 \n [6]. A. Singh, V. Pandey, R. K. Kotnala, and D. Pandey, Phys. Rev. Lett. 101, 2 47602 (2008).  \n[7] G. Catalan and J. F. Scott, Adv. Mater. 21, 2463 (2009).  \n[8]. J. Wang, J. B. 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Run, D. R. Terre;l, and J. H. Scholing, J. Mater. Sci. 9, 1710 (1974)  \n[41]. S. Majumder and G. S .  Bhattacharya, Ceram. Intern. 30, 389 (2004).  \n[42].  G.V. Duong, R Groessinger, and R.S. Turtelli, IEEE Trans Magn 42 3611 (2006) . \n \n \n \n \n \n \n \n \n \n 15 \n \n15 \n \n20 40 60 80(211)(210)(200)(111)(110)(100)\n  Intensity (arb. unit)  \nBragg angle, 2  deg. ) Sample-1\n Sample-2\n Sample-3\n Sample-4 \nFig. 1: X -ray powder diffraction patterns of 0.675BiFeO 3-0.325 BaTiO 3 synthesized with different \nCuO and MnO 2 additive s. Sample 1 correspond to 0 wt % CuO and 0 wt % MnO 2, Sample 2 \ncorrespond to 0.1 wt % CuO and 0.15 wt. % of MnO 2, Sample 3 correspond to 0.2 wt % CuO \nand 0.3 wt % MnO 2 and sample correspond to 0.4 wt % CuO and 0.6 wt % MnO 2.   \n \n 16 \n \n16 \n \n5 m\n5 m(b)\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00102030405060\n  No of grains\nGrain size ( m)\n0 2 4 6 801020304050\n  No of grains\nGrain size (m)(c)(a)\n(d) \nFigure 2: SEM images of (a) S ample -1 and (b) Sample -4. The histogram s of the grain size distributi on \nare shown on the right side of the respective SEM images  \n \n \n \n 17 \n \n17 \n \n-40 -20 0 20 40-40-30-20-10010203040\n-20 -10 0 10 20-0.6-0.4-0.20.00.20.40.6 Sample-4\n Sample-3\n Sample-2\n Sample-1\nM (emu/g)  P (C/cm2)\nElectric Field (kv/cm) Sample-4\n Sample-3\n Sample-2\n Sample-1\n   \nMagnetic Field (kOe)(a)(b) \n          \n Figure 3 (a) Polarization -electric field hysteresis loops of sample -1 to sample -4. The \nmagnetization –magnetic field hysteresis loops of the correspondin g samples are shown in (b).  \n \n 18 \n \n18 \n \n20 30 40 50 60\n-20 -10 0 10 20-60-40-200204060-20 -10 0 10 20-4-3-2-101234\nBaFe 12O19\nBragg's angle , 2(deg.)Perovskite Phase(a)\n+\n++ ++ (*)\n**** Intensity (a.u.)\n  \nBaFe 12O19\n*\n(+)+\n(c)\n M (emu/g)\nMagnetic Field (kOe)M (emu/g)\n   \n Magnetic Field (kOe)(b) \nFigure 4  (a) shows the XRD patterns of the powder collected on the magnetic bar. For sake of \nconvenience the Y axis is plotted on a logarithmic scale. The XRD pattern in the bottom panel of \n(a) is t hat of barium hexaferrite BaFe 12O19.  (b) shows the M-H loops of magnetically separated \npowder particles collected on the bar magnetic . (c) shows the M -H curve of the barium \nhexaferrite BaFe 12O19. \n 19 \n \n19 \n \n20 40 60 80 100\n30 35 40 45 50-20 -10 0 10 20-0.15-0.10-0.050.000.050.100.15\n-20 -10 0 10 20-4-3-2-101234\n-20 -10 0 10 20-60-40-200204060Intensity (a.u.)\nM (emu/g) M (emu/g)M (emu/g)\nBragg's angle, 2 (deg.)\n Intensity (a.u.)\nBragg's angle, 2 (deg.)(a)\n(b)Bi2Fe4O9\n \n \nMagnetic Field (kOe)Magnetic Field (kOe)\n   \nMagnetic Field (kOe)(c)\n(d)\n(e)\n  \n \n   \n ++\n****Pervoskite  Phase   (+)\nBaFe 12O19   \n  \nBaFe 12O19  (*)\n*+ \nFigure 5, (a) shows the XRD pat terns of 0.9BiFeO 3-0.1BaTiO 3. (b) shows the XRD patterns of \nthe powder particles attached on the bar magnet (top), and  barium hexaferrite (bottom). The Y-\nscale in (b) is shown on a logarithmic scale for easy visualization of the weak hexaferrite peaks. \n(c) –(e)  shows the M-H loop s of 0.9BiFeO 3-0.1BaTiO 3, magnetic rich phase, and BaFe 12O19, \nrespectively.  \n 20 \n \n20 \n \n-45 -30 -15 0 15 30 45-30-1501530\n0.0 0.2 0.4 0.6 0.8 1.02022242628\n-15-10 -5 051015-3-2-10123\n0.0 0.2 0.4 0.6 0.8 1.01416182022  P (C/cm2)\nElectric Field (kv/cm) 0.0 T\n 0.5 T\n 1.0 T(a)\n(c)\n  P (C/cm2)\nMagnetic Field (Tesla)(b)  P (C/cm2 )\nElectric Field (kv/cm) 0.0 T\n 0.5 T\n 1.0 T(d)\n  rx102)\nMagnetic Field (Tesla) \nFigure 6 (a) shows the polarization – electric field hysteresis loops of sample -4 in the presence of \nmagnetic field at room te mperature . The maximum polarization as a function of magnetic field is \nshown in (b). (c) shows the  (sub coercive electric field ) of sample -4 in the presence of magnetic \nfield at room temperature (c) Va riation of polarization in the presence of magnetic field, ( \nobserved by  part a), (d) variation of magneto electric coupling coefficient with magnetic field  \n(calculated by using part b), (e) variation of dielectric constant with magnetic field  (calculat ed \nby using part b)  \n 21 \n \n21 \n  \n \n \n \n \n \n \n \n-8000 -4000 0 4000 8000-0.6-0.30.00.30.6 Unpoled Pellet\n Poled @ 3 kV/mm\n \n  M(emu/g)\nMagnetic Field (Oe)\n \n \nFig. 7  Magnetization hysteresis curves of sample 4 before and after electric poling.  \n "    },    {        "title": "0908.1809v1.Evolution_of_critical_scaling_behavior_near_a_ferromagnetic_quantum_phase_transition.pdf",        "content": "arXiv:0908.1809v1  [cond-mat.str-el]  12 Aug 2009Evolution of critical scaling behavior near a ferromagneti c quantum phase transition\nN. P. Butch∗and M. B. Maple\nDepartment of Physics and Institute for Pure and Applied Phy sical Sciences,\nUniversity of California, San Diego, La Jolla, CA 92093\n(Dated: November 15, 2018)\nMagnetic critical scaling in URu 2−xRexSi2single crystals continuously evolves as the ferromag-\nnetic critical temperature is tuned towards zero via chemic al substitution. As the quantum phase\ntransition is approached, the critical exponents γand (δ−1) decrease to zero in tandem with\nthe critical temperature and ordered moment, while the expo nentβremains constant. This novel\ntrend distinguishes URu 2−xRexSi2from stoichiometric quantum critical ferromagnets and app ears\nto reflect an underlying competition between Kondo and ferro magnetic interactions.\nPACS numbers: 71.27.+a,75.40.Cx,75.60.Ej\nUnderstanding the quantum critical phenomena found\ninthevicinityofzero-temperaturephasetransitions[1, 2]\nremains a principal challenge in the study of correlated\nelectron materials such as high-temperature supercon-\nductors and heavy fermion compounds. In the vicin-\nity of these quantum phase transitions (QPTs), quantum\nmechanical fluctuations, as opposed to thermal fluctua-\ntions, can give rise to dramatic novel effects, including\nnon-Fermi liquid behavior, that are observed experimen-\ntally in numerous physical systems [3, 4]. In contrast\nto these well-documented phenomena, there are few ex-\nperimental characterizations of the critical scaling of the\norder parameter itself at QPTs. We report a study of\nmagnetic critical scaling in single crystals of the heavy\nfermion ferromagnet URu 2−xRexSi2, which exhibits an\nunprecedented trend as the magnetic ordering tempera-\nture is tuned towards zero via chemical substitution.\nRecently, the properties of ferromagnetic (FM) heavy\nfermion compounds at finite temperature and at QPTs\nhavebeen the subject ofmuch research. Among the best-\nknown examples are ZrZn 2and UGe 2, and for both, the\nCurie temperature TCcanbe suppressedbyapplied pres-\nsure, disappearing discontinuously at a first-order transi-\ntion [5, 6]. Therearefurther indications, both theoretical\n[7] and experimental [8] that FM QPTs are generally dis-\ncontinuous, though this matter is not yet settled. Heavy\nfermion ferromagnets are particularly complex, due to\nthe coexistence of FM and Kondo interactions, and un-\ncertainty whether light or heavy quasiparticles partici-\npate in the FM order.\nAlthough the heavy fermion compound URu 2Si2has\nachievednotorietyforitsunconventionalsuperconductiv-\nity and mysterious hidden order phase, it is less known\nthat the replacement of Ru with either Tc or Re yields\nheavy fermion ferromagnetism [9, 10, 11, 12]. The itin-\nerant FM order is characterized by a small moment\nand lacks discernable anomalies in electrical resistivity\nor specific heat at TC. In polycrystalline samples of\nURu2−xRexSi2, the maximum TC= 38 K occurs at x=\n0.80 [9], and TCdecreases to 0 K for x <0.35, although\nthe FM phase boundary in this range of concentrationshas not yet been accurately determined. Astoundingly,\nfar in the FM ordered phase, non-Fermi liquid (NFL)\nbehavior is observed in the low-temperature bulk prop-\nerties [13]. A recent neutron scattering study identified\nenergy-temperaturescaling in the dynamic magnetic sus-\nceptibility, which seems to be associated with the bulk\nNFL behavior [14]. In light of these provocativefindings,\nit is crucial to properly define the boundary of the FM\nphase and to confirm that the novel behavior observed in\npolycrystalline samples does not arise from orientational\naveraging of grains or the presence of defects.\nIn this study, we measured the bulk magnetization\nof single crystal samples of URu 2−xRexSi2. Systematic\nmeasurements on 0 .20≤x≤0.60 (10-30% Re) and a\nsubsequent magnetic scaling analysis lead to several ex-\nceptional results. First, the magnetic critical behavior,\ninwhichcriticalexponentsdecreasecontinuouslytowards\nzeroalongwith TC, isdissimilartothatofanyknownma-\nterials and unexplained by currently available theoretical\nmodels. Second, the previously identified NFL behavior\nis intimately related to the novel magnetic critical behav-\nior. Finally, the FM phase boundary persists to lower x\nthan previously believed, in close proximity to the hid-\nden order phase, suggesting that these two phases may\neven meet at a common critical point.\nSingle crystals of URu 2−xRexSi2were synthesized via\nthe Czochralskitechnique inatri-arcfurnaceandmagne-\ntizationmeasurementswereperformedinaQuantumDe-\nsignMPMS.Themoderatemagneticanisotropyobserved\nin URu 2Si2increases substantially upon Re substitution,\nranging from about 10 to over 100 at higher concentra-\ntions. Consequently, all measurements were performed\nwith the magnetic field applied along the [001] easy axis.\nDemagnetization corrections were less than 5%.\nRepresentative isotherms of the magnetization as a\nfunction of magnetic field M(H) are shown in Fig-\nure 1 for several different Re concentrations for which\nURu2−xRexSi2orders ferromagnetically. As the Re con-\ncentration increases from x= 0.20 tox= 0.60, sig-\nnatures of strengthening ferromagnetism are apparent:\nthe magnitude of the magnetization grows, the M(H)2\n00.02 0.04 0.06 0.08 0.1 \nx = 0.20 \n1.8K \n2.5K \n3.0K 3.5K \n10K URu 2-x Re xSi 2\n00.05 0.1 0.15 0.2 x = 0.35 \n2K \n5K \n6K 8K \n10K \n12K M (µB / fu)\n00.1 0.2 0.3 0.4 \n0 1 2 3 4 5 6x = 0.60 \n10K \n15K \n20K 25K \n30K \n35K \nH (T) -0.5 00.5 \n-6 0 6  x \n0\n0.15 \n0.20 \n0.25 \n0.30 \n0.35 \n0.50 \n0.60 T ≈ 2 K \nFIG. 1: Magnetization isotherms for several URu 2−xRexSi2\nsamples exhibiting ferromagnetism. As Re concentration in -\ncreases, the M(H) curves reflect an increased TCand mo-\nment, which is shown on the right in a direct comparison\nof low-temperature data across the entire range of xinvesti-\ngated. Data shown for x= 0.3 andx= 0.6 were measured at\n1.8 K and the other data were taken at 2.0 K.\ndata exhibit more curvature, and magnetic hysteresis de-\nvelops. A direct comparison of low-temperature M(H)\ncurves emphasizes the emergence of magnetic order with\nincreasing x, and the pronounced magnetic hysteresis at\nhigherRe concentration. The parentcompoundURu 2Si2\nexhibits linear paramagnetic M(H) in this field range,\nbut in URu 2−xRexSi2, the isotherms gently curve and do\nnot saturate. Despite the clear increase in magnetization\nwith Re concentration, even at x= 0.60, the moment\nis a small fraction of the high-temperature paramagnetic\neffective moment of approximately 3.8 µB.\nIn the absence of anomalies in electrical resistivity and\nspecific heat, the determination of TCat eachxis chal-\nlenging, but is essentialto the determinationofthe nomi-\nnalQPT.TheFM phaseboundaryofURu 2−xRexSi2was\noriginallydelineatedinpartbymean-fieldArrottanalysis\n[9], although it was noted that a modified Arrott analysis\nproduces unusual values of the scaling exponent δ[13].\nIn this study, we have applied a more general FM critical\nscaling approach based on the Arrott-Noakes equation of\nstate [15], which does not assume mean-field behavior.\nThis procedure yields values of TC, the ordered moment\nM0and the magnetic critical exponents β,γ, andδ. The\nthree exponents are typically defined for M,H, and re-\nduced temperature t=(T−TC)\nTCbyM∼tβfort <0,\nM∼H1/δfort= 0,∂M\n∂H=χ∼t−γfort >0, and\nδ−1 =γ/β. Physically, βdescribes how the ordered10 -2 10 -1 10 010 1\n10 -1 10 010 110 210 3M |t| -β\nβ = 0.8 \nγ = 0.18 \nδ = 1.23 URu 2-x Re xSi 2\nx = 0.20 10 -2 10 -1 10 0\n10 010 110 210 3\nH |t| -βδT < T \nc\nβ = 0.8 \nγ = 0.48 \nδ = 1.6 T > T \nc\nx = 0.35 10 010 1\n10 110 210 310 4β = 0.8 \nγ = 1.0 \nδ = 2.25 \nx = 0.60 \n00.02 0.04 0.06 \n0 5 10 91.8K \n2.0K \n2.5K \n3.0K M1/β (µB / fu)1/β\nx = 0.20 00.05 0.1 0.15 \n0 1000 2000   5K \n  6K \n  8K \n10K \n12K \n(H / M) 1/ γ (T fu / µB)1/ γx = 0.35 00.1 0.2 0.3 \n0 10 20 25K \n26K \n27K \n28K \n29K \n30K \nx = 0.60 \nFIG. 2: Arrott-Noakes scaling law plots and modified Arrott\nplots, vertically paired, for several ferromagnetic sampl es of\nURu2−xRexSi2. In the scaling plots, |M|/|t|βvsH/|t|δβ, the\nM(H) data collapse onto two curves for T > T CandT <\nTC. In the modified Arrott plots, |M|1/βvs|H/M|1/γ, the\nM(H) data are linearized and evenly spaced in temperature.\nEach pair of plots yields a consistent set of values of critic al\nexponents and TC.\nmoment grows below TC, with smaller values indicating\nfaster growth; γdescribes the divergence of the magnetic\nsusceptibility at TC, with smaller values yielding sharper\ndivergence; and δdescribesthe curvatureof M(H)atTC,\nwith smaller values reflecting less curvature and slower\nsaturation. The exponent δandTCare first estimated\nby identifying the isotherm with constant power-law be-\nhavior over the widest range of H. The entire analysis\nrequiresthe simultaneoussatisfactionofthreeconditions:\nscaling in |M|/|t|βvsH/|t|δβ, i.e., the collapse of M(H)\ndata onto 2 diverging curves, for t >0 andt <0; mod-\nified Arrott plots of |M|1/βvs|H/M|1/γ, i.e., linearized\nM(H) isothermsevenly spaced in t; and, crucially, a con-\nsistent value of TCbetween scaling, modified Arrott, and\npower-law fitting methods.\nFigure 2 exhibits the results of the scaling analysis for\ntheM(H) data shown in Fig. 1, which has been applied\nsuccessfully to samples with 0 .20≤x≤0.60. Plotting\nscaled magnetization |M|/|t|βvs scaled field H/|t|δβcol-\nlapses the M(H) data within the critical Trange onto\ntwo curves, corresponding to T > T CandT < T C. In\nthe modified Arrottplots, the M(H) isothermsaretrans-\nformed into parallel lines evenly spaced in Tin a plot of\n|M|1/βvs|H/M|1/γ, allowing a precise determination of3\nthe values of TC. The value of M0is estimated by ex-\ntrapolating the |M|1/βintercepts to T= 0. The value\nofTCforx= 0.20 approaches 1.8 K, the lowest temper-\nature accessible in this study. The consistency between\nthe data and the Arrott-Noakes equation demonstrates\nthe existence of FM order for 0 .20≤x≤0.60.\nThe evolution of the critical exponents, TC, andM0\nwith Re concentration is presented in Fig. 3. The x-\ndependence of the exponents is linear for all x, while a\nlinearx-dependence of TCandM0persists to x= 0.50.\nWhereas in most quantum critical systems only TCis\ntracked to zero, it is rather remarkable in URu 2−xRexSi2\nthatTC,M0,γand (δ−1) all extrapolate to 0 at\nx= 0.15±0.03. To our knowledge, this is the first\nexample of such behavior. In particular, the continu-\nous evolution of the exponents with xseems to violate\nthe tenet that one set of critical exponents controls the\nfinite-temperature transitions, while possibly another set\nof exponents describes the quantum transition [16]. The\npresently determined critical exponents deviate substan-\ntially from mean-field values ( β= 0.5,γ= 1,δ= 3) and\nthose describingclassicalFM systems, where β <0.5and\nδ >3. Moreover, even for x >0.5, where TC>10 K,\nthe exponents do not assume mean-field or classical val-\nues. Note that the curvature of the isotherms cannot\nbe accounted for by subtracting a paramagnetic compo-\nnent, such as a large heavy-electron Pauli susceptibility\nor a Curie-law susceptibility due to magnetic impurities,\nas these corrections lead to inconsistencies between the\nscaling and modified Arrott analyses.\nThe critical exponents describe the unusual evolution\nof FM order in URu 2−xRexSi2. As shown in Fig. 1, with\ndecreasing x, theM(H)isothermsexhibitdecreasingcur-\nvature, as δ→1. Meanwhile, the value of βremains con-\nstant, intimating that the T-dependence of the moment\nincrease is independent of the magnitude of TCandM0,\nas would typically be expected. However, β= 0.8 is un-\nusually large, suggesting a slower increase of the moment\nthan in classical or mean-field ferromagnets. Because\nβis constant, δ→1 constrains γ→0, which is quite\npeculiar. Typically, γ≥1, and physically, a suppres-\nsion ofγto 0 indicates increasingly sharper curvature in\nthe divergent susceptibility at TC, which can be inter-\npreted as a trend towards a first-order transition at the\nQPT. This notion conflicts with the expectation that the\nphase boundary should terminate at a higher-order crit-\nical endpoint. However, we stress that at this ostensible\nfirst-order endpoint, both TCandM0tend to 0, and in\nthis limit therewould be no discontinuityin the orderpa-\nrameter. A similar suppression is expected in the critical\nexponent associated with the correlation length.\nGenerally, itinerant FM transitions appear to be de-\nscribed well by mean field or classical theories, but there\nare few available experimental assessments of their criti-\ncal behavior near a QPT. Mean-field scaling is reported\nfor both chemically-tuned [17, 18, 19] and pressure-tuned00.5 11.5 22.5 \nURu 2-x Re xSi 2δ\nγ\nβ\nδ - 1 = γ / β\n0510 15 20 25 30 35 \n00.1 0.2 0.3 0.4 0.5 \n0 0.1 0.2 0.3 0.4 0.5 0.6 TC\nM0T (K) M0 (µB/fu) \nxHidden \nOrder Ferromagnetism Non Fermi Liquid \nNFL & \nFIG. 3: Concentration dependence of magnetic critical ex-\nponents and phase diagram of URu 2−xRexSi2. Error bars\ndenote the range of values that satisfy the scaling analysis .\nThe exponents δ−1 andγ,TC, andM0all extrapolate to\nzero near x= 0.15. The x-dependence of the exponents δand\nγis well-described by a linear fit, which also holds true for TC\nandM0forx≤0.5; the FM phase boundary drawn is a guide\nto the eye. The NFL regime, identified in measurements on\npolycrystals [13, 14], extends into the FM phase. The hidden\norder phase boundary follows Ref. [13].\nsystems [5, 20]. In contrast, significant deviation from\nclassical critical exponents has been documented mostly\nin chemical substitution studies [21, 22], and rarely in\nstoichiometric ferromagnets [23]. Nonetheless, the con-\ntinuous suppression of critical exponents is so far unique\nto URu 2−xRexSi2.\nOne important consideration is the possible interplay\nbetween magnetism and the heavy fermion state inferred\nfrom the large electronic specific heat measured in the\nFM phase of URu 2−xRexSi2. Based on the scaling anal-\nysis, a substantial heavy fermion paramagnetic contribu-\ntion toM(H) can be ruled out. This conclusion may\nimply that the itinerant FM moment must arise from a\nheavy quasiparticle band, although it is possible that the\nFM moment arisesfrom a lighter band and that the large\nspecific heat has an alternate source. In fact, both light\nand heavy band models of FM Kondo lattices predict a\nspecific heat enhancement [24, 25]. In either scenario, Re\nsubstitution regulates a delicate balance between Kondo\nand FM interactions.\nThe presence of fluctuations besides those of the or-\nder parameter are known to strongly influence critical\nscaling [26, 27]. Disorder due to Re substitution may\ngive rise to such fluctuations, but quantitative agreement4\nwith theoretical predictions for disordered itinerant fer-\nromagnetism [28] remains tentative. However, the now-\napparent proximity of the FM phase to the hidden order\nsuggeststhat the latter might exert influence overthe de-\nvelopmentoftheformer. Despite thefactthatthehidden\nordertransition can only be determined for x≤0.10[10],\nthe hidden order phase boundary extrapolates to 0 K\nnearx≈0.15, raising the interesting possibility of the\nexistenceofamulticritical point. Whether ornot the two\nordered phases meet, correlations of the hidden order pa-\nrameter persist into the FM phase. These have been ob-\nserved in neutron scattering, as antiferromagnetic corre-\nlations and magnetic excitations associated with the hid-\nden order phase are conspicuous for x= 0.20,x= 0.25,\nand even for x= 0.35 [14]. The effect of these corre-\nlations is uncertain, as the nature of the hidden order is\nstill under debate and it is not known whether long-range\nantiferromagnetism arises at intermediate Re concentra-\ntions, as it does in the Rh case [29].\nOur results offer insight into the origin of the unusual\nNFL behavior associated with the specific heat, occur-\nring in the ordered phase belowTC, a fact difficult to\nexplain in the context of a FM quantum critical point\noriginally identified at x= 0.30 [13], because the NFL\nbehavior is typically expected only outside the ordered\nphase [16]. Based upon the present work, the 0 K insta-\nbility is identified at x≈0.15, such that NFL behavior\nexists only inside the FM phase, pointing to an alterna-\ntive explanation. There is a notable agreement between\nthe values of the critical exponent γdetermined here and\nexponents derived from power-law fits of the bulk mag-\nnetic susceptibility of polycrystalline samples, χ∼T−n\n(0.20≤x≤0.35, 2 K< T <10 K) [13, 30]. This un-\nconventional T-dependence, originally attributed to NFL\nbehavior due to a Griffiths phase [13, 14], seems to ac-\ntually reflect the unconventional critical scaling of the\nmagnetic susceptibility, and strongly insinuates that the\nremarkable NFL behavior and critical exponents share\na common origin. Both NFL behavior [31] and FM or-\nder [24] arise in unconventional Kondo models, imply-\ning that in URu 2−xRexSi2, these seemingly exclusive ef-\nfects have a mutual origin in a novel Kondo scheme, in\nwhich Re substitution tunes the Kondo coupling between\nconduction electrons and uranium local moments. The\nproximity of the hidden order and FM phases raises the\npossibility that tuning of Kondo coupling may have sim-\nilar influence over the stability of the inscrutable hidden\norder phase.\nThe extraordinary evolution of critical behavior in\nURu2−xRexSi2underscores that the suppression of FM\norder is not universal, depending instead on details of\ncompeting interactions. However, similar trends in crit-\nical exponents might be found in other FM systems ex-\nhibiting a smooth suppression of TC. Their investiga-\ntion will help clarify the development of criticality near\ncontinuously-tuned FM QPTs that now defies conven-tional understanding.\nWethank B.T.Yukichforexperimentalassistanceand\nD. Belitz, S. Francoual, J. R. Jeffries, T. R. Kirkpatrick,\nJ. Paglione,Q. Si, and S. J. Yamamotofor helpful discus-\nsions. This work was supported by the U.S. Department\nof Energy (DOE) under Research Grant # DE-FG02-\n04ER46105 and the National Science Foundation under\nGrant No. 0802478.\n∗Present address: Center for Nanophysics and Advanced\nMaterials, University of Maryland, College Park, MD\n20742; Electronic address: nbutch@umd.edu\n[1] S. Sachdev, Science 288, 475 (2000).\n[2] P. Coleman and A. J. Schofield, Nature 433, 226 (2005).\n[3] M. B. Maple, et al., J. Low Temp. Phys. 99, 223 (1995).\n[4] H. v. L¨ ohneysen, A. Rosch, M. Vojta, andP. W¨ olfle, Rev.\nMod. Phys. 79, 1015 (2007).\n[5] M. Uhlarz, C. Pfleiderer, and S. M. Hayden, Phys. Rev.\nLett.93, 256404 (2004).\n[6] C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett. 89,\n147005 (2002).\n[7] A. V. Chubukov, C. P´ epin, and J. Rech, Phys. Rev. Lett.\n92, 147003 (2004).\n[8] Y. J. Uemura, et al., Nature Phys. 3, 29 (2007).\n[9] Y. Dalichaouch, M. B. Maple, M. S. Torikachvili, and A.\nL. Giorgi, Phys. Rev. B 39, 2423 (1989).\n[10] Y. Dalichaouch, et al., Phys. Rev. B 41, 1829 (1990).\n[11] M. S. Torikachvili, et al., Phys. Rev. B 45, 2262 (1992).\n[12] Y. Kohori, et al., Physica B 186–188 , 792 (1993).\n[13] E. D. Bauer, et al., Phys. Rev. Lett. 94, 046401 (2005).\n[14] V. V. Krishnamurthy, et al., Phys. Rev. B 78, 024413\n(2008).\n[15] A. Arrott, and J. E. Noakes, Phys. Rev. Lett. 19, 786\n(1967).\n[16] J. A. Hertz, Phys. Rev. B 14, 1165–1184 (1976).\n[17] K. Yoshimura and Y. Nakamura, Solid State Commun.\n56, 767 (1985).\n[18] D. A. Sokolov, M. C. Aronson, W. Gannon, and Z. Fisk,\nPhys. Rev. Lett. 96, 116404 (2006).\n[19] N. T. Huy, et al., Phys. Rev. Lett. 99, 067006 (2007).\n[20] T. Akazawa, et al., J. Phys. Soc. Jpn. 73, 3129 (2004).\n[21] S. Ubaid-Kassis, and A.Schroeder, Physica B 403, 1325\n(2008).\n[22] Y. Itoh, T. Mizoguchi, and K. Yoshimura, J. Phys. Soc.\nJpn.77, 123702 (2008).\n[23] T. Kida, et al., Europhys. Lett. 84, 27004 (2008).\n[24] N. B. Perkins, J. R. Iglesias, M. D. N´ u˜ nez-Regueiro, a nd\nB. Coqblin, Europhys. Lett. 79, 57006 (2007).\n[25] S. J. Yamamoto and Q. Si, cond-mat/0812.0819v2 (un-\npublished).\n[26] A. S. Ferreira and M. A. Continentino, J. Stat. Mech.\nP05005 (2005).\n[27] D. Belitz, T. R. Kirkpatrick, and T. Vojta, Rev. Mod.\nPhys.77, 579 (2005).\n[28] D. Belitz, T. R. Kirkpatrick, M. T. Mercaldo, and S. L.\nSessions, Phys. Rev. B 63, 174428 (2001).\n[29] M. Yokoyama, et al., J. Phys. Soc. Jpn. 73, 545 (2004).\n[30] E. D. Bauer, Ph.D. dissertation, University of Califor nia,\nSan Diego, 2002.5\n[31] I. Affleck, J. Phys. Soc. Jpn. 74, 59 (2005)."    },    {        "title": "0807.3564v1.Anisotropic_tunneling_magnetoresistance_and_tunneling_anisotropic_magnetoresistance__spin_orbit_coupling_in_magnetic_tunnel_junctions.pdf",        "content": "arXiv:0807.3564v1  [cond-mat.mes-hall]  22 Jul 2008Anisotropic tunneling magnetoresistance and tunneling an isotropic\nmagnetoresistance: spin-orbit coupling in magnetic tunne l junctions\nA. Matos-Abiague and J. Fabian\nInstitute for Theoretical Physics, University of Regensbu rg, 93040 Regensburg, Germany\n(Dated: July 5, 2021)\nThe effects of the spin-orbit interaction on the tunneling ma gnetoresistance of ferromag-\nnet/semiconductor/normal metal tunnel junctions are inve stigated. Analytical expressions for the\ntunneling anisotropic magnetoresistance (TAMR) are deriv ed within an approximation in which the\ndependence of the magnetoresistance on the magnetization o rientation in the ferromagnet originates\nfrom the interference between Bychkov-Rashba and Dresselh aus spin-orbit couplings that appear at\njunction interfaces and in the tunneling region. We also inv estigate the transport properties of ferro-\nmagnet/semiconductor/ferromagnet tunneljunctions ands howthatin suchstructuresthe spin-orbit\ninteraction leads not only to the TAMR effect but also to the an isotropy of the conventional tun-\nneling magnetoresistance (TMR). The resulting anisotropi c tunneling magnetoresistance (ATMR)\ndepends on the absolute magnetization directions in the fer romagnets. Within the proposed model,\ndepending on the magnetization directions in the ferromagn ets, the interplay of Bychkov-Rashba\nand Dresselhaus spin-orbit couplings produces differences between the rates of transmitted and re-\nflected spins at the ferromagnet/seminconductor interface s, which results in an anisotropic local\ndensity of states at the Fermi surface and in the TAMR and ATMR effects. Model calculations for\nFe/GaAs/Fe tunnel junctions are presented. Furthermore, b ased on rather general symmetry con-\nsiderations, we deduce the form of the magnetoresistance de pendence on the absolute orientations\nof the magnetizations in the ferromagnets.\nPACS numbers: 73.43.Jn, 72.25.Dc, 73.43.Qt\nI. INTRODUCTION\nThe tunneling magnetoresistance (TMR) effect is\nobserved in ferromagnet/insulator/ferromagnet hetero-\njunctions, in which the magnetoresistance exhibits a\nstrong dependence on the relative magnetization direc-\ntions in the two ferromagnetic layers and on their spin\npolarizations.1,2,3,4,5Because of this peculiarly strong\nasymmetric behavior of the magnetoresistance, TMR de-\nvices find multiple uses ranging from magnetic sensors to\nmagnetic random access memory applications.2,4\nBeyond the conventional TMR effect, it has been\nobserved that the magnetoresistance in magnetic tun-\nnel junctions (MTJs) may also depend on the abso-\nluteorientation of the magnetizations in the ferromag-\nneticleads.6,7,8,9Thisphenomenoniscalledthetunneling\nanisotropic magnetoresistance (TAMR) effect.6,9A the-\noretical investigation of the tunneling magnetoresistance\nin GaMnAs/GaAlAs/GaMnAs tunnel junctions was re-\nported in Ref. 9 which predicted that, as a result of\nthe strong spin-orbit interaction the tunneling magne-\ntoresistance depends on the angle between the current\nflowdirectionandthe orientationoftheelectrodemagne-\ntization. A difference between the tunneling magnetore-\nsistances in the in-plane (i.e., magnetization in the plane\nofthe magnetic layers)and out-of-planeconfigurationsof\nup to 6% was predicted for large values of the electrode\nspin polarization.9Here we refer to this phenomenon as\ntheout-of-plane TAMR. Recent first principles calcula-\ntions in Fe/MgO/Fe magnetic tunnel junctions (MTJs)\npredict an out-of-plane TAMR ratio of about 44 %.10On\nthe other hand, we refer to an in-plane TAMR effect asthe change in the magnetoresistance when the in-plane\nmagnetization of the ferromagnetic layer(s) is rotated in\nthe plane perpendicular to the direction of the current\nflow.\nIt is remarkable that the TAMR is present even in\nMTJs in which only one ofthe electrodes is magnetic and\nthe conventional TMR is absent. In contrast to the con-\nventional TMR-based devices, which require two mag-\nnetic layersfor their operation, TAMR-based devices can\noperate with a single magnetic lead, opening new possi-\nbilities and functionalities for the operation of spintronic\ndevices. The TAMR may also affect the spin-injection\nfrom a ferromagnet into a non-magnetic semiconductor.\nTherefore, in order to correctly interpret the results of\nspin injection experiments in a spin-valve configuration,\nit is essential to understand the nature, properties, and\norigin of the TAMR effect.\nThe first experimental observation of TAMR\nwas in (Ga,Mn)As/AlOx/Au heterojunctions, in\nwhich an in-plane TAMR ratio of about 2 .7% was\nfound.6Experimental investigations of the in-plane\nTAMR in (Ga,Mn)As/GaAs/(Ga,Mn)As and in\n(Ga,Mn)As/ZnSe/(Ga,Mn)As tunnel junctions, in\nwhich both electrodes are ferromagnetic have also been\nreported.7,8In the case of(Ga,Mn)As/ZnSe/(Ga,Mn)As,\nthe in-plane TAMR ratio was found to decrease with\nincreasing temperature, from about 10% at 2 K to\n8.5% at 20 K.8This temperature dependence of\nthe in-plane TAMR is more dramatic in the case of\n(Ga,Mn)As/GaAs/(Ga,Mn)As, for which a TAMR\nratio of the order of a few hundred percent at 4 K was\namplified to 150 000% at 1.7 K.7This huge amplification\nof the in-plane TAMR was suggested to originate from2\nthe opening of the Efros-Shklovskii gap11at the Fermi\nenergy when crossing the metal-insulator transition.7\nMeasurements of the TAMR in p+−(Ga,Mn)As/n+-\nGaAs Esaki diode devices have also been reported.12\nIn addition to the investigations involving vertical\ntunneling devices the TAMR has also been studied\nin break junctions,13,14nanoconstrictions12,15and\nnanocontacts.16\nBeyond the area of currently low Curie temper-\nature ferromagnetic semiconductors, the TAMR\nhas recently been experimentally investigated in\nFe(001)/vacuum/bcc-Cu(001) tunnel junctions,17\nFe/GaAs/Au MTJs,5,18Co/AlO x/Au MTJs,19, CoPt\nstructures20and in multilayer-(Co/Pt)/AlO x/Pt\nstructures.21\nIn what follows we focus our discussion on the case\nof the in-plane TAMR (for brevity we will refer to\nit as the TAMR effect). We investigate the TAMR\nin ferromagnet/semiconductor/normal metal (F/S/NM)\nandinferromagnet/semiconductor/ferromagnet(F/S/F)\nMTJs. We propose a model in which the two-fold sym-\nmetric magnetoresistance dependence on the orienta-\ntion of the in-plane magnetization in the ferromagnetic\nlayer(s) originates from the interference of Dresselhaus\nand Bychkov-Rashba-like spin-orbit couplings. Such in-\nterference effects have already been investigated in lat-\neral transport in 2D electron systems22,23,24, in spin re-\nlaxation in quantum wells25and quantum dots26, or in\n2D plasmons.27The symmetry, which is imprinted in the\ntunneling probabilitybecomesapparentwhen amagnetic\nmoment is present. Our main results are: i) finding an-\nalytical expressions for evaluating the TAMR in both\nF/S/NM and F/S/F MTJs, ii) prediction and evaluation\nof the ATMR in F/S/F heterojunctions, and iii) deriva-\ntion of a simple phenomenological relation describing the\ndependence of the tunneling magnetoresistance on the\nabsolute orientation of the magnetization(s) of the ferro-\nmagnet(s).\nThe paper is organizedas follows. In Sec. II we present\nthe theoretical model describing the tunneling through a\nMTJ. In a first approximation we consider the case of\nan infinitesimally thin barrier (Sec. IIA), while the finite\nspatial extension of the potential barrier is incorporated\nin a more sophisticated approach discussed in Sec. IIB.\nDetailed solutions and tunneling properties within these\napproximationsaregivenin Appendices Aand B, respec-\ntively. In Sec. III we discuss the TAMR in both F/S/NM\n(Sec. IIIA) and F/S/F (Sec. IIIB) MTJs. The ATMR in\nF/S/F tunnel junctions is investigated in Sec. IV, where\nspecific calculations for model Fe/GaAs/Fe MTJs are\npresented. In Sec. V we develop a phenological model for\ndetermining the dependence ofthe TAMR andATMR on\nthe absolute orientation(s) of the magnetization(s) in the\nferromagnetic lead(s). Finally, conclusions are given in\nSec. VI.II. THEORETICAL MODEL.\nConsider a F/S/F tunnel heterojunction. The semi-\nconductor is assumed to lack bulk inversion symmetry\n(zinc-blende semiconductors are typical examples). The\nbulk inversion asymmetry of the semiconductor together\nwith the structure inversion asymmetry (for the case\nof asymmetric junctions) of the heterojunction give rise\nto the Dresselhaus5,28,29,30and Bychkov-Rashba5,30,31\nSOIs, respectively. The interference of these two spin-\norbit couplings leads to a net anisotropic SOI with a C2v\nsymmetry which is imprinted onto the tunneling mag-\nnetoresistance as the electrons pass through the semi-\nconductor barrier. This was discussed in some details\nin Refs. 5,18 for the case of F/S/NM tunnel junctions.\nHere we generalize the model proposed in Refs. 5,18 to\nthe case of F/S/F tunnel junctions. For such structures\nour model predicts the coexistence of both the TAMR\nand ATMR phenomena.\nWe consider a F/S/F tunnel junction grown in the\nz= [001] direction, where the semiconductor forms\na barrier of width dbetween the left and right ferro-\nmagnetic electrodes. At first we discuss a simplified\nmodel for very thin barriers. In that case the bar-\nrier can be approximated by a Dirac delta function and\nthe SOI reduced to the plane of the barrier. In what\nfollows we will refer to this model as the Dirac delta\nmodel (DDM). A second model in which Slonczewski’s\nproposal3,5for ferromagnet/insulator/ferromagnet tun-\nnel junctions is generalized to the case of ferromag-\nnet/semiconductor/ferromagnet junctions by including\nthe Bychkov-Rashba and Dresselhaus SOIs will be re-\nferred to as the Slonczewski spin-orbit model (SSOM).\nA. Dirac delta model (DDM)\nWe consider here the case of a very thin tunneling bar-\nrier. Assuming that the in-plane wave vector k/bardblis con-\nserved throughout the heterostructure, one can decouple\nthe motion along the growth direction ( z) from the other\nspatial degrees of freedom. The effective model Hamilto-\nnian describing the tunneling across the heterojunction\nreads\nH=H0+HZ+HSO. (1)\nHere\nH0=−/planckover2pi12\n2m0d2\ndz2+V0dδ(z), (2)\nwithm0the bare electron mass, and V0anddthe high\nand width, respectively, of the actual potential barrier\n[here modelled with a Dirac delta function δ(z)] along\nthe growth direction ( z= [001]) of the heterostructure.\nThe spin splitting due to the exchange field in the left\n(z <0) and right ( z >0) ferromagnetic regions is given3\nby\nHZ=−Θ(−z)∆l\n2nl·σ−Θ(z)∆r\n2nr·σ.(3)\nHere ∆ land ∆ rrepresent the exchange energy in the\nleft and right ferromagnets, respectively, and Θ( z) is the\nHeaviside step function. The components of the vector\nσare the Pauli matrices, and nj= (cosθj,sinθj,0) with\nj=l,ris a unit vector defining the in-plane magneti-\nzation direction in the left ( j=l) and right ( j=r)\nferromagnets with respect to the [100] crystallographic\ndirection. The Zeeman splitting in the semiconductor\ncan be neglected.\nWe note that in recent experiments with Fe/GaAs/Au\ntunnel junctions,18the reference axis was taken as the\n[110] direction. Therefore, it is convenient to express\nthe magnetization direction relative to the [110] axis by\nintroducing the angle shifting φj=θj−π/4 (j=l,r).\nOne can then write nj= [cos(φj+π/4),sin(φj+π/4),0]\nwithφjgiving the magnetization direction in the left\n(j=l) and right ( j=r) ferromagnets with respect to\nthe [110] crystallographic direction.\nWithin the DDM, the spin-orbit interaction through-\nout the semiconductor barrier (including the interfaces)\ncan be written as\nHSO= (w·σ)δ(z), (4)\nwith the effective spin-orbit coupling field\nw= (−¯αky+ ¯γkx,¯αkx−¯γky,0). (5)\nHere ¯αand ¯γrepresent effective values of the Bychkov-\nRashba and linearized Dresselhaus parameters, respec-\ntively, and kxandkyrefer to the xandycomponents\nof the wave vector k. In terms of the usual Dresselhaus\nparameter γ, the linearized Dresselhaus parameter can\nbe approximated5as ¯γ≈γQ, where Q= 2m0V0d//planckover2pi12\nstands for the strength of the effective wave vector in the\nbarrier.\nThe scattering states in the left ( z <0) and right ( z >\n0) ferromagnetic regions are given by\nΨ(l)\nσ=eikσzχ(l)\nσ√kσ+rσ,σe−ikσzχ(l)\nσ+rσ,−σe−ik−σzχ(l)\n−σ,(6)\nand\nΨ(r)\nσ=tσ,σeiκσzχ(r)\nσ+tσ,−σeiκ−σzχ(r)\n−σ,(7)\nrespectively. Here we have introduced the wave vector\ncomponents\nkσ=/radicalBigg\n2m0\n/planckover2pi12/parenleftbigg\nE+σ∆l\n2/parenrightbigg\n−k2\n/bardbl, (8)\nand\nκσ=/radicalBigg\n2m0\n/planckover2pi12/parenleftbigg\nE+σ∆r\n2/parenrightbigg\n−k2\n/bardbl, (9)withk/bardbl=/radicalBig\nk2x+k2ydenoting the length of the wave\nvector component corresponding to the free motion in\nthex−yplane. The spinors\nχ(j)\nσ=1√\n2/parenleftbigg1\nσei(φj+π/4)/parenrightbigg\n(j=l,r),(10)\ncorrespond to a spin parallel ( σ=↑) or antiparallel\n(σ=↓) to the magnetization direction nj= [cos(φj+\nπ/4),sin(φj+π/4),0] in the left ( j=l) and right ( j=r)\nferromagnets.\nThe reflection and transmission coefficients can be\nfound by imposing appropriate boundary conditions and\nsolving the corresponding system of linear equations (for\ndetails see Appendix A). The transmissivity of an in-\ncoming spin- σparticle can then be evaluated from the\nrelation\nTσ(E,k/bardbl) = Re[κσ(|tσ,σ|2+κ−σ|tσ,−σ|2)].(11)\nExplicit analytical expressions for the transmission coef-\nficients (tσ,σandtσ,−σ) are given in Appendix A.\nB. Slonczewski spin-orbit model (SSOM)\nWe give a generalization of the Slonczewski model3\nfor ferromagnet/insulator/ferromagnet tunnel junctions\nto the case in which the insulator barrier is replaced by\na zinc-blende semiconductor. Unlike in the DDM, now\nthe spatial extension of the potential barrieris taken into\naccount. The model Hamiltonian is\nH=H0+HZ+HBR+HD, (12)\nwhere\nH0=−/planckover2pi12\n2∇/bracketleftbigg1\nm(z)∇/bracketrightbigg\n+V0Θ(z)Θ(d−z).(13)\nThe electron effective mass m(z) is assumed to be m=\nmcin the central (semiconductor) region and m=m0\nin the ferromagnets. The exchange splitting in the ferro-\nmagnets is now given by\nHZ=−Θ(−z)∆l\n2nl·σ−Θ(z−d)∆r\n2nr·σ.(14)\nThe Dresselhaus SOI can be written as5,29,30,32,33\nHD= (kxσx−kyσy)∂\n∂z/parenleftbigg\nγ(z)∂\n∂z/parenrightbigg\n,(15)\nwherexandycorrespond to the [100] and [010] direc-\ntions,respectively. TheDresselhausparameter γ(z)hasa\nfinite value γin thesemiconductorregion,wherethe bulk\ninversion asymmetry is present, and vanishes elsewhere.\nNote that because of the step-like spatial dependence of\nγ(z), the Dresselhaus SOI [Eq. (15)] implicitly includes\nboth the interface and bulk contributions.5,29,334\nThe Bychkov-Rashba SOI is given by34\nHBR= [αlδ(z−zl)−αrδ(z−zr)](kxσy−kyσx),(16)\nand arises due to the ferromagnet/semiconductor inter-\nface inversion asymmetry.5Hereαl(αr) denotes the SOI\nstrength at the left (right) interface zl= 0 (zr=d).\nFor the small voltages considered here (up to a hundred\nmV), the Bychkov-Rashba SOI inside the semiconductor\ncan be neglected.\nThez-components of the scattering states in the left\nand right ferromagnets have the same form as in Eqs. (6)\nand (7), respectively.\nIn the central (semiconductor) region (0 < z < d) we\nhave\nΨ(c)\nσ=/summationdisplay\ni=±(Aσ,ieqiz+Bσ,ie−qiz)χ(c)\ni,(17)\nχ(c)\n±=1√\n2/parenleftbigg1\n±eiξ/parenrightbigg\n. (18)\nThe angle ξis defined through the relation tan( ξ) =\n−ky/kx. We have also used the notation\nq±=q0/radicalbigg\n1∓/parenleftBig2mcγk/bardbl\n/planckover2pi12/parenrightBig2, (19)\nwhere\nq0=/radicalbigg\n2mc(V0−E)\n/planckover2pi12+k2\n/bardbl, (20)\nis the length of the zcomponent of the wave vector in\nthe barrier in the absence of SOI.\nThe expansion coefficients in Eqs. (6), (7), and (17 can\nbe found by applying appropriatematching conditions at\neachinterfaceandbysolvingthe correspondingsystemof\nlinear equations (for details, see Appendix B). Once the\nwave function is determined, the particle transmissivity\ncan be calculated from Eq. (11). Approximate analyti-\ncal expressions for the transmission coefficients tσ,σand\ntσ,−σare given in Appendix B.\nIII. TUNNELING ANISOTROPIC\nMAGNETORESISTANCE (TAMR)\nThe magnetoresistance of a tunnel junction can be ob-\ntained by evaluating the current through the device or\nthe conductance. The current flowing along the hetero-\njunction is given by\nI=e\n(2π)3/planckover2pi1/summationdisplay\nσ=↑,↓/integraldisplay\ndEd2k/bardblTσ(E,k/bardbl)[fl(E)−fr(E)],\n(21)\nwherefl(E) andfr(E) are Fermi-Dirac distributions\nwith chemical potentials µlandµrin the left and rightEFV0\nFe Au GaAs \nz zlzry\nx\nzn\nφFe GaAs Au \nzlzr[001][110] [-110]\n[110] φn[-110] \n[110] ( ) (0) TAMR (0) R R \nRφ−=(a) (b) \n(c) \nFIG.1: (color online). (a)SchematicsofaFe/GaAs/AuMTJ.\nThe magnetization direction in the ferromagnet is specified\nby the vector n. (b) Schematics of the potential profile of the\nheterojunction along the [001] direction. (c) Definition of the\nTAMR in F/S/NM junctions.\n(metallic or ferromagnetic) leads, respectively. For the\ncase of zero temperature and small voltages, the Fermi-\nDirac distributions can be expanded in powers of the\nvoltageVbias. To first order in Vbiasone obtains fl(E)−\nfr(E)≈δ(E−EF)Vbiaswithδ(x) the Dirac delta func-\ntion and EFthe Fermi energy. One then obtains the\nfollowing approximate expression for the conductance\nG=/summationdisplay\nσ=↑,↓Gσ, Gσ=e2\n(2π)3/planckover2pi1/integraldisplay\nd2k/bardblTσ(EF,k/bardbl).(22)\nWe note that although similar, the expression above\ndiffers from the linear response conductance. In our\ncase, the transmissivity Tσ(EF,k/bardbl) depends on the\nBychkov-Rashba parameter ¯ α. Recent first-principles\ncalculations35have shown that the spin-orbit coupling\nfield is different for different bands, therefore the effec-\ntive value of ¯ αis energy dependent. By applying an ex-\nternal voltage the energy window relevant for tunneling\ncan be changed, resulting in voltage-dependent values of\n¯α. Consequently, the conductance in Eq. (22) depends,\nparametrically, on the applied voltage.\nA. TAMR in ferromagnet/semiconductor/normal\nmetal tunnel junctions\nThe tunneling properties of F/S/NM junctions can be\nobtained as a limit case of the models proposed in Sec.\nII for F/S/F tunnel junctions by taking φl=φr=φand\n∆ras the Zeeman splitting in the normal metal region.\nIn the present case l,c, andrrefer to the ferromagnetic\n(left), semiconductor (central), and normal metal (right)\nregions, respectively.\nThe TAMR in F/S/NM tunnel junctions refers to the\nchanges of the tunneling magnetoresistance ( R) when5\nvarying the magnetization direction nlof the magnetic\nlayer with respect to a fixed axis. Here we assume the\n[110]crystallographicdirectionasthe referenceaxis. The\nTAMR is then given by5\nTAMR [110](φ) =R(φ)−R(0)\nR(0)=G(0)−G(φ)\nG(φ).(23)\nSince in a F/S/NM tunnel junction only one electrode is\nmagnetic, the conventional TMR effect is absent.\nAn alternative to the magnetoresistance, which refers\nto the chargetransport, is the spin polarization efficiency\nof the transmission characterized by the tunneling spin\npolarization2,4\nP=I↑−I↓\nI, (24)\nwhereIσis the charge current corresponding to the spin-\nσchannel and Iis the total current. The changes in\nthe tunneling spin polarization when the magnetization\nof the ferromagnet is rotated in-plane can then be char-\nacterized by the tunneling anisotropic spin polarization\n(TASP), which is defined as5\nTASP [110](φ) =P(0)−P(φ)\nP(φ). (25)\nTaking into account that the Zeeman splitting in\nthe normal metal is small we can approximate κσ≈\nκ−σ. Then for the DDM the conductance is given by\nEq. (A19). It follows from Eqs. (23) and Eq. (A19) that\nthe TAMR is given by\nTAMR [110](φ) =Ganiso(0)−Ganiso(φ)\nGiso+Ganiso(φ).(26)\nFor junctions in which the Bychkov-Rashba and Dressel-\nhaus SOIs can be considered as small perturbations, the\nanisotropy is small and Ganiso(φ)≪/Giso. In addition,\nthe spin-orbit contribution G′=G′\n↑+G′\n↓[see Eqs. (A8)-\n(A10)]) to the isotropic part of the conductance is also\nmuch smaller than the contribution G(0)=G(0)\n↑+G(0)\n↓,\ncorresponding to the system in the absence of the SOI,\ni.e.,G′≪G(0). Thereforeonecandropthe contributions\nG′andGanisofrom the denominator in Eq. (26). The\nTAMR can then be approximated as\nTAMR [110](φ)≈Ganiso(0)−Ganiso(φ)\nG(0).(27)\nThe substitution of Eq. (A17) into Eq. (27) leads to\nTAMR [110](φ)≈\ne2\nh/angbracketleftBig\ng2↑k2\n/bardbl/angbracketrightBig\n↑+/angbracketleftBig\ng2↓k2\n/bardbl/angbracketrightBig\n↓\nG(0)λαλγ[cos(2φ)−1].(28)\nHere we have introduced the dimensionless SOC param-\netersλα= 2m0¯α//planckover2pi12andλγ= 2m0¯γ//planckover2pi12. The functions\ng2↑andg2↓are given by Eq. (A18).kxky\n[110] [-110] \nkxky\nFIG. 2: (color online). (a) Schematics of the anisotropy of\nthe spin-orbit magnetic field w(k/bardbl). Thin arrows represent\na vector plot of the SOC field w. The solid line is a polar\nplot of the SOC field strength |w(k/bardbl)|for a fixed value of\nk/bardbl=|k/bardbl|. When the magnetization of the ferromagnet points\nalong the [-110] direction [see thick blue (dark) arrow], th e\ndirection of the strongest SOC field is parallel to the incide nt\nmajority spins which easily tunnel through the barrier. On\nthe contrary, for a magnetization direction [110] [see thic k\ngreen (light) arrow], the strongest SOC field is perpendicul ar\ntotheincidentspins andthetunnelingbecomes less favorab le.\nThe net result is a spin valve effect whose efficiency depends\nupon the absolute orientation of the magnetization and give s\nrise to the TAMR.\nThe expression above gives the angular dependence of\nthe TAMR and is consistent with the angular depen-\ndence experimentally observed in Fa/GaAs/Au tunnel\njunctions.18It also suggests that bias-induced changes\nof the sign of the Bychkov-Rashba parameter ¯ αresult\nin the inversion (change of sign) of the TAMR (such\nan inversion has been experimentally observed.18) Fur-\nthermore, one can see from Eq. (28) that the amplitude\nof the TAMR is governed by the product λαλγ∝¯α¯γ\nand the averages ∝angb∇acketleftg2σk2\n/bardbl∝angb∇acket∇ightσ(σ=↑,↓). When ¯ α¯γ= 0,\nthe two-fold TAMR is suppressed (the suppression of the\nTAMR was also observed in Ref.18), i.e., as long as other\nanisotropiceffects such as uniaxial strain are not present,\nthe Bychkov-Rashba (or Dresselhaus) SOI alone cannot\nexplaintheexperimentallyobserved C2vsymmetryofthe\nTAMR. The TAMR vanishes also if the spin polarization\nof both electrodes becomes sufficiently small. In such a\ncasekF,↑≈kF,↓andg2σvanishes [see Eq. (A18)], result-\ning in the suppression of the TAMR. On the contrary,\nEq. (28) predicts an increase of the TAMR amplitude\nfor F/S/NM tunnel junctions whose constituents exhibit\nlarge values of ¯ α¯γas well as a large spin polarization in\nthe magnetic electrode.\nA simple, intuitive explanation of the origin of the uni-\naxialanisotropyoftheTAMR canbe obtainedbyinvesti-\ngating the dependence of the effective spin-orbit coupling\nfieldw(k/bardbl) [see Eq. (5)], i.e., the effective magnetic field\nthat the spins feel when traversing the semiconducting6\nbarrier. A schematics of the anisotropy of the spin-orbit\nfieldw(k/bardbl) is shown in Fig. (2), where the thin arrows\nrepresent a vector plot of w(k/bardbl), while the solid line is a\npolar plot of the field amplitude |w(k/bardbl)|for a fixed value\nofk/bardbl=|k/bardbl|. The spin-orbit field is oriented in the [110]\n([-110]) direction at the points of low (high) spin-orbit\nfield, where the field amplitude |w|reaches a minimum\n(maximum). When the magnetization in the ferromag-\nnet points along the [-110] direction, the direction of the\nhighest spin-orbit field is parallel to the incident, major-\nity spins which are then easily transmitted through the\nbarrier. On the other hand, for a magnetizationdirection\n[110], the highest spin-orbit field is perpendicular to the\nincident spins (to both the majority and minority spins)\nand the transmission becomes less favorable than in the\ncase the magnetization is in the [-110] direction. This\nspin-orbit induced difference in the tunneling transmis-\nsivities depending on the magnetization direction results\nin the uniaxial anisotropy of the TAMR.36Furthermore,\nthe magnetization direction dependence of the transmis-\nsion and reflection of the incident spins should be re-\nflected in the local density of states at the interfaces of\nthe barrier. Within the DDM the left (F/S) and right\n(S/NM) interfaces are merged into a single plane and one\ncan not distinguish between them. A more detailed view\nof the role of the interfaces requires the use of the SSOM.\nIt turns out (this will be shownlatterin this section) that\nthe F/S interface plays a major role in the TAMR phe-\nnomenon while the S/NM interface appears irrelevant.\nThis is intuitively expected since the exchange splitting\nin the ferromagnet is much larger than the Zeeman split-\nting in the normal metal. Consequently, the spin-valve\neffect at the F/S interface is much stronger than in the\nS/NM interface.\nThe local density of states reflect also the uniaxial\nanisotropy of the TAMR with respect to the magneti-\nzation orientation in the ferromagnet. In fact, one can\nintroduce the anisotropic, local density of states (AL-\nDOS) through the definition\nALDOS [110](z,φ) =LDOS(z,0)−LDOS(z,φ)\nLDOS(z,φ),(29)\nwhere\nLDOS(z,φ) =/summationdisplay\nσ=↑,↓/integraldisplaydk/bardbl\n(2π)2|Ψσ(z,φ,kσF)|2,(30)\nis the total local density of states at position zand evalu-\nated at the Fermi surface determined by the Fermi wave\nvectors\nkσF=/radicalBigg\n2m0\n/planckover2pi12/parenleftbigg\nEF+σ∆l\n2/parenrightbigg\n−k2\n/bardbl.(31)\nSince we are interested only in propagating states, we\nmay restrict the possible values of k/bardblto the interval\n[0,kσ\nmax], withkσ\nmaxgiven by Eq. (A16). Since the spin-40 -20 0 20 40 -40 -20 020 40 \nαl (eV Å 2)αr (eV Å 2)\n0.9950 \n0.9970 \n0.9990 \n1.001 \n1.003 \n1.005 R[-110]  / R [110] \nFIG. 3: (color online). Values of the ratio R[¯110]/R[110]as a\nfunction of the interface Bychkov-Rashba parameters αland\nαr.\nsplitting in the normal metal region is negligibly small,\nκσ≈κ−σ. It follows from Eqs. (7) and (11) that\nTσ(E,k/bardbl)∝ |Ψ(r)\nσ|2, (32)\nand, therefore, the conductance [see Eq. (22)] is related\nto the LDOS at z=dasGσ(φ)∝LDOS(d,φ). One then\nobtains that\nTAMR [110](φ)≈ALDOS [110](z=d,φ).(33)\nFor a numerical illustration we consider an epitaxial\nFe/GaAs/Au heterojunction similar to that used in the\nexperimental observations reported in Ref. 18. We use\nthe value mc= 0.067m0for the electron effective mass\nin the central (GaAs) region. The barrier width and\nhight(measuredfromthe Fermienergy)are, respectively,\nd= 80˚A andVc= 0.75 eV, corresponding to the ex-\nperimental samples in Ref. 18. For the Fe layer a Stoner\nmodel with the majority and minority spin channels hav-\ning Fermi momenta kF↑= 1.05×108cm−1andkF↓=\n0.44×108cm−1,37respectively, is assumed. The Fermi\nmomentum in Au is taken as κF= 1.2×108cm−1.38\nWe consider the case of relatively weak magnetic fields\n(specifically, B= 0.5 T). At high magnetic fields, say,\nseveral Tesla, our model is invalid as it does not in-\nclude cyclotron effects which become relevant when the\ncyclotron radius approaches the barrier width.\nThe Dresselhaus spin-orbit parameter in GaAs is γ≈\n24 eV˚A3.5,30,32On the other hand, the values of the in-\nterface Bychkov-Rashba parameters αl,αr[see Eq. (16)]\nare not know for metal-semiconductor interfaces. Due\nto the complexity of the problem, a theoretical estima-\ntion of such parameters requires first-principles calcula-\ntions including the band structure details of the involved\nmaterials,35which is beyond the scope of the present pa-\nper. Here we assume αlandαras phenomenological pa-\nrameters which must be understood as the values of the\ninterface Bychkov-Rashba parameters at the ferromag-\nnet/semiconductor and semiconductor/normal metal in-\nterfaces,respectively,averagedoveralltherelevantbands7\n0° 90° 180° 270° 360° 0.0 0.1 0.2 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 \n  TAMR [110]  (%) \nφ135 mV \n90 mV \n50 mV (b) \n  TAMR [110]  (%) \n- 90 mV \n- 50 mV [010] [-1-10] [100] \n(a) \nFIG. 4: (color online). Angular dependence of the TAMR in\na Fe/GaAs/Au MTJ. Solid and dashed curves represent the\nresults obtained within theSSOMand theDDM, respectively.\n(a) for voltages -90 and -50 mV. (b) for voltages 50, 90, and\n135 mV.\ncontributingtothetransportacrossthecorrespondingin-\nterfaces. In order to investigate how does the degree of\nanisotropydependonthesetwoparametersweperformed\ncalculations of the ratio R[¯110]/R[110](which is a measure\nof the degree of anisotropy18) as a function of αlandαr\nby using the spin-orbit Slonczewski model described in\nSec. IIB. The results are shown in Fig. 3, where one can\nappreciate that the size of this ratio (and, consequently,\nof the TAMR) is dominated by αl. This is because the\nZeeman splitting in Au is very small compared to the\nexchange splitting in Fe and, consequently, the spin flips\nmainly when crossingthe ferromagnet/semiconductorin-\nterface. Then, since the values of the TAMR are not very\nsensitive to the changes of αrwe can set this parameter,\nwithout loss of generality, to zero. This leaves αlas a\nsingle fitting parameter when comparing to experiment.\nThe values of the phenomenological parameter αlwere\ndetermined in Refs. 5,18 by fitting the theory to the ex-\nperimental value of the ratio R[−110]/R[110]and a very\nsatisfactory agreement between theory and experiment\nwas achieved. This fitting at a single angle was enough\nfor the theoretical model to reproduce the complete an-\ngular dependence of the TAMR, demonstrating the ro-\nbustness of the model.\nAssuming that the interface Bychkov-Rashba param-\neterαlis voltage dependent and performing the fitting\nprocedure for different values of the bias voltage the bias\ndependence of αlcan be extracted.5,18Here we use the\nsame values of αlreported in Refs. 5,18 for computing\nthe angular dependence of the TAMR at different val-\nues of the bias voltage. The results are shown in Fig. 4,\nwhere the dashed and solid lines correspond to calcula-0° 90° 180° 270° 360° -4 -2 02\n50 mV 90 mV 135 mV \nφ\n  TASP [110]  (%) \n- 90 mV \n- 50 mV [010] [-1-10] [100] \nFIG. 5: (color online). Angular dependence of the TASP in\na Fe/GaAs/Au MTJ for different values of the bias voltage.\ntions within the DDM and the SSOM, respectively. An\noverall agreement between the two models can be appre-\nciated. The TAMR exhibits an oscillatory behavior as a\nfunction of the magnetizationdirection [see alsoEq.(28)]\nand can be inverted by changing the bias voltage [com-\npare Figs. 4(a) and (b)]. This bias induced inversion of\nthe TAMR was experimentally observed in Fe/GaAs/Au\ntunnel junctions18and was explained to occur as a con-\nsequence of a bias induced change in the sign of the ef-\nfective Bychkov-Rashba parameter. Preliminary ab ini-\ntiocalculations35for Fe/GaAs structures suggest that\nthe Bychkov-Rashba parameters associated with differ-\nent bands may have different values and even change the\nsign. Thus, the effective value of the interface Bychkov-\nRashba parameter αlwill depend on which bands are the\nones that mainly contribute to the transport across the\nFe/GaAsinterface(atlowtemperaturethosearetheones\nwhich have the appropriate symmetry and lie inside the\nvoltage window around the Fermi energy). For different\nvalues of the bias voltage different set of bands will be\nrelevant to transport. On the other hand, to different set\nof bands correspond different effective values of the in-\nterfaceBychkov-Rashbaparameter. Consequently, αlbe-\ncomesstronglydependent on the biasvoltage. The above\nanalysis leads to the conclusion that the inversion of the\nTAMR originates from the bias induced sign change of\nthe effective Bychkov-Rashba SOI at the Fe/GaAs inter-\nface, as proposed in Refs. 5,18.\nThe sign change of the effective Bychkov-Rashba pa-\nrameter has also influence on the tunneling spin polar-\nization, resulting in the bias induced inversion (change\nof sign) of the TASP as shown in Fig. 5. Bias in-\nduced changes of the sign of the tunneling spin polar-\nization in Fe/GaAs/Cu MTJs has also been reported.17\nThe anisotropy of the tunneling spin polarization, which\nalso exhibits a two-fold symmetry, indicates that the\namount of transmitted and reflected spin at the inter-\nfaces depends on the magnetization direction in the Fe\nlayer, resulting in an anisotropic local density of states\nat the Fermi surface. This is consistent with previous\nworks6,7,21inwhichtheanisotropyofthedensityofstates\nwith respect to the magnetization direction was related8\n0° 90° 180° 270° 360° -0.4 -0.3 -0.2 -0.1 0.0 \n0 50 100 150 200 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1  20 Å\n 80 Å\n 140 Å\n 200 Å\nφ\n  TAMR [110]  (%) [010] [-1-10] [100] \n(a) \n d (Å)\n  TAMR [110] (φ = 90°) (%) (b) \nFIG. 6: (color online). (a) Angular dependence of the TAMR\ninaFe/GaAs/AuMTJfordifferentvaluesofthebarrierwidth\ndand a bias voltage Vbias=−90 mV. (b) Amplitude of the\nTAMR (at φ= 90◦) as a function of the barrier width. The\nTAMR curves were obtained by using the SSOM.\nto the origin of the TAMR. In fact, our model calcula-\ntions reveal that the TAMR, ALDOS [see Eq. (33)], and\nTASP all exhibit a C2vsymmetry with the same kind of\nangular dependence (compare Figs. 4 and 5).\nA system parameter that can influence the size of the\nTAMR is the width of the barrier. The angular depen-\ndence of the TAMR calculated within the SSOM for the\ncase ofVbias=−90 meV is displayed in Fig. 6(a) for dif-\nferent valuesofthe barrierwidth d. As onewouldexpect,\nthechangesinthebarrierwidthdonotaffectthetwo-fold\nsymmetry of the TAMR but only its amplitude, whose\nabsolute value is predicted to increase when increasing\nthe width of the barrier [see Fig. 6(b)].\nWe remark that our model neglects the contribution of\nthe spin-orbit-induced symmetries of the involved bulk\nstructures. Say, Fe exhibits a four-fold anisotropy, which\nshouldbereflectedinthetunnelingdensityofstates. The\nfact that this is not seen in the experiment suggests that\nthis effect is smaller than the two-fold symmetry consid-\nered in our model.\nB. TAMR in\nferromagnet/semiconductor/ferromagnet tunnel\njunctions\nOur discussion in Sec. IIIA suggests that in the case\nof a F/S/F tunnel junction, the magnetoresistance will\ndepend on the absolute direction of the magnetization\nin each of the ferromagnets. In the left and right elec-\ntrodes, the magnetization directions with respect to theEFV0\nFe Fe GaAs \nz zlzry\nx\nznl\nφFe GaAs Fe \nzlzr[001] [110]nr[-110] \nφnl\n[110] nr\nθ[-110] (a) (b) \n[110] ( , ) ( ,0) TAMR ( ,0) R R \nRθ φ θ \nθ−=(c) \n[110] ( ) ( ) ATMR ( ) AP P \nPR R \nRφ φ \nφ−=\nFIG. 7: (color online). (a) Schematics of a Fe/GaAs/Fe MTJ.\nThe magnetization direction in the left (right) ferromagne t is\nspecifiedbythevector nl(nr). (b)Schematicsofthepotential\nprofile of the heterojunction along the [001] direction. (c)\nDefinition of theTAMR and ATMR in F/S/F junctions. Here\nRP(φ) =R(0,φ) andRAP(φ) =R(180◦,φ).\n[110] crystallographic direction are given by the angles\nφ=φlandφr, respectively. For convenience we intro-\nduce the angle θ=φr−φl, describing the magnetiza-\ntion direction in the right ferromagnet relative to that in\nthe left ferromagnet (see Fig. 7). Different values of the\ntunneling magnetoresistance are expected to occur when\nin-plane rotating the magnetizations of both ferromag-\nnets at the same time, while keeping the relative angle θ\nfixed. Thus, the expression for the TAMR in F/S/NM\njunctions [see Eq. (23)] can now be generalized as\nTAMR [110](θ,φ) =R(θ,φ)−R(θ,0)\nR(θ,0)=G(θ,0)−G(θ,φ)\nG(θ,φ).\n(34)\nFollowing the same procedure as in Sec. IIIA, one can,\nin principle, obtain analytical expressions for the TAMR\nin a F/S/F junction. It turns out however that in the\ngeneral case defined by Eq. (34) the resulting relations\nare quite lengthy and not much simpler than the more\naccurate expressions obtained within the SSOM (see Ap-\npendix B). Therefore, we omit here the expressions re-\nsultingfromtheDDM andshowonlytheresultsobtained\nwithin the SSOM.\nThe dependence of the TAMR on the angles θandφis\nshown in Fig. 8 for the case of a Fe/GaAs/Fe MTJ. The\nwidth of the barrier is d= 80˚A. Two different cases,\ncorresponding to ¯ α >0 (a) and ¯ α <0 (b), are consid-\nered. In both cases, the absolute value of the TAMR\nreaches its maximal amplitude when the magnetization\nof the left electrode is parallel to the [-110] direction (i.e.,\nφ= 90◦,270◦) and the one of the right electrode is per-\npendicular to it (i.e., θ= 90◦,270◦). This is because this\nconfiguration, in our parabolic model, correspondsto the\ncase of the stronger structure inversion asymmetry and,\nconsequently, to the largest absolute value of the effec-9\n0° 90° 180° 270° 360° 0° 90° 180° 270° 360° \n(b) αr = 52 eV Å 2αl = 42 eV Å 2αr = 32 eV Å 2\nTAMR [110]  [%] φθ-0.8500 \n-0.6137 \n-0.3775 \n-0.1412 \n0.09500 TAMR [110]  [%] αl = 42 eV Å 2\n(a) \n0° 90° 180° 270° 360° 0° 90° 180° 270° 360° \nφθ-1.168 \n-0.8197 \n-0.4715 \n-0.1232 \n0.2250 \nFIG. 8: (color online). Calculated TAMR defined in Eq. (34)\nin a Fe/GaAs/Fe MTJ as a function of the angles φandθ.\n(a) forαl= 42 eV ˚A2andαr= 32 eV ˚A2(i.e., ¯α >0). (b)for\nαl= 42 eV ˚A2andαr= 52 eV ˚A2(i.e., ¯α <0).\ntive Bychkov-Rashba parameter ¯ α. We also note that at\na fixed value of θthe TAMR has a two-fold symmetry\nwith respect to φand vice versa. It is clear, however,\nthat the angles θandφplay different roles in the sym-\nmetry of the TAMR, which manifest in the lack of mirror\nsymmetry with respect to the axis θ=φ. We have in-\nvestigated some traces of the TAMR displayed in Fig. 8.\nThe results are shown in Fig. 9 where we present polar\nplots of the TAMR as a function of φfor fixed values of\nθ. The solid lines correspond to the calculations within\nthe SSOM. The meaning of the dashed lines will be ex-\nplained in Sec. V. The orientation of the symmetry axis\nof the two-fold symmetric TAMR is determined by the\nrelative angle θrather than by the relative values of the\ninterface Bychkov-Rashba parameters (note for the same\nθ, the symmetry axisdoesnot changeits orientationwith\nαr). The amplitude of the TAMR is bigger for the case\nofαr= 52 eV ˚A2than for αr= 32 eV ˚A2and shows a\nstrong dependence on θ.\nIV. ANISOTROPIC TUNNELING\nMAGNETORESISTANCE (ATMR)\nWe now consider the SOI effects on the TMR for\nthe case of a F/S/F tunnel junction. The conven-\ntionalTMR effect in ferromagnet/insulator/ferromagnet-0.8 -0.4 0.0 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° -0.8 \n-0.4 \n0.0 \n-1.2 -0.8 -0.4 0.0 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° -1.2 \n-0.8 \n-0.4 \n0.0 \n-0.8 -0.4 0.0 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° -0.8 \n-0.4 \n0.0 -0.8 -0.4 0.0 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° -0.8 \n-0.4 \n0.0 \n-1.2 -0.8 -0.4 0.0 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° -1.2 \n-0.8 \n-0.4 \n0.0 \n-0.8 -0.4 0.0 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° -0.8 \n-0.4 \n0.0 TAMR [110] (φ) [%] αr = 52 eV Å 2\nθ = 3π /4 θ = 3 π/4 θ = π/2 θ = π/2 θ = π/4 θ = π/4 \n(f) (c) (e) (d) \n(b) \n TAMR [110] (φ) [%]  TAMR [110] (φ) [%]  TAMR [110] (φ) [%]  TAMR [110] (φ) [%]  TAMR [110] (φ) [%] (a) αr = 32 eV Å 2 \n   \nFIG. 9: (color online). Calculated angular dependence of th e\nTAMR in a Fe/GaAs/Fe MTJ for fixed values of the angle θ\nbetween the electrode and counterelectrode magnetization s.\nThe figures correspond to polar plots of the traces θ=π/4,\nθ=π/2 andθ= 3π/4 of Fig. (8). Solid lines correspond\nto the calculations within the SSOM while dashed lines are\nobtained from the phenomenological model by using Eqs. (54)\n- (56).\n(F/I/F) tunnel junctions relies on the dependence of\nthe magnetoresistance across the junction on the rela-\ntive magnetization directions in the different ferromag-\nnetic layers and their spin polarizations3,5and is usually\ndefined as\nTMR =RAP−RP\nRP, (35)\nwhereRP(RAP) is the magnetoresistance measured\nwhen the magnetization of the left and right ferromag-\nnetic layers are parallel (antiparallel).\nIn the case of F/S/F heterojunctions, the interference\nbetween Bychkov-Rashba and Dresselhaus SOIs leads to\nanisotropic effects. Consequently, the conventional TMR\nin F/S/F junctions will depend not only on the relative\nbut also on the absolute magnetization directions in the\nferromagnets, resulting in an anisotropic magnetoresis-\ntance. For quantifying this phenomenon we use the fol-\nlowing anisotropic generalization of the TMR,\nATMR [110]=RAP(φ)−RP(φ)\nRP(φ)=GP(φ)−GAP(φ)\nGAP(φ),\n(36)\nwhich now accounts for the magnetoresistance depen-\ndence on the absolute magnetization orientations with10\nrespecttothe[110]crystallographicdirection(seeFig.7).\nFurthermore, the efficiency ηof the anisotropic effects on\nthe tunneling magnetoresistance can be defined as\nη=ATMR [110](φ)−ATMR [110](0)\nATMR [110](0).(37)\nAsimplifiedapproximateexpressionfortheATMRcan\nbe found within the DDM by following the same proce-\ndure as in Sec. IIIA. The result is\nATMR [110]≈\nTMR−e2\nhG(0)\nAP\n/parenleftbig\nλ2\nα+λ2\nγ/parenrightbig\n2/summationdisplay\nσ=↑,↓/angbracketleftBig\n(gP\n1σ−gAP\n1σ)k2\n/bardbl/angbracketrightBig\nσ\n+λαλγcos(2φ)/summationdisplay\nσ=↑,↓/angbracketleftBig\n(gAP\n2σ−gP\n2σ)k2\n/bardbl/angbracketrightBig\nσ\n,(38)\nwhere TMR = ( G(0)\nP−G(0)\nAP)/G(0)\nAPis the conventional\nTMR in the absence of SOI, and the functions gP\n1σ,gP\n2σ,\ngAP\n1σ, andgAP\n2σare given by the Eqs. (A26),(A27), (A33),\nand (A34), respectively. Assuming that the spin-orbit\neffects on the conventional TMR are small one can ap-\nproximate the ATMR efficiency as\nη≈e2\nhG(0)\nAPλαλγ[cos(2φ)−1]\nTMR/summationdisplay\nσ=↑,↓/angbracketleftBig\n(gP\n2σ−gAP\n2σ)k2\n/bardbl/angbracketrightBig\nσ.\n(39)\nNote that the angular dependence of the efficiency ηis\nsimilar to that of the TAMR given in Eq. (28).\nIt follows from Eq. (38) that when both Bychkov-\nRashba and Dresselhaus SOIs are present (i.e., when\nλαλγ∝¯α¯γ∝negationslash= 0), the magnetoresistance becomes\nanisotropic, resulting in the ATMR effect here predicted.\nOne can see also that this effect exhibits a two-fold sym-\nmetry with a cos(2 φ) angular dependence. Unlike the\nTAMR, when ¯ α¯γ= 0 the ATMR becomes isotropic but\ndoes not vanish. However, in such a limit the efficiency\nηof the ATMR do vanish [see Eq. (39)], indicating the\nabsence of anisotropy. Like for the TAMR, changes of\nthe sign of ¯ αresult in the inversion (change of sign) of\nthe efficiency ηof the ATMR. On the other hand, since\nthe TMR contribution in Eq. (38) is usually the dom-\ninant one, the bias-induced changes of the sign of the\nBychkov-Rashba parameter ¯ αwill not, in general, cause\nthe inversion (change of sign) of the ATMR. However,\nthe fact that the TMR contribution may change sign in\ndependence ofthe applied voltage39,40,41can result in the\ninversion of the ATMR.\nWe performed calculations within the SSOM of the an-\ngular dependence of the ATMR for a Fe/GaAs/Fe MTJ\nwith a fixed value of the left-interface Bychkov-Rashba\nparameter ( αl= 42 eV ˚A2) corresponding to a bias volt-\nageVbias=−90 mV . The results are shown in Figs. 10\n(a) and (b) for the parameters αr= 32 eV ˚A2and\nαr= 52 eV ˚A2. The angular dependence of the ATMR19.3 19.4 19.5 19.6 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° 19.3 \n19.4 \n19.5 \n19.6 \n18.4 18.5 18.6 \n0° 45° 90° \n135° \n180° \n225° \n270° 315° 18.4 \n18.5 \n18.6  ATMR [110]  (%) \nαl = 42 eV Å 2\nαr = 32 eV Å 2(a) ATMR [110]  (%) \nαl = 42 eV Å 2\nαr = 52 eV Å 2(b) \nFIG. 10: (color online). Angular dependence of the ATMR\ndefined in Eq. (36) for a Fe/GaAs/Fe MTJ. (a) for αl= 42\neV˚A2andαr= 32 eV ˚A2(i.e., ¯α >0). (b)for αl= 42 eV ˚A2\nandαr= 52 eV ˚A2(i.e., ¯α <0).\nis consistent with Eq. (38). The orientation of the sym-\nmetry axis of the ATMR is determined by the sign of\nthe effective value ¯ α. On the other hand, the amplitude\nof the ATMR shows a behavior opposite to the one cor-\nresponding to the TAMR (see Fig. 9), in the sense that\nnow the amplitude of the anisotropy of the TMR is big-\nger for the case αr= 32 eV ˚A2than for αr= 52 eV ˚A2\nand, unlike for the case of the TAMR (see Fig. 9), the\norientation of the symmetry axis of the ATMR can be\nflipped by varying the value of αr.\nV. PHENOMENOLOGICAL MODEL OF TAMR\nAND ATMR\nIn order to explain the origin of the angular depen-\ndenceoftheTAMR[seeEq.(28)]inF/S/NMjunctions, a\nphenomenologicalmodel based on rather general symme-\ntry considerationswas proposedin Ref.18 and elaborated\nin moredetails inRef.5. Hereweextend this phenomeno-\nlogical model to the case of F/S/F tunnel junctions.\nFor a given k/bardbl, there are three preferential directions\nin the system: i) the magnetization direction nlin the\nleft ferromagnet, ii) the magnetizationdirection nrin the\nright ferromagnet, and iii) the direction of the effective\nspin-orbit field w. A scalar quantity such as the trans-\nmissivity for the σspin channel can be expanded in a\nseries of the all possible scalars one can form with the\nvectorsnl,nr, andw. Thus, to the second order in |w|,11\nwe have\nTσ=T(0)\nσ+T(2)\nσ. (40)\nNotethat becauseofsymmetryreasons,5thelinearin |w|\nterms vanishes after averaging over k/bardbland, therefore, we\nhave omitted them in Eq. (40). The zero order terms\nhave the general form\nT(0)\nσ=a(0)\n1σ+a(0)\n2σ(nl·nr), (41)\nwherea(0)\n1σanda(0)\n2σare isotropic expansion coefficients.\nTaking into account that nl= [cos(φ+π/4),sin(φ+\nπ/4),0] andnr= [cos(θ+φ+π/4),sin(θ+φ+π/4),0],\nthe general angular dependence of T(0)can be extracted\nfrom Eq. (41). The result is\nT(0)\nσ=a(0)\n1σ+a(0)\n2σcos(θ). (42)\nThis equation describes the dependence of the transmis-\nsivity on the relative angle θbetween the magnetizations\nin the left and right ferromagnetic electrodes in the ab-\nsence of SOI and is consistent with previous results.3,5\nThe second order contribution T(2)can be cast in the\nfollowing general form\nT(2)\nσ= [a(2)\n1σ+a(2)\n2σ(nl·nr)]|w|2+a(2)\n3σ|(nl·w)|2\n+a(2)\n4σ|(nr·w)|2+a(2)\n5σ(nl·w)(nr·w),(43)\nwith the expansion coefficients a(2)\niσ(i= 1,2,...,5) being\nangular independent. To second order in the SOI, the\nconductance is then given by\nGσ=e2\nh/parenleftBig\n∝angb∇acketleftT(0)\nσ∝angb∇acket∇ightk/bardbl+∝angb∇acketleftT(2)\nσ∝angb∇acket∇ightk/bardbl/parenrightBig\n, (44)\nwhere∝angb∇acketleft...∝angb∇acket∇ightk/bardbldenotes average over k/bardbland evaluation at\nthe Fermi energy EF.\nTaking into account Eqs. (5), (42), (43), and (44) and\nperforming the corresponding angular averaging in k-\nspace one obtains the following, general form of the total\nconductance G=G↑+G↓,\nG(θ,φ) =G1+G2cos(θ)+G3cos(2φ)\n+G4cos(2φ+2θ)+G5cos(2φ+θ),(45)\nwhere\nG1=e2\nh×\n/summationdisplay\nσ=↑,↓/angbracketleftbigg\na(0)\n1σ+|w|2\n2(2a(2)\n1σ+a(2)\n3σ+a(2)\n4σ)k2\n/bardbl/angbracketrightbigg\nσ,(46)\nG2=e2\nh/summationdisplay\nσ=↑,↓/angbracketleftbigg\na(0)\n2σ+|w|2\n2(2a(2)\n2σ+a(2)\n5σ)k2\n/bardbl/angbracketrightbigg\nσ,(47)\nG3=e2\nh/summationdisplay\nσ=↑,↓/angbracketleftBig\na(2)\n3σwxwyk2\n/bardbl/angbracketrightBig\nσ, (48)G4=e2\nh/summationdisplay\nσ=↑,↓/angbracketleftBig\na(2)\n4σwxwyk2\n/bardbl/angbracketrightBig\nσ, (49)\nand\nG5=e2\nh/summationdisplay\nσ=↑,↓/angbracketleftBig\na(2)\n5σwxwyk2\n/bardbl/angbracketrightBig\nσ. (50)\nIn the equations above the average ∝angb∇acketleft...∝angb∇acket∇ightσhave the same\nmeaningas in Eq.(A15). We note that for systemswhich\nare isotropic in the absence of the SOI these equations\nare quite general. They are valid up to second order in\nthe strength of the spin-orbit coupling field regardless of\nthe specific form of w.\nOne can see that for the case of a F/S/NM junction or\nforthecaseofparalleland antiparallelconfigurationsin a\nF/S/F junction, the general angular dependence given in\nEq. (45) is consistent with the corresponding expressions\nobtained in Appendix A. We want to stress, however,\nthat the relations presented in Appendix A are the result\nof a specific approximation (the DDM), while Eqs. (45)\n- (50) are general, and valid for any model or approx-\nimation (as far as the system without SOI is isotropic,\nthe effective spin-orbit field is weak, and nl,nr, andw\nlie in a plane perpendicular to the current flow). More\ngeneral relations corresponding to arbitrary orientations\nofnl,nr, andwwill be given elsewhere.42\nThe general form of the TAMR in F/S/NM junctions\nfollows from Eqs. (5), (23) and (45). Proceeding as in\nSec. IIIA one finds\nTAMR [110](φ)≈\ne2\nh1\nG(0)\n/summationdisplay\nσ=↑,↓/angbracketleftBig\n−(a(2)\n3σ+a(2)\n4σ+a(2)\n5σ)k2\n/bardbl/angbracketrightBig\nσ\n×\nλαλγ[cos(2φ)−1], (51)\nwhich is consistent with the corresponding expression\nfound within the DDM [see Eq. (28)]. Similarly, one ob-\ntains for the TMR and ATMR in F/S/F junctions the\nfollowing general expressions,\nTMR =e2\nh1\nG(0)\nAP/summationdisplay\nσ=↑,↓/angbracketleftBig\n2a(0)\n3σk2\n/bardbl/angbracketrightBig\nσ,(52)\nand\nATMR [110](φ) =\nTMR+e2\n2h(λ2\nα+λ2\nγ)\nG(0)\nAP/summationdisplay\nσ=↑,↓/angbracketleftBig\n(4a(2)\n2σ+2a(2)\n5σ)k2\n/bardbl/angbracketrightBig\nσ\n+e2\nh1\nG(0)\nAP\n/summationdisplay\nσ=↑,↓/angbracketleftBig\n2a(2)\n4σk2\n/bardbl/angbracketrightBig\nσ\nλαλγcos(2φ),(53)\nrespectively. Note that Eq. (53) is consistent with our\nprevious result in Eq. (38).12\nThe TAMR in a F/S/F exhibits a more complicated\nangular dependence, which has the general form\nTAMR [110](θ,φ) =\nA(θ)[1−cos(2φ)]+B(θ)sin(2φ)\nG1+G2cosθ+A(θ)cos(2φ)−B(θ)sin(2φ),(54)\nwhere\nA(θ) =G3+G4cos(2θ)+G5cosθ, (55)\nand\nB(θ) =G4sin(2θ)+G5sinθ. (56)\nAtthephenomenologicallevel,theexpansionconstants\nG1,...G5in Eqs. (45) and (54) can be determined from\nconductance measurements (or conductance theoretical\nevaluation) at selected values of θandφ. For example,\nthe values of the phenomenological constants can be de-\ntermine as follows\nG1=G(π/2,0)+G(π/2,π/2)\n2, (57)\nG2=G(0,π/4)−G1, (58)\nG5=G1−G(π/2,π/4), (59)\nG3=G(π/4,0)−G1−√\n2\n2(G2+G5),(60)\nand\nG4=G1+G3−G(π/2,0). (61)\nIt is worth noting that although we have referred to\nthe coefficients G1,...,G5as constants, strictly speaking,\nthese parameters may exhibit a dependence on the rela-\ntive angle θ. Such a dependence may originate from the\nfact that the averages containing the components of the\nspin-orbit field [see Eqs. (46) - (50)] are, in principle, θ-\ndependent. However, as shown below, this effect is weak\nfor the system here considered.In order to check the validity of the general angular\ndependence given in Eqs. (45) and (54) we have com-\nputed the expansion coefficients from Eqs. (57) - (61) by\nusing the SSOM. We then evaluate the TAMR by us-\ning the phenomenological relation given in Eq. (54) and\ncompare the results (see dashed lines in Fig. 9) with the\nfull angular dependence obtained from the SSOM (solid\nlines in Fig. 9). The agreement is satisfactory although\nsome small discrepancies appear. We attribute these dis-\ncrepancies to the fact that when determining the coeffi-\ncientsG1,...,G5from Eqs. (57) - (61) we have ignored,\nas discussed above, that these coefficients are weakly θ-\ndependent.\nVI. CONCLUSIONS\nWehaveinvestigatedthe spin-orbitinducedanisotropy\nof the tunneling magnetoresistance of F/S/NM and\nF/S/F MTJs. By performing model calculations we have\nshown that the two-fold symmetry of the TAMR effect\nmay arise from the interplay of Bychkov-Rashba and\nDresselhaus SOIs. The spin-orbit interference effects in\nF/S/F MTJs lead to the anisotropy of the conventional\nTMR. Thus, the ATMR depends on the absolute mag-\nnetization directions in the ferromagnets. The magne-\ntoresistance dependence on the absolute orientation of\nthe magnetization in the ferromagnets is deduced from\ngeneral symmetry considerations.\nAcknowledgments\nWe are grateful to M. Gmitra for useful hints and dis-\ncussions and to D. Weiss and J. Moser for discussions on\nTAMR related experiments. This work was supported\nby the Deutsche Forschungsgemeinschaft(DFG) through\nthe Sonderforschungsbereich (SFB) 689.\nAPPENDIX A\nTheeigenfunctionsoftheHamiltonian givenbyEq.(1)\nobey the following matching conditions\nΨ(l)\nσ(0−) = Ψ(r)\nσ(0+);/planckover2pi12\n2m0dΨ(l)\nσ\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=0−+(V0d+w·σ)Ψ(l)\nσ(0−) =/planckover2pi12\n2m0dΨ(r)\nσ\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=0+, (A1)\nwhich, with the scattering states in Eqs.(6) and (7), lead\nto a system of 4 linear equations for determining the co-\nefficients rσ,σ,rσ,−σ,tσ,σ, andtσ,−σ. The transmissioncoefficients are found to be given by\ntσ,σ=−4d2√kσ(k−σ+κ−σ+iQ)/parenleftbig\n1+ei(φr−φl)/parenrightbig\nΩ13\n+8id√kσ(U·Sσ,σ)\nΩ, (A2)\nand\ntσ,−σ=4d√kσd(k−σ+κσ+iQ)/parenleftbig\n1−ei(φr−φl)/parenrightbig\nΩ\n−8i(U·Sσ,−σ)\nΩ, (A3)\nwhere\nΩ = Ω +(−)Ω−(+)−Ω+(+)Ω−(−),(A4)\nwith\nΩ±(λ) =d(k±σ+κλσ+iQ)/parenleftBig\n1±λei(φr−φl)/parenrightBig\n+ 2i(U·S±σ,λσ). (A5)\nThe vectors Sσ,σ′(σ′=±σ) are given by\nSσ,σ′=χ(l)†\nσσχ(r)\nσ′, (A6)\nwhileQ= 2m0V0d//planckover2pi12andU= (2m0d//planckover2pi12)w.\nThe transmissivity for an incident particle with spin σ\nis given by Eq. (11). For the case of a F/S/NM junction\nwe can approximate κ=κσ≈κ−σand the transmissiv-\nity reduces to\nTσ= Re/bracketleftbig\nκ/parenleftbig\n|tσ,σ|2+|tσ,−σ|2/parenrightbig/bracketrightbig\n.(A7)\nIn order to obtain a simplified analytical expression for\nthe TAMR in F/S/NM junctions, we consider the case\nin which the effective spin-orbit field is small. In such a\ncase one can expand Eq. (A7) in powers of w=|w|and\nobtain the conductance from Eq. (22). The result, up to\nsecond order in w, reads\nGσ≈Giso\nσ+Ganiso\nσ. (A8)\nThe isotropic part of the conductance is given by\nGiso\nσ=G(0)\nσ+G′\nσ, (A9)\nwhere\nG(0)\nσ=e2\nh/angbracketleftbigg4|kF,σ|κF\n|Aσ|2/angbracketrightbigg\nσ, (A10)with\nA±σ=kF,±σ+κF+iQ, (A11)\nis the conductance in absence of the SOI and\nG′\nσ=−e2\n2h(λ2\nα+λ2\nγ)∝angb∇acketleftg1σk2\n/bardbl∝angb∇acket∇ightσ,(A12)\nis the isotropic contribution induced by the SOI. Here we\nhave denoted kFσ=kσ(EF,k/bardbl),κF=κ(EF,k/bardbl) and\ng1σ=8κF|kF,σ|\n|Aσ|2|A−σ|2/parenleftbigg2Re[AσA−σ]\n|Aσ|2−1/parenrightbigg\n−g2σ.(A13)\nWe have also introduced dimensionless SOC parameters\nas\nλα=2m0\n/planckover2pi12¯α, λγ=2m0\n/planckover2pi12¯γ. (A14)\nThe function g2is defined below in Eq. (A18). The av-\nerage\n∝angb∇acketleft...∝angb∇acket∇ightσ=1\n2π/integraldisplaykσmax\n0... k/bardbldk/bardbl, (A15)\nwhere\nkσ\nmax= min/parenleftBigg/radicalBigg\n2m0\n/planckover2pi12/parenleftbigg\nEF+σ∆l\n2/parenrightbigg\n,/radicalbigg\n2m0\n/planckover2pi12EF/parenrightBigg\n,\n(A16)\ndenotesthemaximumvaluesof k/bardblforwhich wehaveinci-\ndent and transmitted propagating states. In the average\ndefined in Eq. (A15) the corresponding angular integra-\ntion over k/bardbl[see Eq. (22)] has already been performed.\nThe spin-orbit induced anisotropy of the conductance\nis determined by the relation\nGaniso\nσ(φ) =−e2\nhλαλγ∝angb∇acketleftg2σk2\n/bardbl∝angb∇acket∇ightσcos(2φ),(A17)\nwhere\ng2σ=4κF|kF,σ|/parenleftbig\n|A−σ|2|Aσ−A−σ|2−4Im[A−σ](|A−σ|2Im[Aσ]−|Aσ|2Im[A−σ])/parenrightbig\n|Aσ|4|A−σ|4. (A18)\nThe total conductance can then be written as\nG(φ)≈Giso+Ganiso(φ), (A19)withGiso=Giso\n↑+Giso\n↓andGaniso(φ) =Ganiso\n↑(φ) +\nGaniso\n↓(φ).14\nWe now consider the case of a F/S/F tunnel junction.\nFor the general case of arbitrary orientations of the mag-\nnetization in the left and right ferromagnets, the ana-\nlytical expressions for the conductance are very lengthy\nand we therefore omit them here. Simpler expressions\nare found, however, for the particular cases of parallel\n(P) and antiparallel (AP) configurations. If we consider\nthat the ferromagnetic electrodes are made of the same\nmaterial, then kσ=κσand the transmissivity reduces to\nTσ= Re/bracketleftbig\nkσ|tσ,σ|2+k−σ|tσ,−σ|2/bracketrightbig\n. (A20)\nFollowing the same procedure as above, the following ex-\npression for the conductance in the parallel configuration\nis found\nGP≈Giso\nP+Ganiso\nP(φ). (A21)\nThe isotropic part is given by\nGiso\nP=G(0)\nP+G′\nP, (A22)\nwhere\nG(0)\nP=e2\nh/summationdisplay\nσ=↑,↓/angbracketleftBigg\n4k2\nF,σ\n4k2\nF,σ+Q2/angbracketrightBigg\nσ, (A23)and\nG′\nP=−e2\n2h(λ2\nα+λ2\nγ)/summationdisplay\nσ=↑,↓∝angb∇acketleftgP\n1σk2\n/bardbl∝angb∇acket∇ightσ.(A24)\nThe anisotropic contribution reads\nGaniso\nP(φ) =e2\nhλαλγ/summationdisplay\nσ=↑,↓∝angb∇acketleftgP\n2σk2\n/bardbl∝angb∇acket∇ightσcos(2φ).(A25)\nThe functions gP\n1σandgP\n2σare given by\ngP\n1σ=8kF,σ(2kF,σ+kF,−σ)Q2−32k3\nF,σkF,−σ/parenleftBig\n4k2\nF,σ+Q2/parenrightBig2/parenleftBig\n4k2\nF,−σ+Q2/parenrightBig−gP\n2σ,\n(A26)\nand\ngP\n2σ=4kF,σ(kF,−σ−kF,σ)\n(4k2\nF,σ+Q2)3(4k2\nF,−σ+Q2)/bracketleftbig\n16kF,−σk3\nF,σ−12kF,σ(kF,σ+kF,−σ)Q2+Q4/bracketrightbig\n, (A27)\nrespectively.\nOn the other hand, for the case of antiparallel config-\nuration one obtains the following expressions\nGAP≈Giso\nAP+Ganiso\nAP(φ), (A28)\nwhere\nGiso\nAP=G(0)\nAP+G′\nAP, (A29)\nG(0)\nAP=e2\nh/summationdisplay\nσ=↑,↓/angbracketleftBigg\n4kF,σ(kF,−σ−kF,σ)\n(kF,−σ+kF,σ)2+Q2/angbracketrightBigg\nσ,(A30)\nand\nG′\nAP=−e2\n2h(λ2\nα+λ2\nγ)/summationdisplay\nσ=↑,↓∝angb∇acketleftgAP\n1σk2\n/bardbl∝angb∇acket∇ightσ.(A31)\nThe anisotropic contribution for the antiparallel case\nreads\nGaniso\nAP(φ) =e2\nhλαλγ/summationdisplay\nσ=↑,↓∝angb∇acketleftgAP\n2σk2\n/bardbl∝angb∇acket∇ightσcos(2φ),(A32)and the functions gAP\n1σandgAP\n2σare given by\ngAP\n1σ=8kF,σ(2kF,−σ+kF,σ)Q2\n/bracketleftBig\n(kF,σ+kF,−σ)2+Q2/bracketrightBig3−gAP\n2σ\n+8kF,σ(kF,σ−2kF,−σ)(kF,σ+kF,−σ)2\n/bracketleftBig\n(kF,σ+kF,−σ)2+Q2/bracketrightBig3,(A33)\nand\ngAP\n2σ=4kF,σ(kF,σ−kF,−σ)\n/bracketleftBig\n(kF,σ+kF,−σ)2+Q2/bracketrightBig2,(A34)\nrespectively.\nIn the derivation of Eqs. (A24), (A25), (A31), and\n(A32) we assumed that the spin orbit parameters ¯ αand\n¯γare independent of the relative magnetization orienta-\ntions in the ferromagnets. Strictly speaking the values\nof ¯αand ¯γareθ-dependent and therefore to the parallel\nand antiparallel configurations correspond slightly differ-\nent sets of effective spin orbit parameters. This effect\nturns out to be negligible for the system here investi-\ngated and we therefore omit it.15\nIn both the parallelandantiparallelcases, the averages\nhave the same meaning as in Eq. (A15) but now with\nkσmaxdefined as\nkσ\nmax=/radicalBigg\n2m0\n/planckover2pi12/parenleftbigg\nEF+σ∆l\n2/parenrightbigg\n.(A35)\nIt is worth noting that for a junction with structure\ninversion symmetry the averageBychkov-Rashbaparam-\neter vanishes. One may think that since we are consid-\nering a structure with leads of the same material the\nanisotropic term in the conductance corresponding to\nthe parallel configuration (for the parallel configuration\nthestructurebecomessymmetricunderspatialinversion)\nwill vanishes. This is so for strictly symmetric under\nspatial inversion structures. In practice, however, the in-\nterfaces may not be identical, e.g., one of the interfaces\nmay be epitaxial while the other not. Another possibil-\nity is to consider different terminations of the semicon-\nductor at the left and right interfaces. For the case ofFe/GaAs/Fe structures, for example, one of the Fe/Gas\ninterface may be Ga terminated while the other may be\nAs terminated.43,44This interface-induced asymmetry is\nenough for the average Bychkov-Rashba parameter to\nhave a sizable value.\nAPPENDIX B\nHere we discuss the details about the solutions of the\nSSOM and provide approximate expressions for the tun-\nneling coefficients.\nBy requiring the probability flux conservation across\nthe interfaces one obtains that the eigenfunctions of the\nHamiltonianinEq.(12)shouldfulfilthefollowingbound-\nary conditions\nΨ(i)\nσ(zij) = Ψ(j)\nσ(zij), (B1)\n/planckover2pi12\n2mi/bracketleftbigg\n1−2miγi\n/planckover2pi1(kxσx−kyσy)/bracketrightbiggdΨ(i)\nσ\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=zij−/planckover2pi12\n2mj/bracketleftbigg\n1−2mjγj\n/planckover2pi1(kxσx−kyσy)/bracketrightbiggdΨ(j)\nσ\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nz=zij=\n−αij(kxσy−kyσx)Ψ(i)\nσ(zij), (B2)\nwherei,j=l,c,randzlc= 0 and zcr=dare the lo-\ncations of the left and right interfaces, respectively. The\ninterface Bychkov-Rashba parameters are introduced as\nαlc=αl,αcr=−αr, while the Dresselhaus parameter is\nγc=γin the semiconductor and vanishes elsewhere (i.e.,\nγl=γr= 0).\nApplying the above boundary conditions to the scat-\ntering states given in Section IIB one obtains a system\nof 8 linear equations for determining the 8 unknown ex-\npansion coefficients. The exact expressions for the trans-mission coefficients tσ,σandtσ,−σare quite cumbersome.\nHowever, simplified analytical expressions for tσ,σand\ntσ,−σare found in the limit q0d≫1. In such a case one\nfinds the following approximate relations for the tunnel-\ning coefficients,\ntσ,σ=−Dσ,σ\nD;tσ,−σ=Dσ,−σ\nD, (B3)\nwhereD=f−(−)f+(−)−f−(+)f+(+), with\nf±(λ) =±iαrk/bardblQ\n2V0/parenleftBig\nσei(φr−ξ+π/4)+λe2iξ/parenrightBig\n+d\n2/parenleftbiggm0\nm±λq∓λ−iκ±σ/parenrightbigg/parenleftBig\n1−λσei(φr+ξ+π/4)/parenrightBig\n. (B4)\nand\n1\nm±=1\nmc/parenleftbigg\n1±2mcγk/bardbl\n/planckover2pi12/parenrightbigg\n. (B5)\nFurthermore, we have\nDσ,σ=2m0d\nm+q−f−(+)g−−2m0d\nm−q+f−(−)g+,(B6)and\nDσ,−σ=2m0d\nm+q−f+(−)g−−2m0d\nm−q+f+(+)g+.(B7)\nIn these equations we introduced the notation16\ng±=id√kσ/bracketleftBig/parenleftBig\nf0∓h1−m0d\nm±q∓/parenrightBig/parenleftbig\n1±σei(φl+ξ+π/4)/parenrightbig\n∓h2/parenleftbig\n1∓σei(φl+ξ+π/4)/parenrightbig/bracketrightBig\ne−q±d\nh2\n2+/parenleftBig\nf0−h1−m0d\nm+q−/parenrightBig/parenleftBig\nf0+h1−m0d\nm−q+/parenrightBig , (B8)\nwheref0=i(kσ+k−σ)d/2,\nh1=iσd\n2(kσ−k−σ)cos(φl+ξ+π/4)+αlk/bardblQ\nV0sin(2ξ),\n(B9)\nand\nh2=−σd\n2(kσ−k−σ)sin(φl+ξ+π/4)−iαlk/bardblQ\nV0cos(2ξ).\n(B10)\nIt is worth noting that the approximate expressions\nfor the tunneling coefficients here provided are valid up\nto first order in exp( −q±d). This approximation is ap-\npropriate for treating junctions with high and not too\nthin potential barriers. 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Matter 16, 7581 (2004)."    },    {        "title": "2012.10075v1.Tunable_ferromagnetism_at_non_integer_filling_of_a_moiré_superlattice.pdf",        "content": "Page 1 of 12  \n Tunable ferromagnetism at non-integer filling of a moiré  superlattice \nGuorui Chen1+, Aaron L. Sharpe2,3+, Eli J. Fox3,4+, Shaoxin Wang1, Bosai Lyu5,6, Lili \nJiang1, Hongyuan Li1,7, Kenji Watanabe8, Takashi Taniguchi9, Michael F. \nCrommie1,7,10, M. A. Kastner3,4,11, Zhiwen Shi5,6, David Goldhaber-Gordon3,4*, Yuanbo \nZhang6,12,13*, Feng Wang1,9,14* \n \n1Department of Physics, University of California at Berkeley, Berkeley, CA 94720 , \nUSA. \n2Department of Applied Physics, Stanford University, Stanford, CA 94305, USA. \n3Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator \nLaboratory, Menlo Park, CA 94025, USA.  \n4Department of Physics, Stanford University, Stanford, CA 94305, USA. \n5Key Laboratory of Artificial Structures and Quantum Control (Ministry of \nEducation), Shenyang National Laboratory for Materials Science, School of Physics \nand Astronomy, Shanghai Jiao Tong University, Shanghai 200240 , China. \n6Collaborative Innovation Center of Advanced Microstructures, Nanjing, Jiangsu \n21009 3, China. \n7Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA \n94720, USA.  \n8Research Center for Functional Materials, National Institute for Materials Science, 1-\n1 Namiki, Tsukuba 305-0044, Japan.  \n9International Center for Materials Nanoarchitectonics, National Institute for \nMaterials Science,  1-1 Namiki, Tsukuba 305 -0044, Japan.  \n10Kavli Energy NanoSciences Institute at the University of California, Berkeley and \nthe Lawrence Berkeley National Laboratory, Berkeley, CA 94720 , USA. \n11Department of Physics, Massachusetts Institute of Technology, Cambridge, MA \n02139, USA. \n12State Key Laboratory of Surface Physics and Department of Physics, Fudan   \nUniversity, Shanghai 200433, China. \n13Institute for Nanoelectronic Devices and Quantum Computing, Fudan University, \nShanghai 200433, China. \n14Kavli Energy NanoSciences Institute at the University of California, Berkeley and \nthe Lawrence Berkeley National Laboratory, Berkeley, CA 94720 , USA. \n \n \n \n+These authors contributed equally to this work. \n*Correspondence to: fengwang76@berkeley.edu, zhyb@fudan.edu.cn, goldhaber-\ngordon@stanford.edu   Page 2 of 12  \n The flat bands resulting from moiré  superlattices in magic -angle twisted bilayer \ngraphene (MATBG) and ABC -trilayer graphene  aligned with hexagonal boron \nnitride (ABC -TLG/hBN) have been shown to give rise to fascinating correlated \nelectron phenomena such as correlated insulators (1, 2) and superconductivity (3, \n4). More  recen tly, orbital magnetism associated with correlated Chern insulators  \nwas found in this class of layered structures centered at integer multiples of n0, the \ndensity corresponding to one electron per moiré  superlattice unit cell (5–7). Here \nwe report the experimental observati on of ferromagnetism at fractional  filling of \na flat Chern band in an ABC -TLG/hBN moiré  superlattice . The ferromagnetic \nstate exhibits prominent ferromagnetic hysteresis behavior with  large anomalous \nHall resistivity  in a broad region of densities, centered in the valence mini band at  \nn = -2.3 n0. This ferromagnetism depends very sensitively on the control \nparameters in the moiré  system: n ot only the magnitude of the anomalous Hall \nsignal, but also the sign of the hysteretic  ferromagnetic response  can be modulated \nby tuning the carrier density and displace ment field. Our discovery of electrically  \ntunable ferromagnetism in a moiré  Chern band at non -integer filling highlights \nthe opportunities for explor ing new correlated ferromagneti c states  in moiré  \nheterostructures.  \n  Page 3 of 12  \n  Moiré  superlattices in van der Waals heterostructures ha ve emerged as a \npowerful platform for exploring novel correlated electron physics. The electronic \nstructure of the moiré  miniband can be conveniently controlled by the stacking \nsequence of layers and by electric fields applied normal to the layers( 8), and the moiré  \nbandwidth can be made small enough that electron correlations predominate( 9, 10). In \naddition, the electron density can be tuned across an entire band using gates, something \nthat is much more difficult in more conventional correlated electron materials. In \ngraphene-based moiré  superlattices, such control has enabled discovery of correlated \ninsulators, superconductivity, and Chern insulators with orbital ferromagnetism( 1–7, \n11–15); Mott insulators and generalized Wigner crystals have been reported in \ntransition metal dichalcogenide moiré  superlattices( 16–18). \n Here we report the observation of ferromagnetism at non -integer filling of a \nmoiré  superlattice band of ABC-TLG/hBN. We perform magneto-transport \nmeasurements to map out the phase diagram of an ABC-TLG/hBN heterostructure as a \nfunction of the carrier density n, the vertical displacement field D, and the vertical \nmagnetic field B. In addition to the orbital ferromagnetism and a Chern insulator state, \nwe previously identified( 6), at n = -n0 (one hole per moiré  unit cell), we discover \nanother region of ferromagnetism when the topological Chern band is partially filled \nbetween n = -2.1  n0 and -2.7  n0 and D ranges from -0.45 V/nm to -0.55 V/nm. The \nferromagnetic state exhibits hysteresis and shows a zero-field anomalous Hall (AH) \nsignal as large as -4.8  k at n = -2.3 n0 and D = -0.48 V/nm. The large AH signal is \naccompanied by a peak in the longitudinal resistivity as a function of n. In addition, the \nlongitudinal resistivity of the magnetic states increase slightly when reducing \ntemperature, suggesting that the states showing ferromagnetism are insulating. We \nfurther find that the ferromagnetism depends sensitively on the carrier density and \nvertical displacement field: not only the magnitude of the AH signal, but also the sign \nof the ferromagnetic hysteresis can be controlled by the carrier density and \ndisplacement field.  \n The ABC-TLG/hBN moiré  heterostructures are fabricated following the method \ndescribed in ( 6). We first identify the ABC-TLG domain using scanning near-field \ninfrared nanoscopy and then isolate the ABC domain from adjacent ABA regions using \natomic force microscope cutting( 19, 20). The isolated ABC-TLG is encapsulated in \nexfoliated hBN layers, where one hBN layer is aligned with the ABC-TLG to form the Page 4 of 12  \n moiré  superlattice( 21). For magneto-transport measurements, the ABC-TLG/hBN \nheterostructures are fabricated into a Hall bar geometry with a metal top gate and a \ndoped silicon bottom gate. During fabrication, the ABC-TLG often relaxes to the lower \nenergy ABA state, but after fabrication we find that the ABC stacking is stable.  An \noptical image of the Hall bar device and the Hall measurement configuration are \ndisplayed in Fig. 1A. Figure 1B shows a schematic side-view of the dual-gated device. \nThe top and bottom gate voltages ( Vt and Vb) allow us to independently control the \ncarrier density and the miniband bandwidth and topology of the ABC-TLG/hBN \nheterostructure: the doping relative to the charge neutrality point is set by 𝑛 = (𝐷 b−\n𝐷t)/𝑒, and the miniband bandwidth and topology are controlled by the applied vertical \ndisplacement field 𝐷 = (𝐷 b+ 𝐷 t)/2. Here 𝐷b= +𝜀 b(𝑉b− 𝑉b0)/𝑑 b and 𝐷t=\n−𝜀 t(𝑉t− 𝑉 t0)/𝑑t are the vertical displacement field below and above the ABC-\nTLG/hBN moiré  superlattice, respectively, 𝜀b(t) and 𝑑b(t) are the dielectric constant \nand thickness of the bottom (top) dielectric layers, and 𝑉b(t)0 is the effective offset in the \nbottom (top) gate voltage caused by environment-induced carrier doping. The \nlongitudinal resistance is 𝑅xx=𝑉xx\n𝐼 and Hall resistance is 𝑅yx=𝑉yx\n𝐼. All measurements \nare performed at 0.07 K and with an AC excitation current I = 0.5 nA rms  unless \notherwise noted.  \n The bandwidth and topology of the moiré  minibands in ABC-TLG/hBN can be \ncontrolled by the vertical displacement field D. Figures 1 C, D and E display the \ncalculated single-particle bandstructure in ABC-TLG/hBN for the K valley at D = 0.5 \nV/nm, 0 V/nm, -0.5  V/nm, respectively( 6). At D = 0.5  V/nm a finite bandgap separates \nthe conduction and valence miniband. The highest-energy valence miniband is flat with \na bandwidth as narrow as 20  meV, and it is topologically trivial with C = 0. At D = 0 \nV/nm, the conduction and valence moiré  minibands become broader and overlapping \nso that the band gap closes. At D = -0. 5 V/nm, a bandgap reopens, with band topology \nchanged relative to positive D field. Specifically, the highest-energy valence miniband \nbecomes topologically nontrivial with a finite Chern number and a narrow bandwidth \nof 20  meV. The electronic states in the K’ valley are time reversed pairs of those in the \nK valley, and the topological minibands have opposite Chern numbers. The valence \nminiband,  because it has a smaller bandwidth, should have stronger e-e interactions \ncompared to the conduction miniband( 22–26). We are therefore not surprised that the \ninteresting phenomena reported here occur with hole, rather than electron, doping.  Page 5 of 12  \n  This electronic bandstructure of the moiré  minibands can be conveniently tuned \nby the D field in ABC-TLG/hBN heterostructure. Figure 1F displays the longitudinal \nresistance Rxx as a function of band filling n / n0 and displacement field D in the \ndevice( 27) (n and D are linearly transformed from the gate voltages, as shown in Fig. \nS1). In addition to the band insulating states (characterized by resistance peaks) at the \ncharge neutral point ( n = 0) and the fully filled point ( n = -4n0), correlated insulator \nstates also emerge at 1/4 filling ( n = -n0) and 1/2 filling ( n = -2n0) of the hole minibands \n(i.e. one and two holes per moiré  unit cell, respectively). (2) These correlated states \nappear at finite displacement field | D|, presumably because they require narrowing of \nthe moiré  minibands. \n To explore ferromagnetism and evidence for band topology, we turn to \nmagneto-transport measurements. Figure 2A and 2B show the longitudinal resistance \nat B = 0 T as a function of the band filling of topological ly trivial and non-trivial valence \nminibands at D = 0.4V/nm and -0.47 V/nm, respectively. In the trivial flat band (at D \n= 0.4 V/nm), strong Mott insulator states are present at n = -n0 and n = -2  n0. In the \ntopological flat band (at D = -0.47 V/nm), w eaker resistance peaks are observed at n = \n- n0, n = -2 n0, and n = -2.3 n0. Figure 2C and D display the corresponding two -\ndimensional plots of the Hall resistance Ryx as a function of the hole doping and the \nperpendicular magnetic field B (normal to ABC-TLG/hBN  layers ). In the trivial flat \nband (at D = 0.4 V/nm), the Hall resistance signal tends to be rather small for all hole \ndoping (Fig. 2C) and no ferromagnetic signature is observed. (The large signals at n = \n- n0 and n = -2 n0 fillings are artifacts coming from crosstalk from the large Rxx of the \ncorrelated insulator states ; they do not change sign when the magnetic field is reversed.) \nWith displacement field reversed D = -0.47 V/nm, the behavior is strikingly different: \nat carrier densities around n = -n0 and n = -2.3 n0, the Hall resistance Ryx is very large at \nweak magnetic fields and persists down to zero magnetic field. Ryx switch es sign \nabruptly at a magnetic field close to,  but away from,  zero (Fig. 2D; field is swept from \npositive to negative). Such behavior is characteristic of a ferromagnetic state. Indeed, \nthe state at n = -n0 has been previously( 6) shown to be a C = 2 Chern insulator state. (28) \nThe Chern insulator at n = - n0 can be understood as arising from a single fully-filled \nvalley-polarized Chern band( 26). The state around n = -2.3  n0, in contrast, is a \ncompletely new magnetic state that has not been predicted and emerges, surprisingly, \nat non -integer filling of the moiré  miniband.  Page 6 of 12  \n To establish the region of parameter space where the new magnetic state exists, we \napplied B = 0.2 T to favor polarization, and track Ryx as a function of n and D (Fig. 3 A). \nIn an oval region of parameter space centered around n = -2.3 n0 and D = -0.48 V/nm \n(outlined by the dashed line in Fig. 3A, which is discussed in more detail below), Ryx is \nlarge and negative.  Figure. 3B shows the corresponding two -dimensional plots of Rxx \nas a function of n / n0 and D at a weak magnetic field B = 0.2 T. This oval region also \nshows enhanced Rxx compared with the surrounding region, as shown in Fig. 3B. This \nsuggests that the ferromagnetic phase in this range of parameters is always insulating.  \nNext we examine in detail the B field and temperature dependence of the new \nmagnetic state at which the AH has the largest magnitude ( n = -2.3n0 and D = -0.48 \nV/nm). We measure the anti-symmetrized Hall resistivity 𝜌yx(B,up) =\n𝑅yx(B,up)−𝑅 yx(−B,down)\n2, 𝜌yx(𝐵, down) =𝑅yx(B,down )−𝑅 𝑦𝑥(−B, up)\n2 and symmetrized \nlongitudinal resistivity 𝜌xx(𝐵,up) =𝑅xx(B,up)+𝑅 xx(−B,down)\n2∙𝑊\n𝐿, 𝜌xx(B, down) =\n𝑅xx(B,down )+𝑅 xx(−B, up)\n2∙𝑊\n𝐿, where up/down indicates the direction in which the magnetic \nfield was being swept, the channel width W = 1 m, and the channel length L = 4 m. \n(Throughout this paper, we use 𝜌yx when presenting AH signals as functions of the \nmagnetic field to eliminate longitudinal resistance contributions, but we use the directly \nmeasured Ryx when presenting color maps of the dependence of Hall resistance on n \nand D.) When B is swept over the narrow range between -0.1 to 0.1 Tesla, the Hall \nresistivity yx shows a clear AH signal with strong ferromagnetic hysteresis (Fig. 3C). \nThe AH signal reaches 𝜌yxAH = -4. 8 k with a coercive field of Bc = 0.04 T at the base \ntemperature of T = 0.07  K. 𝜌yxAH decreases monotonically with increasing temperature, \nreaching almost zero at T = 1.6 K (Fig. 3E). We note that 𝜌yxAH at n = -2.3n0 exhibits the \nopposite sign  compared to that of the Chern insulator at n = -n0 (27)(Fig. S2 shows a \ndirect comparison between these two magnetic states).   \nFigure 3D shows that the overall longitudinal resistivity of the magnetic state \nincreases as temperature is lower ed. Figure 3E and F display the AH signal 𝜌yxAH and \nlongitudinal conductivity xx (derived from the measured resistivities) as functions of \nthe temperature at B = 0. As the temperature is lowered, the ferromagnetic state \nstrengthens, as evinced by increasing 𝜌yxAH and the conductivity xx decreases. Th e Page 7 of 12  \n temperature dependences of xx and xx suggest that the ferromagnetic state is \ninsulating. However, we do not observe well-defined Arrhenius behavior (see plot of \nlog xx versus 1/ T in Fig. S5C).  \n Next we examine the evolution of the AH signal as a function of D and n. We \nsweep the field up and down over a small range of magnetic fields and measure yx. \nFigure 4A shows the D-dependen ce at fixed doping of n = -2.5 n0. Beginning at D = -\n0.49 V/nm, where there is no nonlinearity, the hallmark of AH, the AH grows with less \nnegative D as one approaches the center of the oval. Upon tuning away from the center \nof the oval, the AH signal decreases quickly and becomes zero at D = -0.38 V/nm. \nSurprisingly, the AH reappears at D = -0.37 V/nm, accompanied by a switch in sign of \nthe hysteresis loop. It then decreases and becomes very small again for D ≳ -0.35 V/nm.   \n The magnitude and sign of the ferromagnetic hysteresis also depends sensitively \non the carrier density n for a fixed D = -0.39 V/nm (Fig. 4B). At this displacement field, \nthe AH signal reaches a first maximum at n = -2.48  n0. With less negative n, the AH \nsignal decreases quickly. Then the AH signal changes sign, and although the hysteresis \ndisappears near n = -2.31  n0, the non-linear component, which is the best measure of \nthe AH signal,  continues to grow as n is made less negative. Additional data of this type \nare shown in the supplemental material, showing that the AH signal vanishes before n \nreaches -2 n0.  \n To visualize the dramatic change of the ferromagnetic state with n and D, we \nindicate the parameter space with negative and positive 𝜌yxAH in Fig. 4C with blue and \nred squares, respectively, where the intensity of the color indicates the magnitude of \nthe anomalous Hall signal. Values of ( n, D) where AH is zero are indicated by circles. \nA variety of addition al magnetic field sweeps for various values of n and D are shown \nin Fig. S4 for the same sample studied in Fig. 4 and in Fig. S7 for a second sample( 27).  \nWe note that the hysteresis loops of the magnetic state can be rather complicated close \nto the oval boundaries, at which the AH vanishes. Some hysteresis loops show multiple \nintermediate jumps in yx (Ryx), and some hysteresis loops appear at non-zero magnetic \nfields (Fig. 4 and Fig. S4 ).  \nWe have shown that ferromagnetism can emerge not only at integer filling of a \nminiband in an ABC-TLG/hBN moiré  superlattice( 6) but also in a region centered \nabout non-integer filling, behavior which is unique among the reported magnetic states Page 8 of 12  \n in moiré  systems. As with AH in other moiré  systems, the system appears to be \ninsulating at zero field and the magnitude of the AH is large, indicating that it is unlikely \nthat the AH arises from some scattering mechanism as in ferromagnetic metals. We \ntherefore assume that, while the ferromagnetic state at non-integer filling in ABC-\nTLG/hBN as yet exhibits significant longitudinal resistance and no quantization of AH, \nthe large AH arises, as it does in the other systems, from a combination of band topology \nand e-e interactions. Interactions break valley/spin degeneracy, and give a gap between \nthe bands with opposite Chern numbers. The observed magnetic state appears only \nwhen the valence miniband has been tuned by the applied displacement field to be \ntopologically non-trivial, suggesting the importance of band topology for realizing the \nmagnetic state.  \nThe ferromagnetic hysteresis in Hall resistance at non-integer filling can change \nsign with field effect control of D and n, perhaps indicating that different ferromagnetic \nstates can be stabilized in different regions of parameter space( 29). Similar behavior \nhas been observed in a second device( 27) (See Fig. S6-8). This reversal of the sign of \nAH as a function of carrier density is similar to that recently reported for \nferromagnetism centered at n = 3n0 in twisted monolayer-bilayer graphene( 30). These \nauthors report that the Hall resistance saturates near h/2e2 on both sides of the sign \nreversal, where h is P lanck’s constant and e is the charge of an electron, while the \nlongitudinal resistance Rxx ≤ 1 k, suggesting that on the two sides of the sign reversal \nthere are Chern insulators with Chern numbers C = ± 2.   The sign reversal of the AH \nwith carrier density reported here can then be explained as follows: with an applied \nexternal magnetic field, we control the preferred magnetization direction, as the \nmagnetic moment wants to align with the field. But Chern number is not simply related \nto magnetization: the magnetization of an orbital Chern insulator reverses as the \nchemical potential passes through the gap( 29). Therefore the magnetization of each \nvalley, and hence the magnetization for a Chern insulator of each Chern number, \nreverses as a function of carrier density, yielding the measured reversal of AH \nresistance. The sign of the AH is not reported to be sensitive to D in twisted monolayer-\nbilayer graphene( 30) as it is for the non-integer filling ferromagnetic state in the present \nwork. However, it is predicted that the sign of the magnetization of an orbital Chern \ninsulator of a given Chern number also depends on gap size, the bandwidth of the bands, \nand the valley splitting( 29). Tuning D strongly affects these parameters in ABC-Page 9 of 12  \n TLG/hBN, therefore it is perhaps not surprising that, for fixed chemical potential, we \nsee a sign flip in the measured AH resistance as a function of D, as well. Twisted \nmonolayer-bilayer graphene has also been found to be magnetic in a region centered \nnear n = n0 but this state does not exhibit a sign change of AH as a function of carrier \ndensity( 30, 31).  \nThus both the large AH and its sign reversal likely arise from a correlation-\ninduced gap between bands with opposite Chern numbers.  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MacDonald, The Curious Magnetic Properties of Orbital \nChern Insulators. arXiv:2001.05084 [cond-mat]  (2020) (available at \nhttp://arxiv.org/abs/2001.05084). \n30.  H. Polshyn, J. Zhu, M. A. Kumar, Y. Zhang, F. Yang, C. L. Tschirhart, M. Serlin, \nK. Watanabe, T. Taniguchi, A. H. MacDonald, A. F. Young, Nonvolatile \nswitching of magnetic order by electric fields in an orbital Chern insulator. \narXiv:2004.11353 [cond-mat]  (2020) (available at \nhttp://arxiv.org/abs/2004.11353). \n31.  S. Chen, M. He, Y.-H. Zhang, V. Hsieh, Z. Fei, K. Watanabe, T. Taniguchi, D. H. \nCobden, X. Xu, C. R. Dean, M. Yankowitz, Electrically tunable correlated and \ntopological states in twisted monolayer-bilayer graphene. arXiv:2004.11340 \n[cond -mat]  (2020) (available at http://arxiv.org/abs/2004.11340). \n \nACKNOWLEDGMENT S  Page 12 of 12  \n We acknowledge the help in the bandstructure calculations and helpful discussions with \nYa-Hui Zhang and T. Senthil. Funding:  G.C., H. L, M.C., and F.W. were supported as \npart of the Center for Novel Pathways to Quantum Coherence in Materials, an Energy \nFrontier Research Center funded by the U.S. Department of Energy, Office of Science, \nBasic Energy Sciences. A.S. acknowledges support from an ARCS Foundation \nFellowship, a Ford Foundation Predoctoral Fellowship, and a National Science \nFoundation Graduate Research Fellowship. E.F., and D.G.- G.’s work on this project \nwere supported by the U.S. Department of Energy, Office of Science, Basic Energy \nSciences, Materials Sciences and Engineering Division, under Contract No. DE-AC02-\n76SF00515. Dilution fridge support: Low-temperature infrastructure and cryostat \nsupport were funded in part by the Gordon and Betty Moore Foundation through Grant \nNo. GBMF3429. Part of the sample fabrication was conducted at Nano-fabrication  \nLaboratory at Fudan University. Y.Z. acknowledges financial support from  \nNational Key Research Program of China (grant nos. 2016YFA0300703,   \n2018YFA0305600), NSF of China (grant nos. U1732274, 11527805 and  \n11421404), Shanghai Municipal Science and Technology Commission (grant  \nnos. 18JC1410300 and 2019SHZDZX01) and Strategic Priority Research  \nProgram of Chinese Academy of Sciences (grant no. XDB30000000). K.W. and T.T. \nacknowledge support from the Elemental Strategy Initiative conducted by the MEXT, \nJapan,  Grant Number JPMXP0112101001, JSPS KAKENHI Grant Number \nJP20H00354 and the CREST(JPMJCR15F3), JST.  Author contributions:  F.W.,  Y.Z. \nand D.G.- G. supervised the project. G.C. fabricated samples. G.C., A.S. and E.F. \nperformed transport measurements. G.C., S.W., B.L., L.J., H.L. and Z.S. prepared \ntrilayer graphene and performed near-field infrared and AFM measurements. K.W. and \nT.T. grew hBN single crystals. G.C., A.S., E.F., D.G. -G., Y.Z. and F.W. analyzed the \ndata. G.C.,  A.S., M.A.K., D.G. -G. and F.W. wrote the paper, with input from all authors. \nCompeting interests:  M.A.K. is chair of the Basic Energy Sciences Advisory \nCommittee. Basic Energy Sciences provided funding for this work. Data and \nmaterials availability:  The data that support the findings of this study are available \nfrom the corresponding authors upon reasonable request. -0.10.00.1\n \n Energy (eV)..\nFig.1. Device structure, band structure and longitudinal resistance of ABC -TLG/hBN . \n(A) Optical image of ABC -TLG/hBN Hall bar device and measurement configuration. White \nscale bar: 5 mm. (B) Schematic cross -sectional view of the dual -gated device shown in (A). \nThe red atoms represent one unit cell of ABC -TLG. (C,D andE) Calculated single -particle \nband structure of ABC -TLG/hBN in the K valley for displacement fields D= 0.5, 0 and -0.5 \nV/nm. At D= 0.5 V/nm a finite bandgap separates the conduction and valence bands. The \nhighest energy valence miniband is topologically trivial with C= 0. The bandgap closes at D\n= 0 V/nm. At D= -0.5 V/nm the band inverts to generate a new bandgap.  The highest energy \nvalence miniband becomes topologically nontrivial with a finite Chern number. The \nminibands of the K’ valley are time-reversed pairs of the K valley minibands and have \nopposite Chern numbers. (F) Color plot of the measured longitudinal resistance Rxx= Vxx/ I\nas a function of band filling factor n/ n0and vertical displacement field Dat T= 1.5 K. In \naddition to the band insulating states at the charge neutrality point ( n= 0) and the fully filled \npoint ( n= -4 n0), correlated insulator states emerge at 1/4 filling ( n= -n0) and 1/2 filling ( n= \n-2 n0) at finite |D|.Si (bottom gate)SiO2top gate\n𝑉𝑡\n𝑉𝑏hBN (aligned)hBN (misaligned)\nIVyxVxx\nGC≠ 0C= 0D= 0.5 V/nm\nD= -0.5 V/nm\nK’ K GA\nBC F\nD= 0 V/nm\nKD\nE .\n...\n..\n..\n-0.10.00.1\n \n Energy (eV)\n-0.10.00.1\n \n Energy (eV)\n-4 -3 -2 -1 0 1 2 3 4-0.6-0.4-0.20.00.20.40.6D (V/nm)\nn/n0Rxx(kW)0 40\n-3 -2 -1050100\n  Rxx (k)\nn / n0\n-3 -2 -1-101B (T)\nn / n0\nRyx(k)-8 8\nD= 0.4 V/nm\n-3 -2 -10510\n  Rxx (k)\nn / n0\n-3 -2 -1-101B (T)\nn / n0 D= -0.47 V/nm\nRyx(k)-8 8B\nDA\nC\nFig. 2. Longitudinal and Hall resistances at selected values of D.(Aand B) Longitudinal \nresistance as a function of band filling at D= 0.4V/nm and -0.47 V/nm, respectively. (Cand \nD)  Corresponding color plots of the Hall resistance as a function of band filling and \nperpendicular magnetic field. For D= 0.4 V/nm in (C), the Hall resistance tends to be small \nfor all doping and no ferromagnetic signature is observed. (The relatively large signals at n\n= -n0and n= -2n0in (C) are artifacts coming from crosstalk from the large Rxx, evinced by \nthe observation that they do not change sign when the magnetic field is reversed.) For D = -\n0.47V/nm in (D), the Hall resistance Ryxis very large at weak magnetic fields and can \npersist down to zero magnetic field at n= -n0and n= -2.3 n0(denoted by the two dashed \nvertical lines). The non-zero values of Ryxat B= 0 represent AH signals from ferromagnetic \nstates at n= -n0and n= -2.3 n0. The state at n = -n0has been identified as a C= 2 Chern \ninsulator state ( two oblique solid lines) with orbital ferromagnetism ( 6). The AH at n= -2.3\nn0results from a new magnetic state that emerges at non-integer filling of the topological \nhole miniband.D= 0.4 V/nm\nB = 0 TD= -0.47 V/nm\nB= 0 TFig. 3. Anomalous Hall effect at non -integer filling. (Aand B) Color plots of Ryxand Rxx\nas functions of band filling and displacement field, respectively, for the valence miniband at \nB= 0.2 T and T= 0.07 K. The dashed oval outlines a region with large Ryx signals at very \nsmall field between n= -2.1 n0and -2.6 n0and between D= -0.4 and -0.53 V/nm (more \ndetail is given in Fig. 4). The oval region also shows enhanced Rxxin (B), suggesting that the \nferromagnetic phase is insulating. ( Cand D) Magnetic -field dependent antisymmetrized ρyx\nand symmetrized ρxxat different temperatures for the ferromagnetic state with largest AH at \nthe center of the oval region ( n= -2.3 n0and D= -0.48 V/nm). The AH signal shows clear \nferromagnetic hysteresis in (C), reaching 𝜌yxAH= -4.8 kWand a coercive field of Bc= 0.04 T \nat T= 0.07 K . (Eand F) The evolution of 𝜌yxAHand sxxas a function of temperature at n= -\n2.3 n0and D= -0.48 V/nm . The amplitude of 𝜌yxAHdecreases with increasing T, vanishing \naround 1.6 Kelvin, while sxxincreases with increasing T.C E\n0-2-4\n  \nrAH\nyx (kW)\n0.0 0.5 1.0 1.52468\nT (K)\n \nsxx (e2/h)\nD F\n-505\n T = 0.07 K\n 0.23 K\n 0.45 K\n 0.71 K\n 1.0 K\n 1.3 K\n 1.6 K\n \n  \nryx (kW)\n-0.1 0.0 0.1345\n \nrxx (kW)\nB (T)\nB= 0 T\nB\n-0.6-0.5-0.4D (V/nm)\n40 0Rxx(kW)\n-8Ryx(kW)A\nB= 0.2 T\nT= 0.07 K\n8\n-3 -2 -1-0.6-0.5-0.4D (V/nm)\nn / n0Fig. 4. Tunable anomalous Hall effect at non -integer filling. (A) The D-dependent \nferromagnetism at n= -2.5 n0. The AH resistance is maximal at D= -0.48 V/nm, becomes \nzero at D= -0.38 V/nm, and then reappears with a sign change at D= -0.37 V/nm. The ryx\nsignals at D= -0.49, -0.48 and -0.43 V/nm are manually multiplied by a factor of 0.1 for \nclarity. (B) The n-dependent ferromagnetism at a fixed D= -0.39 V/nm. The AH signal \nreaches a maximum at n= -2.48 n0, becomes nearly zero at n= -2.38 n0, and  reappears \nwith a sign change at n = -2.31 n0. Although the hysteresis is gone at n= -2.2 n0, the \nstrongly nonlinear dependence indicates a large AH; AH vanishes at less negative n. Each \ncurve in (A and B) is offset for clarity, and the dashed lines denote zero ryxsignal. (C)The \nindividual squares represent measured AH resistance, with color scale shown in the lower \nleft corner. The circles represent data with no measureable AH. B CA\n-3.0 -2.5 -2.0 -1.5-0.6-0.5-0.4\nn / n0\n D (V/nm)\n0.5 -5𝜌yxAH(kW)\n0\n-0.1 0.0 0.1-1.5-1.0-0.50.00.51.01.5\n-0.49 V/nm-0.48 V/nmx 0.1\nx 0.1-0.39 V/nm-0.38 V/nm\n-0.43 V/nm-0.37 V/nm\n  \nryx (kW)\nB (T)-0.36 V/nm\nx 0.1n = -2.5 n0\n-0.1 0.0 0.1-2-1012\n-2.38n0-2.20n0\n  \nryx (kW)\nB (T)-2.31n0\n-2.42n0\n-2.48n0\n-2.63n0D = -0.39 V/nmSupplementary Materials for \nTunable ferromagnetism at non-integer filling of a moiré  superlattice \nGuorui Chen+, Aaron L. Sharpe+, Eli J. Fox+, Shaoxin Wang, Bosai Lyu, Lili Jiang, \nHongyuan Li, Kenji Watanabe, Takashi Taniguchi, Michael F. Crommie,  M. A. \nKastner,  Zhiwen Shi, David Goldhaber-Gordon*, Yuanbo Zhang*, Feng Wang* \n \n+These authors contributed equally to this work. \n*Corresponding author. Email: fengwang76@berkeley.edu, zhyb@fudan.edu.cn, \ngoldhaber-gordon@stanford.edu \n \n \nThis PDF file includes: \nSupplementary Text \nFigs. S1 to S8  \nReferences and Notes  Contents \n1. Gate map of the ABC-TLG/hBN  device \n2. Comparison of AH between n = - n0 and n = -2.3 n0 \n3. n- and D-dependence of the strongest AH signals \n4. Additional anomalous Hall data \n5. Discussion on the origin of observed AH at non -integer filling \n6. Data for the second device \n \nSupplementary Text \n1. Gate map of the ABC-TLG/hBN  device \n The color plot of longitudinal resistance  Rxx as a function of n/n0 and D in Fig. 1F \nis calculated from the initially measured gate map of Rxx as a function of Vt and Vb in \nFig. S1.  From the capacitance model and quantum Hall effect, the fully filled point at \nn = ± 4n0 corresponds to the carrier density of n = ± 2.1× 1012 cm-2, from which the moiré  \nperiod is estimated to be 15 nm and the twist angle between ABC-TLG and hBN is zero \n(2).   \nFig. S1.  Gate map of ABC-TLG/hBN.  Color plot of R xx as a function of Vt and V b measured at T  \n= 1.5 K.  The data of Fig. 1F in the main text are created from these based on the relation between \nn, D and Vt, Vb discussed in the main text. \n \n2. Comparison of AH between n = - n0 and n = -2.3 n0 \n To compare the magnetic states in ABC- TLG/hBN, we plot Ryx as a function of B \nat n = - n0 for the Chern insulating state and n = -2.3n0 for the non -integer filling state \nat different temperatures side by side in Fig. S2. Here we select different D so that the \nstrongest AH is shown for both states.  The AH of the non -integer filling magnetic state \nat n = -2.3  n0 has smaller amplitude than the AH of the Chern insulator state at n = - n0, \nand they have opposite signs. The opposite sign AH signals might be an indication that \nthe net valley polarizations are opposite for n = - n0 and n = -2.3  n0. There are several \njumps in the hysteretic loop at n = -2.3  n0 suggesting a complex magnetic domain \nstructure.  The AH signal at n = -2.3  n0 disappears at T = 1.6 K, lower than that of the \nChern insulator state at n = - n0, which persists to 3.5 K. In addition, the AH signal of \nthe Chern insulator has a well quantized value at finite magnetic field and shows a \nquantum Hall dispersion governed by the Streda relation  n = CeB/h for C = 2. The AH \nat n = -2.3 n0, however, does not become quantized even at high magnetic field.  \n \nFig. S2. Ferromagnetism at n = - n0 and n = -2.3 n0. Ryx as a function of B field at different \ntemperatures for ( A) n = - n0 and (B) n = -2.3 n 0. The D field is selected to be -0.5 V/nm in ( A) and \n-0.48 V/nm in (B) to yield the largest  𝑅yxAH for the same device. \n \n3. n- and D-dependence of the strongest AH signals \n  Near D = -0.48 V/nm and n = -2.3 n0, where the AH signal is largest (see Fig. 4C \nin the main text), the longitudinal and Hall resistivities are quite sensitive to D and n. \nFig. S3A shows xx, 𝜌yxAH and Bc, the coercive field, for densities near this point. \nWhereas there is a large peak in xx at the integer filling of n = -2 n0, at most a very \nweak AH signal is detected there. On the other hand, the AH signal and the coercive \nfield are maximum near n = -2.3 n0. Figure S3B shows that there is a similar sensitivity \nto D in this region. Note that at this value of D we do not see the sign change of AH \ndiscuss in the text, seen at less negative D. \n \nFig. S3.  n- and D-dependence of ferromagnetism at non-integer filling.  (A) n- and (B) D-\ndependence of AH of the non-integer filling centered at n = -2.3 n0 and D = -0.48 \nV/nm at T = 0.07 K.  \n \n4. Additional anomalous Hall data \n We present here a sampling of the many magnetic field sweeps taken to generate \nthe data points in Fig. 4C in the main text. Figure S4 shows two values of n, which \ndisplay sign changes as D is varied and two values of D that show sign changes as n is \nvaried. The AH hysteretic loops at non-integer filling can be rather complicated at some \npoints in the parameter space of n and D. Some loops,  such as D = -0.48 V/nm in Fig. \nS4B and n = -2.65 n0 in Fig. S4D, show multiple intermediate jumps, and some \nhysteresis loops, such as that at D = -0.48 V/nm and n = -2.5 n0 in Fig. S4B and those \nat n = -2.70 n0 and -2.61 n0 at D = -0.45 V/nm in Fig. S4D,  appear at non-zero magnetic \nfields,  suggesting a complex domain structure. \n \nFig. S4. Additional Ryx data.  Ryx as a function of magnetic field at ( A) n = -2.3 n0, and (B) n = -2.5 \nn0 for various D, and ( C) D = -0.39 V/nm and ( D) D = -0.46 V/nm for various n. Each curve is offset \nfor clarity, and the dashed line with the same color as the data denotes Ryx = 0 for that data set.  The \ndashed line is not located at the center of the corresponding AH loop due to the cross-talk from Rxx. \nThe Ryx signals for D = -0.52, -0.48, -0.43 and -0.4 V/nm in (A) and D = -0.49, -0.48 and -0.43 \nV/nm in (B) are manually multiplied by a factor of 0.1. Data of this kind are used to generate the \ndata points in Fig. 4C of the main text. \n \n \n5. Discussion on the origin of observed AH at non -integer filling \n By plotting the behavior of the conductivity tensor xx and yx, we can get more \ninsight into the mechanism of the observed AH at non -integer filling. The AH effect \ncan either be intrinsic, arising from Berry curvature of the filled bands, or extrinsic, \nresulting from scattering mechanisms. We suggest AH at the non -integer filling is \nintrinsic. First, the AH is only observed for the sign of the D field which leads to the \nChern band according to our band structure calculation, and not at the other sign which \nis predicted to lead to a topological trivial band. Second, skew scattering, one of the \nextrinsic mechanisms leading to AH in magnetic metals, gives rise to a linear \nrelationship between yx and xx. Figure S5A and B clearly show nonlinear \nrelationships between yx and xx when either density or temperature are varied (32). It \nis more challenging to rule out side jump scattering, another extrinsic mechanism. Both \nof these mechanisms, however, can be effective only in metallic materials and, as \nshown in Fig. 2B of the main text, the AH occurs where the resistance is quite high. \nFurthermore the longitudinal conductivity increases with increasing temperature, as \nshown in Fig. S5C.  Third, the largest anomalous Hall angle yx/xx is close to unity , \nwhich is much larger than previously reported extrinsic or intrinsic AH materials (≤ 0.2)  \n(32,33)  except for Chern insulators (5–7,34–36), which also suggests the intrinsic \nmechanism. \n Fig. 5D shows yx and xx as functions of n at D = -0.48 V/nm at B = 0 T. The large \nyx is accompanied with a drop of xx, which is one would expect for a Chern insulator. \nWe believe that the absence of quantization of the former and non-zero value of the \nlatter probably results from the presence of domains. \n \nFig. S5.  Behavior of the conductivity tensor at B = 0. (A) The longitudinal conductivity xx is \nplotted parametrically against the Hall conductivity yx for a series of measurements at different \ntemperatures with the density and D field fixed at n = -2.3 n 0 and D  = -0.48 V/nm. ( B) xx is \nplotted parametrically against yx for a series of measurements at different densities at T  = 0.07 K, \nwith D = -0.48 V/nm. ( C) Arrhenius plot of xx on a log scale versus 1/ T. (D) xx and yx as \nfunctions of carrier density, from the same data as in (B), showing the emergence of a dip in xx \naccompanied by the yx maximum around n = -2.3n 0. \n \n6. Data for the second device \n The ferromagnetism at the non -integer filling is reproduced in the second ABC-\nTLG/hBN device. Fig. S6 A and B shows the optical image and schematic side view of \nthe second device. Figure S6 C shows Rxx as a function of n and D at T = 5 K. The ABC-\nTLG is aligned with the top hBN at zero twist angle. T he Chern band is, therefore, at \npositive D with weak resistance peaks at n = - n0 and -2  n0, and the topological trivial \nband is at negative D with Mott strong insulating resistance peaks at n = - n0 and -2 n0.    \n \n \nFig. S6.  Basic characterization of the second device.  (A) Optical image the second device. Scale \nbar: 5 m. (B) The illustration of the unit cell of ABC-TLG aligned with the top hBN layer. ( C) \nThe Rxx as a function of carrier density and vertical displacement field at T = 5.2 K.  \n \n Based on the knowledge from the first device discussed in the main text, we focus \non the valence minibands on the non -trivial side of D. In Fig. S7A we indicate values \nof n and D for which Ryx had been measured using red-, blue- and gray-filled squares , \ncorresponding to positive, negative and zero  AH, respectively. Fig. S7B shows some \ntypical AH hysteresis loops at D = 0.3 7 V/nm for various n. These show the sign change \nof AH discussed in the main text with reference to Fig. 4. Figures S7 C, D and E show \nthe evolution of the AH with carrier density.  In Fig. S7C we plot Rxy for as a function \nof n for the two directions of field sweep, which shows the hysteresis. Taking the \ndifference of these two sweeps shows the size and sign of the AH as displayed in Fig. \nS7D. Figure S7E shows how the coercive field varies with n. \n \n \nFig. S7.  Non-integer filling ferromagnetism in the second device.  (A) The background color \nscale gives R xx as a function of n / n0 and D for the non-trivial side. The red- and blue-filled \nsquares correspond to positive and negative AH where blue-white-red correspond to 𝑅yxAH = -2 \nk to 0 to 2k. (B) The evolution of AH hysteresis at different doping for D = 0.37 V/nm. From \ntop to bottom: n = -2.14 n0, -2.37n 0, -2.48n 0, -2.53n 0, -2.62n 0, -2.81n 0, manual offset of 6 k , 4 \nk, 2 k, 0, -2 k, -4 k, -6 k. (C) Ryx at B = 0 T for up and down sweeps of B  field as a \nfunction of carrier density for D  = 0.36 V/nm. ( D and E) The 𝜌yxAH and Bc as a function of carrier \ndensity for D = 0.36 V/nm. \n \n Fig. S8 compares the n- and D-dependent longitudinal resistivity and AH behavior \nin device 1 (discussed in the main text) and device 2. The qualitative behaviors are \nsimilar: (1) The ferromagnetic state  emerges in the doping range of -2.2 to -2.8  n0. (2) \nBoth negative and positive ferromagnetic hysteresis are present and they depend \nsensitively on D and n. However, there are significant differences in the detailed \nferromagnetic phase diagram. The magnetic states are more limited in D range, 0.35 to \n0.4 V/nm,  in device 2, compared to a range of about -0.35 to -0.55 V/nm for device 1.  \nThe 𝑅yxAH in device 2 is also smaller than the first device, with the large st AH at ~ 2 \nk. 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Lett.  113, 137201 (2014). \n "    },    {        "title": "1810.01606v1.Selective_Activation_of_an_Isolated_Magnetic_Skyrmion_in_a_Ferromagnet_with_Microwave_Electric_Fields.pdf",        "content": "Selective Activation of an Isolated Magnetic Skyrmion in a Ferromagnet with\nMicrowave Electric Fields\nAkihito Takeuchi1,a)and Masahito Mochizuki2, 3,b)\n1)Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 252-5258,\nJapan\n2)Department of Applied Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555,\nJapan\n3)PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012,\nJapan\nWe theoretically reveal that pure eigenmodes of an isolated magnetic skyrmion embedded in a ferromagnetic\nenvironment can be selectively activated using microwave electric \felds without exciting gigantic ferromagnetic\nresonances, in contrast to conventional methods using microwave magnetic \felds. We also demonstrate that\nthis selective activation of a skyrmion can e\u000eciently drive its translational motion in a ferromagnetic nanotrack\nunder application of an external magnetic \feld inclined from the normal direction. We \fnd that a mode with\ncombined breathing and rotational oscillations induces much faster skyrmion propagation than the breathing\nmode studied previously by Wang et al: [Phys. Rev. B 92, 020403(R) (2015)].\nPACS numbers: 76.50.+g,78.20.Ls,78.20.Bh,78.70.Gq\nA skyrmion crystal, a hexagonally crystalized state\nof magnetic skyrmions1{6, has characteristic resonance\nmodes at microwave frequencies7{10, which give rise to\nintriguing physical phenomena11such as microwave di-\nrectional dichroism12{15, spin-voltage induction16,17, and\nspin-current generation18. When a static magnetic \feld\nHexis applied perpendicular to a thin-plate specimen\nof the skyrmion-hosting magnet, several types of spin-\nwave modes emerge depending on the microwave po-\nlarization7. A microwave magnetic \feld H!normal to\nHz\nHxHex\n/g84\nxyz\nD0+/g39D(t)~\nmiz-1         0         1Csi\n-0.01     0      0.01 -0.700.7\n-0.010-0.005-0.002(a) (b)\n(c) (e) (d) (f)24 sites24 sites/g84=0˃ /g84=30˃ /g84=0˃ /g84=30˃\nFIG. 1. (color online). (a) Schematics of a magnetic bi-\nlayer system hosting skyrmions stabilized by the interfacial\nDzyaloshinskii-Moriya interaction. (b) External magnetic\n\feldHexwhere\u0012is an inclination angle from the normal\ndirection. (c), (d) Color maps of the normal component of\nmagnetizations mz(c) and scalar spin chiralities cs(d) of a\nskyrmion under a perpendicular Hex\feld. In-plane compo-\nnents of the magnetizations ( mx;my) are displayed by arrows.\n(e), (f) Those under an inclined Hex\feld with\u0012=30\u000e.\na)Electronic mail: akihito@phys.aoyama.ac.jp\nb)Electronic mail: masa mochizuki@waseda.jpthe skyrmion plane ( H!\n?) activates the so-called breath-\ning mode in which all the skyrmions constituting the\nskyrmion crystal uniformly expand and shrink in an os-\ncillatory manner. On the other hand, the H!\feld within\nthe skyrmion plane ( H!\nk) activates two types of rotation\nmodes with opposite rotational senses, in which cores of\nall the skyrmions circulate uniformly in counterclockwise\nand clockwise ways.\nIn addition to the crystallized form, skyrmions can\nappear as individual defects in a ferromagnetic state;\nsuch skyrmions are also expected to have peculiar col-\nlective modes19. Isolated skyrmions con\fned in a nano-\nferromagnet are potentially useful for applications20to\nmemory devices21, magnonics devices22{24, spin-torque\noscillators25,26, and microwave sensing devices27. As\nsuch, it is necessary to clarify the microwave-active eigen-\nmodes of a single skyrmion in a ferromagnetic specimen.\nIn addition, it is important to establish a way to ma-\nnipulate isolated skyrmions using microwaves for their\ndevice application. As the microwave \feld H!\n?cannot\nactivate precessions of the magnetizations when the mi-\ncrowave \feld is applied parallel to them, we can activate\npure breathing-type skyrmion oscillations with H!\n?un-\nder a perpendicular Hex\feld without exciting the back-\nground ferromagnetic state. However, once the Hex\feld\nis inclined, the microwave magnetic \feld H!\n?excites huge\nferromagnetic resonances, which drown out the weaker\nskyrmion resonances. Moreover, the microwave energy is\nabsorbed by the sample when exciting the gigantic fer-\nromagnetic resonances, which would inevitably result in\nhigh energy consumption and considerable temperature\nrise. Therefore, a technique to selectively activate an iso-\nlated skyrmion is urgently required.\nIn this Letter, we \frst theoretically show that the\neigenmodes of an isolated skyrmion embedded in a ferro-\nmagnetic environment can be selectively activated with a\nmicrowave electric \feld E!via oscillatory modulation of\nthe Dzyaloshinskii-Moriya interaction (DMI) without ac-arXiv:1810.01606v1  [cond-mat.mes-hall]  3 Oct 20182\ntivating ferromagnetic resonances. We then demonstrate\nthat translational motion of the skyrmion can be driven\nthrough activating its resonance modes using this electric\ntechnique in an inclined Hex\feld. The latter part of the\nresearch was motivated by the recent theoretical work\nby Wangetal: that demonstrated skyrmion propagation\nvia activating the breathing mode with a microwave H!\n?\n\feld28. Our study reveals that skyrmion motion can be\ndriven not only by the breathing mode but also by other\nE!-active modes. Furthermore, we \fnd that a mode with\ncombined clockwise and breathing oscillations can induce\nmuch faster propagation of the skyrmion than the previ-\nously studied breathing mode. Our \fndings pave a new\nway toward e\u000ecient manipulation of isolated skyrmions\nin nano-devices via the application of microwaves.\nWe consider a magnetic bilayer system composed of a\nferromagnetic layer and a heavy-metal layer with strong\nspin-orbit interactions [see Fig. 1(a)], where the spa-\ntial inversion symmetry is broken at their interface, and\nthereby the DMI is active. An inclined magnetic \feld\nHex= (Hx;0;Hz) withHx=Hztan\u0012is applied where \u0012\nis the inclination angle [see Fig. 1(b)]. For 0\u000e<\u0012\u001490\u000e,\ntheHex\feld is inclined toward the positive xdirection.\nThe DMI favors a rotating alignment of the magneti-\nzations, which results in the formation of a Neel-type\nskyrmion. The skyrmion has a circular symmetry un-\nder a perpendicular Hex\feld (\u0012= 0\u000e), but has dis-\nproportionate weight in distributions of the magnetiza-\ntions and scalar spin chiralities [see Fig. 1(c) and (d)].\nTo describe the magnetism in this magnetic bilayer sys-\ntem, we employ a classical Heisenberg model on a square\nlattice with magnetization vectors miwhose norm mis\nunity29,30. The Hamiltonian contains the ferromagnetic-\nexchange interaction, the Zeeman coupling to the mag-\nnetic \felds, and the interfacial DMI:\nH=\u0000JX\n<i;j>mi\u0001mj\u0000[Hex+H(t)]\u0001X\nimi\n+D(t)X\ni[(mi\u0002mi+^x)\u0001^y\u0000(mi\u0002mi+^y)\u0001^x]:(1)\nHereH(t) = (0;0;Hz(t)) and E(t) = (0;0;Ez(t)) rep-\nresent time-dependent magnetic and electric \felds ap-\nplied perpendicular to the sample plane, respectively.\nWe neglect magnetic anisotropies because they do not\nalter the results qualitatively although stability of the\nskyrmions and resonant frequencies of the eigenmodes\nmay be slightly changed. The strength of the interfacial\nDMI can be tuned by applying a gate electric \feld normal\nto the plane via varying the extent of the spatial inversion\nasymmetry31,32. The DMI coe\u000ecient D(t) =D0+\u0001D(t)\nhas two components, namely, a steady component D0and\naE(t)-dependent component \u0001 D(t) =\u0014Ez(t) with\u0014be-\ning the coupling constant. We take J=1 for the energy\nunits and take D0=J=0.27. For the inclined magnetic\n\feld, we take Hz=0.057 with \u0012being a variable. The\nunit conversions when J=1 meV are summarized in Ta-\nble I.Exchange int. J= 1 1 meV\nTime t= 1 0.66 ps\nFrequencyf=!=2\u0019!= 0:01 2.41 GHz\nMagnetic \feld H= 1 8.64 T\nTABLE I. Unit conversion table when J=1 meV.\nWe simulate the magnetization dynamics by numeri-\ncally solving the Landau-Lifshitz-Gilbert equation using\nthe fourth-order Runge-Kutta method. The equation is\ngiven by,\ndmi\ndt=\u0000\rmmi\u0002He\u000b\ni+\u000bG\nmmi\u0002dmi\ndt: (2)\nHere\u000bG(=0.04) and \rmare the Gilbert-damping con-\nstant and the gyrotropic ratio, respectively. The e\u000bective\n\feldHe\u000b\niis calculated as He\u000b\ni=\u0000(1=\rm\u0016h)@H=@mi.\nWe \frst calculate the dynamical electromagnetic and\nmagnetic susceptibilities \u001femand\u001fmm:\n\u001fem(!) =r\u00160\n\u000f0\u0001M!\nz\nEpulse; \u001fmm(!) =\u0001M!\nz\n\u00160Hpulse:(3)\nAfter applying a short pulse Hz(t) orEz(t) with dura-\ntion of \u0001t=1 in the units of J=\u0016h, we trace time pro\fles\nof the net magnetization Mz(t) = (1=N)P\nimzi(t) and\n\u0001Mz(t) =Mz(t)\u0000Mz(0) and obtain the Fourier trans-\nform \u0001M!\nz. Dividing this quantity by an amplitude of\nthe pulseHpulse orEpulse, we obtain these susceptibili-\nties. The calculations are performed using a system of\nN=160\u0002160 sites with periodic boundary conditions.\nFigure 2(a) displays the imaginary parts of the electro-\nmagnetic susceptibilities Im \u001fem(!) for several values of\n\u0012, which describe the response of the magnetizations to\ntheE!\feld. When \u0012= 0\u000e, the spectrum exhibits a sin-\ngle peak corresponding to the breathing mode activated\nby the oscillating DMI under the application of an AC\nelectric \feld. When the Hex\feld is inclined with \u00126= 0\u000e,\ntwo novel modes emerge, one with a higher and one with\na lower frequency than the breathing mode. The inten-\nsities of the novel modes increase, whereas the intensity\nof the original breathing mode is increasingly suppressed\nas\u0012increases.\nThe imaginary parts of the magnetic susceptibilities\nIm\u001fmm(!) in Fig. 2(b) describe the response of the mag-\nnetizations to the H!\feld. We \fnd that only a breath-\ning mode (m-Mode 2) appears when the Hex\feld is\nperpendicular ( \u0012= 0\u000e); however, its intensity decreases\nas\u0012increases. Remarkably, a large ferromagnetic reso-\nnance from the surrouding ferromagnetic magnetizations\nemerges under an inclined Hex\feld, whereas it is totally\nsilent under the perpendicular Hex\feld. We also \fnd a\nnovel mode (m-Mode 1) at lower frequencies.\nIn reality, the skyrmion has another H!-active mode\n(m-Mode 3) at higher frequencies, but it is hidden be-\nhind the gigantic ferromagnetic resonance and thus can-\nnot be seen in the spectra of Im \u001fmm(!). We can see this3\n0 0.04 0.0800.0050.010.015\n00.010.02\n0 0.04 0.080 0.04 0.08/g84=0˃\n/g90/g90Im/g70mmIm/g70em(arb. units)\nIm/g70mc(arb. units)\n00.010.02 Im/g70mm(arb. units)\n/g84=30˃/g84=25˃/g84=15˃/g84=10˃\n/g84=20˃/g84=0˃\n/g84=30˃/g84=25˃/g84=15˃/g84=10˃\n/g84=20˃/g84=0˃\n/g84=30˃/g84=25˃/g84=15˃/g84=10˃\n/g84=20˃\n/g90(a) (b) (c)\nIm/g70emIm/g70mc\ne-Mode 1e-Mode 2\ne-Mode 3m-Mode 3m-Mode2\nm-Mode1m-Mode 2\nm-Mode 1\nFMR/g68/g68=x\n/g68=y\ntotal\nE/g90||zHz\nHxH\n/g84\nxyz\nH/g90||zHxH\n/g84\nxyzHz\nFIG. 2. (color online). Imaginary parts of (a) the electromagnetic susceptibility Im \u001fem(!), (b) the magnetic susceptibility\nIm\u001fmm(!), and (c) the chirality susceptibility Im \u001fmc\n\u000b(!) for several values of \u0012. Here, an inclined magnetic \feld Hex=\n(Hztan\u0012;0;Hz) withHz=0.057 is applied. Three E!-active modes are labeled as e-Modes 1-3 in (a), whereas the three H!-\nactive modes are labeled as m-Modes 1-3 in (b) and (c). The extremely intense mode around !\u00180:05\u00000:06 in (b) is the\nferromagnetic resonance (FMR).\nweak response of the skyrmion to the H!\feld by fo-\ncusing on the vector spin chiralities, si=P\n\rmi\u0002mi+\r\n(\r= ^x;^y). The calculated imaginary parts of the dynam-\nical susceptibilities Im \u001fmc\n\u000b(!) for the\u000b-component of the\nvector spin chirality S\u000b= (1=N)P\nis\u000bi(\u000b=x;y) are\nshown in Fig. 2(c). We \fnd that they coincide with the\nspectra of Im \u001fem(!) in Fig. 2(a). These results indicate\nthat the magnetic method using H!cannot selectively\nactivate the eigenmodes of an isolated skyrmion in the\nferromagnetic specimen; however, the results show that\nthe electrical method using E!can achieve this. This\nelectrical technique is anticipated to play a crucial role\nfor developing future skyrmion-based devices.\nBased on the susceptibility data, we \fnd that an iso-\nlated skyrmion in a ferromagnetic specimen has three\nlow-lying eigenmodes. In Fig. 3, we show simulation re-\nsults of snapshots for \u0012= 30\u000e. It is found that all of\nthese modes have the breathing component, i.e., they\nshow oscillatory expansion and shrinkage. Among these\nthree modes, the higher-frequency mode ( !=0.0664) can\nbe regarded as a pure breathing mode, whereas the other\ntwo modes show distinct behaviors. The lower-frequency\nmode (!= 0:0453) is accompanied by a clockwise rota-\ntion of skyrmion in an elliptical orbit oriented horizon-\ntally against the inclined direction of Hexas shown in\nFig. 3(a). The moderate-frequency mode ( != 0:0513)\nis also accompanied by the clockwise rotation of the\nskyrmion in an elliptical orbit, but its trajectory is ori-\nented vertically against the inclined direction of Hexas\nshown in Fig. 3(b). The higher-lying pure breathing\nmode at!=0.0664 does not show any rotational compo-\nnent [Fig. 3(c)]. The three E!-active modes are referred\n123\n4\n1\n234\n1234\nxy25 sites25 sitest=2030 t=2064 t=2098 t=2132(a) e-Mode 1: CW + Breathing 1 ( /g90=0.0453)\n(b) e-Mode 2: CW + Breathing 2 ( /g90=0.0513)\n(c) e-Mode 3: Breathing ( /g90=0.0664)t=2053 t=2083 t=2112 t=2142\nt=2057 t=2081 t=2105 t=2129\n-1 1 mizCW\nrotation+\nrotation+\nCW\nbreathingbreathingbreathingt - t0Mz2/g83//g90\n12\n34FIG. 3. (color online). Simulated snapshots of the magnetiza-\ntion dynamics for three E!-active eigenmodes (e-Modes 1-3)\nof an isolated skyrmion in the ferromagnetic specimen under\nan inclined Hex\feld where Hz=0.057 and \u0012= 30\u000e.\nto as e-Modes 1, 2, and, 3 [see Fig. 2(a) and Fig 3].\nNext we numerically investigate the translational mo-\ntion of a skyrmion driven by the electrically activated res-\nonance modes under an inclined Hex\feld [see Fig. 4(a)].\nHere the inclination angle of Hexis \fxed at \u0012= 30\u000e.\nThe amplitude of the AC electric \feld Ez(t) =E!\nzsin!t\nis \fxed at \u0014E!\nz= 0:05D0= 0:0135. A recent ex-\nperiment for Ta/FeCoB/TaO xreported a huge E-\feld-\ninduced variation of the interfacial DMI that reaches\n140% when the applied voltage is 10 V. This observation\nsupports the experimental feasibility of the 5% modula-4\n-4 -3 -2 -1 0(/g90=0.0453)-8-6-4-20\n0 0.004 0.008 0.012/g90=0.0513/g90=0.0664\n/g90=0.0453/g90=0.0664\n/g90=0.0513-50\n-11 0 0.04 0.0800.005\n/g84=30°y coordinate\nx coordinate/g90\n0\n-11y coordinate-4 -3 -2 -1 0 -5\n0 0.04 0.0800.01\n/g84=30°\n/g90E/g90 activation\n(trajectory)\n/g78Ez/g90Displacement in x\n0 0.001 0.002 0.003-202Disp. in y\n-8-6-4-20Displacement in x\n-202Disp. in y\nHz/g90Im/g70em\nH/g90 activation\n(trajectory)E/g90 activation\n(drift velocity)\nH/g90 activation\n(drift velocity)(c)\nIm/g70mm(d)\n(e)\n(f)(g)\n(h)vx (m/s)-0.2\n-0.4\n-0.6\n-0.80vx (m/s)-0.2\n-0.4\n-0.6\n-0.80-0.200.2vy (m/s)\n-0.200.2vy (m/s)\n36 sites\n36 sitest=0 t=8ʷ103t=24ʷ103\nxy\ne-Mode 1\ne-Mode 3\ne-Mode 2e-Mode 3e-Mode 2e-Mode 1\nm-Mode 1\nm-Mode 3m-Mode 2m-Mode 3 + FMR\nm-Mode 2m-Mode 1\n~\nxyzHz\nHxHex\n/g84(b)\nmz\n-1         0         1(a)\nFIG. 4. (color online). (a) Illustration of skyrmion propa-\ngation driven by the E!\feld through activating a resonant\nmode of the skyrmion under an inclined Hex\feld. (b) Simu-\nlated snapshots of the skyrmion propagation for the E!-active\nmode with !=0.0513 (e-Mode 2). (c)-(e) Trajectories (c) and\ndrift velocities vx(d) andvy(e) of the propagating skyrmion\ndriven by the E!\feld for three di\u000berent resonant modes.\n(f)-(h) Those of the propagating skyrmion driven by the H!\n\feld. All the simulations were performed with an inclined\nHex\feld where Hz=0.057 and \u0012= 30\u000e.\ntion assumed here. Figure 4(b) shows simulated snap-\nshots of the skyrmion motion when the E!\feld with\n!=0.0513 is applied, which indeed displays propagation\nin the negative xdirection.\nThe trajectories of the propagating skyrmion during a\ntime period from t=0 tot=5000 are shown in Fig. 4(c) for\nthree di\u000berent E!-active modes at !=0.0453, 0.0513, and\n0.0664. They were obtained by tracing the center-of-mass\ncoordinate ( jx;jy) of the topological-charge distribution:\nj\r=X\ni=(ix;iy)i\rcs(ix;iy)=X\ni=(ix;iy)cs(ix;iy) (4)with\ncs=1\n8\u0019[mi\u0001(mi+^x\u0002mi+^y) +mi\u0001(mi\u0000^x\u0002mi\u0000^y)]:\n(5)\nWe \fnd that the direction and velocity of the motion\nvary depending on the skyrmion resonance mode. For\nall the modes, the skyrmion moves mainly in the nega-\ntivexdirection. However, the trajectories for e-Modes\n2 and 3 are slightly slanted toward the negative ydirec-\ntion; meanwhile, the trajectory for e-Mode 1 is perfectly\nparallel to the xaxis. Interestingly, despite the slanted\ntrajectory for e-Mode 2, its traveling distance along the\nxaxis is identical to that of the trajectory for e-Mode 1,\nwhich is directed perfectly along the xaxis. It can also\nbe seen that the directions of the skyrmion movement for\ne-Modes 2 and 3 are the same, even though their trav-\neling distances are di\u000berent. The traveling distance for\ne-Mode 3 is much shorter than that for e-Mode 2 because\nof the smaller intensity of the latter mode, as can be seen\nin the inset of Fig. 4(c).\nIn Fig. 4(d) and (e), we plot the velocities v=(vx;vy)\nof the skyrmion for three di\u000berent E!-active modes (see\nthe right vertical axes) as functions of the strength of the\ntime-dependent DMI, i.e., \u0014E!\nz. The values are calcu-\nlated from the simulated displacements of the skyrmion\nin thexandydirections for a time period from t=2000\ntot=10000 (see the left vertical axes) by assuming J= 1\nmeV anda=5\u0017A withabeing the lattice constant. It can\nbe seen that the velocities are proportional to the square\nof\u0014E!\nz, and they are on the order of 10\u00001m/s. In fact,\nthe traveling speed of the skyrmions achieved using this\ntechnique turns out to be relatively slow compared with\nthe speed achieved in techniques based on electric-current\ninjection33{40. However, the present technique has an ad-\nvantage: it is free from the Joule heating, and thus the\nenergy consumption and the temperature rise could be\nsigni\fcantly suppressed.\nWe \fnally study the skyrmion motion driven by AC\nmagnetic \felds H!under an inclined Hex\feld with\nHz=0.057 and \u0012= 30\u000e. The H!is applied perpendicular\nto the skyrmion plane, which is given by H!\nzsin!twith\nH!\nz= 0:05Hz= 0:00285. The simulated trajectories\nand velocities v=(vx;vy) are shown in Fig. 4(f) and (g),\nrespectively. We \fnd that the trajectories are again ori-\nented almost in the negative xdirection; however, for the\nferromagnetic resonance mode with !=0.0664 the trajec-\ntory is slanted toward the positive ydirection, which con-\ntrasts with the case of the E!-active mode. Noticeably,\nthe higher-frequency mode with !=0.0664 has a much\nfaster propagation of the skyrmion than the other two\nmodes. However, the usage of this mode is not ener-\ngetically e\u000ecient because this mode is not an eigenmode\nof the isolated skyrmion but a very intense resonance of\nthe vast ferromagnetic magnetizations, which necessarily\nleads to large energy consumption and considerable rise\nof device temperatures.\nIn summary, we have theoretically found that reso-\nnance modes of an isolated sykrmion in a ferromagnet can5\nbe activated by application of AC electric \felds through\noscillatory variation of the interfacial DMI. The advan-\ntage of this method compared with conventional methods\nusing AC magnetic \felds is that we can selectively excite\nskyrmions without activating gigantic ferromagnetic res-\nonances; this results in a signi\fcant suppression of both\nenergy consumption and temperature rise. Our result re-\nvealed that among the three E!-active modes, the mode\nwith combined clockwise and breathing oscillations in-\nduces much faster skyrmion propagation than the pre-\nviously studied breathing mode. Our \fndings will pave\na new way toward the e\u000ecient manipulation of isolated\nskyrmions and thus will be useful for future skyrmion-\nbased devices.\nThis work was supported by JSPS KAKENHI (Grant\nNo. 17H02924), Waseda University Grant for Special Re-\nsearch Projects (Project Nos. 2017S-101, 2018K-257),\nand JST PRESTO (Grant No. JPMJPR132A).\n1A. N. Bogdanov and D.A. Yablonskii, Sov. Phys. JETP 68, 101\n(1989).\n2A. Bogdanov and A. Hubert, J. Mag. Mag. Mat. 138, 255 (1994).\n3S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch, A.\nNeubauer, R. Georgii, and P. B oni, Science 323, 915 (2009).\n4X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y.\nMatsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010).\n5N. Nagaosa and Y. Tokura, Nat. 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Phys. 13,\n162 (2017)."    },    {        "title": "0809.3567v1.Quantized_Conductance_of_a_Single_Magnetic_Atom.pdf",        "content": "arXiv:0809.3567v1  [cond-mat.mtrl-sci]  21 Sep 2008Quantized Conductance of a Single Magnetic Atom\nN. N´ eel, J. Kr¨ oger,∗and R. Berndt\nInstitut f¨ ur Experimentelle und Angewandte Physik,\nChristian-Albrechts-Universit¨ at zu Kiel, D-24098 Kiel, Germany\n(Dated: November 1, 2018)\nA single Co atom adsorbed on Cu(111) or on ferromagnetic Co is lands is contacted with non-\nmagnetic W or ferromagnetic Ni tips in ascanning tunnelingm icroscope. Whenthe Co atom bridges\ntwo non-magnetic electrodes conductances of 2e2/h are found. With two ferromagnetic electrodes\na conductance of e2/h is observed which may indicate fully spin-polarized trans port.\nPACS numbers: 68.37.Ef,72.25.Ba,73.23.Ad,73.63.-b,73. 63.Rt\nThe conductance of nanometer-sized contacts may be\ndecomposed into contributions of transport eigenchan-\nnels according to G= G0/summationtextn\ni=1τi, where G 0= 2e2/h\nis the quantum of conductance (-e: electron charge, h:\nPlanck’s constant), and τiis the transmission probabil-\nity of the ith channel [1, 2]. The factor 2 in the quantum\nof conductance is due to spin degeneracy. In contacts in-\nvolving magnetic electrodes the spin degeneracy oftrans-\nport channels may be lifted. Each spin-polarized channel\nthenmaycontributeuptoG 0/2tothetotalconductance.\nA quantized conductance of G 0/2 is expected when a\nfully spin-polarized current is transmitted with a proba-\nbility of 1 through a spin-polarized transport channel.\nThese conditions appear difficult to fulfill. Neverthe-\nless, experimental observations of conductance quanti-\nzation in units of G 0/2 have repeatedly been reported\n[3, 4, 5, 6, 7, 8]. These conductances were observed with\n[4, 6] or without [3, 5, 7, 8] external magnetic fields,\nfor ferromagnetic [3, 4, 5, 6, 7] and non-magnetic elec-\ntrodes [7, 8]. On the other hand, the absence of non-\ninteger conductance quantization has also been inferred\nfrom experimental results [9, 10]. Untiedt et al.showed\nthat contaminantslike H 2orCO modify the conductance\nand could, in the case of CO adsorption on Pt electrodes,\ngive rise to a conductance of G 0/2 [10]. A considerable\nvariety of model calculations [11, 12, 13, 14, 15] have\nbeen performedand supportthe notionthat conductance\nquantization in units of G 0/2 is not expected from the\ninvestigated ferromagnetic contacts. It should be noted,\nhowever, that the modeling performed so far did not in-\nclude geometrical relaxations of the contacts although\nthe importance of the detailed atomic arrangement has\nbeen emphasized [11, 15, 16, 17].\nThe contradictory conclusions reached from the vari-\nous experiments may be related to a lack of characterisa-\ntion of the atomic details of the junction. This problem\nmay be reduced by using a cryogenic scanning tunnel-\ning microscope to probe the conductance of clean single-\natom contacts in ultra-high vacuum. Here we apply this\napproach to investigate prototypical junctions. A sin-\ngle magnetic atom on a spin-polarized island or a non-\nmagnetic substrate is contacted with non-magnetic and\nferromagnetic tips. We find that the conductance of asingle Co atom is G 0when two non-magnetic electrodes\nareused. With ferromagneticelectrodes the conductance\nis G0/2. Conductances of ≈0.9G0are observed for a\ncombination of a non-magnetic and a ferromagnetic elec-\ntrode. In contrast to previous experiments, the contact\ngeometry and chemistry are characterised by imaging of\nthe contact area prior to and after conductance measure-\nments. We hint that the observed conductance may be\nrelated to the detailed geometry and bonding at the con-\ntact.\nThe experiments were performed using a home-built\nscanning tunneling microscope operated at 7K and in\nultra-high vacuum with a base pressure of 10−9Pa. The\nCu(111) surface as well as chemically etched W and Ni\ntips werecleaned by argonion bombardment and anneal-\ning. Tungsten and Ni tips were fabricated from 0 .25mm\nthick wire of 99 .99% purity. Cobalt deposition onto\nCu(111) was performed at room temperature using an\nelectron beam evaporator and a Co evaporant of 99 .99%\npurity. Single Co atoms were deposited onto the cold\nsample surface through openings in the cryostat shields.\nThe tip magnetization direction is dictated by the shape\nanisotropy since magnetocrystalline anisotropy is small\nin Ni. Therefore, the tip is magnetized along its axis\nleadingtoamagnetizationperpendiculartothesubstrate\nsurface [18]. Cobalt islands on Cu(111) are well studied\n[19, 20, 21] and identified as single-domain ferromagnetic\nexhibiting a perpendicular magnetization with strong co-\nercivity and remanence [21].\nFigure 1 shows a pseudo-three-dimensional scanning\ntunneling microscopy (STM) image of a Co island and\nsingle Co atoms adsorbed on Cu(111) at 7K. The Co is-\nlandexhibitsatriangularshapeandathicknessoftwoCo\nlayers as expected. On top of the island a single Co atom\nwas adsorbed. These surface structures together with a\nnon-magnetic W or a ferromagnetic Ni tip provide four\ncontact configurations. The W or Ni tip may contact a\nCo atom adsorbedon the non-magnetic substrate surface\nor on a ferromagnetic Co island.\nCleanliness of the tip as well as of the Cu(111) surface\nand the surface of adsorbed Co islands was monitored\nby spectroscopy of the differential conductance (d I/dV).\nThe Shockley-type surface state of Cu(111) was observed2\nFIG. 1: (Color online) Pseudo-three-dimensional represen -\ntation of a constant-current STM image of a triangular Co\nisland adsorbed on Cu(111). Single Co atoms that are ad-\nsorbed on the substrate surface and on top of the island ap-\npear as protrusions (sample voltage: V= 100mV, current:\nI= 100pA, size: 24 .5nm×24.5nm).\nas a sharp step-like onset of d I/dV, while Co islands ex-\nhibited occupied as well as unoccupied dstates as pro-\nnounced peaks in spectra of d I/dV[21]. Therefore, the\npresence of contaminants, in particular of hydrogen, ad-\nsorbed on the substrate or the islands can be safely ruled\nout. Moreover, hydrogen adsorbed on Co islands has\nrecently been shown to desorb by using tunneling cur-\nrents exceeding 20nA [22]. In the contact experiments\nreported here, currents of the order of 10 µA are passed\nthrough the islands.\nFigure 2 displays the conductance of a single Co atom\nbetween different combinations of electrodes, as a func-\ntion of the displacement ∆ zof the microscope tip. For\nsmall tip displacements the conductance varies exponen-\ntiallywith thedisplacementasexpected forthetunneling\nregime(denoted1inFig.2). Inatransitionregion(2)the\nconductance rapidly increases. Finally, a smaller varia-\ntion of the conductance occurs in the contact region (3).\nTo define a contactconductance, Gc, we approximatethe\nconductance data in the transition and contact regions\nby straight lines. Their point of intersection defines the\ncontact conductance. This definition has previously been\nusedforcontactstonoblemetalatomsandmoleculesand\nwas found to reproduce their expected contact conduc-\ntances [23]. Conductance values obtained from Co atoms\naccording to this procedure are summarized in Table I.\nWhile for the W – Co – Cu junction the contact con-1.5\n1.0\n0.5\n0.0G (2e2/ h)\n1.00.80.60.40.20.0\n∆z (Å)123Gc\nFIG. 2: (Color online) Conductance Gvs. tip displacement\n∆zrecorded from a single Co atom with different electrode\ncombinations. W or Ni tips are used to contact a single Co\natom on a Cu substrate or on a Co island. Squares: W – Co\n– Cu, lozenges: W – Co – Co, triangles: Ni – Co – Cu, circles:\nNi – Co – Co. Tunneling (1), transition (2) and contact (3)\nregions of the conductance curves are indicated. Linear fits\nto the data in the transition and contact regions are used\nto define a contact conductance Gc. ∆z= 0 corresponds to\nV= 100mV and I= 500nA prior to opening the feedback\nloop of the microscope. The conductance curves for the W\n– Co – Cu, W – Co – Co, and Ni – Co – Cu contacts were\nvertically offset by 0 .6G0, 0.4G0, and 0.2G0, respectively.\nTABLE I: Contact conductances Gcof a single Co atom for\ndifferent electrode combinations. Uncertainty margins refl ect\nthe standard deviations of the contact conductances observ ed\nin repeated experiments\nTip–Atom–Surface Materials Gc(G0)\nW – Co – Cu 1 .03±0.02\nW – Co – Co 0 .88±0.02\nNi – Co – Cu 0 .85±0.02\nNi – Co – Co 0 .48±0.02\nductance is ≈G0, which is in agreement with single-Co\nconductance measured on Cu(100) [24], we observe lower\ncontactconductanceswith ferromagneticelectrodes. The\nmoststrikingresultisobtainedwhenaCoatomadsorbed\nto a Co island is contacted by aferromagnetictip. In this\ncase the contact conductance is ≈G0/2. Combinations\nof non-magnetic and ferromagnetic electrodes lead to Co\natom conductancesof0 .88G0(W –Co – Co) and 0 .85G0\n(Ni – Co – Cu).\nWe note that conductance curves acquired for voltages\n|V| ≤0.1V exhibited the same characteristics as pre-\nsented in Fig.2. Voltages |V|>0.1V led to an enhanced3\nmobility of Co atoms adsorbed on Co islands. We expe-\nrienced the Co atom to change its adsorption site - either\nto the tip or to an adjacent site on the island - during tip\nexcursion toward the adsorbed atom.\nThese results clearly show that the conductance of a\nferromagnetic Ni – Co – Co contact is close to G 0/2. At\npresent, it is not clear whether this value is due to trans-\nportthroughacombinationofpartiallyopen channelsor,\nmost excitingly, a single fully spin-polarized channel. In\nany event, the experimental result appears to contradict\ncurrent modeling results on ballistic transport through\nmagnetic constrictions which do not indicate conduc-\ntance quantization in units of G 0/2.\nWe suggest that relaxations of the atomic positions\nin the contact region may be at the origin of this dis-\ncrepancy. Recent theoretical work by H¨ afner et al.[15]\nlends support to this interpretation. The conductance\nof atomic-size Co and Ni contacts and the spin polariza-\ntion of the current were reported to be very sensitive to\nthe contact geometry. In particular, for electrode sepa-\nrations in the tunneling range, H¨ afner et al.calculated\na conductance of essentially G 0/2 reflecting a spin po-\nlarization of the current of nearly 100%. For smaller\nelectrode separations at contact, however, the spin po-\nlarization was predicted to drop sharply to zero over a\nrange of ≈0.5˚A. Relaxations of atomic positions ow-\ning to adhesive forces, which have been found for other\nsingle-atom [25] and single-molecule [26] contacts, were\nnot included in these calculations. They may shift the\nrange of distances where spin polarization is lost. This\nscenario, which remains to be analyzed by detailed calcu-\nlations, wouldimply that theobservedG 0/2conductance\nis due to a fully spin-polarized channel.\nIn conclusion, weobservedaconductanceofG 0/2from\nsingle Co atoms between ferromagnetic electrodes. The\ncontacted atom as well as the status of the electrodes\nwere characterized by imaging and spectroscopy of the\natom and the hosting surface. The observed conduc-\ntance reduction from ≈1G0to≈G0/2 appears to con-\ntradict available modeling results for transport through\nmagnetic atoms. It calls for calculations which take into\naccount the detailed structure and bonding of the junc-\ntion.\nWe thank M. Brandbyge(Technical University of Den-\nmark) for preliminary calculations and discussions. Fi-\nnancial support by the Deutsche Forschungsgemeinschaft\nthrough SFB 668 is acknowledged.\n∗Electronic address: kroeger@physik.uni-kiel.de[1] R. Landauer, IBM J. Res. Dev. 1, 223 (1957).\n[2] M. B¨ uttiker, IBM J. Res. Dev. 32, 63 (1988).\n[3] J. M. Costa-Kr¨ amer, Phys. Rev. B 55, R4875 (1997).\n[4] T. Ono, Y. Ooka, H. Miyajima, and Y. Otani, Appl.\nPhys. Lett. 75, 1622 (1999).\n[5] F. Elhoussine, S. M´ at´ efi-Tempfli, A. Encinas, and L. Pi-\nraux, Appl. Phys. Lett. 81, 1681 (2002).\n[6] M. Shimizu, E. Saitoh, H. Miyajima, and Y. Otani, J.\nMagn. Magn. Mater. 239, 243 (2002).\n[7] V. Rodrigues, J. Bettini, P. C. Silva, and D. Ugarte,\nPhys. Rev. Lett. 91, 096801 (2003).\n[8] D. M. Gillingham, I. Linington, C. M¨ uller, and J. A. C.\nBland, J. Appl. Phys. 93, 7388 (2003).\n[9] C. Sirvent, J. G. Rodrigo, S. Vieira, L. Jurczyszyn, N.\nMingo, and F. Flores, Phys. Rev. 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B¨ oricke, E. Lægsgaard, I. Stens-\ngaard, A. Ruban, J. K. Nørskov, and F. Besenbacher,\nSurf. Sci. 387, 86 (1997).\n[21] O. Pietzsch, A. Kubetzka, M. Bode, and R. Wiesendan-\nger, Phys. Rev. Lett. 92, 057202 (2004).\n[22] M. Sicot, O. Kurnosikov, O. A. O. Adam, H. J. M.\nSwagten, and B. Koopmans, Phys. Rev. B 77, 035417\n(2008).\n[23] J. Kr¨ oger, N. N´ eel, and L. Limot, J. Phys.: Condens.\nMatter20, 223001 (2008).\n[24] N. N´ eel, J. Kr¨ oger, L. Limot, K. Palotas, W. A. Hofer,\nand R. Berndt, Phys. Rev. Lett. 98, 016801 (2007).\n[25] L. Limot, J. Kr¨ oger, R. Berndt, A. Garcia-Lekue, and\nW. A. Hofer, Phys. Rev. Lett. 94, 126102 (2005).\n[26] N. N´ eel, J. Kr¨ oger, L. Limot, T. Frederiksen, M. Brand -\nbyge, and R. Berndt, Phys. Rev. Lett. 98, 065502 (2007)."    },    {        "title": "1403.6197v1.Unconventional_critical_scaling_of_magnetization_in_uranium_ferromagnetic_superconductors_UGe__2__and_URhGe.pdf",        "content": "arXiv:1403.6197v1  [cond-mat.supr-con]  25 Mar 2014APS/123-QED\nUnconventional critical scaling of magnetization\nin uranium ferromagnetic superconductors UGe 2and URhGe∗\nNaoyuki Tateiwa1,†Yoshinori Haga1, Tatsuma D. Matsuda1,2, Etsuji Yamamoto1, and Zachary Fisk1,3\n1Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai,\nNaka, Ibaraki 319-1195, Japan\n2Department of Physics, Tokyo Metropolitan University,\nHachioji, Tokyo 192-0397, Japan\n3University of California, Irvine, California 92697, USA\n(Dated: January 22, 2014)\nWe report a dc magnetization study of the critical phenomeno n around the ferromagnetic transi-\ntion temperature TCin high-quality single crystals of uranium ferromagnetic s uperconductors UGe 2\nand URhGe. The critical exponents, βfor the temperature dependence of the magnetization below\nTC,γfor the magnetic susceptibility, and δfor the magnetic isothermal at TChave been deter-\nmined with a modified Arrott plot, a Kouvel-Fisher plot, and t he scaling analysis. Magnetization in\nthe ferromagnetic state has strong uniaxial magnetic aniso tropy in the two compounds. However,\nthe universality class of the critical phenomena do not belo ng to the three dimensional (3D) Ising\nsystem. Although the values of βin UGe 2and URhGe are close to those in the 3D magnets, the\nvalues of γare close to unity, that expected from the mean field theory. S imilar critical exponents\nhave been reported previously for the 3D Ising ferromagnet U Ir where superconductivity appears\nunder high pressure. The critical behavior may be limited to a very narrow Ginzburg critical region\nof ∆TG∼1 mK because of the strong itinerant character of the 5 felectrons in the ferromagnetic\nsuperconductor UCoGe where the mean field behavior of the mag netization has been reported. The\nunconventional critical scaling of magnetization in UGe 2, URhGe and UIr cannot be explained via\nprevious approaches to critical phenomena. The ferromagne tic correlation between the 5 felectrons\ndiffers from that in the 3D Ising system and this difference may be a key point for the understanding\nof the ferromagnetism where superconductivity emerges.\nPACS numbers: 75.40.-s,75.50.Cc,74.25.Ha\nI. INTRODUCTION\nThe coexistence of superconductivity and ferromag-\nnetism, considered as a theoretical possibility over 50\nyears ago by Ginzburg[1], has been found in the ura-\nnium compounds UGe 2[2, 3], URhGe[4], and UCoGe[5].\nExtensivetheoreticalandexperimental studieshavebeen\ncarried out[6]. Since the middle of the 1970s, the coex-\nistence has been found in the 4 f-localized systems such\nas ErRh 4B4[7, 8], Chevrel compound HoMo 6S8[9], and\nboron carbide superconductor ErNi 2B2C[10]. The fer-\nromagnetism and superconductivity of the systems are\ncarried by different electrons: fanddelectrons respec-\ntively, and the states compete each other. A characteris-\ntic featurein the uraniumferromagneticsuperconductors\nisthatthesame5 felectronsoftheuraniumatomsarere-\nsponsible for both long-range ordered states. Interesting\nphysical phenomena such as anomalous enhancement of\nthe upper critical field Hc2for the superconducting state\nunderhigh magneticfieldmayoriginatefromcooperative\ninterplay between the two phases[6].\nCritical ferromagnetic fluctuations have been thought\n∗Phys. Rev. B 89, 064420 (2014).\n†Electronic address: tateiwa.naoyuki@jaea.go.jpto induce unconventional superconductivity in the vicin-\nity of a quantum phase transition[11]. Spin fluctua-\ntion theories reveal the importance of the dimensional-\nity of the spin-fluctuation spectrum for the unconven-\ntional superconductivity[12–14]. In particular, longitu-\ndinal spin fluctuations play an important role for spin-\ntriplet superconductivity as was experimentally shown\nin a recent NMR experiment on UCoGe[15]. Indeed, the\nferromagnetic phase has strong Ising-type anisotropy in\nthe uranium ferromagneticsuperconductors[5, 6, 16]. Al-\nthough many studies have been done on the supercon-\nductivity and its related phenomena, a systematic and\ncomplete description of the ferromagnetic criticality has\nnot been made for the uranium ferromagnetic supercon-\nductors.\nFerromagnetism in the uranium ferromagnetic super-\nconductorsis carriedby the mobile5 felectrons[17]. Itin-\nerant ferromagnetism of the 5 felectrons may have mag-\nnetic properties differing from thosein intermetallic com-\npounds of 3 dtransition metals. The relaxation rate for\nthe magnetization density Γ in UGe 2and UCoGe does\nnot exhibit the linear Landau damping (Γ ∝q) charac-\nteristic of the itinerant ferromagnetism described by self-\nconsistent renormalized spin fluctuation (SCR) theory\n[18–20]. Dual nature of the 5 felectrons between itiner-\nant and localized characters has been suggested in Muon\nspin rotation spectroscopy and macroscopic experiments2\nTABLE I: Critical exponents β,γ,δfor different universality\nclasses[24].\nβ γ δ\nMean field 0.5 1 3\n3D Heisenberg 0.367 1.388 4.78\n3D XY 0.345 1.316 4.81\n3D Ising 0.326 1.238 4.80\n2D Ising 0.125 1.75 15\nin UGe 2[21–23]. It is necessary then to investigate par-\nticular features in the itinerant ferromagnetism of the 5 f\nelectrons.\nIn this paper, we present detailed dc magnetization\nstudies of UGe 2and URhGe to investigate the classical\ncritical phenomenon associated with the ferromagnetic\ntransition. UGe 2shows a second order phase transi-\ntion from the paramagnetic to the ferromagnetic (FM1)\nphases at TC= 52.6 K and URhGe orders ferromagnet-\nically at TC= 9.5 K[2–4, 6]. Superconductivity appears\nin the high pressure FM1 phase in UGe 2. URhGe has a\nsuperconducting transition with transition temperature\nTsc= 0.2 K at ambient pressure. We study the critical\nphenomena of the ferromagnetic states in the two com-\npounds where the superconductivity appears at low tem-\nperatures. It is found that the universality class for the\nferromagnetic transitions in UGe 2and URhGe does not\nbelong to the three dimensional (3D) Ising class expected\nfromthe stronguniaxialanisotropyin the magnetization.\nWe find a unique scaling relation which may be inherent\nto the uranium ferromagnetic superconductors.\nIn the vicinityofasecond-ordermagneticphasetransi-\ntion with Curie temperature TC, the divergence of corre-\nlation length ξ=ξ0|1−T/TC|−νleads to universal scal-\ning laws for spontaneous magnetization MSand initial\nsusceptibility χ.νis the critical exponent. The math-\nematic definitions of exponents from magnetization can\nbe described as follows[24].\nχ(T)−1∝ |t|−γ′\n(T < T C),|t|−γ(TC< T) (1)\nMS(T)∝ |t|β(T < T C) (2)\nµ0H∝MS1/δ(T=TC) (3)\nHere,tdenotes the reduced temperature t= 1−T/TC.\nParameters β,γ,γ′andδare the critical exponents. Ta-\nble I shows the theoretical critical exponents for various\nmodels.\nII. EXPERIMENT\nHigh-quality single crystal samples of UGe 2and\nURhGe have been grown by Czochralski pulling in a\ntetra arc furnace[16, 25]. Magnetization was mea-\nsured in a commercial superconducting quantum inter-\nference (SQUID) magnetometer (MPMS, Quantum De-\nsign). The internal magnetic field µ0Hwas determinedby subtracting the demagnetization field DMfrom the\napplied magnetic field µ0Hext:µ0H=µ0Hext-DM.\nThe demagnetizing factor Dwas calculated from the\nmacroscopic dimensions of the sample. The magnetic\nfield was applied along the magnetic easy aandc-axes\nof the orthorhombic structure of UGe 2and URhGe, re-\nspectively. We have determined the critical exponents\nin the compounds using a modified Arrott plot, critical\nisothermanalysis,a Kouvel-Fisherplot, and scalinganal-\nysis.\nIII. RESULTS\nA. Modified Arrott plot and critical isotherm\nConventional methods to determine the critical expo-\nnents and the critical temperature involve the use of Ar-\nrott plots. Isotherms plotted in the form of M2vs.H/M\nconstitute a set of parallel straight lines around TC. The\nplot assumes that the critical exponents follow mean-\nfield theory ( β= 0.5,γ= 1.0, and δ= 3.0). The H/M\nvs.M2plots in UGe 2and URhGe do not yield straight\nlines around TC, indicating that the mean field model\nis not valid. So, we have re-analyzed the magnetization\nisothermswiththeArrott-Noakesequationofstatewhich\nshould hold in the asymptotic critical region[26].\n(H/M)1/γ= (T−TC)/T1+(M/M1)1/β(4)\n, where T1andM1are material constants. In the\ncorresponding modified Arrott plot, the data for UGe 2\nand URhGe are represented in the form of M1/βversus\n(H/M)1/γas shown in Figure 1 (a) and (b). The val-\nuesβandγare chosen in such a way that the isotherms\nyield as closely as possible a linear behavior. A best fit\n10x1015\n5\n0M1//s98 (A/m)1//s98\n3020100\n(H/M)1//s103(b)\nURhGe\n T  = 8.6 K\n10.4 K∆T  = \n0.2 K4x1015\n0M1//s98 (A/m)1//s98\n3020100\n(H/M)1//s103(a)\n T  = 48.5 K\n 57.5 K∆T  = \n1.0 KUGe2\n \nFIG. 1: (Color online) Modified Arrott plot of magnetization\nin (a)UGe 2for 48.5 K ≤T≤57.5 K and 0.1 T ≤µ0H≤\n6.0 T and in (b) URhGe for 8.6 K ≤T≤10.4 K and 0.1 T\n≤µ0H≤2.0 T. Blues lines show fits to the data with the\nequation (4).3\n105M (A/m)\n0.1 1\n/s1090H (T)UGe2\nT  = 48.5 K\n 57.5 K∆T  = \n1.0 K(a)\n 52.5 K\n δ =4.163x1044567M (A/m)\n0.1 1\n/s1090H (T)URhGe\nT  = 8.6 K\n 10.4 K 9.45 K∆T  = \n0.1 K(b)\n  δ =4.41\nFIG. 2: (Color online) Magnetic field dependence of magne-\ntization in (a)UGe 2from 48.5 K to 57.5 K and in (b) URhGe\nfrom 8.6 K to 10.4 K. Dotted points indicate the critical\nisotherm at 52.5 K and 9.45 K for UGe 2and URhGe, respec-\ntively. Blues lines show fits to eq. (3) to obtain the critical\nexponent δ.\nof equation (4) to the data in UGe 2for 47.0 K < T <\n57.5 K and 0.1 T < µ0H <6 T yields TC= 52.6±0.1 K,\nβ= 0.334 ±0.002, and γ= 1.05±0.05. The fit of the\ndata in URhGe for 8.5 K < T <10.4 K and 0.1 T < µ0H\n<2.0 T yields TC= 9.44±0.02 K,β= 0.303 ±0.003,\nandγ= 1.02±0.03. The obtained critical exponents\nare shown in Table. II.\nThe third critical exponent δcan be determined from\nthe critical isotherm at TCaccording to the eq (3) as\nshown in Figure 2 (a) and (b). From fits to the isotherms\nat 52.5 K in UGe 2and at 9.48 K in URhGe with eq.\n(3), the value of δwas obtained as δ= 4.16±0.02 for\nUGe2and 4.41 ±0.02 for URhGe. These values are\nlower than that expected for 3D Ising ferromagnet ( δ=\n4.80). The value of δcan be calculated from βandγ\nusing Widom scaling relation δ= 1+γ/β[27]. The value\nofδwas estimated as 4.15 ±0.05 for UGe 2and 4.37 ±\n0.05 for URhGe. The values are consistent with those\ndetermined from the critical isotherms.\nIn UGe 2and URhGe, the values of the critical expo-\nnentβfor the magnetization are close to those in the 3D\nferromagnets. Meanwhile, the values of the exponents\nγfor the magnetic susceptibility and δfor the critical\nisotherms are smaller than those expected for the 3D\nIsing model.\nB. Kouvel-Fisher method\nThe critical exponents βandγcan be more accurately\ndetermined by the Kouvel-Fisher (KF) relations[28].\nIn the modified Arrott plot, the straight lines intersect\ntheM1/β-axis in the ferromagnetic state at the values\nMs1/βwhereMsis the spontaneous magnetization andin the paramagnetic state at χ−1/γ. The obtained tem-\nperature dependences of the spontaneous magnetization\nMsand the initial magnetic susceptibility χare shown\nin upper panels of Figure 3 (a) and (b). Solid lines rep-\nresent fits to the data with Eq. (2) and (1) for Ms(T)\nandχ−1(T), respectively. The KF method is based on\nfollowing two equations:\nMS(T)[dMS(T)/dT]−1= (T−TC−)/β(T) (5)\nχ−1(T)[dχ−1(T)/dT]−1= (T−TC+)/γ(T) (6)\nEq. (5) and (6) are derived from Eq. (4) in the limit\nH→0 forT <and> TC, respectively. The quanti-\ntiesβ(T) andγ(T) are identical with the critical values\nβandγ, respectively, in the limit T→TC. Accord-\ning to the equations, the values of βandγcan be de-\ntermined from the slope of Ms(T)[dMS(T)/dT]−1and\nχ−1(T)[dχ−1(T)/dT]−1-plots,respectively,at TCandthe\nintersection with the T-axis yields TCas shown in low\npanels of Figure 3 (a) and (b). Solid lines represent the\nfits to the data with Eq. (5) and (6). The exponents\nfor UGe 2are determined as β= 0.331 ±0.002 and γ\n= 1.03±0.02 with TC= (TC++TC−)/2 = 52.60 ±\n0.02 K by the KF method. The exponents for URhGe\nare determined as β= 0.303 ±0.002 and γ= 1.01±\n0.02 with TC= 9.47±0.01 K by the KF method. The\ncritical exponents in the two compounds are consistent\nwith those determined in the modified Arrott plot. The\n5\n0-1  (d /T)-1  (K) \n60 50 \nTemperature (K) -20 \n-10 \n0MS (d MS/T)-1  (K) \nTC = \n 52.60 K\n \n = \n  0.331\n \n   =\n  1.0310 \n5\n0 -1 4x10 4\n2\n0MS (A/m) (b) URhGe\nTC\nFM PM \n1\n0-1  (d /T)-1  (K) \n11 10 9 8\nTemperature (K) -5 \n0MS (d MS/T)-1  (K) \nTC = \n 9.47 K \n \n = \n  0.303 \n \n   = \n  1.01 20 \n10 \n0 -1 10x104\n0MS (A/m) (a)UGe2\nTC\nFM PM \nFIG. 3: (Color online) [Upper panels] Temperature depen-\ndence of the spontaneous magnetization Ms(T) and the in-\nverse of the initial magnetic susceptibility χ−1determined\nfrom the modified Arrott plot and [Lower panels] Kouvel-\nFisher plots for Ms(T) andχ−1in (a) UGe 2and (b) URhGe.4\n1.1\n1.0/s103eff \n0.01 0.1\nt = |(T-TC)/TC|URhGeUGe2(b)0.35\n0.30/s98eff \n0.01 0.1\nt = |(T-TC)/TC|URhGeUGe2(a)\nFIG. 4: (Color online) Effective exponents (a) βeffbelowTC\nand (b)γeffaboveTCas a function of reduced temperature t\n[=|(T−TC)/TC|] in UGe 2and URhGe.\nvalues of the critical exponent γin UGe 2and URhGe are\nalso close to unity in the KF method.\nThecriticalexponentssometimesshowvarioussystem-\natictrendsorcrossoverphenomenaasoneapproaches TC.\nThis occurs if a magnetic system is governed by various\ncompeting couplings or disorders. To check this possibil-\nity, it is useful to obtain effective exponents βeffandγeff\nas follows.\nβeff(t) =d[lnMs(t)]/d(lnt), (7)\nγeff(t) =d[lnχ−1(t)]/d(lnt) (8)\nThe effective exponents βeffandγeffas a function of re-\nduced temperature tin UGe 2and URhGe are plotted in\nFigure 4 (a) and (b). Both βeffandγeffshow a mono-\ntonict-dependence in the asymptotic critical region, sug-\ngesting that the obtained exponents are not those that\nhappen to appear around a crossoverregion between two\nuniversality classes as observed in Ni 3Al[29].\nC. Scaling theory\nWe want to know whether the set of the critical ex-\nponents are the same below and above TC. Analysis\nwith scaling theory can determine separately the values\nγ’ (T < T C) andγ(TC< T). Theory predicts the\nexistence of a reduced equation of state close to the fer-\nromagnetic transition temperature[24]:\nM(µ0H,t) =|t|βf±(µ0H/|t|β+γ) (9)\n, wheref+forTC< Tandf−forT < T Care regular\nanalytical functions. Defining the renormalized magneti-\nzation as m≡ |t|−βM(µ0H,t) and the renormalized field\nash≡H|t|−(β+γ), the scaling equation is rewritten as m105M/|t| β (A/m)\n1 10 100 1000\n(µ0H)/|t|( β+γ) (T)URhGe\n \nT <TC \n (b)\n \nTC = 9.47 K\nβ = 0.302\nγ ' = 1.00 ( T < TC)\nγ   = 1.02 ( TC < T )T  >TC\n 104105M/|t| β (A/m)\n1101001000\n(µ0H)/|t|( β+γ) (T) T <TC \n \n UGe2\n (a)\nTC = 52.79 K\nβ = 0.329\nγ ' = 1.00 ( T < TC)\nγ   = 1.02 ( TC < T ) T  >TC \n \n \nFIG. 5: (Color online) Scaled magnetization as a function\nof renormalized field following Eq. (9) below and above the\ncritical temperature TCfor (a) UGe 2and for (b) URhGe.\nSolid lines show best fit polynomials. The magnetization dat a\nin the temperature range t=|(T−TC)/TC|<0.1 are shown.\n=f±(h). This equation implies that M(µ0H,t)/|t|βas\na function of µ0H/|t|β+γproduces two universal curves:\none forT < T Cand the other for T > T Cif the cor-\nrectβ,γ, andtvalues are chosen. Figure 5 (a) and (b)\nshow the scaled magnetization as a function of renor-\nmalized field following Eq. (9) at different tempera-\ntures below and above TCin UGe 2and URhGe, respec-\ntively. The magnetization data in the temperature range\nt=|(T−TC)/TC|<0.1 are shown. All data points\nfall on two curves in the two compounds. The scaling\nanalysis yields TC= 52.79±0.02 K,β= 0.329±0.002,\nγ′= 1.00±0.02 forT < T C, andγ= 1.02±0.02 forTC\n< Tin UGe 2. The analysis yields TC= 9.47±0.01 K,\nβ= 0.302 ±0.001,γ′= 1.00±0.01 forT < T C, andγ\n= 1.02±0.01 forTC< Tin URhGe.\nIn previous neutron scattering experiments on\nUGe2[18, 30], the value of βwas determined as 0.36(1)\nfrom the temperature dependence of the magnetic scat-\ntering intensity below TCand mean field-like behavior of\nthemagneticsusceptibility(1 /χ∝T)wasobservedabove\nTC. These are consistent with the present study. An im-5\nTABLE II: Critical exponents β,γ, andδ, ferromagnetic transition temperature TC, and spontaneous magnetic moment µsin\nuranium ferromagnetic superconductors UGe 2, URhGe, UIr[34, 35], and UCoGe[5, 19].\nTC(K) β γ′(T < T C)γ(TC< T) δ µ s(µB/U)\nUGe2 1.46\nModified Arrott 52.6 ±0.1 0.334 ±0.002 1.05 ±0.05\nKouvel-Fisher 52.60 ±0.02 0.331 ±0.002 1.03 ±0.02\nScaling 52.79 ±0.02 0.329 ±0.002 1.00 ±0.02 1.02 ±0.02\nln(M) vs. ln(µ0H) 4.16±0.02\nURhGe 0.42\nModified Arrott 9.44 ±0.02 0.303 ±0.002 1.02 ±0.03\nKouvel-Fisher 9.47 ±0.01 0.303 ±0.002 1.01 ±0.02\nScaling 9.47 ±0.01 0.302 ±0.001 1.00 ±0.01 1.02 ±0.01\nln(M) vs. ln(µ0H) 4.41±0.02\nUIr[34, 35] 45.15 0.355(50) 1.07(10) 4.01(5) 0.5\nUCoGe[5, 19] 2.5 ∼Mean field type ∼ 0.05\nportantresultofthescalinganalysisisthatsetofthecrit-\nical exponents in UGe 2and URhGe are the same above\nand below TC. The values of γare close to unity below\nand above TC.\nThe value of α, the critical exponent for the specific\nheat (C(T)∝|t|α), is estimated as ∼0.3 for UGe 2and\nURhGe using the Rushbrooke scaling relation ( α+2β+\nγ= 2)[31]. This suggests the failure of the mean field\ntheory (α= 0) where the specific heat does not show\na divergent behavior at the transition temperature. In\nthe theory, the contribution from the critical magnetic\nfluctuation to the specific heat becomes zero ( Cmag= 0)\naboveTC. Meanwhile, the values of the magnetic specific\nheatCmagand the thermal expansion αmagremain sig-\nnificant ( Cmag>0,αmag>0) in the paramagneticphase\nfor the temperature range t[= (T−TC)/TC]<0.1∼0.2\nin UGe 2(Fig. 4(a) and Fig. 7 in Ref. 32) and URhGe\n(Fig. 3 in Ref. 33). This suggests the development of\ncritical fluctuations above TC. The experimental obser-\nvations suggest that the mean field theory is insufficient\nto describe the thermodynamic quantities around TC.\nIV. DISCUSSION\nTable II summarizes the critical exponents β,γ, and\nδ, ferromagnetic transition temperature TC, and spon-\ntaneous magnetic moment µsin uranium ferromagnetic\nsuperconductors. As mentioned in the introduction,\nthe universality class for the 3D Ising model was ex-\npected from strong uniaxial anisotropy in the ferromag-\nnetic magnetization of UGe 2and URhGe. However, the\npresent study suggests that the universality class in the\ncompounds does not belong to any known class. While\nthe values of βare close to those in the 3D magnets, the\nvalues of γare close to unity, that expected from the\nmean field theory.\nThe critical exponents in the ferromagnetic compound\nUIr with TCof 46 K at ambient pressure are shown\nin Table II[34, 35]. Superconductivity has been found\nat high pressure in the ferromagnetic phase in UIr[36,37]. Although the magnetization shows strong uniaxial\nanisotropy in the ferromagnetic state in UIr, the univer-\nsality class of the critical phenomenon does not belong to\nthe 3D Ising class[34]. The values ofthe criticalexponent\nβandγare close to those in UGe 2and URhGe. These\nresults suggest a new universality class for the ferromag-\nnetic transition in the uranium ferromagnetic supercon-\nductors. In particular, the T-linear behavior of χmay be\na characteristic feature. We discuss several possibilities\nfor the unconventional critical scaling in these uranium\nferromagnetic superconductors.\n(1) A common feature in UGe 2, URhGe and UIr is\nthat the crystal structure can be regarded as coupled\nchains of the uranium atoms (“zigzag structure”) run-\nning along the a-axis in the orthorhombic structure of\nUGe2and URhGe, and along the b-axis in the mono-\nclinic structure of UIr[6, 36]. The magnetic moments\nalign parallel to the chain direction in UGe 2and per-\npendicular to the direction in the latter two compounds.\nThe nearest neighbor magnetic exchange interaction Jij\n=Jfor bonds along the chain direction may differ from\nthat (Jij=rJwithr >0) for bonds perpendicular to\nthe direction. The magnetic structure can be mapped\nonto the anisotropic 3D Ising model. The critical ex-\nponents in the uranium ferromagnetic superconductors\nare not reproduced even when the spatial anisotropic ex-\nchange interaction is introduced[38]. Also, the present\nresults are not consistent with numerical calculations on\nthe anisotropic next nearest neighbor 3D Ising (ANNNI)\nmodel[39].\n(2) Next, we discuss the itinerancy of the mobile 5 f\nelectrons. The long-range interactions of the delocalized\nmagnetic moments yield the mean field theory-like be-\nhavior even very close to TC. It is necessary to know\nwhether the analyses were done in the asymptotic criti-\ncalregionwhoseextentcanbeestimatedbytheGinzburg\ncriterion[40–42].\n∆TG=TCk2\nB/[32π2(∆C)2ξ06] (10)\nHere, ∆Cis the jump of the specific heat at the ferro-\nmagnetic transition temperature and ξ0is the magnetic6\ncorrelation length. The Ginzburg criterion characterizes\nthe temperature range where the mean field treatment\ndoes not hold. The stronger the itinerant character of\nthe electron becomes, the narrower the asymptotic crit-\nical region. For example, the value of ∆ TGin itinerant\nferromagnet ZrZn 2with the spontaneous magnetic mo-\nmentµs= 0.16µB/f.u. and the magnetic correlation\nlengthξ0=33˚Awasestimatedas∆ TG=0.4mK[42,43].\nThe experimentally determined critical exponents in the\ntemperature range t=|(T−TC)/TC|<0.1 are of the\nmean field type. The value of ∆ TGin UGe 2is estimated\nas∼100 K using the neutron scattering and specific heat\ndata[3, 18]. Therefore, our data are collected inside the\nasymptoticcriticalregionwherethe meanfield treatment\nfails. We cannot estimate ∆ TGfor URhGe and UIr since\nthere has been no report on the correlation length ξ0\nin the two compounds. The analyses for URhGe and\nfor UIr in Ref. 34 suggest that the temperature ranges\nof the asymptotic critical region t[=|(T−TC)/TC|] are\nlarger than 0.1 in the two compounds. The values of\nthe exponent βin UGe 2, URhGe and UIr are extremely\nsmaller than that ( β= 0.5) in the mean field theory.\nThe observed T-linear behavior of 1/ χdoes not indicate\nthat the analyses were done outside the asymptotic criti-\ncal region. Furthermore, the present analyses suggest no\nasymmetry in the temperature range of the asymptotic\ncritical region in UGe 2and URhGe.\nIntheferromagneticsuperconductorUCoGe, thespon-\ntaneous magnetic moment is 0.05 µB/U, one order of\nmagnitude smaller than those in UGe 2and URhGe[2,\n4, 5]. The magnetic correlation length is estimated as\nξa,b\n0= 86˚A along the aandbdirections and ξc\n0= 32\n˚A along the cdirection[19]. The stronger itinerant char-\nacter of the 5 felectrons suggests a narrower asymptotic\ncritical region in UCoGe. Indeed, the value of ∆ TGis es-\ntimated as less than ∆ TG∼1 mK from the specific heat\nand the neutron scattering data[6, 19]. The experimen-\ntally observed critical phenomenon is expected to be of\nmean field type. The H/Mvs.M2(Arrott-) plots yield\nstraight lines around TC[5, 44]. The critical behavior is\nmasked by the strong itinerant character of the 5 felec-\ntrons in UCoGe.\n(3) Even though a localized moment system, the uni-\nversality class of the magnetic phase transition depends\non the range of the exchange interaction J(r). It is noted\nthat the critical exponents for 3D Heisenberg, XY, Ising\nmodels and 2D Ising models in Table I are of short-\nrange type, i.e., the magnetic interaction falls off rapidly\nwith distance. Fisher et al.performed a renormaliza-\ntion group theory analysis of systems with the exchange\ninteraction of a form J(r)∼1/rd+δ, wheredis the di-\nmension of the system and δis the range of exchange\ninteraction[45]. Calculations showed that such a model\nfor long-range interactions can hold for δ <2. The ex-\nponentγis given as γ= Γ{δ,d,n}, where Γ is a known\nfunction [Eq. (9) of Ref. 45] and nis the dimension\nof the order parameter. This theory has been examined\nfor different sets of {d:n}(d,n= 1, 2, 3), following aprocedure similar to Ref. 46 which reported the critical\nphenomenon in Pr 0.5Sr0.5MnO3[46]. We do not find a\nreasonable solution of δthat reproduces the critical ex-\nponents in UGe 2, URhGe and UIr.\n(4) Classical dipole-dipole interaction affects the criti-\ncal phenomenon. The case in gadolinium ( TC= 292.7 K,\nµs= 7.12µB/Gd) has been extensively studied[47]. The\neffect of the dipole-dipole interaction in UGe 2, URhGe\nand UIr may be small since the strength of the effect\nis proportional to the square of the spontaneous mag-\nnetic moment µs[48]. The critical exponents in the ura-\nnium ferromagnetic superconductors are not consistent\nwith those of critical phenomenon associated with the\nisotropic or anisotropic dipole-dipole interaction[49, 50].\n(5) In metallic ferromagnets, the mean square ampli-\ntude of the local spin density S2\nLpersists in the para-\nmagnetic state above TCand temperature change in its\namplitude is the origin of the Curie-Wiess behavior in\nthe magnetic susceptibility[20]. The SCR theory gives\nχ−1(T)∝(T−TC)2andMS2(T)∝(T4/3\nC−T4/3) around\nTC,i.e,γ= 2,β= 1/2, and δ= 3, which are not\nconsistent with those in UGe 2, URhGe and UIr. Some\nweak ferromagnets of the 3 dtransition metal such as\nMnSi or Co 2CrGa show an anomalous critical isotherm\n(µ0H∝M1/δ) with a value of δclose to 5[51, 52]. The\nvalue is larger than that ( δ= 3.0) in the SCR or the\nmean field theory. The behavior has been explained with\nthe spin fluctuation theory by taking into account the\nzero point fluctuation under requirements of the total\nspin amplitude conservation (TAC) and the global con-\nsistency (GC)[53, 54]. Although the values of δin UGe 2,\nURhGe and UIr are larger than that in the mean field\ntheory,the T-linearbehaviorof1/ χisnotconsistentwith\nthe theory ( γ= 2). We hope that the uniaxial magnetic\nanisotropy will be taken into account in the spin fluctu-\nation theory.\nAs discussed in (1)-(5), the anomalous critical expo-\nnents in UGe 2, URhGe and UIr cannot be explained\nwith previous approaches to critical phenomenon. The\npresent study suggests that the ferromagneticcorrelation\nbetween the 5 felectrons differs from that in the 3D Ising\nsystem and this difference is a key point for the under-\nstanding of the ferromagnetism where superconductivity\nemerges. Finally, we propose following two viewpoints\non the ferromagnetism in the uranium superconductors\nfor future studies on the anomalous critical scaling.\n(i) We mention the dual nature of the 5 felectrons be-\ntween itinerant and localized characters in UGe 2[21–23].\nThe non-Landau damping for the magnetization density\nΓ in UGe 2and UCoGe has been discussed theoretically\nbased on the duality[55, 56]. Theoretical models for the\nsuperconductivity in the ferromagnetic state in URhGe\nand the antiferromagnetic state in UPd 2Al3have been\ndeveloped based on the duality model[57–59]. The du-\nality may be a key point for the co-existence of the fer-\nromagnetism and the superconductivity. The correlation\nlength of the itinerant component with a magnetic mo-\nment of∼0.02µB/U was estimated as ξ= 84˚A in UGe 27\nby the Muon spin rotation spectroscopy. This value is\nsignificantly larger than that found in the neutron scat-\ntering experiment whose main contribution comes from\nthe localized component since the magnetic scattering\nintensity is proportional to the square of the magnetic\nmoment[18]. A novel type of critical phenomenon may\nappear due to the two correlation lengths as well as a\nHund-like coupling between the two components.\n(ii) UGe 2has a tricritical point where the paramagnet\nto ferromagnet transition changes from a second-order to\na first order phase transition when driven toward the fer-\nromagnetic QCP by applying external pressure[60], sim-\nilar to several itinerant ferromagnets such as ZrZn 2[61],\nCo(S1−xSex)2[62], and MnSi[63]. This change of the\nnature of the transition has been regarded as a phe-\nnomenon specific to the quantum phase transition[64].\nRecently, the pressure effect on the ferromagnetic tran-\nsition has been re-considered from different points of\nview. When the ferromagnetic transition temperature\nis strongly pressure-dependent, the magneto-elastic in-\nteraction or the critical fluctuation of the order parame-\nter provides development of the first order instability at\nthe phase transition[65–67]. Neutron Larmor diffraction\nstudy reveals that magneto-elastic coupling is strength-\nened at the pressures where superconductivity appears\nin UGe 2[68]. Mineev shows that the order parame-\nter fluctuations give rise to the logarithmic increase of\nthe specific heat near TCin the uranium ferromagnetic\nsuperconductors[65]. As mentioned before, the magnetic\nspecific heat Cmagand the magnetic thermal expansion\nαmagin UGe 2and URhGe show the anomalous tempera-\nturedependencejustabove TC[32,33]. Futuretheoretical\nstudy is necessary to determine for the effect of the order\nparameter fluctuations on the magnetization around TC.\nV. CONCLUSION\nA dc magnetization study has been done of the critical\nphenomenon around the ferromagnetic transition tem-\nperature in high-quality single crystals of uranium ferro-magnetic superconductor UGe 2and URhGe. We have\ndetermined the critical exponents, βfor the magneti-\nzation,γfor the magnetic susceptibility, and δfor the\nmagnetic isotherm at the transition temperature with a\nmodified Arrott plot, a Kouvel-Fisher plot, and the scal-\ning analysis. Although the magnetization shows strong\nuniaxialmagneticanisotropy,the universalityclassofthe\ncritical phenomenon does not belong the three dimen-\nsional (3D) Ising system. In the asymptotic critical re-\ngion, the values of βin UGe 2and URhGe are close to\nthose in 3D magnets but the susceptibility χshows a\nmean field-like behavior (1/ χ∝T). Similar critical ex-\nponents have been reported previously in 3D Ising ferro-\nmagnet UIr where the superconductivity appears under\nhigh pressure. The critical behavior may be limited to\na very narrow Ginzburg critical region of ∆ TG∼1 mK\nbecause of the strong itinerant character of the 5 felec-\ntrons in the ferromagnetic superconductor UCoGe. The\nanomalous critical exponents in UGe 2, URhGe and UIr\ncannot be explained via previous approaches to the crit-\nical phenomena. We suggest that this unconventional\ncritical scaling of magnetization is inherent in the ura-\nnium ferromagnetic superconductors and it reflects a pe-\nculiar feature of the ferromagnetism of the 5 felectrons\nwhere superconductivity emerges.\nVI. ACKNOWLEDGMENTS\nWe acknowledge discussions with Drs. H. Sakai, K.\nKubo, N. Metoki, Y. Tokunaga and K. Kaneko. We\nalso thank Dr. S. Kambe and Prof. V. P. Mineev\nfor giving suggestive comments which improve this pa-\nper. 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Ser. 273, 012085 (2011)."    },    {        "title": "1605.07319v3.Superconductivity_in_Uranium_Ferromagnets.pdf",        "content": "Superconductivity in Uranium Ferromagnets\nV.P.Mineev1;2\n1Commissariat a l'Energie Atomique, UGA, INAC-FELIQS, 38000 Grenoble, France;\n2Landau Institute for Theoretical Physics, 119334, Moscow, Russia\n(Dated: February 10, 2017)\nThe theoretical description and the survey of physical properties of superconducting states in\nthe uranium ferromagnetic materials are presented. On the basis of microscopic theory is shown\nthat the coupling between the electrons in these ferromagnetic metals by means of magnetization\n\ructuations gives rise the triplet pairing superconducting state and the general form of the order\nparameter dictated by the symmetry is established. The theory allows to explain some speci\fc\nobservations including peculiar phenomenon of reentrant superconductivity in URhGe in magnetic\n\feld perpendicular to the direction of spontaneous magnetization.\nIn addition we describe several particular topics relating to uranium superconducting ferromag-\nnets: (i) critical magnetic relaxation in dual localized-itinerant ferromagnets, (ii) phase transition\nto ferromagnetic state in Fermi liquid and UGe 2, (iii) superconducting ordering in ferromagnetic\nmetals without inversion symmetry.\nKey words: ferromagnetism, superconductivity\nPACS numbers: 74.20.Mn, 74.20.Rp, 74.70.Tx, 74.25.Dw, 75.40.Gb\nCONTENTS\nI. Introduction\nII. Order parameters, symmetry of states and quasiparticle spectrum\nA. Symmetry of superconducting states in orthorhombic ferromagnets\nB. Superconducting states in UCoGe\nC. Quasiparticles spectrum in a ferromagnet superconductor with triplet pairing\nIII. Superconducting states in microscopic weak coupling theory\nA. Triplet pairing by spin-\ructuations exchange\nB. Magnetic susceptibility of an orthorhombic ferromagnet\nC. Pairing amplitudes\nD. Critical temperature of phase transition to the paramagnetic superconducting state in UCoGe\nE. Phase transition from the paramagnetic to ferromagnetic superconducting state in UCoGe\nF. Superconducting states in orthorhombic ferromagnets\nG. Equal-spin-pairing states\nH. Equal-spin-pairing states near critical temperature\nIV. Physical properties\nA. Critical temperature\nB. Upper critical \feld parallel to c-axis in UCoGe\nC. Upper critical \feld in URhGe\nD. Zeros in spectrum and speci\fc heat at low temperatures\nV. Reentrant superconductivity in URhGe\nA. Phase transition in an orthorhombic ferromagnet under magnetic \feld perpendicular to spontaneous magne-\ntization\nB. Susceptibilities\nC. Superconducting state in vicinity of the \frst order transition\nD. Concluding remarks\nVI. Critical magnetic relaxation in uranium ferromagnets\nA. Critical magnetic relaxation in ferromagnets\nB. Magnetic relaxation in dual localized-itinerant ferromagnets\nC. Concluding remarks\nVII. Anisotropy of nuclear magnetic relaxation and the upper critical \feld in UCoGe\nA. Nuclear magnetic relaxation rate\nB. Upper critical \feld anisotropy\nVIII. First order phase transition to ferromagnet state in UGe 2\nA. Phase transition to ferromagnetic state in Fermi liquid theory\nB. Magneto-elastic mechanism of development of the \frst order type instability\nC. Speci\fc heat near the Curie temperaturearXiv:1605.07319v3  [cond-mat.supr-con]  9 Feb 20172\nD. First-order type transition in UGe 2\nE. Concluding remarks\nIX. Superconducting order in UIr\nX. Conclusion\nBibiliography\nI. INTRODUCTION\nSuperconducting and ferromagnetic ordering are usually antagonists each other. The reason is that the exchange\n\feld exceeds the paramagnetic limiting \feld or the \feld of the Cooper pairs depairing in singlet superconductors.\nNevertheless, the singlet superconductivity can coexist with ferromagnetism when the critical temperature of transition\nto superconducting state surpasses the Curie temperature as it is the case in so called ternary compounds actively\ninvestigated in nineteen eighties. The coexistence reveals itself in a form known as the Anderson-Suhl or crypto-\nferromagnetic superconducting state [1, 2] characterized by the formation of a periodic domain-like magnetic structure.\nThe structure period or domain size \u0015is larger than the interatomic distance and smaller than the superconducting\ncoherence length \u00180what weakens the depairing e\u000bect of the exchange \feld leading to an e\u000bective averaging of it to\nzero.\nThe co-existence of superconductivity and ferromagnetism recently discovered [3{6] in several uranium compounds\nUGe 2, URhGe, UCoGe, UIr possesses quite di\u000berent properties. In the \frst two compounds the Curie temperatures\nTCurie is more than the order of magnitude higher than their critical temperatures for superconductivity Tsc(Fig.1a,b).\nIn UCoGe the ratio TCurie=Tscat ambient pressure is about four (Fig.1c). This fact and also that the upper critical\n\feld at low temperatures strongly exceeds the paramagnetic limiting \feld in the \frst three compounds (see reviews\n[7{9]) indicate that here we deal with the Cooper pairing in the triplet state. In UIr the upper critical \feld is smaller\nthan the paramagnetic limiting \feld [6]. This, however, can be due to the low specimen quality caused by impurities,\ninhomogeneities etc, inasmuch the impurities in unconventional superconductors strongly suppress the upper critical\n\feld.\nFerromagnetism does not suppress the superconductivity with triplet pairing, hence, there is no reason for the\nformation of a cryptomagnetic state. Indeed, no traces of a space modulation of magnetic moments directions on the\nscale smaller than the coherence length has been revealed [4, 10{12]. On the other hand, the neutron depolarization\nmeasurements on UGe 2down to 4.2 K (that is in the ferromagnet but not superconducting region) establish, that the\nmagnetic moment strictly aligned along the a-axis, with a typical domain size in the bc-plane of the order 4 :4\u000210\u00004cm\n[13] that is about two orders of magnitude larger than the largest superconducting coherence length in the b-direction\n\u0018b\u00197\u000210\u00006cm. Similar size of domains has been recently measured in UCoGe.[14]\nSo, it is natural to consider these ferromagnetic superconductors as triplet superconductors similar to super\ruid\nphases of He3. It must be kept in mind, however, that unlike to the liquid helium which is completely isotropic neutral\nFermi liquid here we deal with superconductivity developing in strongly anisotropic ferromagnetic metals. Namely,\nUGe 2, URhGe, UCoGe have an orthorhombic structure with magnetic moment oriented along the a-axis in the \frst\nof these compounds and along the c-axis in the last two of them (Fig.2). UIr has monoclinic PbBi-type structure\n(space group P21) without inversion symmetry with magnetic moment oriented along [10 \u00161] direction.[15]\nThe magnetic moments in UGe 2[16], in URhGe [17] and in UCoGe [18] are mostly concentrated around uranium\nions. AtT= 0 they are 1 :4\u0016B;0:4\u0016b;0:07\u0016B, correspondingly. Although these values are signi\fcantly smaller than\nthe moment per uranium atom deduced from the susceptibility above TCurie : 2:8\u0016B;1:8\u0016B;1:5\u0016B, correspondingly,\nthis is still not su\u000ecient to treat the uranium compounds as completely itinerant ferromagnets. They are rather\ndual localized-itinerant ferromagnets. The most weakly delocalized material is UGe 2[16, 19] and the most itinerant\nis UCoGe[20, 21].\nThe local nature of magnetism in uranium ferromagnetic compounds put forward as the most plausible pairing\nmechanism the interaction between the conduction electrons by means of spin waves in the system of localized\nmoments. The \frst such type model has been applied to the superconducting antiferromagnet UPd 2Al3[22] and then\nto the reentrant superconductivity in ferromagnetic URhGe [23].\nThe general form of the order parameters of superconducting states in orthorhombic ferromagnets dictated by the\nsymmetry was found in the papers [24{26]. Then, the corresponding microscopic description of triplet superconductiv-\nity based on the pairing interaction due to exchange of magnetization \ructuations in the orthorhombic ferromagnets\nwith strong magnetic anisotropy has been developed [27, 28]. This approach has allowed to explain an interplay of\nthe pressure dependence of the Curie temperature and the critical temperature of the superconducting transition\nand the magnetic \feld dependence of pairing interaction for \feld orientations parallel or perpendicular to the direc-3\ntion of spontaneous magnetization. The latter, in its turn, allows to explain peculiar phenomenon of the reentrant\nsuperconductivity in URhGe [29] in magnetic \feld along the bdirection.\nHere we present the survey of the theory and the physical properties of the superconducting uranium compounds.\nStarting from the symmetry of superconducting states valid for a multi-band orthorhombic ferromagnet we limit\nourselves by the description of a simplest two-band spin-up, spin-down superconducting ferromagnet. The structure of\nthe superconducting order parameters and the quasiparticle spectrum are derived. Then we expose corresponding weak\ncoupling microscopic theory of pairing due to exchange by magnetic \ructuations in strongly anisotropic media with\northorhombic symmetry. The theory reproduces the superconducting order parameter dictated by the symmetry. The\nassumptions made in the previous treatments are demonstrated explicitly. Then we discuss low temperature speci\fc\nheat, upper critical \feld and other concrete properties of orthorhombic ferromagnet superconducting materials.\nSpecial attention is paid to the peculiar phenomenon of reentrant superconductivity in URhGe [30]. In frame of\nthe Landau phenomenological theory it is demonstrated that a magnetic \feld perpendicular to the direction of easy\nmagnetization decreases the Curie temperature and at strong enough \feld the phase transition between anisotropic\nferromagnetic and paramagnetic states is changed from the second to the \frst-order. The pairing interaction strongly\nincreases in vicinity of the transition stimulating the reentrance to the superconducting state suppressed by the orbital\nmechanism.\nWe show that the magnetic \feld along the direction of spontaneous magnetization suppresses the longitudinal\n\ructuations of magnetization. This allows to explain the peculiar phenomena of \feld direction dependence of nuclear\nmagnetic resonance relaxation [31] and sharp anisotropy of the upper critical \feld [9, 32] in UCoGe.\nIn addition we describe several particular topics related to the superconductivity in uranium ferromagnets.\nFirst, we discuss the origin of non-Landau damping of critical magnetic \ructutions in ferromagnets with dual\nlocalized-itinerant nature of f-electrons [33, 34].\nThen, we consider the phase transition to ferromagnetic state in UGe 2[35] and in Fermi liquid.\nFinally, we expose the general structure of superconducting ordering in ferromagnetic metals without inversion\nsymmetry as it is in UIr.\nII. ORDER PARAMETERS, SYMMETRY OF STATES AND QUASIPARTICLE SPECTRUM\nA. Symmetry of superconducting states in orthorhombic ferromagnets\nWe shall consider two band ferromagnetic metal with the electron spectra\n\"\"(k) =\u0018\"(k) +\u0016; \"#(k) =\u0018#(k) +\u0016 (1)\nfor the spin-up and the spin-down bands ( see Fig.3), where \u0018\"; \u0018#are the energies counted from the chemical potential\n\u0016.\nThe spin-triplet superconducting state arising in a ferromagnetic metal consists of the spin-up, spin-down and\nspin-zero Cooper pairs described by the matrix order parameter [36]\n\u0001\u000b\f(k;r) =\u0012\n\u0001\"(k;r) \u00010(k;r)\n\u00010(k;r) \u0001#(k;r)\u0013\n= \u0001\"(k;r)j\"\"i + \u00010(k;r)(j\"#i +j#\"i) + \u0001#(k;r)j##i\n= (d(k;r)\u001b)i\u001by=\u0012\n\u0000dx(k;r) +idy(k;r)dz(k;r)\ndz(k;r)dx(k;r) +idy(k;r);\u0013\n; (2)\nwhere \u0001\"(k;r), \u0001#(r;k;r), \u0001 0(k;r) are the amplitudes of spin-up, spin-down and zero-spin of superconducting order\nparameter depending on the Cooper pair centre of gravity coordinate rand the momentum kof pairing electrons.\n\u001b= (\u001bx;\u001by;\u001bz) are the Pauli matrices. Equivalently, the order parameter can be written as the complex vector\nd(k;r) =1\n2\u0002\n\u0000\u0001\"(k;r)(^x+i^y) + \u0001#(k;r)(^x\u0000i^y)\u0003\n+ \u00010(k;r)^z: (3)\nHere and in what follows ^ x;^y;^zare the unit vectors along the corresponding coordinate axes.\nWe consider a ferromagnetic orthorhombic crystal with strong spin-orbital coupling \fxing the spontaneous magne-\ntization along one of the symmetry axis of the second order chosen as the z-direction. Its point symmetry group or\nblack-white group consists from the rotation on angle \u0019aroundzaxis and the rotations on angle \u0019aroundxandy\ndirections combined with the operation of time inversion Rwhich changes the direction of spontaneous magnetization\nto the opposite one\nD2(Cz\n2) = (E;Cz\n2;RCx\n2;RCy\n2): (4)4\nBeing not interested of possible translation invariance breaking at the transition into superconducting state we do\nnot consider the full space group of the normal state. Besides the point operations the symmetry group of the normal\nstate includes the group of gauge transformations U(1)\nGFM=U(1)\u0002D2(Cz\n2) =U(1)\u0002(E;Cz\n2;RCx\n2;RCy\n2): (5)\nThe superconducting states with di\u000berent critical temperatures are described by the basis functions of di\u000berent\nirreducible co-representations of the symmetry group of the normal state. There are only two di\u000berent co-\nrepresentations AandBof the group GFM[24, 26]. The vector order parameters Eq.(3)\ndA(k;r);dB(k;r)\nof these states are determined by the amplitudes\n\u0001\"\nA(k;r) =^kx\u0011\"\nx(r) +i^ky\u0011\"\ny(r);\n\u0001#\nA(k;r) =^kx\u0011#\nx(r) +i^ky\u0011#\ny(r); (6)\n\u00010\nA(k;r) =^kz\u00110\nz(r);\n\u0001\"\nB(k;r) =^kz\u0010\"\nz(r);\n\u0001#\nB(k;r) =^kz\u0010#\nz(r); (7)\n\u00010\nB(k;r) =^kx\u00100\nx(r) +i^ky\u00100\ny(r):\nHere and in what follows ^kx;^ky;^kzare the components of the unit vector of momentum ^k=k=jkj. For the\nstate A the pair of scalar complex order parameter amplitudes for spin-up pairing have the common phase\n(\u0011\"\nx;\u0011\"\ny) = (j\u0011\"\nxj;j\u0011\"\nyj)ei'\". The spin-down pair also have the common phase ( \u0011#\nx;\u0011#\ny) = (j\u0011#\nxj;j\u0011#\nyj)ei'#. And the\nzero-spin amplitude has its own phase \u00110\nz=j\u00110\nzjei'0. We assume that three phases '\";'#;'0either coincide among\nthemselves '\"='#='0=', or di\u000ber on\u0006\u0019. The same property takes place for the spin-up, spin-down and\nzero-spin order parameter components of the B state.\nOne can check that the order parameter dA(k;r) is invariant in respect to all the transformations of the group GA\nwhich is isomorphic to the black-white group of the normal state D2(Cz\n2) but contains the combined elements of time\ninversion and gauge transformations\nGA= (E;Cz\n2;e2i'RCx\n2;e2i'RCy\n2): (8)\nFor instance, the element RCx\n2, combining transformations ^ y!\u0000 ^y;^z!\u0000 ^z;^ky!\u0000 ^ky;^kz!\u0000 ^kzand complex\nconjugation of the order parameter, transforms it to itself to within the phase factor e\u0000i'such that:\ne2i'RCx\n2dA(k;r) =dA(k;r):\nThe order parameter dB(k;r) is invariant in respect to all transformations of the group\nGB=D2(E) = (E;Cz\n2ei\u0019;e2i'RCx\n2ei\u0019;e2i'RCy\n2): (9)\nImportant to note that the \fve-component \u0011\"\nx; \u0011\"\ny; \u0011#\nx; \u0011#\ny; \u00110\nzorder parameter of the A-state and the four-\ncomponent \u0010\"\nz; \u0010#\nz;\u00100\nx; \u00100\nyorder parameter of the B-state found from the pure symmetry considerations include the\nzero-spin components. In other words they are not equal-spin-pairing states , consisting of the Cooper pairs with\nthe opposite spins. This fact will be explained below in the frame of microscopic approach.\nWriting Eqs.(6) and(7) we were limited by the simplest form of the superconducting states order parameters. In\ngeneral, one must take into account the following complications.\n(i) Each term in Eqs.(6) and(7) can contain k2\nx;k2\ny;k2\nzdependent factors invariant in respect all rotations of the\northorhombic group[24].\n(ii) Eqs.(6) and(7) are written as they should be for a two-band spin-up and spin-down ferromagnet. For a multi-\nband ferromagnet spin-up, spin-down and spin-zero parts of the order parameter should consist of several terms\nrelating to di\u000berent bands.\n(iii) Also, if necessary, the higher order harmonics (higher powers of ^kl\nx^km\ny^kn\nz) of the same symmetry as the linear\nin respect of components ^kin the Eqs.(6) and(7) can be taken into account [24].5\nB. Superconducting states in UCoGe\nUnlike to URhGe and UGe 2, where the superconducting state arises only in the ferromagnetic state, the phase\ndiagram of UCoGe in Fig.1c includes ferromagnetic (FM+SC) and paramagnetic (SC) superconducting states [37].\nThe symmetries of all the states shown in Fig.1c obey to the usual for a second order phase transition subordination\nrules [38]. Namely, the group of symmetry of the ferromagnetic superconducting state A Eq.(8)\nGFM+SC= (E;Cz\n2;e2i'RCx\n2;e2i'RCy\n2) (10)\nis the subgroup of the group of symmetry of the ferromagnetic state Eq.(5)\nGFM=U(1)\u0002D2(Cz\n2) =U(1)\u0002(E;Cz\n2;RCx\n2;RCy\n2); (11)\nand the group of symmetry of the paramagnetic superconducting state\nGSC= (E;Cz\n2;Cx\n2;Cy\n2) +e2i'R\u0002(E;Cz\n2;Cx\n2;Cy\n2): (12)\nIn its turn both of these groups are the subgroups of the group of the paramagnetic normal state\nGN=U(1)\u0002f(E;Cz\n2;Cx\n2;Cy\n2) +R\u0002(E;Cz\n2;Cx\n2;Cy\n2)g: (13)\nThe order parameter of the paramagnetic superconducting state looks like the order parameter of super\ruid3He-B\nphase [36]\nd(k) =kx\u0011x^x+ky\u0011y^y+kz\u0011z^z: (14)\nAt phase transition to the superconducting ferromagnet state the exchange \feld lifts the Kramers degeneracy between\nthe spin-up and the spin-down electron states. Hence, the unitary order parameter (14) transforms into the nonunitary\norder parameter of the type A superconducting ferromagnet state (6):\nkx\u0011x^x+ky\u0011y^y+kz\u0011z^z\n=1\n2[(kx\u0011x\u0000iky\u0011y)(^x+i^y) + (kx\u0011x+iky\u0011y)(^x\u0000i^y)] +kz\u0011z^z=1\n2\u0002\n\u0000\u0001\"(k)(^x+i^y) + \u0001#(k)(^x\u0000i^y)\u0003\n+ \u00010(k)^z\n=)1\n2(kx\u0011\"\nx(r)\u0000iky\u0011\"\ny(r))(^x+i^y);1\n2(kx\u0011#\nx(r) +iky\u0011#\ny(r))(^x\u0000i^y); kz\u00110\nz(r)^z: (15)\nSimilar considerations of the phases symmetry and the order parameter transformation can be made for the ferro-\nmagnetic superconducting state B.\nExperimentally a phase transition from the paramagnetic superconducting state to the ferromagnetic supercon-\nducting state in UCoGe has never been revealed.\nC. Quasiparticles spectrum in a ferromagnet superconductor with triplet pairing\nWe consider two band ferromagnetic metal with the electron spectra \u0018\"(k); \u0018#(k) for the spin-up and the spin-down\nbands.\nEven in the absence of external \feld in a ferromagnet there is an internal \feld Hintacting on the electron charges.\nThe internal magnetic \feld in all uranium superconducting ferromagnets is larger than the lower critical \feld Hc1\n(see, for instance, the paper [14]). Hence, the Meissner state is absent and the superconducting state is always the\nAbrikosov mixed state with space inhomogeneous distributions of the order parameter and the internal magnetic \feld.\nIn this case to \fnd the elementary excitation energies one has to solve the coupled systems of the Gor'kov and the\nMaxwell di\u000berential equations. Some simpli\fcations appear at low temperatures. Here, owing to Hint<< Hc2one\ncan work in the London approximation such that the internal magnetic \feld Hint(r) =rotA(r) is a slow function of\ncoordinates. In the inter-vortex space the order parameter is constant, and the electron (hole) momenta acquire a\nDoppler shift k\u0006mvs(r) due to non-zero velocity of the super\ruid component\nvs(r) =~\n2m\u0012\nr'+2e\n~cA(r)\u0013\n:\nThen, the Gor'kov equations are\n\u0012\ni!n\u00001\n2(\u0018\"++\u0018#+)\u001b0\u00001\n2(\u0018\"+\u0000\u0018#+)\u001bz\u0000i(d\u001b)\u001by\ni\u001by(d\u0003\u001b)i!n+1\n2(\u0018\"\u0000+\u0018#\u0000)\u001b0+1\n2(\u0018\"\u0000\u0000\u0018#\u0000)\u001bz\u0013\u0012^G\u0000^F\n\u0000^Fy\u0000~G\u0000k;\u0000!\u0013\n=\u0012\n\u001b00\n0\u001b0;\u0013\n; (16)6\nwhere\n\u0018\";#\u0006(k) =\u0018\";#(k\u0006mvs)\u0019\u0018\";#(k)\u0006kvs: (17)\nAti!n!Ethe equality of the determinant of this system to zero gives the energy of elementary excitations\nE=kvs+s\n1\n2(\u00182\n\"+\u00182\n#) + (dd\u0003)\u0006r\n1\n4h\n\u00182\n\"\u0000\u00182\n#+ 2i(d\u0002d\u0003)zi2\n\u0000(i(d\u0002d\u0003)z)2+ (i(d\u0002d\u0003)2): (18)\nThe excitations energies acquire the simplest form in the so called equal-spin-pairing state, when d0= 0\nE\"=kvs+q\n\u00182\n\"+dd\u0003+i(d\u0002d\u0003)z=kvs+q\n\u00182\n\"+ \u00012\n\"; (19)\nE#=kvs+q\n\u00182\n#+dd\u0003\u0000i(d\u0002d\u0003)z=kvs+q\n\u00182\n#+ \u00012\n#: (20)\nIt is instructive also to write the energy of excitations in nonunitary superconducting state [36]\nE=kvs+p\n\u00182+dd\u0003\u0006ji(d\u0002d\u0003j: (21)\narising from a paramagnetic normal state where \u0018\"=\u0018#=\u0018. In all the cases the Kramers degeneracy is lifted.\nLet us look now what kind of pairing interaction gives rise the A and B superconducting states in ferromagnets\nwith orthorhombic symmetry.\nIII. SUPERCONDUCTING STATES IN MICROSCOPIC WEAK COUPLING THEORY\nA. Triplet pairing by spin-\ructuations exchange\nThe interactions between two electrons is assumed to be due to the attraction of one electron by the magnetic\npolarization cloud of the other. Unlike to super\ruid He3, where the pairing of atoms originates from the magnetic\npolarization in the isotropic Fermi liquid, pairing of electrons in a ferromagnetic metal occurs in an anisotropic media\ndue to polarization of the electron liquid and the localized moments as well.\nSo, we consider the pairing originating from the attraction\nHelm=\u00001\n2\u00162\nBI2Z\nd3rd3r0Si(r)\u001fij(r\u0000r0)Sj(r0) (22)\nbetween the electrons with magnetic moments \u0016Bby means the electron-magnon interaction in ferromagnetic media\nwith orthorhombic symmetry. Here,\nS(r) = y\n\u000b(r)\u001b\u000b\f \f(r)\nis the operator of the electron spin density, \u001fij(r) is the media susceptibility, Iis an exchange constant.\nTransforming the interaction Hamiltonian in the momentum representation and leaving only the odd in respect of\nkandk0parity terms we obtain from Eq.(22) after some straightforward algebra [39] the BCS Hamiltonian for triplet\npairing\nHpairing =1\n2X\nkk0V\u000b\f\r\u000e(k;k0)ay\n\u000b(k)ay\n\f(\u0000k)a\r(\u0000k0)a\u000e(k0); (23)\nwhere\nV\u000b\f\r\u000e(k;k0) =Vij(k;k0)(i\u001bi\u001by)\u000b\f(i\u001bj\u001by)y\n\r\u000e(24)\nand\nVij(k;k0) =\u0000\u00162\nBI2\u00121\n2Tr^\u001fu(k;k0)\u000eij\u0000\u001fu\nij(k;k0)\u0013\n(25)7\nis expressed through the odd part of the media static susceptibility\n^\u001fu(k;k0) =\u001fu\nij(k;k0) =1\n2[\u001fij(k\u0000k0)\u0000\u001fij(k+k0)]:\nThe critical temperature or the upper critical \feld are determined as the eigenvalue of the linear equation for the\norder parameter [36]\n\u0001\u000b\f(k;q) =\u0000TX\nnX\nk0V\f\u000b\u0015\u0016(k;k0)G\u0015\r(k0;!n)G\u0016\u000e(\u0000k0+q;\u0000!n)\u0001\r\u000e(k0;q): (26)\nHere the matrix of order parameter in the momentum representation is\n^\u0001(k;q) =Z\n^\u0001(k;r)eiqrd3r=\u0012\n\u0001\"(k;q) \u00010(k;q)\n\u00010(k;q) \u0001#(k;q)\u0013\n; (27)\nG\u0015\r(k0;!n) is the matrix of normal metal Green function. In absence of external \feld or when the external\nmagnetic \feld is strictly parallel to the spontaneous magnetization it is diagonal\n^Gn=\u0012\nG\"0\n0G#\u0013\n; (28)\nwhere\nG\";#=1\ni!n\u0000\u0018\";#\nk: (29)\nMatrix equation (26) is the system of coupled linear equations for the order parameter components\n\u0001\"(k;q) =\u0000TX\nnX\nk0\b\nV\"\"(k;k0)G\"G\"\u0001\"(k0;q) +V\"#(k;k0)G#G#\u0001#(k0;q) +V\"0(k;k0)\u0002\nG#G\"+G\"G#\u0003\n\u00010(k0;q)\t\n;(30)\n\u0001#(k;q) =\u0000TX\nnX\nk0\b\nV#\"(k;k0)G\"G\"\u0001\"(k0;q) +V##(k;k0)G#G#\u0001#(k0;q) +V#0(k;k0)\u0002\nG#G\"+G\"G#\u0003\n\u00010(k0;q)\t\n;(31)\n\u00010(k;q) =\u0000TX\nnX\nk0\b\nV0\"(k;k0)G\"G\"\u0001\"(k0;q) +V0#(k;k0)G#G#\u0001#(k0;q) +V00(k;k0)\u0002\nG#G\"+G\"G#\u0003\n\u00010(k0;q)\t\n:(32)\nHere the arguments in the Green functions products are as in the matrix equation (26). For instance\nG\"G\"=G\"(k0;!n)G\"(\u0000k0+q;\u0000!n):\nThe pairing amplitudes found from Eqs. (25) are\nV\"\"(k;k0) =Vxx+Vyy+iVxy\u0000iVyx=\u0000\u00162\nBI2\u001fu\nzz; (33)\nV##(k;k0) =Vxx+Vyy\u0000iVxy+iVyx=\u0000\u00162\nBI2\u001fu\nzz; (34)\nV\"#(k;k0) =\u0000Vxx+Vyy+iVxy+iVyx=\u0000\u00162\nBI2(\u001fu\nxx\u0000\u001fu\nyy\u00002i\u001fu\nxy); (35)\nV#\"(k;k0) =\u0000Vxx+Vyy\u0000iVxy\u0000iVyx=\u0000\u00162\nBI2(\u001fu\nxx\u0000\u001fu\nyy+ 2i\u001fu\nxy); (36)\nV00(k;k0) =Vzz=\u0000\u00162\nBI2(\u001fu\nxx+\u001fu\nyy\u0000\u001fu\nzz)=2; (37)\nV\"0(k;k0) = (V0\"(k;k0))\u0003=\u0000Vxz+iVyz=\u0000\u00162\nBI2(\u001fu\nxz\u0000i\u001fu\nyz); (38)\nV#0(k;k0) = (V0#(k;k0))\u0003=Vxz+iVyz=\u0000\u00162\nBI2(\u0000\u001fu\nxz\u0000i\u001fu\nyz): (39)\nOne can see that the equations for \u0001\", \u0001#and \u00010are entangled. Moreover, the entanglement still exists even in\nthe case of strong spin-up and spin-down bands splitting which allows to omit all the terms including combinations\nG#G\"+G\"G#corresponding to the interband pairing. Ignoring the inter-band pairing [40], we \fnd\n\u0001\"(k;q) =\u0000TP\nnP\nk0\b\nV\"\"(k;k0)G\"G\"\u0001\"(k0;q) +V\"#(k;k0)G#G#\u0001#(k0;q)\t\n; (40)\n\u0001#(k;q) =\u0000TP\nnP\nk0\b\nV#\"(k;k0)G\"G\"\u0001\"(k0;q) +V##(k;k0)G#G#\u0001#(k0;q)\t\n; (41)\n\u00010(k;q) =\u0000TP\nnP\nk0\b\nV0\"(k;k0)G\"G\"\u0001\"(k0;q) +V0#(k;k0)G#G#\u0001#(k0;q)\t\n: (42)8\nWe see, that the equation determining the order parameter component \u00010corresponding to the pairing of particles\nwith the opposite spins still exists. According to Eq. (42) the pairing with the opposite spins is induced by the\npairing terms with parallel spins. Thus, in general, a superconducting state in a ferromagnetic metal is not\nequal-spin-pairing state. This property originates from a spin-orbit coupling. Indeed, in what follows, we shall see\nthat the pairing amplitudes V0\"; V0#arise due to the spin-orbital terms in the ferromagnet gradient energy which are\npresumably small. So, with good accuracy one can work with Eqs. (40) and (41) corresponding to equal-spin-pairing\nsuperconductivity neglecting by the amplitude \u00010induced by the pairing of electrons with spin up-up and down-down\nspins. This case, we deal with two-band superconducting state similar to the A2state of super\ruid3He [41]. This\nproperty is supported by the recent low temperature thermal conductivity measurements under magnetic \feld[42].\nBoth the spin up-up V\"\"and the spin down-down V##pairing amplitudes are determined by the susceptibility\ncomponent parallel to the direction of spontaneous magnetization \u001fu\nzzwhich strongly exceeds the susceptibility along\nthe other crystallographic directions. On the other hand, the spin-up and the spin-down Cooper pairs interact each\nother due to the susceptibility anisotropy \u001fxx6=\u001fyywhich gives rise common for the spin-up and the spin-down bands\nphase transition to the A 2-type state. The susceptibility anisotropy does not exist in the exchange approximation\nand completely determined by the spin-orbit coupling. Even in isotropic liquid3He the spin-orbit coupling leads to\nthe entanglement between the spin-up and the spin-down order parameters [43]. However, in view of smallness of\nspin-orbit interaction, the entanglement between the spin-up and the spin-down components is practically absent,\nwhat results in two subsequent phase transitions under magnetic \feld: \frst to the spin-up A1state and then to the\nmixed spin-up and spin-down A2super\ruid state [41].\nLet us now \fnd the susceptibility.\nB. Magnetic susceptibility of an orthorhombic ferromagnet\nWe search for the static magnetic susceptibility following phenomenological approach. The starting point is the\nLandau free energy of an orthorhombic ferromagnet in magnetic \feld H(r)\nF=Z\ndV(FM+Fr); (43)\nwhere in expressions for condensation energy density\nFM=\u000bzMz2+\u000byM2\ny+\u000bxMx2+\fzMz4+\fxyMx2My2+\fyzMz2My2+\fxzMz2Mx2\u0000MH; (44)\nand gradient energy density\nFr=\r\u000b\f\nij@M\u000b\n@xi@M\f\n@xj(45)\nwe bear in mind the orthorhombic anisotropy. The zdirection is always chosen along the spontaneous magnetization,\nhence in URhGe and in UCoGe x;y;z coordinates are directed along the a;b;c crystallographic axis, and in UGe 2\nalongb;c;a axis.\n\u000bz=\u000bz0(T\u0000Tc); \u000bx>0; \u000by>0; (46)\nTcis the Curie temperature, and\n\r\u000b=\f\nij=0\n@\r\u000b\nxx0 0\n0\r\u000b\nyy0\n0 0\r\u000b\nzz1\nA; \u000b =x;y;z (47)\nand\n\r\u000b6=\f\nij=\rij\nij=0\n@0\rxy\rxz\n\rxy0\ryz\n\rxz\ryz01\nA: (48)\nThe corresponding energy density in the exchange approximation is\nFexchange\nM +Fexchange\nr =\u000bM2+\fM4\u0000MH +\rij@M\n@xi@M\n@xj: (49)9\nwhere the matrix \rijis\n\rij=0\n@\rxx0 0\n0\ryy0\n0 0\rzz1\nA (50)\nsuch that the gradient energy is determined only by 3 constants instead 12 constants involved in general case taking\ninto account the relativistically small interactions.\nLet us take the magnetic \feld as the sum of the constant part in zdirection and the coordinate dependent small\naddition\nH(r) =\u000eHx(r)^x+\u000eHy(r)^y+ (Hz+\u000eHz(r))^z: (51)\nBy variation of functional (43) in respect to the components of magnetization we arrive to the equations\n2\u000bxMx+ 2\fxyM2\nyMx+ 2\fxzM2\nzMx\u00002\rx\nij@2Mx\n@xi@xj\u0000\rxy@2My\n@x@y\u0000\rxz@2Mz\n@x@z=\u000eHx;\n2\u000byMy+ 2\fxyM2\nxMy+ 2\fyzM2\nzMy\u00002\ry\nij@2My\n@xi@xj\u0000\rxy@2Mx\n@x@y\u0000\ryz@2Mz\n@y@z=\u000eHy; (52)\n2\u000bzMz+ 4\fzM3\nz+ 2\fxzM2\nxMz+ 2\fyzM2\nyMz\u00002\rz\nij@2Mz\n@xi@xj\u0000\rxz@2Mx\n@x@z\u0000\ryz@2My\n@y@z=Hz+\u000eHz:\nThe equilibrium magnetization projections are determined by the equations:\nMx= 0; M y= 0; (53)\nM2\nz=\u0000\u000bz\n2\fz+Hz\n4\fzMz: (54)\nThe \frst and the last term in this expression correspond to the spontaneous and the induced part of magnetization\nalongzdirection.\nTaking magnetization as the sum of the constant part and the coordinate dependent small addition\nM(r) =Mz^z+\u000eMx(r) +\u000eMy(r) +\u000eMz(r); (55)\nwe obtain from the Eqs.(52) the linear equations for the Fourier components of \u000eM(k):\n2(\u000bx+\fxzM2\nz+\rx\nijkikj)\u000eMx(k) +\rxykxky\u000eMy(k) +\rxzkxkz\u000eMz(k) =\u000eHx(k);\n\rxykxky\u000eMx(k) + 2(\u000by+\fyzM2\nz+\ry\nijkikj)\u000eMy(k) +\ryzkykz\u000eMz(k) =\u000eHy(k); (56)\n\rxzkxkz\u000eMx(k) +\ryzkykz\u000eMy(k) + 2(\u000bz+ 6\fzM2\nz+\rz\nijkikj)\u000eMz(k) =\u000eHx(k):\nThe coupling between the magnetization components in Eqs.(56) are due to the anisotropy terms originating from\nthe small relativistic interactions. Hence, solving Eqs.(56) one can neglect by all the products of the terms like\n\rxykxky\ryzkykzetc and obtain:\n\u001fxx=\u000eMx\n\u000eHx\u00191\n2(\u000bx+\fxzM2z+\rx\nijkikj); (57)\n\u001fyy=\u000eMy\n\u000eHy\u00191\n2(\u000by+\fyzM2z+\ry\nijkikj); (58)\n\u001fzz=\u000eMz\n\u000eHz\u00191\n2(\u000bz+6\fzM2z+\rz\nijkikj); (59)\n\u001fxy=\u000eMx\n\u000eHy=\u000eMy\n\u000eHx\u0019\u0000\rxykxky\n4(\u000bx+\fxzM2z+\rx\nijkikj)(\u000by+\fyzM2\nz+\ry\nijkikj); (60)\n\u001fxz=\u000eMx\n\u000eHz=\u000eMz\n\u000eHx\u0019\u0000\rxzkxkz\n4(\u000bx+\fxzM2z+\rx\nijkikj)(\u000bz+6\fzM2z+\rz\nijkikj); (61)\n\u001fyz=\u000eMy\n\u000eHz=\u000eMz\n\u000eHy\u0019\u0000\ryzkykz\n4(\u000by+\fyzM2z+\ry\nijkikj)(\u000bz+6\fzM2z+\rz\nijkikj): (62)\nThe expressions for the susceptibility components depend from the wave vector through the combinations like\n\rk2. They are found at wave vectors much smaller than the inverse interatomic distance a\u00001. The corresponding\nwave vector dependent terms in the susceptibility components found from an appropriate microscopic model will be\ngiven by the combinations of trigonometric functions like \rsin2ka=a2. These combinations at small kreproduce our10\nphenomenological expressions and at k\u0018=kFthey are of the order \r=a2as the combinations in the phenomenological\ntheory. This means that our formulas for susceptibilities are still qualitatively valid at large wave vectors transfer\nk\u0018=kFdetermining pairing interaction. The Fermi momenta kFat di\u000berent Fermi surface points are di\u000berent, hence,\nkFis a function of direction in the reciprocal space possessing full orthorhombic symmetry.\nThe odd part of z-component of susceptibility is found as\n\u001fu\nzz(k;k0) =1\n2[\u001fzz(k\u0000k0)\u0000\u001fzz(k+k0)]\n=\rijkik0\nj\n(\u000bz+ 6\fzM2z+\rij(kikj+k0\nik0\nj))2\u0000(2\rijkik0\nj)2: (63)\nAccording to Eqs.(33) and (34) the pairing interaction is mostly determined by this formula. The situation is similar\nto the case of weak coupling singlet pairing, where the zero-frequency limit of phonon propagator plays the role of\nthe potential for the phonon-mediated attraction between the electrons. We are interested in the pairing interaction\ninside the ferromagnetic state where \u000bz+ 6\fzM2\nzhas a \fnite positive magnitude. At the Curie temperature this\ncombination is equal to zero and \u001fu\nzz(k;k0) diverges at the coincident arguments corresponding the Cooper pairing.\nThis is inevitable property of a model with static interaction. To avoid this pairing interaction divergency D. Fay\nand J .Appel [44] in their theory of p-wave superconductivity in an itinerant ferromagnet have introduced a cuto\u000b\ndepending from the distance from the ferromagnetic phase transition. As result the critical temperature of phase\ntransition to the superconducting state having a \fnite value both in the ferromagnetic and the paramagnetic state\nproved to be equal to zero at transition between them. This misleading property does not take place in a model\ntaking into account a retardation e\u000bect in the pairing interaction.\nAt \fnite\u000bz+6\fzM2\nzwe can keep only the angular dependence of interaction in the numerator of Eq.(63) neglecting\nby the angular dependence of kFand the orthorhombic symmetry terms in denominator \rz\nij(kikj+k0\nik0\nj)\u00192\rzk2\nF\nas well as by all the higher angular harmonics of interaction determined by the last term in denominator [40]. The\ncalculations without these simpli\fcations are much more cumbersome but do not give rise the qualitatively di\u000berent\nresults. Hence, we obtain\n\u001fu\nzz(k;k0)\u0018=\rz\nijk2\nF\na2z^ki^k0\nj; (64)\nwhere\naz=\u000bz+ 6\fzM2\nz+ 2\rzk2\nF= 2\fz(3M2\nz\u0000M2\nz0) + 2\rzk2\nF; (65)\nwhereMzis the solution of Eq.(54) and Mz0=Mz(Hz= 0) = (\u0000\u000bz=2\fz)1=2. At temperatures noteceably smaller\nthan the Curie temperature one can use experimental values for the \feld dependent magnetization Mz(Hz) and its\nalmost temperature independent spontaneous part Mz0=Mz(Hz= 0).\nFound in similar manner the odd part of the susceptibility xandy-components are\n\u001fu\nxx(k;k0)\u0018=\rx\nijk2\nF\na2x^ki^k0\nj; \u001fu\nyy(k;k0)\u0018=\ry\nijk2\nF\na2y^ki^k0\nj; (66)\nwhere\nax=\u000bx+\fxzM2\nz+ 2\rxk2\nF; ay=\u000by+\fyzM2\nz+ 2\ryk2\nF: (67)\nAll o\u000b-diagonal components of susceptibility are linear in respect of anisotropy terms determined by the spin-orbital\ncoupling:\n\u001fu\nxy(k;k0)\u0018=\rxyk2\nF\n4~ax~ay(^kx^k0\ny+^k0\nx^ky); (68)\n\u001fu\nxz(k;k0)\u0018=\rxzk2\nF\n4~ax~az(^kx^k0\nz+^k0\nx^kz); (69)\n\u001fu\nyz(k;k0)\u0018=\ryzk2\nF\n4~ay~az(^ky^k0\nz+^k0\ny^kz); (70)\n~ax=\u000bx+\fxzM2\nz; ~ay=\u000by+\fyzM2\nz; ~az=\u000bz+ 6\fzM2\nz= 4\fzM2\nz+Hz\n2Mz: (71)\nHere we have completely neglected by the quartic terms in respect of the wave vector components. They have the\nsame symmetry as Eqs. (68)-(70) but strongly complicate the corresponding expressions.11\nC. Pairing amplitudes\nThe equations (33)-(39) express the pairing amplitudes through the susceptibility components in a ferromagnetic\nmetal with arbitrary symmetry. The explicit formula for the susceptibility component in an orthorhombic ferromagnet\nare found in the previous section. So, the pairing amplitudes in this particular case are\nV\"\"(k;k0) =V##(k;k0) =\u0000\u00162\nBI2\u001fu\nzz=\u0000\u00162\nBI2k2\nF\rz\nij^ki^k0\nj\n4[\fz(3M2z\u0000M2\nz0)+\rzk2\nF]2=\u0000V1ij^ki^k0\nj; (72)\nV\"#(k;k0) =\u0000V2ij^ki^k0\nj+iV3(^kx^k0\ny+^ky^k0\nx); (73)\nV#\"(k;k0) = (V\"#(k;k0))\u0003; (74)\nV00(k;k0) =\u0000W1ij^ki^k0\nj; (75)\nV\"0(k;k0) = (V0\"(k;k0))\u0003=\u0000W2(^kx^k0\nz+^kz^k0\nx) +iW3(^ky^k0\nz+^kz^k0\ny); (76)\nV#0(k;k0) = (V0#(k;k0))\u0003=\u0000(V\"0(k;k0))\u0003(77)\nHere, the constants are\nV1ij=\u00162\nBI2k2\nF\rz\nij\na2z=\u00162\nBI2k2\nF\rz\nij\n4[\fz(3M2z\u0000M2\nz0)+\rzk2\nF]2; (78)\nV2ij=\u00162\nBI2k2\nF\u0010\rx\nij\na2x\u0000\ry\nij\na2y\u0011\n; V 3=\u00162\nBI2k2\nF\rxy\n4~ax~ay; (79)\nW1ij=\u00162\nBI2k2\nF\n2\u0010\rx\nij\na2x+\ry\nij\na2y\u0000\rz\nij\na2z\u0011\n; W 2=\u00162\nBI2k2\nF\rxz\n4~ax~az; W 3=\u00162\nBI2k2\nF\ryz\n4~ay~az: (80)\nThe pairing interaction between the particles in the same spin-up or spin-down band plays the most important role.\nThe corresponding amplitude originates from the odd part of the magnetic susceptibility component \u001fu\nzzwhich is the\nlargest and the temperature and the magnetic \feld dependent component of susceptibility.\nThe amplitudes V1ijandW1ijare determined mainly by the exchange interaction. The amplitude V2ijis equal to\nzero in the exchange approximation. It has, however, non-negligible magnitude corresponding to the strong enough\northorhombic anisotropy of susceptibility \u001fxx6=\u001fyy. The amplitudes V3; W 2; W 3are determined by the spin-orbit\nterms in the gradient energy of an orthorhombic ferromagnet, and we will treat them as the smallest amplitudes.\nD. Critical temperature of phase transition to the paramagnetic superconducting state in UCoGe\nThe equations (30)-(32) are also applicable to determination of the critical temperature of the phase transition from\nthe normal to the paramagnetic superconducting state taking place in UCoGe at high pressures (see Fig.1c). This\ncase an internal magnetic \feld is absent, hence, the normal state Green functions for the spin-up and the spin-down\nelectrons are equal G\"=G#=Gand the order parameter is homogeneous in space. So, the equations take the form\n\u0001\"(k) =\u0000TX\nnX\nk0\b\nV\"\"(k;k0)\u0001#(k0) +V\"#(k;k0)\u0001#(k0) + 2V\"0(k;k0)\u00010(k0)\t\nG(k0;!n)G(\u0000k0;\u0000!n); (81)\n\u0001#(k) =\u0000TX\nnX\nk0\b\nV#\"(k;k0)\u0001\"(k0) +V##(k;k0)\u0001#(k0) + 2V#0(k;k0)\u00010(k0)\t\nG(k0;!n)G(\u0000k0;\u0000!n); (82)\n\u00010(k) =\u0000TX\nnX\nk0\b\n2V0\"(k;k0)\u0001\"(k0) +V0#(k;k0)\u0001#(k0) +V00(k;k0)\u00010(k0)\t\nG(k0;!n)G(\u0000k0;\u0000!n): (83)\nSubstitution the paramagnetic state order parameter components (see Chapter IIB)\n\u0001\"=\u0000kx\u0011x+iky\u0011y;\u0001#=kx\u0011x+iky\u0011y;\u00010=kz\u0011z^z (84)\nto these equations gives rise 5 equations for 3 amplitudes \u0011x;\u0011y;\u0011z. Two of these equations coincide with two others,\nso, the 3 independent equations are\n(\u0015\u00001\u0000g1x+g2x)\u0011x+g3y\u0011y+ 2w2z\u0011z= 0;\ng3x\u0011x+ (\u0015\u00001\u0000g1y\u0000g2y)\u0011y+ 2w3z\u0011z= 0; (85)\n2w2x\u0011x+ 2w3y\u0011y+ (\u0015\u00001\u0000w1z)\u0011z= 0:12\nHere\ng1x=V1xxh^k2\nxN0(k)i; g2x=V2xxh^k2\nxN0(k)i; g1y=V1yyh^k2\nyN0(k)i; g2y=V2yyh^k2\nyN0(k)i; w1z=W1zzh^k2\nzN0(k)i;\ng3x=V3h^k2\nxN0(k)i; g3y=V3h^k2\nyN0(k)i;\nw2x=W2h^k2\nxN0(k)i; w2z=W2h^k2\nzN0(k)i; w3z=W3h^k2\nzN0(k)i; w3y=W3h^k2\nyN0(k)i\nare the constants of pairing interaction, the angular brackets mean the averaging over the Fermi surface, N0(k) is the\nangular dependent density of electronic states at the Fermi surface. The function\n\u0015(T) = 2\u0019TX\nn\u001501\n!n= ln\"\nT;\nwhere\"=2\r\"0\n\u0019, ln\r= 0:577 is the Euler constant, and \"0is an energy cuto\u000b for the pairing interaction. The critical\ntemperature of phase transition to the paramagnetic superconducting state is\nTsc=\"exp\u0012\n\u00001\ng\u0013\n; (86)\nwhere the constant gis the maximal eigen value of the system of equations (85).\nE. Phase transition from the paramagnetic to ferromagnetic superconducting state in UCoGe\nThe equations (81)-(83) are also applicable for the determination of the critical temperature of phase transition\nwhich has to separate the paramagnetic superconducting and the ferromagnetic superconducting states in UCoGe (see\nFig.1c) but has not been revealed experimentally. This case, the Green function in this system is the Green function\nof the paramagnetic superconducting state\nG(k;!n) =\u0000i!n+\u0018k\n!2n+\u00182\nk+\u00112x^k2x+\u00112y^k2y+\u00112z^k2z: (87)\nSubstitution the ferromagnetic state order parameter components (see Chapter IIB)\n\u0001\"=\u0000kx\u0011\"\nx+iky\u0011\"\ny;\u0001#=kx\u0011#\nx+iky\u0011#\ny;\u00010=kz\u00110\nz^z (88)\nto the equations (81)-(83) gives rise 5 equations for 5 amplitudes \u0011\"\nx;\u0011\"\ny;\u0011#\nx;\u0011#\ny;\u00110\nz. The maximum eigen value of\nthis system determines the critical temperature of phase transition from the paramagnetic to the ferromagnetic\nsuperconducting state.\nThis phase transition occurs in the itinerant electron subsystem. Mathematically it is described by the smooth\ndevelopment of an inequality in the spin-up and the spin-down amplitudes of the order parameter that is by the\ndevelopment of spontaneous magnetic moment of pure superconducting nature. Simultaneously the magnetization\nnot related to the itinerant electron subsystem but to the subsystem of localized moments will appear. Its emergence\ninduced by the superconducting electrons magnetic moment is similar to the crossover between paramagnetic and fer-\nromagnetic normal states under an external magnetic \feld. One can expect that, due to the magnetization smallness,\nbelow the transition line the superconductor still remains in the Meissner state.\nThere is an other possible scenario of phase transition from the paramagnetic to ferromagnetic superconducting\nstate. It is realized when the transition driving force is the ordering in the subsystem of localized moments. This case\nthe subsystem of superconducting electrons is tuned up to ferromagnetic superconducting state due to the emergency\nof spontaneous magnetization.\nThe proper theory of phase transition from the paramagnetic to the ferromagnetic superconducting state must\ninclude the e\u000bect of emergency of supercurrents and the \feld dependence of magnetization which can be important in\nview of divergency of magnetic susceptibility near the Curie temperature. A satisfactory treatment of this phenomenon\nis at the moment absent [45].\nF. Superconducting states in orthorhombic ferromagnets\nLet us \fnd now what kind of superconducting state emerges at phase transition from the normal ferromagnetic to\nthe superconducting ferromagnetic state. Performing the Taylor expansion of Eqs.(30)-(32) in powers of qup to the13\nsecond order and and then transforming them to the coordinate representation, that means simple substitution\nq!D=\u0000irr+ 2eA(r); (89)\nwe obtain equations\n\u0001\"(k;r) =TP\nnRd3k0\n(2\u0019)3V1ij^ki^k0\nj\u0010\nG\"G\"+1\n2G\"@2G\"\n@k0\nl@k0mDlDm\u0011\n\u0001\"(k0;r) (90)\n+TP\nnRd3k0\n(2\u0019)3[V2ij^ki^k0\nj\u0000iV3(^kx^k0\ny+^k0\nx^ky)]\u0010\nG#G#+1\n2G#@2G#\n@k0\nl@k0mDlDm\u0011\n\u0001#(k0;r)\n+TP\nnRd3k0\n(2\u0019)3[W2(^kx^k0\nz+^kz^k0\nx)\u0000iW3(^ky^k0\nz+^kz^k0\ny)]\u0010\nG\"G#+1\n2G\"@2G#\n@k0\nl@k0mDlDm+G#G\"+1\n2G#@2G\"\n@k0\nl@k0mDlDm\u0011\n\u00010(k0;r);\n\u0001#(k;r) =TP\nnRd3k0\n(2\u0019)3(V2ij^ki^k0\nj+iV3(^kx^k0\ny+^k0\nx^ky))\u0010\nG\"G\"+1\n2G\"@2G\"\n@k0\nl@k0mDlDm\u0011\n\u0001\"(k0;r) (91)\n+TP\nnRd3k0\n(2\u0019)3V1ij^ki^k0\nj\u0010\nG#G#+1\n2G#@2G#(\n@k0\nl@k0mDlDm\u0011\n\u0001#(k0;r)\n+TP\nnRd3k0\n(2\u0019)3[\u0000W2(^kx^k0\nz+^kz^k0\nx)\u0000iW3(^ky^k0\nz+^kz^k0\ny)]\u0010\nG\"G#+1\n2G\"@2G#\n@k0\nl@k0mDlDm+G#G\"+1\n2G#@2G\"\n@k0\nl@k0mDlDm\u0011\n\u00010(k0;r);\n\u00010(k0;r) =TP\nnRd3k0\n(2\u0019)3[W2(^kx^k0\nz+^kz^k0\nx) +iW3(^ky^k0\nz+^kz^k0\ny)]\u0010\nG\"G\"+1\n2G\"@2G\"\n@k0\nl@k0mDlDm\u0011\n\u0001\"(k0;r)\n+TP\nnRd3k0\n(2\u0019)3[\u0000W2(^kx^k0\nz+^kz^k0\nx) +iW3(^ky^k0\nz+^kz^k0\ny)]\u0010\nG#G#+1\n2G#@2G#\n@k0\nl@k0mDlDm\u0011\n\u0001#(k0;r) (92)\n+TP\nnRd3k0\n(2\u0019)3W1ij^ki^k0\nj\u0010\nG\"G#+1\n2G\"@2G#\n@k0\nl@k0mDlDm+G#G\"+1\n2G#@2G\"\n@k0\nl@k0mDlDm\u0011\n\u00010(k0;r):\nHere, as before, the arguments in the Green functions products are\nG\"G\"=G\"(k0;!n)G\"(\u0000k0;\u0000!n); G\"@2G\"\n@k0\nl@k0m=G\"(k0;!n)@2G\"(\u0000k0;\u0000!n)\n@k0\nl@k0m; :::\nIn the single domain approximation in the absence of external \feld H= 0 or at the external \feld directed along\nthe axis of spontaneous magnetization ^ zthe order parameter components are the z-coordinate independent and the\nlong derivatives are\nDx=\u0000i@\n@x; Dy=\u0000i@\n@y+2e\nc(H+Hint)x: (93)\nHere, we have introduced the internal electromagnetic \feld corresponding to the spontaneous magnetization Hint=\n4\u0019M and ignore the di\u000berence between the external \feld and the magnetic \feld induced inside the media by the\nexternal \feld.\nTaking into account the wave vector dependence of pairing interaction in the equations (90)-(92) we can choose the\nsuperconducting order parameter as the following linear combinations of the momentum direction projections on the\ncoordinate axis\n\u0001\"(k;r) =^kx\u0011\"\nx(r) +i^ky\u0011\"\ny(r) +^kz\u0010\"\nz(r);\n\u0001#(k;r) =^kx\u0011#\nx(r) +i^ky\u0011#\ny(r) +^kz\u0010#\nz(r);\n\u00010k;r) =^kx\u00100\nx(r) +i^ky\u00100\ny(r) +^kz\u00110\nz(r):\nOne can check that substitution of these expressions to Eqs. (90)-(92) leads to two independent systems of the\ndi\u000berential equations\n\u0011\u000b(r) =A\u000b\f\u0011\f(r); \u0010\u000b(r) =B\u000b\f\u0010\f(r) (94)\nfor the components of vectors\n\u0011\u000b(r) = (\u0011\"\nx(r);\u0011#\nx(r);\u0011\"\ny(r);\u0011#\ny(r);\u00110\nz(r)) (95)\nand\n\u0010\u000b(r) = (\u0010\"\nz(r);\u0010#\nz(r);\u00100\nx(r);\u00100\ny(r)) (96)\ncorresponding to two di\u000berent superconducting states with di\u000berent critical temperatures relating to\nco-representations A and B. Thereby, the derived microscopic equations con\frm the conclusions made\nin Chapter II from the pure symmetry considerations.14\nG. Equal-spin-pairing states\nIn what follows we will work with Eqs. (40) and (41) corresponding to the equal-spin-pairing superconductivity.\nThis case, the four component state A is\n\u0001\"(k;r) =^kx\u0011\"\nx(r) +i^ky\u0011\"\ny(r); (97)\n\u0001#(k;r) =^kx\u0011#\nx(r) +i^ky\u0011#\ny(r) (98)\nand the two component state B is\n\u0001\"(k;r) =^kz\u0010\"\nz(r); (99)\n\u0001#(k;r) =^kz\u0010#\nz(r): (100)\nThe corresponding equations (94) for the critical temperatures of A and B states are determined by the following 4x4\nand 2x2 matrices\nA\u000b\f=0\nBBBBBBB@g\"\n1x\u0015+L\"\n1xg#\n2x\u0015+L#\n2x+iL#\n3yxiL\"\n1xy\u0000g#\n3y\u0015+iL#\n2xy\u0000L#\n3y\ng\"\n2x\u0015+L\"\n2x\u0000iL\"\n3yxg#\n1x\u0015+L#\n1xg\"\n3y\u0015+iL\"\n2xy+L\"\n3yiL#\n1xy\n\u0000iL\"\n1yxg#\n3x\u0015\u0000iL#\n2yx+L#\n3xg\"\n1y\u0015+L\"\n1yg#\n2y\u0015+L#\n2y+iL#\n3xy\n\u0000g\"\n3x\u0015\u0000iL\"\n2yx\u0000L\"\n3x\u0000iL#\n1yxg\"\n2y\u0015+L\"\n2y\u0000iL\"\n3xyg#\n1y\u0015+L#\n1y1\nCCCCCCCA; (101)\nB\u000b\f= \ng\"\n1z\u0015+L\"\n1zg#\n2z\u0015+L#\n2z\ng\"\n2z\u0015+L\"\n2zg#\n1z\u0015+L#\n1z!\n: (102)\nHere,\ng\"\n1x=V1xxh^k2\nxN\"\n0(k)i=\u00162\nBI2k2\nF\rz\nxxh^k2\nxN\"\n0(k)i\n4 [\fz(3M2z\u0000M2\nz0) +\rzk2\nF]2(103)\nis one of the constants of pairing interaction, the angular brackets mean the averaging over the Fermi surface, N\"\n0(k)\nis the angular dependent density of electronic states at the Fermi surface of the band \". Correspondingly\ng#\n2x=V2xxh^k2\nxN#\n0(k)i; g#\n3x=V3h^k2\nxN#\n0(k)i: (104)\nAll the other constants of pairing interaction are obtained by the obvious substitutions x$yand\"$# orx!z.\nThe function\n\u0015(T) = 2\u0019TX\nn\u001501\n!n= ln\u000f\nT; (105)\nwhere\u000f=2\r\"0\n\u0019, ln\r= 0:577 is the Euler constant, and \"0is an energy cuto\u000b for pairing interaction. We assume here\nthat it has the same value for both bands.\nThe \frst type of di\u000berential operators is de\fned as follows\nL\"\n1x=1\n2V1xxTX\nnZd3k\n(2\u0019)3^k2\nxG\"(k;!n)D\"; (106)\nandL#\n2yand the other operators with same structure are obtained by obvious substitutions ( x!y;z), (1!2) and\n(\"!#), but similar operator with index 3 is\nL\"\n3x=1\n2V3TX\nnZd3k\n(2\u0019)3^k2\nyG\"(k;!n)D\"; (107)15\nhere,\nD\"=@2G\"(\u0000k;\u0000!n)\n@k2xD2\nx+@2G\"(\u0000k;\u0000!n)\n@k2yD2\ny: (108)\nThe second type of operators is\nL\"\n1xy=1\n2V1xxTX\nnZd3k\n(2\u0019)3^kx^kyG\"(k;!n)@2G\"(\u0000k;\u0000!n)\n@kx@ky(DxDy+DyDx); (109)\nandL\"\n2yxand the others operators of this type are obtained by obvious substitutions ( x!y), (1!2), (\"!#). Similar\noperators with index 3 is de\fned as\nL\"\n3xy=1\n2V3TX\nnZd3k\n(2\u0019)3^kx^kyG\"(k;!n)@2G\"(\u0000k;\u0000!n)\n@kx@ky(DxDy+DyDx); (110)\nH. Equal-spin-pairing states near critical temperature\nAs we already mentioned the internal \feld acting on the electron charges in the uranium ferromagnets is much\nsmaller than the upper critical \feld at zero temperature. In this case the gradient terms produce only small of order\nofO(Hint\nHc2(T=0)) correction to the eigenvalues of linear di\u000berential equations (94) for the order parameter components.\nThen, these equations are transformed into the algebraic equations:\n\u0011\"\nx= (g\"\n1x\u0011\"\nx+g#\n2x\u0011#\nx+g#\n3y\u0011#\ny)\u0015;\n\u0011#\nx= (g\"\n2x\u0011\"\nx+g#\n1x\u0011#\nx\u0000g\"\n3y\u0011\"\ny)\u0015; (111)\n\u0011\"\ny= (g\"\n1y\u0011\"\ny+g#\n2y\u0011#\ny\u0000g#\n3x\u0011#\nx)\u0015;\n\u0011#\ny= (g\"\n2y\u0011\"\ny+g#\n1y\u0011#\ny+g\"\n3x\u0011\"\nx)\u0015:\nfor the A state and\n\u0010\"\nz= (g\"\n1z\u0010\"\nz+g#\n2z\u0010#\nz)\u0015; (112)\n\u0010#\nz= (g\"\n2z\u0010\"\nz+g#\n1z\u0010#\nz)\u0015:\nfor the B state. Taking into account that the constants of interaction with indices 1,2 strongly exceed the constants\nwith index 3 originating from the spin-orbit terms in the gradient energy\ng1; g2>> g 3;\nwe come to three independent systems of equations for the x, y and z components of the order parameter in two band\nsuperconductor\n\u0011\"\nx= (g\"\n1x\u0011\"\nx+g#\n2x\u0011#\nx)\u0015;\n\u0011#\nx= (g\"\n2x\u0011\"\nx+g#\n1x\u0011#\nx)\u0015; (113)\n\u0011\"\ny= (g\"\n1y\u0011\"\ny+g#\n2y\u0011#\ny)\u0015;\n\u0011#\ny= (g\"\n2y\u0011\"\ny+g#\n1y\u0011#\ny)\u0015; (114)\nand\n\u0010\"\nz= (g\"\n1z\u0010\"\nz+g#\n2z\u0010#\nz)\u0015;\n\u0010#\nz= (g\"\n2z\u0010\"\nz+g#\n1z\u0010#\nz)\u0015: (115)\nThus, in the exchange approximation for the energy of magnetic inhomogeneity we have three di\u000berent superconducting\nstates ( ^kx\u0011\"\nx;^kx\u0011#\nx);(^ky\u0011\"\ny;^ky\u0011#\ny) and ( ^kz\u0010\"\nz;^kz\u0010#\nz) with di\u000berent critical temperatures de\fned by the determinants\nof Eqs. (113), (114) and (115).16\nIV. PHYSICAL PROPERTIES\nA. Critical temperature\nAssuming that the largest critical temperature corresponds to the ( ^kx\u0011\"\nx;^kx\u0011#\nx) superconducting state the zero of\ndeterminant of the system (113) yields the BCS-type formula\nT=\" exp\u0012\n\u00001\ng\u0013\n; (116)\nwhere the constant of interaction\ng=g\"\n1x+g#\n1x\n2+s\n(g\"\n1x\u0000g#\n1x)2\n4+g\"\n2xg#\n2x (117)\nis the function of temperature and magnetic \feld. Thereby the formula (116) is, in fact, an equation for the determi-\nnation of the critical temperature of the transition to the superconducting state. Let us look on it in the easiest case\nof a single-band (say spin-up) superconducting state when g=g\"\n1x.\nIn URhGe the transition to the superconducting state occurs at temperature much lower than the Curie temperature.\nHence, one can neglect by the temperature dependence of the constant of interaction. Then the critical temperature\nis determined by\nln\"\nTsc\u0018=1\ng\"\n1x/\u0000\n\u000b0zTc+\rzk2\nF\u00012\n\u00162\nBI2\rzxxk2\nFh^k2xN\"\n0(k)i; (118)\nwhere we have used Eqs. (103), (65) for g\"\n1xin the absence of magnetic \feld. The Curie temperature Tcin URhGe is\nan increasing function of pressure (see Fig.1b). Pressure dependence of all other quantities in the right hand side in\nthis equation is unknown. In assumption that the right hand side as whole is also increased with pressure we see that\nthis should be accompanied by the slow decrease of the temperature of transition to the superconducting state. And\nvice versa, when the right hand side decreases with pressure, the Tsc(P) is grow up. The \frst obviously corresponds\nto the observed pressure dependences Tc(P) andTsc(P) in URhGe, the second one to the situation in UCoGe( see\nFig.1c). In the latter case, of course, this argumentation is applicable to the pressure dependences in the region where\nTscis signi\fcantly smaller than Tc.\nWe do not consider here UGe 2where the superconducting state arises in the phase diagram region below the line\nof the \frst-order transition from the paramagnet to the ferromagnet state.\nB. Upper critical \feld parallel to c-axis in UCoGe\nThe upper critical \feld Hc2(T) parallel to the axis of spontaneous magnetization in UCoGe [46] possesses the de\fnite\nupward curvature in the slope near the critical temperature (Fig.4). The natural explanation of this phenomenon is\nthat the critical temperature itself is a function of external \feld. Indeed, near the critical temperature Tscthe upper\ncritical \feld is\nHc2=ATsc(Tsc\u0000T); (119)\nwhereA\u0019\u001e0\nv2\nFis a constant and the critical temperature is the function of pairing amplitude Tsc=\u000fexp(\u00001\ng). Again,\nassuming that the largest critical temperature corresponds to the ( ^kx\u0011\"\nx;^kx\u0011#\nx) superconducting state, we have in the\nsingle band approximation\nln\"\nTsc=1\ng\"\n1x/\u0002\n\fz(3M2\nz\u0000M2\nz0) +\rzk2\nF\u00032: (120)\nAt temperatures well below the Curie temperature the magnetization is almost temperature independent. On the\nother hand the UCoGe magnetic moment under the \feld in cdirection quite rapidly increases [47]. At \felds about\n1 TeslaMz=Mz(H) is about twice larger than Mz0=Mz(H= 0). Hence, in accordance with (120), the magnetic\n\feld increase decreases the constant of interaction g\"\n1xand the critical temperature Tsc(g\"\n1x).17\nThe temperature dependence of the upper critical \feld (120) can be rewritten as the \feld dependence of the\ntemperature of phase transition to the superconducting state Torb\nscdetermined by the orbital e\u000bect and by the \feld\ndependence of pairing interaction g\"\n1x=g\"\n1x(H):\nTorb\nsc=Tsc(g\"\n1x)\u0000H\nATsc(g\"\n1x): (121)\nObviously, the \feld dependence Tsc(g\"\n1x(H)) not only shifts down the linear \feld dependence of Torb\nsc(H) but also\ncreates an upward curvature in accordance with the experimental data shown in Fig.4.\nIn URhGe the temperature dependence of the upper critical \feld parallel to spontaneous magnetization (see Fig.5)\ndoes not reveal an upward curvature [48]. Unlike to UCoGe in this material the change of magnetic moment in the\n\feldHzsmaller than 1 Tesla is negligibly small [49]. Hence, the \feld dependence of the pairing constant plays no\nrole.\nC. Upper critical \feld in URhGe\nThe superconducting critical temperature in all uranium ferromagnets increases with the sample quality as it should\nbe in unconventional superconducting states where the Tsc(l) dependence from electron mean free path at l > \u0018 0is\ndescribed by [36]\nTsc\u0019Tsc0\u0000\u0019vF\n8l(122)\nand the zero temperature upper critical \feld increases with the sample purity as square of the critical temperature\nHc2\u0019\u001e0\n\u0019\u00182\n0/T2\nsc: (123)\nThe latter relation has been demonstrated by measurements of upper critical \feld in URhGe on the samples of di\u000berent\nquality [48] (Fig.5).\nAn other peculiar property revealed by Hardy and Huxley [48] is the temperature dependence of the upper critical\n\feld anisotropy. The ratio of Hc2(T) along the caxis to that along the baxis is independent of temperature. However,\nthe ratio of Hc2(T) parallel to the aaxis divided by the value along the baxis (orcaxis) increases linearly by\napproximately 20% as the temperature is decreased from Tcto zero (Fig.6). This behavior is consistent with a choice\nof an equal-spin-paired gap having a line node in the bcplane. Namely, working in the single-band approximation\nand taking the order parameter amplitude at H= 0 as\n\u0001\"(k;r) =\u0011\"\nxkx;\none can show [50] that the solutions of the linear Gor'kov equations corresponding to the maximal upper critical \feld\nfor di\u000berent \feld directions are\nHkb \u0001\"(k;r)\u0018A(H;T)(kx+ikz) 0(x;z) +B(H;T)(kx\u0000ikz) 2(x;z); (124)\nHkc \u0001\"(k;r)\u0018A(H;T)(kx+iky) 0(x;y) +B(H;T)(kx\u0000iky) 2(x;y); (125)\nHka \u0001\"(k;r)\u0018kx 0(y;z); (126)\nwhere n(x;y) are the Landau functions of particle with charge 2 ein a magnetic \feld, nis the Landau level number,\nand the coe\u000ecients A(H;T) andB(H;T) are the functions of magnetic \feld and temperature. We see, that the\nsolutions for the \feld along the cand thebaxes have the same structure and di\u000ber from the solution for the \feld along\ntheadirection, what naturally explains the observed temperature dependence of the upper critical \feld anisotropy.\nThis property is still valid in a multi-band superconductor with equal-spin-pairing if we assume (as we did in the\nsingle band case) that our superconducting state is the particular A-state such that the order parameter spin-up and\nspin-down amplitudes at zero \feld in di\u000berent bands have the form:\n\u0001\"(k;r) =\u0011\"\nxkx;\u0001#(k;r) =\u0011#\nxkx: (127)\nThus, the observed behavior of the temperature dependence of upper critical \feld anisotropy strongly points on the\npreferable order parameter structure in URhGe.18\nD. Zeros in spectrum and speci\fc heat at low temperatures\nAs we already pointed out even in the absence of an external \feld in a ferromagnetic superconductor there is an\ninternal \feld Hintacting on the electron charges. The internal magnetic \feld in all uranium ferromagnets is larger\nthan the lower critical \feld Hc1. Hence, the Meissner state is absent and the superconducting state is always the\nAbrikosov mixed state with space inhomogeneous distributions of the order parameter and the internal magnetic \feld.\nAt low temperatures, when due to Hint<< Hc2the distance between vortices is much larger than the core radius,\none can separate the speci\fc heat inputs arising from the vortex cores and the inter-vortex space.\nIt is usually accepted to operate with the ratio C=T =\rwhich is in a normal metal directly proportional to\nthe electron density of states. The cores contribution to the speci\fc heat is due to the almost gapless excitations\nlocalized in the vortex cores. Hence, due to the vortex cores \rkeeps a \fnite value in the superconducting state at low\ntemperatures\n\rv\u0019Hint\nHc2\rN; (128)\nwhere\rNis the normal state value of \r.\nAnother contribution to the density of states originates from so called the Volovik e\u000bect [51] taking place in the\ninter-vortex space. This case, the energy of excitations is given by Eqs. (19) and (20). In the absence an additional\nphase transition inside of superconducting state the order parameter of the superconducting state belonging to A\nco-representation is given by Eqs. (97), (98), or to B co-representation is given by Eqs.(99), (100). The A-state order\nparameter is equal to zero in isolated points kx=ky= 0, hence the inter-vortex space contribution to the density of\nstates is given by [51]\n\rA\niv\u0019Hint\nHc2ln\u0012Hc2\nHint\u0013\n\rN: (129)\nThe B-state order parameter is equal to zero at line kz= 0, hence the inter-vortex space contribution to the density\nof states is given by [51]\n\rB\niv\u0019r\nHint\nHc2\rN: (130)\nAs we pointed out in the previous Chapter the mixing of xandycomponent of the order parameter in A-state is in\nfact quite small, owing to the smallness of V3amplitude of pairing. So, the gap in A state spectrum is almost equal to\nzero either at line kx= 0, or at line ky= 0. Due to this reason the inter-vortex contribution to the density of states\nin the A-state can be given by the same square-root formula as for the B state.\nThe Eqs.(129), (130) are applicable to the defect free superconducting crystals. In presence of inhomogeneities\ncreated by impurities, dislocations, domain walls the gap in the quasiparticle spectrum is suppressed in \fnite vicinity\nof the order parameter zeros [36] as well as in the \fnite vicinity of the inter-domain walls. As result, the zero energy\ndensity of states acquires a \feld independent contribution. At high enough impurity concentration the square root\n\feld dependence can also be modi\fed [52].\nQualitatively the total low temperature \r0value for superconducting ferromagnets at moderate impurities amount\nis described by\n\r0=\rdw+\riv+\rv\u0019 \na+r\nHint\nHc2+Hint\nHc2!\n\rN; (131)\nwhere constant a<< 1.\nOne can estimate magnitude of internal \feld as\nHint=const\u0016u\na3uu; (132)\nwhere\u0016uis the magnetic moment per uranium atom, auuis the distance between nearest neighbor uranium atoms. The\ninter-uranium distances in UCoGe, URhGe and UGe 2are close to each other. On the other hand, the corresponding\nzero temperature magnetic moments 0.05 \u0016B, 0.4\u0016Band\u0016Bare quite di\u000berent what determines the di\u000berence in\nHintin these materials. The indeterminacy is introduced by the unknown pre-factors in Eq. (132). Another way to\ndetermine the internal \feld is just to take it equal to the external \feld along the direction of spontaneous magnetization\nthat suppresses ferromagnet many domain structure.19\nThe internal \feld estimated in review [9] is about 100G for UCoGe, 800G for URhGe and 2800 G for UGe 2\naccording to higher value of magnetic moment in this material. The zero temperature upper critical \feld directed\nalong spontaneous magnetization in UCoGe is 1.2 T , in UGe 2it is approximately 2.2 T. Known value of Hc2for\nURhGe is 0.6 T has been measured, however, in low RRR=21 single crystal. So, one can expect that the real value of\nlow temperature upper critical \feld in URhGe is roughly the same as in UCoGe. Thus, the \feld depend part of ratio\n\r0=\rNfound making use Eq. (131) approximately is 0.1 for UCoGe, 0.3 for URhGe and 0.5 for UGe 2. Corresponding\nexperimentally established values are presented in Fig.7.\nV. REENTRANT SUPERCONDUCTIVITY IN URhGe\nURhGe has a peculiar property. At low enough temperature the magnetic \feld about 1.3 Tesla directed along the\nb-axis suppresses the superconducting state [48] but at much higher \feld of about 10 Tesla the superconductivity is\nrecreated and exists till the \feld about 13 Tesla.[30] The maximum of the superconducting critical temperature in\nthis \feld interval is \u00190:4K. In the same \feld interval the material transfers from the ferromagnet to the paramagnet\nstate by means of the \frst-order type transition. The superconducting state exists not only inside of the ferromagnetic\nstate but also in the paramagnetic state separated from the ferromagnetic state by the phase transition of the \frst\norder (Fig.8).\nThe observation of the abrupt collapse of spontaneous magnetization under a strong enough external \feld along the\nbaxis has been reported already in the \frst publication about magnetic \feld-induced superconductivity in ferromagnet\nURhGe.[30] Recently, the \frst-order character of transition has been con\frmed by the direct observation of hysteresis\n[53] in the Hall resistivity in the vicinity of the transition \feld HR\u001812:5T.\nIn this Chapter we develop the Landau type phenomenological description of the ferromagnet-paramagnet phase\ntransition under an external magnetic \feld perpendicular to spontaneous magnetization. We \fnd the components of\nmagnetic susceptibility determining the superconducting pairing interaction and show that the magnetic susceptibility\ncorresponding to longitudinal magnetic \ructuations strongly increases in the vicinity of the \frst-order transition\nstimulating the reentrance of the superconducting state.\nA. Phase transition in orthorhombic ferromagnet under magnetic \feld perpendicular to spontaneous\nmagnetization\nThe Landau free energy of an orthorhombic ferromagnet in magnetic \feld H(r) is\nF=Z\ndV(FM+Fr); (133)\nwhere in\nFM=\u000bzMz2+\fzMz4+\u000ezM6\nz\n+\u000byM2\ny+\u000bxMx2+\fxyM2\nxM2\ny+\fyzMz2My2+\fxzMz2Mx2\u0000MH; (134)\nwe bear in mind the orthorhombic anisotropy and also the term of the sixth order in powers of Mz. The density of\ngradient energy is taken in the exchange approximation,\nFr=\rij@M\n@xi@M\n@xj: (135)\nHere,x;y;z are the coordinates pinned to the a;b;c crystallographic directions correspondingly, a;b;c ,\n\u000bz=\u000bz0(T\u0000Tc0);\u000bx>0; \u000by>0 (136)\nand\n\rij=0\n@\rxx0 0\n0\ryy0\n0 0\rzz1\nA: (137)\nIn constant magnetic \feld H=Hy^ythe equilibrium magnetization projections along the x;ydirections are obtained\nby minimization of free energy (134) in respect of Mx;My\nMx= 0; My=Hy\n2(\u000by+\fyzM2z): (138)20\nSubstituting these expressions back to (134) we obtain\nFM=\u000bzMz2+\fzM4\nz+\u000ezM6\nz\u00001\n4H2\ny\n\u000by+\fyzM2z; (139)\nthat gives after expansion of the denominator in the last term,\nFM=\u0000H2\ny\n4\u000by+ ~\u000bzMz2+~\fzMz4+~\u000ezM6\nz+:::; (140)\nwhere\n~\u000bz=\u000bz0(T\u0000Tc0) +\fyzH2\ny\n4\u000b2y; (141)\n~\fz=\fz\u0000\fyz\n\u000by\fyzH2\ny\n4\u000b2y(142)\n~\u000ez=\u000ez+\f2\nyz\n\u000b2y\fyzH2\ny\n4\u000b2y(143)\nWe see that under a magnetic \feld perpendicular to the direction of spontaneous magnetization the Curie temper-\nature decreases as\nTc=Tc(Hy) =Tc0\u0000\fyzH2\ny\n4\u000b2y\u000bz0: (144)\nThe coe\u000ecient ~\fzalso decreases with Hyand reaches zero at\nHcr\ny=2\u000b3=2\ny\f1=2\nz\n\fyz: (145)\nAt this \feld under ful\fllment the condition,\n\u000bz0\fyzTc0\n\u000by\fz>1 (146)\nthe Curie temperature (144) is still positive and at\nHy>Hcr\ny\nphase transition from a paramagnetic to a ferromagnetic state becomes the transition of the \frst order (Fig.9). The\npoint (Hcr\ny;Tc(Hcr\ny)) on the line paramagnet-ferromagnet phase transition is a tricritical point.\nThe minimization of the free energy Eq. (140) gives the value of the order parameter in the ferromagnetic state,\nM2\nz=1\n3~\u000ez[\u0000~\fz+q\n~\f2z\u00003~\u000bz~\u000ez]: (147)\nThe minimization of the free energy in the paramagnetic state,\nFpara=\u000byM2\ny\u0000HyMy (148)\nin respectMygives the equilibrium value of magnetization projection on axis yin paramagnetic state,\nMy=Hy\n2\u000by: (149)\nSubstitution back in Eq. (148) yields the equilibrium value of free energy in the paramagnetic state,\nFpara=\u0000H2\ny\n4\u000by: (150)21\nOn the line of the phase transition of the \frst order from the paramagnetic to ferromagnetic state determined by the\nequations [54]\nFM=Fpara;@FM\n@Mz= 0 (151)\nthe order parameter Mzhas the jump (Fig.10) from\nM?2\nz=\u0000~\fz\n2~\u000ez: (152)\nin the ferromagnetic state to zero in the paramagnetic state. Its substitution back in equation FM=Fparagives the\nequation of the \frst-order transition line,\n4~\u000bz~\u000ez=~\f2\nz; (153)\nthat is\nT?=T?(Hy) =Tc0\u0000\fyzH2\ny\n4\u000b2y\u000bz0+~\f2\nz\n4\u000bz0~\u000ez: (154)\nThe corresponding jump of My(see Fig.10) is given by\nM?\ny=Mferro\ny\u0000Mpara\ny =Hy\n2(\u000by+\fyzM?2\nz)\u0000Hy\n2\u000by: (155)\nB. Susceptibilities\nIn a perpendicular \feld magnetic susceptibilities along all directions are found in the same manner as this has been\nperformed in the Section IIIB for the case parallel \feld. In the ferromagnetic state T <T?they are\n\u001ff\nxx(k)\u0018=1\n2(\u000bx+\fxzM2z+\fxyM2y+\rijkikj);\n\u001ff\nyy(k)\u0018=1\n2(\u000by+\fyzM2z+\rijkikj);\n\u001ff\nzz(k)\u0018=1\n2(\u000bz+6\fzM2z+15\u000ezM4z+\fyzM2y+\rijkikj)=1\n2(4\fzM2z+12\u000ezM4z+\rijkikj): (156)\nIn the paramagnetic state T >T?they are:\n\u001fp\nxx(k)\u0018=1\n2(\u000bx+\fxyM2y+\rijkikj);\n\u001fp\nyy(k)\u0018=1\n2(\u000by+\rijkikj);\n\u001fp\nzz(k)\u0018=1\n2(~\u000bz+\rijkikj)=1\n2(\u000bz0(T\u0000Tc(Hy))+\rijkikj): (157)\nThe spin-triplet pairing interaction is expressed through the odd part of the susceptibility components :\n\u001fu\nii(k;k0) =1\n2[\u001fii(k\u0000k0)\u0000\u001fii(k+k0)]; i =x;y;z: (158)\nThus for the ferromagnetic state T <T?we have\n\u001ffu\nxx(k;k0)\u0018=\rijk2\nF\n(af\nx)2^ki^k0\nj; \u001ffu\nyy(k;k0)\u0018=\rijk2\nF\n(af\ny)2^ki^k0\nj; \u001ffu\nzz(k;k0)\u0018=\rijk2\nF\n(af\nz)2^ki^k0\nj; (159)\nwhere\naf\nx=\u000bx+\fxzM2\nz+\fxyM2\ny+ 2\rk2\nF;\naf\ny=\u000by+\fyzM2\nz+ 2\rk2\nF;\naf\nz= 4\fzM2\nz+ 12\u000ezM4\nz+ 2\rk2\nF: (160)22\nHereMz(Hy);My(Hy) are the equilibrium values of the magnetization components. It is instructive to compare the\nobtained expressions for the odd parts of the susceptibility components with Eqs.(64)-(67) found at \u000ez= 0; Hy= 0\nbutHz6= 0.\nFor the paramagnetic state T >T?they are\n\u001fpu\nxx(k;k0)\u0018=\rijk2\nF\n(ap\nx)2^ki^k0\nj; \u001fpu\nyy(k;k0)\u0018=\rijk2\nF\n(ap\ny)2^ki^k0\nj; \u001fpu\nzz(k;k0)\u0018=\rijk2\nF\n(ap\nz)2^ki^k0\nj; (161)\nap\nx=\u000bx+\fxyM2\ny+ 2\rk2\nF;\nap\ny=\u000by+ 2\rk2\nF;\nap\nz= 2(\u000bz0(T\u0000Tc(Hy)) + 2\rk2\nF: (162)\nThus, at the \frst-order transition from the paramagnetic to the ferromagnetic state the components of susceptibility\nabruptly change their values.\nAs we have seen in Chapter IIIA the pairing interaction in the ferromagnetic state is mostly determined by the odd\npart of the zcomponent of susceptibility\n\u001ffu\nzz(k;k0)\u0018=\rijk2\nF\n4(2\fzM2z+ 6\u000ezM4z+\rk2\nF)2^ki^k0\nj: (163)\nThe equilibrium magnetization Mz(Hy) decreases with magnetic \feld Hy(see Fig.10). One can expect, that the jump\nofMzon the \frst-order transition is much smaller than the low temperature magnetization at zero \feld Hy= 0 :\nMzjHy=0;T=0>>M?\nz: (164)\nThen according to the equation (163) the \u001ffu\nzzon the line of the \frst order strongly exceeds its own initial value at\nHy= 0 what stimulates the reentrance of superconductivity near the \frst-order transition.\nC. Superconducting state in vicinity of the \frst order transition\nThe suppression of the Curie temperature by the magnetic \feld perpendicular to spontaneous magnetization leads\nto e\u000bective increase of pairing interaction. This e\u000bect can in principle compensate the suppression of superconductivity\nby the orbital depairing. In URhGe the Curie temperature is much higher than Tsc. Hence, the orbital e\u000bect succeeds\nto suppress the superconducting state ( Hb\nc2(T= 0)\u00191:3Tsee [48]) much before the e\u000bect of decreasing of the\nCurie temperature and the stimulation of pairing intensity reveals itself. But at \felds higher than 10 Tesla the latter\ne\u000bect starts to overcome the orbital depairing and the superconducting state recreates. The critical temperature of\nsuperconducting transition begins grow up and approaches to the line of the \frst order transition from ferromagnetic\nto paramagnetic state and intersects it [30, 53]. Here we have a look what is going on with line of superconducting\nphase transition at the intersection with the line of ferromagnet-paramagnet \frst order phase transition T?(Hy).\nIf the external \feld orientated along b-axis, that is perpendicular to the exchange \feld h(see Fig.11), it is natural\nto choose the spin quantization axis along the direction of the total magnetic \feld h^z+Hy^y. Then the normal state\nmatrix Green function is diagonal\n^Gn=\u0012\nG\"0\n0G#\u0013\n; (165)\nwhere\nG\";#=1\ni!n\u0000\u0018\";#\nk\u0006\u0016Bq\nh2+H2y: (166)\nAll the formula obtained in the Chapter IIIA are valid. The only modi\fcation is that one need to work with the\nsusceptibility tensor written in the new coordinate frame\n\u001fij!~\u001fij=Ril\u001flmRjm; (167)23\nwhere\n^R=0\n@1 0 0\n0 cos'\u0000sin'\n0 sin'cos'1\nA (168)\nis the matrix of rotation around ^ xdirection on the angle given by\ntan'=Hy\nh:\nFor simplicity, one can work with the equal-spin-pairing superconductivity neglecting by the amplitude \u00010. Also,\nthe gradient energy of orthorhombic ferromagnet Eq.(135) was taken in the exchange approximation such that the\npairing amplitude V3= 0. Unlike to the case of parallel \feld, here we neglect by the orbital e\u000bects ignoring\nthe order parameter coordinate dependence. This case the critical temperature of transition in superconducting state\nTsc(Hy) is determined from the self-consistency equations\n\u0001\"(k) =\u0016BI2TP\nnP\nk0n\u0002\n\u001ffu\nzz(k;k0) cos2'+\u001ffu\nyy(k;k0) sin2'\u0003\nG\"\n1G\"\n2\u0001\"(k0)\n+\u0002\n(\u001ffu\nxx(k;k0)\u0000\u001ffu\nyy(k;k0)) cos2'+ (\u001ffu\nxx(k;k0)\u0000\u001ffu\nzz(k;k0)) sin2'\u0003\nG#\n1G#\n2\u0001\"(k0)o\n; (169)\n\u0001#(k) =\u0016BI2TP\nnP\nk0n\u0002\n(\u001ffu\nxx(k;k0)\u0000\u001ffu\nyy(k;k0)) cos2'+ (\u001ffu\nxx(k;k0)\u0000\u001ffu\nzz(k;k0)) sin2'\u0003\nG\"\n1G\"\n2\u0001\"(k0;q)\n+\u0002\n\u001ffu\nzz(k;k0) cos2'+\u001ffu\nyy(k;k0) sin2'\u0003\nG#\n1G#\n2\u0001#(k0)o\n: (170)\nHere,G\"\n1=G\"(k0;!n),G\"\n2=G\"(\u0000k0;\u0000!n) and similarly for the G#\n1andG#\n2Green functions given by Eq.(166). In\nthe ferromagnetic state near the \frst order transition the angle '\u0019\u0019=4 , and the susceptibilities are determined by\nEqs. (159) and(160).\nIn the paramagnetic state the susceptibilities are determined by Eqs. (161) and(162). The angle '=\u0019=2,\nG\";#\npara=1\ni!n\u0000\u0018k\u0006\u0016BHy(171)\nand one can work with the equations for (\u0001\";\u0001#) independently from equation for \u00010[27]\n\u0001\"(k;q) =\u0016BI2TP\nnP\nk0n\n\u001fpu\nyy(k;k0)G\"\n1G\"\n2\u0001\"(k0) + (\u001fpu\nxx(k;k0)\u0000\u001fpu\nzz(k;k0))G#\n1G#\n2\u0001#(k0;q)o\n; (172)\n\u0001#(k;q) =\u0016BI2TP\nnP\nk0n\n(\u001fpu\nxx(k;k0)\u0000\u001fpu\nzz(k;k0))G\"\n1G\"\n2\u0001\"(k0;q) +\u001fpu\nyy(k;k0)G#\n1G#\n2\u0001#(k0;q)o\n:(173)\nAs we mentioned already the components of susceptibility undergo a \fnite jump at \frst order phase transition\nfrom ferromagnetic to paramagnetic state. The Fermi surfaces of split spin-up and spin-down electron bands, and the\naverage density of states on them also undergo an abrupt changes. Finally, the structure of equations for determination\nof the critical temperature of transition in superconducting state Tsc(Hy) is quite di\u000berent on both sides of the\nferromagnet-paramagnet phase transition. So, the line of Tsc(Hy) should undergo a jump at the intersection of line\nof the \frst order phase transition T?(Hy). The experiment [30] clearly demonstrates an abrupt fall of the critical\ntemperature at this transition.\nD. Concluding remarks\nMaking use the phenomenological description of the phase diagram in URhGe under a magnetic \feld along baxis\nperpendicular to the direction of spontaneous magnetization we have found that the phase transition between the\nferromagnetic and the paramagnetic states under a strong enough magnetic \feld perpendicular to the direction of\neasy magnetization changes its order from the second to the \frst order type. The reentrance of superconductivity is\nexplained by the strong increase in magnetic susceptibility in the vicinity of the \frst-order transition in comparison24\nwith its zero-\feld value. The reentrant superconductivity near the \frst-order transition line T?(Hy) exists both in\nferromagnet and paramagnet state. The critical temperature of the transition to the superconducting state undergoes\nan abrupt fall down at the intersection with the line of the ferromagnet-paramagnet phase transition.\nThe suppression of the Curie temperature by a magnetic \feld perpendicular to the spontaneous magnetization\nincreases the pairing interaction. This e\u000bect compensates the suppression of superconductivity by the orbital depairing.\nIn UCoGe, where the Curie temperature does not strongly exceed the temperature of transition to superconducting\nstate, this mechanism stimulates the upturn of the upper critical \feld for the \feld along bdirection above 5 Tesla\nshown in Fig.12. The enhancement of superconductivity in UCoGe is accompanied by an enhancement of nuclear\nrelaxation rate caused by the increase of magnetic susceptibility in process of approach to the Curie temperature in\n\feld parallel to baxis [55].\nTheadirection is magnetically much harder than the bone:\u000bx\u001d\u000by. Hence, the suppression of the Curie\ntemperature by magnetic \feld Hxis much less e\u000bective and practically unobservable in available magnetic \felds [55].\nHowever, one can expect the similar development of superconductivity stimulation at much higher \felds along the a\ndirection.\nIn the vicinity of the phase transition of the \frst order caused by the magnetic \feld along the b-axisHy\u001912T\nthe application of some \feld along the aaxis increasing the magnitude of the total magnetic \feld H=q\nH?2\ny+H2x\nintroduces negligible changes in the pairing interaction that was in the absence of the \feld along the hard a-axis. At\nthe same time the orbital upper critical \feld in the adirection is one and a half times larger than in the bdirection [48].\nThis roughly explains the stability of reentrant superconductivity in URhGe up to the \felds H=q\nH?2\ny+H2x\u001930\nTesla [56].\nIt is known that in the presence of an external \feld along the direction of spontaneous magnetization ^ zthe line of\nthe \frst order transition T?(Hy) spreads to two surfaces of the \frst order transition T?(Hy;\u0006Hz). At these surfaces\nthe jump in ferromagnet spontaneous magnetization decreases with increase of jHzjand disappears completely on\nsome lines beginning at the tricritical point Tc(Hcr\ny;Hz= 0). It was suggested [56] that these lines are \fnished at zero\ntemperature in some quantum critical points on the ( Hy;Hz) plane. The quantum critical magnetic \ructuations have\nbeen put forward as a source stimulating superconductivity in the vicinity of the line of the ferromagnet-paramagnet\n\frst-order phase transition. This idea looks plausible. One can remark, however, that in general the tricritical line\nTcr(Hy;Hz) can never reach zero temperature or simply be dislocated far enough from the superconducting region\non the phase diagram.\nMotivated by the idea of critical \ructuations the recent measurements by Y.Tokunaga et al [57] have demonstrated\nthe enormous increase of NMR relaxation rate in URh 0:9Co0:1Ge atHy\u001913 Tesla. These experiments have been\nperformed at temperature 1.6 Kelvin that is in the region near the second-order phase transition between the ferro-\nmagnetic and the paramagnetic states. The reentrant superconductivity appears at much lower temperatures near\nthe line of the \frst-order transition where the role of critical \ructuations is certainly less important.\nHere we have demonstrated that the reentrant superconductivity in URhGe can arise even in the absence of critical\n\ructuations due to a drastic increase in the longitudinal susceptibility in the vicinity on the \frst-order transition line\nfrom the paramagnet to the ferromagnet state.\nVI. CRITICAL MAGNETIC RELAXATION IN URANIUM FERROMAGNETS\nA. Critical relaxation in ferromagnets\nThe excitations in magnetic systems are measured by the neutron scattering. The inelastic magnetic neutron\nscattering intensity\nI(Q;!) =A(ki;kf)(\u000e\u000b\f\u0000^Q\u000b^Q\f)jF(Q)j2S\u000b\f(q;!) (174)\nis related to the dynamical structure factor\nS\u000b\f(q;!) =Z1\n\u00001dtei!thM\u000bq(t)M\f\u0000q(0)i\nwhich is a wave vector - frequency dependent magnetic moments correlation function [58] related by the \ructuation-\ndissipation theorem to the imaginary part of susceptibility\nS\u000b\f(q;!) =2\n1\u0000exp(\u0000!\nT)\u001f00\n\u000b\f(q;!): (175)25\nThe total wave-vector transfer is Q=q+\u001c,\u001cis a reciprocal lattice vector and qlies in the \frst Brillouin zone. ^Q\u000bis\nthe direction cosine of Qalong coordinate axis \u000b.F(Q) is the magnetic form factor measured by the elastic neutron\nscattering. We put the Planck constant ~= 1. To be compatible with neutron scattering literature we shall use in\nthis chapter the notation qfor the wave vectors.\nFor each crystallographic direction one can \ft the imaginary part of susceptibility as\n\u001f00(q;!)\n!=A\n!2+ \u00002q; (176)\nthat is to express it through the experimentally measured amplitude Aand the width \u0000 qof dynamical form-factor.\nThen the Kramers-Kronig relation allows to \fnd the real part of static susceptibility\n\u001f(q) =\u001f0(q;0) =1\n\u0019Z\u001f00(q;!)\n!d!=A\n\u0000q: (177)\nIn the absence of walls and spin-orbital coupling the magnetization is a conserved quantity, hence, in a Heisenberg\nferromagnet above Curie temperature the only mechanism leading to the magnetization relaxation is the spin di\u000busion\nthat results in [58, 59]\nS(q;!) =2!\u001f(q)\n1\u0000exp(\u0000!\nT)\u0000q\n!2+ \u00002q; (178)\nsuch that line width of quasi elastic scattering\n\u0000q=Dq2(179)\nis determined by the di\u000busion coe\u000ecient D. Theq2law dependence was observed in a wide temperature range above\nTcin Ni and Fe ( see Ref.[60] and references therein) reducing at T=Tcto \u0000/q2:5dependence according to\npredictions of mode-mode coupling theory [61].\nIn weak itinerant ferromagnets above Curie temperature another mechanism of dissipationless relaxation can dom-\ninate with structure factor given by the same Eq. (178) but with the linewidth determined by equality [62, 63]\n\u001f(q)\u0000q=\u001fP!(q); (180)\nwhere\u001fPis the noninteracting Pauli susceptibility, !(q) is the Landau damping frequency equal to2\n\u0019qvFfor the\nspherical Fermi surface. The linear in wave vector line width was observed in MnSi [64], however, in the other weak\nitinerant ferromagnets MnP [65] and Ni 3Al [66] the linewidth q-dependence is closer to the dynamic scaling theory\npredictions [61].\nB. Magnetic relaxation in dual localized-itinerant ferromagnets\nThe magnetic susceptibility in uranium ferromagnets along easy axis is much larger than in a direction perpendicular\nto it. In UGe 2the easy axis is along the acrystallographic direction. Neutron scattering measurements reported in\nthe paper [33] with scattering wave vector qparallel to the crystal aaxis revealed no extra scattering relative to the\nbackground while for the qparallel to the c-axis a strongly temperature dependent contribution was found as it should\nbe according to Eq.(174). However, unlike both Eqs (179) and (180) \u0000 qdoes not vanish as q!0 for temperatures\ndi\u000berent from Tc. The same result was found in UCoGe [68] for the scattering with qparallel to the a-axis because\nthe easy axis in this material is along c-direction [69]. Hence, in uranium ferromagnets there is some mechanism of\nthe uniform magnetization relaxation to the equilibrium.\nThe magnetization in an electron gas relaxes due to the spin-\rip processes caused by the spin-orbit coupling either\nbetween the electrons [70] or by the spin-orbit coupling between the itinerant Bloch electron spins with potential of ions\nin vibrating lattice [71, 72]. Both mechanisms produce so tiny homogeneous relaxation rate, that it is unobservable in\nferromagnetic materials, while the relaxation rate \u0000 q=0found in UGe 2[33] is of the order of several Kelvin. Hence, the\nmagnetization relaxation in this material is due to some di\u000berent mechanism. In what follows, for the concreteness,\nwe shall discuss mostly UGe 2.\nMagnetic susceptibility of single UGe 2crystals has been measured by several groups [73{75]). The easy axis\nmagnetization at zero temperature was found 1.43 \u0016B=f.u. that in the case of itinerant ferromagnetism corresponds\nto completely polarized single electron band. On the other hand the neutron scattering measurements of the magnetic\nform factor [16] shows that: (i) the shape of its qdependence is not distinguishable from the wave vector dependences26\nof the form factors of free U3+or U4+ions, (ii) its low temperature value at q!0 coincides with the magnetization\nmeasured by magnetometer with accuracy of the order of 1 percent. Thus, practically whole magnetic moment both\nin paramagnetic and in ferromagnetic states concentrated at uranium atoms [76]\nThe static magnetic properties of UGe 2are well described in [75] in terms of crystal \feld splitting of the U4+\nstate, which is the3H4term of the 5 f2con\fguration of localized electrons, despite of the presence of the itinerant\nelectrons \flling the bands formed by two 7 s, one 6dand one 5furanium and also germanium orbitals. So, UGe 2is\nactually a dual system where local and itinerant states of f-electrons coexist. The3H4term of each atom of UGe 2\nin paramagnetic state mostly consists of superposition of three quasi-doublets and three singlets arising from the\nstate with a \fxed value of total momentum J= 4 split by the crystal \feld [75]. The temperature decrease causes\nthe change in probabilities of populations of crystal \feld states revealing itself in temperature dependence of the\nmagnetic moment. The degeneracy removal of the ground state formed by the lower quasi-doublet allows the system\nto order magnetically with the ordered moment of \u00181:5\u0016Btwice smaller than the Curie-Weiss moment deduced from\nsusceptibility above the Curie temperature. The itinerant electron subsystem formed by 7 s, 6dand partly 5 felectrons\nis also present providing about 0 :02\u0016Blong range magnetic correlations as demonstrated by muon spin relaxation\nmeasurements [77, 78]. All mentioned observations as well the theoretical treatment [75] unequivocally point on the\nlocal nature of UGe 2ferromagnetism. This means that the quasi-elastic neutron scattering occurs mostly on the\n\ructuations of magnetization in the localized moments subsystem.\nThe interaction between localized and itinerant electron subsystems leads to the magnetization relaxation measured\nby neutron scattering in paramagnetic and ferromagnetic state of UGe 2. This type of relaxation can be considered\nas an analog of spin-lattice relaxation well known in physics of nuclear magnetic resonance [79]. In our case the\nmagnetization created by the local moments of uranium atoms giving the dominate contribution to\nthe neutron scattering plays the role of \"spin\" subsystem, whereas the itinerant electrons present\nthe \"lattice\" degrees of freedom absorbing and dissolving \ructuations of magnetization. Unlike to the NMR\nrelaxation determined by the nucleus and the electrons magnetic moments interaction the spin-lattice relaxation\nbetween the localized and conducting electrons is determined by spin-spin exchange processes and\nhas no relativistic smallness typical for NMR relaxation. A deviation of magnetization from the equilibrium value\nrelaxes by transfer to the itinerant electrons. According to this, we shall treat the magnetization almost completely\ndetermined by the local moments of uranium atoms as not conserved quantity [34].\nThe process of the easy axis magnetization relaxation to equilibrium is described by the Landau-Khalatnikov kinetic\nequation [80]\n@M\n@t=\u0000A\u000eF\n\u000eM: (181)\nHere\nF=Z\ndV\u0012\n\u000b0(T\u0000Tc)(T)M2+\rij@M\n@xi@M\n@xj\u0000MH\u0013\n(182)\nis the energy of the order parameter \ructuations above the Curie temperature in a quasi stationary magnetic \feld\nalong the easy axis. The gradient energy in orthorhombic crystal written in the exchange approximation is determined\nby three nonzero constants \rxx;\ryy;\rzz. Thex;y;z axes are pinned to the b;c;a directions. The kinetic equation can\nbe rewritten as\n@M\n@t+riji=\u0000M\u000b\n\u001c+AH; (183)\nwhere\u001c\u00001= 2A\u000b0(T\u0000Tc),\nji=\u00002A\rij@M\n@xj(184)\nare the components of the spin di\u000busion currents.\nSubstituting in Eq.(183) M=mq!ei(qr\u0000!t)andH=hq!ei(qr\u0000!t)we obtain susceptibility\n\u001f(q;!) =mq!\nhq!=A\n\u0000i!+ \u0000q: (185)\nThe width of quasi-elastic scattering for qk^caxis is\n\u0000q= 2A\u0002\n\u000b0(T\u0000Tc) +\ryyq2\nc\u0003\n: (186)27\nBelow the Curie temperature in the ferromagnetic state the equilibrium magnetization M=M0(T) and energy of\n\ructuations is\nF=Z\ndV\u0012\n2\u000b0(Tc\u0000T)(M\u0000M0)2+\rij@M\n@xi@M\n@xj\u0000(M\u0000M0)H\u0013\n(187)\nBy the similar derivation we obtain for the susceptibility the same expression as Eq.(185) with the following width of\nquasielastic scattering\n\u0000q= 2A\u0002\n2\u000b0(Tc\u0000T) +\ryyq2\nc\u0003\n: (188)\nC. Concluding remarks\nExperimentally there were determined two independent values: the width \u0000 qand the amplitude A=\u001f(q)\u0000qof\ndistribution given by Eq. (178) (see Fig.13). Here, we have established the following. The line width of quasi-\nelastic neutron scattering near the Curie temperature proves to be a linear function of T\u0000Tc. The absolute value\nof the derivative jd\u0000qc=dTjin a ferromagnetic region is roughly twice as large as the corresponding derivative in a\nparamagnetic region. The dependence of the wave vector qcis parabolic. All of these \fndings are in qualitative\ncorrespondence with the experimental observations reported in the paper [33] (see Fig.14a,b).\nAt the same time there has been found [33] (see Fig.14d) that the product \u001f(q)\u0000qis temperature independent\nabove Curie temperature but reveals the fast drop below Tc. Such type behavior means that below Tcthe decrease of\nsusceptibility \u001f(q) with temperature proves to be much faster than it is in the accordance with the mean \feld theory.\nThis unusual behavior of susceptibility in UGe 2far enough from the critical region deserves a special investigation.\nMany experimental observations point to the local nature of magnetism in uranium ferromagnetic superconductors.\nThe interaction between localized and itinerant electron subsystems gives rise to a speci\fc mechanism of magnetiza-\ntion relaxation similar to \"spin-lattice\" relaxation known in physics of nuclear magnetic resonance. This relaxation\ndetermined by the exchange spin-spin coupling is much faster than NMR relaxation supported by much weaker inter-\naction between electron and nuclei magnetic moments. The phenomenological description of quasi-elastic magnetic\nrelaxation is based on speci\fc for heavy fermionic ferromagnet uranium compounds property that magnetization\nsupported by the moments located at uranium atoms is not a conserved quantity. As result the line width of quasi\nelastic neutron scattering at q!0 acquires non-vanishing value at all temperatures besides the Curie temperature.\nThis conclusion has been con\frmed by the microscopic analysis [81] where was shown explicitly that near the\nferromagnetic instability in a system consisting of localized and itinerant electrons the magnetization concentrating\nmostly on the localized subsystem is not conserved.\nVII. ANISOTROPY OF NUCLEAR MAGNETIC RESONANCE RELAXATION AND THE UPPER\nCRITICAL FIELD IN UCoGe\nThe nuclear magnetic resonance measurements on59Co nucleus in UCoGe show that the magnetic \feld along\nthec-axis strongly suppresses the magnetic \ructuations along this direction [31, 82]. Also, there was revealed that\nthe upper critical \feld value in this superconducting material drops abruptly at small \feld declinations from both\ntheaand thebcrystallographic directions towards the direction of the spontaneous magnetization c[9, 32]. So,\nthe component of magnetic \feld along the caxis very e\u000bectively suppresses the superconducting state. As we have\nseen, the triplet pairing in uranium ferromagnet superconductors mostly provided by the longitudinal \ructuations of\nmagnetization with the amplitude proportional to the odd part of the susceptibility \u001fu\nzz. Here we shall demonstrate\nthat both mentioned phenomena have the same origin and they are explained by the strong increase of magnetization\n[47, 83] and the corresponding decrease of di\u000berential susceptibility (59) in UCoGe in magnetic \feld along c-axis.\nA. Nuclear magnetic resonance relaxation rate\nNuclear spin-lattice relaxation rate measured in a \feld along the \u000bdirection is expressed in terms of the imaginary\npart of the dynamic susceptibility along the \fand\rdirections perpendicular to \u000bas\n1\nT\u000b\n1/TX\nk\u0014\njA\f\nhfj2\u001f00\n\f(k;!)\n!+jA\r\nhfj2\u001f00\n\r(k;!)\n!\u0015\n: (189)28\nAt low temperatures 1 =T1forHkcis more than order magnitude smaller those measured in the other two \feld\ndirections [82]. So, if we are interested in the relaxation rate in the \feld tilted on angle \u0012in respect to the baxis in\nthebcplane, such that \u0012is noticeably smaller than \u0019=2, we can use the following expression\n1\nT1(\u0012)/TX\nkjAz\nhfj2\u001f00\nzz(k;!)\n!cos2\u0012: (190)\nIn assumption that the \ructuations of the hyper\fne e-m \feld on the Co sites are determined by the \ructuations of\nmagnetization of the subsystem of localized moments as we did discussing the neutron scattering relaxation rate we\ncan use the formula\n\u001f00\nzz(q;!)\n!=A\n!2+ \u00002\nk;\u0000k= 2A(a+\rijkikj); (191)\nwhere\na=\u000bz+\fyzM2\ny+ 6\fzM2\nz= 2\fz(3M2\nz\u0000M2\nz0): (192)\nTo obtain the last equality, as in the Eq.(65), we have used the equilibrium condition\n2\u000bz+ 2\fyzM2\ny+ 4\fzM2\nz=Hz\nMz:\nHere\nMz=Mz(Hy;Hz) =Mz(Hcos\u0012;Hsin\u0012); Mz0=Mz(H;0) (193)\nare the equilibrium components of magnetization in the \feld\nH=Hy^y+Hz^z=Hcos\u0012^y+Hsin\u0012^z: (194)\nAt arbitrary temperatures below the Curie temperature one can use experimental values for \feld dependent magne-\ntizationMz(Hy;Hz).\nThe NMR measurements are done at the frequencies !<< \u0000q, hence, the relaxation rate is determined as\n1\nT1(\u0012)/TjAhfj2Acos2\u0012Zd3k\n(2\u0019)3\u00002\nk: (195)\nFor the simplicity one can calculate the converging integral in spherical approximation\nZd3k\n(2\u0019)3\u00002\nk\u0019Z1\n04\u0019k2dk\n(2\u0019)3(2A)2(a+\rk2)2=1\n32\u0019A2p\na\r3:\nThus, keeping only the \feld dependent part in Eq.(195) we obtain\n1\nT1(\u0012)/cos2\u0012pa=1\u0000H2\nz\nH2p\n2\fz(3M2z\u0000M2\nz0): (196)\nThe measurements of the NMR relaxation rate dependence from the magnetic \feld orientation was performed in\nrelatively small \felds H < 3:5 Tesla [31]. This case, Mz0=Mz(H;0) is almost independent from the \feld H=Hy\nand one can take it approximately equal to the spontaneous magnetization Mz0(0;0). On the other hand, Mz=\nMz(Hy;Hz)\u0019Mz(0;Hz) quickly grows with increase of Hz. For instance, in \feld Hz= 1 Tesla the magnetization\nMz(0;Hz) is twice large than it is at Hz= 0 [47]. Hence, according to Eq.(196), taking H= 3:5 Tesla and Hz= 1\nTesla, we obtain\n1\nT1(Hz= 1T)\u00190:41\nT1(Hz= 0): (197)\nIn comparison with this estimation the relaxation rate1\nT1(Hz) measured at T= 1:7 Kelvin [31] experiences faster\nfalling with growing of the Hzcomponent of magnetic \feld ( Hcin Fig.15). This is not astonishing in view of roughness\nof our approximations made at the Eq.(196) derivation.\nThus, the NMR relaxation rate dependence from the magnetic \feld along the easy magnetization axis originates\nfrom the corresponding \feld dependence of the longitudinal component of magnetic susceptibility.29\nB. Upper critical \feld anisotropy\nThe anomalous upper critical \feld anisotropy in UCoGe [9, 32] also \fnds the natural explanation in terms of strong\n\feld dependence of the longitudinal susceptibility determining the constant of pairing interaction. We have already\ndemonstrated this dependence in the case of magnetic \feld parallel to the direction of the spontaneous magnetization\nthat is along the caxis (see chapter IVB). If the magnetic \feld is directed along the bcrystallographic axis the critical\ntemperature in neglect of orbital e\u000bects is determined by Eqs. (169) and (170). In a \feld directed in the bcplane\nH=Hy^y+Hz^zthe equations (169) and (170) keep their form but the Green functions now are\nG\";#=1\ni!n\u0000\u0018\";#\nk\u0006\u0016Bq\n(h+Hz)2+H2y(198)\nand the angle is given by\ntan'=Hy\nh+Hz:\nThe susceptibilities are\n\u001fu\nzz(k;k0)\u0018=\rijk2\nF\n4 [\fz(3M2z\u0000M2\nz0) +\rk2\nF]2^ki^k0\nj; (199)\n\u001fu\nyy(k;k0)\u0018=\rijk2\nF\n[\u000by+\fyzM2z+ 2\rk2\nF]2^ki^k0\nj; (200)\nwhere\nMz=Mz(Hy;Hz) =Mz(Hcos\u0012;Hsin\u0012); Mz0=Mz(H;0)\nare the equilibrium components of magnetization in the \feld H=Hy^y+Hz^z.\nAs usual one can neglect the \u001fu\nyyin comparison with \u001fu\nzz. Then, as in the chapter IVB, assuming that the largest\ncritical temperature corresponds to the ( ^kx\u0011\"\nx;^kx\u0011#\nx) superconducting state, we have in the single band approximation\nln\"\nTsc=1\ng\"\n1x/\u0002\n\fz(3M2\nz\u0000M2\nz0) +\rzk2\nF\u00032\ncos2': (201)\nThere are no experimental data about the low temperature behavior of Mz=Mz(Hy;Hz) as function of both its\narguments. All the measurements have been performed in the \feld directed along the crystallographic axies a;b;c\n[47, 83]. However, looking at the data [83] in the strong \feld - low temperature region, where the phenomenon of\nthe strong upper critical \feld anisotropy has been revealed, one can expect that an increase Hzat \fxedHystrongly\nincreasesMz. Also a decrease of Hycausing the increase of Curie temperature Tc(Hy) increases Mzas well. Thus, a\nmagnetic \feld declination from the boradirection toward caxis leads to the increase of Mz=Mz(Hy;Hz), hence,\nto the sharp drop in the constant of pairing interaction described by Eq.(201). This explains the low-temperature -\nhigh \feld upper critical \feld anisotropy observed in UCoGe [9, 32] (see Fig.16).\nVIII. FIRST ORDER PHASE TRANSITION TO FERROMAGNET STATE IN UGe 2\nThe pressure-temperature phase diagrams of several weak ferromagnets exhibit similarity. The transition from the\nparamagnetic to the ferromagnetic states at ambient pressure occurs by means of the second-order phase transition.\nThe phase transition temperature decreases with pressure increase such that it reaches the zero value at some pressure\nP0. In a pressure interval below P0the ordered ferromagnetic moment disappears discontinuously. Thus at high\npressures and low temperatures the ferromagnetic and the paramagnetic states are divided by the \frst-order type\ntransition whereas at higher temperatures and lower pressures this transition is of the second-order. Such type of\nbehavior is typical for MnSi [84{87], UGe 2[88, 89] (see Fig.17), ZrZn 2[90]. The same behavior has been established\nin the ferromagnetic compounds Co(Si 1\u0000xSex)2[91] and (Sr 1\u0000xCax)RuO 3[87] where the role of governing parameter\nplays the concentration of Se and Ca correspondingly.30\nA. Phase transition to ferromagnetic state in Fermi liquid theory\nThe phase transition from paramagnetic to itinerant ferromagnetic state is usually considered in frame of Stoner\ntheory where it is the transition of the second order [62]. Some time ago, Belitz, Kirkpatrick, and Vojta (BKV)\nhave argued that the phase transition in clean itinerant ferromagnets is of \frst-order at low temperatures, due to the\ncorrelation e\u000bects that lead to a logarithmic term in the free energy density expansion in powers of dimensionless\nmagnetization M [92]:\nE=E0+\u000bM2+\fM4+vM4lnjMj+::: (202)\nIndeed, at positive coe\u000ecient vthe e\u000bective fourth order term in this formula is negative at small M , and the\ntransition to ferromagnet state is of the \frst-order.\nThe logarithm correction to the fourth order term has the long story. For the \frst time it was calculated in 1970\nby S.Kanno [93] in the dilute Fermi gas model in the second order in respect to dimensionless gas parameter kFa,\nwherekFis the Fermi momentum related to the total density\nn=n\"+n#=k3\nF\n3\u00192\nanda > 0 is the s-wave scattering length. In general, to solve the phase transition problem at T= 0, one must\ncalculate the Fermi-gas energy density\nE(x) =3\n5n\"Ff(M)\nas a function of the dimensionless spin polarization (magnetization)\nM=n\"\u0000n#\nn\"+n#\nat givenkFa. Here\"F=k2\nF=2m.\nIn the \frst order in kFathe ferromagnetic phase transition is of the second-order and occurs [94] at kFa=\u0019=2. The\nsecond-order perturbation theory predicts a \frst-order phase transition [95, 96] at kFa= 1:054, consistent with the\nBKV argument. However, since the critical gas parameter is expected to be of order O(1), perturbative predictions\nmay be unreliable. The nonperturbative e\u000bects are studied by He and Huang [97] by summing the particle-particle\nladder diagrams to all orders in the gas parameter. The theory predicts a second-order phase transition, which\nindicates that ferromagnetic transition in Fermi liquid occurs not according to BKV scenario. The predicted [97]\ncritical gas parameter kFa= 0:858 is in good agreement with the recent quantum Monte Carlo result kFa= 0:86 for\na nearly zero-range potential [98].\nSo, the \frst order phase transition in UGe 2cannot be explained in frame of isotropic Fermi liquid theory even if\nwe will forget that this compound presents the strongly anisotropic ferromagnetic metal with magnetization mostly\nsupported by the magnetic moments localized at uranium atoms.\nFinally, it should be noted that an isotropic ferromagnetic Fermi liquid is unstable in respect to the transversal\ninhomogeneous deviations of magnetization [99, 100]. So, the problem about the isotropic Fermi liquid phase transition\ninto ferromagnetic state has only academic interest.\nB. Magneto-elastic mechanism of development of the \frst order type instability\nThe magneto-elastic mechanism of development of the \frst order type instability has been put forward in the paper\n[101] where it was demonstrated that the change of transition character from the second to the \frst-order takes place\nat large compressibility and at strong enough steepness of the exchange interaction dependence from the interatomic\ndistance. This can be considered in frame of the Landau theory of the phase transitions. Namely, in neglect the shear\ndeformation the free energy density near the phase transition to the Ising type ferromagnetic state has the following\nform\nF=\u000b0(T\u0000Tc)M2+\fM4+K\n2\"2\u0000q\"M2: (203)\nHere,Mis the magnetization density, \"is the relative volume change, Kis the bulk modulus. The coe\u000ecient qis\nrelated to the Curie temperature pressure dependence as\nq=\u000b0dTc\nd\"=\u0000\u000b0KdTc\ndP: (204)31\nAt \fxed pressure, that is when the specimen volume changes are not accompanied by pressure changes in environment\nmedia@F\n@\"= 0, the deformation is determined by square of magnetization \"=q\nKM2what yields\nF=\u000bM2+\u0012\n\f\u0000q2\n2K\u0013\nM4: (205)\nHence, atq2\n2K>\fthe phase transition changes its character from the second to the \frst order. This inequality can\nbe rewritten through the measurable parameters as\nK\u0001C\nTc\u0012dTc\ndP\u00132\n>1; (206)\nwhere we used the formal expression \u0001 C=\u000b2\n0\n2\fTcfor the speci\fc heat jump at phase transition of the second order.\nThe magneto-elastic interaction also produces another general mechanism for instability of second-order phase\ntransition toward to the discontinuous formation of ferromagnetic state from the paramagnetic one. For the \frst\ntime it was pointed out by O.K.Rice [102] who has demonstrated that at small enough distance from the volume\ndependent critical temperature Tc(V), where the speci\fc heat Cfl(\u001c)\u0018\u001c\u0000\u000b; \u001c=T\nTc(V)\u00001, tends to in\fnity due to\nthe critical \ructuations, the system bulk modulus K=\u0000V@P\n@V=V@2FV\n@V2, expressed through the free energy density\nF=F0+Ffl; Ffl\u0018\u0000Tc\u001c2\u0000\u000bstarts to be negative\nK=K0\u0000ACfl(\u001c)V2\nTc\u0012@Tc\n@V\u00132\n=K0\u0000AK2\n0Cfl(\u001c)\nTc\u0012@Tc\n@P\u00132\f\f\f\f\f\n\u001c!0<0; (207)\nthat contradicts to thermodynamic stability of the system. In reality, at temperature decreasing before the temper-\nature corresponding to K= 0 is reached the system undergoes the \frst-order transition, such that to jump over the\ninstability region directly in the ferromagnetic state with \fnite magnetization and related to it striction deformation.\nThis transition is similar to the jump over the region with @P=@V > 0 on the van der Waals isotherm at the liquid-gas\ntransition.\nThe condition of the \frst order instability (207) can be written in similar to Eqn.(206) form\nK0Cfl(\u001c)\nTc\u0012@Tc\n@P\u00132\n>1: (208)\nUnlike to Eq. (206) this formula demonstrates that the \frst-order instability is inevitable due to in\fnite increase of\n\ructuation speci\fc heat.\nThe striction interaction can change the shape of the free energy singularity in respect to its form at \fxed volume.\nMore elaborate treatment [103] taking into account this e\u000bect leads to the following condition of the \frst order\ninstability\n1\nTc4\u0016K\n3K+ 4\u0016f00(x)\u0012@Tc\n@P\u00132\n>1: (209)\nHere the function f(x) determines the \ructuation part of free energy F=\u0000Tcf\u0010\nT\u0000Tc\nTc\u0011\n,\u0016is the shear modulus.\nUsually, the left hand side in Eqn. (208) is quite small and the transition of the \frst order occurs at temperature\nT?close to the critical temperature where \ructuation speci\fc heat is large enough. It means that the temperature\ndi\u000berenceT?\u0000Tcis much smaller than the critical temperature Tc. The latent heat at this transition\nq\u0019Cfl(T?\u0000Tc) (210)\nproves to be extremely small. So, the \frst-order phase transition is practically indistinguishable from the second-order\none and called weak \frst order phase transition or the phase transition of the \frst order closed to the\nsecond order .\nAccording to Eqs. (206), (208) the magneto-elastic mechanism e\u000bectively leads to the \frst-order transition when\nthe critical temperature is strongly pressure dependent. This is the case in all mentioned above materials. To check\nthe criteria (206), (208) one must calculate the mean \feld jump and \ructuation part of the speci\fc heat near Curie\ntemperature for each particular material. To be concrete, here, we will do these calculations for UGe 2[35] characterized\nby strong magnetic anisotropy and by the precipitous drop of the critical temperature at pressure increase near 14-15\nkbar [3].32\nC. Speci\fc heat near the Curie temperature\nUGe 2is the orthorhombic crystal with ferromagnetic order at ambient pressure found below Tc= 53K. Magnetic\nmeasurements reveal a very strong magnetocrystalline anisotropy [104] with abeing the easy axis. We shall denote it\naszdirection. As in the previous Chapter we shall take into account only the easy axis order parameter \ructuations.\nAbove the Curie temperature they are determined by deviation of system free energy\nF=Z\nd3r\u001a\n\u000bM2+\fM4+\rijriMrjM\u00001\n2@2M(r)\n@z2ZM(r0)d3r0\njr\u0000r0j\u001b\n(211)\nfrom the equilibrium value. \u000b=\u000b0(T\u0000Tc). Here, the gradient terms are written taking into account the orthorhombic\nanisotropy\n\rij=0\n@\rxx0 0\n0\ryy0\n0 0\ryy1\nA;\nwhere the x;y;z axes are pinned to the b;c;a directions. The last nonlocal term in Eq. (211) corresponds to\nmagnetostatic energy [105] \u0000MH\u0000H2=8\u0019, where internal magnetic \feld Hexpressed in terms of magnetization\ndensity by means of the Maxwell equations\nrotH= 0; div (H+ 4\u0019M) = 0:\nWe shall use the following estimations for the coe\u000ecients in the Landau free energy functional\n\u000b0=1\nm2n; (212)\n\f=Tc\n2(m2n)2n; (213)\n\rx\u0019\ry\u0019\rz\u0019Tca2\nm2n: (214)\nHere,mis the magnetic moment per uranium atom at zero temperature, m= 1:4\u0016Bat ambient pressure [16], n=a\u00003\nis the density of uranium atoms, which can be approximately taken equal to cube of the inverse nearest-neighbor\nuranium atoms separation a= 3:85\u0017A[19].\nThe mean \feld magnetization and the jump of speci\fc heat are\nM2=\u0000\u000b\n2\f= (mn)2Tc\u0000T\nTc(215)\n\u0001C=Tc\u000b2\n0\n2\f=n: (216)\nThe experimentally found speci\fc heat jump \u0001 Cexp\u001910J\nmolK\u00191 per uranium atom [19] is in the remarkable\ncorrespondence with Eq.(216).\nFor calculation of the \ructuation speci\fc heat we use the Fourier representation of the quadratic in the order\nparameter part of Eq.(211)\nF=X\nk\u0000\n\u000b+\rijkikj+ 2\u0019k2\nz=k2\u0001\nMkM\u0000k; (217)\nwhereMk=R\nM(r)e\u0000ikrd3r. The last term in this expression corresponds to magnetostatic energy [35, 105].\nThe corresponding thermodynamic potential and the speci\fc heat found in the similar uniaxial segnetoelectric\nmodel are [106]\n\n =\u0000T\n2X\nkln\u0019T\n\u000b+\rijkikj+ 2\u0019k2z=k2; (218)\nCfl0=T2\u000b2\n0\n2(2\u0019)3Zdkxdkydkz\n[\u000b+\rijkikj+ 2\u0019^k2z]2: (219)33\nProceeding to spherical coordinates and performing integration over modulus kwe come to\nCfl0=T2\u000b2\n0\n32\u00192Z1\n0d\u0010Z2\u0019\n0d'\n(\u000b+ 2\u0019\u00102)1=2(\r?+\u00102(\rz\u0000\r?))3=2: (220)\nHere,\r?(') =\rxcos2'+\rysin2'. At critical temperature \u000b= 0 and the integral diverges. Hence, performing\nintegration over \u0010with logarithmic accuracy we obtain\nCfl0=T2\nc\u000b2\n0\n32\u0019p\n2\u0019\r3=2ln2\u0019\n\u000b\u0019n\n32\u0019r\nTc\n2\u0019m2nln2\u0019m2n\nT\u0000Tc; (221)\nwhere\n1\n\r3=2=1\n2\u0019Z2\u0019\n0d'\n\r3=2\n?('):\nThe used condition \u000b\u001c2\u0019atTc= 10Kis realized at\nT\u0000Tc\nTc<2\u0019m2n\nTc\u00190:015: (222)\nIn view of roughness of the parameter estimation the region of logarithmic increase of speci\fc heat can be in fact\nbroader.\nThe calculation taking into account the interaction of \ructuations in the formally similar uniaxial segnetoelectric\nmodel has been performed by Larkin and Khmelnitskii [107]. In our notations the expression for the \ructuation\nspeci\fc heat at const pressure obtained in this paper is\nCfl=31=3T2\nc\u000b2\n0\n16\u0019\r2=3\nLK\r3=2\u0012\nln2\u0019\n\u000b\u00131=3\n: (223)\nHere\rLK=3Tc\fp\n32\u0019\r3=2is the e\u000bective constant of interaction. Using the Eqs. (212)-(214) one can rewrite Eq. (223) as\nCfl\u0019n\n10\u0012Tc\n2\u0019m2n\u00131=6\u0012\nln2\u0019m2n\nT\u0000Tc\u00131=3\n: (224)\nThe power of the logarithm (ln\u000bx\n2\u0019)1=3is the quite slow function slightly exceeding unity, hence, in the temperature\nregion given by inequality (222) one may estimate the \ructuation speci\fc heat as\nCfl\u0019n\n5: (225)\nWe see that the \ructuation speci\fc heat is smaller than the mean \feld jump given by Eqn. (216). Hence, to check\nthe \frst order phase transition instability in UGe 2one must to proceed with the criterium (206).\nD. First-order type transition in UGe 2\nThe Curie temperature in UGe 2falls monotonically with increasing pressure from 53 K at ambient pressure and\ndrops precipitously above 15 Kbar.[3] The average value of the critical temperature derivative can be estimated as\n@Tc\n@P\u001940Kelvin\n14kbar= 4\u000210\u000025cm3(226)\nFor the bulk modulus we have\nK=\u001ac2\u00191011erg=cm3; (227)\nwhere we have substituted typical sound velocity c\u0019105cm=sec and used known [108] density value \u001a= 10:26g=cm3.\nThus, we have for the combination Eq.(206)\nKn\nTc\u0012@Tc\n@P\u00132\n= 0:2: (228)\nAtTc\u001910Kthe pressure derivative of the critical temperature is much higher (and its square is even more higher)\nthan its average value given by Eq. (226). So, we come to conclusion that at critical temperature of the order 10 K\nthe criterium (206) is ful\flled and the phase transition of the second order turns into the \frst order one.34\nE. Concluding remarks\nThe magneto-elastic interaction provides development of the \frst-order instability at the phase transition to the\nordered state in a ferromagnet. However, actual temperature interval of this instability development is negligibly\nsmall and the \frst-order transition looks almost indistinguishable from the second order one. The particular feature\nof the anisotropic ferromagnet UGe 2is the precipitous drop of the Curie temperature as the function of pressure near\n14-15 kbar. Due to this property at about these pressures the second order phase transition (or very weak transition\nof the \frst-order) to the ferromagnet state turns into the real \frst-order type transition.\nAt low temperatures according to the Nernst law and the Clausius-Clapeyron relation\ndTc\ndP=v1\u0000v2\ns1\u0000s2\f\f\f\f\nT!0!1 (229)\nthe drop of transition temperature with pressure begins to be in\fnitely fast. It means that weak \frst order transition\nhas the tendency to be stronger and stronger as temperature decreases. Hence, the e\u000bect of magneto-elastic interaction\nor, more generally, of the order parameter interaction with the elastic degrees of freedom at arbitrary type of ordering\nraises the doubts upon the existence of quantum critical phenomena.\nIX. SUPERCONDUCTING ORDER IN FERROMAGNETIC UIr\nUIr has the monoclinic PbBi-type structure without inversion symmetry shown in Fig.18. The magnetic property\nat ambient pressure is Ising-like ferromagnetic with the Curie temperature Tc= 46K. The magnetic susceptibility\nfollows the Curie-Weiss law with an e\u000bective moment \u0016eff= 2:4\u0016B=Uwhile the ordered moment is 0 :5\u0016B=U. The\npressure-temperature phase diagram of UIr consists of a low pressure phase FM1, a high pressure magnetic phase\nFM2 and a superconducting phase as shown in Fig.19a. The discrete change of the ordered moment indicates that\nthe FM1-FM2 transition is of the \frst-order. The FM2-nonmagnetic transition is of the second-order [110].\nFor UIr, it is still not clear whether the superconductivity coexists with magnetic ordering or not [110]. However, if\nit is the case, here we deal with unique situation when superconducting state arises in material with broken space and\ntime parity. Hence, it is instructive to describe the symmetry and the order parameter of this type superconductivity.\nThe group of symmetry of the normal nonmagnetic state\nGN= (E;C 2b)\u0002R\u0002U(1) (230)\nincludes the point symmetry group C2= (E;C 2b), whereC2bis the rotation around the b-axis on angle \u0019(see Fig.19),\nthe time inversion Rand the gauge group U(1). In the FM2 state the time inversion is broken and the symmetry\ngroup\nGFM= (E;RC 2b)\u0002U(1) (231)\nincludes now the combination of rotation on angle \u0019around the b-axis and the operation Rchanging the direction of\nspontaneous magnetization, lying in the ( a;c) plane, to the opposite. Finally, in the superconducting state coexisting\nwith FM2 state the gauge invariance is broken and the symmetry group is\nGFM+SC= (E;e2i'RC2b): (232)\nThe space parity is broken, hence the magnetic pairing interaction inevitably includes the Dzyaloshinskii-Moriya-\ntype propagators. [39] As result the superconducting order parameter consists of sum of triplet and singlet parts\n^\u0001 =i(d\u001b)\u001by+id0\u001by: (233)\nThe triplet part has the usual form\nd(k;r) =1\n2\u0002\n\u0000(^x+i^y)\u0001\"(k;r) + (^x\u0000i^y)\u0001#(k;r)\u0003\n+ \u00010(k;r)^z; (234)\nhowever, unlike the orthorhombic crystals we considered in Chapter II, the coordinate axis of nonunitary supercon-\nducting ordering does not coincide with monoclinic crystallographic directions. Namery, here ^ zis the unit vector\naligned parallel to spontaneous magnetization lying in ( a;c) plane (direction [1 ;0;\u00161]), ^xis the unit vector directed\nalongb-axis, and ^y= ^z\u0002^x.\n\u0001\"(k;r) =kx\u0011\"\nx(r) +iky\u0011\"\ny(r);\u0001#(k;r) =kx\u0011#\nx(r) +iky\u0011#\ny(r);\u00010(k;r) =kz\u00110\nz(r); (235)35\nwherekx;ky;kzare projections of momentum on the axis ^ x;^y;^zdetermined above. The singlet part of the order\nparameter is\nd0(k;r) =F\u00110(r); (236)\nwhereFis some function of k2\nx;k2\ny;k2\nz.\nX. CONCLUSION\nThe treatment of properties of uranium compounds developed in the present paper is based on the symmetry of\nsuperconducting states with triplet pairing in the orthorhombic ferromagnets. The phenomenological considerations\nwere supported by the microscopic calculations performed in frame of the weak coupling superconductivity theory.\nWe have operated with pairing interaction expressed through the frequency-independent magnetic susceptibility of\nanisotropic ferromagnetic media. This approach reproduces the structure of superconducting states found on pure\nsymmetry ground and allows to explain qualitatively many experimental observations.\nThe topic, that was out our attention, is the ARPES experiments and the band structure calculations which are still\nin not complete agreement. We can recommend the two recent papers reporting the ARPES studies in URhGe [111]\nand in UGe 2and UCoGe [112], the comparison with the band structure calculations and the vast list of references to\nthe previous studies.\nAcknowledgements\nI am grateful to A. Huxley, J.-P. Brison, S. Raymond, K. Ishida, D. Aoki, M. Zhitomirsky and J. Flouquet for\nthe discussions of physics of uranium ferromagnets. I am also indebted to M. Sadovskii who invited me to give the\nlectures at Winter Theoretical School \"Kourovka-2016\" what has obviously helped this paper emergence.\n[1] Maple M B J. 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(M.Taupin, unpublished (2016)).42\nH (тесла) \n  Т(К)                                                                                                             \nFIG. 5: The temperature dependence of the applied \feld at which superconductivity is destroyed for two URhGe crystals\nwithRRR = 34 andRRR = 21 for \felds applied parallel to the caxis.[48]43\n \n \nFIG. 6: The temperature dependence of the ratio of the upper critical \felds parallel to di\u000berent axes in URhGe.[48]44\n  \nFIG. 7: (Color online) Speci\fc heat divided by temperature C=T =\rin (a) UGe2, (b) URhGe, and (c) UCoGe. (d) Scaled\nresidual\r0=\rN-value as a function of ordered moment. [9]45\n            \n \nFIG. 8: (Color online) Temperature-\feld phase diagram for Hkb-axis in URhGe. PM, FM and PPM denote paramagnetic,\nferromagnetic and strongly polarized paramagnetic states, repectively. SC and RSC denote superconducting and reentrant\nsuperconducting states, respectively.[53]46\n       \n \nFIG. 9: (Color online) ( H;T) phase diagram of an uniaxial ferromagnet under magnetic \feld perpendicular to sponta-\nneous.magnetization. PM and FM denote paramagnetic and ferromagnetic states, respectively. Red line is the Curie tem-\nperature line. Blue line is the line of the \frst-order transition.47\n \n \nFIG. 10: (Color online) Magnetic \feld dependences of Mz(Hy) andMy(Hy) with the jumps at the \frst-order transition.48\n \n \nFIG. 11: (Color online) Magnetic \feld Hydirected perpendicular to the exchange \feld h.49\n \n \nFIG. 12: (Color online) Field-temperature phase diagram of URhGe and UCoGe for the \feld along the b-axis.[9]50\n \n \nFIG. 13: The detected neutron scattering (normalized to a \fxed incident beam monitor count) is shown as a function of\nenergy transfer at Q= (0;0;1:04) reciprocal lattice units, just above Tc, at 48 K and at 1.5 K [33].51\n \n \nFIG. 14: (Color online). Panel (a) shows the qdependence ( qkc) of \u0000 qat three di\u000berent temperatures. Panel (b) shows\nthe temperature dependence of \u0000 qatq= (0;0;0:04). Panel (c) shows the qdependence of the product \u001f(q)\u0000qat di\u000berent\ntemperatures above Tc. Panel (d) shows the temperatures dependence of \u001f(q)\u0000qatq= (0;0;0:04) above and below Tc. [33]52\n \n \nFIG. 15: (Color online). Plot of the 1 =T1againstHc[31].53\n \n \nFIG. 16: (Color online). Angular dependence of Hc2at 0:1Kin UCoGe. Tsc(H= 0)\u00190:6K[9].54\n \n \nFIG. 17: The schematic phase diagram of UGe2. Thick lines denote \frst order transitions and \fne lines second order transitions.\nThe dashed line is a crossover. Dots mark the positions of critical (tricritical) points. The region where superconductivity\noccurs is shaded.[33]55\n \n \nFIG. 18: The monoclinic structure of UIr. The arrows indicate the direction of spontaneous magnetization. [109]56\n \n  \nFIG. 19: (a) PT phase diagram of UIr. Solid and open circles are determined by the resistivity and magnetization measure-\nments, respectively. The superconductivity is observed near the critical pressure where the FM2 phase disappears. (b) Pressure\ndependence of the ordered moment along [1 ;0;\u00161] at 2 K. Solid circles are the ordered moments M0determined from the Arrott\nplot. Open circles are the residual magnetizations Mr. [110]"    },    {        "title": "1505.01402v1.Magnetization_Process_of_the_n_type_Ferromagnetic_Semiconductor__In_Fe_As_Be_Studied_by_X_ray_Magnetic_Circular_Dichroism.pdf",        "content": "arXiv:1505.01402v1  [cond-mat.mtrl-sci]  6 May 2015Magnetization Process of the n-type Ferromagnetic Semiconductor (In,Fe)As:Be\nStudied by X-ray Magnetic Circular Dichroism\nS. Sakamoto,1L. D. Anh,2P. N. Hai,2G. Shibata,1\nY. Takeda,3M. Kobayashi,4Y. Takahashi,1T. Koide,5M. Tanaka,2and A. Fujimori1\n1Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n2Department of Electrical Engineering and Information Syst ems,\nThe University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan\n3Synchrotron Radiation Research Unit, Japan Atomic Energy A gency (JAEA), Sayo-gun, Hyogo 679-5148, Japan\n4Department of Applied Chemistry, The University of Tokyo, B unkyo-ku, Tokyo 113-8656, Japan\n5Photon Factory, Institute of Materials Structure Science,\nHigh Energy Accelerator Research Organization (KEK), Tsuk uba, Ibaraki 305-0801, Japan\n(Dated: July 31, 2018)\nIn order to investigate the mechanism of ferromagnetic orde ring in the new n-type magnetic\nsemiconductor (In,Fe)As co-doped with Be, we have performe d X-ray absorption spectroscopy and\nX-ray magnetic circular dichroism (XMCD) studies of ferrom agnetic and paramagnetic samples.\nThe spectral line shapes suggest that the ferromagnetism is intrinsic originating from Fe atoms\nincorporated into the Zinc-blende-type InAs lattice. The m agnetization curves of Fe measured by\nXMCD were well reproduced by the superposition of a Langevin function representing superparam-\nagnetic (SPM) behavior of nano-scale ferromagnetic domain s and aT-linear function representing\nCurie-Weiss paramagnetism even much above the Curie temper atures. The data at 20 K showed\na deviation from the Langevin behavior, suggesting a gradua l establishment of macroscopic ferro-\nmagnetism on lowering temperature. The existence of nano-s cale ferromagnetic domains indicated\nby the SPM behavior suggests spatial fluctuations of Fe conce ntration on the nano-scale.2\nI. INTRODUCTION\nSince the discovery of ferromagnetism in (In,Mn)As [1] and (Ga,Mn)A s [2], the Mn-doped III-V ferromagnetic\nsemiconductors (FMS) have been intensively studied, driven by the scientific interest of how ferromagnetism is\nrealized by such a small amount of magnetic impurities [3, 4] and by fut ure technological applications utilizing\nboth the charge and spin degrees of freedom [5–7]. Although their electronic structures and the mechanism of the\nferromagnetism still remain highly controversial [8–13], it is widely belie ved that carriers mediate ferromagnetic\ninteraction between spatially separated magnetic ions. This kind of f erromagnetism is called carrier-induced ferro-\nmagnetism, and its microscopicelucidation and utilization have been th e main subjects in the field of semiconductor\nspintronics. In the case of Mn-doped III-V compounds, however , there is a limitation that only p-type materials\nare available because the Mn2+ions substituting the group-III elements work not only as local mag netic moments\nbut also as acceptors providing the system with hole carriers.\nRecently, Hai et al.[14–18] have synthesized a new n-type III-V FMS, namely, Fe- and Be-codoped InAs. Usually,\nthesubstitutionofFe3+ionsfortheInsitesdoesnotprovidechargecarriers,resultinginC urie-Weissparamagnetism\n[19]. However, Hai et al. have shown that by co-doping (In,Fe)As with Be atoms, which ente r the interstitial sites\nand actasdoubledonors,electronsareintroducedintothe syste mandmediate ferromagneticinteractionbetweenFe\natoms. Here, unlike the Mn-doped materials, (In,Fe)As:Be has the a dvantagethat one can control the concentration\nof magnetic ions and that of charge carriers independently.\nIn order to reveal the nature of the ferromagnetism of this new F MS, it is necessary to characterize the electronic\nand magnetic structure on the microscopic level. For this purpose, we have performed X-ray absorption spec-\ntroscopy (XAS) and X-ray magnetic circular dichroism (XMCD) meas urements. XMCD is defined as the difference\nbetween the absorption of right- and left-handed circularly polariz ed X-rays and is a powerful method to probe the\nelement-specific electronic and magnetic properties of compounds by utilizing the absorption edges of constituent\nelements. Furthermore, one can exclude extrinsic contributions t o magnetism such as magnetic contaminations and\nthe diamagnetic response of the substrate in the case of thin films a nd that of the host material in the case of\nFMS. Here, such diamagnetic contributions prohibit the extraction of the intrinsic paramagnetic component [20],\nwhich is important to characterize the complex magnetism of FMS. In addition, XMCD sum rules [21, 22] make it\npossible to obtain the orbital and spin magnetic moments separately . A recent XMCD study [23] have shown that\nthe orbital moment of (In,Fe)As is significantly larger than that of F e metal. Since Fe3+(d5) does not have orbital\nmagnetic moment, the observed finite orbital magnetic moment wou ld represent the charge-transfer d6Lstates of3\nFe3+, whereLdenotes a ligand hole.\nIn the previous study [15], ferromagnetic domains of ∼10µm were observed at sufficiently low temperatures well\nbelow Curie temperature ( TC) by magneto-optical imaging, explaining the hysteretic behavior of magnetization.\nAt high temperatures much above TC, however, magnetization still strongly depends on magnetic field, w hich\nimplies the existence of rather small ferromagnetic domains of ∼nm acting as a superparamagnet. Under such\ncircumstances, it is important to know how the ferromagnetism evo lves as a function of temperature and magnetic\nfield on a more microscopic level, i.e., on the nano-meter scale. In this p aper, we have performed a careful analysis\nof XMCD data in order to address the issue of the evolution of magne tism from high to low temperatures.\nII. EXPERIMENT\nThree samples In 0.95Fe0.05As:Be (sample 1), In 0.9Fe0.1As:Be (sample 2) and In 0.95Fe0.05As (sample 3) without\nBe were synthesized using the low-temperature molecular beam epit axy (LT-MBE) method as follows. The 20-30\nnm-thick (In,Fe)As layers were grown on InAs(001) substrates a t 240◦C after growing 50 nm-thick InAs buffer\nlayers at 500◦C. In order to prevent surface oxidation, the samples were cover ed by sub-nm-thick amorphous\nAs capping layers. The amount of Be was estimated to be 2 .6×1019cm−3. Samples 1 and 2 were found to\nbe ferromagnetic, whose Curie temperatures were 15 K and 40 K, r espectively, determined by the Arrott plot\nof visible magnetic circular dichroism (vis-MCD) intensities, while sample 3 was paramagnetic. Careful sample\ncharacterizations by transmission electron microscopy (TEM), en ergy dispersive X-ray spectroscopy (EDX), and\nthree dimensional atom probe (3DAP) proved that there were no F e related precipitates or secondary phases in the\nsamples [15]. Experiments were done at the undulator beamline BL-16 A2 of Photon Factory (PF), High Energy\nAccelerator Research Organization (KEK). Temperature was var ied from 20 K to 260 K and magnetic field from 0\nT to 5 T. The total electron-yield (TEY) mode was used. The directio n of the incident X-rays and magnetic field\nwas perpendicular to the sample surfaces, and XMCD spectra were obtained by changing the polarization of X-rays\nwhile the direction of magnetic field was fixed. Backgrounds of the sp ectra were assumed to be the summation of\na linear function and two-step inverse tangent functions represe nting the Fe L2,3edge jumps [24].\nIII. RESULTS\nThe XAS and XMCD spectra of (In,Fe)As with and without Be are show n in Fig. 1 together with the spectra\nof Fe metal (ferromagnetic) and γ-Fe2O3(ferrimagnetic) as references. Clear XMCD signals were obtained f or the4\n740730720710700\nPhoton Energy [eV]-0.8-0.6-0.4-0.20.00.2XMCD [arb. unit]\n714 712 710 708 706 704\nPhoton Energy [eV]-4-3-2-101XAS [arb. unit]\n730720710700\nPhoton Energy [eV](a) Fe L2,3 XAS\nγ-Fe2O3\nProfeta et al.\n(c) Fe L3 edge  5T,20K\n 5T 125K\n 1T,262K\n XAS\n       2nd deriv.Fe metal\nChen et al.(b) Fe L2,3 XMCD\nFe 10% w/ Be\nFe 5% w/ Beαβ\nα\nβµ0H=5T, T=20K intrinsic\n extrinsic\nα'\nXAS\nXMCD(In,Fe)As\n Fe 10%Fe 5% w/o Be\nµ0H=1T, T=260KXAS = σ + + σ -\nXMCD = σ + - σ -\nx 0.5\nx 0.5(In,Fe)Asα\nβ\nFIG. 1. XAS (a) and XMCD (b) spectra of (In,Fe)As in compariso n with those of metallic Fe [24] and γ-Fe2O3[25]. The\nXAS and XMCD spectra have been normalized so that the peak int ensities of XAS are equal to one. For the (In,Fe)As\nsample with 10% Fe, the spectra have been decomposed into the intrinsic and extrinsic components as depicted with red\ndashed and blue dashed curves, respectively. (c) Fe L3-edge XAS and XMCD spectra of the 10% Fe-doped (In,Fe)As samp le\nat different temperatures and magnetic fields. The XMCD spect ra as well as XAS spectra have been normalized to its peak\nheight. In order to emphasize weak structures, the second de rivative of the XAS spectra with reversed sign are also shown\nby gray solid curve.\nferromagnetic (In,Fe)As samples and virtually no signal was detect ed for the paramagnetic (In,Fe)As sample as\nexpected. The XAS spectra at the Fe L3edges are mainly composed of two structures αandβ. The small shoulder\nstructure βcould be attributed to an extrinsic origin of Fe oxides at the surface [23], considering that its energy is\nclose to the peak energy of γ-Fe2O3and that its intensity changes in non-systematic ways with varying m agnetic\nfield and temperature, as shown in Fig. 1(c). This probably resulted from the change of the beam position between\ndifferent scans and inhomogeneous distribution of Fe oxides on the s urface. Utilizing this non-systematic spectral\nchange, we have deduced intrinsic XAS and XMCD spectra in the follow ing manner [26]. First, using two spectra\nS1and S2with different intensities of structure β, the intrinsic spectrum S intwas obtained as S int∝S1−pS2, where\npwas chosen so that structure βvanished. Then, the extrinsic spectrum S extwas obtained as S ext∝S1−qSint,\nwhereqwas chosen so that line shape of S extagreed with that of γ-Fe2O3. The decomposed intrinsic and extrinsic\nspectra for the 10% Fe-doped sample are shown in Figs. 1(a) and 1( b). It was difficult to apply the same procedure5\nto the 5% Fe-doped sample because structure βis less prominent and the intensity of structure βdid not change\nsignificantly between different scans. However, using the intrinsic a nd extrinsic spectra for the 10% sample, we have\nestimated the mixing ratio of the intrinsic and extrinsic components, that is, how much oxides exist at the surface.\nThus we could obtain the correct values of the magnetic moments fo r the 5% Fe-doped sample as well as 10%\nFe-doped one using XMCD sum rules [21, 22]. We note that, except fo r the extrinsic shoulder structures β, the line\nshapes of the XAS spectra were independent of the Fe and Be cont ents and shared a broad single-peak structure.\nThis means that the local electronic structures of Fe atoms is basic ally the same between the three samples, notably\nbetween the ferromagnetic and paramagnetic ones. The result th at multiplet structures reminiscent of the localized\n3dsystems [27] are absent suggests that the delectrons of Fe are relatively delocalized [28, 29] probably through\nthe strong hybridization between the Fe 3 dorbitals and the ligand s,porbitals.\nThe deduced intrinsic XAS and XMCD spectra of (In,Fe)As:Be resemb le those of Fe metal at a first glance,\nand therefore, one might suspect that the ferromagnetism origin ates from Fe metal precipitates. However, this\npossibility can be ruled out from following reasons. First, the XAS spe ctra of the paramagnetic sample showed the\nsame spectral line shape as those ofthe ferromagneticsamples. I f the observedspectral line shapes of(In,Fe)As were\ndue to Fe metal particles, the XMCD intensity should have been comp arable among the three (In,Fe)As samples\nincluding the paramagnetic one. Second, looking at the magnified spe ctra of the Fe L3edge presented in Fig. 1(c),\nthe peak positions of the XAS and XMCD do not coincide with each othe r unlike the XAS and XMCD spectra of\nFe metal [30]. Corresponding to the peak in the XMCD spectra, the re is a tiny shoulder α′in the XAS spectra ∼\n0.5 eV below its peak position α. This shoulder is clearly seen in the second derivative of the XAS spec tra with\nrespect to photon energy, but less prominent in the spectra of Fe metal [30]. From these observations, we conclude\nthat the ferromagnetism in this system is distinct from that of Fe me tal and of the intrinsic origin arising from Fe\natoms incorporated into the Zinc-blende-type InAs lattice.\nIn order to obtain the magnetic moment of Fe for the ferromagnet ic samples, we have applied the XMCD sum\nrules [21, 22]:\nmorb=−4/integraltext\nL2,3∆σdω\n3/integraltext\nL2,3σdωnh, (1)\nmspin+7mT=−6/integraltext\nL3∆σdω−4/integraltext\nL2,3∆σdω\n/integraltext\nL2,3σdωnh, (2)\nσ=σ++σ−,∆σ=σ+−σ−, (3)\nwheremorbandmspinare the orbital and spin magnetic moments in units of µB/atom, respectively, and mTis the6\nTABLE I. Spin and orbital magnetic moments of Fe in (In,Fe)As :Be and Fe metal.\nmorb/mspinmorbmspin\nIn0.95Fe0.05As:Be 0.088±0.016 0.22* 2.49*\nIn0.9Fe0.1As:Be 0.075±0.015 0.21* 2.80*\nIn0.95Fe0.05As:Be [23] 0.065±0.014 0.10 1.60\nFe bcc [24] 0.043±0.001 0.085 1.98\n*measured at µ0H= 5 T and T= 20 K\nexpectation value of the magnetic dipole operator which is negligibly sm all for an atomic site with high symmetry\nsuch asTdorOh[31].σ+andσ−denote absorption cross sections for X-rays with positive and neg ative helicity,\nrespectively, nhthe number of 3 dholes. We applied these equations to the decomposed intrinsic spect ra instead of\nthe raw spectra. The area of the intrinsic XAS spectra, namely the denominator of the equations, were estimated\nto be∼20% smaller than that of the raw XAS spectra for the 10% Fe-doped sample, and ∼5% smaller for the\n5% Fe-doped one. As for the XMCD spectra, the extrinsic compone nts were almost absent and did not contribute\nto the area of the XMCD spectra and hence to the numerator of Eq s. (1) and (2). Note that the existence of the\nextrinsic components in the XAS spectra only changes the calculate d spin moment mspinand orbital moment morb\nby the same factor while keeping the ratio morb/mspinunchanged.\nTable I summarizes the orbital and spin moments of (In,Fe)As:Be ded uced using the XMCD sum rules and those\nof bcc Fe metal [24]. For the calculation of the moments, we assume d the number of electrons in the Fe 3 dstates\nto be 5 and the correction factor for the spin sum rule to be 0.685 fo r Fe3+[32]. Being consistent with the previous\nreport [23], the ratio of the orbital moment to the spin moment morb/mspin, which is not affected by the assumption\nofnh= 5 and the existence of the extrinsic component in the XAS spectra , is larger than that of Fe metal and is\npositive. This can be explained if the d6Lconfiguration is mixed into the ground state due to charge transfe r from\nligand orbitals to the Fe 3 dorbitals.\nMagnetization ( M) per Fe atom at various magnetic fields and temperatures has thus been deduced from the\nXMCD intensities and are plotted in Figs. 2(a) and 2(b). Here, the ho rizontal axis is the effective magnetic field\ncorrected for the demagnetizing field of the thin films although this c orrection is small in the case of the diluted\nmagnetic systems. In the present experimental set-up, hyster esis, which was observed in the previous study [23],\nwas too small to be detected. It is worth noting that the M-Hcurves were almost identical between 5% and 10%\nFe-doped samples even though the TC’s were different. At low magnetic fields, a steep increase of magnetiz ation was\nobserved even above TCand, at high magnetic fields, linear increase of magnetization was obs erved even at the low\ntemperatureof20Kwheretheferromagneticcomponentwouldbe saturated. Thisbehaviorcanbeunderstoodiftwo7\n543210\nMagnetic Field [T]3\n2\n1\n03\n2\n1\n0Magnetization [ µB/Fe]\n543210\nMagnetic Field [T]\n3\n2\n1\n0\n0.120.080.040\nµ0H/T  [T/K]0.120.080.040\nµ0H/T  [T/K]3\n2\n1\n0\n250200150100500\nTempearture [K]200\n150\n100\n50\n0200\n150\n100\n50\n0χT [µBK/T]\n250200150100500\nTemperature [K] 20K 150K\n 40K 200K\n110K  260K\n125K 20K 150K\n 60K 200K\n100K 260KFe 10%Fe 5%Fe 5%  Fe 10%(a) InFeAs:Be (b)\nTCTC(c) (d)\n(e) (f)\nFe 5% Fe 10%\nFIG. 2. XMCD intensities of (In,Fe)As:Be samples with 5% and 10% Fe doping. (a), (b) Magnetic field dependence of\nmagnetization deduced from the Fe L2,3-edge XMCD intensities at various temperatures. (c), (d) Th e same data as (a)\nand (b) plotted against µ0H/T. At the low temperature of 20 K, the magnetization shows a cle ar deviation from the high\ntemperature data. In panel (a)-(d), solid curves represent the fitting curves by the summation of the superparamagnetic\nLangevin function and the paramagnetic linear function. Th e paramagnetic components are separately shown by dashed\nlines. (e), (f) Slope of the magnetization curves at zero mag netic field against µ0H/T, namely ∆M/∆/parenleftbigµ0H\nT/parenrightbig\n|H→0=χTas\na function of temperature.\ncontributions to the magnetization are superimposed, namely, sup erparamagnetism from nano-scale ferromagnetic\ndomains (FM/SPM domains) and paramagnetism from isolated magnet ic moments. Therefore, we have fitted the\nfollowing summation of the Langevin function L( ξ) and aT-linear function to the experimental XMCD intensity:\nM= 5xL/parenleftbiggµµ0H\nkBT/parenrightbigg\n+(1−x)Cµ0H\nT+TA, (4)\nL(ξ) = coth( ξ)−1\nξ, (5)\nwherex(0≤x≤1) denotes the fraction of Fe atoms participating in the superpara magnetism or ferromagnetism,\nµthe total magnetic moment per nano-scale FM/SPM domain, Cthe Curie constant of the Fe3+ion with 5 µB,\nandTAthe Weiss temperature representing the antiferromagenetic inte raction between paramagnetic Fe3+ions.\nFigures 2(c) and 2(d) show the magnetization curves at various te mperatures replotted as functions of µ0H/T.\nThey approximately fall onto a single curve, which is a fingerprint of s uperparamagnetism, except for the data at\n20 K close to TC. The deviation may represent the onset of macroscopic ferromag netism at 20 K. Figures 2(e) and8\n1.0\n0.8\n0.6\n0.4\n0.2\n0x\n400\n300\n200\n100\n0/s109  [/s109B/cluster]\n4x1019\n3\n2\n1\n0N [ cm-3]\n250200150100500\nTemperature [K](In,Fe)As:Be(a) Fraction of FM/SPM Fe\n(b) Magnetic moment per \nFM/SPM domain\n(c) Density of FM/SPM domains\n Fe 5%\n Fe 10%FM SPMPM \nFM SPM\nFM SPM\nFIG. 3. (a) Fraction xof Fe atoms participating in ferromagnetism or superparama gnetism to all the Fe atoms. (b) Total\nmagnetic moment µinside one FM/SPM domain, obtained by fitting Eq. (4) to the da ta. (c) Density of FM/SPM domain\nNestimated from the fitting parameters xandµ.\n2(f) show the slopes of magnetization at zero magnetic field with res pect toµ0H/T, i.e.∆M/∆/parenleftBig\nµ0H\nT/parenrightBig\n|H→0=χT.\nIf the system follows the Langevin law of superparamagneitsm, χTshould be constant. Hence the drop of χT\naround 50 K on decreasing temperature indicates the gradual est ablishment of macroscopic ferromagnetism from\nthe nano-scale superparamagnetism.\nTheM-Hdata were fitted by Eq. (4) assuming TA= 30 K, which is nearly equal to the Weiss temperature of\nparamagnetic (Ga,Fe)As without carrier doping (= 32 K) [19], and the results of the fit are shown in Figs. 2(a)-2(d)\nbysolidcurves. Inthefigures, theparamagneticcomponentsare separatelyshownbydashedlines. Notethatafinite\nvalueofTAlargerthan30Kwasnecessarytofittothedata. Figures3(a)and 3(b)showthetemperaturedependence\nof fitting parameters xandµ. The fraction of Fe atoms participating in the ferromagnetism or su perparamagnetism\n(x) isestimatedtobe 40%-60%andincreasingasthe temperatureisde creased. ThemagneticmomentperFM/SPM\ndomain ( µ) are200-300 µB, which correspondsto 40-60Fe atoms, andthey decreaseasthe temperatureis decreased.\nInterestingly, the estimated values are very similar between the tw o samples. The estimated density of FM/SPM\ndomains Nare shown in Fig. 3(c) and found to be ∼1019cm−3, comparable to the density of Be atoms.9\nFIG. 4. Schematic picture of the formation of nano-scale FM d omains in (In,Fe)As:Be. There exist FM Fe-rich domains and\nparamagnetic isolated Fe ions. Here the density of Fe atoms a nd the size and the number of FM domains reflect the obtained\nvalue for the 10% Fe-doped sample.\nIV. DISCUSSION\nThere may be two possible scenarios to explain the formation of nano -scale FM/SPM domains observed for\n(In,Fe)As. One is the bound magnetic polaron (BMP) model [33, 34], w here relatively localized carriers stabilize\nferromagneticspinalignmentextendingoverthe scaleofnm. Inthis model, the polaronsoverlapwith eachotherand\nform macroscopic ferromagnetic domains at the critical concentr ation where the system satisfies the Mott criterion\nfor insulator-to-metal transition. Second is the formation of Fe r ich regions during the growth while maintaining\nthe zinc-blende crystal structure, often discussed in the conte xt of spinodal decomposition [35, 36], as schematically\nshown in Fig. 4. Considering the metallic behavior of (In,Fe)As [15], in pa rticular, the small effective mass of\nelectrons in the conduction band [16], the BMP model is less likely beca use it predicts a heavy effective mass for\nthe charge carriers, and the second scenario is more probable. As noted above, the M-Hbehaviors are almost\nidentical between the two ferromagnetic samples with different Fe c ontents, meaning one FM/SPM domain consists\nof 40-60Fe atoms for both samples. If the first scenario is correc t and the distribution of Fe atoms is rather uniform,\nelectrons are bound to charged impurities such as Be2+to form magnetic polarons and there should be difference in\nthe total magnetic moment of a FM/SPM domain between the two sam ples because the number of Fe atoms within\nthe same Bohr radius of electrons should be different. On the other hand, in the case of the second scenario, where\nFe-rich FM/SPM domains are formed in the InAs host, the observed independence of magnetic moment on the Fe\ncontent can be explained if Fe-rich regions with optimized number of F e atoms are formed. Suppose an extreme\ncase where Fe atoms occupy all the In sites inside the FM/SPM region s, the radius of the domain containing 40-60\nFe atoms is estimated to be ∼1 nm, which is difficult to detect by conventional x-ray diffraction. Fr om the fact\nthat the density of FM/SPM domains are comparable to that of Be at oms, one can speculate that the FM Fe-rich\ndomains were preferentially formed around Be impurities. Actually, it was reported that the co-doping of shallow\nimpurities affects the distribution or aggregation of magnetic ions [37 ].10\nWe shall discuss below how the magnetism in this system behaves with t emperature. As can be seen from Figs.\n3(a) and 3(b), the lower the temperature becomes, the more Fe a toms participate in ferromagnetism but the smaller\nthe average size of FM/SPM domains becomes. This behavior can be u nderstood if there exists size distribution\nof Fe rich regions. As described in Fig. 4, at high temperatures, only large Fe-rich regions are ferromagnetic. On\nlowering temperature, smaller Fe-rich regions becomes ferromagn etic as described in the middle panel of Fig. 4, and\nhence the averagemagnetic moment of FM/SPM domains ( µ) decreases[Fig. 3(b)]. On the other hand, the fraction\nof Fe atoms participating in ferromagnetism or superparamagnetis m (x) increases [Fig. 3(a)] and the density of\nFM/SPM domains ( N) increases [Fig. 3(c)]. Further decreasing the temperature below TCwould result in the\noverlap of the FM/SPM domains as shown in the left panel of Fig. 4, an d the macroscopic ferromagnetism, which\ncannot be described by the Langevin law, starts to be stabilized. Th e reason for the drop of χTat 20 K [Figs. 2(e)\nand 2(f)] insteadofa divergencewith the emergenceofmacrosco picferromagnetismmaybe that the ferromagnetism\nis soft in (In,Fe)As. For example, let us write the magnetization of a s ystem where superparamagnetism and soft\nferromagnetism coexist as\nM(H)\nMsat= (1−y)L(µµ0H\nkBT)+ytanh(H/H0), (6)\nwhere the first term and second term represent the superparam agnetism and the soft ferromagnetism, respectively.\nydenotes the fraction of ferromagnetism and H0is the saturation field. The second term is just an empirical\nexpression[38], but onecanassumethatthe magnetizationisindepe ndent oftemperatureaslongasthe temperature\nis sufficiently low. Using this expression, χTcan be written as\nχT\nMsat= (1−y)µ0µ\n3kB+yT\nH0. (7)\nTherefore, if µ0µ/3kBis larger than T/H0, i.e.H0>3kBT/µ0µ,χTwould show a drop with the emergence of\nferromagnetic term.\nAt the present stage, one can conclude that the largemagnetizat ion of the present (In,Fe)As:Be samples above TC\noriginatesfromthenano-scaleFMdomains. Thepreviouslyobserve dferromagneticdomainof ∼10µmatsufficiently\nlow temperature would contain a large number of ( ∼107assuming the domain size of 10 µm×10µm×20nm) the\nnano-scale FM domains revealed in the present work, and would resu lt from ferromagnetic coupling between the\nFM/SPM domains and/or magnetic anisotropy of the FM/SPM domains .11\nV. CONCLUSION\nWe have performed XMCD measurements on the new n-type ferromagnetic semiconductors (In,Fe)As:Be with\ndifferent Fe and Be contents. Despite the fact that TCwas different between the samples, the XAS and XMCD line\nshapes and magnetization data were almost identical. The analysis of the spectral line shapes and the magnetic\nmoment using sum rules suggest that the ferromagnetism is intrinsic originating from the Fe atoms incorporated\ninto the Zinc-blende-type InAs lattice, and not from segregated s econdary phases. From the magnetization curves\nat various temperatures even much above TC, a SPM-like steep increase in the low Hregion and a paramagnetic\nlinear increase in the high Hregion were observed, which implies the co-existence of FM domains o f∼nm size\nand paramagnetic isolated Fe3+ions. The fraction of Fe atoms participating in the ferromagnetism o r superpara-\nmagnetism was estimated to be 40-60% and the magnetic moment per FM/SPM domain to be 200-300 µB. 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Ohta\nDepartment of Physics, Chiba University, Chiba 263-8522, J apan\n(Dated: December 1, 2018)\nWe study the ground-state properties of the double-chain Hu bbard model coupled with ferro-\nmagnetic exchange interaction by using the weak-coupling t heory, density-matrix renormalization\ngroup technique, and Lanczos exact-diagonalization metho d. We determine the ground-state phase\ndiagram in the parameter space of the ferromagnetic exchang e interaction and band filling. We find\nthat, in high electron density regime, the spin gap opens and the spin-singlet dxy-wave-like pairing\ncorrelation is most dominant, whereas in low electron densi ty regime, the fully-polarized ferromag-\nnetic state is stabilized where the spin-triplet py-wave-like pairing correlation is most dominant.\nPACS numbers: 71.10.Fd, 71.10.Hf, 74.20.Mn, 74.20.Rp\nI. INTRODUCTION\nFerromagnetism in itinerant electron systems has in-\ncreasingly been understood since the Hubbard model\nwas introduced in 1963.1,2It is known that subtle in-\nterplay between quantum many-body effects and spin-\nindependent Coulomb interactions plays a crucial role in\ngenerating ferromagnetic orders in some solids;3a vari-\nety of origins such as Nagaoka-Thouless mechanism,4,5\nflat-band,6,7orbital degeneracy,8,9,10three-site ring ex-\nchange interaction,11,12etc., have so far been proposed.\nIn addition to ferromagnetism, (possibly) spin-triplet su-\nperconductivity was recently discovered in the metallic\nferromagnets UGe 2,13URhGe,14and ZrZn 2.15Conse-\nquently, the relation between superconductivity and fer-\nromagnetism has been of special interest in the field of\nstrongly correlated systems. From the theoretical point\nof view, the occurrence of superconductivity in ferromag-\nnetic materials is naturally explained by the formation\nof Cooper pairs with parallel spins, namely, spin-triplet\npairs.16\nAmong the origins of ferromagnetism mentioned\nabove, only the three-site ring exchange interaction acts\non a couple of electrons. It yields a ferromagnetic spin\ncorrelation, which in turn produces an attractive effect\nbetweenthem. Onemayeasilyimaginethataspin-triplet\nsuperconductivity is realized if the attractive interaction\nbetween electrons can survive against the other effects.\nRecently, we have confirmed that this mechanism actu-\nally works in a fairly simple correlated electron system;\nthe system consists of two Hubbard chains coupled with\nzigzag bonds and has a unique structure of hopping inte-\ngrals.17,18,19Inthismodel, thespin-tripletpairingofelec-\ntrons occurs between the inter-chain neighboring sites. If\nthe ferromagnetic correlation between the two chains areessential for the spin-triplet superconductivity, we may\nbe allowedto mimic our originalmodel by a double-chain\nHubbard model coupled with ferromagnetic exchange in-\nteraction. This new model is much easier to analyze than\nthe original one due to the reduction of quantum fluctu-\nations. Therefore, the introduction of this model will en-\nable us to investigate the spin-triplet superconductivity\nin more detail for wide range of parameters.\nIn this paper, we thus study the double-chainHubbard\nmodel coupled with ferromagnetic exchange interaction.\nWe use the weak-coupling bosonization/renormalization\ngroup (RG) analyses,20the density-matrix renormaliza-\ntion group (DMRG) technique, and the Lanczos exact-\ndiagonalization method.22We thereby determine the\nground-state phase diagram: we find that, in the high\nelectron density regions, the spin gap opens and the\ndxy-wave-like pairing occurs, while in the low electron\ndensity regions, the system is in the metallic state with\nfull spin polarization, where the py-wave-like spin-triplet\npairing correlationbecomes the most dominant. We note\nthat this model can be regarded as a single-chain model\nwith two degenerate orbitals if we assume the ferromag-\nnetic exchange interaction to be identified with the intra-\natomicHund’srulecoupling. Thissingle-chainmodelhas\nso far been studied both analytically23,24,25and numer-\nically26,27. It has been proposed that the Haldane gap\nstate is realized at half filling. It has also been pointed\nout that the system remains gapful for low hole doping\nregions.28\nOur paper is organized as follows. In Sec. II, we de-\nfine the double-chain Hubbard model coupled with fer-\nromagnetic exchange interaction. In Sec. III, we analyze\nthe model using the weak-coupling theory and derive the\npairing order parameter. In Sec. IV, we calculate several\nquantities with the DMRG and Lanczos methods and\npresentthecalculatedresults. Wealsocheckiftheresults2\nFIG. 1: Schematic representation of the double-chain Hub-\nbard model coupled with ferromagnetic exchange interactio n.\nNo hopping process is allowed between the two chains and no\nexchange interactions exist along the chains.\nare consistent with the weak-coupling results. Section V\ncontains summary and conclusions.\nII. MODEL\nWe study the double-chain Hubbard model coupled\nwith ferromagnetic exchange interaction (see Fig. 1), the\nHamiltonian of which is defined by\nH=t/summationdisplay\nrx,ry,σ/parenleftBig\nc†\nrx,ry,σcrx+1,ry,σ+H.c./parenrightBig\n+U/summationdisplay\nrx,rynrx,ry,↑nrx,ry,↓\n−J/summationdisplay\nrxSrx,1·Srx,2 (1)\nwherec†\nrx,ry,σ(crx,ry,σ) is the creation (annihilation) op-\nerator of an electron with spin σ(=↑,↓) at siterxon\nlegry(= 1,2),nrx,ry,σ=c†\nrx,ry,σcrx,ry,σis the density\noperator, and Srx,ryis the spin-1\n2operator. tis the\nnearest-neighbor hopping integral along the chain, Uis\nthe onsite Coulomb interaction, and J(>0) is the ferro-\nmagnetic exchange interaction between two sites on each\nrung. Note that no hopping process is allowed between\nthe two chains and thus the two bands are degenerate.\nIII. WEAK-COUPLING THEORY\nWe first consider the weak-coupling regime where only\nthe low-energy excitations near the Fermi points are cru-\ncial. Thus far, the weak-coupling theory has been well-\nconstructed for two-band models.24,30,31,32,33,34,35,36,37\nWe further develop the approach to analyze the Hamil-\ntonian (1). Assuming a linearization of the dispersion\nrelations in the vicinity of the Fermi level, we introduce\nthe field operators of right- and left-going electrons as\nψp,σ,±(x) =1√\nL/summationdisplay\nkxeikxc±\np,σ(k) (2)\nwherec±\np,σis the Fourier transform of combined operator\n(crx,1,σ±crx,1,σ)/√\n2 for the right-going ( p= +) and\nleft-going (p=−) electrons. Lis the chain length where\nthe lattice spacing is set to be unity. Using these fieldoperators, the Hamiltonian can be rewritten as a sum of\nthe linearized kinetic energy and interaction terms. We\nthus obtain\nH=/integraldisplay\ndxH0+/integraldisplay\ndxHI\nH0=vF/summationdisplay\np,σ,ζψ†\np,σ,ζ/parenleftbigg\n−ipd\ndx/parenrightbigg\nψp,σ,ζ\nHI=1\n4/summationdisplay\np,σ/summationdisplay\nζ′gǫ,¯ǫ\n1⊥ψ†\np,σ,ζ1ψ†\n−p,−σ,ζ2ψp,−σ,ζ4ψ−p,σ,ζ3\n1\n4/summationdisplay\np,σ/summationdisplay\nζ′gǫ,¯ǫ\n2⊥ψ†\np,σ,ζ1ψ†\n−p,−σ,ζ2ψ−p,−σ,ζ4ψp,σ,ζ3\n1\n4/summationdisplay\np,σ/summationdisplay\nζ′gǫ,¯ǫ\n/bardblψ†\np,σ,ζ1ψ†\n−p,σ,ζ2ψp,σ,ζ4ψ−p,σ,ζ3.(3)\nwhereǫ=ζ1ζ3and ¯ǫ=ζ1ζ2. The primed summa-\ntion overζi(i= 1,2,3,4) is restricted by a relation\nζ1ζ2ζ3ζ4= +1, which comes from the momentum con-\nservation condition in the transverse components. The\ncoupling constants gǫ,¯ǫ\n1⊥andgǫ,¯ǫ\n2⊥are related to the origi-\nnal parameters in the Hamiltonian (1):\ng++\n1⊥=U, g+−\n1⊥=−J\n4, g−−\n1⊥=−J\n2,\ng++\n2⊥=U, g+−\n2⊥=−J\n4, g−−\n2⊥=−J\n2.(4)\nFor the SU(2) symmetric case, we can choose\ng++\n/bardbl=g++\n1⊥−g++\n2⊥(5)\ng−+\n/bardbl=g−+\n1⊥−g−+\n2⊥(6)\ng+−\n/bardbl=g+−\n1⊥−g−−\n2⊥(7)\ng−−\n/bardbl=g−−\n1⊥−g+−\n2⊥. (8)\nIn this Section III, we consider the case away from half\nfilling. Hence, the Umklapp term g3gives no contribu-\ntion and the Fermi velocity renormalization due to the\nforward-scattering term g4may be neglected.\nA. Bosonization\nUsing the Abelian bosonization method,29we intro-\nduce eight chiral bosonic fields φp\nµrwhereµrefers to the\ncharge (ρ) and spin ( σ) sectors; meanwhile, rrefers to\nthe even (+) and odd ( −) sectors. The bosonic fields\nsatisfy the commutation relations [ φ±\nµr(x),φ±\nµ′r′(x′)] =\n±i(π/4)sgn(x−x′)δµ,µ′δr,r′and [φ+\nµr(x),φ−\nµ′r′(x′)] =\ni(π/4)δµ,µ′δr,r′. We then define a new set of chiral\nbosonic fields as\nφp,s,ζ=φp\nρ++ζφp\nρ−+sφp\nσ++sζφp\nσ− (9)\nwherep=±,s=±, andζ=±. The chiral bosons obey\nthe commutation relations [ phip,s,ζ(x),phip,s′,ζ′(x′)] =3\n±i(π/4)sgn(x−x′)δp,p′δs,s′and [φ+,s,ζ(x),φ−,s′,ζ′(x′)] =\ni(π/4)δs,s′δζ,ζ′. The field operators of the right-moving\nand left-moving electrons (2) are then written as\nψp,σ,ζ=ησ,ζ√\n2πaexp(ipkFx+ipφp,s,ζ) (10)\nwheres= +(−) forσ=↑(↓). The Majorana fermions\nησ,ζ, known as Klein factors, are introduced to ensure\nthe proper anticommutation relations between fermion\nfields with different band and spin indices. They obey\n{ησ,ζ,ησ′,ζ′}= 2δσ,σ′δζ,ζ′. Itisgenerallymoreconvenient\ntotradethechiralbosonfieldspairwiseforaconventional\nbosonic phase field φand its dual field θ, so that we also\nintroduce the bosonic fields given by\nφµ,r=φ+\nµ,r+φ−\nµ,r (11)\nθµ,r=φ+\nµ,r−φ−\nµ,r, (12)\nwhere the operator Π µr(x) =∂xθµr/πis a canonical con-\njugate variable to φµr, which satisfies [ φµr(x),Πµr(x′)] =\niδ(x−x′)δµ,µ′δr,r′.\nNow, we can rewrite the noninteracting term of the\nHamiltonian (3) as\nH0=vF\n2π/summationdisplay\nµ=ρ,σ/summationdisplay\nr=±/bracketleftBig\n(∂xθµ,r)2+(∂xφµ,r)2/bracketrightBig\n.(13)\nOn the other hand, the interacting term is more com-\nplicated, containing many products of the Klein fac-\ntors such as ˆΓ≡η↑,+η↓,+η↑,−η↑,−,ˆhσ≡ησ,+ησ,−, and\nˆhζ′≡η↑,ζη↓,ζ. However, it is known that their eigenval-\nues are Γ = ±1,hσ=±i, andh′\nζ=±i.37Thus, if we\nadopt the following convention Γ = +1, hσ=i,h′\nζ=iζ,\nthe interacting term is reduced to\nHI=/summationdisplay\nµ=ρ,σ/summationdisplay\nr=±gµ,r\n2π2∂xφ+\nµ,r∂xφ−\nµ,r+\n1\n2π2/bracketleftbig/parenleftbig\ng+−\n1⊥−g−−\n2⊥/parenrightbig\ncos2φρ−cos2φσ−\n+/parenleftbig\ng−+\n1⊥−g−+\n2⊥/parenrightbig\ncos2θρ−cos2θσ−\n+g++\n1⊥cos2φσ+cos2φσ−+g+−\n1⊥cos2φρ−cos2φσ+\n−g−+\n1⊥cos2θρ−cos2φσ++g−−\n1⊥cos2φσ+cos2θσ−\n−g−+\n2⊥cos2θρ−cos2φσ−+g−−\n2⊥cos2φρ−cos2θσ−/bracketrightbig\n(14)\nwith\ngρ+=−g++\n1⊥+2g++\n2⊥−g−−\n1⊥+g+−\n2⊥(15)\ngρ−=−g++\n1⊥+2g++\n2⊥+g−−\n1⊥−g+−\n2⊥(16)\ngσ+=−g++\n1⊥−2g−−\n1⊥(17)\ngσ−=−g++\n1⊥+2g−−\n1⊥. (18)\nIf we use the notation with gµ,r,H0can also be written\nas\nH′\n0=/summationdisplay\nµ,ruµ,r\n2π/bracketleftBig\nKµ,r(∂xθµ,r)2+K−1\nµ,r(∂xφµ,r)2/bracketrightBig\n(19)with the critical exponents\nKµ,r=/radicalBigg\n2πvF−gµ,r\n2πvF+gµ,r(20)\nand the renormalized Fermi velocity\nuµ,r=vF/radicalBigg\n1−/parenleftbigggµ,r\n2πvF/parenrightbigg2\n. (21)\nB. Renormalization-group analysis\nBy treating the interaction perturbatively, we derive\nthe renormalization group (RG) equations in the one-\nloop level as follows:\n∂g++\n1⊥\n∂S=−2/parenleftbig\ng++\n1⊥/parenrightbig2−2g−+\n1⊥g−+\n2⊥\n−2/parenleftbig\ng+−\n1⊥/parenrightbig2+2g+−\n1⊥g−−\n2⊥(22)\n∂g−+\n1⊥\n∂S=−2g−+\n1⊥g++\n2⊥−2g−+\n2⊥g++\n1⊥−4g−+\n1⊥g−−\n1⊥\n+2/parenleftbig\ng−+\n1⊥g+−\n2⊥+g−−\n1⊥g−+\n2⊥/parenrightbig\n(23)\n∂g1⊥\n−−\n∂S=−2/parenleftbig\ng−+\n1⊥/parenrightbig2−2/parenleftbig\ng−−\n1⊥/parenrightbig2\n+2g−+\n1⊥g−+\n2⊥−2g+−\n1⊥g−−\n2⊥(24)\n∂g+−\n1⊥\n∂S=−2/parenleftbig\n2g+−\n1⊥−g−−\n2⊥/parenrightbig\ng++\n1⊥+2g+−\n1⊥g++\n2⊥\n−2g−−\n1⊥g−−\n2⊥−2g+−\n1⊥g+−\n2⊥(25)\n∂g++\n2⊥\n∂S=−/parenleftbig\ng++\n1⊥/parenrightbig2−/parenleftbig\ng−+\n1⊥/parenrightbig2−/parenleftbig\ng−+\n2⊥/parenrightbig2+/parenleftbig\ng−−\n2⊥/parenrightbig2(26)\n∂g−+\n2⊥\n∂S=−2g++\n1⊥g−+\n1⊥−2g++\n2⊥g−+\n2⊥+2g−+\n2⊥g+−\n2⊥(27)\n∂g−−\n2⊥\n∂S=−2g−−\n1⊥g+−\n1⊥+2g++\n2⊥g−−\n2⊥−2g−−\n2⊥g+−\n2⊥(28)\n∂g+−\n2⊥\n∂S=−/parenleftbig\ng−−\n1⊥/parenrightbig2−/parenleftbig\ng+−\n1⊥/parenrightbig2\n+/parenleftbig\ng−+\n2⊥/parenrightbig2−/parenleftbig\ng−−\n2⊥/parenrightbig2(29)\nwhereS= lnl/(πvF) is the RG time and lis the scaling\nquantity. We note that the couplings g−−\n1⊥andg+−\n2⊥play\nimportant roles here. In general, those couplings are ir-\nrelevant when the hopping processes between two chains4\nare relevant24,32,33,34,36. However, there is no transverse\nhopping term in our model, so that we have to start from\nthe picture of degenerate bands.\nBy solving the RG equations (22)-(29), we obtain the\nrelations\ng+−\n1⊥,g−−\n2⊥,−g−−\n1⊥,−g+−\n2⊥≫g++\n2⊥≫g++\n1⊥≫0 (30)\nandg1⊥\n−+=g−+\n2⊥= 0 forU/greaterorsimilarJ. In this case, the pa-\nrametersKρ−,Kσ+, andKσ−are scaled as Kρ−→0,\nKσ+→0, andKσ−→ ∞. We therefore can simplify the\nsecond term of the phase Hamiltonian (14) as\n/parenleftbig\ng+−\n1⊥−g−−\n2⊥/parenrightbig\ncos2φρ−cos2φσ−\n+g++\n1⊥cos2φσ+cos2φσ−+g+−\n1⊥cos2φρ−cos2φσ+\n+g−−\n1⊥cos2φσ+cos2θσ−+g−−\n2⊥cos2φρ−cos2θσ−.\n(31)\nNote that the phase variables θρ−andθσ+are not in-\ncluded in this term. Taking g++\n1⊥>0 into account, we\nfind that the fields φρ−andφσ+are locked at ∝an}bracketle{tφρ−∝an}bracketri}ht=\nπ\n2I1+πI2and∝an}bracketle{tφσ+∝an}bracketri}ht=π\n2(I1+ 1) +πI3whereInare\nintegers.\nLet us then turn to the σ−mode that remains to be\nstudied. The effective Hamiltonian of the σ−mode takes\nthe form\nHφσ−=/integraldisplay\ndx/bracketleftBig\nKσ−(∂xθσ−)2+K−1\nσ−(∂xφσ−)2\n+gφcos2φσ−+gθcos2θσ−] (32)\nwheregφ=g+−\n1⊥−g−−\n2⊥−g++\n1⊥andgθ=g−−\n2⊥−g−−\n1⊥. We\nhereadopt aset ofthe variables( φρ−,φσ+) = (πI1,π/2+\nπI2); however, it leads to a physically equivalent result if\nwe chose another set ( φρ−,φσ+) = (π/2+πI1,πI2). For\nthe Hamiltonian (32), there are three RG equations as\nfollows:\ndKσ−\ndl=y2\nθ−K2\nσ−y2\nσ (33)\ndyφ\ndl= (2−2Kσ−)yφ (34)\ndyθ\ndl=/parenleftbig\n2−2K−1\nσ−/parenrightbig\nyθ. (35)\nwhereyα=gα/(πvF) withα=θ,φ. Since|yθ| ≫ |yφ|\nandKσ−>1, the parameters are renormalized to yθ→\n∞,yφ→0, andKσ−→ ∞. As a results, we find that\nthreemodesarelockedas φρ−=π\n2I1+πI2,φσ+=π\n2(I1+\n1)+πI3, andθσ−=π\n2(I1+1)+πI4. Therefore, there is\na gap in all the spin excitations.\nC. Order parameters\nIn order to determine the dominant correlation, we in-\ntroduce the orderparametersofthe superconducting cor-\nrelations for four kinds of pairing symmetry, which areFIG. 2: Symmetry of the pairing state in the double-chain\nHubbard model. The sign of the order parameter is also\nshown.\nshown in Fig. 2. They are given in terms of the phase\nfields as follows:\nOS1=/summationdisplay\np=±,ζ=±ψp,↑,ζψ−p,↓,−ζ\n∼2\nπaeiθρ+[cosθσ−cosφσ+cosφρ−\n−isinθσ−sinφσ+sinφρ−] (36)\nOT1=/summationdisplay\np=±,ζ=±ζψp,↑,ζψ−p,↓,−ζ\n∼2\nπaeiθρ+[isinθσ−cosφσ+cosφρ−\n−cosθσ−sinφσ+sinφρ−] (37)\nOS2=/summationdisplay\np=±,ζ=±ζpψp,↑,ζψ−p,↓,−ζ\n∼2i\nπaeiθρ+[isinθσ−sinφσ+cosφρ−\n+cosθσ−cosφσ+sinφρ−] (38)\nOT2=/summationdisplay\np=±,ζ=±pψp,↑,ζψ−p,↓,−ζ\n∼2i\nπaeiθρ+[cosθσ−sinφσ+cosφρ−\n+isinθσ−cosφσ+sinφρ−]. (39)\nOne can easily find that the inter-chain diagonal singlet\npairing state, which corresponds to the order parameter\nOS2, is only of the quasi-ordering superconducting in-\nstability when φρ−andθσ−are locked. The asymptotic\nbehavior of the S2 correlation function is given by\n/angbracketleftBig\nO†\nS2(r)OS2(0)/angbracketrightBig\n∼1\nr1/2Kρ+. (40)\nand the other interchain pairing correlations decay expo-\nnentially. We also mention that all the intrachain pair-5\ning correlations decay exponentially due to the locked\nmodes. If we assume a S2 ground state O†\nS2|0∝an}bracketri}htand ap-\nply the interacting term of Hamiltonian (14) to the state,\nwe obtain\nHIO†\nS2|0∝an}bracketri}ht= (g−−\n1⊥−g+−\n1⊥−g−−\n2⊥+g+−\n2⊥)O†\nS2|0∝an}bracketri}ht.(41)\nSinceg−−\n1⊥<0,g+−\n1⊥>0,g−−\n2⊥>0, andg+−\n2⊥<0, we find\nthat the S2 pairing state can gain much energy.\nTheotherpossibleorderisthe4 kFcharge-densitywave\n(CDW) correlation,ofwhich theorderparameterisgiven\nby\nO4kF=/summationdisplay\np,σ,σ′ψ†\np,σ,+ψ†\np,σ′,−ψ−p,σ′,−ψ−p,σ,+\n∼1\na2π2cos(4kFx+2φρ+)(cos2φσ++cos2φσ−),\n(42)\nwhere the cos2 φσ+term comes from the cases where the\nspins are parallel σ=σ′, and the cos2 φσ−term comes\nfrom the cases where the spins are antiparallel σ=−σ′.\nThe latter component cos2 φσ−decays exponentially be-\ncause the field θσ−is locked. Thus, the 4 kF-CDW corre-\nlation shows a power-low behavior like\n/angbracketleftBig\nO†\n4kF(r)O4kF/angbracketrightBig\n∼1\nr2Kρ+. (43)\nBy comparing Eq.(43) with Eq.(40), we find that the\ndxy-wave-likeS2 pairing correlation is most dominant for\nKρ+>0.5, whereas the 4 kF-CDW is most dominant for\nKρ+<0.5. This is the same as that of the standard\ntwo-band model.33\nIV. NUMERICAL RESULTS\nNext, we turn to the intermediate to strong coupling\nregime. We employ the Lanczos and DMRG methods\nto obtain energies and physical quantities in the ground\nstate and low-lying excited states. In order to carry out\nourcalculations,weconsider N(=N↑+N↓)electronsina\nsystem with length L(containing 2 Lsites). The electron\ndensity is given by n=N/L. Note that the number of\nelectrons must be taken as N= 4lwithl(>1) being\nan integer to maintain the total spin of the unpolarized\nground state as S= 0.\nFor static quantities, we use the DMRG method with\napplying the open-end boundary conditions for precise\ncalculations. We study systems with length up to L=\n128 and keep up to m≈2400 density-matrix eigen-\nstates in the DMRG procedure. In this way, the dis-\ncarded weights are typically of the order 10−8∼10−7\nand the ground-state energy is obtained in the accuracy\nof∼10−7t. All the calculated energies are extrapolated\nto the limit m→ ∞. For dynamical quantities, we use\nthe Lanczos method for small clusters with the periodic\nboundary conditions. The system size is assumed as\nL= 8, i.e., 8 ×2 ladder.01\n0.1 1 10 100/CB/BP/CB\n/D1/CP/DC/C2\nFIG. 3: (Color online) Calculated values of the total-spin\nquantum number Sas a function of JforU= 20 (circles), 40\n(squares), and ∞(crosses). L= 48 and n= 0.5 are assumed.\nA. Spin polarization\nApproaching from the strong-coupling regime U≫t,\nwe may anticipate the presence of the fully-polarized fer-\nromagnetic state. When the two chains are uncoupled,\ni.e.,J= 0, the state can be interpreted as the Nagaoka\nstate. Generally, the appearance of the Nagaoka state\nis limited to large- Uand low-nrange. However, the\nfully-polarizedregionwouldbespreadintosmaller- Uand\nhigher-nrange if a finite Jis introduced. This is because\nthe polarized electrons can gain the kinetic energy with-\nout loss of the exchange interaction between the chains.\nLet us start with investigating how the ferromagnetic\nphase appears in the parameter space ( U,J). We can\nfind it by calculating the expectation value of total-spin\noperator Sin the ground state, which is defined by\n/angbracketleftbig\nS2/angbracketrightbig\n=/summationdisplay\nij∝an}bracketle{tSi·Sj∝an}bracketri}ht=S(S+1).(44)\nFor a fully-polarized state, one will obtain the total-spin\nquantum number S=Smax=N/2, i.e.,S/Smax= 1. In\nFig. 3, we show the total spin Snormalized with respect\ntoSmaxas a function of Jfor several values of U. The\nsystem size and filling are fixed at L= 48 andn= 0.5,\nrespectively. We calculate Sfor systems with length L=\n24, 36, and 48, and confirm that the size dependence is\nnegligible.\nAtU= 20, with increasing J, we find a transition\nfrom the paramagnetic state to the fully-polarized state\natJc1∼4.5. Moreover,wefindthatthesystemgoesback\nto the paramagnetic state at Jc2∼8.5. This is because\nthe formation of the local spin-singlet bound states gains\nmore energy for very large values of JsinceJincreases\nthe gain in the spin-singlet binding energy as shown in\nSec.IVB3. We also find that the regionwith the full spin\npolarization broadens with increasing U. In the limit of\nU=∞,Jc1→0 andJc2→ ∞. We note that both of\nthe transitions are discontinuous, i.e., of the first-order\nin the thermodynamic limit. Thus, the critical transition\npoints can be determined in the parameter space ( U,J),6\n/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s32/s32/s99/s40/s76 /s41\n/s49/s47 /s76/s32/s110 /s32/s61/s32/s48/s46/s53/s44/s32 /s85 /s32/s61/s32/s52/s44/s32 /s74 /s32/s61/s32/s49\n/s32/s110 /s32/s61/s32/s48/s46/s53/s44/s32 /s85 /s32/s61/s32/s52/s44/s32 /s74 /s32/s61/s32/s52\n/s32/s110 /s32/s61/s32/s48/s46/s53/s44/s32 /s85 /s32/s61/s32/s105/s110/s102/s105/s110/s105/s116/s121/s44/s32 /s74 /s32/s61/s32/s52\n/s32/s110 /s32/s61/s32/s49/s44/s32 /s85 /s32/s61/s32/s48/s44/s32 /s74 /s32/s61/s32/s52\nFIG. 4: (Color online) ∆ c(L) as a function of the inverse sys-\ntem length 1 /L. Lines are the fits with the 2nd-order poly-\nnomials.\nwhich will be given as a ground-state phase diagram in\nSec.IVF.\nB. Excitation gaps\nTo ascertain the presence of the lowest excitation gap\nin the charge, spin, and pairing sectors, we calculate the\ncharge gap, spin gap, and binding energy in the ther-\nmodynamic limit. We study several chains with lengths\nup toL= 64 and then perform the finite-size scaling\nanalysis based on the size-dependence of each quantity.\n1. Charge gap\nThe charge gap ∆ cis defined by\n∆c= lim\nL→∞∆c(L), (45)\n∆c(L) =E0\nL(N↑+1,N↓+1)+E0\nL(N↑−1,N↓−1)\n−2E0\nL(Ne)\nwhereE0\nL(N↑,N↓) denotes the ground-state energy of a\nchain of length LwithN↑spin-up and N↓spin-down\nelectrons. In Fig. 4, we plot the charge gap ∆ c(L) as\na function of the inverse system length 1 /Lfor several\nparameter sets.\nAt half filling ( n= 1), we can easily expect the system\nto be a Mott insulator for U >0. However, we also find\nthe system is insulating even for U= 0 ifJis finite. This\nis associated with the fact that a localized spin-triplet\npair is formed on each rung. If an electron is add to the\nrung, an effective on-rung repulsive interaction Ueff=\nU+J\n2acts. Thus, we obtain the effective Hamiltoniannear half filling\nHn∼1\neff=−t/summationdisplay\nrx,σ(c†\nrx,σcrx,σ+H.c.)\n+/parenleftbigg\nU+J\n2/parenrightbigg/summationdisplay\nrx,σnT\nrxnrx,σδ∝an}bracketle{tSzrx∝an}bracketri}ht,±1\n2(46)\nwherecrx,σ=crx,1,σcrx,2,σis the annihilation operator\nof an electron with spin σat rungrx,nrx,σ=c†\nrx,σcrx,σ\nis the number operator, Sz\nrx=nrx,↑−nrx,↓is thez-\ncomponent of total spin, and nT\nrx=T†\nrxTrxwith the spin-\ntriplet operator Trx=crx,1,σcrx,2,σor1√\n2(crx,1,σcrx,2,¯σ+\ncrx,1,¯σcrx,2,σ.\nLet us then turn to the case away from half filling.\nTherearetwopoints to be noted here: oneis whether the\nsystem is insulating at quarter filling ( n= 0.5) as is the\ncaseofthe Hubbardand t−Jladder; the otheriswhether\nthe phase separation occurs in the fully-polarized state.\nAs seen in Fig. 4, all the results for n= 0.5 are smoothly\nextrapolated to zero in the thermodynamic limit 1 /L→\n0. AtU=∞andJ= 4, the system is in the fully-\npolarized state. We therefore conclude that the system\nis metallic in the entire region except n= 1.\n2. Spin gap\nThe spin gap ∆ sis defined by\n∆s= lim\nL→∞∆s(L), (47)\n∆s(L) =E0\nL(N↑+1,N↓−1)−E0\nL(N↑,N↓).\nIn Fig. 5, we plot the spin gap ∆ s(L) as a function of\ninverse system length 1 /Lat (a)n= 1 and (b) n= 0.75.\nForU= 4,n= 1 andU= 1,n= 0.75, values of ∆ s(L)\nseem to be smoothly extrapolated to zero when 1 /L→0,\nwhich is in contrast to the weak-coupling analysis. For\nthe other cases, however, the extrapolated lines reach\nzero at finite values of 1 /L. This is unphysical and so\nit may be a good guess that ∆ s(L) is fairly flat around\n1/L= 0, as seen in the results for U= 4,n= 1 and\nU= 1,n= 0.75. Unfortunately, we cannot treat the\nlarge enough systems to confirm if this is the case. Nev-\nertheless, ifweassumethatthe 1 /L-dependenceof∆ s(L)\nreflects the spinon band structure near the Fermi point,\nthe flat behavior of ∆ s(L) may rather imply the presence\nofaspingapfulstate. Sincewecannotdetectthespingap\nless than ∼10−6within the present calculations, the ex-\nistence of a very small spin gap, i.e., ∆( L→ ∞)/lessorsimilar10−6,\nis conceivable.\nTo determine if the spin gap is present or absent, we\nalso calculate the equal-time spin-spin correlation func-\ntion\nS(|rx−r′\nx|,ry,r′\ny) =∝an}bracketle{tSz\nrx,rySz\nr′x,r′y∝an}bracketri}ht−∝an}bracketle{tSz\nrx,ry∝an}bracketri}ht∝an}bracketle{tSz\nr′x,r′y∝an}bracketri}ht,\n(48)\nwhereSz\nrx,ryis the z component of the spin operator of\nan electron at site rxon rungry. In Fig. 5, we present7\n0 0.05 0.100.10.20.30.40.5\n U = 1, J = 1\n U = 4, J = 1\n U = 4, J = 4 U = 1, J = 1\n U = 4, J = 1\n U = 4, J = 4n = 1\n(a)\n020406010-610-510-410-310-210-1\n U = 1, J = 1\n U = 4, J = 1 U = 1, J = 1\n U = 4, J = 1(c)\n0204060(d)0 0.05 0.1n = 0.75\n(b)/A1\n/D7/CY/CW /CB/DE\n/D6\n/DC/BN /BD/CB/DE\n/D6/BC\n/DC/BN /BE/CX /A0 /CW /CB/DE\n/D6\n/DC/BN /BD/CX/CW /CB/DE\n/D6/BC\n/DC/BN /BE/CX/CY\n/BD /BP/C4 /BD /BP/C4/CY /D6/DC\n/A0 /D6\n/BC/DC\n/CY /CY /D6/DC\n/A0 /D6\n/BC/DC\n/CY\nFIG. 5: (Color online) Upper panels: ∆ s(L) as a function of\ninverse system size for (a) n= 1 and (b) n= 0.75. Solid lines\nare the polynomial fits to the data for the finite-size scaling\nanalysis. Lower panels: Semilogarithmic plot of the magni-\ntude of the spin-spin correlation function S(|rx−r′\nx|,1,2′) at\nJ= 1 for (c) n= 1 and (d) n= 0.75. The data are fitted\nwith a function S(|rx−r′\nx|,1,2)≃exp“\n−|rx−r′\nx|\nξ”\n.\na semilogarithmic plot of |S(|rx−r′\nx|,1,2)|as a function\nof the distance |rx−r′\nx|for (c)n= 1 and (d) n= 0.75.\nNote that the long-range behavior of |S(|rx−r′\nx|,1,1)|is\nalmost the same as that of |S(|rx−r′\nx|,1,2)|. The results\ncan be fitted with a function exp( −|rx−r′\nx|/ξ) and thus\nthe exponential decay of the correlation functions is con-\nfirmed for all the parameter sets. The correlation lengths\nare estimated as ξ= 10.35(9.74) forU= 1(4) atn= 1;\nξ= 14.29(12.99) forU= 1(4) atn= 0.75. According to\nthe Tomonaga-Luttinger liquid theory, the spin-spin cor-\nrelation can decay exponentially only when there is a gap\nin all the magnetic excitations. Consequently, it would\nbe rather appropriate to conclude that there exists quite\nsmall spin gap ( /lessorsimilar10−6). The correlationlengths seem to\nbe much longerthan those of other standard spin-gapped\nsystems, e.g., ξ= 3.19 in the two-leg isotropic Heisenberg\nsystem. However, it has been found that in the zigzag\nHeisenbergchain, the correlationlengths increaserapidly\nwith decreasing binding energy.38Thus, the very large\nvalues ofξmay reflect an exponentially small spin gap.0246810-0.5-0.4-0.3-0.2-0.10\nU = 20(c)\nFully-polarized\n         region0 0.05 0.1-0.6-0.5-0.4-0.3-0.2-0.10\n  U = 4, J = 1\n  U = 4, J = 4\n  U = 20, J = 4\n  U = 20, J = 5(a)\n01234-0.2-0.10\nU = 4(b)\nFIG. 6: (Color online) Upper panel: ∆±\nB(L) as a function\nof inverse system length 1 /Latn= 0.5. Solid lines are\nthe polynomial fits to the data for finite-size scaling analy-\nsis. Solid (empty) symbols denote the binding energy of elec -\ntrons (holes). Lower panels: Values of ∆±\nB(L) extrapolated\nto 1/L→0 plotted as a function of Jfor (b)U= 4 abd (c)\nU= 10.\n3. Binding energy\nThe binding energy ∆±\nBis defined by\n∆±\nB= lim\nL→∞∆±\nB(L), (49)\n∆±\nB(L) =E0\nL(N↑±1,N↓±1)+E0\nL(N↑,N↓)\n−2E0\nL(N↑±1,N↓)\nwhere the + ( −) sign corresponds to the binding energy\nof electrons (holes). In Fig. 6 (a), we plot the binding\nenergy ∆±\nB(L) as a function of the inverse system length\n1/Latn= 0.5 for several sets of parameters. For all the\nparameter sets, |∆±\nB(L)|is found to decrease monoton-\nically as a function of 1 /L, so that we can extrapolate\n|∆±\nB(L)|to the thermodynamic limit systematically. We\nperform a least-squares fit of |∆±\nB(L)|to a polynomial in\n1/Land obtain the extrapolated values. We note that\nthe binding energies of electrons and holes are extrapo-\nlated to the same value at 1 /L→0.\nIn Fig. 6 (b), the extrapolated values of ∆±\nBforU= 4\nas a function of Jare shown. When Uis small, the sys-\ntem is expected to be in the spin-singlet superconducting\nphase, as discussed in Sec. III. Hence, the binding en-\nergyis determined by an energyofthe spin-singlet bound\nstate. In analogywith the Haldane gap, we expect a scal-\ning ∆±\nB∝Jt/U2from perturbation assuming the double8\n/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49\n/s49 /s49/s48/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49\n/s49 /s49/s48/s40/s97/s41/s32 /s110 /s32/s61/s32/s48/s46/s56/s55/s53\n/s32/s32/s68 /s40/s124 /s114\n/s120 /s114/s39\n/s120 /s124/s41/s32/s83/s49\n/s32/s83/s50\n/s32/s84/s49\n/s32/s84/s50/s40/s98/s41/s32 /s110 /s32/s61/s32/s48/s46/s53\n/s32/s32\n/s32/s83/s49\n/s32/s83/s50\n/s32/s84/s49\n/s32/s84/s50\n/s40/s99/s41/s32 /s110 /s32/s61/s32/s48/s46/s51/s55/s53\n/s32/s32/s68 /s40/s124 /s114\n/s120 /s114/s39\n/s120 /s124/s41\n/s124/s114\n/s120 /s114/s39\n/s120 /s124/s32/s83/s49\n/s32/s83/s50\n/s32/s84/s49\n/s32/s84/s50/s40/s100/s41/s32 /s110 /s32/s61/s32/s48/s46/s49/s50/s53\n/s32/s32\n/s124/s114\n/s120 /s114/s39\n/s120 /s124/s32/s83/s49\n/s32/s83/s50\n/s32/s84/s49\n/s32/s84/s50\nFIG. 7: (Color online) Log-log plot of the pair-correlation\nfunctions Dα(|rx−r′\nx|) calculated for L= 128 at (a) n=\n0.825,U= 10,J= 2, (b) n= 0.5,U= 10,J= 4, and (c)\nn= 0.25,U= 10,J= 5.\noccupancy is sufficiently excluded. The origin of this en-\nergygainisalsointerpreted asthe double exchangeinter-\naction in our model. For small J, this estimation seems\nnot to work well. However, the double occupancy is in-\ncreasingly excluded with increasing Jand thus |∆±\nB(L)|\nincreases linearly with JatJ/greaterorsimilar3. In Fig. 6 (c), the ex-\ntrapolated values of ∆±\nBforU= 20 as a function of Jare\nshown. Since the double occupancy is almost excluded,\n|∆±\nB|increases linearly with Jeven ifJis small. It is\nnotable that the binding energy is strongly suppressed in\nthe fully-polarized phase. This phase is regarded as the\nspin-triplet superconducting one, where the binding en-\nergy is determined by an energy of the spin-triplet bound\nstate, which is scaled as |∆±\nB| ∝J−Jc1. Generally, an\nenergy of the spin-triplet bound state is much lower than\nthat of the spin-singlet bound state.39\nC. Superconducting correlation\nFor our model, we can consider four kinds of supercon-\nducting correlations as mentioned in Sec.III. In order to\nestimate them numerically, we define the corresponding\npair correlation functions as\nDα(|rx−r′\nx|) =/angbracketleftbig\n∆†\nα(rx)∆α(r′\nx)/angbracketrightbig\n(50)with\n∆S1(rx) =crx,1,↑crx,2,↓−crx,1,↓crx,2,↑(51)\n∆T1(rx) =crx,1,↑crx,2,↓+crx,1,↓crx,2,↑(52)\n∆S2(rx) =crx,1,↑crx+1,2,↓−crx,1,↓crx+1,2,↑(53)\n∆T2(rx) =crx,1,↑crx+1,2,↓+crx,1,↓crx+1,2,↑,(54)\nwhich are calculated by the DMRG method. The cal-\nculated results for three sets of parameters are shown in\nFig. 7. For n= 0.825,U= 10, and J= 4 [Fig. 7(a)],\nthe S2 pairing correlation is clearly the most dominant\none, which shows a power-law length dependence. The\nratio of the decay is estimated to be ∼r−0.7, which leads\ntoK+\nρ= 0.714. This is consistent with the bosonization\nresult for the spin gapful state. At n= 0.5,U= 10,\nJ= 4 [Fig. 7(b)], the system has a S= 0 ground state\nbut is somewhat closer to the fully-polarized ferromag-\nnetic phase. The S2 pairing correlation is still the most\ndominant one but the T1 pairing correlation becomes\nmuch enhanced at a short distance |rx−r′\nx|/lessorsimilar10. This\nis because the formation of a local spin-triplet bound\nstate on a rung can gain some energies. We note that\nthe T2 pairing correlation is also enhanced, reflecting a\ntendency to the fully-polarized ferromagnetic state. The\ndecay ratio of the S2 correlation is ∼r−0.8, which leads\ntoK+\nρ= 0.625. This value agrees well with our di-\nrect estimation of K+\nρ(See Sec.IVE). If the system fur-\nther approaches the fully-polarized ferromagnetic phase\n[Fig. 7(c)], the change in the correlation functions at\nshort ranges becomes more prominent. Surprisingly, we\nfind that the decay lengths of all the correlations are\nalmost unchanged as far as Jis fixed in the S= 0\nground state. In the fully-polarized ferromagnetic phase\n[Fig. 7(d)], the T1 pairing correlation is the most domi-\nnantoneandonlytheS2pairingcorrelationiscompeting.\nThe decay ratio of both correlation functions is ∼r−1,\nwhich leads to K+\nρ= 0.5.\nD. Anomalous Green function\nLet us now determine the pairing symmetry in the S2\nsuperconducting state. To this end, we calculate the one-\nparticle anomalous Green function,40which exhibits the\nexcitations of the Bogoliubov quasiparticles in the BCS\ntheory. Therefore, we can see how the nodes appear in\nthe superconducting gapfunction. The anomalousGreen\nfunction is defined by\nGky,k′y(qx,z) =\n/angbracketleftBig\nψNe−2\n0/vextendsingle/vextendsingle/vextendsinglecqx,ky,↑1\nz−H+E0c−qx,k′y,↓/vextendsingle/vextendsingle/vextendsingleψNe\n0/angbracketrightBig\n(55)\nwhere/vextendsingle/vextendsingle/vextendsingleψNe\n0/angbracketrightBig\ndenote the wavefunctionofthe groundstate\nwithNe/2up-spin and Ne/2down-spinelectrons, and E09\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s48 /s49 /s50 /s51 /s52 /s53/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54\n/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54/s45/s48/s46/s54/s48/s46/s48/s48/s46/s54 /s32/s32/s107\n/s120/s32/s61/s32/s48\n/s32/s70 \n/s107\n/s121/s107\n/s121/s39/s40/s107\n/s120 /s44/s32 /s41\n/s107\n/s120/s32/s61/s32 /s47/s52\n/s32/s32/s107\n/s120/s32/s61/s32 /s47/s50\n/s32/s32/s107\n/s120/s32/s61/s32/s51 /s47/s52\n/s32/s32/s107\n/s120/s32/s61/s32\n/s32/s32\n/s107\n/s120/s32/s61/s32/s48\n/s32/s32\n/s107\n/s120/s32/s61/s32 /s47/s52\n/s32/s32\n/s107\n/s120/s32/s61/s32 /s47/s50\n/s32/s32\n/s107\n/s120/s32/s61/s32/s51 /s47/s52\n/s32/s32\n/s107\n/s120/s32/s61/s32/s32/s32/s107\n/s120/s32/s61/s32 /s47/s52\n/s32/s32/s107\n/s120/s32/s61/s32 /s47/s50\n/s32/s32/s32/s107\n/s120/s32/s61/s32 /s51 /s47/s52\n/s32/s32\n/s107\n/s120/s32/s61/s32 /s47/s52\n/s32/s32\n/s107\n/s120/s32/s61/s32 /s47/s50/s32\n/s32/s32\n/s107\n/s120/s32/s61/s32 /s51 /s47/s52\nFIG. 8: Left panel: Anomalous Green’s function at ky= 0,\nk′\ny=π,n=Ne/N= 5/8,U= 10,J= 2, and N= 16. Right\npanel: Same as the left panel but at ky=π,k′\ny= 0.\nis chosen as the average value of E0\nL(N↑−1,N↓−1) and\nE0\nL(N↑,N↓). We then estimate the spectral function as\nFky,k′y(kx) =−1\nπImGky,k′y(qx,ω+iη) (56)\nwithη= 0+and its frequency integral as\nFky,k′y(kx) =/angbracketleftBig\nψNe−2\n0/vextendsingle/vextendsingle/vextendsingleckx,ky,↑c−kx,k′y,↓/vextendsingle/vextendsingle/vextendsingleψNe\n0/angbracketrightBig\n.(57)\nWe calculate the anomalous Green function (55) for the\nladder with length L= 8 by using the Lanczos method.\nWe should note that the BCS theory is not readily ap-\nplicable for (quasi) one-dimensional system; actually,\nFky,k′\ny(qx) is not long-range ordered with a logarithmic\ndecay as a function of L. However, apart from this pref-\nactor, we can naively expect that Fky,k′y(qx) calculated\non finite-size systems should provide information on the\npairing symmetry at intermediate distances.\nThe calculated results for Fky,k′y(kx,ω) are shown in\nFig. 8. We can observe pronounced low-energy peaks\nat|kx|=π\n4for allkyandk′\ny, which are the nearest\nto the Fermi momenta kF(=5\n16π), and much smaller\npeaks at higher energies for other momenta. We also\nfind that the spectral weight vanishes at qx= 0,qx=π,\nandky=k′\ny. The weights of the peaks appear to\nbe similar to the BCS form of the condensation am-\nplitude, which has a maximum value at the Fermi mo-\nmenta. We may thus assume that the superconduct-\ning ground state in strongly correlated electron sys-\ntems can be characterized by a gap function, which is\ndirectly proportional to the frequency-integrated func-\ntionFky,k′y(qx).40,41The function Fky,k′y(kx) obeys the02468100.50.550.60.650.7\nU = 10\nU = 20/C3\n/AQ/C2\n/D2 /BP /BC /BM /BH\nFIG. 9: (Color online) DMRG results for Kρ+as a function\nofJ. The results for U= 10 and 20 at n= 0.5 are shown.\nDotted line is guide for eyes.\nrelationFky,k′y(kx) =−Fk′y,ky(kx) andFky,k′y(kx) =\n−Fky,k′\ny(−kx). This indicates the formation of the dxy-\nwave like pairing state in our system, which is consistent\nwith the weak-coupling results shown in Sec.IIIC.\nE. Tomonaga-Luttinger liquid parameter\nFor the calculation of the TL parameter, we use a re-\ncently proposed method of the DMRG technique.42As\nmentioned in Sec. IIIC, the TL parameter K+\nρdeter-\nmines the long-range decays of the dxy-like (S2) pair-\ning and 4kF-CDW correlations in the metallic TL-liquid\nground state, whereas the parameter K−\nρis scaled as\nK−\nρ→0. For the double-chain model, we define the\nTL parameter as\nKρ±=π\n2lim\nq→0∂N±(q)\n∂q(58)\nwith\nN±(q) =1\nL/summationdisplay\nrx,r′xeiq(rx−r′\nx)/angbracketleftBig\nn±\nrxn±\nr′x/angbracketrightBig\n.(59)\nwheren±\nrx=nrx,1±nrx,2. In Fig. 9, we show the calcu-\nlated results for Kρ+as a function of JforU= 10and 20\natn= 0.5. ForU= 10,K+\nρincreasesmonotonouslywith\nincreasingJ, which is consistent with the monotonous\nincrease in the binding energy with respect to Jin the\nstrong-coupling regime. For U= 20, the behaviors for\nJ/lessorsimilar4 andJ/greaterorsimilar9 are quite similar to those for U= 10,\nalthough the values are somewhat smaller due to the ef-\nfects of the Umklapp scattering. However, we estimate\nthe TL parameter as K+\nρ= 0.5 for 5/lessorsimilarJ/lessorsimilar8. This\nregime corresponds to the fully-polarized ferromagnetic\nphase and the TL parameter should be the same as that\nof a spinless fermion system. We thus find Kρ>1/2 in10\n0 0.5 10246/C2/D2\n/CB/D4/CX/D2/B9/D8/D6/CX/D4/D0/CT/D8 /CB/BV/B4 /CB /BP /CB/D1/CP/DC\n/B5/A1/D7\n/BP /BC/A1/BU\n/BO /BC\n/CB/D4/CX/D2/B9/D7/CX/D2/CV/D0/CT/D8 /CB/BV/B4 /CB /BP /BC/B5/A1/D7\n/BQ /BC/A1/BU\n/BO /BC\n/CD /BP /BH /CD /BP /BD/BC /CD /BP /BE/BC\nFIG. 10: (Color online) Phase diagram of our double-chain\nHubbard model obtained from the DRMG calculations. The\nphase boundary between the fully-polarized ferromagnetic\nphase and dxy-like superconducting (SC) phase is shown for\nU= 5, 10, and 20.\nthe entire region except in the fully-polarized ferromag-\nnetic phase and we confirm that the S2-type supercon-\nducting correlation is most dominant.\nF. Phase diagram\nIn Fig. 10, we show the phase diagram of our model\nin the parameter space ( n,J) forU= 5, 10, and 20.\nThe phase boundary is determined by the calculated re-\nsults for the total-spin quantum number. At higher elec-\ntron concentrations from the phase boundary, the sys-\ntem is characterized as the S2-type spin-singlet super-\nconducting state, and at lower concentrations from the\nphase boundary, the system is characterized as the fully-\npolarized ferromagnetic state and simultaneously as the\nT1-typespin-tripletsuperconductingstate. Wenotethat\nthe fully-polarized ferromagnetic phase is spread to the\nhigher concentration range with increasing U, and it oc-\ncupies the entire parameter region ( n <1,J >0) at\nU=∞. This result is connected to the Nagaoka state4\natU→ ∞.\nV. SUMMARY AND DISCUSSION\nWe have studied the ground-state properties of the\ndouble-chain Hubbard model coupled with ferromagnetic\nexchange interaction by using the weak-coupling the-\nory, DMRG technique, and the Lanczos method. Wehavethereby determined the ground-statephase diagram\nin the parameter space of the ferromagnetic exchange\ninteraction and band filling. We have found that, in\nhigh electron density regime, the spin gap opens and\nthe spin-singlet dxy-wave-like pairing correlation is most\ndominant, and in low electron density regime, the fully-\npolarized ferromagnetic state is stabilized and simulta-\nneously the spin-triplet py-wave-like pairing correlation\nbecomes most dominant.\nHere, let us make some comment on what happens\nif some additional terms are introduced to our model.\nFirst, we consider the case where the hopping integral\nbetween the two chains t⊥is added. In this case, the\ncouplingsg−−\n1⊥andg+−\n2⊥become irrelevant, which leads\nto a collapse of the pairing state between different bands,\ni.e.,ky= 0 andπ. Thus, the spin-singlet superconduc-\ntivity is suppressed. Since the term t⊥induces the anti-\nferromagnetic interaction on each rung, the spin-triplet\nsuperconductivitymay alsobe suppressed. Next, we con-\nsider the case where the intersite Coulomb interaction\nbetween the two chains V⊥is added. In this case, we\ncaneasilyimaginethat thespin-tripletsuperconductivity\nis strongly suppressed because the dy-wave-like pairing\nstate is formed on rung. The double exchange interac-\ntion for the dxy-wave-likepairingstate is alsosuppressed.\nFinally, let us discuss possible relationship between\nthe present model and the model of two Hubbard chains\ncoupled with zigzag bonds where the spin-triplet super-\nconductivity has been shown to occur.17,18,19The two\nmodels in the strong-coupling regime have the follow-\ning features in common: (i) the superconductivity occurs\nnear the region of ferromagnetism, (ii) there is a compe-\ntition between the spin-singlet and spin-triplet pairings,\nand (iii) the spin-gap is quite small or vanishes. These\nsituations may well suggest that the spin-triplet pairing\ncould be dominant in the present model near the phase\nboundarybetweenthespin-singletsuperconductivityand\nferromagnetism. 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Department of Physics and Electronics, School of Science, Beijing University \nof Chemical Technology, Beijing 100029, China  \n4. Technische Universität Dresden, 01062 Dresden, Germany  \n5. High Magnetic Field Laboratory, Hefei Institutes of Physical Science , Chinese \nAcademy of Sciences, Hefei 2300 31, People's Republic of China  \n \nAbstract:  \nWe present a comprehensive structural characterization of ferromagnetic SiC single \ncrystals  induced by Ne ion irradiation . The ferromagnetism has been confirmed by \nelectron spin resonanc e and possible transition metal impurities can be excluded to \nbe the origin of the observed ferromagnetism. Using X -ray diffraction and Rutherford \nbackscattering/channeling spectroscopy, we estimate the damage to the crystallinity \nof SiC which mutually  influences the ferromagnetism in SiC.  Page | 2 \n  \nI. Introduction:  \nDefect -induced ferromagnetism  in materials which do  not contain  partially filled 3d or \n4f electrons  has recently entered the focus  of condensed matter research [1]. It \nchallenges the mJ paradigm for magn etism, where m refers the local moment and J \nstands for the interaction between the local moments. Experimentally,  defect -induced \nferromagnetism was observed in many materials, including graphite  [2-5] and  various \noxides [6-12]. SiC single crystal s are emerging as another candidate for this \ninvestigation and ha ve been shown to be ferromagnetic after particle irradiation  [13, \n14] or after aluminum doping  [15]. The magnetization in SiC is found to sensitively \ndepend on the fluence of ion or neutron irradiatio n. Compared with other materials , \nSiC is commercially available at large scale with the microelectronic quality grade  \n[16]. In this paper, we present a systematic structural investigation in correlation with \nthe magnetic properties of SiC prepared by Ne io n irradiation. A possible Fe, Co or Ni \ncontamination in SiC is excluded by particle  induced X -ray emission (PIXE) and by \nAuger electron spectroscopy (AES) measurements . The appearance of \nferromagnetism has been confirmed by electron spin  resonance  (ESR) spectroscopy . \nMeasurements of X -ray diffraction  (XRD ) and Rutherford backscattering/channeling \nspectroscopy  (RBS/C)  reveal the damage to the crystallinity of SiC which can \nextinguish  the ferromagnetism in SiC.  \nII. Experimental methods:  \nThe 6H-SiC(0001 ) wafer  purchased from KMT corporation (Hefei, China)  is one -side \npolished and semi -insulating.  The same wafer was cut into smaller pieces which \nwere  implanted with Ne ions at an energy of 140 keV. The ion fluences were 5×1013, Page | 3 \n 1×1014, 5×1014, 1×1015 cm-2, and conseq uently the samples will be named  as 5E13, \n1E14, 5E14 and 1E15, respectively. Magnetometry was performed using a MPMS -XL \nmagnetometer from Quantum Design. The ferromagnetic resonance was measured \nat 9.46 GHz by an electron paramagnetic resonance spectromete r (Bruker ELEXSYS \nE500 ).  PIXE  was performed using 3 MeV protons with a broad beam  of 1 mm2, while  \nAES was done by  a scanning Auger electron spectrometer Microlab 310F (Fisons \nInstruments). In order  to evaluate the crystalline variation after ion irradiati on, \nsynchrotron radiation XRD  was carried out  at the Rossendorf beamline (BM20) at the \nESRF with an X-ray wavelength of 0.1078 nm.  As a complementary method, RBS/C \nwith 1.7 MeV He+ was used to  quantitatively determine the ion -implantation -induced \natomic d isorder of the Si sublattice in 6H -SiC.  \nIII. Results and discussion:  \n-2500 0 2500-2-1012@300 K\n  \n  \nField (Oe)Magnetization (emu/g)  Virgin\n  5E13\n  1E14\n  5E14\n  1E15(a)\n-4000 -2000 02000 4000\n  \n   Virgin\n 5E13\n0 1600 3200Intensity (arb. u.)\n  \n100 K300 K(b) sample 1E14\nField (Oe)\n \nFig. 1: (a) Magnetization vs. field measured at 300 K for all samples: the diamagnetic background from \nthe substrate has been subtracted.  The magnetization was calculated by assum ing a thin layer of 460 \nnm thickness. The inset shows the magnetization measured for the virgin sample (black line) and Page | 4 \n sample 5E13 without subtracting the background. (b) ESR spectra of sample 1E14 at temperatures  \nbetween 100 and 300 K : a broad resonance peak appears indicating ferromagnetic resonance . The \nblack line  indicate s the resonance centers which slightly change with temperature.  \n \nA. Magnetic properties  \nThe magnetic properties of Ne irradiated samples at di fferent temperature s have \nbeen reported in our previous paper  [13]. The inset of Fig. 1(a) shows the \nmagnetization at 300 K measured for  sample 5E13 and the virgin SiC from the same \nwafer. The virgin SiC  only shows diamagnetism, while sample 5E13 presents a \nferromagnetic hysteresis. Fig. 1 (a) shows the magnetization at 300 K for all samples \nafter subtracting the diamagnetic background. The saturation magnetization shows a \nnonlinear dependence with increasing Ne fluence (defect  concentration ) and reach es \nits maximum for sample 1E14.  \nThe appearance of the ferromagnetism in ion irradiated SiC is further confirmed by \nESR. We have measured virgin SiC, sample 1E14 and one paramagnetic SiC in a \nbroad field range. Only for sample 1E14, a broad ESR peak, and a shift  of the center  \nof the resonance at lower temperatures are clearly observed (Fig . 1b). Even at 300 K, \nthe ESR signal shows a very large shift from the free -electron position  (around 3 300 \nOe), show ing that th e sample is ferromagnetic and  has Tc above 300 K. The g-factor \nincreases slightly with decreasing temperatures.    \nB. Scrutinizing possible magnetic contaminations  \nFor the investigation of defect -induced ferromagnetism, the crucial issue is to \ncarefully examine if the ferromagnetic contamination is actuall y responsible for the Page | 5 \n observed ferromagnetism. The magnetization [Fig. 1(a)] shows a clear dependence \non the ion fluence, however the magnetic contamination s are expected to be random \nor to increase with ion fluence  if they are due to ion implantation . We further applied \nPIXE and AES to scrutinize Fe/Co/Ni elements in our studied SiC specimens.  Figure \n2 show s the PIXE spectra for virgin SiC and 5E13.  In the spectrum, the narrow peak \nis from Si K -line X-ray emission. The broad peak is due to the Bremsstrahlu ng \nbackground.  No transition metal impurities were detected within  the detection limit  of \naround  1 µg/g.   \n0 2 4 6 8 10 12 14100101102103104\n  Yield (counts)\nEnergy (keV) Virgin 6H-SiC\n 5E13Si K-edge\nFe/Co/Ni K-edge\n \nFig. 2 PIXE spectrum for the virgin SiC wafer and a ferromagnetic sample. Within the detection limit, \nwe do  not observe a ny Fe, Co or Ni contamination.  \n \nTo exclude contamination introduced during implantation and handling  of the \nsamples, we applied the surface sensitive AES method to the ferromagnetic pieces. \nFor most elements, the detection limit of AES is in the range of 0.01-0.1 at %, which is \naround 1012-1013 atom/cm2 by assuming a detection depth of 2 nm. We measured \nboth the front and back sides for virgin and ferrom agnetic SiC after Ne irradiatio n and \ndid not observe any signal from  Fe/Co/Ni  within the detection limit  (not shown ).  Page | 6 \n C. Structural properties  \nIn order to correlat e magnetization with structure , we performed XRD and RBS/C to \ncheck the crystalline deformation and defects.  \nFigure 3 depicts the θ–2θ scans at SiC(000  12) recorded for Ne ion irradiated 6H -\nSiC. A  virgin sample is also shown for comparison.  All curves show a main sharp \ndiffraction peak at the diffraction angle θ = θ Bragg coming from the sample volum \nbelow  the irradiated part. The diffraction peaks  at the low angle side of the main \nBragg peak, (θ<θ Bragg) are the characteristic s of an expansion gradient of the lattice \nalong SiC[0001] . The elastic strain in the near surface region (Δd/d) can be deduced \nfrom the position of this satellite peak. Except for the largest fluence applied here \n(1E15), a fring e pattern is clearly visible. This feature is due to the presence of the \nnon-homogeneous lattice expansion  along the depth and arises from interferences \nbetween x -ray beams diffracted from regions with the same lattice spacing . The \nmaximum value of the nor mal strain (Δd/d) M is estimated to be roughly proportional to \nthe fluence.  Furthermore, we also performed the corresponding reciprocal spacing \nmapping  (RSM) around  SiC(000  12) (see Figure 4). Only a vertical streak coming \nfrom the irradiated region of the  crystals is observed. No broadening in the horizontal \ndirection (equivalent to a rocking curve) is measured as compared with that obtained \nfrom virgin crystals. These observations indicate that the defects are randomly \ndistributed point defects or small d efect clusters  [18]. \nFor the sample irradiated with the largest fluence of 1E15 cm-2, the absence of the \nsatellite peak shows that the near -surface region of this sample is heavily damaged, \ni.e. no more crystalline. Such a signal is consistent with a stron gly defective \ncrystalline lattice characterized by the presence of extended defects : amorphous Page | 7 \n clusters or layer [ 18]. The conclusions from XRD observation are  fully consistent with \nthe observation by positron annihilation spectroscopy, which reveals the c lustering of \nvacancies  [13].  \n0.75 0.76 0.77 0.78 0.79 0.80 0.8110-310-210-1100101102103104105106107\n  Intensity (a.u.)\n1/d (1/ Å)Resolution=0.0177 °\n Virgin\n 5E13\n 1E14\n 5E14\n 1E15\n \nFig. 3 XRD 2θ/θ scans of the virgin and Ne irradiated 6H -SiC recorded in the vicinity of the (000  12) \nreflection.     \n Page | 8 \n \n \nFig. 4 Reciprocal space mapping of SiC(000  12). There is no indication of broadening along q x. The \nstreak labeled CTR is the crystal truncation ro d. The streak labeled PSD is due to the transmittance \nfunction of the position sensitive detector.  \n \nThe RBS/C spectra alo ng the SiC[0001] axis are shown in Fig. 5 for virgin and for Ne \nirradiated samples. The random spectrum is assumed equivalent to a comp letely \namorphized SiC, while the virgin spectrum corresponds to an essentially damage -\nfree crystal. The minimum yield (χmin), the ratio of the channeling  spectrum to the \nrandom spectrum, is 2.3% for the virgin SiC crystals at the surface region. For Page | 9 \n sample 5E13 (not shown) , the channeling spectrum is only slightly higher than the \nvirgin sample. A peak in the damage depth around 150 nm is becoming visible for \nfluences of 5×1014 and 1×1015 /cm2. Note that for sample  1E15  χmin is 93%, indicating \nthe almost full amorphization  at the damage peak. This is in agreement with the \nthreshold displacement per atom ( DPA) values  for amorphization  reported  in \nliterature  [19]. DPA presents the irradiation effect considering both ion fluence and \nenergy [20].  \n200 400 600 800 1000\n  \n Yield (arb. u.)a  Random\nb  Virgin\nc  1E14\nd  5E14\ne  1E15\nEnergy (keV)SiCa\ne\nd\nc\nbDepth (nm)0 100 200\n \nFig. 5: RBS/C spectra for 6H-SiC implanted with Ne ions. A random spectrum and a channeling \nspectrum from a virgin sample are also included for comparison. The dashed  (dotted)  line indicates \nthe position of the damage peaks  for Si (carbon) . A rough depth scale is given.  \n024\n    (d/d0) (%)\n04080\n(c) Magnetization\n     at 5 K(b) Disorder\n min (%)(a) Strain\n0.01 0.1 1012\n M (emu/g)\nDPA\n Page | 10 \n Fig. 6: Evolution of the saturation magnetization , the Si -sublattice disorder and the maximum  strain  \nwith DPA. The gray area indicates the DPA range for optimizing the ferromagnetism.  \nIV. Summary  \nCombining  both structural and magnetic properties  of ion irradiated SiC , we are able \nto conclude  a mutual  relation between defect concentration  and magnetization. As \nshown in Fig. 6, there is a narrow DPA window, within which the sample is  \nferromagneti c. The initial introduction of str uctural disorder  (mainly point defects)  \nleads to pronounced magnetization. Further in creasing disorder induces \nagglomeration of point defects and finally amorphizes the host SiC crystal , \nconsequently  the ferromagnetism nearly drops to zero .  \nAcknowledgemen t \nThe work was financially  supported by the Helmholtz Association  (VH-NG-713).   Y. \nWang thanks the China Scholarship Council for supporting his stay at HZDR.   \n \nReference  \n1. J. M. D. Coey, P . Stamenov, R . D. Gunning, M . Venkatesan, K Paul, New J. Phys. 12, 0 53025 \n(2010).  \n2. P. Esquinazi, D. Spemann, R. Höhne, A. Setzer, K. -H. Han, and T. Butz, Phys. Rev. Lett. 91, \n227201  (2003).  \n3. Z. He, X. Yang, H. Xia, X. Zhou, M. Zhao, Y. Song, and T. Wang, Carbon 49, 1931 (2011) . . \n4. X. Yang , Z. He, W . Li, H . Xia, Y . Song, X . Zhou, X . Liu, M . Zhao, T . Wang, and K . Hou, J. Appl. \nPhys. 109, 083933 (2011).  \n5. J. Barzola -Quiquia, W. Böhlmann, P. Esquinazi, A. Schadewitz, A. Ballestar, S. Dusari, L. \nSchultze -Nobre, and B. K erstin g, Appl. Phys. Lett. 98, 192511 (2011) . \n6. M. Venkatesan, C. B. Fitzgerald, and J. M. D. Coey, Nature 430, 630 (2004).  Page | 11 \n 7. Q. Xu, H . Schmidt, S . Zhou, K . Potzger, M . Helm, H . Hochmuth, M . Lorenz, A . Setzer, P . \nEsquinazi, C . Meinecke, and M . Grundmann, Appl.  Phys. Lett. 92, 082508 (2008).  \n8. M. Khalid, M. Ziese, A. Setzer, P. Esquinazi, M. Lorenz, H. Hochmuth, M. Grundmann, D. \nSpemann, T. Butz, G. Brauer, W. Anwand, G. Fischer, W. A. Adeagbo, W. Hergert, and A. \nErnst, Phys. Rev. B 80, 035331 (2009 ).  \n9. X. Zuo , S. Yoon, A . Yang, W . Duan, C . Vittoria, and V . Harris, J. Appl. Phys. 105, 07C508 (2009).  \n10. C. Araujo, M . Kapilashrami, X . Jun, O. D. Jayakumar, S . Nagar, Y . Wu, C . Århammar, B . \nJohansson, L . Belova, R . Ahuja, G. Gehring, and K. V. Rao, Appl. Phys. Lett. 96, 2 32505 (2010).  \n11. H. Thakur, P. Thakur, R . Kumar, N. B. Brookes, K. K. Sharma , A. P. Singh , Y. Kumar, S. Gautam , \nand K. H. Chae , Appl. Phys. Lett. 98, 192512 (2011).  \n12. C. Gómez -Polo, S. Larumbe, and J. M. Pastor, J. Appl. Phys. 113, 17B511 (2013).  \n13. L. Li, S. Pr ucnal, S. D. Yao, K. Potzger, W. Anwand, A. Wagner, and S. Zhou , Appl. Phys. Lett. \n98, 222508 (2011).  \n14. Y. Liu, G. Wang, S. Wang, J. Yang, L. Chen, X. Qin, B. Song, B. Wang, and X. Chen, Phys . Rev. \nLett. 106, 087205 (2011).  \n15. B. Song, H. Bao, H. Li, M. Lei, T.  Peng, J. Jian, J. Liu, W. Wang, W. Wang, and X. Chen, J. Am. \nChem. Soc. 131, 1376 (2009).   \n16. J. R. Jenny, D. P. Malta, S. G. Müller, A. R. Powell, V. F. Tsvetkov, H. M. D. Hobgood, R. C. \nGlass, and C. H. Carter: J. Electron. Mater. 32 , 432 (2003).   \n17. P. Esqu inazi, J. Barzola -Quiquia, D. Spemann, M. Rothermel, H. Ohldag, N. Garc´ıa, A. Setzer, \nT. Butz, J. Magn. Magn. Mater. 322 , 1156  (2010) . \n18. A. Debelle, L. Thome,  D. Dompoint, A. Boulle, F. Garrido, J. Jagielski and D. Chaussende , J. \nPhys. D: Appl. Phys. 43 , 455408  (2010) . \n19. W.J. Weber, L.M. Wang, N. Yu, and N.J. Hess, Mater. Sci. Eng. A 253, 62 (1998).  \n20. J. Ziegler, J. Biersack, and U. Littmark, The Stopping and Range of Ions in Matter  (Pergamon, \nNew York, 1985).  Page | 12 \n Fig. captions  \nFig. 1: (a) Magnetization vs. field m easured at 300 K for all samples: the diamagnetic background from \nthe substrate has been subtracted. The magnetization was calculated by assum ing a thin layer of 460 \nnm thickness. The inset shows the magnetization measured for the virgin sample (black line ) and \nsample 5E13 without subtracting the background. (b) ESR spectra of sample 1E14 at temperatures \nbetween 100 and 300 K: a broad resonance peak appears indicating ferromagnetic resonance. The \nblack line indicates the resonance centers which slightly cha nge with temperature.  \nFig. 2 PIXE spectrum for the virgin SiC wafer and a ferromagnetic sample. Within the detection limit, \nwe do  not observe any Fe, Co or Ni contamination.  \nFig. 3 XRD 2θ/θ scans of the virgin and Ne irradiated 6H -SiC recorded in the vici nity of the (000  12) \nreflection.     \nFig. 4 Reciprocal space mapping of SiC(000  12). There is no indication of broadening along q x. The \nstreak labeled CTR is the crystal truncation rod. The streak labeled PSD is due to the transmittance \nfunction of the pos ition sensitive detector.  \nFig. 5: RBS/C spectra for 6H -SiC implanted with Ne ions. A random spectrum and a channeling \nspectrum from a virgin sample are also included for comparison. The dashed  (dotted)  line indicates \nthe position of the damage peaks  for Si  (carbon) . A rough depth scale is given.  \nFig. 6: Evolution of the saturation magnetization, the Si -sublattice disorder and the maximum strain \nwith DPA. The gray area indicates the DPA range for optimizing the ferromagnetism.  \n "    },    {        "title": "1608.03359v2.Magnetoresistance_generated_from_charge_spin_conversion_by_anomalous_Hall_effect_in_metallic_ferromagnetic_nonmagnetic_bilayers.pdf",        "content": "arXiv:1608.03359v2  [cond-mat.mes-hall]  23 Nov 2016Magnetoresistance generated from charge-spin conversion by anomalous Hall effect in\nmetallic ferromagnetic/nonmagnetic bilayers\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan\nA theoretical formulation of magnetoresistance effect in a m etallic ferromagnetic/nonmagnetic bi-\nlayeroriginated fromthecharge-spinconversionbytheano malous Halleffectispresented. Analytical\nexpressions of the longitudinal and transverse resistivit ies in both nonmagnet and ferromagnet are\nobtained by solving the spin diffusion equation. The magneto resistance generated from charge-spin\nconversion purely caused by the anomalous Hall effect in the f erromagnet is found to be proportional\nto the square of the spin polarizations in the ferromagnet an d has fixed sign. We also find additional\nmagnetoresistances in both nonmagnet and ferromagnet aris ing from the mixing of the spin Hall\nand anomalous Hall effects. The sign of this mixing resistanc e depends on those of the spin Hall\nangle in the nonmagnet and the spin polarizations of the ferr omagnet.\nPACS numbers: 85.75.-d, 72.25.Ba, 73.43.Qt, 72.25.Mk\nI. INTRODUCTION\nIt was recently found that the resistance of a bilayer\nconsisting of an insulating ferromagnet and a metallic\nnonmagnet depends on the magnetization direction even\nthough no current flows in the ferromagnet [1–5]. This\nnew type of magnetoresistance effect, called spin Hall\nmagnetoresistance, was explained theoretically by using\nthe diffusive spin transport theory in the nonmagnet [6],\ndespite the existence ofother contributionsbeing implied\n[7–9]. The key idea of the spin Hall magnetoresistance\nwasthe charge-spinconversion[10–17]causedbythe spin\nHall effect [18–20] originating from the spin-orbit inter-\naction in nonmagnetic heavy metals. The spin Hall mag-\nnetoresistance was observed also in a metallic ferromag-\nnetic/nonmagnetic bilayer [21–24].\nA metallic ferromagnet shows the anomalous Hall ef-\nfect [25] generating another electric voltage in the di-\nrection perpendicular to both the magnetization and an\nexternal electric field. Note that the transverse electric\ncurrent generated by this Hall voltage is spin-polarized\nbecauseofthespin-dependenttransportpropertiesinfer-\nromagnets [26,27]. In other words, the anomalous Hall\neffect generates the electric and spin currents simultane-\nously. Then, a new magnetoresistance effect that origi-\nnates from the charge-spin conversion by the anomalous\nHall effect, in addition to this conventional Hall voltage,\nis expected, as in the case of the spin Hall magnetore-\nsistance. This new type of magnetoresistance effect can\nbe distinguished from the conventional anomalous Hall\neffect because the former is characterized by the physical\nquantities related to the spin-dependent transport in a\nferromagnet.\nIn this paper, a theoretical formulation is given for\nthe magnetoresistance effect generated from the charge-\nspin conversion caused by the anomalous Hall effect in a\nferromagnetic/nonmagneticbilayer. Solving the spin dif-\nfusion equation and using the spin-dependent Landauer\nformula, analytical expressions of the transverse resistiv-F\nNdF\ndNxyzmV\nFIG. 1: Schematic view of the system under consideration.\nAn external voltage is applied along the xdirection, gener-\nating the external electric field Ex. Magnetoresistance in the\ntransversedirectionismeasuredfrom theelectric voltage flow-\ning along the ydirection in the nonmagnet or ferromagnet,\nwhereas thelongitudinal one is detectedfrom theelectric v olt-\nage along the xdirection.\nities in both nonmagnet and ferromagnet are obtained,\nboth of which have angular dependence that is the same\naswith the planarHall effect. One ofthesemagnetoresis-\ntances originates from the charge-spin conversion caused\npurely by the anomalous Hall effect in the ferromagnet.\nThiscontributionhasfixedsignbecauseitisproportional\ntothe squareofthe spinpolarizationsofthe ferromagnet.\nAnother magnetoresistance effect arises from the mixing\nof the spin Hall and anomalous Hall effects. The sign of\nthis term depends on those of the spin Hall angle in the\nnonmagnet and the spin polarizations of the ferromag-\nnet. The longitudinal resistivity is also investigated and\nfound to also has the mixing term.\nThe paper is organized as follows. The electric and\nspin currents, as well as the spin accumulation, in a fer-\nromagnetic/nonmagneticbilayer are calculated in Sec. II\nby solving the spin diffusion equation. The magnetore-\nsistance effect due to the charge-spin conversion by the\nspin Hall and anomalous Hall effects is studied in Sec.\nIII. The conclusion is summarized in Sec. IV.2\nII. CHARGE AND SPIN TRANSPORT IN A\nFERROMAGNETIC/NONMAGNETIC BILAYER\nIn this section, we derive the analytical formulas of the\nelectric and spin currents, as well as the spin accumula-\ntion, in a ferromagnetic/nonmagnetic bilayer.\nA. Electric current\nThe system we consider is schematically shown in Fig.\n1, where the ferromagnet (F) and nonmagnet (N) are at-\ntached at the xy-plane,z= 0. In the following, we use\nthe subscripts F and N to distinguish between the quan-\ntities related to the ferromagnet and nonmagnet. The\nthicknesses of the ferromagnet and nonmagnet along the\nz-direction are denoted as dFanddN, respectively. The\nunit vectorpointing in the magnetization direction of the\nferromagnet is m. An external voltage is applied along\nthex-direction. The electric current density in the non-\nmagnet is driven by the external electric field Exand the\ninverse spin Hall effect, and is given by [12]\nJci,N=σN\ne∂i¯µN−ϑσN\neǫijα∂jδµN,α,(1)\nwherei=x,y,zrepresents the spatial direction of the\ncurrent flow, whereas αis used to represent the direction\nof the spin polarization. The spin-polarization direction\nof the spin accumulation in the nonmagnet is determined\nby the boundary condition of the spin current at the fer-\nromagnetic/nonmagnetic interface, as discussed below.\nThe conductivity and spin Hall angle of the nonmag-\nnet areσNandϑ, respectively. The elementary charge\nise=|e|. The Levi-Civita asymmetric tensor is ǫijk\n(ǫ123= +1). The electrochemical potential ¯ µand spin\naccumulation δµare defined using the spin-dependent\nelectrochemical potential ¯ µν(ν=↑,↓) as ¯µ= (¯µ↑+¯µ↓)/2\nandδµ= (¯µ↑−¯µ↓)/2, respectively. The electrochemi-\ncal potential in the nonmagnet is related to the external\nelectric field as ¯ µN=eExx[12]. On the other hand, the\nspin accumulation of the nonmagnet varies along the z\ndirection. The electric current density along the zdi-\nrection then becomes zero, guaranteeing the open circuit\nboundary condition. The electric current densities in the\nnonmagnet along the xandydirections are explicitly\ngiven by\nJcx,N=σNEx+ϑσN\ne∂zδµN,y, (2)\nJcy,N=−ϑσN\ne∂zδµN,x. (3)\nOn the other hand, the electric current density in the\nferromagnet is given by [27]\nJci,F=σF\ne∂i¯µF+σAH\neǫiαkmα∂k¯µF\n+βσF\ne∂iδµF+ζσAH\neǫiαkmα∂kδµF,(4)whereσAHis the conductivity of the anomalous Hall ef-\nfect. The spin polarizations of σFandσAHare denoted\nasβandζ, respectively [27]. The second and fourth\nterms in Eq. (4) indicate that the anomalous Hall ef-\nfect generates an electric current perpendicular to both\nthe external current and magnetization. We assume that\nthe penetration depth of the transverse spin current in\nthe ferromagnet is sufficiently short due to the large ex-\nchange interaction between the conduction electrons and\nthe magnetization, for simplicity [28–33]. The spin po-\nlarizations of the spin current and accumulation in the\nferromagnet then become parallel to the magnetization\nm. Using the conservation law of the electric current,\n∂iJci,F= 0, and applying the open-circuit boundary con-\ndition along the zdirection, Jcz,F= 0, we find that the\nelectrochemical potential and spin accumulation in the\nferromagnet are related as\n¯µF=−βδµF+eExx+σAH\nσFmyeExz,(5)\nwhere the second term is the conventional electric po-\ntential, whereas the third term arises from an internal\nelectric field under the open-circuit boundary condition.\nThe electric current densities in the ferromagnet along\nthexandydirections are\nJcx,F=σF/bracketleftBigg\n1+/parenleftbiggσAH\nσFmy/parenrightbigg2/bracketrightBigg\nEx\n−(β−ζ)σAH\nemy∂zδµF,(6)\nJcy,F=σAH/bracketleftbigg\nmz−/parenleftbiggσAH\nσF/parenrightbigg\nmxmy/bracketrightbigg\nEx\n+(β−ζ)σAH\nemx∂zδµF.(7)\nThelasttermsin Eqs. (6)and (7) representthe contribu-\ntions to the longitudinal and transverse electric currents\nfrom the charge-spin conversion caused by the anoma-\nlous Hall effect. In the following, we will calculate the\nmagnetoresistance effect due to these terms.\nB. Spin current and spin accumulation\nThe spin accumulation in the nonmagnet obeys the\ndiffusion equation given by [12,34]\n∂2\nzδµN,α=δµN,α\nℓ2\nN, (8)\nwhereℓNis the spin diffusion length of the nonmagnet.\nThe boundary conditions of Eq. (8) is given by the spin\ncurrent density at the boundary, where the spin current\ndensity is related to the spin accumulation via [12]\nJsiα,N=−/planckover2pi1σN\n2e2∂iδµN,α−/planckover2pi1ϑσN\n2e2ǫiαk∂k¯µN.(9)3\nTo simplify the notation, we introduce the unit vectors\neαrepresenting the direction of the spin polarization,\nand define δµN=δµN,αeαandJsz,N=Jszα,Neα. The\nopen-circuit boundary condition along the zdirection is\nJsz,N=0atz=−dN. On the other hand, we denote the\nspin current density at the ferromagnetic/nonmagnetic\ninterface ( z= 0) asJF/N\nsz. The spin accumulation in the\nnonmagnet is then obtained from Eq. (8) as\nδµN=2e2ℓN\n/planckover2pi1σNsinh(dN/ℓN)/bracketleftbigg\n−/planckover2pi1ϑσN\n2eExcosh/parenleftbiggz\nℓN/parenrightbigg\ney\n−/parenleftbigg\nJF/N\nsz−/planckover2pi1ϑσN\n2eExey/parenrightbigg\ncosh/parenleftbiggz+dN\nℓN/parenrightbigg/bracketrightbigg\n(10)\nThe spin accumulation in the ferromagnet also obeys the\ndiffusion equation with the spin diffusion length ℓF. Note\nthatthe spincurrentdensityinthe ferromagnetisrelated\nto the spin accumulation as [27]\nJsi,F=−/planckover2pi1σF\n2e2∂iδµF−/planckover2pi1σAH\n2e2ǫiαkmα∂kδµF\n−/planckover2pi1βσF\n2e2∂i¯µF−/planckover2pi1ζσAH\n2e2ǫiαkmα∂k¯µF.(11)\nSimilar to the nonmagnet, we define δµF=δµFmwhere\nthe vector notation in boldface represents the direction\nof the spin polarization, which in the ferromagnet is as-\nsumed to be parallel to the magnetization. Then, the\nspin accumulation in the ferromagnet is given by\nδµF=2e2ℓF\n/planckover2pi1σ∗\nFsinh(dF/ℓF)/braceleftbigg\n−/planckover2pi1(β−ζ)σAHmy\n2eExcosh/parenleftbiggz\nℓF/parenrightbigg\n+/bracketleftbigg\nm·JF/N\nsz+/planckover2pi1(β−ζ)σAHmy\n2eEx/bracketrightbigg\ncosh/parenleftbiggz−dF\nℓF/parenrightbigg/bracerightbigg\n,\n(12)\nwhereσ∗\nF= (1−β2)σF.\nC. Spin current at ferromagnetic/nonmagnetic\ninterface\nThespin currentatthe ferromagnetic/nonmagneticin-\nterface is determined by the spin-dependent Landauer\nformula given by [35]\nJF/N\nsz=−1\n2πS/bracketleftbigg(1−γ2)g\n2m·(δµF−δµN)m\n−grm×(δµN×m)−giδµN×m],(13)\nwhereg=g↑↑+g↓↓is the sum of the dimensionless in-\nterface conductances of spin-up and spin-down electrons,\nandγ= (g↑↑−g↓↓)/gis its spin polarization. The con-\nductance gis related to the interface resistance rvia\nr= (h/e2)S/g, whereSis the cross-section area of the\ninterface. The real and imaginary parts of the mixingconductance are denoted as grandgi, respectively. Sub-\nstituting Eqs. (10) and (12) at z= 0 into Eq. (13), we\nfind that Eq. (13) is rewritten as\nJF/N\nsz=−/planckover2pi1g∗my\n2e/bracketleftbigg(β−ζ)σAH\ngFtanh/parenleftbiggdF\n2ℓF/parenrightbigg\n−ϑσN\ngNtanh/parenleftbiggdN\n2ℓN/parenrightbigg/bracketrightbigg\nExm\n+/planckover2pi1ϑσN\n2etanh/parenleftbiggdN\n2ℓN/parenrightbigg\nEx[m×(ey×m)Re\n+ey×mIm]gr+igi\ngN+(gr+igi)coth(dN/ℓN).\n(14)\nHere, we define gF/S=hσ∗\nF/(2e2ℓF),gN/S=\nhσN/(2e2ℓN), and\n1\ng∗=2\n(1−γ2)g+1\ngFtanh(dF/ℓF)+1\ngNtanh(dN/ℓN).\n(15)\nSubstituting Eq. (14) into Eqs. (10) and (12), the longi-\ntudinal and transverse electric current densities are cal-\nculated from Eqs. (2), (3), (6), and (7).\nIII. MAGNETORESISTANCE EFFECT DUE TO\nCHARGE-SPIN CONVERSION\nIn this section, we study the magnetoresistance effect\noriginated from the charge-spin conversion by the spin\nHall and anomalous Hall effects. We first focus on the\ntransverse resistivity because the transverse voltage has\nbeen usually measured in the experiments of the anoma-\nlous Hall effect [25]. We also study the longitudinal re-\nsistivity for generality because the longitudinal voltage\nhas been measured in the experiments of the spin Hall\nmagnetoresistance [3].\nA. Transverse resistivity\nWe define the averaged transverse electric current\ndensity in the nonmagnet from Eq. (3) as Jcy,N=\n1\ndN/integraltext0\n−dNJcy,Ndz. Then, the transverse resistivity in the\nnonmagnet is defined as [6]\nρT\nN≡ −Jcy,N/Ex\nσ2\nN= ∆ρN,ymxmy+∆ρ′\nN,ymz,(16)\nwhere ∆ρN,yand ∆ρ′\nN,ywithρN= 1/σNare\n∆ρN,y\nρN=ℓN\ndN/braceleftbigg\nϑ2/bracketleftbigg\nRegr+igi\ngN+(gr+igi)coth(dN/ℓN)−g∗\ngN/bracketrightbigg\ntanh/parenleftbiggdN\n2ℓN/parenrightbigg\n+ϑ(β−ζ)σAHg∗\nσNgFtanh/parenleftbiggdF\n2ℓF/parenrightbigg/bracerightbigg\ntanh/parenleftbiggdN\n2ℓN/parenrightbigg\n.\n(17)4\n∆ρ′\nN,y\nρN=−ϑ2ℓN\ndNImgr+igi\ngN+(gr+igi)coth(dN/ℓN)tanh2/parenleftbiggdN\n2ℓN/parenrightbigg\n.\n(18)\nIn the following, we neglect terms related to giby assum-\ninggr≫gi, for simplicity [36]. Similarly, we define the\naveraged transverse electric current density in the ferro-\nmagnet from Eq. (7) as Jcy,F=1\ndF/integraltextdF\n0Jcy,Fdz. Then,\nthe transverse resistivity in the ferromagnet is\nρT\nF≡ −Jcy,F/Ex\nσ2\nF=−ρF/parenleftbiggσAH\nσF/parenrightbigg\nmz+ρF/parenleftbiggσAH\nσF/parenrightbigg2\nmxmy\n+∆ρF,ymxmy,\n(19)\nwhere ∆ρF,ywithρF= 1/σFis\n∆ρF,y\nρF=ℓF\n(1−β2)dF/braceleftBigg/bracketleftbigg(β−ζ)σAH\nσF/bracketrightbigg2/bracketleftbigg\n2−g∗\ngFtanh/parenleftbiggdF\n2ℓF/parenrightbigg/bracketrightbigg\n+ϑ(β−ζ)σNσAHg∗\nσ2\nFgNtanh/parenleftbiggdN\n2ℓN/parenrightbigg/bracerightbigg\ntanh/parenleftbiggdF\n2ℓF/parenrightbigg\n.\n(20)\nThe first term in Eq. (19) is the transverseresistivitydue\nto the conventional anomalous Hall effect, whereas the\nsecond term is originated from the internal electric field\ndue to the open-circuit condition of the electric current\nalong the zdirection. On the other hand, the third term\nor, equivalently, Eq. (20), represents the contribution\nfrom the charge-spinconversioncausedby the anomalous\nHall effect.\nEquations(17)and(20)indicatethatthetransversere-\nsistivity shows the angular dependence that is the same\nas with the planar Hall effect in ferromagnets [37]. How-\never, the dependences of Eqs. (17) and (20) on the thick-\nnessesdNanddFare scaled by the spin diffusion lengths,\nand thus these effects can be separated from the con-\nventional planar Hall effect by measuring the thickness\ndependence of total transverse resistivity. The terms in\nEqs. (17) and (20) are classified into three contributions.\nThe first one is the spin Hall magnetoresistance [6] de-\nscribed by the first term of Eq. (17). This term arises\npurely from the spin Hall effect, and is finite even when\nthe ferromagnet does not show the anomalous Hall ef-\nfect. The second contribution is the first term of Eq.\n(20) originating from the charge-spin conversion purely\ncaused by the anomalous Hall effect. This term is finite\neven when the spin Hall effect is absent. These first and\nsecond contributions have fixed signs because these are\nproportional to the square of the spin Hall angle ϑand\nthe spin polarization β−ζ, respectively. On the other\nhand, the third contribution corresponds to the second\nterms of Eqs. (17) and (20), which originate from the\ncharge-spin conversion generated by the mixing of the\nspin Hall and anomalous Hall effects. In other words,\nthe spin current generated by the anomalous Hall effect\nis converted to the electric current by the inverse spin\nHall effect or vice versa. The sign of the third contribu-\ntion depends on those of ϑandβ−ζ.Although all terms in Eqs. (17) and (20) have the\nangular dependence that is the same as with the planar\nHall effect, their origins are different. The angular de-\npendence of the spin Hall magnetoresistance arises from\nthe absorption of the spin current having the spin polar-\nization perpendicular to the magnetization at the ferro-\nmagnetic/nonmagnetic interface [6]. Thus, the amount\nof the spin Hall magnetoresistance depends on the mix-\ning conductance. On the other hand, the magnetoresis-\ntance due to the anomalous Hall effect arises from the\nspin current polarized along the direction parallel to the\nmagnetization, and thus, is independent of the mixing\nconductance. Let us explain the origins of the magne-\ntoresistances in Eqs. (17) and (20), except the spin Hall\nmagnetoresistance, with the help of schematic views in\nFig. 2. Figure 2(a) schematically shows the generation\nof the transverse electric current by the charge-spin con-\nversion purely caused by the anomalous Hall effect. The\nanomalous Hall effect generates the spin current in the\ndirection m×Exex. Therefore, the spin current flows\nin thezdirection when the magnetization has a finite\nycomponent my. The anomalous Hall effect then con-\nverts this spin current to the electric current in the di-\nrection perpendicular to both the spin current and the\nmagnetization, i.e., m×(m×Exex) direction. Thus,\nwhen the xcomponent of the magnetization mxis also\nfinite, the electric current along the ydirection is gen-\nerated in the ferromagnet, as shown in Fig. 2(a). This\nterm corresponds to the first term of Eq. (20). The\nspin current generated by the anomalous Hall effect is\nalso converted to the electric current in the nonmagnet\nby the inverse spin Hall effect, as schematically shown in\nFig. 2(b), which corresponds to the second term in Eq.\n(17). On the other hand, Fig. 2(c) schematically shows\nthe conversion of the spin current generated in the non-\nmagnet to the electric current in the ferromagnet. The\nspin Hall effect generates the spin current in the zdirec-\ntion, whose polarization points to the ydirection. This\nspin current can be injected into the ferromagnet when\nmyis finite because only the spin currentpolarized in the\nmagnetization direction can survive inside the ferromag-\nnet. Then, the anomalous Hall effect converts this spin\ncurrent into the electric current flowing in the direction,\nm×(ey×Exex)∝m×ez. Therefore, the electriccurrent\nin theydirection is generated when mxis finite, which\ncorrespondsto the second term ofEq. (20). These physi-\ncalpictures indicate that both the xandycomponentsof\nthe magnetization should be finite to generate the trans-\nverse electric current. Therefore, the magnetoresistances\nin Eqs. (17) and (20) show the angular dependence of\nmxmy.\nFigure 3(a) shows the dependence of the transverse\nresistivity in the nonmagnet, given by Eq. (17), on\nits thickness dNby the solid line. A factor 1 /[1 +\n(ρNdF)/(ρFdN)] is multiplied to Eq. (17) in this figure\nto account for the shunt effect [24]. The values of the\nparameters are taken from typical experiments and first-\nprinciples calculations for CoFeB, NiFe, W, and so on5\nEx\nspin current by AHE \nin m×Ex direction\nelectric current by AHE \nin m×( m×Ex) directionm m F FEx\nN N\nelectric current by inverse SHE \nin m×( m×Ex) directionspin current by AHE \nin m×Ex direction(a) (b)\nm FEx\nNelectric current by AHE \nin m×( ey×Ex) direction\nspin current by SHE \nin ey×Ex direction(c)\nFIG. 2: Schematic pictures of the generation of the transver se (y) electric currents by the charge-spin conversions. (a) The\nanomalous Hall effect (AHE) generates the spin current in the m×Exexdirection, and converts the spin current into the electric\ncurrent in the m×(m×Exex) direction. The nonmagnet does not play any role in this situ ation, and thus is represented by\nthe dashed line. (b) The spin current generated by the anomal ous Hall effect is converted to the electric current by the inv erse\nspin Hall effect (SHE) in the nonmagnet. (c) The spin current g enerated by the spin Hall effect is injected into the ferromag net,\nand is converted to the electric current by the anomalous Hal l effect.\nthickness of nonmagnet, dN (nm)0\n-0.10.1\n0 2 4 6 8 10 12 14 16 18 20 ΔρN, y/ρN (10 -3 )\ntotal magnetoresistancespin Hall effect\nmixing term(a)\nthickness of ferromagnet, dF (nm)0\n-0.10.10.20.3\n0 2 4 6 8 10 12 14 16 18 20 ΔρF, y/ρF (10 -3 )\ntotal magnetoresistanceanomalous Hall effect\nmixing term(b)\n0.20.30.4\nFIG. 3: (a) Dependence of the transverse resistivity in the n onmagnet normalized by ρN, given by Eq. (17), on the thickness dN\n(solid line). The dotted (red) and dashed (blue) lines repre sent the contribution from the spin Hall magnetoresistance and the\nmixing term of the spin Hall and anomalous Hall effect, which c orrespond to the first and second term of Eq. (17), respective ly.\n(b) Dependence of the transverse resistivity in the ferroma gnet normalized by ρF, given by Eq. (20), on the thickness dF(solid\nline). Similar to (a), the first and second terms in Eq. (20) ar e shown by the dotted (red) and dashed (blue) lines, respecti vely.\nThe factors considering the shunt effect are multiplied to bo th Eqs. (17) and (20).\n[24,27,36,38–41]; ρF= 300 Ωnm, ℓF= 5 nm, β= 0.75,\nr= 0.25 kΩnm2,γ= 0.50,σAH/σF= 0.015,ζ= 1.5,\nρN= 1500 Ωnm, ℓN= 1 nm, ϑ= 0.1, andgr/S= 15\nnm−2. The thickness of the ferromagnet is fixed to\ndF= 2 nm. The first and second terms of Eq. (17) are\nalso shown by the dotted and dashed lines, respectively.\nThe transverse resistivity in the ferromagnet, given by\nEq. (20), multiplied by the factor 1 /[1+(ρFdN)/(ρNdF)],\nas a function of the thickness dFis also shown in Fig.\n3(b) by a solid line, where dNis fixed to 2 nm. These\nfigures indicate that the magnetoresistance effect on the\nsame order of the spin Hall magnetoresistance can be\nexpected due to the charge-spin conversion caused by\nthe anomalous Hall effect. It was recently shown that\nthe anisotropicmagnetoresistanceand the planarHall ef-\nfect in the W/CoFeB/MgO heterostructure is one order\nof magnitude smaller than the spin Hall magnetoresis-\ntance [23]. Therefore, the magnetoresistance effect stud-\nied in the present work is considered to be measurable,\nalthough it has the same angular dependence with theplanar Hall effect.\nB. Longitudinal resistivity\nThe longitudinal resistivity in the nonmagnet is de-\nfined from Eq. (2) as\nρL\nN≡/parenleftbiggJcx,N\nEx/parenrightbigg−1\n≃ρN+∆ρ(0)\nN,x+∆ρ(1)\nN,x/parenleftbig\n1−m2\ny/parenrightbig\n+∆ρ(2)\nN,xm2\ny,\n(21)\nwhere ∆ρ(k)\nN,x(k= 1,2,3) are, respectively, given by\n∆ρ(0)\nN,x\nρN=−2ϑ2ℓN\ndNtanh/parenleftbiggdN\n2ℓN/parenrightbigg\n, (22)\n∆ρ(1)\nN,x\nρN=ϑ2ℓN\ndNRegr+igi\ngN+(gr+igi)coth(dN/ℓN)tanh2/parenleftbiggdN\n2ℓN/parenrightbigg\n,\n(23)6\n∆ρ(2)\nN,x\nρN=ℓN\ndN/bracketleftbiggϑ2g∗\ngNtanh/parenleftbiggdN\n2ℓN/parenrightbigg\n−ϑ(β−ζ)g∗σAH\ngFσNtanh/parenleftbiggdF\n2ℓF/parenrightbigg/bracketrightbigg\ntanh/parenleftbiggdN\n2ℓN/parenrightbigg\n.\n(24)\nWe note that ∆ ρ(0)\nN,xand ∆ρ(1)\nN,xcorrespond to the spin\nHall magnetoresistance derived for a ferromagnetic in-\nsulator/nonmagnetic metal bilayer [6]. The first term\nin ∆ρ(2)\nN,xis originated from the charge-spin conversion\npurely by the spin Hall effect, and is finite when the fer-\nromagnet is metallic. On the other hand, the second\nterm in ∆ ρ(2)\nN,xis originated from the spin current gener-\nated by the anomalous Hall effect, which is converted to\nthe electric current in the nonmagnet by the inverse spin\nHall effect.\nSimilarly, the longitudinal resistivity in the ferromag-\nnet is defined from Eq. (6) as\nρL\nF≡/parenleftbiggJcx,F\nEx/parenrightbigg−1\n≃ρF−ρF/parenleftbiggσAH\nσF/parenrightbigg2\nm2\ny+∆ρF,xm2\ny,\n(25)\nwhere ∆ρF,xis given by\n∆ρF,x\nρF=−ℓF\n(1−β2)dF/braceleftBigg/bracketleftbigg(β−ζ)σAH\nσF/bracketrightbigg2/bracketleftbigg\n2−g∗\ngFtanh/parenleftbiggdF\n2ℓF/parenrightbigg/bracketrightbigg\n+ϑ(β−ζ)σNσAHg∗\nσ2\nFgNtanh/parenleftbiggdN\n2ℓN/parenrightbigg/bracerightbigg\ntanh/parenleftbiggdF\n2ℓF/parenrightbigg\n.\n(26)\nAs in the case of the transverse resistivity, Eq. (26) has\ntwo terms, where the first term arises from the charge-\nspin conversion purely caused by the anomalous Hall ef-\nfect [42], whereas the second term comes from the spin\ncurrent generated by the spin Hall effect in the nonmag-\nnet.\nWe note that the magnitudes of the transverse and\nlongitudinal resistivities in the ferromagnet that origi-\nnate from the charge-spin conversion, given by Eqs. (20)\nand (26), are equivalent. This is because both the lon-\ngitudinal and transverse currents due to this conversion,\ngiven by the last terms in Eqs. (6) and (7), respectively,\ndepend on the same term ( ∝∂zδµF), except the angulardependence. Therefore, the dependence of Eq. (26) on\nthe ferromagnetic thickness is the same with Fig. 3(b),\nexcept the sign difference. On the other hand, the longi-\ntudinal and transverse currents in the nonmagnet given\nby Eqs. (2) and (3) depend on different components of\nδµN,α. Thus, the magnitudes of the longitudinal and\ntransverseresistivities are, in general, different, as can be\nseen in Eqs. (17) and (21). However, when we compare\nthe longitudinal resistivities for my= 0 and my=±1,\nas done in the experiments [24], we found that\nρL\nN(my= 0)−ρL\nN(my=±1) = ∆ρ(1)\nN,x−∆ρ(2)\nN,x,(27)\nwhich is identical to the transverse resistivity, given by\nEq. (17), and therefore its thickness dependence is given\nby Fig. 3(a).\nIV. CONCLUSION\nIn conclusion, the magnetoresistance effect in a metal-\nlic ferromagnetic/nonmagneticbilayer due to the charge-\nspin conversion by the anomalous Hall effect is predicted\ntheoretically. The magnetoresistance generated from\ncharge-spin conversion purely caused by anomalous Hall\neffect in the ferromagnet is proportional to the square of\nthe spin polarizations in the ferromagnet and has a fixed\nsign. 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SPIE\n9931, 99310W (2016)."    },    {        "title": "0907.2415v1.Localized_ferromagnetic_resonance_force_microscopy_in_permalloy_cobalt_films.pdf",        "content": "arXiv:0907.2415v1  [cond-mat.mtrl-sci]  14 Jul 2009Localized ferromagnetic resonance force microscopy in\npermalloy-cobalt films\nE. Nazaretski, I. Martin, K. C. Cha, E. A. Akhadov, and R. Movsho vich\nLos Alamos National Laboratory, Los Alamos, NM 87545\nYu. Obukhov, D. V. Pelekhov, and P. C. Hammel\nDepartment of Physics, Ohio State University, Columbus OH 4 3210,\nAbstract\nWe report Ferromagnetic Resonance Force Microscopy (FMRFM ) experiments on a justaposed\ncontinuous films of permalloy and cobalt. Our studies demons trate the capability of FMRFM\nto perform local spectroscopy of different ferromagnetic mat erials. Theoretical analysis of the\nuniform resonance mode near the edge of the film agrees quanti tatively with experimental data.\nOur experiments demonstrate the micron scale lateral resol ution in determining local magnetic\nproperties in continuous ferromagnetic samples.\nPACS numbers: 07.79.Pk, 07.55.-w, 76.50.+g, 75.70.-i\n1Magnetic resonance force microscopy (MRFM) is attracting increa sing attention as a re-\nsult of its high spin sensitivity and excellent spatial resolution in param agnetic and nuclear\nspin systems.[1, 2, 3, 4, 5, 6] MRFM studies on microfabricated and c ontinuous ferromag-\nnetic samples have been also performed. [7, 8, 9, 10, 11] Here we re port FMRFM experi-\nments performed on a non-overlapping permalloy (Py) and cobalt (C o) continuous films and\ndemonstrate the capability of FMRFM to spectroscopically identify t he distinct magnetic\nproperties of two adjacent ferromagnetic films. We quantitatively model the resulting force\nsignal strength and compare it with the experimental data.\nThe permalloy-cobalt sample is schematically shown in Fig. 1. A 20 nm thic k Ti film\nwas uniformly applied onto the surface of a 100 µm thick Si (100) wafer. 20 nm of Co\nwas deposited into a rectangular area (2.5 ×5 mm) defined in photoresist followed by\nthe lift-off. A complimentary rectangular area of 20 nm thick Py was s imilarly defined\nand deposited. The entire structure was then coated with a 20 nm t hick layer of Ti. The\ninterface between the Co and Py regions was examined in a scanning e lectron microscope\n(SEM) and revealed a gap whose width varies between 3 and 6 µm along the entire length of\nthe sample (see SEM image in Fig. 1). An approximately 1.7 ×1.7 mm2piece was cut and\nglued to the stripline resonator of the FMRFM apparatus and the film plane was oriented\nperpendicular to the direction of the external magnetic field Hext. For FMRFM studies\nwe used the cantilever with the spherical magnetic tip (see SEM image in Fig. 1) and its\nspatial field profile has been carefully characterized [12]. More det ails on the experimental\napparatus can be found in Ref. [13]\nIn Fig. 2 we show the evolution of the FMRFM signal as a function of th e lateral position\nand applied magnetic field. The cantilever was scanned across the int erface between Co and\nPy, in the region indicated by arrows in Fig. 1. The FMRFM signal was re corded in two\ndifferent regions of Hextwhich correspond to Py and Co resonance fields for the microwave\nfrequency of fRF=9.35 GHz. Insets in Fig. 2 show the evolution of the FMRFM spectra a s\na function of lateral position. The signal, reminiscent of those repo rted earlier in [14], is\ncomprised of two distinctive contributions. The first, a negative sig nal which occurs at lower\nvalues of Hextis a localized resonance originating from the region of the sample right under\nthe cantilever tip where the probe field is strong and positive. The se cond contribution is\npositive and is observed at higher values of Hext. This signal arises from a larger region of\n2sample remote from the tip which, therefore, experience a weak ne gative tip field; we will\nlabel this the “uniform resonance”. As seen in Fig. 2, at the beginnin g of the lateral scan\nthe negative (lower field) resonance structure is present only in th e Co spectrum (see inset\na)). Near a lateral position of 9 µm we see no localized signals (with negatively shifted Hext)\nfor either the Py or the Co signals. However upon scanning further over the Py film, the\nPy resonance begins to show a localized signal, while the Co signal cont inues to show only\na uniform (positively shifted Hext) signal (inset b)).\nWe analyze the uniform contribution to the FMRFM signals considering the case when\nthe entire dynamic magnetization mis constant and the resonance field is only weakly\naffected by the probe. This approximation is valid for the large probe -sample distances\n(insets b) and c) in Fig. 2). The frequency of the uniform resonanc e in a thin film can be\nwritten as ωRF/γ=Hext−4πMs, where 4 πMsis the saturation magnetization and γis the\ngyromagnetic ratio. FMRFM spectra shown in b) and c) insets in Fig. 2 yield the values of\n4πMs= 8052 G for Py and 4 πMs= 15013 G for Co respectively.\nForquantitative analysisoftheFMRFMdataitisimportanttohavean accurateestimate\nof the probe-sample separation. Magnetic Force Microscopy (MFM ) measurements were\nused to calibrate the probe-sample separation. The cantilever was scanned across the Py\n- Co interface and changes in its resonance frequency were recor ded. The gradient of the\nMFM force for a semi-infinite film can be written as follows:\n∂F\n∂z= 4mpMsLx(x2−3z2)\n(x2+z2)3, (1)\nwheremp= 7×10−9emu is the probe magnetic moment [12] and Lis the film thickness. z\nis the probe-film distance and xis the lateral position with respect to the film edge ( x≥0).\nMFM data were acquired at H ext= 18255 G, thus, both films were saturated. The MFM\ndata and the fit to Eq. 1 are shown in Fig. 3a, yielding the tip-sample se paration z≈4.4\nµm and the films boundaries ( x≤8µm for Co and x≥11µm for Py).\nThe tip field suppresses the uniform FMR mode in the region under the tip, and ac-\ncording to Obukhov et al.[15] the magnitude of the suppression depends on the tip-sample\nseparation. It is described as partial suppression at distances z≫/radicalBig2mp\nπMsLα0(α0is the first\nzero of the Bessel function J0(α0) = 0) and full suppression at z≪/radicalBig2mp\nπMsLα0. The region\nof suppressed magnetization is confined to a region of radius r=√\n2z. FMRFM data dis-\ncussed here were taken at the boundary of these two regions, th us we consider the regime of\n3full suppression, however we introduce the magnitude of the supp ression as a fit parameter.\nFerromagnetic resonance excitation generates a precessing tra nsverse magnetization m, thus\nreducing Mz; the change of MzisδMz=/radicalBig\nM2s−m2−Ms≈ −m2/2Ms. Here we modulate\nthe amplitude of mwith a 100% modulation depth at the cantilever resonance frequenc y.\nThe FMRFM force exerted on a cantilever is F=−/integraltextLm2/2Ms·∂Hp/∂zdr′, where integra-\ntion is performed over the entire film area. The total FMRFM force c lose the edge of the\nfilm is well approximated by\nF=−m2\n2MsL/parenleftBigg4xzmp\n(x2+z2)2−β/integraldisplay\nSθ(x′)∂Hp\n∂z(x−x′)dr′/parenrightBigg\n, (2)\nwhere the first term describes the force between the probe and t he semi-infinite film and\nthe second term represents the force between the probe and th e area of radius r=√\n2z\nunder the tip. The Heaviside function θ(x′) represents the fact that the film is positioned\natx′≥0 and the dimensionless parameter βquantifies the degree of suppression of the\nuniform FMR mode. In Fig. 3b we plot the experimental data extract ed from Fig. 2 and\ncorresponding fits using Eq. 2. Fig. 3b demonstrates good qualitat ive and quantitative\nagreement between theory and experiment and demonstrates th e validity of the model. It\nis important to mention that in our model we assume the dynamic magn etization mto be\nconstant throughout the film. However, mmay vary due to the change of the demagnetizing\nfielde.g. −4πMsfarfromthefilmboundaryand −2πMsatthefilmboundary. Ourestimates\nshow that mchanges from a constant value in the film down to zero at the film edge . The\nlength scale of this change is πMsL/∆H≈1µm (∆His the linewidth of the uniform\nresonance), small compared to the probe-sample distance thus o nly weakly affecting the fits\nshown in Fig. 3b.\nThe spatial resolution of the uniform FMR mode shown in Fig. 3b is comp arable to\nthe MFM lateral resolution depicted in Fig. 3a and is determined by the probe-sample\nseparation of z ≈4.4µm. However, it can be further improved by tracking the intensity\nof the FMRFM signal at values of H extlower than that of the uniform FMR mode (insets\na) and d) in Fig. 2). In Fig. 3c we show the FMRFM force acquired at H ext= 17960 G\nfor Co and H ext= 11150 G for Py respectively (values of H extare schematically indicated\nwith dotted lines in Fig. 2). The contribution to the FMRFM signal at low er values of H ext\noriginates from the localized region of the sample under the probe. A s seen in Fig. 3c the\nlateral resolution is on the order of 3 µm (10% - 90% change in localized signal intensity)\n4and is determined by the FMR resonance linewidth and the spatial pro file of the FMR\nmode under the tip. Further theoretical and numerical analysis is r equired to understand\nthe evolution of the FMR modes excited under the probe in the prese nce of a strongly\ninhomogeneous tip field and boundaries of the sample.\nIn conclusion, we have obtained local FMR spectra in justaposed fe rromagnetic samples\nand our quantitative model for the spatial variation of the uniform mode agrees well with\nexperimental data. We have demonstrated spectroscopic imaging of Py and Co semi-infinite\nfilms with the spatial resolution for the tip induced resonance of ≈3µm.\nThe work performed at Los Alamos was supported by the US Depart ment of Energy, and\nCenter for Integrated Nanotechnologies at Los Alamos and Sandia National Laboratories.\nThe work at OSU was supported by the US Department of Energy th rough grant DE-FG02-\n03ER46054.\n5[1] D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature 360, 563 (1993)\n[2] D. Rugar and O. Z¨ uger and S. Hoen and C. S. Yannoni and H. M. Vieth and R. D. Kendrick,\nScience264, 1560 (1994)\n[3] Z. Zhang, M. L. Roukes, and P. C. Hammel, J. Appl. Phys. 80, 6931 (1996)\n[4] D. Rugar, R. Budakian, H. J. Mamin, and W. Chui, Nature 430, 329 (2004)\n[5] C. L. Degen, Q. Lin, A. Hunkeler, U. Meier, M. Tomaselli, a nd B. H. Meier, Phys. Rev. Lett.\n94, 207601 (2005)\n[6] C. L. Degen, M. Poggio, H. J. Mamin, C. T. Rettner, and D. Ru gar, Proc. Natl. Acad. Sci.\n106, 1313 (2009)\n[7] Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys. Lett. 68, 2005 (1996)\n[8] G. de Loubens, V.V. Naletov, O. Klein, J. Ben Youssef, F. B oust, and N. Vukadinovic, Phys.\nRev. Lett. 98, 127601 (2007)\n[9] Yu. Obukhov, D. V. Pelekhov, J. Kim, P. Banerjee, I. Marti n, E. Nazaretski, R. Movshovich,\nS. An, T. J. Gramila, S. Batra, P. C. Hammel, Phys. Rev. Lett. 100, 197601, (2008)\n[10] E. Nazaretski, D. V. Pelekhov, I. Martin, M. Zalalutdin ov, D. Ponarin, A. Smirnov, P. C.\nHammel, and R. Movshovich, Phys. Rev. B 79, 132401 (2009)\n[11] E. Nazaretski, J. D. Thompson, M. Zalalutdinov, J. W. Ba ldwin, B. Houston, T. Mewes, D.\nV. Pelekhov, P. Wigen, P. C. Hammel, and R. Movshovich, J. App l. Phys.101, 074905 (2007)\n[12] E. Nazaretski, E. A. Akhadov, I. Martin, D. V. Pelekhov, P. C. Hammel, and R. Movshovich,\nAppl. Phys. Lett. 92, 214104 (2008).\n[13] E. Nazaretski, T. Mewes, D. Pelekhov, P. C. Hammel, and R . Movshovich, AIP Conf. Proc.\n850, 1641 (2006)\n[14] E. Nazaretski, D. V. Pelekhov, I. Martin, M. Zalalutdin ov, J. W. Baldwin, T. Mewes, B.\nHouston, P. C. Hammel, and R. Movshovich, Appl. Phys. Lett. 90234105 (2007)\n[15] Yu. Obukhov, D. V. Pelekhov, E. Nazaretski, R. Movshovi ch, P. C. Hammel, Appl. Phys.\nLett.94, 172508 (2008)\n6Figure Caption\nFigure 1: Schematic of the Co - Py sample. The arrows mark the scan range for spectra\nshown in Fig. 2. The SEM image on the right shows the gap between Py a nd Co. The SEM\nimage on the left depicts the cantilever tip.\nFigure 2: FMRFM force image as a function of Hextand the lateral position. We show\nthe Co and Py forces in the upper and lower panels respectively. Ins ets a) - d) demonstrate\nthe evolution of the FMRFM signal as a function of lateral position ind icated on the left-\nhand side of each inset. Vertical dashed lines show the boundaries o f the Co and Py films.\nThe horizontal dashed-doted lines are drawn through the values o f Hext= 18255 G for\nCo and H ext= 11296 G for Py respectively and correspond to the uniform reson ance (see\nFig. 3b). The horizontal dotted lines at H ext= 17960 G and H ext= 11150 G for Co and\nPy respectively, mark the localized FMRFM signals. Experimental par ameters: T = 11 K,\nfRF=9.35 GHz, probe-sample distance ≈5.6µm.\nFigure 3: a) MFM data acquired at H ext=18255 G, solid line is the fit to Eq. 1. b)\nFMRFM force data for the uniform ferromagnetic resonance (FMR ) modes. H ext= 18255\nG for Co (squares) and H ext= 11296 G for Py (circles). Solid and dashed lines are fits of\nEq. 2 to the data. Fit parameters: m/Ms= 0.0014, β= 0.65 for Co and m/Ms= 0.0028, β\n= 0.5 for Py. c) FMRFM force for the localized (close to the probe) FM R mode acquired at\nHext= 17960 G for Co (squares) and H ext= 11150 G for Py (circles). The lateral resolution\nis better than 3 µm.\n7100 m□Si /c109\nTi\n20□nm□Co\n20□nm□Py‘0 m’/c109 ‘19 m’/c109\n2 m/c109 2 m/c109z\nx\nFIG. 1:\na) b)\nd)\nc)\nFIG. 2:\n8FIG. 3:\n9"    },    {        "title": "1803.03486v1.Non_local_coupling_between_antiferromagnets_and_ferromagnets_in_cavities.pdf",        "content": "Non-local coupling between antiferromagnets and ferromagnets in cavities\n\u001fyvind Johansen\u0003and Arne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: October 9, 2018)\nMicrowaves couple to magnetic moments in both ferromagnets and antiferromagnets. Although\nthe magnons in ferromagnets and antiferromagnets radically di\u000ber, they can become entangled\nvia strong coupling to the same microwave mode in a cavity. The equilibrium con\fguration of\nthe magnetic moments crucially governs the coupling between the di\u000berent magnons because the\nantiferromagnetic and ferromagnetic magnons have the opposite spins when their dispersion relations\ncross. We derive analytical expressions for the coupling strengths and \fnd that the coupling between\nantiferromagnets and ferromagnets is comparable to the coupling between two ferromagnets. Our\n\fndings reveal a robust link between cavity spintronics with ferromagnets and antiferromagnets.\nMagnets and photons can couple strongly and co-\nherently on the quantum level [1]. This coupling has\nbeen observed as hybridizations between ferromagnetic\nmagnons and either microwave resonators [2], microwave\nphotons [3], or optical photons [4{7]. Together, these\n\fndings constitute the birth of cavity spintronics , a\nnew interdisciplinary \feld with roots in spintronics,\ncavity quantum electrodynamics and quantum optics.\nShortly after the initial observations of magnon-photon\nhybridization, experiments also achieved ultrastrong cou-\npling between magnons and microwaves [8{11]. One can\ntune the coupling strength by changing the temperature\nof the system [12, 13]. The cooperativity of the hy-\nbridization, namely, the ratio of the coupling strength\nto the loss rates, is a measure of the coherence in the\nsystem and can be as large as 107[10]. The transmission\nand re\rection coe\u000ecients of a cavity are measures of the\nmagnon-photon coupling. Brillouin light scattering [14]\nor spin pumping [15{17] can provide additional informa-\ntion. Although the focus thus far has been placed on the\nhybridization between ferromagnetic magnons and pho-\ntons, theory predicts that there should also be a signif-\nicant coupling between antiferromagnetic magnons and\nphotons [18]. Recently, a robust coupling between mi-\ncrowave photons and antiferromagnetic \ructuations in\nan organic magnet has been observed [19].\nWhen two ferromagnets or ferrimagnets couple to the\nsame cavity mode or the same photons, a non-local in-\nteraction between the magnons emerges [20, 21]. The\nquantized magnetic \feld of the cavity mode or photons\nmediates this indirect coupling. In turn, this coupling fa-\ncilitates the coherent transport of magnons over macro-\nscopic distances.\nThe magnons in ferromagnets and antiferromagnets\nstrongly di\u000ber. Nevertheless, we will demonstrate that\na clear non-local coupling can also arise between the\nmagnons in an antiferromagnet and those in a ferromag-\nnet. This link opens the door toward long-range spin\ncommunication between antiferromagnetic and ferromag-\nnetic spintronic devices. Such a connection has so far\n\u0003Corresponding author: oyvinjoh@ntnu.nobeen elusive because the vanishing magnetization of the\nantiferromagnet renders it invisible to dipolar interac-\ntions with the ferromagnet. The need for and interest in\nsuch a long-range coupling between dissimilar magnetic\nmaterials have increased with the recent emergence of\nantiferromagnetic spintronics [22{26]. Our \fndings open\na path toward integrating these new components with\nexisting ferromagnetic ones.\nHp\nSA\nSBSC\nLxyzH0\nAF\nF\nFIG. 1. A microwave cavity consisting of two perfect conduc-\ntors separated by a distance L. The magnons in an antifer-\nromagnet (with sublattice spins SAandSB) and those in a\nferromagnet (with spins SC) couple via the quantized mag-\nnetic \feld Hp. At equilibrium, the magnetic moments are\ncollinear along the zaxis. Detection of the coupling is pos-\nsible when the frequencies of the magnons are similar. This\ncan be achieved by applying an external magnetic \feld H0\nalong the equilibrium axis of the magnetic moments.\nTo study the cavity-mediated coupling between an an-\ntiferromagnet and a ferromagnet, we consider the setup\ndepicted in Fig. 1. The Hamiltonian consists of four\ncomponents:\nH=HAF+HF+Hph+Hm-ph; (1)\nwhereHAFdescribes the antiferromagnet; HF, the fer-\nromagnet;Hph, the cavity modes; and Hm-ph, the cou-\npling of the magnons in both the antiferromagnet and\nthe ferromagnet to the cavity modes. We consider anarXiv:1803.03486v1  [cond-mat.mes-hall]  9 Mar 20182\nantiferromagnet described by the Hamiltonian\nHAF=JX\nhi;ji2AFSi\u0001Sj+j\rjX\ni2AFH0\u0001Si\n\u0000Kk\n2X\ni2AF(Sz\ni)2+K?\n2X\ni2AF(Sx\ni)2; (2)\nwhereJ >0 is the exchange coupling between the spins\nS,\ris the gyromagnetic ratio, H0=H0^ zis an exter-\nnal magnetic \feld (in units of Tesla) along the easy axis,\nandKk>0 andK?\u00150 are the easy- and hard-axis\nanisotropy constants, respectively. For the ferromagnet,\nwe consider energy contributions from the Zeeman and\ndipolar interactions. It will become clear that the pho-\ntons couple only to homogeneous magnetic excitations.\nWe can therefore use a simpli\fed expression for the dipo-\nlar interaction, resulting in a ferromagnetic Hamiltonian\nof\nHF=j\rjX\ni2FH0\u0001Si+\u00160\n2Z\nVFdVX\ni=x;y;zNiM2\ni:(3)\nHere,Nx;y;z are the elements of the diagonal demagneti-\nzation tensor, VFis the volume of the ferromagnet, \u00160is\nthe vacuum permeability, and Miis thei-th component\nof the magnetization. The magnetization is related to\nthe spins by M=\u0000j\rjP\ni2FSi=VF.\nNow, let us rewrite the two magnetic Hamiltonians\nin terms of magnon operators. We perform a Holstein-\nPrimako\u000b transformation [27] of the spin operators Si,\nretaining terms of up to second order in the magnon op-\nerators in the resulting Hamiltonian. Following this pro-\ncedure, we perform a Fourier transformation from spatial\ncoordinates into a momentum representation. To diago-\nnalize the antiferromagnetic and ferromagnetic Hamilto-\nnians, we also perform Bogoliubov transformations.\nFor the antiferromagnet, the Bogoliubov transfor-\nmation is four-dimensional in the general case, when\nK?6= 0. In terms of the initial Holstein-Primako\u000b\nmagnon annihilation (creation) operators a(y)\nqandb(y)\nq\nwithin sublattices A and B, respectively (see Fig. 1), the\nBogoliubov transformation into the diagonal eigenmode\nmagnons\u000b(y)\nqand\f(y)\nqis [28]\n0\nBB@\u000bq\n\fy\n\u0000q\n\u000by\n\u0000q\n\fq1\nCCA=0\nBB@u\u000b;av\u000b;bv\u000b;au\u000b;b\nv\u0003\n\f;au\u0003\n\f;bu\u0003\n\f;av\u0003\n\f;b\nv\u0003\n\u000b;au\u0003\n\u000b;bu\u0003\n\u000b;av\u0003\n\u000b;b\nu\f;av\f;bv\f;au\f;b1\nCCA0\nBB@aq\nby\n\u0000q\nay\n\u0000q\nbq1\nCCA:(4)\nHere, qis the wave vector of the magnons. To deter-\nmine the elements of the Bogoliubov transformation, we\nimpose the requirement that the operators \u000b(y)\nqand\f(y)\nq\nmust satisfy bosonic commutation relations as well as the\nrelations [\u000bq;HAF] = ~!\u000b\u000bqand [\fq;HAF] = ~!\f\fq.\nHere,!\u000b;\fare the eigenfrequencies of the antiferromag-\nnet. The excitations in the antiferromagnet are then\nHAF=X\nq\u0000\n~!\u000b\u000by\nq\u000bq+~!\f\fy\nq\fq\u0001\n: (5)Let us de\fne the frequencies !E=~JsZ,!H=j\rjH0,\n!k=~sKk, and!?=~sK?, wheresis the spin number\nandZis the number of nearest neighbors in the anti-\nferromagnet; then, we can express the eigenfrequencies\nas\n!2\n\u0006\u0019!E\u0000\n2!k+!?\u0001\n+!2\nH\n\u0006q\n!2\nE!2\n?+ 4!2\nH!E\u0000\n2!k+!?\u0001\n; (6)\nwhere!\u000b=!+and!\f=!\u0000. For simplicity, we\nhave here included only terms to the lowest order in the\nanisotropy frequencies !k=?, as these are typically much\nsmaller than the exchange frequency !E. We also assume\nthat the wavelength of the magnons is much larger than\nthe atomic length scale.\nIn the long-wavelength limit ( q=0), the quantized\nferromagnetic Hamiltonian can be expressed as [28, 29]\nHF=AFcyc+BF\u0000\ncc+cycy\u0001\n; (7)\nwhere AF=~[!H+!M(Nxz+Nyz)=2] and\nBF=~!MNxy=4. Here, we have de\fned !M=j\rj\u00160Ms,\nwhereMsis the saturation magnetization of the ferro-\nmagnet, and Nij=Ni\u0000Nj. We have also introduced\nthe ferromagnetic magnon annihilation (creation)\noperatorc(y)from the initial Holstein-Primako\u000b trans-\nformation for the ferromagnet. This Hamiltonian can be\ndiagonalized through a Bogoliubov transformation de-\n\fned by\u0011=uFc+vFcy. Imposing bosonic commutation\nrelations on \u0011(y)yields the constraint u2\nF\u0000v2\nF= 1. The\nelements of the Bogoliubov transformation are [29]\nvF= 2BF=q\n(AF+~!\u0011)2\u00004B2\nF; (8a)\nuF=vF(AF+~!\u0011)=(2BF): (8b)\nThe energy of the \u0011magnons is ~!\u0011=p\nA2\nF\u00004B2\nF.\nWe assume a thin-\flm geometry with Nx= 1 and\nNy=Nz= 0. The resonance frequency is then\n!\u0011=p\n!H(!H+!M).\nNow, let us consider the cavity. We \frst introduce\na simple geometry. We will later argue that the sim-\nple geometry also captures the essential physics in more\ncomplex cavities. We describe the microwave cavity as\ntwo perfect conductor plates separated by a distance L\nin thexdirection. The plates are located at x= 0 and\nx=L. The boundary conditions near the conducting\nplates cause a quantization of the electromagnetic modes\nin thexdirection. The allowed wave numbers in this\ndirection are n\u0019=L (n= 1;2;:::). We assume that the\nmicrowaves propagate only in the direction perpendicu-\nlar to the conducting plates, i.e., along the xaxis. The\nresulting quantized magnetic \feld is\nHp(r) =X\nn;\u0015icos\u0010n\u0019x\nL\u0011\u0012~!n\u00160\nV\u00131=2\n\u0002\u0010\npn;\u0015^ x\u0002^ e\u0015\u0000py\nn;\u0015^ x\u0002^ e\u0003\n\u0015\u0011\n: (9)3\nHere,p(y)\nn;\u0015is the boson annihilation (creation) operator\nassociated with a cavity mode with quantum numbers\nnand\u0015.\u0015denotes the polarization, and ^ e\u0015is the po-\nlarization unit vector. Vis the volume of the cavity,\n!n=n\u0019c=L is the cavity mode frequency, and cis the\nspeed of light. The cavity Hamiltonian is\nHph=X\nn;\u0015~!n\u0012\npy\nn;\u0015pn;\u0015+1\n2\u0013\n: (10)\nThe cavity modes can couple to the magnons in both\nantiferromagnets [18] and ferromagnets [2, 3]. The in-\nteraction is represented by a Zeeman term between the\nquantized magnetic \feld Hpin Eq. (9) and the spins in\nthe magnetic materials. This interaction is described by\nthe magnon-photon Hamiltonian Hm-ph.\nWe use a circular polarization basis, \u0015=\u0006and\n^ e\u0015=^ e\u0006= (^ z\u0006i^ y)=p\n2, for the cavity modes. Since the\nspeed of light is much greater than the group velocity of\nthe magnons, the cavity modes only couple to magnons\nof approximately jqj= 0. For simplicity, we assume that\nthe cavity only couples to the q=0modes. We also\nuse the rotating wave approximation (RWA) to simplify\nthe magnon-photon interaction. In the RWA, rapidly os-\ncillating terms, e.g., pairs of creation operators or pairs\nof annihilation operators, are disregarded because their\naverage contributions quickly become small.\nWe tune the system size such that the n= 1 cavity\nmode has a frequency near the crossing of the ferromag-\nnetic and antiferromagnetic dispersion relations. The\ncoupling between the ferromagnet and the antiferromag-\nnet is signi\fcant only near this crossing region. There-\nfore, we disregard any contributions from the high-energy\nn>1 cavity modes and \u000bmagnons of the antiferromag-\nnet. There are also three-particle (and higher-order) in-\nteractions in which pairs of magnon creation/annihilation\noperators couple to the cavity mode, i.e., terms such as\n\fy\fypn;\u0015and\f\fpy\nn;\u0015. Such couplings to the n= 2 cav-\nity mode, for example, oscillate su\u000eciently slowly to be\nsigni\fcant in the RWA. However, these higher-order con-\ntributions scale more weakly with the number of spins\nin the ferromagnet/antiferromagnet ( NF/AF) than the\ntwo-particle interactions do. Such terms can therefore\nbe safely disregarded.\nWith these approximations and simpli\fcations, the\nmagnon-photon interaction Hamiltonian reduces to\nHm-ph =~\u0000\n~gAF\fy\u0000~gF\u0011y\u0001\n(p++p\u0000) + H.c.:(11)\nHere, we have introduced the ferromag-\nnetic coupling strength ~ gF =gF(uF+vF)\nand the antiferromagnetic coupling strength\n~gAF =gAF(u\f;a\u0000v\f;b+v\f;a\u0000u\f;b)=p\n2, where\ngF/AF =j\rjp\n~!p\u00160sNF/AF=(4V), with!p=!n=1\nbeing the microwave frequency. In addition, the\ncoupling strength in a cavity depends on the over-\nlap of the cavity mode with the magnetic materials\n[15]. More general geometries are described by sub-\nstitutinggF/AF!\u0018F/AFgF/AF, where\u0018F/AF is ageometrical overlap factor of order one for the ferromag-\nnet/antiferromagnet [13]. In the following illustrations,\nwe consider a perfect overlap between the cavity mode\nand each magnetic material; i.e., we set \u0018F,AF = 1. In\ngeneral, the elements of the Bogoliubov transformation\nfor the antiferromagnetic magnons ( u\f;a,v\f;b,v\f;a, and\nu\f;b) are complicated functions of the parameters of the\nantiferromagnetic Hamiltonian expressed in Eq. (2).\nWe therefore present the coupling strengths between\nthe antiferromagnetic magnons and the cavity for only\ntwo particular scenarios. In the \frst scenario, the\nantiferromagnet is uniaxial ( !?= 0), and we \fnd that\nthe coupling strength is\n~gAF\ngAF\u0019\u0012!k\n8!E\u00131=4\n: (12)\nIn the second scenario, the antiferromagnet has an easy-\nplane anisotropy such as that in NiO, where O(!?)\u0018\nO(p!k!E) [30, 31]. The coupling strength is then\n~gAF\ngAF\u0019!Hp2!E\u0000\n2!E!k\u0000!2\nH\u00011=4: (13)\nThe expressions given in (12) and (13) are valid to the\nlowest order in !k=!E.\nWe consider two geometries. In the \frst case, the\nmagnetic moments of the ferromagnet and antiferromag-\nnet are perpendicular to the propagation direction of the\ncavity modes, as in Fig. 1. In this geometry, which is\nthe one we have considered so far, the magnons couple\nto both circular polarizations with the same strength,\nas seen in Eq. (11). This is because, in the RWA and\nto leading order in the spin \ructuations, the component\nof the quantized magnetic \feld along the magnetic mo-\nments does not in\ruence the dynamics of the magnetic\nmoments. Since the magnets experience an oscillating\nmagnetic \feld along only one axis, they e\u000bectively couple\nto a cavity mode that is linearly polarized. The coupling\nof antiferromagnetic and ferromagnetic magnons requires\nthat they couple to the same cavity mode with quantum\nnumbersnand\u0015. Linearly polarized microwaves couple\nto magnons with di\u000berent spins and can therefore couple\nequally strongly to both antiferromagnetic and ferromag-\nnetic magnons. Consequently, this geometry enables a\nmagnon-magnon interaction between an antiferromagnet\nand a ferromagnet.\nIn the second geometry, the magnetic moments are\ncollinear with the propagation direction of the cavity\nmodes. In such a geometry, the coupling strengths\nstrongly depend on the cavity mode polarization. Each\nmagnon couples to the microwave mode with the same\nspin. At the crossing, the antiferromagnetic and ferro-\nmagnetic magnons therefore couple to di\u000berent polariza-\ntion states \u0015of the cavity modes. Cavity modes with\ndi\u000berent polarizations \u0015=\u0006are independent. This ge-\nometry therefore does not lead to a magnon-magnon cou-\npling. This lack of indirect coupling can be circumvented4\nwhen at least one of the magnon modes is squeezed or\nhybridized (i.e., it has a non-integer spin [28]). In this\ncase, the magnons have \fnite, but dissimilar, couplings\nto both polarizations of the cavity mode. Although this\ngenerates a \fnite magnon-magnon coupling, this coupling\nwill typically be rather weak because the magnons still\nprimarily couple to di\u000berent cavity mode polarizations.\nWe therefore focus on the former geometry, illustrated in\nFig. 1. In this geometry, there are equally strong cou-\nplings to both polarizations and, consequently, a stronger\ncoupling between the magnons.\nMore complex cavity structures can be e\u000bectively re-\nduced to our planar cavity geometry when there is only\none mode with a frequency near the crossing point of\nthe magnon dispersion relations. The magnon-magnon\ncoupling is then dominated by this mode. The overlap\nfactors\u0018F,AF depends on the details of the cavity struc-\nture.\nWe now wish to determine the coupling strength be-\ntween the ferromagnetic and antiferromagnetic magnons.\nWe consider a dispersive regime, in which the frequency\nof the cavity mode is slightly detuned from the crossing\npoint between the ferromagnetic and antiferromagnetic\nmagnon dispersion relations. We assume that this detun-\ning \u0001 =!p\u0000!\u0011=!p\u0000!\fis much larger than the cou-\npling strengths ~ gF,AF and that the frequency of the cav-\nity mode is su\u000eciently close to the crossing point where\n!\u0011=!\f. We can then determine the magnon-magnon\ncoupling by performing a unitary transformation of the\nHamiltonian given in Eq. (1) and then applying second-\norder degenerate perturbation theory, with ~ gF,AF as the\nperturbation parameters. We thus compute that at the\ncrossing point between the antiferromagnetic and ferro-\nmagnetic magnon dispersion relations, the transformed\nHamiltonian describing the excitations is\nH0=~\u0014\n!\f\u00002~g2\nAF\n\u0001\u0015\n\fy\f+~\u0014\n!\u0011\u00002~g2\nF\n\u0001\u0015\n\u0011y\u0011\n+~\u0014\n!p+~g2\nF+ ~g2\nAF\n\u0001\u0015\u0010\npy\n\u0000p\u0000+py\n+p+\u0011\n+ 2~~gF~gAF\n\u0001\u0000\n\fy\u0011+\u0011y\f\u0001\n(14)\nto second order in ~ gF,AF. The \frst three terms in Eq.\n(14) describe shifts in the frequencies of both the magnon\nand cavity modes, whereas the last term describes a dis-\npersive coupling between the ferromagnetic and antifer-\nromagnetic magnons.\nIt is instructive to \fnd numerical estimates of the ef-\nfective coupling strengths for di\u000berent combinations of\nmaterials. We consider a scenario in which the volume\nof each of the magnetic materials constitutes 1% of the\nvolume of the cavity; this magnitude is similar to that in\nprevious experiments involving ferromagnets in spherical\ncavities [20, 21]. As ferromagnets, we consider yttrium-\niron-garnet (YIG) and Co-Fe alloys. YIG is commonly\nused in cavity spintronics due to its high spin density and\nincredibly low damping. We also consider a ferromag-\nnetic Co-Fe alloy because it can have a damping as lowTABLE I. Exchange and anisotropy frequencies of the anti-\nferromagnets.\nMaterial !E(1012s\u00001)!k(1012s\u00001)!?(1012s\u00001)\nMnF 2[38] 9.3 1.5 \u000110\u00001-\nNaNiO 2[36] 8.4 \u000110\u000016.2 \u000110\u00002-\nNiO [30, 31] 1.7 \u00011022.3 \u000110\u000031.3 \u000110\u00001\nTABLE II. Coupling strengths for di\u000berent combinations of\nferromagnets and antiferromagnets.\nMaterials (AF / F) ~ gAF=(2\u0019) (GHz) ~gF=(2\u0019) (GHz)\nMnF 2/ YIG 0.31 2.30\nMnF 2/ Co-Fe 0.33 7.67\nNaNiO 2/ YIG 0.44 1.28\nNaNiO 2/ Co-Fe 0.50 5.03\nNiO / YIG 0.17 2.03\nNiO / Co-Fe 0.15 6.88\nas\u001810\u00004[32] and has an even higher spin density than\nYIG, enabling stronger coupling to the cavity mode. The\nspin density sNF=VFin YIG is\u00182\u00011022cm\u00003[33]. In a\nCo-Fe alloy, it is possible to achieve a spin density as high\nas\u00182\u00011023cm\u00003, where we have estimated the magni-\ntude of the spin density as Ms=\u0016B, with\u0016Bbeing the\nBohr magneton. The saturation magnetizations of these\nferromagnetic materials are \u00160Ms= 0:247 T for YIG\n[34] and\u00160Ms\u00192:4 T for the Co-Fe alloy [32]. As anti-\nferromagnets, we consider the uniaxial antiferromagnetic\nmaterials MnF 2and NaNiO 2as well as the easy-plane an-\ntiferromagnetic material NiO. Estimates of the spin den-\nsities in MnF 2, NaNiO 2and NiO yield\u00181022cm\u00003[35],\n\u00185\u00011022cm\u00003[36] and\u00181023cm\u00003[37], respectively.\nThe exchange and anisotropy frequencies of the antifer-\nromagnetic materials are listed in Table I. Using these\nparameters, and choosing the n= 1 frequency mode of\nthe cavity to lie at the crossing point between the anti-\nferromagnetic and ferromagnetic dispersion relations, we\nobtain the coupling strengths given in Table II.\nOur results reveal that the coupling strength of the\ncavity with the antiferromagnet is typically an order of\nmagnitude lower than that with the ferromagnet. This\nis due to the weak interaction between antiferromag-\nnets and magnetic \felds, which gives rise to a suppres-\nsion factor from the high exchange energy, as seen in\nEqs. (12) and (13). Importantly, it is still possible to\nachieve a sizeable magnon-magnon coupling between the\nantiferromagnet and the ferromagnet. If we detune the\ncavity from the crossing point of the magnon disper-\nsion relations by \u0001 = 5~ gF, then the magnon-magnon\ncoupling becomes \u00192~gAF=5. For MnF 2/Co-Fe, this\nwould correspond to a coupling strength of 132 MHz;\nfor NaNiO 2/Co-Fe, the corresponding value is 200 MHz,\nand for NiO/YIG, it is 68 MHz. These magnitudes are\nsimilar to the coupling strength observed between two\nYIG spheres [20, 21]. The ine\u000eciency of the coupling5\nbetween the antiferromagnet and the cavity is compen-\nsated for by the high frequency at the crossing point be-\ntween the magnon dispersion relations, with a propor-\ntionality of ~ gF/AF/p!. The magnon dispersion rela-\ntions cross at 148 GHz ( H0=4.2 T) for MnF 2/Co-Fe, at\n36 GHz (H0=0.6 T) for NaNiO 2/Co-Fe, and at 100 GHz\n(H0= 3:5 T) for NiO/YIG.\nAlthough the linewidths of many ferromagnets have\nbeen extensively studied, there are very few similar mea-\nsurements available for antiferromagnets. It is possible\nthat the linewidth may become larger in antiferromag-\nnets since the exchange energy often dominates. If this\nis the case, the cooperativity (which is a measure of\nthe coherence within the system) will be lower in sys-\ntems involving antiferromagnets than in systems consist-\ning solely of ferromagnets.\nIn summary, we have shown that despite having op-posite spins, antiferromagnetic magnons and ferromag-\nnetic magnons can be coupled inside a microwave cavity.\nThis coupling is strongest when the magnetic moments\nin the antiferromagnet and ferromagnet are perpendicu-\nlar to the propagation direction of the cavity modes at\nequilibrium. In this geometry, the cavity mode couples to\nboth the antiferromagnetic and ferromagnetic magnons,\nresulting in a robust non-local magnon-magnon coupling.\nThe magnitude of this coupling is similar to that of the\nnon-local interaction between two ferromagnets.\nAcknowledgments. The authors would like to thank\nMathias Kl aui for valuable input. This work was sup-\nported by the Research Council of Norway through\nits Centres of Excellence funding scheme, Project No.\n262633 \\QuSpin\" and Grant No. 239926 \\Super Insu-\nlator Spintronics,\" as well as by the European Research\nCouncil via Advanced Grant No. 669442 \\Insulatronics.\"\n[1] O. O. Soykal and M. E. Flatt\u0013 e, Phys. Rev. Lett. 104,\n077202 (2010).\n[2] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen-\nstein, A. Marx, R. Gross, and S. T. B. Goennenwein,\nPhys. Rev. Lett. 111, 127003 (2013).\n[3] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-\nami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603\n(2014).\n[4] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Ya-\nmazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura,\nand Y. Nakamura, Phys. Rev. Lett. 116, 223601 (2016).\n[5] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys.\nRev. Lett. 117, 123605 (2016).\n[6] J. A. Haigh, A. 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Ross, Spin Dynamics in an Antiferromagnet , Ph.D.\nthesis, Technische Universit at M unchen (2013)."    },    {        "title": "1309.4195v1.Giant_magnetoresistance_in_the_junction_of_two_ferromagnets_on_the_surface_of_diffusive_topological_insulators.pdf",        "content": "arXiv:1309.4195v1  [cond-mat.mes-hall]  17 Sep 2013Giant magnetoresistance in the junction of two ferromagnet s on the surface of\ndiffusive topological insulators\nKatsuhisa Taguchi1, Takehito Yokoyama2, and Yukio Tanaka1\n1Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan\n2Department of Physics, Tokyo Institute of Technology, Toky o, 152-8551, Japan\n(Dated: March 27, 2018)\nWe reveal the giant magnetoresistance induced by the spin-p olarized current in the ferromagnet\n(F1)/topological insulator (TI)/ferromagnet (F 2) junction, where two ferromagnets are deposited on\nthe diffusive surface of the TI. We can increase and reduce the value of the giant magnetoresistance\nby tuning the spin-polarized current, which is controlled b y the magnetization configurations. The\nproperty is intuitively understood by the non-equilibrium spin-polarized current, which plays the\nrole of an effective electrochemical potential on the surfac e of the TI.\nPACS numbers: 72.25.-b, 73.43.Qt, 75.47.-m,\nSpintronics aims to control the charge transport by\nusing spin degrees of freedom[1–5], and also to manipu-\nlate the spin-polarization by using the charge current[6–\n11]. The former is applicable for the giant magnetore-\nsistance (GMR) and tunnel magnetoresistance (TMR)\nin the junction of a metal or an insulator sandwiched\nby two ferromagnets. The resistance depends on the di-\nrection of the magnetizations. The latter, the control of\nthe spin-polarization, has been discussed in metals, semi-\nconductors, in the presence of the spin-orbit interaction,\nwhich can easily induce spin polarization by the electric\nfield. Spin-orbit interaction, which plays a dominant role\ninthemanipulationofthespinpolarization,canalsoplay\nan important role in the control of the charge transport.\nTopological insulators (TIs) are candidate materials\nwhere the spin-orbit interaction enhances the magnitude\nof both charge and spin currents, where we can also\nexpect anomalous electromagnetic phenomena stemmed\nfromtopologicalfeaturesofbulkelectronicproperties[12–\n14]. Nowadays,toclarifytheelectromagneticphenomena\nspecific to TIs is believed to serve as a guide to fabri-\ncate future magneto-electronic devices. TIs are gapped\ninbulkandhavegaplesssurfacestate, whichisdubbed as\nthe helical surface state, in which the spin configuration\nand momentum are locked by spin-orbit interaction[12–\n14].\nIn the helical surface state, persistent pure spin-\npolarized current without dissipation is generated. This\nspin-polarized current could be useful for GMR and\nTMR, which lead to novel magnetic devices. Preexisting\nworks[15–18] have predicted anomalous charge transport\nfeatures by the spin-polarized current in the junction of\nferromagnets on the surface of TIs in the ballistic trans-\nport regime. However, the actual charge transport on\nthe surface of TIs may be in the diffusive regime with\nimpurity scattering. Thus, a study of charge transport\ndue to the helical surfaces state is necessary in a realistic\nsituation.\nIn this letter, we study charge transport between\ntwo-dimensional ferromagnet (F 1)/topological insulator(TI)/ferromagnet (F 2) junction, where two ferromagnets\nare deposited on the surface of TI. We find anomalous\nproperties of GMR, which have never appeared in con-\nventional metallic ferromagnet junctions. The GMR\nincludes not only the Ohmic resistance term but also\nanomalous one (defined in Eq.(16)) depending on two\nmagnetizations. The later term is proportional to the\nspin polarization, which stems from the injected and ex-\ntracted spin-polarized current. By tuning the configura-\ntion of magnetizations in the F 1and F2, these two terms\ncancel each other and gigantic reduction ofmagnetoresis-\ntance becomes possible. The present feature may serve\nas a guide to fabricate future magneto-electronic devices\nbased on TIs.\nHereafter, we will consider a ferromagnet\nF1/TI/ferromagnet F 2junction on the surface of\nTI in the presence of the applied charge current (Fig.1\n(a)). We assume that the directions of magnetizations\nof F1and F 2are along the y-axis and the direction\nof the charge current is along the x-axis. Then, the\ncharge current induces the spin polarization syby the\nspin-orbit interaction on the surface of TI, and syobeys\nthe spin-diffusion equation as[19, 20]\n˙sy=3D\n2∇2\nxsy−sy\nτ+vF\n2e∇xNe, (1)\nwhereNe,D≡vFℓ/2,vF,ℓ, andτare charge den-\nsity, diffusion constant, Fermi velocity, mean free path,\nand transport relaxation time, respectively. The charge\ncurrent drives the spatial gradient of the charge density\n∇xNeas\nj=−D∇Ne+evF(ˆz×s). (2)\nIn the following discussion, we simply assume that the\napplied current does not have spatial dependence and\nis independent of x(j=Iˆx). The charge current, in\naddition, generates the spin polarization on the surface\nofTIbythespin-polarizedcurrent. Thespinpolarization\nisdetermined bytheboundaryconditionsatthe interface2\nof the left-side ( x=−L/2) between F 1and TI and the\nright side ( x=L/2) between TI and F 2, where Lis\nthe distance between the two electrodes. We use the\nboundary conditions at both interfaces as follows: The\nspin-polarized currentη1I\neflows from F 1into TI at the\nleft interface x=−L/2, where η1is the the degree of\nspin polarization of the magnetization of F 1[19].η1=\n1(−1) is defined such that the direction of magnetization\nis positive (negative) along the y-axis and the current\nis fully spin-polarized. At the right interface, the spin-\npolarized currentη2I\neis extracted from TI into F 2.η2\nrepresents the direction of the magnetization F 2andη2\nis defined in the same way. On the other hand, the spin\ncurrentjy\ns,xon the surface of TI is given by\njy\ns,x=−3\n2D∇xsy−1\n2evFNe. (3)\nThe first and the second terms represent the non-\nequilibrium and equilibrium component of spin-polarized\ncurrents, respectively. The equilibrium component does\nnot appear explicitly in the boundary condition[21, 22].\nThus, the boundary conditions are given by [19]\n−3\n2D∇xsy/vextendsingle/vextendsingle\nx=−L\n2=η1I\ne, (4)\n−3\n2D∇xsy/vextendsingle/vextendsingle\nx=L\n2=η2I\ne. (5)\nThe spin polarization is obtained from Eqs. (1)-(5) as\nsy=sy\n1+sy\n2, (6)\nwith\nsy\n1=I\n2evFκℓ/braceleftbig\nη1cosh[κ(L\n2−x)]−η2cosh[κ(L\n2+x)]/bracerightbig\nsinh(κL),\n(7)\nsy\n2=−I\n2evF, (8)\nandκ≡2\nℓ/radicalBig\n2\n3.sy\n1depends on the position x(−L/2≤\nx≤L/2),η1andη2, and it decays exponentially with\nthe diffusion length scale of κ−1from the boundaries.\nsy\n2is only proportional to Iindependent of x. When\nη2= 0, the spin polarization is consistent with that in\nRef. 19[23]. The physical meaning of sy\n1is the spin polar-\nization due to the spin-polarized current, which is given\nby−3\n2D∇xsy[Eqs. (4)-(5)]. The expression of sy\n2means\nthat the spin polarization generated by the spin-orbit in-\nteraction on the surface of TI[24], which is independent\nof the magnetizations.\nThe spin polarization syis plotted in Fig. 1 (b)-(c).\nFirst, wediscussthe casewheremagnetizationsofF 1and\nF2are antiparallel to each other with ( η1,η2) = (−1,1).\nIn the antiparallel magnetization configuration, the mag-\nnitudes of sydecay from the interface at x=±L/2, andI(a)\nTIF1 F2y/Hat\nx/Hatz/Hat\n/Minus1.0/Minus0.5 0.0 0.5 1.0/Minus1.6/Minus1.4/Minus1.2/Minus1.0/Minus0.8/Minus0.6\n2x/Slash1Lsy/LBracket1I/Slash1/LParen1evF/RParen1/RBracket1\n/Minus1.0/Minus0.5 0.0 0.5 1.0/Minus1.0/Minus0.50.0\n2x/Slash1Lsy/LBracket1I/Slash1/LParen1evF/RParen1/RBracket1L/Slash1/ScriptAltL/Equal1L/Slash1/ScriptAltL/Equal2L/Slash1/ScriptAltL/Equal10\nL/Slash1/ScriptAltL/Equal1\nL/Slash1/ScriptAltL/Equal2L/Slash1/ScriptAltL/Equal10(b)\n(c)\nFIG. 1: (Color online) (a) A schematic illustration of the\nF1/TI/F 2junction, where the arrow from F 1to F2shows\nthe applied charge current and the blue arrows represent the\ndirection of the magnetization in F 1and F 2. (b)-(c) The spin\npolarization syfor various magnetization configurations with\nL/ℓ= 1 (solid line), 2 (dashed line), and 10 (dotted line),\nwhereLis the length between two ferromagnets and ℓis the\nmean free path on the surface of TI. The direction of the\nmagnetization F 2is antiparallel (b) or parallel (c) to that of\nF1(inset (b) or (c)).\nsyhas a nonzero value for all x.syhas a strong spa-\ntial dependence for −L/2< x < L/ 2 withL/ℓ∼1 and\nL/ℓ∼2, while it becomes almost constant except for\nthe vicinity of x=±L/2 withL/ℓ∼10.syis an even\nfunction of xfor allLin the antiparallel magnetization\nconfiguration (see Fig. 1 (b)), while, in the parallel case\n[(η1,η2) = (−1,−1)],syis not an even function of x(see\nFigs. 1 (c)), because sy\n1andsy\n2(=const) are an odd and\neven function of x, respectively.\nsycontributestothevoltagedrop VbetweenF 1andF2\non the surface of TI due to the spin-orbit interaction[19].3\nVis defined by [25]\nV=−1\ne2ν/integraldisplayL/2\n−L/2dNe\ndxdx, (9)\nwhereν=µ\n2πv2\nFis the density of states on the surface\nstate of TI at chemical potential µ=vFkF. From Eq.\n(2), the voltage drop can be expressed by\nV=VO+Vs, (10)\nwith\nVO=2\ne2ℓνvF/integraldisplayL/2\n−L/2dx(I+evFsy\n2),(11)\nVs=2\neℓν/integraldisplayL/2\n−L/2dxsy\n1. (12)\nIn the above, the voltage drop VO, which is given by\nthe charge current and constant spin polarization due\ntoI, obeys the Ohm’s law. The spin polarization sy\n2in\nthe second term of VOis negative in the presence of the\nchargecurrent flowing towardspositive x-axis, and hence\nsy\n2has an opposite contribution to the first term of VOin\nEq. (11). On the other hand, the voltage drop Vsin Eq.\n(12) is given by the spin polarization sy\n1, which depends\non the magnetization configuration [see Eq. (7)]. Since\nVsisdeterminedbythemagnetizationconfigurations,the\nresulting Vsonly depends on the boundary condition of\nspin-polarized currentand no more obeys the Ohm’s law.\nVscan be regarded as a boundary term and is obtained\nasVs=η1−η2\ne2νvFI. Finally, the total voltage drop Vis given\nby\nV=1\ne2νvF/parenleftbiggL\nℓ+η1−η2/parenrightbigg\nI. (13)\nThe first term in Eq. (13) obeys the Ohm’s law and is\nproportional to the length Lof TI. The second (third)\nterm is independent of L, and is proportional to η1(η2).\nThe coefficient1\ne2νvFis proportional to the Sharvin re-\nsistance in the two-dimension[26].\nThe magnetoresistance V/I=1\ne2νvF(L\nℓ+η1−η2) de-\npends on the configurations of the magnetizations in F 1\nand F2:\nRσσ′=RO+Rs,σσ′, (14)\nRO=L\ne2νvFℓ, (15)\nRs,σσ′=1\ne2νvF(η1−η2) (16)\nwhereσandσ′denote spin indices with σ=↑(↓) and\nσ′=↑(↓).η1= 1(−1) andη2= 1(−1) correspond\ntoσ=↑(↓) andσ′=↑(↓), respectively. The magne-\ntoresistance has two terms: Ohmic resistance term ( RO)\nand the boundary term ( Rs,σσ′).ROdoes not depend/Minus1 0 10.7\n0\n/Minus0.7\n2x/Slash1Ls1y/LBracket1I/Slash1/LParen1ev F/RParen1/RBracket1\n/Minus1 0 10.7\n0\n/Minus0.7\n2x/Slash1Ls1y/LBracket1I/Slash1/LParen1ev F/RParen1/RBracket1(a) (b)\n/Minus1 0 10.40.60.8\n2x/Slash1Ls1y/LBracket1I/Slash1/LParen1ev F/RParen1/RBracket1(c) (d)\n/Minus1 0 1/Minus0.4\n/Minus0.6\n/Minus0.8\n2x/Slash1Ls1y/LBracket1I/Slash1/LParen1ev F/RParen1/RBracket1\n2610/Minus0.8/Minus0.40MR ratioL/Slash1/ScriptAltL(e)\n1 2 3 4/Minus100/Minus50050100\nL/Slash1/ScriptAltLMR ratio\nFIG. 2: (Color online) The spin polarization, sy\n1, for antipar-\nallel [(a) and (b)] and parallel [(c) and (d)] magnetization\nconfigurations with L/ℓ= 2. The schematic magnetiza-\ntion configurations in F 1/TI/F 2junctions are shown in the\ninsets. (e) Magnetoresistance (MR) ratio as a function L.\nThe blue solid (red dashed) line represents MR ↓=R↓↓−R↓↑\nR↓↑\n(MR↑=R↑↑−R↑↓\nR↑↓). The enlarged plot of MR ↑is shown by\nthe dashed line in the inset.\non the magnetization configurations. Rs,σσ′depends on\nthe magnetization configurations. For the parallel con-\nfiguration, the boundary term is absent. On the other\nhand, for the antiparallel configurations, the boundary\nterm exists. Moreover, in the antiparallel configuration,\nthe boundary term has a negative (positive) sign when\nthe direction of the magnetization F 1and F 2are nega-\ntive (positive) and positive (negative) along the y-axis,\nrespectively. Whether Rs,σσ′is zero or nonzero can be\nseen from the difference of the spatial dependences of sy\n1\nforL/ℓ= 2 in Fig. 2 (a)-(d). sy\n1is even (odd) function of\nxin antiparallel (parallel) configuration (see inset in Fig.\n2(a)-(d)). Since Rs,σσ′isobtainedby Vs/I∝/integraltextL/2\n−L/2dxsy\n1,\nit is nonzero (zero) in the antiparallel (parallel) config-\nurations of the magnetizations. The sign of the Rs,σσ′\nterm can be seen from the sign of sy\n1in Fig. 2 (c) and\n(d), and is positive (negative) in R↑↓(R↓↑).\nThese properties of the magnetoresistance are\nquite different from those of the conventional4\nmagnetoresistance[1–5]: The magnetoresistance is\nonly proportional to the relative angle between the\ndirections of the two magnetizations (i.e., R↑↑=R↓↓\nandR↑↓=R↓↑). The magnitude of the magnetore-\nsistance is usually given by R↑↑< R↑↓. On the other\nhand, the magnetoresistance in Eqs. (14)-(16) behaves\nin a drastically different way. There are two conditions\nto determine the magnitudes of magnetoresistance\nRσσ′: Whether the magnetization configuration is\nparallel or antiparallel, and whether the direction of the\nmagnetizations F 1and F 2are positive and negative or\nnegative and positive along the y-axis in the antiparallel\nconfiguration, respectively. Rσσ′satisfies R↑↑=R↓↓\nandR↓↑< R↑↓, and we can obtain R↓↑< R↑↑< R↑↓\nfrom Eqs. (14)-(16). Thus, the relation R↓↑< R↑↑is\ndifferent from that of the conventional case, although\nR↑↑=R↓↓andR↑↑< R↑↓are the same properties as\nthat of the conventional magnetoresistance. Moreover,\nR↓↑is lower than the Ohmic resistance, and becomes\nzero for L/ℓ= 2[19]. The anomalous property of the\nmagnetoresistance, R↓↑, could be applicable for an\ncharge transport with low Joule heating.\nWe will consider the magnetoresistance ratio, MR ↓\n(MR↑), when the direction of the magnetization F 2is\nfree and F 1is fixed along the negative (positive) y-axis.\nThese magnetoresistance ratios are given by\nMR↓≡R↓↓−R↓↑\nR↓↑=2\nL/ℓ−2, (17)\nMR↑≡R↑↑−R↑↓\nR↑↓=−2\nL/ℓ+2. (18)\nMR↓and MR ↑depend on L/ℓ. The value of MR ↓is neg-\native for L <2ℓand is positive for L >2ℓ, and diverges\natL/ℓ= 2 (see Fig. 2(b)) since R↓↑is zero. The value of\nMR↑is negative for all L/ℓ(see the inset of Fig. 2(b)).\nThe magnitude of MR ↑and MR ↓are 49% and 1000% in\nL/ℓ= 2.1, respectively. The gigantic magnetoresistance\nratio MR ↓stems from the boundary term Rs,σσ′in the\nsurface state of the TI.\nFinally, we show that the value of Vscan be intuitively\nunderstood from the boundary conditions of the spin-\npolarized current. sy\n1ofVscan be rewritten by Eqs. (1)\nand (2) as[27]\nsy\n1=3\n4τD∇2\nxsy\n1. (19)\nBy using the above equation, Vsin Eq. (12) is given\nby the integration of the spatial gradient of the non-\nequilibrium spin-polarized current −3\n2D∇xsy, and the\nvalue of Vsis proportional to the difference of the non-\nequilibrium spin-polarized current at the interfaces:\nVs=−1\neνvF/integraldisplayL/2\n−L/2∇x/parenleftbigg\n−3\n2D∇xsy\n1/parenrightbigg\ndx (20)\n=−1\neνvF/parenleftbigg\n−3\n2D∇xsy\n1/vextendsingle/vextendsinglex→L\n2\nx→−L\n2/parenrightbigg\n=η1−η2\ne2νvFI.(21)Intheelectromagnetism,the voltagedropisgeneratedby\nelectrochemical potential φas/integraltext\ndx∇xφand the electric\nfieldisgivenby −∇xφ. Therefore,wecanregardthenon-\nequilibrium spin-polarized current as an electrochemical\npotential and its gradient as an effective electric field,\nwhich drives the anomalous voltage drop in the diffusive\nsurface of TIs.\nIn conclusion, we studied the anomalous magnetore-\nsistance in the F 1/TI/F 2junction by using the spin-\ndiffusion equation on the surface of TIs. It is found\nthat the magnetoresistance includes the boundary term,\nwhich is absent in the usual ferromagnetic junction. The\nboundary resistance is triggered by the spin polariza-\ntion of the current, which is induced by both the spin-\npolarized current injected from F 1into the TI and ex-\ntracted from the TI into F 2. The boundary resistance\nplays the role of the reduction (enhancement) to the\nOhmic term when the directions of the two magnetiza-\ntionsarenegative(positive) andpositive(negative)along\nthey-axis. Recently, a fine quality film of the TI comes\nto be able to fabricate[28, 29]. Also, it has been demon-\nstrated that the ferromagnetism at ambient temperature\ncan be induced in the Mn-doped Bi 2Te3by the magnetic\nproximity effect through deposited Fe overlayer[30]. By\nusing the film of the TI, the boundary resistance can\nbe experimentally demonstrated and would be applica-\nble for a charge transport with low energy consumption\nin future.\nThis work was supported by a Grant-in-Aid for Young\nScientists (B) (No. 22740222, No. 23740236)) and by\na Grant-in-Aid for Scientific Research on Innovative Ar-\neas ”Topological Quantum Phenomena” (No. 22103005,\nNo. 25103709) from the Ministry of Education, Culture,\nSports, Science and Technology, Japan (MEXT). K.T.\nacknowledges the support by the JSPS.\n[1] I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau,\nF. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J.\nChazelas, Phys. Rev. Lett. 61, 2472 (1988).\n[3] M. Julliere, Phys. Lett. A 54, 225 (1975).\n[4] S. Maekawa and U. G¨ afvert, IEEE Trans. Magn. 18, 707\n(1982).\n[5] J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meser-\nvey, Phys. Rev. Lett. 74, 3273 (1995).\n[6] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n[7] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.\nRev. Lett. 78, 1335 (1997).\n[8] M. I. D’yakonov and V. I. Perel’, Phys. Lett. A 35, 459\n(1971); JETP Lett. 13, 467 (1971).\n[9] S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301,\n1348 (2003).\n[10] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 1910 (2004).5\n[11] J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth ,\nPhys. Rev. Lett. 94, 047204 (2005).\n[12] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045\n(2010).\n[13] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057\n(2011).\n[14] Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).\n[15] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev.\nB,81, 121401(R) (2010).\n[16] S. Mondal, D. Sen, K. Sengupta, and R. Shankar, Phys.\nRev. B.82, 045120 (2010).\n[17] B. D. Kong, Y. G. Semenov, C. M. Krowne, and K. W.\nKim, App. Phys. Lett. 98, 243112 (2011).\n[18] M. J. Ma, M. B. A. Jalil, S. G. Tan, Y. Li, and Z. B. Siu\nAIP Advances 2, 032162 (2012)\n[19] A. A. Burkov and D. G. Hawthorn, Phys. Rev. Lett. 105,\n066802 (2010).\n[20] P. Schwab, R. Raimondi, and C. Gorini, EPL (Euro-\nphysics Letters), 93, 67004 (2011).\n[21] A. G. Mal’shukov, L. Y. Wang, C. S. Chu, K. A. Chao,\nPhys. Rev. Lett. 95, 146601 (2005); V. M. Galitski, A.\nA. Burkov, and S. Das Sarma, Phys. Rev. B 74, 115331\n(2006); O. Bleibaum, Phys. Rev. B 73, 035322 (2006); Y.\nTserkovnyak, B. I. Halperin, A. A. Kovalev, A. Brataas,\nPhys. Rev. B 76, 085319 (2007).\n[22] S. Takahashi and S. Maekawa, J. Phys. Soc. Jpn. 77,\n031009 (2008).\n[23]sy\n1can be also expressed as sy\n1=√2\n3I\nevF\nsinh(2√2\n3L\nℓ)[η1cosh(/radicalBig\n2\n3L−2x\nℓ)−η2cosh(/radicalBig\n2\n3L+2x\nℓ)].\n[24] S. Raghu, S. B. Chung, X. -L. Qi, and S.-C. Zhang, Phys.\nRev. Lett. 104, 116401 (2010).\n[25] Derivation of the voltage drop: By using the charge\ndensityNe(µ) =e/summationtext\nkf(ǫk−µ), the difference δN≡\nNe(µ+eV)−Ne(µ) at low temperature is given by\nδNe=e/summationtext\nk[f(ǫk−µ−eV)−f(ǫk−µ)] =e2νV.δN\nis also written, when there is the voltage drop Vbe-\ntweenx=−L\n2andx=L\n2, asδNe=Ne(x=−L\n2:\nµ+eV)−Ne(x=L\n2:µ) =−/integraltextL\n2\n−L\n2dx′dNe(x)\ndx′.\n[26] A. M. Zagoskin, Quantum Theory of Many-Body Sys-\ntem: Techniques andApplications (Springer-Verlag, New\nYork, 1998).\n[27] The spin-diffusion equation in Eq. (1) is rewritten by Eq .\n(2) as3D\n2∇2\nxsy−2\nτsy−I\neℓ= 0. Tosubstitute sy=s′y−Iτ\n2eℓ\ninto the above equation, we obtain3D\n2∇2\nxs′y=2\nτs′y. Now\ns′yis equal to sy\n1, because of −Iτ\n2eℓ=sy\n2ands′y=sy+\nIτ\n2eℓ=sy−sy\n2from Eq. (6).\n[28] A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Phys.\nRev. Lett. 109, 066803 (2012).\n[29] A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando, Adv.\nMater.24, 5581 (2012).\n[30] I. Vobornik, U. Manju, J. Fujii, F. Borgatti, P. Torelli , D.\nKrizmancic, Y. S. Hor, R. J. Cava, and G. Panaccione,\nNano Lett. 11, 4079 (2011)."    },    {        "title": "0801.0145v1.Lecture_Notes_on_Semiconductor_Spintronics.pdf",        "content": "Lecture Notes on Semiconductor Spintronics\nTomasz Dietl\nInstitute of Physics, Polish Academy of Sciences and ERATO Semiconductor\nSpintronics Project, Japan Science and Technology Agency, al. Lotnik\u0013 ow 32/46,\nPL-02668 Warszawa, Poland and Institute of Theoretical Physics, Warsaw\nUniversity, Poland dietl@ifpan.edu.pl\nAbstract: These informal lecture notes describe the progress in semicon-\nductor spintronics in a historic perspective as well as in a comparison to\nachievements of spintronics of ferromagnetic metals. After outlining motiva-\ntions behind spintronic research, selected results of investigations on three\ngroups of materials are presented. These include non-magnetic semiconduc-\ntors, hybrid structures involving semiconductors and ferromagnetic metals,\nand diluted magnetic semiconductors either in paramagnetic or ferromagnetic\nphase. Particular attention is paid to the hole-controlled ferromagnetic sys-\ntems whose thermodynamic, micromagnetic, transport, and optical properties\nare described in detail together with relevant theoretical models.\n1 Why Spintronics?\nThe well-known questions fuelling a broad interest in nanoscience are: will it\nstill be possible to achieve further progress in information and communication\ntechnologies simply by continuing to miniaturize the transistors in micropro-\ncessors and the memory cells in magnetic and optical discs? How to reduce\npower consumption of components in order to save energy and to increase\nbattery operation time? How to integrate nowadays devices with biological\nmolecules and functionalities?\nSince 70s, the miniaturization by obeying Moore's law has persistently\nlead to an exponential increase in the quantity of information that can be\nprocessed, stored, and transmitted per unit area of microprocessor, memory,\nand \fberglass, respectively. A modern integrated circuit contains now one bil-\nlion transistors, each smaller than 100 nm in size, i.e., a \fve hundred times\nsmaller than the diameter of a human hair. The crossing of this symbolic\n100 nm threshold at the outset of the 21st century ushered in the era of nan-\notechnology. As the size of transistors decreases, their speed increases, and\ntheir price falls. Today it is much less expensive to manufacture one transis-\ntor than to print a single letter. Despite the series of successes that industrialarXiv:0801.0145v1  [cond-mat.mtrl-sci]  30 Dec 20072 Tomasz Dietl\nlaboratories have scored over the past 40 years in surmounting one technical\nand physical barrier after another, there is a prevalent sense that in the near\nfuture a qualitative change is now in store for us in terms of the methods of\ndata processing, storing, encoding, and transmission. For this reason, govern-\nments in many countries are \fnancing ambitious interdisciplinary programs\naimed at insuring active participation in the future development of nanotech-\nnology.\nAmong the many proposals for where to take such research, the \feld of\nspintronics, i.e., electronics aimed at understanding electron spin phenomena\nand at proposing, designing, and developing devices to harness these phenom-\nena, is playing a major role. The hopes placed in spintronics are founded on the\nwell-known fact that since magnetic monopoles do not exist, random magnetic\n\felds are signi\fcantly weaker than random electric \felds. For these reasons,\nmagnetic memories are non-volatile, while memories based on an accumulated\nelectric charge (dynamic random access memory, or DRAM) require frequent\nrefreshing.\nOne of the ambitious goals in the spintronics \feld is to create magnetic\nrandom access memory (MRAM), a type of device that would combine the\nadvantages of both magnetic memory and dynamic random access memory.\nThis requires novel methods of magnetizing memory cells and reading back\nthe direction of such magnetization , which would not involve any mechani-\ncal systems. Another important step along this path would be the ability to\ncontrol magnetization isothermally, by means of light or electric \feld. Modern\ndevices expend relatively large amounts of energy on controlling magnetiza-\ntion ( i.e., storing data), as they employ Oersted magnetic \felds generated by\nelectric currents.\nThe development of more \"intelligent\" magnetization control methods\nwould also make it possible to build spin transistors, devices composed of\ntwo layers of ferromagnetic conductors separated by non-magnetic material.\nIt stands to reason that if carriers injected into the non-magnetic layer preserve\ntheir spin direction, then the electric conductivity depends on the relative di-\nrection of the magnetization vectors in the ferromagnetic layers. This could\no\u000ber a means of producing an energy-conserving and fast switching device, as\nit would allow current to be controlled without changing the carrier concentra-\ntion. An obvious prerequisite for such a transistor to operate is the e\u000ecient\ninjection of spin-polarized carriers made of ferromagnetic material into the\nnon-magnetic area. Also, there should be no processes that could disrupt the\nspin polarization. Simultaneously, researchers are seeking ways of generating,\namplifying, and detecting spin currents: here, the underlying conviction is that\nthe movement of electrons with opposite spins does not entail any losses, yet\ncan carry information. This would lay the foundations for the development\nof low-power devices, characterized by signi\fcantly reduced heat dissipation.\nAnother important issue is to develop methods for injecting spin-polarized\ncarriers into semiconductors. Apart from the possibility of designing the mag-\nnetization sensors and spin transistors, polarized carrier injection could proveLecture Notes on Semiconductor Spintronics 3\nto be useful as a method for the fast modulation of semiconductor lasers and\nwould allow surface-emission lasers to work in a single mode fashion.\nPerhaps the most important intellectual challenge to be faced in spintron-\nics is to create a hardware for quantum information science. Researchers over\nthe world have joined e\u000borts to lay the theoretical foundations for this new\ndiscipline [1], one notable example being the Horodecki family from Gda\u0013 nsk\n[2]. Experiments conducted by David Awschalom's group in Santa Barbara\nshow that spin degrees of freedom are of particular importance as they main-\ntain their phase coherence signi\fcantly longer than orbital degrees of freedom\ndo [3]. Electron spin is therefore much more suitable than electron charge for\nputting into practice modern ideas for performing numerical computations\nusing the superposition and entanglement of quantum states. Spin nanostruc-\ntures might consequently alter the basic principles not only in the design of\nelectronic elements, but also in the very computer architecture that has been\nin use for half a century. It is noteworthy that quantum encoders are already\nnow being sold and installed: such devices use the polarization of light to\nencode the transmitted information, and the unauthorized interception and\nreading of this information appears to be impossible.\nToday's research on spin electronics involves virtually all material families.\nThe most advanced are studies on magnetic multilayers. As demonstrated in\n80s by groups of Albert Fert [4] in Orsay and Peter Gr unberg [5] in J ulich,\nthese systems exhibit giant magnetoresistance (GMR ). According to theory\ntriggered by these discoveries and developed by J\u0013 ozef Barna\u0013 s from Pozna\u0013 n\nand co-workers [6], GMR results from spin-dependent scattering at adjacent\ninterfaces between non-magnetic and magnetic metals, which changes when\nthe magnetic \feld aligns magnetization of particular layers. Since 90s, the\nGMR devices have been successfully applied in reading heads of high-density\nhard-discs. Recent works focuss also on spin-dependent tunnelling via an oxide\n\flm. Remarkably, for the case of crystalline MgO sandwiched between contacts\nof amorphous Fe-Co-B layers, the di\u000berence between tunnelling resistance for\nanti-parallel and parallel orientations of magnetization , the TMR , reaches\na factor of three at 300 K [7, 8, 9]. Moreover, the magnetization direction\ncan be switched by an electric current below 106A cm\u00002[10], opening the\ndoors for a direct magnetization writing by current pulses. Last but not least\nsuch structures can be used for injecting highly polarized spin currents to\nsemiconductors, such as GaAs [11].\nThese informal lecture notes on semiconductor spintronics exploit and up-\ndate author's earlier reviews [12, 13, 14, 15, 16, 17, 18], where more system-\natic references to original papers can be found. Particular attention is paid\nhere to those results of research on spin properties of semiconductors, which\nappear relevant in the context of disruptive classical and quantum informa-\ntion and communication technologies. First part of the paper shows brie\ry\nhow spin e\u000bects speci\fc to non-magnetic semiconductors can be exploited\nin spintronic devices. This is followed by a presentation of chosen properties\nof hybrid semiconductor/ferromagnetic metal structures. The main body of4 Tomasz Dietl\nthe paper is devoted to diluted magnetic semiconductors (DMS ), especially\nto materials exhibiting the ferromagnetic order, as they combine complemen-\ntary resources of semiconductor materials and ferromagnetic metals. Here, the\nfundamental research problem is to identify the extent to which the methods\nthat have been so successfully applied to controlling the density and degree of\nspin polarization of carriers in semiconductor structures might be employed to\ncontrol the magnetization magnitude and direction. Apart from the possibil-\nity of designing the aforementioned magnetoresistive sensors and spin align-\ners, ferromagnetic semiconductors are the materials of choice for spin current\nampli\fcation and detection. Furthermore, their outstanding magnetooptical\nproperties can be exploited for fast light modulation as well as optical isola-\ntors, perhaps replacing hybrid structures consisting of paramagnetic DMS ,\nsuch as (Cd,Mn)Te, and a permanent magnet.\nIn the course of the years semiconductor spintronics has evolved into a\nrather broad research \feld. These notes are by no means exhaustive and,\nmoreover, they are biased by author's own expertise. Fortunately, however, in\na number of excellent reviews the issues either omitted or only touched upon\nhere has been thoroughly elaborated in terms of content and references to the\noriginal papers. For instance, the progress in fabrication and studies of spin\nquantum gates of double quantum dots has been described by van Viel et\nal. [19]. A comprehensive survey on spin-orbit e\u000bects and the present status\nof spin semiconductor transistors has been completed by \u0014Zuti\u0013 c, Fabian, and\nDas Sarma [20]. Finally, Jungwirth et al. [21] have reviewed various aspects\nof theory of (Ga,Mn)As and related materials. Excellent reviews on the entire\nsemiconductor spintronics are also available [22, 23].\n2 Non-magnetic Semiconductors\n2.1 Overview\nThe beginning of spintronic research on non-magnetic semiconductors can be\ntraced back to the detection of nuclear spin polarization in Si illuminated by\ncircularly polarized light reported in late 60s by Georges Lampel at Ecole\nPolytechnique [24]. Already this pioneering experiment involved phenomena\ncrucial for semiconductor spintronics: (i) the spin-orbit interaction that allows\nfor transfer of orbital (light) momentum to spin degrees of freedom and (ii) the\nhyper\fne interaction between electronic and nuclear spins. Subsequent exper-\nimental and theoretical works on spin orientation in semiconductors, carried\nout in 70s mostly by researchers around Ionel Solomon in Ecole Polytechnique\nand late Boris P. Zakharchenya in Io\u000be Institute, were summarized in a by\nnow classic volume [25].\nMore recently, notably David Awschalom and his co-workers \frst at IBM\nand then at Santa Barbara, initiated the use of time resolved optical magneto-\nspectroscopies that have made it possible to both temporally and spatially ex-\nplore the spin degrees of freedom in a wide variety of semiconductor materialsLecture Notes on Semiconductor Spintronics 5\nand nanostructures [26]. The starting point of this experimentally demanding\ntechnique is the preparation of spins in a particular orientation by optically\npumping into selected electronic states. The electron spin then precesses in an\napplied or molecular magnetic \feld produced by electronic or nuclear spins.\nThe precessing magnetic moment creates a time dependent Faraday rotation\nof the femtosecond optical probe. The oscillation and decay measure the e\u000bec-\ntive Land\u0013 e g-factor, the local magnetic \felds, and coherence time describing\nthe temporal dynamics of the optically injected spins.\nPresent spintronic activities focuss on two interrelated topics. The \frst is\nto exploit Zeeman splitting and spin-orbit interactions for spin manipulation.\nTo this category belongs, in particular, research on spin \flters and detectors,\non the Datta-Das transistor [20], on optical generation of spin currents [27]\nand on the spin Hall e\u000bect [28]. The other topic is the quest for solid-state\nspin quantum gates that would operate making use of spin-spin exchange [29]\nand/or hyper\fne interactions [30]. An important aspect of the \feld is a dual\nrole of the interactions in question in non-magnetic semiconductors: from one\nhand they allow for spin functionalities, on the other they account for spin\ndecoherence and relaxation, usually detrimental for spin device performance.\nThis, together with isotope characteristics, narrows rather severely a window\nof material parameters at which semiconductor spin devices might operate.\n2.2 Spin relaxation and dephasing\nOwing to a large energy gap and the weakness of spin-orbit interactions,\nespecially long spin life times are to be expected in the nitrides and oxides.\nFigure 1 depicts results of time-resolved Faraday rotation, which has been\nused to measure electron spin coherence in n-type GaN epilayers [31]. Despite\ndensities of charged threading dislocations of 5 \u0002108cm\u00002, this coherence\nyields spin lifetimes of about 20 ns at temperatures of 5 K, and persists up to\nroom temperature.\nFigure 2 presents a comparison of experimental and calculated magnetore-\nsistance (MR) of a ZnO:Al thin \flm containing 1 :8\u00011020electrons per cm3\n[32]. Here, spin e\u000bects control quantum interference corrections to the classical\nDrude-Boltzmann conductivity. A characteristic positive component of MR,\nsignalizing the presence of spin-orbit scattering, is detected below 1 mT at low\ntemperatures. This scattering is linked to the presence of a Rashba -like term\n\u0015soc(s\u0002k) in thekphamiltonian of the wurzite structure, \frst detected in n-\nCdSe in the group of the present author [33]. As shown in Fig. 2, a quite good\ndescription of the \fndings is obtained with \u0015so= 4:4\u000110\u000011eV cm, resulting\nin the spin coherence time 1 ns, more than 104times longer than the momen-\ntum relaxation time. Importantly, this low decoherence rate of wide-band gap\nsemiconductors is often coupled with a small value of the dielectric constant\nthat enhances characteristic energy scales for quantum dot charging as well as\nfor the exchange interaction of the electrons residing on the neighboring dots.\nThis may suggest some advantages of these compounds for fabrication of spin6 Tomasz Dietl\neffectively have a three-parameter fit, with the resulting\ncurves shown as solid lines in Fig. 3 ~b!.\nAccording to Elliot-Yafet18spin-scattering theory, mo-\nmentum scattering plays a crucial role in the dephasing, andscattering from the charged threading dislocations shouldlead to spin decoherence or dephasing via local spin-orbitinteractions. Although momentum scattering is strongly en-hanced by the presence of these defects, as seen in the carriermobility,\n14the measured spin coherence appears to be robust\nto their presence. A possible explanation for this surprisingbehavior lies in the fact that the valence-band spin-orbit cou-pling is ;100 times weaker than in GaAs,\n16resulting in a\ndecoupling of momentum- and spin-scattering processes.\nThe temperature dependence of t2is shown in Fig. 4 ~a!at\na variety of magnetic fields for the sample with lowest dop-\ning. At high temperatures ( T.25K), t2decreases mono-\ntonically with increasing temperature. Further, t2is either\nconstant or increasing with increasing magnetic field @Fig.\n4~b!#in this temperature range. This dependence is expected,\nand agrees qualitatively with the existence of anisotropic in-ternal magnetic fields such as caused by either the non-\ncentrosymmetric crystalline structure ~D’yakonov-Perel !19or\nthe strong spontaneous piezoelectric fields ~Rashba effect !20\nobserved in GaN. Below T525K this behavior changes\nqualitatively. Whereas at low magnetic fields t2still in-\ncreases with decreasing temperature, at higher fields it goesthrough a field-dependent maximum and subsequently de-creases. This indicates the onset of an additional field- andtemperature-dependent spin scattering ~or dephasing !mecha-\nnism. The same magnetic field induced scattering mecha-nism is observed in all samples, and is depicted on a log-log\nscale in Fig. 4 ~c!atT55 K, revealing a simple power law\nbehavior,\nt2}B2a, with alargest close to the MIT.\nSummarizing the experimental features of t2, we find\nthat they are all in qualitative agreement with results ob-tained in GaAs:\n3~i!the zero-field spin lifetime is longest at\nintermediate doping, i.e. in the vicinity of the MIT. ~ii!An\nadditional spin-dephasing mechanism evolves at low tem-peratures and nonzero magnetic field. It is enhanced withdecreasing temperature and increasing field. ~iii!This field-\ninduced suppression of the spin lifetime is strongest in thevicinity of the MIT.\nIn order to understand this additional spin scattering\nmechanism, we consider possible origins for the dephasing\nof\nt2. Field-induced spin relaxation is known to arise from\ninhomogeneous dephasing due to a Gaussian distribution ofgfactors, resulting in a simple inverse relationship between\nt2andB(a51).21This is not found experimentally in all\nsamples. Additionally, spin-scattering mechanisms such asthe Elliot-Yafet or D’yakanov-Perel mechanisms are un-\nFIG. 3. ~a!TRFR rotation taken at T55 K for the sample with\nn53.531016cm23at magnetic fields of B50.02, 0.05, 0.1, 0.4,\n0.8, 1.0, 1.5, 4, and 6 T ~from top to bottom !.~b!Selected enve-\nlopes plotted on a semilog scale for samples with n53.5\n31016cm23~B50.4, 1.5, 4, and 6 T !andn5931017cm23~B\n50.5, 1.5, and 5 T !. Solid lines are fits to the data ~see text !.\nFIG. 4. ~a!Spin scattering time t2vs temperature at various\nfields for the sample with n53.531016cm23.~b!The magnetic-\nfield dependence of t2for the same sample. ~c!t2vs magnetic field\non a log-log plot at T55 K for all samples. Solid lines are power-\nlaw fits to t2}B2a, yielding a50.52 ~open squares !, 0.93 ~filled\ntriangles !, and 0.13 ~open circles !.RAPID COMMUNICATIONS\nSPIN COHERENCE AND DEPHASING IN GaN PHYSICAL REVIEW B 63121202 ~R!\n121202-3\nFig. 1. Spin scattering time \u001c2of n-GaN at various magnetic \felds (a), temperatures\n(b) (n= 3:5\u00021016cm\u00003), and electron concentrations at 5 K (c) (after Beschoten\net al. [31]).\nquantum gates. Another material appealing in this context is obviously Si,\nand related quantum structures, in which the interfacial electric \feld controls\nthe magnitude of the Rashba term [34] and material containing no nuclear\nspins can be obtained.\n2.3 An example of spin \flter\nTurning to the case of narrow-gap semiconductors we note that strong spin-\norbit e\u000bects speci\fc to these systems results, among other things, in a large\nZeeman splitting of the carrier states, which can be exploited for fabrication\nof e\u000ecient spin \flters. As an example, we consider quantum point contacts\npatterned of PbTe quantum wells embedded by Bi-doped Pb 0:92Eu0:08Te bar-\nriers [35, 36]. Owing to biaxial strain, the fourfold L-valley degeneracy of the\nconduction band in PbTe is lifted, so that the relevant ground-state 2D sub-\nband is formed of a single valley with the long axis parallel to the [111] growth\ndirection. As discussed recently [36], the paraelectric character of PbTe results\nin e\u000ecient screening of Coulomb scattering potentials, so that signatures ofLecture Notes on Semiconductor Spintronics 7\nLecture Notes on Semiconductor Spintronics 7\n02x10-34x10-36x10-3\n3 6 00 0.0004\n01x10-72x10-7lso=4.4x10-11eVcm\n0.01%n-ZnO\nT (K):  \n0.05\n0.4\n1\n1.5\n2\n5\n \n \n \nMagnetic field (mT)Sheet resistance changes ( W)\n(h/e2)B/Btr\nFig. 2. Resistance changes in the magnetic field for n-ZnO (symbols) compared to\ncalculations (solid lines) within the weak localization th eory for the 2D case. Curves\nare vertically shifted for clarity (after Andrearczyk et al. [32]).\n0 1 2 3 4220200180160140120\nMAGNETIC FIELD [T]GATE VOLTAGE [mV]\nFig. 3. Transconductance dG/dV g(gray scale) showing dependence of 1D sub-\nbands on the magnetic field and gate voltage for PbTe nanocons triction of a wide\n(Pb,Eu)Te/PbTe/(Pb,Eu)Te quantum well (after Grabecki et al. [35]).\nFig. 2. Resistance changes in the magnetic \feld for n-ZnO (symbols) compared to\ncalculations (solid lines) within the weak localization theory for the 2D case. Curves\nare vertically shifted for clarity (after Andrearczyk et al. [32]).\nballistic transport can be observed despite of signi\fcant amount of charged\ndefects in the vicinity of the channel. At the same time, the electron density\ncan be tuned over a wide range by biasing a p-n junction that is formed be-\ntween the p+interfacial layer and the n-type quantum well [36]. Furthermore,\na rather large magnitude of electron spin splitting for the magnetic \feld along\nthe growth direction, corresponding to the Land\u0013 e factor jg\u0003j\u0019 66, can serve\nto produce a highly spin-selective barrier. According to results displayed in\nFig. 3, spin-degeneracy of the quantized conductance steps starts to be re-\nmoved well below 1 T, so that it has become possible to generate entirely\npolarized spin current carried by a number of 1D subbands [35].\n3 Hybrid Structures\n3.1 Overview\nThe hybrid nanostructures, in which both electric and magnetic \feld are\nspatially modulated, are usually fabricated by patterning of a ferromagnetic\nmetal on the top of a semiconductor or by inserting ferromagnetic nanopar-\nticles or layers into a semiconductor matrix. In such devices, the stray \felds8 Tomasz Dietl\nLecture Notes on Semiconductor Spintronics 7\n02x10-34x10-36x10-3\n3 6 00 0.0004\n01x10-72x10-7lso=4.4x10-11eVcm\n0.01%n-ZnO\nT (K):  \n0.05\n0.4\n1\n1.5\n2\n5\n \n \n \nMagnetic field (mT)Sheet resistance changes ( W)\n(h/e2)B/Btr\nFig. 2. Resistance changes in the magnetic field for n-ZnO (symbols) compared to\ncalculations (solid lines) within the weak localization th eory for the 2D case. Curves\nare vertically shifted for clarity (after Andrearczyk et al. [32]).\n0 1 2 3 4220200180160140120\nMAGNETIC FIELD [T]GATE VOLTAGE [mV]\nFig. 3. Transconductance dG/dV g(gray scale) showing dependence of 1D sub-\nbands on the magnetic field and gate voltage for PbTe nanocons triction of a wide\n(Pb,Eu)Te/PbTe/(Pb,Eu)Te quantum well (after Grabecki et al. [35]).\nFig. 3. Transconductance dG=dV g(gray scale) showing dependence of 1D sub-\nbands on the magnetic \feld and gate voltage for PbTe nanoconstriction of a wide\n(Pb,Eu)Te/PbTe/(Pb,Eu)Te quantum well (after Grabecki et al. [35]).\ncan control charge and spin dynamics in the semiconductor. At the same\ntime, spin-polarized electrons in the metal can be injected into or across the\nsemiconductor [37, 38]. Furthermore, the ferromagnetic neighbors may a\u000bect\nsemiconductor electronic states by the ferromagnetic proximity e\u000bect even\nunder thermal equilibrium conditions. Particularly perspective materials in\nthe context of hybrid structures appear to be those elemental or compound\nferromagnets which can be grown in the same reactor as the semiconductor\ncounterpart.\n3.2 Spin injection\nIt is now well established that e\u000ecient spin injection from a ferromagnetic\nmetal to a semiconductor is possible provided that semiconductor Sharvin\nresistance is comparable or smaller than the di\u000berence in interface resistances\nfor two spin orientations. Often, to enhance the latter, a heavily doped or\noxide layer is inserted between the metal and as-grown semiconductor. In this\nway, spin current reaching polarization tens percents has been injected form\nFe into GaAs [39, 11]. At the same time, it is still hard to achieve TMR\nabove 10% in Fe/GaAs/Fe trilayer structures without interfacial layer [40],\nwhich may suggest that the relevant Schottky barriers are only weakly spin\nselective.Lecture Notes on Semiconductor Spintronics 9\nThe mastering of spin injection is a necessary condition for the demonstra-\ntion of the Datta-Das transistor [41], often regarded as a \rag spintronic device.\nIn this spin FET, the orientation of the spins \rowing between ferromagnetic\ncontacts, and thus the device resistance, is controlled by the Rashba \feld\ngenerated in the semiconductor by an electrostatic gate. Recently, a current\nmodulation up to 30% by the gate voltage was achieved in a Fe/(In,Ga)As/Fe\nFET at room temperature [42]. This important \fnding was obtained for a\n1\u0016m channel of narrow gap In 0:81Ga0:19As, in which TMR achieved 200%,\nindicating that the destructive role of the Schottky barriers got reduced. Fur-\nthermore, an engineered interplay between the Rashba and Dresselhaus e\u000bects\n[43, 44] resulted in a spin relaxation time long comparing to spin precession\nperiod and the dwell time.\n3.3 Search for solid-state Stern-Gerlach e\u000bect\nThe ferromagnetic component of hybrid structures can also serve for the gen-\neration of a magnetic \feld. This \feld, if uniform, produces a spin selective\nbarrier that can serve as a local spin \flter and detector. A non-homogenous\n\feld, in turn, might induce spatial spin separation via the Stern-Gerlach (S-\nG) mechanism. Figure 4(a) presents a micrograph of a Stern-Gerlach device,\nwhose design results from an elaborated optimization process [45]. A local\nmagnetic \feld was produced by NiFe (permalloy, Py) and cobalt (Co) \flms.\nThe micromagnets resided in deep groves on the two sides of the wire, so that\nthe 2D electron gas in the modulation-doped GaAs/AlGaAs heterostructure\nwas approximately at the center of the \feld, and the in\ruence of the com-\npeting Lorentz force was largely reduced. Hall magnetometry was applied in\norder to visualize directly the magnetizing process of the two micromagnets\nin question.\nAs shown in Fig. 5, a current increase in counters was detected when a\n\feld gradient was produced by an appropriate cycle of the external magnetic\n\feld at 100 mK. The range of magnetic \felds where the enhancement was\nobserved corresponded to the the presence of the \feld gradient according to\nthe Hall magnetometry, which also showed that Py magnetization diminished\nalmost twofold prior to a change in the direction of the external magnetic \feld.\nThis e\u000bect, associated with the formation of closure domains in soft magnets,\nexplained why the current changes appeared before the \feld reversal. The\nrelative change \u0001Iof counter current depended on VG,\u0001I=I increased from\n0:5% at zero gate voltage to 50% close to the threshold. Furthermore, for VG\nabout\u00000:8 V\u0001Iwas negative. It was checked that results presented in Fig. 5\nwere unaltered by increasing the temperature up to 200 mK and independent\nof the magnetic \feld sweep rate.\nTheoretical studies [45] of the results shown in Fig. 5 demonstrated that\nsemiconductor nanostructures of the kind shown in Fig. 4 can indeed serve\nto generate and detect spin polarized currents in the absence of an external\nmagnetic \feld. Moreover, the degree and direction of spin polarization at low10 Tomasz Dietl\n0\n1\n2\n3\nx (μm)4\n5\n600.5\nBy(T)\n−0.2\n0\n y (μm) 0.2\n70.10.20.30.40.5\nT\n−0.8 −0.6 −0.4 −0.2 0\nVG (V)051015I1 ,  I2  (nA)\n7μ mPy\nCo\nT = 100 mK(a)\n(c) (d)E1\n2\nI1I1\n2I(b)\nFig. 4. (a) Scanning electron micrograph of the spin-\flter device. Fixed AC volt-\nageV0is applied between emitter (E) and \"counters\" (1), (2); VGis the DC gate\nvoltage. The external in-plane magnetizing \feld ( Bk) is oriented as shown. (b) The\nin-plane magnetic \feld By(wider part of the channel is in front) calculated for half-\nplane, 0.1\u0016m thick magnetic \flms separated by a position dependent gap W(x)\nand magnetized in the same directions (saturation magnetization as for Co). (c) By\ncalculated for antiparallel directions of micromagnet magnetization s. (d) Counter\ncurrentsI1andI2as a function of the gate voltage at V0= 100\u0016V andBk= 0;\nupper curve (shown in gray) was collected during a di\u000berent thermal cycle and after\nlonger infra-red illumination (after Wr\u0013 obel et al. [45]).\nelectron densities can easily be manipulated by gate voltage or a weak external\nmagnetic \feld. While the results of the performed computations suggest that\nthe spin separation and thus Stern-Gerlach e\u000bect occurs under experimen-\ntal conditions in question, its direct experimental observation would require\nincorporation of spatially resolved spin detection.\n4 Diluted Magnetic Semiconductors\n4.1 Overview\nThis family of materials encompasses standard semiconductors, in which a\nsizable portion of atoms is substituted by such elements, which produce local-\nized magnetic moments in the semiconductor matrix. Usually, magnetic mo-Lecture Notes on Semiconductor Spintronics 11\n−0.1 −0.05 0 0.05 0.1\nB|| (T)3.43.6I1 (nA)\n−0.1 −0.05 0 0.05 0.1\nB|| (T)77.2I1 (nA)\n−0.1 −0.05 0 0.05 0.1\nB|| (T)22.2I1 (nA)\n−0.1 −0.05 0 0.05 0.1\nB|| (T)0.40.6field down\nfield upVG = − 0.2 V VG = − 0.6 V \nVG = − 0.8 V VG = − 0.88 V \nFig. 5. The counter current I1of as a function of the in-plane magnetic \feld for\nvarious gate voltages for the device shown in Fig. 4. After Wrobel et al. [45].\nments originate from 3d or 4f open shells of transition metals or rare earths\n(lanthanides), respectively, so that typical examples of diluted magnetic semi-\nconductors (DMS ) are Cd 1\u0000xCoxSe, Ga 1\u0000xMnxAs , Pb 1\u0000xEuxTe and, in a\nsense, Si:Er. A strong spin-dependent coupling between the band and local-\nized states accounts for outstanding properties of DMS . This coupling gives\nrise to spin-disorder scattering, giant spin-splittings of the electronic states,\nformation of magnetic polarons, and strong indirect exchange interactions\nbetween the magnetic moments, the latter leading to collective spin-glass, an-\ntiferromagnetic or ferromagnetic spin ordering. Owing to the possibility of\ncontrolling and probing magnetic properties by the electronic subsystem or\nvice versa, DMS have successfully been employed to address a number of\nimportant questions concerning the nature of various spin e\u000bects in various\nenvironments and at various length and time scales. At the same time, DMS\nexhibit a strong sensitivity to the magnetic \feld and temperature as well as\nconstitute important media for generation of spin currents and for manip-\nulation of localized or itinerant spins by, e.g., strain, light, electrostatic or\nferromagnetic gates. These properties, complementary to both non-magnetic\nsemiconductors and magnetic metals, open doors for application of DMS as\nfunctional materials in spintronic devices.\nExtensive studies of DMS started in 70s, particularly in the group of\nRobert R. Ga l\u0018 azka in Warsaw, when appropriately puri\fed Mn was em-\nployed to grow bulk II-VI Mn-based alloys by various modi\fcations of the\nBridgman method [46]. Comparing to magnetic semiconductors, such as Eu\nchalcogenides (e.g., EuS) and Cr spinels (e.g., CdCr 2Se4) investigated earlier\n[47], DMS exhibited smaller defect concentrations and were easier to dope by\nshallow impurities. Accordingly, it was possible to examine their properties\nby powerful magnetooptical and magnetotransport techniques [12, 46, 48, 49].\nSince, in contrast to magnetic semiconductors, neither narrow magnetic bands12 Tomasz Dietl\nnor long-range magnetic ordering a\u000bected low-energy excitations, DMS were\nnamed semimagnetic semiconductors. More recently, research on DMS have\nbeen extended toward materials containing magnetic elements other than Mn\nas well as to III-VI, IV-VI [50] and III-V [51] compounds as well as group\nIV elemental semiconductors and various oxides [52]. In consequence, a va-\nriety of novel phenomena has been discovered, including e\u000bects associated\nwith narrow-bands and magnetic phase transformations, making the border-\nline between properties of DMS and magnetic semiconductors more and more\nelusive.\nA rapid progress of DMS research in 90s stemmed, to a large extend, from\nthe development of methods of crystal growth far from thermal equilibrium,\nprimarily by molecular beam epitaxy (MBE), but also by laser ablation. These\nmethods have made it possible to obtain DMS with the content of the magnetic\nconstituent beyond thermal equilibrium solubility limits [53]. Similarly, the\ndoping during MBE process allows one to increase substantially the electrical\nactivity of shallow impurities [54, 55]. In the case of III-V DMS [51], in which\ndivalent magnetic atoms supply both spins and holes, the use of the low-\ntemperature MBE (LT MBE) provides thin \flms of, e.g., Ga 1\u0000xMnxAs with\nxup to 0.07 and the hole concentration in excess of 1020cm\u00003, in which\nferromagnetic ordering is observed above 170 K [56]. Remarkably, MBE and\nprocesses of nanostructure fabrication, make it possible to add magnetism to\nthe physics of semiconductor quantum structures. Particularly important are\nDMS , in which ferromagnetic ordering was discovered, as discussed in some\ndetails later on.\n4.2 Magnetic impurities in semiconductors\nA good starting point for the description of DMS is the Vonsovskii model,\naccording to which the electron states can be divided into two categories: (i)\nlocalized magnetic d or f shells and (ii) extended band states built up of s, p,\nand sometimes d atomic orbitals. The former give rise to the presence of local\nmagnetic moments and intra-center optical transitions. The latter form bands,\nmuch alike as in the case of non-magnetic semiconductor alloys. Indeed, the\nlattice constant of DMS obeys the Vegard low, and the energy gap Egbetween\nthe valence and the conduction band depends on xin a manner qualitatively\nsimilar to non-magnetic counterparts. According to the Anderson model, the\ncharacter of magnetic impurities in solids results from a competition between\n(i) hybridization of local and extended states, which tends to delocalized mag-\nnetic electrons and (ii) the on-site Coulomb interactions among the localized\nelectrons, which stabilizes the magnetic moment in agreement with Hund's\nrule.\nFigure 6 shows positions of local states derived from 3d shells of transi-\ntion metal (TM) impurities in respect to the band energies of the host II-VI\nand III-V compounds. In Fig.The levels labelled \"donors\" denote the ion-\nization energy of the magnetic electrons (TM2+!TM3+or dn!dn\u00001),Lecture Notes on Semiconductor Spintronics 13\nwhereas the \"acceptors\" correspond to their a\u000enity energy (TM2+!TM1+\nor dn!dn+1). The di\u000berence between the two is the on-d-shell Coulomb\n(Hubbard) repulsion energy Uin the semiconductor matrix. In addition, the\npotential introduced by either neutral or charged TM can bind a band carrier\nin a Zhang-Rice-type singlet or hydrogenic-like state, respectively. Such bound\nstates are often experimentally important, particularly in III-V compounds,\nas they correspond to lower energies than the competing d-like states, such as\npresented in Fig. 6.\nFerromagnetic semiconductors\nD(0/+)A(0/-)Energy [eV]4\n2\n-20\nCdTeCdSeCdSZnTeZnSeZnSZnO\nd1d2d3d4d5d6d7d8d9Sc Ti V Cr Mn Fe Co Ni Cu4\nd1 d2  d3d4   d5  d6\nInN\n-22\n0A(0/-)\nD(0/+)\nAlN\nAlSbAlAsAlPInSbInAsInPTi V Cr Mn Fe Co\nGaN\nGaSbGaAsGaPEnergy (eV)\nFigure 1. Approximate positions of transition metals levels relative to the conduction and valence band edges of II–VI (left panel) and III–V\n(right panel) compounds. By triangles the dN/dN−1donor and by squares the dN/dN+1acceptor states are denoted (adapted from [43]).\na Zhang–Rice polaron (charge transfer) state, which then\ngives rise to an additional level in the band gap [45–49].Furthermore, if the d\nN/dN−1donor state resides above the\nbottom of the conduction band, the ground state corresponds to\na hydrogenic-like level dN−1+e located below the band edge, as\nobserved in CdSe:Sc [50]. Similarly, if the acceptor state liesunder the top of the valence band, the ground state corresponds\nto a hydrogenic-like acceptor d\nN+1+h, not to the dNstate.\nImportantly, band carriers introduced by such magnetic ionscan mediate exchange interactions between the parent spins.\nObviously, energies of hydrogenic-like states follow the band\nedges, and by no means are described by the internal referencerule. This appears to be the situation of the Mn related levels in\nthe gap of III–V compounds [49], the case discussed in detail\nbelow.\n3.2. Mn in II–VI compounds\nIt is well established that Mn is divalent in II–VI compounds,\nand assumes the high spin d\n5configuration characterized by\nS=5/2a n dg=2.0 [51–53]. Indeed, according to figure 1,\nthe Mn ions neither introduce nor bind carriers, but give rise\nto the presence of the localized spin in II–VI DMS. The spin\ndependent hybridization between anion p and Mn d statesleads to the superexchange, a short-range antiferromagneticcoupling among the Mn moments. In order to take the\ninfluence of this interaction into account, it is convenient to\nparametrize the dependence of magnetization on the magneticfield in the absence of the carriers, M\no(H), by the Brillouin\nfunction, in which two empirical parameters, the effective spin\nconcentrationxeffN0<xN 0and temperature Teff>T ,t a k e\nthe presence of the superexchange interactions into account\n[52, 53]. The dependencies xeff(x)andTAF(x)≡Teff(x)−T\nhave been determined for a number of Mn-based II–VIDMS. Importantly, the antiferromagnetic superexchange canbe overcompensated by ferromagnetic interactions mediated\nby band holes [26], the theoretical prediction confirmed\nsubsequently by the observation of ferromagnetic ordering inp-type II–VI DMS [15, 16].0.45 0.50 0.55 0.60 0.65-3-2-10\nInSbAlSbGaSb\nInAsInPGaAs AlAsGaP AlPInNGaNEnergy (eV)\nLattice constant (nm)\nFigure 2. Experimental energies of Mn levels in the gap of III–V\ncompounds according to in respect to valence-band edges, whoserelative positions are taken from [151] (after [49]).\n3.3. Mn in III–V compounds\nFigure 2, taken from [49], shows the energetic position of\nthe Mn impurity level in III–V compounds, as evaluated byvarious authors from measurements of optical spectra andactivation energy of conductivity. Apriori , the Mn atom,\nwhen substituting a trivalent metal, may assume either of twoconfigurations: (i) d\n4or (ii) d5plus a weakly bound hole,\nd5+h. Accordingly, the experimentally determined energies\ncorrespond to either d4/d5or d5+h/d5levels.\nIt appears to be a general consensus that the Mn acts as an\neffective mass acceptor (d5+h) in the case of antimonides and\narsenides. Such a view is supported by the relatively smallMn concentrations leading to the insulator-to-metal transition,which according to the Mott criterion n\n1/3aB=0.26, points\nto a relatively large extension of the effective Bohr radius aB.\nMoreover, the ESR studies of GaAs:Mn reveal, in addition\nto the well known spectrum of Mn d5with the Land ´e factor\n379\nFig. 6. Approximate positions of transition metals levels relative to the conduction\nand valence band edges of II-VI (left panel) and III-V (right panel) compounds.\nBy triangles the dN/dN\u00001donor and by squares the dN/dN+1acceptor states are\ndenoted (adapted from Langer et al. [57] and Zunger [58]).\nIn the case of Mn, in which the d shell is half-\flled, the d-like donor state\nlies deep in the valence band, whereas the acceptor level resides high in the\nconduction band, so that U\u00197 eV according to photoemission and inverse\nphotoemission studies. Thus, Mn-based DMS can be classi\fed as charge trans-\nfer insulators, Eg< U. The Mn ion remains in the 2+ charge state, which\nmeans that it does not supply any carriers in II-VI materials. However, it\nacts as a hydrogenic-like acceptor in the case of III-V antimonides and ar-\nsenides, while the corresponding Mn-related state is deep, presumably due to\na stronger p-d hybridization, in the case of phosphides and nitrides. According\nto Hund's rule the total spin S= 5=2 and the total orbital momentum L= 0\nfor the d5shell in the ground state. The lowest excited state d\u00035corresponds\ntoS= 3=2 and its optical excitation energy is about 2 eV. Thus, if there\nis no interaction between the spins, their magnetization is described by the\nparamagnetic Brillouin function. In the case of other transition metals, the\nimpurity-induced levels may appear in the gap, and then compensate shallow\nimpurities, or even act as resonant dopant, e.g., Sc in CdSe, Fe in HgSe or\nCu in HgTe. Transport studies of such systems have demonstrated that inter-14 Tomasz Dietl\nsite Coulomb interactions between charged ions lead to the Efros-Shklovskii\ngap in the density of the impurity states, which makes resonant scattering to\nbe ine\u000ecient in semiconductors [59]. Furthermore, spin-orbit interaction and\nJahn-Teller e\u000bect control positions and splittings of the levels in the case of\nions withL6= 0. If the resulting ground state is a magnetically inactive singlet\nthere is no permanent magnetic moment associated with the ion, the case of\nFe2+, whose magnetization is of the Van Vleck-type at low temperatures.\n4.3 Exchange interaction between band and localized spins\nThe important aspect of DMS is a strong spin-dependent coupling of the ef-\nfective mass carriers to the localized d electrons, \frst discovered in (Cd,Mn)Te\n[60, 61] and (Hg,Mn)Te [62, 63]. Neglecting non-scalar corrections that can\nappear for ions with L6= 0, this interaction assumes the Kondo form,\nHK=\u0000I(r\u0000R(i))sS(i); (1)\nwhereI(r\u0000R(i)) is a short-range exchange energy operator between the\ncarrier spin sand the TM spin localized at R(i). When incorporated to the\nkpscheme, the e\u000bect of HKis described by matrix elements huijIjuii, where\nuiare the Kohn-Luttinger amplitudes of the corresponding band extreme. In\nthe case of carriers at the \u0000point of the Brillouin zone in zinc-blende DMS\n, the two relevant matrix elements \u000b=hucjIjuciand\f=huvjIjuviinvolve\ns-type and p-types wave functions, respectively. There are two mechanisms\ncontributing to the Kondo coupling [48, 64, 65]: (i) the exchange part of the\nCoulomb interaction between the e\u000bective mass and localized electrons; (ii)\nthe spin-dependent hybridization between the band and local states. Since\nthere is no hybridization between \u00006and d-derived (e gand t 2g) states in zinc-\nblende structure, the s-d coupling is determined by the direct exchange . The\nexperimentally determined values are of the order of \u000bNo\u00190:25 eV, where\nNois the cation concentration, somewhat reduced comparing to the value\ndeduced from the energy di\u000berence between S1 states of the free singly ionized\nMn atom 3d54s1,\u000bNo= 0:39 eV. In contrast, there is a strong hybridization\nbetween\u00008and t 2gstates, which a\u000bects their relative position, and leads to a\nlarge magnitude of j\fNoj\u00191 eV. If the relevant e\u000bective mass state is above\nthe t 2glevel (the case of, e.g., Mn-based DMS ), \f <0 but otherwise \fcan\nbe positive (the case of, e.g., Zn 1\u0000xCrxSe [66]).\n4.4 Electronic properties\nE\u000bects of giant spin splitting\nIn the virtual-crystal and molecular-\feld approximations, the e\u000bect of the\nKondo coupling is described by HK=IM(r)s=g\u0016B, where M(r) is magne-\ntization (averaged over a microscopic region around r) of the localized spins,Lecture Notes on Semiconductor Spintronics 15\nandgis their Land\u0013 e factor. Neglecting thermodynamic \ructuations of magne-\ntization (the mean-\feld approximation) M(r) can be replaced by Mo(T;H),\nthe temperature and magnetic \feld dependent macroscopic magnetization of\nthe localized spins available experimentally. The resulting spin-splitting of\ns-type electron states is given by\n\u0016h!s=g\u0003\u0016BB+\u000bMo(T;H)=g\u0016B; (2)\nwhereg\u0003is the band Land\u0013 e factor. The exchange contribution is known as the\ngiant Zeeman splitting, as in moderately high magnetic \felds and low tem-\nperatures it attains values comparable to the Fermi energy or to the binding\nenergy of excitons and shallow impurities. For e\u000bective mass states, whose pe-\nriodic part of the Bloch function contains spin components mixed up by a spin-\norbit interaction, the exchange splitting does not depend only on the product\nofMoand the relevant exchange integral, say \f, but usually also on the mag-\nnitude and direction of Mo, con\fnement, and strain. Furthermore, because\nof con\fnement or non-zero kthe Bloch wave function contains contributions\nfrom both conduction and valence band, which a\u000bects the magnitude and\neven the sign of the spin splitting [62, 63, 49, 67]. The giant Zeeman splitting\nis clearly visible in magnetooptical phenomena as well as in the Shubnikov-\nde Haas e\u000bect, making an accurate determination of the exchange integrals\npossible, particularly in wide-gap materials, in which competing Landau and\nordinary spin splittings are small.\nThe possibility of tailoring the magnitude of spin splitting in DMS struc-\ntures o\u000bers a powerful tool to examine various phenomena. For instance, spin\nengineering was explored to control by the magnetic \feld the con\fnement of\ncarriers and photons [68], to map atom distributions at interfaces [69] as well\nas to identify the nature of optical transitions and excitonic states. Further-\nmore, a subtle in\ruence of spin splitting on quantum scattering amplitude\nof interacting electrons with opposite spins was put into evidence in DMS\nin the weakly localized regime in 3D [33], 2D [70, 71], and 1D systems [72].\nThe redistribution of carriers between spin levels induced by spin splitting\nwas found to drive an insulator-to-metal transition [73] as well as to gener-\nate universal conductance \ructuations in DMS quantum wires [72]. Since the\nspin splitting is greater than the cyclotron energy, there are no overlapping\nLandau levels in modulation-doped heterostructures of DMS in the quantum\nHall regime in moderately strong magnetic \felds. This made it possible to\ntest a scaling behavior of wave functions at the center of Landau levels [74].\nAt higher \felds, a crossing of Landau levels occurs, so that quantum Hall\nferromagnet could be evidenced and studied [75]. At the same time, it has\nbeen con\frmed that in the presence of a strong spin-orbit coupling (e.g., in\nthe case of p-type wave functions) the spin polarization can generate a large\nextraordinary (anomalous) Hall voltage [76]. Last but not least, optically [77]\nand electrically controlled spin-injection [78] and \fltering [79] were observed\nin all-semiconductor structures containing DMS .16 Tomasz Dietl\nSpin-disorder scattering\nSpatial \ructuations of magnetization , disregarded in the mean-\feld ap-\nproximation, lead to spin disorder scattering. According to the \ructuation-\ndissipation theorem, the corresponding scattering rate in the paramagnetic\nphase is proportional to T\u001f(T), where\u001f(T) is the magnetic susceptibility\nof the localized spins [12, 80]. Except to the vicinity of ferromagnetic phase\ntransitions, a direct contribution of spin-disorder scattering to momentum re-\nlaxation turns out to be small. In contrast, this scattering mechanism controls\nthe spin lifetime of e\u000bective mass carriers in DMS , as evidenced by studies\nof universal conductance \ructuations [81], line-width of spin-\rip Raman scat-\ntering [80], and optical pumping e\u000eciency [82]. Furthermore, thermodynamic\n\ructuations contribute to the temperature dependence of the band gap and\nband o\u000b-set. In the case when the total potential introduced by a magnetic ion\nis grater than the width of the carrier band, the virtual crystal and molecular\n\feld approximations break down, a case of the holes in Cd 1\u0000xMnxS. A non-\nperturbative scheme was developed [83, 84] to describe nonlinear dependencies\nof the band gap on xand of the spin splitting on magnetization observed in\nsuch situations.\n4.5 Magnetic polarons\nBound magnetic polaron (BMP), that is a bubble of spins ordered ferromag-\nnetically by the exchange interaction with an e\u000bective mass carrier in a lo-\ncalized state, modi\fes optical, transport, and thermodynamic properties of\nDMS . BMP is formed inside the localization radius of an occupied impurity\nor quantum dot state but also around a trapped exciton, as the polaron forma-\ntion time is typically shorter than the exciton lifetime [85]. The BMP binding\nenergy and spontaneous carrier spin-splitting are proportional to the magni-\ntude of local magnetization , which is built up by two e\u000bects: the molecular\n\feld of the localized carrier and thermodynamic \ructuations of magnetization\n[86, 87, 88, 12]. The \ructuating magnetization leads to dephasing and enlarges\nwidth of optical lines. Typically, in 2D and 3D systems, the spins alone cannot\nlocalized itinerant carriers but in the 1D case the polaron is stable even with-\nout any pre-localizing potential [83]. In contrast, a free magnetic polaron|a\ndelocalized carrier accompanied by a travelling cloud of polarized spins|is\nexpected to exist only in magnetically ordered phases. This is because co-\nherent tunnelling of quasi-particles dressed by spin polarization is hampered,\nin disordered magnetic systems, by a smallness of quantum overlap between\nmagnetization s in neighboring space regions. Interestingly, theory of BMP\ncan readily be applied for examining e\u000bects of the hyper\fne coupling between\nnuclear spins and carriers in localized states.Lecture Notes on Semiconductor Spintronics 17\n4.6 Exchange interactions between localized spins\nAs in most magnetic materials, classical dipole-dipole interactions between\nmagnetic moments are weaker than exchange couplings in DMS . Direct d-d\nor f-f exchange interactions, known from properties of magnetic dimmers, are\nthought to be less important than indirect exchange channels. The latter in-\nvolve a transfer of magnetic information via spin polarization of bands, which\nis produced by the exchange interaction or spin-dependent hybridization of\nmagnetic impurity and band states. If magnetic orbitals are involved in the\npolarization process, the mechanism is known as superexchange , which is\nmerely antiferromagnetic and dominates, except for p-type DMS . If fully oc-\ncupied band states are polarized by the sp-d exchange interaction, the result-\ning indirect d-d coupling is known as the Bloembergen-Rowland mechanism.\nIn the case of Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction, the d-d\ncoupling proceeds via spin polarization of partly \flled bands, that is by free\ncarriers. Since in DMS the sp-d is usually smaller than the width of the rele-\nvant band (weak coupling limit) as well as the carrier concentration is usually\nsmaller than those of localized spins, the energetics of the latter can be treated\nin the continuous medium approximation, an approach referred here to as the\nZener model. Within this model the RKKY interaction is ferromagnetic, and\nparticularly strong in p-type materials, because of a large magnitudes of the\nhole mass and exchange integral \f. It worth emphasizing that the Zener model\nis valid for any ratio of the sp-d exchange energy to the Fermi energy. Finally,\nin the case of systems in which magnetic ions in di\u000berent charge states coexist,\nhopping of an electron between magnetic orbitals of neighboring ions in dif-\nfering charge states tends to order them ferromagnetically. This mechanism,\ndoubted the double exchange , operates in manganites but its relevance in\nDMS has not yet been found.\nIn general, the bilinear part of the interaction Hamiltonian for a pair of\nspinsiandjis described by a tensor ^J,\nHij=\u00002S(i)^J(ij)S(j); (3)\nwhich in the case of the coupling between nearest neighbor cation sites in the\nunperturbed zinc-blende lattice contains four independent components. Thus,\nin addition to the scalar Heisenberg-type coupling, Hij=\u00002J(ij)S(i)S(j),\nthere are non-scalar terms (e.g., Dzialoshinskii-Moriya or pseudo-dipole).\nThese terms are induced by the spin-orbit interaction within the magnetic ions\nor within non-magnetic atoms mediating the spin-spin exchange . The non-\nscalar terms, while smaller than the scalar ones, control spin-coherence time\nand magnetic anisotropy . Typically, J(ij)\u0019\u00001 meV for nearest-neighbor\npairs coupled by the superexchange , and the interaction strength decays\nfast with the pair distance. Thus, with lowering temperature more and more\ndistant pairs become magnetically neutral, Stot= 0. Accordingly, the temper-\nature dependence of magnetic susceptibility assumes a modi\fed Curie form,\n\u001f(T) =C=T\r, where\r <1 and both Cand\rdepend on the content of the18 Tomasz Dietl\nmagnetic constituent x. Similarly, the \feld dependence of magnetization is\nconveniently parameterized by a modi\fed Brillouin function BS[89],\nMo(T;H) =Sg\u0016BNoxeffBS[Sg\u0016BH=kB(T+TAF)]; (4)\nin which two x{ andT{dependent empirical parameters, xeff<xandTAF>\n0, describe the presence of antiferromagnetic interactions.\n4.7 Magnetic collective phenomena\nIn addition to magnetic and neutron techniques [90], a variety of optical and\ntransport methods, including 1 =fnoise study of nanostructures [81], have suc-\ncessfully been employed to characterize collective spin phenomena in DMS .\nUndoped DMS belong to a rare class of systems, in which spin-glass freezing\nis driven by purely antiferromagnetic interactions, an e\u000bect of spin frustration\ninherent to the randomly occupied fcc sublattice. Typically, in II-VI DMS ,\nthe spin-glass freezing temperature Tgincreases from 0.1 K for x= 0:05 to\n20 K atx= 0:5 according to Tg\u0018x\u000e, where\u000e\u00192, which re\rects a short-\nrange character of the superexchange . For xapproaching 1, antiferromagnetic\ntype III ordering develops, according to neutron studies. Here, strain imposed\nby the substrate material{the strain engineering{can serve to select domain\norientations as well as to produce spiral structures with a tailored period [91].\nParticularly important is, however, the carrier-density controlled ferromag-\nnetism of bulk and modulation-doped p-type DMS described next.\n5 Properties of Ferromagnetic Semiconductors\n5.1 Overview\nSince for decades III-V semiconductor compounds have been applied as\nphotonic and microwave devices, the discovery of ferromagnetism \frst in\nIn1\u0000xMnxAs [92] and then in Ga 1\u0000xMnxAs by Hideo Ohno and collabora-\ntors in Sendai [93] came as a landmark achievement. In these materials, sub-\nstitutional divalent Mn ions provide localized spins and function as acceptor\ncenters that provide holes which mediate the ferromagnetic coupling between\nthe parent Mn spins [94, 95, 96]. In another technologically important group\nof semiconductors, in II-VI compounds, the densities of spins and carriers\ncan be controlled independently, similarly to the case of IV-VI materials, in\nwhich hole-mediated ferromagnetism was discovered by Tomasz Story et al.\nin Warsaw already in the 80s [97]. Stimulated by the theoretical predictions\nof the present author [94], laboratories in Grenoble and Warsaw, led by late\nYves Merle d'Aubign\u0013 e and the present author, joined e\u000borts to undertake\ncomprehensive research dealing with carrier-induced ferromagnetism in II-IV\nmaterials containing Mn. Experimental studies conducted with the use ofLecture Notes on Semiconductor Spintronics 19\nmagnetooptical and magnetic methods led to the discovery of ferromagnetism\nin 2D and [54] 3D II-VI materials [55] doped by nitrogen acceptors.\nGuided by the growing amount of experimental results, the present author\nand co-workers proposed a theoretical model of the hole-controlled ferromag-\nnetism in III-V, II-VI, and group IV semiconductors containing Mn [98, 99].\nIn these materials conceptual di\u000eculties of charge transfer insulators and\nstrongly correlated disordered metals are combined with intricate properties\nof heavily doped semiconductors, such as Anderson-Mott localization and de-\nfect generation by self-compensation mechanisms. Nevertheless, the theory\nbuilt on Zener's model of ferromagnetism and the Kohn-Luttinger kptheory\nof the valence band in tetrahedrally coordinated semiconductors has quan-\ntitatively described thermodynamic, micromagnetic, transport, and optical\nproperties of DMS with delocalized or weakly localized holes [98, 99, 21, 100],\nchallenging competing theories. It is often argued that owing to these stud-\nies Ga 1\u0000xMnxAs has become one of the best-understood ferromagnets. Ac-\ncordingly, this material is now employed as a testing ground for various ab\ninitio computation approaches to strongly correlated and disordered systems.\nMoreover, the understanding of the carrier-controlled ferromagnetic DMS has\nprovided a basis for the development of novel methods enabling magnetization\nmanipulation and switching.\n5.2 p-d Zener model\nIt is convenient to apply the Zener model of carrier-controlled ferromagnetism\nby introducing the functional of free energy density, F[M(r)]. The choice of\nthe local magnetization M(r) as an order parameter means that the spins\nare treated as classical vectors, and that spatial disorder inherent to magnetic\nalloys is neglected. In the case of magnetic semiconductors F[M(r)] consists\nof two terms,F[M(r)] =FS[M(r)] +Fc[M(r)], which describe, for a given\nmagnetization pro\fle M(r), the free energy densities of the Mn spins in the\nabsence of any carriers and of the carriers in the presence of the Mn spins, re-\nspectively. A visible asymmetry in the treatment of the carries and of the spins\ncorresponds to an adiabatic approximation: the dynamics of the spins in the\nabsence of the carriers is assumed to be much slower than that of the carriers.\nFurthermore, in the spirit of the virtual-crystal and molecular-\feld approxi-\nmations, the classical continuous \feld M(r) controls the e\u000bect of the spins\nupon the carriers. Now, the thermodynamics of the system is described by the\npartition function Z, which can be obtained by a functional integration of the\nBoltzmann factor exp( \u0000R\ndrF[M(r)]=kBT) over all magnetization pro\fles\nM(r) [87, 88]. In the mean-\feld approximation (MFA), a term correspond-\ning to the minimum of F[M(r)] is assumed to determine Zwith a su\u000ecient\naccuracy.\nIf energetics is dominated by spatially uniform magnetization M, the spin\npart of the free energy density in the magnetic \feld Hcan be written in the\nform [101]20 Tomasz Dietl\nFS[M] =ZM\n0dMoh(Mo)\u0000MH: (5)\nHere, h(Mo) denotes the inverse function to Mo(h), where Mois the avail-\nable experimentally macroscopic magnetization of the spins in the absence of\ncarriers in the \feld hand temperature T. In DMS , it is usually possible to pa-\nrameterizeMo(h) by the Brillouin function that, according to Eq. 4, takes the\npresence of intrinsic short-range antiferromagnetic interactions into account.\nNearTCand forH= 0,Mis su\u000eciently small to take Mo(T;h) =\u001f(T)h,\nwhere\u001f(T) is the magnetic susceptibility of localized spins in the absence of\ncarriers. Under these conditions,\nFS[M] =M2=2\u001f(T); (6)\nwhich shows that the increase of FSwithMslows down with lowering tem-\nperature, where \u001f(T) grows. Turning to Fc[M] we note that owing to the giant\nZeeman splitting of the bands proportional to M, the energy of the carriers,\nand thusFc[M], decreases with jMj,Fc[M]\u0000Fc[0]\u0018\u0000M2. Accordingly, a\nminimum ofF[M] at non-zero Mmay develop in H= 0 at su\u000eciently low\ntemperatures signalizing the appearance of a ferromagnetic order.\nThe present authors and co-workers [98] found that the minimal hamil-\ntonian necessary to describe properly e\u000bects of the complex structure of the\nvalence band in tetrahedrally coordinated semiconductors upon Fc[M] is the\nLuttinger 6\u00026kpmodel supplemented by the p-d exchange contribution\ntaken in the virtual crystal and molecular \feld approximations,\nHpd=\fsM=g\u0016B: (7)\nThis term leads to spin splittings of the valence subbands, whose magnitudes|\nowing to the spin-orbit coupling|depend on the hole wave vectors kin a\ncomplex way even for spatially uniform magnetization M. It would be tech-\nnically di\u000ecult to incorporate such e\u000bects to the RKKY model, as the spin-\norbit coupling leads to non-scalar terms in the spin-spin Hamiltonian. At the\nsame time, the indirect exchange associated with the virtual spin excitations\nbetween the valence subbands, the Bloembergen-Rowland mechanism, is au-\ntomatically included. The model allows for biaxial strain, con\fnement, and\nwas developed for both zinc blende and wurzite materials [99]. Furthermore,\nthe direct in\ruence of the magnetic \feld on the hole spectrum was taken\ninto account. Carrier-carrier spin correlation was described by introducing a\nFermi-liquid-like parameter AF[94, 54, 96], which enlarges the Pauli suscep-\ntibility of the hole liquid. No disorder e\u000bects were taken into account on the\nground that their in\ruence on thermodynamic properties is relatively weak\nexcept for strongly localized regime. Having the hole energies, the free energy\ndensityFc[M] was evaluated according to the procedure suitable for Fermi\nliquids of arbitrary degeneracy. By minimizing F[M] =FS[M]+Fc[M] with\nrespect to Mat givenT,H, and hole concentration p, Mn spin magnetization\nM(T;H) was obtained as a solution of the mean-\feld equation,Lecture Notes on Semiconductor Spintronics 21\nM(T;H) =xeffNog\u0016BSBS[g\u0016B(\u0000@Fc[M]=@M+H)=kB(T+TAF)];(8)\nwhere peculiarities of the valence band structure, such as the presence of var-\nious hole subbands, anisotropy, and spin-orbit coupling, are hidden in Fc[M].\nNear the Curie temperature TCand atH= 0, where Mis small, we expect\nFc[M]\u0000Fc[0]\u0018\u0000M2. It is convenient to parameterize this dependence by\na generalized carrier spin susceptibility ~ \u001fc, which is related to the magnetic\nsusceptibility of the carrier liquid according to ~ \u001fc=AF(g\u0003\u0016B)2\u001fc. In terms\nof ~\u001fc,\nFc[M] =Fc[0]\u0000AF~\u001fc\f2M2=2(g\u0016B)2: (9)\nBy expanding B S(M) for small Mone arrives to the mean-\feld formula for\nTC=TF\u0000TAF, whereTFis given by\nTF=xeffNoS(S+ 1)AF~\u001fc(TC)\f2=3kB: (10)\nFor a strongly degenerate carrier liquid j\u000fFj=kBT\u001d1, as well as ne-\nglecting the spin-orbit interaction ~ \u001fc=\u001a=4, where\u001ais the total density-\nof-states for intra-band charge excitations, which in the 3D case is given by\n\u001a=m\u0003\nDOSkF=\u00192\u0016h2. In this case and for AF= 1,TFassumes the well-known\nform, derived already in 40s in the context of carrier-mediated nuclear ferro-\nmagnetism [102]. In general, however, ~ \u001fchas to be determined numerically\nby computingFc[M] for a given band structure and degeneracy of the carrier\nliquid. The model can readily be generalized to various dimensions as well as\nto the case, when Mis not spatially uniform in the ground state.\nThe same formalism, in addition to TCand Mn magnetization M(T;H), as\ndiscussed above, provides also quantitative information on spin polarization\nand magnetization of the hole liquid [99]. Furthermore, it can be exploited\nto describe chemical trends as well as micromagnetic, transport, and optical\nproperties of ferromagnetic DMS , the topics discussed in the subsequent\nsections.\n5.3 Curie Temperature { Chemical Trends\nLarge magnitudes of both density of states and exchange integral speci\fc to\nthe valence band make TFto be much higher in p-type than in n-type mate-\nrials with a comparable carrier concentration. Accordingly, in agreement with\ntheoretical evaluations [94], no ferromagnetic order was detected above 1 K\nin n-(Zn,Mn)O:Al, even when the electron concentration exceeded 1020cm\u00003\n[103]. At the same time, theoretical calculations carried out with no adjustable\nparameters explained satisfactorily the magnitude of TCin both (Ga,Mn)As\n[98, 104] and p-type (Zn,Mn)Te [55]. Furthermore, theoretical expectations\nwithin the p-d Zener model are consistent with chemical trends in TCvalues\nobserved experimentally in (Ga,Mn)Sb, (Ga,Mn)P, (In,Mn)As, (In,Mn)Sb,\n(Ge,Mn), and p-(Zn,Be)Te though e\u000bects of hole localization [99, 55] preclude\nthe appearance of a uniform ferromagnetic order with a univocally de\fned TC22 Tomasz Dietl\nvalue in a number of cases. In addition to localization, a competition between\nlong-range ferromagnetic interactions and intrinsic short-range antiferromag-\nnetic interactions [100], as described by TAF>0 andxeff< x, may a\u000bect\nthe character of magnetic order [105]. It appears that the e\u000bect is more rel-\nevant in II-VI DMS than in III-V DMS where Mn centers are ionized, so\nthat the enhanced hole density at closely lying Mn pairs may compensate\nantiferromagnetic interactions [98]. In both groups of materials the density\nof compensating donor defects appear to grow with the Mn concentration\n[95, 55]. In the case of (Ga,Mn)As the defect involved is the Mn interstitial\n[106], which can be driven and passivated at the surface be low temperature\nannealing [107].\nAccording to evaluations carried out by the present author and co-workers\n[98] room temperature ferromagnetism could be observed in a weakly compen-\nsated (Ga,Mn)As containing at least 10% of Mn. At the same time, because\nof stronger p-d hybridization in wide band-gap materials, such as (Ga,Mn)N\nand (Zn,Mn)O, TC>300 K is expected already for x= 5%, provided that\nthe hole concentration would be su\u000eciently high. However, it was clear from\nthe beginning [98] that the enhancement of the hole binding energy by p-d\nhybridization as well as a limited solubility of magnetic constituent together\nwith the e\u000bect of self-compensation may render the fabrication of high tem-\nperature ferromagnetic DMS challenging. Nevertheless, a number of group has\nstarted the growth of relevant systems, the e\u000bort stimulated even further by a\nnumber of positive results as well as by numerous theoretical papers suggest-\ning, based on ab initio computations, that high temperature ferromagnetism\nis possible in a large variety of DMS even without band holes. Today, however,\na view appears to prevail that the high temperature ferromagnetism, as evi-\ndenced by either magnetic, magnetotransport or magnetooptical phenomena,\nresults actually from the presence of precipitates of known or so-far unknown\nferromagnetic or ferrimagnetic nanocrystals containing a high density of mag-\nnetic ions. At the same time, it becomes more and more clear that the ab initio\ncomputations in question su\u000bered from improper treatment of correlation and\ndisorder, which led to an overestimation of tendency towards a ferromagnetic\norder. It seems at the end that, as argued initially [94, 98], the delocalized or\nweakly localized holes are necessary to stabilize a long-range ferromagnetic or-\nder in tetrahedrally coordinated DMS with a small concentration of randomly\ndistributed magnetic ions.\n5.4 Micromagnetic properties\nMagnetic anisotropy\nAs the energy of dipole-dipole magnetic interactions depends on the dipole\ndistribution, there exists the so-called shape anisotropy. In particular, for thin\n\flms, the di\u000berence in energy density corresponding to the perpendicular and\nin-plane orientation of magnetization M is given byLecture Notes on Semiconductor Spintronics 23\nE=\u0016oM2=2; (11)\nwhich leads to the anisotropy \feld \u0016oHA=\u0016oMof about 60 mT for\nGa0:95Mn0:05As.\nAlready early studies of the ferromagnetic phase in (In,Mn)As [108] and\n(Ga,Mn)As [109] demonstrated the existence of magnetic anisotropy , whose\ncharacter and magnitude implied a sizable contribution of a microscopic ori-\ngin. Magneto-crystalline anisotropy is usually associated with the interaction\nbetween spin and orbital degrees of freedom of the magnetic ion d-electrons.\nAccording to the model advocated here, these electrons are in the d5con\fgu-\nration. For such a case the orbital momentum L= 0, so that e\u000bects stemming\nfrom the spin-orbit coupling are expected to be rather weak. It was, however,\nbeen noted that the interaction between the localized spins is mediated by the\nholes that have a non-zero orbital momentum l= 1 [98]. An important aspect\nof the p-d Zener model is that it does take into account the anisotropy of the\ncarrier-mediated exchange interaction associated with the spin-orbit coupling\nin the host material [98, 99, 110].\n0.1 1 100.00.20.40.60.81.0 TR / TCx = 5.3%\nεεεεxx = -0.27%\n<100>[001]\n<110>\n \nHole concentration    [ 1020cm-3 ][001]\n<110>\nFig. 7. Experimental (full points) and computed values (thick lines) of the ratio\nof the reorientation to Curie temperature for the transition from perpendicular to\nin-plane magnetic anisotropy . Dashed lines mark expected temperatures for the\nreorientation of the easy axis between h100iandh110iin-plane directions (after\nSawicki et al. [111]).\nA detail theoretical analysis of anisotropy energies and anisotropy \felds in\n\flms of (Ga,Mn)As was carried out for a number of experimentally important\ncases within the p-d Zener model [99, 110]. In particular, the cubic anisotropy\nas well as uniaxial anisotropy under biaxial epitaxial strain were examined as\na function of the hole concentration p. Both shape and magneto-crystalline24 Tomasz Dietl\nanisotropies were taken into account. The perpendicular and in-plane orien-\ntation of the easy axis is expected for the compressive and tensile strain, re-\nspectively, provided that the hole concentration is su\u000eciently small. However,\naccording to theory, a reorientation of the easy axis direction is expected at\nhigher hole concentrations. Furthermore, in a certain concentration range the\ncharacter of magnetic anisotropy is computed to depend on the magnitude\nof spontaneous magnetization , that is on the temperature. The computed\nphase diagram for the reorientation transition compared to the experimental\nresults for a \flm is shown in Fig. 7. In view that theory is developed with\nno adjustable parameters the agreement between experimental and computed\nconcentrations and temperature corresponding to the reorientation transition\nis very good. Furthermore, the computed magnitudes of the anisotropy \feld\nHu[99] are consistent with the available \fndings for both compressive and\ntensile strain.\nAccording to the discussion above, the easy axis assumes the in-plane\norientation for typical carrier concentrations in the most thoroughly studied\nsystem (Ga,Mn)As/GaAs. In this case the easy axis is expected to switch\nbetweenh100iandh110iin-plane cubic directions as a function of p[99, 110].\nSurprisingly, however, only the h100ibiaxial magnetic symmetry has so-far\nbeen observed in \flms of (Ga,Mn)As/GaAs at low temperatures. Nevertheless,\nthe corresponding in-plane anisotropy \feld assumes the expected magnitude,\nof the order of 0.1 T, which is typically much smaller than that correspond-\ning to the strain-induced energy of magnetic anisotropy . It is possible that\nanisotropy of the hole magnetic moment, neglected in the theoretical calcula-\ntions [99, 110], stabilizes the h100iorientation of the easy axis.\nIn addition to the cubic in-plane anisotropy, the accumulated data for\nboth (Ga,Mn)As/GaAs and (In,Mn)As/(In,Al)As point to a non-equivalence\nof [110] and [-110] directions, which leads to the in-plane uniaxial magnetic\nanisotropy . Such a uniaxial anisotropy is not expected for D 2dsymmetry of\na Tdcrystal under epitaxial strain [112, 113]. Furthermore, the magnitude\nof the corresponding anisotropy \feld appears to be independent of the \flm\nthickness [114], which points to a puzzling symmetry breaking in the \flm\nbody.\nMagnetic sti\u000bness and domain structure\nAnother important characteristics of any ferromagnetic system is magnetic\nsti\u000bnessA, which describes the energy penalty associated with the local twist-\ning of the direction of magnetization . Remarkably, Adetermines the mag-\nnitude and character of thermodynamic \ructuations of magnetization , the\nspectrum of spin excitations as well as the width and energy of domain walls.\nAn important result is that the magnetic sti\u000bness computed within the 6 \u00026\nLuttinger model is almost by a factor of 10 greater than that expected for a\nsimple spin degenerate band with the heave-hole band-edge mass [115]. This\nenhancement, which stabilizes strongly the spatially uniform spin ordering,Lecture Notes on Semiconductor Spintronics 25\nstems presumably from p-like symmetry of the valence band wave functions,\nas for such a case the carrier susceptibility (the Lindhard function) decreases\nstrongly with q[116].\nThe structure of magnetic domains in (Ga,Mn)As under tensile strain has\nbeen determined by micro-Hall probe imaging [117]. The regions with magne-\ntization oriented along the [001] and [00-1] easy axis form alternating stripes\nextending in the [110] direction. As shown in Fig. 8, the experimentally deter-\nmined stripe width is W= 1:5\u0016m at 5 K for 0.2 \u0016m \flm of Ga 0:957Mn0:043As\non Ga 0:84In0:16As, for which tensile strain of \u000fxx= 0:9% is expected. Accord-\ning to micromagnetic theory, Wis determined by the ratio of the domain\nwall energy to the stray \feld energy. As shown in Fig. 8, the computed value\nwith no adjustable parameters W= 1:1\u0016m [118] compares favorably with the\nexperimental \fnding, W= 1:5\u0016m at low temperatures. However, the model\npredicts much weaker temperature dependence of Wthan that observed ex-\nperimentally, which was linked [118] to critical \ructuations, disregarded in the\nmean-\feld approach.\n0.0 0.2 0.4 0.6 0.8 1.00246810\nGa0.957Mn0.043As\n Domain width [ µm]\nReduced temperature T/TC\nFig. 8. Temperature dependence of the width of domain stripes as measured by\nShono et al. [117] for the Ga 0:957Mn 0:043As \flm with the easy axis along the growth\ndirection (full squares). Computed domain width is shown by the solid line (after\nDietl et al. [118]).\n5.5 Optical properties\nMagnetic circular dichroism\nWithin the Zener model, the strength of the ferromagnetic spin-spin interac-\ntion is controlled by the k\u0001pparameters of the host semiconductor and by the26 Tomasz Dietl\nmagnitude of the spin-dependent coupling between the e\u000bective mass carriers\nand localized spins. In the case of II-VI DMS , detailed information on the ex-\nchange -induced spin-splitting of the bands, and thus on the coupling between\nthe e\u000bective mass electrons and the localized spins has been obtained from\nmagnetooptical studies [12]. A similar work on (Ga,Mn)As [119, 120, 121]\nled to a number of surprises. The most striking was the opposite order of\nthe absorption edges corresponding to the two circular photon polarizations\nin (Ga,Mn)As comparing to II-VI materials. This behavior of circular mag-\nnetic dichroism (MCD ) suggested the opposite order of the exchange -split\nspin subbands, and thus a di\u000berent origin of the sp-d interaction in these\ntwo families of DMS . A new light on the issue was shed by studies of pho-\ntoluminescence (PL) and its excitation spectra (PLE) in p-type (Cd,Mn)Te\nquantum wells [54]. As shown schematically in Fig. 9, the reversal of the order\nof PLE edges corresponding to the two circular polarizations results from the\nMoss-Burstein e\u000bect, that is from the shifts of the absorption edges associated\nwith the empty portion of the valence subbands in the p-type material.\nThe above model was subsequently applied to interpret the magnetoab-\nsorption data for metallic (Ga,Mn)As [120, 99]. More recently, the theory was\nextended by taking into account the e\u000bect of scattering-induced mixing of k\nstates [122]. As shown in Fig. 10, this approach explains the slop of the ab-\nsorption edge as well as its \feld-induced splitting assuming the value of the\np-d exchange energy \fN0=\u00001 eV.\nRecently, the formalisms suitable for description of either interband [99]\nor intraband [123] optical absorption were combined [124] in order to examine\ntheoretically optical (dynamic) conductivity in the whole spectral range up to\n2 eV. Furthermore a possible presence of optical absorption involving defect\nstates was taken into account. In this way, the most general quantitative the-\nory of optical and magnetoptical e\u000bects in magnetic semiconductors available\nto date was worked out. A good quantitative description of experimental data\n[125, 126] was obtained verifying the model. However, some discrepancies in\nthe low photon energy range were detected, which con\frmed the presence of\nquantum localization e\u000bects. At the same time, a disagreement in the high en-\nergy region pointed to the onset of intra-d band transitions. The Faraday and\nKerr rotations were also computed showing a large magnitude and a complex\nspectral dependence in the virtually whole studied photon energy range up to\n2 eV, which suggests a suitability of this material family for magnetooptical\napplications.\n5.6 Charge transport phenomena\nHall e\u000bect in ferromagnetic semiconductors - theory\nThe assessment of magnetic characteristics by means of magnetotransport\nstudies is of particular importance in the case of thin \flms of diluted magnets,\nin which the magnitude of the total magnetic moment is typically small. ForLecture Notes on Semiconductor Spintronics 27\n1640 1660 1680 1700substrateσ-\nσ+50 kOe150 OePLE (a.u.)\nEnergy (meV)#1\n2 K\n500 Oe50 Oe0 Oe\nA. Haury et al.,  Fig. 1σ-σ+\nPLEPL\nFig. 9. Photoluminescence excitation spectra (PLE), that is the photoluminescence\n(PL) intensity as a function of the excitation photon energy intensity, for \u001b+(solid\nlines) and\u001b\u0000(dotted lines) circular polarizations at selected values of the mag-\nnetic \feld in a modulation-doped p-type quantum well of Cd 0:976Mn 0:024Te at 2 K.\nThe photoluminescence was collected in \u001b+polarization at energies marked by the\nnarrowest features. The sharp maximum (vertical arrow) and step-like form (hori-\nzontal arrow) correspond to quasi-free exciton and transitions starting at the Fermi\nlevel, respectively. Note reverse ordering of transition energies at \u001b+and\u001b\u0000for PL\nand PLE (the latter is equivalent to optical absorption). The band arrangement at\n150 Oe is sketched in the inset (after Haury et al. [54]).28 Tomasz Dietl\nOne of the most attractive features of Ga 12xMnxAs is its\nDMS-type behavior, i.e., strong interaction between delocal-\nizeds- andp-type band electrons and localized d-type elec-\ntrons of the magnetic ions ~so called s,p-dexchange inter-\naction !. In particular, this interaction yields band splittings of\nthe order of 100 meV, which corresponds to an effective g\nfactor of a few hundreds.10,15The interaction for the conduc-\ntion band ( s-dexchange !is driven by direct potential ex-\nchange and should always be ferromagnetic ~FM!,21which is\nindeed the case for all DMS examples known so far.10,22On\nthe other hand, valence band p-dexchange is dominated by\nthe kinetic exchange mechanism and can be both ferromag-netic and antiferromagnetic ~AFM !, depending on exchange\nchannels, i.e., available paths for virtual electron jumps be-tween the valence band and dorbitals.\n22,23\nThe energy band structure of a zinc-blende-type diluted\nmagnetic semiconductor in the vicinity of the Gpoint is\nshown in Fig. 2. The exchange interaction induces bandssplitting. The energies of the electronic transitions to the\nconduction band from the heavy holes E\nhh-c, light holes\nElh-c, and spin-orbit split band Eso-cins1ands2polar-\nized light in the vicinity of the Gpoint are as follows.24\nFors1,\nEhh-c,s15Eg13b23a,\nElh-c,s15Eg1b13a,\nEso-c,s15Eg2b13a,\nand for s2,\nEhh-c,s25Eg23b13a,\nElh-c,s25Eg2b23a,\nEso-c,s25Eg1b23a, ~10a!\nwherea51\n6N0ax^2S&,b51\n6N0bx^2S&. ~10b!\nN0aandN0bare the exchange integrals for the conduction\n(s-d) and valence bands ( p-d), and ^S&is the thermody-\nnamic average of the spin Sand is proportional to the mag-\nnetization of the system of magnetic ions.10\nFor Ga 12xMnxAs epilayers ( x’0.03–0.04 !thep-dex-\nchange interaction was found to be antiferromagnetic ( N0b\n,0).15Transmission was measured in the spectral range\n1.4–2.0 eV, at temperatures 2 K ,T,60 K, and magnetic\nfield up to 5 T. Circularly polarized light ( s2ands1) was\nused. Detailed information about this experiment and thesample preparation can be found in Ref. 15.\nThe sample absorption spectra are shown ~by points !in\nFig. 3. Instead of a sharp absorption edge a broad, ramplikeedge is observed. The highest value of the detectable absorp-\ntion coefficient in this experiment was about 10\n4cm21. Un-\nder external magnetic field the absorption edge splits byabout 100 meV. Such strong edge splitting is characteristic\nfors,p-dexchange effects.\n10,24Thes2edge is redshifted\nwith respect to the s1edge, which is attributed to the Moss-\nBurstein shift.15\nFor epilayers with hole concentration ranging between\n1018and 1020cm23, the fact that the top of the valence band\nis empty ~filled with holes !must be taken into account, since\nthe Fermi energy level Fcan be up to about 300 meV below\nthe top of the valence band. Consequently the Moss-Bursteinshift of the absorption edge becomes sizable and is differentfor transitions originating from different valence subbands\nsplit bys,p-dexchange interaction ~Fig. 4 !. It appears that\nin this case, assuming direct transitions, for FM s-dex-\nchange (N\n0a.0) and AFM p-dexchange ( N0b,0),s1FIG. 2. Energy band structure of zinc-blende-type diluted mag-\nnetic semiconductor in the vicinity of the Brillouin zone Gpoint.\nThe right panel shows the allowed interband optical transitionsin the Faraday configuration under external magnetic field.\ns1\nand s2transitions are marked. The bold numbers give the\nrelative intensities.  \n  \n    \n    \n   \n   \nFIG. 3. The absorption coefficient of Ga12xMnxAs (x\n50.032). The experimental data are shown by points ~Ref. 12 !.The\nopen ~closed !symbols correspond to s1(s2) polarization of light.\nThe zero field absorption data ~a!are moved up relative to the\nmeasurements at 5.0 T ~b!for clarity. The results of the model\ncalculations for each polarization are presented by solid lines. Thevalues of p57.0310\n19cm23,N0b50.0~a!,N0b521.0 eV ~b!,\nandsk51/14 Å21were used.J. SZCZYTKO, W. BARDYSZEWSKI, AND A. TWARDOWSKI PHYSICAL REVIEW B 64075306\n075306-4\nFig. 10. Transmission of Ga 0:968Mn 0:032As \flm for two circular light polarizations\nin the Faraday con\fguration in the absence of the magnetic \feld (data shifted up\nfor clarity) and in 5 T at 2 K (points) [120]. Solid lines are calculated for the hole\nconcentration p= 7\u00021019cm\u00003, exchange energy N0\f=\u00001 eV, and allowing for\nscattering-induced breaking of the kselection rules [122].\nthis reason, recent years have witnessed a renewed interest in the nature of\nthe anomalous Hall e\u000bect (AHE), which{if understood theoretically{can serve\nto determine the magnitude of magnetization . Also magnetoresistance, to be\ndiscussed later on, provides information on the magnetism and on the interplay\nbetween electronic and magnetic degrees of freedom.\nThe Hall resistance RHall\u0011\u001ayx=dof a \flm of the thickness dis empiri-\ncally known to be a sum of ordinary and anomalous Hall terms in magnetic\nmaterials [127],\nRHall=R0\u0016oH=d +RS\u0016oM=d: (12)\nHere,R0andRSare the ordinary and anomalous Hall coe\u000ecients, respectively\n(R0>0 for the holes), and M(T;H) is the component of the magnetization\nvector perpendicular to the sample surface. While the ordinary Hall e\u000bect\nserves to determine the carrier density, the anomalous Hall e\u000bect (known also\nas the extraordinary Hall e\u000bect) provides valuable information on magnetic\nproperties of thin \flms. The coe\u000ecient RSis usually assumed to be propor-\ntional toR\u000b\nsheet, whereRsheet(T;H) is the sheet resistance and the exponent\n\u000bdepends on the mechanisms accounting for the AHE.\nIf the e\u000bect of stray magnetic \felds produced by localized magnetic mo-\nments were been dominating, RSwould scale with magnetization Mbut would\nbe rather proportional to R0than toRsheet. There is no demagnetization ef-\nfect in the magnetic \feld perpendicular to the surface of a uniformly magne-Lecture Notes on Semiconductor Spintronics 29\ntized \flm,B=\u0016oH. However, this is no longer the case in the presence of\nmagnetic precipitates, whose stray \felds and AHE may produce an apparent\nmagnetization -dependent contribution the host Hall resistance.\nWhen e\u000bects of stray \felds can be disregarded, spin-orbit interactions\ncontrol totally RS. In such a situation \u000bis either 1 or 2 depending on the\norigin of the e\u000bect: the skew-scattering (extrinsic) mechanism, for which the\nHall conductivity is proportional to momentum relaxation time \u001c, results in\n\u000b\u00191 [127]. From the theory point of view particularly interesting is the\nintrinsic mechanism for the Hall conductivity \u001bAH=RSM=(Rsheetd)2] does\nnot depend explicitly on scattering e\u000eciency but only on the band structure\nparameters [128, 129, 21].\nFor both extrinsic and intrinsic mechanisms, the overall magnitude of the\nanomalous Hall resistance depends on the strength of the spin-orbit interac-\ntion and spin polarization of the carriers at the Fermi surface. Accordingly,\nat given magnetization M, the e\u000bect is expected to be much stronger for\nthe holes than for the electrons in tetrahedrally coordinated semiconductors.\nFor the carrier-mediated ferromagnetism, the latter is proportional to the\nexchange coupling of the carriers to the spins, and varies { not necessarily\nlinearly { with the magnitude of spin magnetization M. Additionally, the\nskew-scattering contribution depends on the asymmetry of scattering rates\nfor particular spin subbands, an e\u000bect which can depend on Min a highly\nnontrivial way. Importantly, the sign of either of the two contributions can\nbe positive or negative depending on a subtle interplay between the orienta-\ntions of orbital and spin momenta as well as on the character (repulsive vs.\nattractive) of scattering potentials.\nRecently, Jungwirth et al. [129] developed a theory of the intrinsic AHE in\np-type zinc-blende magnetic semiconductors, and presented numerical results\nfor the case of (Ga,Mn)As, (In,Mn)As, and (Al,Mn)As. The derived formula\nfor\u001bAHcorresponds to that given earlier [128, 130, 131] in the weak scatter-\ning limit. The intrinsic AHE can also be regarded as a zero-frequency limit\nof\u001bxy(!), where\u001b(!) is the dynamic (optical) conductivity tensor, related\ndirectly the Kerr e\u000bect, widely studied in experimentally and theoretically in\nferromagnetic metals [132]. For the hole concentration psuch that the Fermi\nenergy is much smaller than the spin-orbit splitting \u0001obut larger than the\nexchange splitting hbetween the majority jz=\u00003=2 and minority jz= +3=2\nbands atk= 0,\u0001o\u001dj\u000fFj\u001dh, Jungwirth et al. [129] predict within the\n4\u00024 spherical Luttinger model\n\u001bin\nAH=e2hmhh=[4\u00192\u0016h3(3\u0019p)1=3]: (13)\nHere the heavy hole mass mhhis assumed to be much larger than the light\nhole massmlh, whereas\u001bin\nAHbecomes by the factor of 24=3greater in the\nopposite limit mhh=mlh. In the range h\u001cj\u000fFj\u001c\u0001othe determined value\nof\u001bin\nAHis positive, that is the coe\u000ecients of the normal and anomalous Hall\ne\u000bects are expected to have the same sign. However, if the Fermi level were\napproached the split-o\u000b \u00007band, a change of sign would occur.30 Tomasz Dietl\nA formula for \u001bin\nAHwas also derived [133] from Eq. 4 of Jungwirth et al.\n[129]), employing the known form of the heavy hole Bloch wave functions\nuk;jz[116]. Neglecting a small e\u000bect of the spin splitting on the heavy hole\nwave functions, \u001bin\nAHwas found to be given by the right hand side of Eq. 14\nmultiplied by the factor (16 =9) ln 2\u00001=6\u00191:066 [133].\nIn order to evaluate the ratio of intrinsic and skew-scattering mechanisms,\nthe general theory of the AHE e\u000bect in semiconductors [128, 130, 131, 129]\nwas applied [15]. Assuming that scattering by ionized impurities dominates,\nthis ratio is then given by [134, 135, 131],\n\u001bin\nAH\n\u001bss\nAH=\u0006f(\u0018)(NA+ND)=(prskF`): (14)\nHere,f(\u0018)\u001910 is a function that depends weakly on the screening dimen-\nsionless parameter \u0018; (NA+ND)=pis the ratio of the ionized impurity and\ncarrier concentrations; rsis the average distance between the carriers in the\nunits of the e\u000bective Bohr radius, and `is the mean free path. Similarly, for\nspin-independent scattering by short range potentials, V(r) =V\u000e(r\u0000ri) [130]\nwas applied [133]. Assuming that scattering by ionized impurities is negligible,\n\u001bin\nAH\n\u001bss\nAH=\u00003=[\u0019V\u001a(\"F)kF`]; (15)\nwhere\u001a(\"F) is the density of states at the Fermi level. Of course, the overall\nsign depends on the sign of the scattering potential V.\nIn order to \fnd out which of the two AHE mechanisms operates predomi-\nnantly in p-type tetrahedrally coordinated ferromagnetic semiconductors, we\nnote that scattering by ionized impurities appears to dominate in these heav-\nily doped and compensated materials. This scattering mechanism, together\nwith alloy and spin disorder scattering, limits presumably the hole mobility\nand leads ultimately to the metal-to-insulator transition (MIT). Since at the\nMITrs\u00192 andkF`\u00191 one expects from Eq. 15 that as long as the holes re-\nmain close to the localization boundary the intrinsic mechanism accounts for\nthe AHE. It would be interesting to know how quantum localization correc-\ntions a\u000bect the anomalous Hall conductivity as well as how to extend theory\ntowards the insulator side of the MIT. A work in this direction was reported\n[136].\nObviously, the presence of the AHE makes a meaningful determination of\nthe carrier type and density di\u000ecult in ferromagnetic semiconductors. Usually,\nthe ordinary Hall e\u000bect dominates only in rather high magnetic \felds or at\ntemperatures several times larger than TC. It appears, therefore, that a careful\nexperimental and theoretical examination of the resistivity tensor in wide \feld\nand temperature ranges is necessary to separate characteristics of the spin and\ncarrier subsystems.Lecture Notes on Semiconductor Spintronics 31\nComparison between theoretical and experimental results\nAs mentioned above, because of the dominance of the anomalous Hall term in\nwide temperature and \feld ranges, it is not straightforward to determine the\ncarrier type and concentration in ferromagnetic semiconductors. Only at low\ntemperatures and under very high \felds, the anomalous Hall term saturates,\nso that the ordinary Hall coe\u000ecient can be determined from the remaining\nlinear change of the Hall resistance in the magnetic \feld. Note that although\nmagnetization saturates in relatively low magnetic \felds, the negative MR\nusually persists, and generates the \feld dependence of the anomalous Hall\ncoe\u000ecient.\nMagnetotransport data collected for (Ga,Mn)As in a wide temperature and\n\feld ranges [95, 137] were exploited to test the theory of the AHE [129]. The\nresults of such a comparison are shown in Fig. 11. There is a good agreement\nbetween the theoretical and experimental magnitude of the Hall conductivity.\nImportantly, no signi\fcant contribution from skew scattering is expected for\nthe (Ga,Mn)As sample in question [137], for which ( NA+ND)=p\u00195,rs\u00191:1,\nandkF`\u00190:8, so that\u001bin\nAH=\u001bss\nAH\u001957.\nAnother material for which various contributions to Hall resistance were\nanalyzed is Zn 0:981Mn0:019Te:N containing 1 :2\u00021020holes per cm3[133]. In\nFig. 12,\u001ayx=\u001axx\u0000\u0016B, i.e., the spin dependent Hall angle, is compared to the\nmagnetization measured in a vibrating sample magnetometer [138] for this\n\flm. The normal Hall angle \u0016B=\u0016\u0016oHwas subtracted assuming a constant\nhole mobility \u0016i.e., assigning the conductivity changes entirely to variations\nin the hole concentration. This assumption is not crucial for the present highly\ndoped sample, but it proves to be less satisfactory for the less doped samples.\nAs shown in Fig. 12, a reasonable agreement is found by taking,\n\u001ayx=\u001axx=\u0016B+\u0002M=MS; (16)\nwhereMSis the saturation value of magnetization and \u0002= 0:04 is the ad-\njustable parameter. For the sample in question, the maximum value of hole\npolarization, ( pup\u0000pdown)=(pup+pdown), has been estimated to be of the\norder of 10% [138].\nHere, similarly to the case of (Ga,Mn)As, the sign and magnitude of the\nanomalous Hall coe\u000ecient indicated that the intrinsic mechanism is involved.\nThe value of \u0002was evaluated theoretically from Eq. 13 by adopting parame-\nters suitable for the sample in question, mhh= 0:6mo,\u001axx= 5\u000210\u00003\ncm\nand the saturation value of the splitting h= 41 meV. This leads to \u001bin\nAH=\n13:1 (\ncm)\u00001and\u0002in= 0:065 [133], in a reasonable agreement with the ex-\nperimental value \u0002= 0:04. Since a contribution from the light hole band will\nenhance the theoretical value, it was concluded [133] that the present theory\ndescribes the anomalous hole e\u000bect within the factor of about two.\nIt is important to note that there exist several reasons causing that the\nHall e\u000bect and direct magnetometry can provide di\u000berent information on mag-\nnetization . Indeed, contrary to the standard magnetometry, the AHE does32 Tomasz Dietl\nVOLUME 88, N UMBER 2 0 PHYSICAL REVIEW LETTERS 2 0M AY2002\nHere p/H33527k3\nhh/H208623p2/H2085111/H20849mlh/H20862mhh/H208503/H208622/H20852is the total hole\ndensity and mlh/H33527m/H20862/H20849g112g2/H20850is the light-hole effec-\ntive mass. The lower bound in Eq. (6) is obtained assum-ingm\nlhømhhwhile the upper bound is reached when\nmlh/H33360mhh.\nBased on the above analysis we conclude that the\nBerry phase anomalous velocity can yield a sizableAHE in (III, Mn)V ferromagnets. The dot-dashed linein Fig. 1 shows our analytic results for GaAs effective\nmasses m\nhh/H335270.5mandmlh/H335270.08m. Note that in\nexperiment, anomalous Hall conductances are of order1\n10V21cm21and the effective exchange field\nh/H11011102100meV. According to Eq. (6) larger\nsAHvalues should be expected in systems with larger\nheavy-hole effective masses and in systems with the ratio\nmlh/H20862mhhclose to unity .\nSo far we have discussed the limit of in finitely strong\nspin-orbit coupling with an exchange field that is small\nrelative to the hole Fermi energy. In the opposite lim-its of zero spin-orbit coupling or large h,s\nAHvanishes.\nThis implies that the anomalous Hall conductivity is gener-\nally nonlinear in the exchange field and the magnetization.\nTo explore the intermediate regime we numerically diag-onalized the 6-band Luttinger Hamiltonian [23,31] withthe spin-orbit gap D\nso/H335271eV, and for the GaAs value\nDso/H33527341meV. The results shown in Fig. 1 con firm that\na smaller sAHis expected in systems with smaller Dsoand\nsuggest that both positive and negative signs of sAHcan\noccur in general .\nThe curves in Fig. 1 are obtained by neglecting band\nwarping in III-V semiconductor compounds. The property\nthat the valence bands in these materials are strongly\nnonparabolic, even in the absence of the fieldhand even\n0 50 100 150 200\nh (meV)−20020406080σAH (Ω−1cm−1)∆so→∞\n∆so=1eV\n∆so=341meV\nFIG. 1. Illustrative calculations of the anomalous Hall conduc-\ntance as a function of polarized Mn ions field for hole den-\nsityp/H335270.35 nm21. The dot-dashed curve was obtained as-\nsuming in finitely large spin-orbit coupling. The decrease of\ntheoretical sAHwith decreasing spin-orbit coupling strength is\ndemonstrated for Dso/H335271eV (dashed line) and Dso/H33527341meV\n(solid line).in the large Dsolimit, is accurately captured by intro-\nducing the third phenomenological Luttinger parameterg\n3[23,31]. Our numerical results indicate that warping\n(g2fig 3) leads to an increase of sAH,as seen when\ncomparing the solid curves in Fig. 1 and in the top panelof Fig. 2. The hole-density dependence of s\nAH, illustrated\nin Fig. 2, is qualitatively consistent with the sphericalmodel prediction (6). The numerical data in Fig. 2 are\nalso consistent with the trends for dependence on host\nparameters, highlighted in italics in the preceeding para-graphs, suggesting a large positive AHE coef ficient for\n(Al, Mn)As /H20849m\nhh/H335270.66m,mhh/H20862mlh/H335273.96,g2/H20862g3/H33527\n1.73/H20850, an intermediate positive sAH in (Ga, Mn)As\n/H20849mhh/H335270.5m,mhh/H20862mlh/H335276.05,g2/H20862g3/H335271.42/H20850, and\na relatively weak AHE in (In, Mn)As /H20849mhh/H33527\n0.43m,mhh/H20862mlh/H3352721.5,g2/H20862g3/H335271.08/H20850with a sign\nthat may be sensitive to strain and other details of aparticular sample.\nWe now compare our s\nAHtheory with the experi-\nmental data available in (In, Mn)As and (Ga, Mn)As\n020406080σAH (Ω−1cm −1)p=0.1 nm−3\n0.2 nm−3\n0.35 nm−3\n0 50 100 150 200\nh (meV)-200204060σAH (Ω−1cm −1)\n0 100 200\nh (meV)04080σAH (Ω−1cm −1)(Ga,Mn)As\n(In,Mn)As(Al,Mn)As\nFIG. 2. Full numerical simulations of sAHfor GaAs host (top\npanel), InAs host (bottom panel), and AlAs host (inset) with hole\ndensities p/H335270.1nm21(dotted lines), p/H335270.2nm21(dashed\nlines), and p/H335270.35 nm21(solid lines). Luttinger parameters of\nthe valence bands were obtained from Ref. [32]. Filled circles in\nthe top and bottom panels represent measured AHE [1,3,9]. The\nsaturation mean- fieldhvalues for the two points were estimated\nfrom nominal sample parameters [1,3,9]. Horizontal error bars\ncorrespond to the experimental uncertainty of the Jpdcoupling\nconstant. Experimental hole density in the (Ga, Mn)As sample is\np/H335270.35 nm21; for (In, Mn)As, p/H335270.1nm21was determined\nindirectly from sample ’s transition temperature.\n207208-3 207208-3\nFig. 11. Full numerical simulations of the anomalous Hall conductivity \u001bAHfor\nGaAs host with hole densities p= 1020, (dotted lines), 2 \u00021020(dashed lines),\nand 3:5\u00021020cm\u00003(solid lines). Filled circle represents measured Hall conductiv-\nity (Fig. 2). The saturation mean-\feld value of the splitting hbetween\u00008heavy\nhole subbands was estimated from nominal sample parameters. Horizontal error bar\ncorresponds to the experimental uncertainty of the p \u0000d exchange integral. Experi-\nmental hole density in the (Ga,Mn)As sample is 3 :5\u00021020cm\u00003(after Jungwirth\net al. [129]).\nnot provide information about the magnetization of the whole samples but\nonly about its value in regions visited by the carriers. Near the metal-insulator\nboundary, especially when the compensation is appreciable, the carrier distri-\nbution is highly non-uniform. In the regions visited by the carriers the ferro-\nmagnetic interactions are strong, whereas the remaining regions may remain\nparamagnetic. Under such conditions, magnetotransport and direct magnetic\nmeasurements will provide di\u000berent magnetization values [98]. In particular,\nMSatT!0, as seen by a direct magnetometry, can be much lower than that\nexpected for a given value of the magnetic ion concentration. High magneticLecture Notes on Semiconductor Spintronics 33\nLecture Notes on Semiconductor Spintronics 33\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s48/s46/s49\n/s109\n/s111/s72 /s32/s40/s84/s41/s114\n/s121/s120/s47/s114\n/s120/s120/s32/s45/s32 /s109 /s32/s109\n/s111/s72\nFig. 12. Comparison of the normalized anomalous Hall effect (lines) w ith the nor-\nmalized magnetization M/M S(crosses); from top to bottom: 1.7, 2.8, 4.2, 7, 10, 30,\nand 50 K; the data are shifted for clarity (after [138]).\nFurthermore, the spin-orbit interaction accounts for anis otropic magnetoresis-\ntance (AMR ), that is the dependence of resistance on the angl e between the\ncurrent and magnetization , an effect particularly useful fo r position sensing\nin engines. Under these conditions, even if magnetization r emains in-plane\nthe resistivity tensor may assume a non-diagonal form. This leads to the ap-\npearance of a Hall voltage, a phenomenon known as the planar H all effect\n(PHE ) [139]. The information on the orientation of in-plane magnetization ,\nwhich can be obtained from AMR and PHE is, thus, complementar y to that\nprovided by AHE which is sensitive only to the perpendicular component\nof magnetization . In particular, AMR and PHE can trace the ch aracter of\nin-plane magnetization reorientation at the coercive field and serve to deter-\nmine the corresponding anisotropy fields. Last but not least , AMR and PHE\nFig. 12. Comparison of the normalized anomalous Hall e\u000bect (lines) with the nor-\nmalized magnetization M=M S(crosses); from top to bottom: 1.7, 2.8, 4.2, 7, 10, 30,\nand 50 K; the data are shifted for clarity (after [138]).\n\felds are then necessary to magnetize all localized spins. The corresponding\n\feld magnitude is expected to grow with the temperature and strength of\nantiferromagnetic interactions that dominate in the absence of the holes.\nAnisotropic magnetoresistance and planar Hall e\u000bect\nIn cubic materials the conductivity tensor is diagonal in the absence of an ex-\nternal magnetic \feld. However, non-zero values of strain make the resistance\nto depend on the orientation of current in respect to crystallographic axes.\nFurthermore, the spin-orbit interaction accounts for anisotropic magnetoresis-\ntance (AMR ), that is the dependence of resistance on the angle between the34 Tomasz Dietl\ncurrent and magnetization , an e\u000bect particularly useful for position sensing\nin engines. Under these conditions, even if magnetization remains in-plane\nthe resistivity tensor may assume a non-diagonal form. This leads to the ap-\npearance of a Hall voltage, a phenomenon known as the planar Hall e\u000bect\n(PHE ) [139]. The information on the orientation of in-plane magnetization ,\nwhich can be obtained from AMR and PHE is, thus, complementary to that\nprovided by AHE which is sensitive only to the perpendicular component\nof magnetization . In particular, AMR and PHE can trace the character of\nin-plane magnetization reorientation at the coercive \feld and serve to deter-\nmine the corresponding anisotropy \felds. Last but not least, AMR and PHE\nare sensitive probe of spin anisotropy at the Fermi surface associated with\nthe strain and spin-orbit interaction for non-zero magnetization . The corre-\nsponding theory of AMR was developed by Jungwirth et al. [140] within the\nDrude-Boltzmann formulation of charge transport in solids.\nTo test the theoretical predictions concerning e\u000bects of biaxial strain upon\nAMR , (Ga,Mn)As samples under compressive and tensile strain were studied\nfor longitudinal and two perpendicular orientations of the magnetic \feld in\nrespect to electric current [141]. As show in Fig. 13, above 0.5 T, negative\nmagnetoresistance is observed, whose magnitude is virtually independent of\nexperimental con\fguration. However, the absolute value of resistance \u001ain\nthis range depends on the \feld direction, which is the signature of AMR .\nThese data provide information on processes of the \feld-induced rotation of\nmagnetization for various orientations of the \feld in respect to crystal and\neasy axes. In particular, the values of the \feld corresponding to the resistance\nmaxima are expected to be of the order of the anisotropy \feld.\nIf only spin-orbit e\u000bects were controlled AMR , its magnitude would de-\npend only on the angle between the current and \feld directions. Accord-\ning to Fig. 13, this is not the case since AMR depends also on the direc-\ntions of the \feld and current in respect to crystal axes. It is convenient to\nintroduce AMR op= [\u001axx(Hkx)\u0000\u001axx(Hky)]=\u001axx(Hky) and AMR\nop= [\u001axx(Hkx)\u0000\u001axx(Hkz)]=\u001axx(Hkz), where the current and growth\ndirections are denoted by xandz, respectively, and \u001axxis the longitudinal\nresistivity. Importantly, the sign and order of magnitude of AMR is consistent\nwith theoretical expectations [140, 142]. In particular, the predicted di\u000berence\nin sign of AMR op\u0000AMRipin the case of compressive and tensile strain is\ncorroborated by the data. On the other hand, the dependence of AMR opand\nAMRipat given strain on the current direction appears as challenging. It\nmay result from the lowering of the symmetry of the (Ga,Mn)As \flms from\nthe expected D 2dto C 2v, as discussed in Sec. 3.\nParticularly intriguing are hysteretic resistance jumps observed for samples\nunder compressive strain and for the \feld pointing along the growth direction.\nWe assign this e\u000bect to a large ratio of the anisotropy and coercive \felds, which\nmakes that even a rather small misalignment, and thus a minute in-plane \feld,\ncan result in magnetization switching between in-plane easy directions. TheseLecture Notes on Semiconductor Spintronics 35\n \n \nFig. 13. Left panel: Field-induced changes in resistance of Ga 0:95Mn 0:05As/GaAs\n(compressive strain) lower panel: current along [110]; middle panel: current along\n[100]) and of Ga 0:957Mn 0:043As/(In,Ga)As under tensile strain (lower panel, cur-\nrent alongh110i) for three orientations of the magnetic \feld in respect to current\ndirection at 2 K. Right panel: \feld and temperature dependencies of resistance\nin Ga 0:95Mn 0:05As/GaAs (compressive strain, upper panel) and in tensile strained\nGa0:957Mn 0:043As/(In,Ga)As (lower panel) for magnetic \feld perpendicular to the\n\flm plane. Starting from up, subsequent curves at H= 0 correspond to tempera-\ntures in K: 70, 60, 80, 50, 90, 40, 100, 30, 125, 20, 2, 5, 10, 150, 200, 300 (upper\npanel) and to 50, 60, 40, 70, 30, 80, 90, 20, 100, 2, 10, 5, 125, 150, 200, 300 (lower\npanel). The thick solid lines superimposed on 2 K data in positive magnetic \feld\nside show Kawabata's theory predictions (after Matsukura et al. [141]).\nresults provide, therefore, information on resistance values in a demagnetized\nstate for the studied current directions.\nLow and high \feld magnetoresistance\nApart from AMR , there is a number of other e\u000bects that can produce a sizable\nmagnetoresistance in magnetic semiconductors, especially in the vicinity of the\nlocalization boundary [12], where quantum corrections to Drude-Boltzmann\nconductivity become important. In particular, carrier di\u000busion in the molec-\nular \feld of randomly oriented spin clusters that form above TCshifts the\nmetal-to-insulator transition towards higher carrier concentrations [71]. The36 Tomasz Dietl\nresulting temperature dependent localization may lead to a resistance max-\nimum atTC, which will be destroyed by the magnetic \feld. This accounts\npresumably for the \feld and temperature dependence of resistivity near TC\nvisible clearly in Fig. 13.\nHowever, the negative magnetoresistance hardly saturates even in rather\nstrong magnetic \felds, and occurs also at low temperatures, where the spins\nare fully ordered ferromagnetically according to the Hall e\u000bect data. This\nsurprising observation was explained by the present author and co-workers\n[133, 141] in terms of weak localization orbital magnetoresistance. Indeed, in\nthe regime in question the giant splitting of the valence band makes both spin-\ndisorder and spin-orbit scattering relatively ine\u000ecient. Under such conditions,\nweak localization magnetoresistance can show up at low temperatures, where\nphase breaking scattering ceases to operate. According to Kawabata [143],\n4\u001a=\u001a\u0019\u00004\u001b=\u001b=\u0000nve2Co(e=\u0016hB)1=2=(2\u00192\u0016h); (17)\nwhereCo= 0:605,\u001bis the conductivity, and 1 =2\u0014nv\u00142 depending on\nwhether one or all four hole subbands contribute to the charge transport. For\nthe samples under compressive and tensile strain, the above formula gives\n4\u001a=\u001a=\u00000:13nvand\u00000:25nv, respectively at B= 9 T. These values are to\nbe compared to experimental data of Fig. 13, 4\u001a=\u001a=\u00000:09 and -0.14 at 2 K.\nThe \ftting to Eq. 17 reproduces the data at 2 K quite well (thin solid lines in\nFig. 13 and gives nv= 1:46 and 0.82 for the compressive and tensile samples,\nrespectively, as could be expected for ferromagnetic \flms of (Ga,Mn)As. Since\nnegative magnetoresistance takes over above Bi\u00191 T, we can evaluate a\nlower limit for the spin-disorder scattering time, \u001cs=m\u0003=(eBikFl) = 8 ps\nfor the hole e\u000bective mass m\u0003= 0:7moandkFl= 0:5, wherekFthe Fermi\nmomentum and lthe mean free path.\n5.7 Spin transport phenomena\nTo this category belongs a number of e\u000bects observed in heterostructures of\n(Ga,Mn)As, and important for perspective spintronic devices, such as spin\ninjection of holes [144, 145] and electrons in the Zener diode [146, 147], giant\nmagnetoresistance (GMR ) [148], tunnelling magnetoresistance (TMR ) [149,\n150, 151], tunnelling anisotropic magnetoresistance (TAMR ) [152, 153], and\ndomain wall resistance [154, 155].\nSince in most semiconductor spin transport devices the relevant length\nscale is shorter that the phase coherence length, a formulation of theory in\nterms of the Boltzmann distribution function fis not valid. Recently, theory\nthat combines an empirical tight-binding approach with a Landauer-B uttiker\nformalism was developed [156, 157]. In contrast to the standard kpmethod,\nthis theory describes properly the interfaces and inversion symmetry breaking\nas well as the band dispersion in the entire Brillouin zone, so that the essential\nfor the spin-dependent tunnelling Rashba and Dresselhaus terms as well asLecture Notes on Semiconductor Spintronics 37\nthe tunnelling viakpoints away from the zone center are taken into account.\nThis approach [156, 157], developed with no adjustable parameters, explained\nexperimentally observed large magnitudes of both electron current spin po-\nlarization up to 70% in the (Ga,Mn)As/n-GaAs Zener diode [158] and TMR\nof the order of 300% in a (Ga,Mn)As/GaAs/(Ga,Mn)As trilayer structure\n[151]. Furthermore, theory reproduced a fast decrease of these \fgures with\nthe device bias as well as it indicated that the magnitude of TAMR should\nnot exceed 10% under usual strain conditions.\n5.8 Methods of Magnetization Manipulation\nSince magnetic properties are controlled by band holes, an appealing possi-\nbility is to in\ruence the magnetic ordering isothermally, by light or by the\nelectric \feld, which a\u000bect the carrier concentration in semiconductor struc-\ntures. Such tuning capabilities of the materials systems in question were put\ninto the evidence in (In,Mn)As/(Al,Ga)Sb [159, 160] and modulation doped p-\n(Cd,Mn)Te/(Cd,Mg,Zn)Te [54, 161] heterostructures, as depicted in Figs. 14\nand 15. Actually, these \fndings can be quantitatively interpreted by consid-\nering the e\u000bect of the electric \feld or illumination on the hole density under\nstationary conditions and, therefore, on the Curie temperature in the relevant\nmagnetic layers. Interestingly, according to experimental \fndings and theo-\nretical modelling, photocarriers generated in II-VI systems by above barrier\nillumination destroy ferromagnetic order in the magnetic quantum well resid-\ning in an undoped (intrinsic) region of a p-i-p structure [54, 161] but they\nenhance the magnitude of spontaneous magnetization in the case of a p-i-n\ndiode [161], as shown in Fig. 15.\nAnother method of magnetization manipulation, suitable for low-power\nswitching of bits in magnetic memories, was invoked by Luc Berger [162] and\nJohn Slonczewski [163], who considered since dozen of years magnetization\nreversal by a transfer of spin momentum from the current of spin polarized\ncarriers to localized magnetic moments in ferromagnetic metals. In the case\nof semiconductors, the current-induced magnetization reversal was demon-\nstrated in submicron pillars of (Ga,Mn)As/GaAs/(Ga,Mn)As [164]. Further-\nmore, spin-polarized current was shown to displace magnetic domain walls in\n(Ga,Mn)As with the easy axis perpendicular to the \flm plane [165, 166].\n6 Summary and Outlook\nAs an outcome of the great progress made in the \feld of semiconductor spin-\ntronics in the past few years as reviewed above, spin transistors were for the\n\frst time described in The International Technology Roadmap for Semicon-\nductors: Update 2004 Emerging Research Devices. Here it was also suggested\nthat spin transistors might replace unipolar silicon transistors, which have\nbeen so successfully employed since the 1960s. It is, however, also obvious38 Tomasz Dietl\nof them, such as spin injection as well as giant and\ntunnellingmagnetoresistance,arefamiliarfromparallel\nstudies of magnetic multilayers. Other effects, likeisothermaldrivingofthesystembetweenferromagnetic\nandparamagneticphasesbytheelectricfieldorbythe\nphotonfluxinthedirectionthatcanbeselectedbyanappropriate design of structure, appear unique to\nferromagnetic semiconductors. These striking effects\noffernewtoolsforpatterningmagneticnanostructuresas well as for information writing and processing,\nbeyondheatingeffectsoflightexploitedintheexisting\nmagneto-optical memories. Obviously, however, whilethe potential of ferromagnetic DMS can already beexploitedforthedevelopmentofquantuminformation\nhardware, their practical applications in classical in-\nformation systems have to be preceded by progress inthesynthesisoffunctionalmaterialswith T\nCsurpassingcomfortably the room temperature. Following the\ntheoreticalsuggestion [13,14],anumber ofnitride and\noxide-based DMS, containing Mn or other magneticelements,hasbeenobtained.Theyshowindeedindica-\ntions of ferromagnetism at room temperature [27–29]\nbut further progress in this field requires the develop-ment of growth and characterization methods that\nenables a better control over solubility limits, self-\ncompensations, phase segregations, and precipitationsofothercompounds [27–30].\nOn the theoretical side, it appears that double- and\nsuper-exchange,ratherthantheZener/RKKYmechan-ism, may account for ferromagnetism in systemscontaining magnetic elements other than Mn. Since\nferromagnetic DMS combine intricate properties of\ncharge-transfer insulators and strongly correlated dis-orderedmetalswiththephysicsofdefectandbandstatesARTICLE IN PRESS\nFig.2. MagnetisationhysteresisloopsdeterminedbymeasurementsofanomalousHalleffectatconstanttemperatureof22.5Kfor\nvariousgatevoltagesinfield-effecttransistorwith(In,Mn)Aschannel(afterOhnoetal. [20]).T. Dietl / Journal of Magnetism and Magnetic Materials 290–291 (2005) 14–19 17\nFig. 14. Magnetization hysteresis loops determined by measurements of anomalous\nHall e\u000bect at constant temperature of 22.5 K for various gate voltages in \feld-e\u000bect\ntransistor with (In,Mn)As channel (after Ohno et al. [160]).\nfrom this review that a number of challenges are ahead, so that semiconduc-\ntor spintronics will attract a lot of attentions of the research community in\nthe years to come.\nFrom the device physics perspective, further works on magnetooptical iso-\nlators and modulators as well as on electrically controlled spin current gen-\neration, injection, detection, \fltering, and ampli\fcation, particularly in spin\nbipolar devices [20] are expected. At the same time, further advancement in\nlow-power magnetization switching will allow the development of new gener-\nation magnetic random access memories (MRAM) and, perhaps, extend the\nuse of magnetism towards logics. Last but not least a progress in manipula-\ntion of single electron or nuclear spins in scalable solid-state devices can be\nenvisaged, though a time scale in question is hard to predict.\nSimilarly to other branches of condensed matter physics, breakthrough\nachievements will be triggered by developments of new materials. Further\nprogress in p-type doping and magnetic ion incorporation to standard semi-Lecture Notes on Semiconductor Spintronics 39\n1700 17101.49 K1.651.872.052.192.803.034.2 K0VPL Intensity  (a.u.)a\n \n1700 17101.49 K1.651.882.052.192.822.97 b\n4.2 K-1V\nV\n1700 17100 Vc\n1.5 K\nPhoton energyp-(Cd,Mg,Zn)Te\n(Cd,Mn )Te\n(Cd,Mg,Zn)Te\nn-(Cd,Mg,Zn)Te\nFig. 15. E\u000bect of temperature (a), bias voltage (b), and illumination (c) on pho-\ntoluminescence of structure consisting of modulation doped p-(Cd,Mn)Te quantum\nwell and n-type barrier. Zero-\feld line splitting (marked by arrows) witnesses the\nappearance of a ferromagnetic ordering (a) which does not show up if the quantum\nwell is depleted from the holes by reverse bias of p-i-n diode (b). Low-temperature\nsplitting is enhanced by additional illumination by white light (c), which increases\nhole concentration in quantum well (after Boukari et al. [161]).\nconductors will make it possible to synthesize functional high temperature\nferromagnetic DMS . At the same time, a control over ferromagnetic precip-\nitates in various semiconductors will result in composite materials that will\nbe useful as magnetooptical media and for high density memories. Partic-\nular attention will be paid to insulating ferrimagnetic oxides and nitrides,\nwhich could serve as spin selective barriers up to well above room temper-\nature. Moreover, e\u000borts will be undertaken to convert them into functional\nmagnetic semiconductors by elaboration of puri\fcation methods and master-\ning doping protocols that will produce high mobility electrons and holes in\nthese systems. Another line of research will be devoted to search for nonmag-\nnetic barrier materials in which transmission coe\u000ecients could be electrically\nadjusted to optimize either reading or writing process in MRAM cells. Par-\nticularly prospective appear multiferroic systems, as in these multifunctional40 Tomasz Dietl\nmaterials the coupling between magnetic and electric polarizations o\u000bers new\ndevice paradigms.\nThe important aspect of extensive studies on ferromagnetism in semi-\nconductors discussed in this review is the demonstration of suitability of\nempirically-constrained theoretical methods in quantitative description of a\nlarge body of thermodynamic, micromagnetic, transport, and optical proper-\nties of ferromagnetic semiconductors. In particular, a successful description of\nspintronic e\u000bects in both nonmagnetic and magnetic semiconductors is possi-\nble provided that all peculiarities of the host band structure, especially those\nassociated with the spin-orbit interaction, are carefully taken into account.\nIndeed, as a result of such an e\u000bort (Ga,Mn)As has reached the status of the\nbest understood ferromagnet. At the same time, research on DMS has dis-\nclosed shortcomings of today's computational materials science in predicting\nand elucidating magnetic properties of solids. It appears that this failure of\nab initio methods (prediction of ferromagnetism in systems where it is absent\nand inability to explain its nature in materials where it does exist) originates\nfrom the co-existence of strong correlation with electronic and magnetic dis-\norder in DMS . This calls for novel computation protocols that will be able to\nhandle randomness and correlation on equal footing, also at non-zero temper-\natures, and will allow for the existence of electronic and/or chemical nanoscale\nphase separations. 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Lett.\n96, 096601 (2006)"    },    {        "title": "2111.12344v2.Platform_for_controllable_Majorana_zero_modes_using_superconductor_ferromagnet_heterostructures.pdf",        "content": "arXiv:2111.12344v2  [cond-mat.supr-con]  26 Nov 2021Platform for controllable Majorana zero modes using\nsuperconductor/ferromagnet heterostructures\nGiorgos Livanas1, Nikolaos Vanas1, Manfred Sigrist2, Georgios\nVarelogiannis1\n1Department of Applied Mathematical and Physical Sciences, NTUA, 15780, Athens, Greece\n2Institut f¨ ur Theoretische Physik, ETH, 8093, Z¨ urich, Swi tzerland\nAbstract We propose a novel platform for the creation and manipulation of Ma jorana zero modes consisting of\na ferromagnetic metallic wire placed between conventional superco nductors which are in proximity to ferromag-\nnetic insulators. Our device relies on the interplay of applied supercu rrents and exchange fields emerging from\nthe ferromagnetic insulators. We assert that the proposed supe rconductor/ferromagnet heterostructures exhibit\nenhanced controllability, since topological superconductivity can b e tuned apart from gate voltages applied on the\nferromagnetic wire, also by manipulating the applied supercurrents and/or the magnetisation of the ferromagnetic\ninsulators.\n1 Introduction\nTopological quantum computation is the most promis-\ning answer to the quantum decoherence problem hin-\nderingthedevelopmentofquantumcomputers[ 1].Topo-\nlogicallyprotectedqubitsformedbynon-Abeliananyons\nwould allow to store and process quantum informa-\ntion in decoherence-resilient schemes. Majorana zero\nmodes (MZMs) are arguably the most natural realiza-\ntion of such non-Abelian anyons. Indeed, MZMs are ex-\npected to emerge at the edges of 1D [ 2] and at the vor-\ntices of 2D [ 3] topological p-wave superconductors but\nunfortunately p-wave superconductivity appears to be\nscarce in nature. However, a topological p-wave super-\nconducting state has been demonstrated to emerge in\nheterostructures comprising topological insulators and\nconventional superconductors [ 4], where proximity in-\nduced singlet superconductivity pairs the helical states\nconfined at the boundaries of the topological insulator\nrealising an effectively spinless superconducting state.\nFollowing this seminal work numerous proposals have\nbeen put forward involving superconductor-semicondu\nctor heterostructures [ 5,6,7], non-centrosymmetric su-\nperconductors[ 8],surfacesofheavymetals[ 9],thequan-\ntum anomalous Hall state [ 10] and 2D electron gases\n[11]. Key elements for realising effectively spinless su-\nperconductivity in all the aforementioned proposals, is\nspin momentum locking and a Zeeman energy term,thereby many schemes for realising MZMs rely on ef-\nfective spin orbit coupling emerging from the applica-\ntion of laser beams in ultracold atoms superfluids [ 12],\nnanomagnets in superconducting wires [ 13] or carbon\nnanoribbons [ 14], magnetic tunnel junctions in proxi-\nmized 2D electron gases [ 15], from the helical ordering\nof magnetic adatoms embedded in conventional super-\nconductors [ 16,17,18,19,20] as well as from the curved\ngeometry of carbon nanotubes [ 21] and semiconducting\nwires [22]. Moreover, in most of the above setups for\nengineering MZMs, the required Zeeman energy is pro-\nvided by the application of an external magnetic field.\nSuperconductor/ferromagnetheterostructures(SC-FM)\nsupport the emergence of topological superconductiv-\nity without the requirement of an external magnetic\nfield, as has been already demonstrated for ferromag-\nnetic metal/superconductor heterostructures with spin\nactive interfaces [ 23,24] and ferromagnetically aligned\nmagnetic adatoms chains embedded in conventional su-\nperconductors with spin orbit coupling [ 25]. However,\nno related devices exhibiting enhanced controllability\nof the MZMs, necessary for the future design of braid-\ning operations, has been proposed so far.\nHere we propose such a platform for engineering\ncontrollable TSC and MZMs based on superconduc-\ntor/ferromagnet heterostructures (SC-FM) and appli-\ncationofsupercurrentsandgatevoltages.Theheterostruc-\nturesconsistofaferromagneticmetallicwire(FM)placed2\namong conventional superconductors (SCs) which are\nin proximity to ferromagnetic insulators (FIs)\n(Fig.1).Thegeneralunderlyingmechanismhasalready\nbeenpresentedinourpreviouspublication[ 26]andcon-\nsists of two stages. In the first stage, triplet p-wave cor-\nrelations are induced in the SCs, due to the coexistence\nof conventional superconductivity with applied super-\ncurrents and exchange fields emerging from the FIs. As\nwe show below, the d-vector [ 27] of the induced p-wave\ncorrelations is parallel to polarization of the exchange\nfield of the FIs. In the second stage, these p-wave cor-\nrelations mediate into the FM where an analogous to\nKitaev’schainspinlesssuperconductingstate isrealised\nprovidedthe magnetisationofthe wireand the d-vector\nof the p-wave correlations are misaligned. We demon-\nstrate that our device platform exhibits enhanced con-\ntrollability through the combination of externally ap-\nplied supercurrents and the manipulation of the ex-\nchangefieldsresultingfromtheFIsthatcanbeachieved\nby the use of currents [ 28] or applied gate voltages [ 29,\n30]. We remark that although supercurrents were in-\nvolved in other proposals for engineering MZMs [ 31,32,\n33,34,35,36], in our device they do not constitute sim-\nply an alternative to magnetic fields, but instead they\ncan also be exploited to manipulate the phase of the\ninduced TSC, a property crucial for designing feasible\nbraiding schemes.\n2 Induced p-wave superconductivity\nInitially, we describe how the coexistence of conven-\ntional superconductivity, supercurrents and exchange\nfieldsinducestripletp-wavecorrelationswithad-vector\nparalleltothe polarizationofthe exchangefields. As we\nhave already pointed out in [ 26] these four fields or or-\nder parameters constitute a quartet [ 37] meaning that\nthe presence of any three of them induces the missing\nfourth one. To this end we employ the following Hamil-\ntonian representing a 1D SC,\nH=/integraldisplay\nΨ†/parenleftbigg\n[∂2\nx\n2m+µ]τz+τzh·˜σ+∆τyσy/parenrightbigg\nΨ(1)\nwhereΨ†= (ψ†\n↑,ψ†\n↓,ψ↑,ψ↓)theextendedNambuspinor\nandτand˜σ= (σx,τzσy,σz) are the Pauli matrices\nacting on particle-hole and spin space, respectively, h\nthe exchange field and ∆=|∆|eiJxτzthe singlet su-\nperconducting field. Under the gauge transformation\nΨ(x)→e−iJ/2xτzΨ(x) kinetic term∂2\nx\n2mtransforms to\n∂2\nx\n2m+iJ\n2m∂x−J2\n8mτzwhereiJ\n2m∂xis a current term\nand−J2\n8mτzcan be absorbed in the chemical poten-\ntialµ. Thus, Hamiltonian Eq. 1takes the form H=/integraltext\ndxΨ†/parenleftbigg\n[∂2\nx\n2m−µ]τz+iJ\n2m∂x+τzh·˜σ+|∆|τyσy/parenrightbigg\nΨ. For\nperiodic boundary conditions, a Fourier transformation\nto momentum kspace leads to the energy bands of the\nwireEs,±(k) =−sh+J\n2mk±/radicalBig/parenleftbigk2\n2m−µ/parenrightbig2+∆2, where\ns=±, with + and −corresponding to different spin\nconfigurations depending on the orientation of the h.\nFor translationally invariant systems p-wave triplet\ncorrelations derive from equation\n<∆p>∝1\nβΣiωnTr{kτxd·˜σiσyˆG(k,iωn)},(2)\nwhereˆG(k,iωn) the Matsubara Green’s function with\nωn= (2n+ 1)π/βthe fermion frequencies and β∝\n1/Tthe inverse of temperature. The above expression\nsimplifies to\n<∆p>∝/summationdisplay\nk/summationdisplay\nmnF(Em)\n[U(k)†kτxd·˜σiσyU(k)]mm,(3)\nwheremthe energy band index, U(k) the transforma-\ntion matrix which diagonalises the Hamiltonian matrix\nˆH(k) while [] mmdenotes the mdiagonal term of the\ncorresponding matrix. Based on the above analysis it is\nstraightforward to derive the expression\n<∆p,d/bardblh>∝/summationdisplay\nkk∆/radicalBig\n∆2+(k2\n2m−µ)2·\n/bracketleftbigg\nnF(Es,−)−nF(Es′,+)−[nF(Es′,−)−nF(Es,+)/bracketrightbigg\n(4)\nfor the triplet correlations with d∝bardblh. Since forh= 0,\nnF(Es,±) =nF(Es′,±), triplet correlations vanish, <\n∆p,d/bardblh>= 0. Similarly, for J= 0 we get <∆p,d/bardblh>=\n0, since the expression k(nF(Es,−)+nF(Es′,+)\n−[nF(Es,+)+nF(Es′,−)]) is odd in momenta k. How-\never, when both handJare finite, both spin and spa-\ntial inversion symmetries break, leading to a finite sum\nfor triplet correlations < ∆p,d/bardblh>. Regarding corre-\nlations< ∆p,d⊥h>, matrix [U(k)†kτxd⊥h·˜σiσyU(k)]\nhas no diagonal elements, therefore they vanish. Thus,\nonly whenhandJare both finite in a conventional su-\nperconductor ∆,< ∆p,d/bardblh>correlations are induced.\nFrom Eq. 4it becomes apparentthat <∆p,d/bardblh>corre-\nlations emerge from the imbalance among the two con-\nfigurations ψ†\nk,sψ†\n−k,s′andψ†\nk,s′ψ†\n−k,sof the interband\nsinglet pairing (See Sup. Mat.). Moreover, since J→\n−Jleads toEs,±(k)→Es,±(−k) we get<∆p,d/bardblh>→\n−< ∆p,d/bardblh>or equivalently the phase of the triplet\norder parameter changes by π. Most importantly, there\nare no threshold values for handJin order for <3\nΔkE\n+\n-SCb)\nΔpE\nkFMc)\nFig. 1 a) Device for engineering controllable MZMs based\non superconductor-ferromagnet heterostructures. A ferro -\nmagnetic metallic wire (FM) is placed between two conven-\ntional superconductors (SCs) where supercurrents (blue ar -\nrows) are applied in an anti-parallel configuration. Exchan ge\nfields are induced in the SCs due to proximity with the ferro-\nmagnetic insulators (FIs). The optimal configuration is whe n\nthe magnetisations of the FIs (black arrows) is perpendicul ar\nto that of the FM (white arrow) and anti-parallel with each\nother. The application of supercurrent Jand exchange field\nhxin the SC induces an equal spin triplet pairing component\n(blue dots) which mediates in the FM resulting in the emer-\ngence of MZMs γa= (ψ†+ψ) (green dot) and γb=i(ψ†−ψ)\n(yellow dot) at the edges of the wire. b) The energy bands\nof a system with an applied current Jand ahxexchange\nfield. The singlet superconducting field ∆pairs electron from\ndifferent bands. The application of Jandhxresults in an\nimbalance of singlet pairing between the two spin configura-\ntions, and therefore an equal spintriplet pairing componen t is\ninduced. However, the induced triplet superconducting fiel d\nstillpairs electrons from different bands.c) The inducedeq ual\nspin triplet superconducting correlations ∆pmediate in the\nFM where an exchange field hzis present. In the spin con-\nfiguration of the energy bands of FM, ∆ppairs electrons of\nthe same band, i. e. intraband pairing, realising an effectiv ely\nspinless superconducting state.\n∆p,d/bardblh>correlationsto be induced. Therefore the par-\nticular mechanism is valid even for small values of these\nparameters which leave conventional superconductivity\nunaffected.\n3 Topological superconductivity in\nsuperconductor-ferromagnet heterostructures\nEvidently, topological superconductivity cannot be re-\nalised simply by applying a supercurrent in a supercon-\nducting wire under the presence of an exchange field,\nsince the emergent triplet pairing correlations are par-allel to the polarization of the exchange field and there-\nfore connect different energy bands of the wire, i.e. we\nget interband triplet pairing. In order to realise MZMs\nwe need intraband superconductivity. In our device in-\ntraband superconductivity emerges in the FM where\nthe triplet correlations induced in the conventional SCs\nmediate by means of the proximity effect. Necessary\ncondition for the emergence of intraband superconduc-\ntivity is that the exchange field hof the wire and the d\nvectoroftheinducedtripletcorrelationsaremisaligned.\nTherefore, a topological superconducting state is re-\nalised in a FM separating two conventional SCs when\nsupercurrents are applied in an antiparallel configura-\ntion and the exchange fields are perpendicular to the\nmagnetisation of the wire and antiparallel to each other\n(see Fig. 1). The antiparallel configuration guarantees\nthatthesupercurrentandtheexchangefieldvanishinto\nthe wire and therefore no < ∆p,d/bardblhFM>correlations\nare induced there. Although this is the optimal config-\nuration, we show that TSC is realised even for small\nmisalignment angles among the magnetisations of the\nFM and the FIs and even for not exactly anti-parallel\nsupercurrents and exchange fields (see Fig: 2).\nIn order to illustrate numerically the above argu-\nments we employ the following lattice Bogoliubov de\nGennes equation, H=/summationtext\ni,jΨ†\niHi,jΨj, which describes\nSC-FM presented in Fig. 1, with\nHi,j= (tfi,j+µi)τz+τzhi·˜σ+∆iτyσy+Ji·gi,j(5)\nwhereΨ†\ni= (ψ†\n↑,i,ψ†\n↓,i,ψ↑,i,ψ↓,i) is the Nambu spinor,\nτand˜σ= (σx,τzσy,σz) the Pauli matrices acting on\nparticle-holeandspin spacerespectively, fi,j=δj,i±ˆx+\nδj,i±ˆyandgi,j= (±iδj,i±ˆx,±iδj,i±ˆy) the even and\nodd in spatial inversion, respectively, functions con-\nnecting nearest neighbours lattice points. Moreover, t\nis the hopping integral, µithe chemical potential, hi\nthe exchange field, ∆ithe conventional superconduct-\ning field and Jithe supercurrent term, on site i. Hamil-\ntonian Eq. 5describes the SC-FM-SC heterostructure\nconsidering the appropriate values of ∆i,µi,hiandJi\nfor each region. The proximity to the FIs is modelled\nby considering a finite exchange field hover the SCs\nemerging from the interfacial exchange interaction be-\ntween the localized magnetic moments of the FIs and\nthe SCs conduction band electrons. MZMs are antici-\npated to emerge localised at the edges of the wire for\n|2t−/radicalbig\n(hFM)2−∆′2|<|µFM|<|2t+/radicalbig\n(hFM)2−∆′2|\nwhere∆′is the induced by proximity singlet supercon-\nductingfieldovertheFM[ 26].FromFig. 2aweconclude\nthat the hopping term along the wire is renormalised\ntot′≃0.85t, due to the coupling of the wire to the\nSCs [38]. Figs.2b),c) and d) demonstrate the signif-\nicant robustness of our device against deviations from4\n0 1 2 3 4 5\nµFM/t-1.5-1-0.500.511.5E/Eg,0a)\n0 0.2 0.4 0.6 0.8 1\n|Jl|/|Jr|-2-1012E/Eg,1b)\n-1 -0.5 0 0.5 1hx \u0000 \u0001\n/\u0002\u0003 \u0004 \u0005-2-1012E/Eg,-1c)\n0 0.2 0.4 0.6 0.8 1\nθ/π-2-1012E/Eg,0d)\nFig. 2a) The low energy spectrum of the device as a func-\ntion of the chemical potential of the wire µFM/t. Eigenen-\nergies are normalised with respect to energy gap Eg,0for\nµFM= 0 andhFM/t= 2.4. b) The energy gap in the wire\nforhFM/t=µFM/t= 2.4 with respect to the ratio of the\nmagnitude of the supercurrents |Jl|,|Jr|in the left and the\nright SC, respectively. The supercurrents flow in the opposi te\ndirection. The energy spectrum is normalised to the energy\ngapEg,1corresponding to Jl=−Jr. c) Low energy exci-\ntation spectrum as a function of the ratio hx,l/hx,rof the\nexchange fields along the x-axis in the left and right SC. The\nantiparallel configuration leads to the maximum energy gap.\nd) The dependence of the excitation spectrum on the rota-\ntion angleθofhrover the right SC in the x−zplane, where\nforθ= 0hr=−hlalong thex-axis. Energies in c) are nor-\nmalised toEg,−1corresponding to hx,l=−hx,r, while in d)\nare normalised to Eg,0forθ= 0. The device appears to be\nmore robust against deviations from the optimal anti-paral lel\nconfiguration of the exchange fields.\nthe optimal antiparallel alignment of the magnetisation\nin the FIs and the antiparallel configuration of the su-\npercurrents in the SCs.\n4 Controlling Majorana zero modes\nTo demonstrate the controllability of our setup we con-\nsider a junction between two ferromagnetic wires FM1\nand FM2 as depicted in Fig. 3. The conventional su-\nperconductors in this configuration are separated by an\ninsulatorinorderforthesupercurrentsadjacentofFM1\nand FM2 to be controlled independently. First, we con-\nsider a sinusoidal variation J=J0cos(φ) of the mag-\nnitude of the supercurrents applied in the SCs that are\nadjacent to the FM2. The supercurrents always flow in\nopposite directions. In Fig. 3is presented the supercur-\nrents configuration for φ=πwhile in Fig. 4a) is shown\nthe low-energy spectrum of this junction with respect\nto parameter φ. The FM wires are coupled through the\ninsulating region with a hopping term tc= 0.5t. For\n−π/2< φ < π/ 2, the superconducting states in thetwo FMs acquire the same phase (0- Josepshon junc-\ntion) and MZMs γ1,blocalised in the right edge of FM1\nandγ2,alocalised at the left edge of the FM2 inter-\nact with each other giving rise to an ordinary fermionic\nstate with finite energy. The other two MZMs γ1,aat\nthe left edge of FM1 and γ2,bat the right edge of FM2\nremain unpaired and therefore stick at zero energy. For\nφ=±π/2 the energy gap in FM2 closes and MZMs\nγ2,aandγ2,bfuse whileγ1,aandγ1,bremain uncoupled.\nRemarkably for π/2< φ <3π/2, for which the super-\ncurrent flow is reversedon both sides of FM2, the phase\nof the induced p-wave superconductivity in FM2 changes\nbyπleading to the formation of a π-Josepshon junction\namong the FM wires. Since the reversal of the flow of\nthe supercurrent leads to an interchange of the posi-\ntion ofγ2,aandγ2,b, andγ2,bcannot couple with γ1,b,\nthe heterostructure hosts four MZMs reflected in the\nlow-energy quasi-particle spectrum. Note on the same\nfigure that the four MZMs are stabilised even for very\nsmall values of the supercurrent.\nFM\nFig. 3 Setup of two ferromagnets (FMs) separated by an\ninsulator (I) in proximity with conventional superconduct ors\n(SCs)and ferromagnetic insulators(FIs). The junction isc on-\ntrolled by varying the supercurrents (blue arrows) and/or r o-\ntating the magnetisation (black arrow) of the FIs adjacent t o\nFM2 by angle θ.\nNext we investigate the same junction by rotat-\ning the magnetisation and therefore the exchange field\nof the FIs adjacent to FM2, in the x-z plane, h=\nhzsin(θ) +hxcos(θ) whereθis the angle from the x-\naxis. Since we require that the xcomponent of the ex-\nchange fields of FIs adjacent to FM2, point at the op-\nposite direction for every θ, the magnetisation in the\nupper FI rotates clockwise while in the lower FI rotates\ncounter-clockwise (Fig. 3). The supercurrent in the two\nFMs flows along the same direction, therefore for θ= 0\nthe junction is formed by two identical copies of FMs\nembedded in conventional SCs. According to our mech-\nanism topological SC emerges only when the magneti-5\n0 100 200 300 400 500 600 7 \u0006 \u0007800z0\n0 \b \t1|Ψ\n\n|/|Ψmax|0 100 200 300 400 500 600 700 80000.51|Ψz|/|Ψmax|\n0 100 200 300 400 500 600 700 800X0\n\u000b \f \r1|Ψ\n\u000e|/|Ψmax|0 100 200 300 400 500 600 700 80000.51|Ψz|/|Ψmax|θ=0.55π\nFig. 4 Low-energy quasiparticle energy spectrums for the\nsetup presented in Fig. 3, a) with respect to parameter φcon-\ntrolling the supercurrents ( J=J0cos(φ)) in the conventional\nSCs adjacent to FM2 and b) with respect to θcorresponding\nto the angle of FIs magnetisation in the x-z plane. Eigenen-\nergies are normalised to the energy gap Egap,πforφ=π\nandθ=πrespectively. The second lowest eigenenergy (black\nline) is multiplied by 10 to demonstrate clearly the splitti ng\nof the MZMs. The spectral function |Ψz|of the two lowest\neigenenergies of the heterostructure, c) for θ= 0.45πand d)\nforθ= 0.55π. The particular angles are indicated in figure\nb) by the dashed red line. The function is normalised to the\nmaximum value |Ψmax|. Forθ= 0.45π, MZMs localised in\nthe right edge of FM1 and left edge of FM2 form a fermionic\nstate with finite energy E/t= 0.03. Forθ= 0.55π, four\nMZMs emerge. Since in FM2 the FIs magnetisation is not\nin the optimal anti-parallel configuration, MZMs appear to\nbe less localised. In the upper c) plot the left and the right\npart of the function is normalised independently in order to\ncompare the localisation length of MZMs in the two FMs.\nsation of the FIs and that of the ferromagnetic wire is\nmisaligned. Indeed, for θ=π/2 the energy gap at the\nwire closes (Fig. 4b)). Remarkably for π/2<θ<3π/2\nfor which the xcomponent of the magnetisation of the\nFIs adjacent to FM2 is opposite to that of the FIs adja-\ncenttoFM1, aπ-Josepshon junction is formed andfour\nMajorana zero modes emerge localised at the edges of\nthe two wires. In Fig. 4c) we display the spectral func-\ntion|Ψz|of the two lowest eigenvalues for θ= 0.45πandθ= 0.55π. We remark that small rotations of the\nFIs magnetisation can be achieved by the application\nof gate voltages.\n5 Conclusions\nWepresentanoveldeviceplatformforengineeringMZMs\nbased on SC-FM heterostructures. Key elements of the\nunderlyingmechanismarethe externallyapplied super-\ncurrents and the emergent exchange fields, due to prox-\nimity of the SCs with FIs. Developments in the fabrica-\ntion of SC-FI heterostructures [ 39] and ferromagnetic\nmetallic Co [ 40,41] or Fe [ 42] nanonwires embedded\nin conventional SCs, enhance the experimental feasibil-\nity of our proposal. Several works have demonstrated\nboth theoretically [ 43,44] and experimentally [ 45,46,\n47] the emergence of exchange fields in conventional\nSCs in proximity to FIs, e.g. in EuS/Al and EuO/Al\nheterostructures. Finally, our platform does not require\nexternally applied magnetic fields, yet it facilitates the\ncontrol of the emergent MZMs, either through the ap-\nplied supercurrents or the manipulation of FIs mag-\nnetisation by use of currents or applied gate voltages.\nIn this way we circumvent the screeningeffects problem\nthat undermines schemes for braiding MZMs based on\nsemiconductors.\nThis research is carried out in the context of the\nproject”Majoranafermionsfrominducedquartetstates”\n(MIS5049433)underthecallforproposals”Researchers’\nsupport with an emphasis on young researchers- 2nd\nCycle”. The project is co-financed by Greece and the\nEuropean Union (European Social Fund- ESF) by the\nOperationalProgrammeHumanResourcesDevelopment,\nEducation and Lifelong Learning 2014-2020.”\nAuthor contribution statement Dr.GiorgosLivanashas\nconducted the analysis and wrote the initial version of\nthe manuscript, Nikos Vanas contributed to numeri-\ncal calculations, Prof. Giorgos Varelogiannis and Prof.\nManfredSigrist haverevisedand wrotethe final version\nof the manuscript. The work was conceptualized by Dr.\nGiorgos Livanas and Prof. Giorgos Varelogiannis.\nReferences\n1. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S.\nDas Sarma, Non-Abelian anyons and topological quantum\ncomputation, Rev. Mod. Phys. 80, 1083 (2008) .\n2. A.Y. Kitaev, Unpaired Majorana fermions in quantum\nwires,Phys.-Usp. 44, 131-136 (2001) .\n3. D.A. Ivanov, Non-Abelian Statistics of Half-\nQuantum Vortices in p-Wave Superconductors,\nPhys. Rev. Lett. 86, 268 (2001) .6\n4. Liang Fu and C. L. 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Materials 1, 054402, (2017) ."    },    {        "title": "0709.1759v2.Two_Ferromagnetic_Phases_in_Spin_Fermion_Systems.pdf",        "content": "arXiv:0709.1759v2  [cond-mat.str-el]  24 Mar 2008Two ferromagnetic phases in spin-Fermion systems\nNaoum Karchev\nDivision of Material Physics, Graduate School of Engineeri ng Science,\nOsaka University, Toyonaka, Osaka 560-8531, Japan\nDepartment of Physics, University of Sofia, 1126 Sofia, Bulga ria\nWe consider spin-Fermion systems which obtain their magnet ic properties from a system of local-\nizedmagneticmomentsbeingcoupledtoconductingelectron s. Thedynamicaldegrees offreedom are\nspin-soperators of localized spins and spin-1 /2 Fermi operators of itinerant electrons. We develop\nmodified spin-wave theory and obtain that system has two ferr omagnetic phases. At the character-\nistic temperature T∗, the magnetization of itinerant electrons becomes zero, an d high temperature\nferromagnetic phase ( T∗< T < T C) is a phase where only localized electrons give contributio n to\nthe magnetization of the system. An anomalous increasing of magnetization below T∗is obtained\nin good agrement with experimental measurements of the ferr omagnetic phase of UGe2.\nPACS numbers: 75.30.Et, 71.27.+a, 75.10.Lp, 75.30.Ds\nThis Brief Report is inspired from the experimental\nmeasurements of the ferromagnetic phase of UGe2which\nreveal the presence of an additional phase line that lies\nentirely within the ferromagnetic phase. The character-\nistic temperature of this transition T∗, which is below\nthe Curie temperature TC, decreases with pressure and\ndisappears at a pressure close to the pressure at which\nnew phase of coexistence of superconductivity and ferro-\nmagnetism emerges[1, 2, 3].Strong anomaly in resistivity\nis observed at T∗[4]. The additional phase transition\ndemonstrates itself and through the change in the Tde-\npendenceoftheorderedferromagneticmoment[2, 5, 6, 7].\nThemagnetizationshowsananomalousenhancementbe-\nlowT∗. An anomaly is found in the heat capacity at the\ncharacteristic temperature T∗[8]. Theoretically, it was\nassumed that the interplay of charge-density wave and\nspin-density wave fluctuations is the origin of anomalous\nproperties[9]. Alternatively, it was proposed that the un-\nusual phase diagramis result ofnovel tuning of the Fermi\nsurface topology by the magnetization[10].\nOur objective is spin-Fermion systems, which obtain\ntheir magneticpropertiesfroma systemoflocalizedmag-\nnetic moments and itinerant electrons. It is obtained\nthat the true magnons in these systems, which are the\ntransversal fluctuations corresponding to the total mag-\nnetization, arecomplicatemixtureofthe transversalfluc-\ntuations of the spins of localized and itinerant electrons.\nThe magnons interact with different spins in a differ-\nent way, and magnons’ fluctuations suppress the or-\ndered moments of the localized and itinerant electrons\nat different temperatures. As a result, the ferromag-\nnetic phase is divided onto two phases: low tempera-\nture phase 0 < T < T∗, where all electrons contribute\nthe ordered ferromagnetic moment, and high tempera-\nture phase T∗< T < T C, where only localized spins form\nmagnetic moment. To describe the two phases, a modi-\nfied spin-wave theory is developed. We have reproduced\ntheoretically the anomalous temperature dependence of\nthe ordered moment, known from the experiments with\nUGe2[2, 5, 6, 7].\nThe spin-fermion model is known as s−d(ors−f).The model appears in the literature also as the ferro-\nmagnetic Kondo Lattice model (FKLM) or the double\nexchange model (DEM). The exact results for the spin-\nFermion model are reported in [11].\nThe dynamical degrees of freedom are spin- soperators\nof localized spins and spin-1 /2 Fermi operators of itiner-\nant electrons. We consider a theory with Hamiltonian\nh=H−µN=−t/summationdisplay\n/angbracketleftij/angbracketright/parenleftbig\nc+\niσcjσ+h.c./parenrightbig\n−µ/summationdisplay\nini\n−Jl/summationdisplay\n/angbracketleftij/angbracketrightSi·Sj−J/summationdisplay\niSi·si (1)\nwheresν\ni=1\n2/summationtext\nσσ′c+\niστν\nσσ′ciσ′, with the Pauli matrices\n(τx,τy,τz), is the spin of the conduction electrons, Si\nis the spin of the localized electrons, µis the chemical\npotential, and ni=c+\niσciσ. The sums are over all sites of\na three-dimensional cubic lattice, and /angbracketlefti,j/angbracketrightdenotes the\nsum over the nearest neighbors. The Heisenberg term\n(Jl>0) describes ferromagnetic Heisenberg exchange\nbetween nearest-neighbors localized electrons. The term\nwhich describes the spin-Fermion interaction ( J >0) is\nknown as a Kondo interaction ( J=JK) in the ferromag-\nnetic Kondo model or as a Hund’s term in the double\nexchange model ( J=JHandJl<0).\nWe represent the Fermi operators in terms of\nthe Schwinger bosons ( ϕi,σ,ϕ+\ni,σ) and slave Fermions\n(hi,h+\ni,di,d+\ni). The Bose fields are doublets ( σ= 1,2)\nwithout charge, while Fermions are spinless with charges\n1 (di) and -1 ( hi).\nci↑=h+\niϕi1+ϕ+\ni2di, c i↓=h+\niϕi2−ϕ+\ni1di,\nni= 1−h+\nihi+d+\nidi, sν\ni=1\n2/summationdisplay\nσσ′ϕ+\niστν\nσσ′ϕiσ′,\nϕ+\ni1ϕi1+ϕ+\ni2ϕi2+d+\nidi+h+\nihi= 1 (2)\nNext, we make a change of variables, introducing Bose\ndoublets ζiσandζ+\niσ[12]\nζiσ=ϕiσ/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig−1\n2,2\nζ+\niσ=ϕ+\niσ/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig−1\n2, (3)\nwhere the new fields satisfy the constraint ζ+\niσζiσ= 1. In\nterms of the new fields the spin vectors of the itinerant\nelectrons have the form\nsν\ni=1\n2/summationdisplay\nσσ′ζ+\niστν\nσσ′ζiσ′/bracketleftbig\n1−h+\nihi−d+\nidi/bracketrightbig\n(4)\nWhen, in the ground state, the lattice site is empty,\nthe operator identity h+\nihi= 1 is true. When the lat-\ntice site is doubly occupied, d+\nidi= 1. Hence, when\nthe lattice site is empty or doubly occupied the spin\non this site is zero. When the lattice site is neither\nempty nor doubly occupied ( h+\nihi=d+\nidi= 0), the\nspin equals si= 1/2ni,where the unit vector nν\ni=/summationtext\nσσ′ζ+\niστν\nσσ′ζiσ′(n2\ni= 1) identifies the local orientation\nof the spin of the itinerant electron. Let us average the\nspin of electrons in the subspace of the Fermions ( d+\ni, di)\nand (h+\ni, hi) (to integrate the Fermions out in the path\nintegral approach) and introduce the notation\nm=1\n2/parenleftbig\n1−< h+\nihi>−< d+\nidi>/parenrightbig\n.(5)\nOne obtains si=mniwheres2\ni=m2. Hence, the ampli-\ntude of the spin vector mis an effective spin of the itin-\nerant electrons accounting for the fact that some sites, in\nthe ground state, are doubly occupied or empty.\nIt is more convenient to use the rescaled Bose fields\nξiσ=√\n2mζiσ, ξ+\niσ=√\n2mζ+\niσ(6)\nwhich satisfy the constraint ξ+\niσξiσ= 2m. Let us intro-\nduce the vector,\nMν\ni=1\n2/summationdisplay\nσσ′ξ+\niστν\nσσ′ξiσ′M2\ni=m2.(7)\nThen, the spin-vector of itinerant electrons can be writ-\nten in the form\nsi=1\n2mMi/parenleftbig\n1−h+\nihi−d+\nidi/parenrightbig\n(8)\nThe contribution of itinerant electrons to the total mag-\nnetization is <sz\ni>. Accounting for the definition of m\n(see Eq.5), one obtains <sz\ni>=<Mz\ni>.\nThe Hamiltonian is quadratic with respect to the\nFermions di,d+\niandhi,h+\ni, and one can average in the\nsubspace of these Fermions (to integrate them out in the\npath integral approach). As a result, we obtain an effec-\ntive theory of two spin-vectors SiandMiwith Hamilto-\nnian\nheff=−Jl/summationdisplay\n/angbracketleftij/angbracketrightSi·Sj−Jit/summationdisplay\n/angbracketleftij/angbracketrightMi·Mj−J/summationdisplay\niSi·Mi\n(9)\nThe first term is the term which describes the exchange\nof localized spins in the Hamiltonian Eq.(1). The secondterm is obtained integrating out the Fermions. It is cal-\nculated in the one loop approximation and in the limit\nwhen the frequency and the wave vector are small. For\nthe effective exchange constant Jit, at zero temperature,\nwe obtained\nJit= (10)\nt\n6m21\nN/summationdisplay\nk/parenleftBigg3/summationdisplay\nν=1coskν/parenrightBigg\n/bracketleftbig\n1−θ(−εh\nk)+θ(−εd\nk)/bracketrightbig\n−2t2\n3m2sJ1\nN/summationdisplay\nk/parenleftBigg3/summationdisplay\nν=1sin2kν/parenrightBigg\n/bracketleftbig\n1−θ(−εh\nk)−θ(−εd\nk)/bracketrightbig\nwhereNis the number of lattice’s sites, εh\nkandεd\nkare\nFermions’ dispersions,\nεh\nk= 2t(coskx+cosky+coskz)+stJ/2+µ(11)\nεd\nk=−2t(coskx+cosky+coskz)+stJ/2−µ,\nand wave vector kruns over the first Brillouin zone of\na cubic lattice. The third term in Eq.(9) is obtained\nfrom the last term in the Hamiltonian Eq.(1) using the\nrepresentation Eq.(8) for the spin of itinerant electrons\nand Eq.(5).\nWe are going to study the ferromagnetic phase of the\ntwo-spin system Eq.(9) with Jl>0, Jit>0, andJ >0.\nTo proceed we use the Holstein-Primakoff representation\nof the spin vectors Sj(a+\nj,aj) andMj(b+\nj, bj), where\na+\nj, ajandb+\nj, bjare Bose fields. In terms of these\nfields and keeping only the quadratic terms, the effective\nHamiltonian Eq.(9) adopts the form\nheff=sJl/summationdisplay\n/angbracketleftij/angbracketright(a+\niai+a+\njaj−a+\njai−a+\niaj)\n+mJit/summationdisplay\n/angbracketleftij/angbracketright(b+\nibi+b+\njbj−b+\njbi−b+\nibj) (12)\n−J/summationdisplay\ni(√sm[a+\nibi+b+\niai]−sb+\nibi−ma+\niai)\nIn momentum space representation, the Hamiltonian\nreads\nheff=/summationdisplay\nk/parenleftbig\nεa\nka+\nkak+εb\nkb+\nkbk−γ(a+\nkbk+b+\nkak)/parenrightbig\n,\n(13)\nwhere the dispersions are given by equalities,\nεa\nk= 2sJlεk+mJ, εb\nk= 2mJitεk+sJ,(14)\nεk= 3−coskx−cosky−coskz, andγ=J√sm.\nTo diagonalize the Hamiltonian, one introduces Bose\nfieldsαk, α+\nk, βk, β+\nk,\nak= cosθkαk+sinθkβk, bk=−sinθkαk+cosθkβk\n(15)\nwith coefficients of transformation,\ncosθk=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\n1+εa\nk−εb\nk/radicalBig\n(εa\nk−εb\nk)2+4γ2\n,(16)3\nand sinθk= (1−cos2θk)1/2. The transformed Hamilto-\nnian\nheff=/summationdisplay\nk/parenleftBig\nEα\nkα+\nkαk+Eβ\nkβ+\nkβk/parenrightBig\n,(17)\nwhere\nE±\nk=1\n2/bracketleftbigg\nεa\nk+εb\nk±/radicalBig\n(εa\nk−εb\nk)2+4γ2/bracketrightbigg\n(18)\n, andEα\nk=E+\nk, Eβ\nk=E−\nk. With positive exchange con-\nstantsJl>0, Jit>0, andJ >0, the bose fields’ dis-\npersions are positive εa\nk>0, εb\nk>0 for all wave vectors\nk. As a result, Eα\nk>0 andEβ\nk≥0 withEβ\n0= 0. Near\nthe zero wave vector, Eβ\nk≈ρk2where the spin-stiffness\nconstant is ρ= (s2Jl+m2Jit)/(s+m). Hence, βkis\nthe long-range (magnon) excitation in the two-spin ef-\nfective theory, while αkis a gapped excitation with gap\nEα\n0= (s+m)J.\nThe dimensionless magnetization per lattice site of the\nsystemMis a sum of the magnetization of the local-\nized electrons Ml=< Sz\ni>and the magnetization\nof the itinerant electrons Mit=< sz\ni>=< Mz\ni>,\n(M=Ml+Mit). By means of the Holstein-Primakoff\nrepresentation the magnetizations adopts the form Ml=\ns−1/N/summationtext\nk< a+\nkak>, Mit=m−1/N/summationtext\nk< b+\nkbk>.\nFinally, by means of the transformation Eq.(15) one can\nrewriteMlandMitin terms of the Bose functions of the\nexcitations αk-nα\nkandβk-nβ\nk\nMl=s−1\nN/summationdisplay\nk/bracketleftBig\ncos2θknα\nk+ sin2θknβ\nk/bracketrightBig\n,(19)\nMit=m−1\nN/summationdisplay\nk/bracketleftBig\nsin2θknα\nk+ cos2θknβ\nk/bracketrightBig\n.\nThe magnetization of the system is\nM=s+m−1\nN/summationdisplay\nk/bracketleftBig\nnα\nk+nβ\nk/bracketrightBig\n.(20)\nThe magnon excitation βkin the effective theory\nEq.(9) is a complicate mixture of the transversal fluc-\ntuations of the spins of localized and itinerant electrons\nEq.(15). As a result, the magnons’ fluctuations suppress\nin a different way the magnetic order of these electrons.\nQuantitatively,thisdependsonthecoefficientscos θkand\nsinθkin Eqs.(19). If the spin-Fermion interaction is very\nstrong,J≫JitandJ≫Jl, one can calculate the coef-\nficients approximately using approximate expressions for\ndispersions Eqs. (14): εa\nk≈mJ,andεa\nk≈sJ. As a re-\nsult, one obtains cos2θk≈m/(m+s). For large J, the\ngap of the αexcitation is very big, Eα\n0= (s+m)J, and\none can drop this excitation in the calculations. Then,\nthe approximate expressions for magnetization satisfy\nMl/s=Mit/m, which means that the strong spin-\nFermion interaction aligns the magnetic orders of theitinerant and localized electrons so strong that they be-\ncome zero at one and just the same temperature. The\nresult is different if the spin-Fermion interaction is rela-\ntively small. The magnetization depends on the dimen-\nsionless temperature T/Jand dimensionless parameters\ns, m, Jl/JandJit/J. We consider a theory with spin\nof the localized electrons s= 1 and calculate the param-\neters of the effective theory Eq.(9) in one Fermion-loop\napproximation for density of Fermions n= 0.4 and mi-\ncroscopic parameter J/t= 12.4. The result is m= 0.2\nandJit/J= 0.1. Finally, we set Jl/J= 0.5. For these\neffective parameters, the functions M(T/J),Ml(T/J),\nandMit(T/J) are depicted in Fig.1 The green line is\nthe magnetization of the localized electrons, the red line\nis the magnetization of the itinerant electrons, and the\nblue line is the total magnetization. The figure shows\nthat the magnetic order of itinerant electrons (red line)\nis suppressed first, at temperature T∗/J= 0.5603. Once\nsuppressed, the magnetic order cannot be restored at\ntemperatures above T∗because of the increasing effect\nof magnon fluctuations. Hence, the magnetization of the\nitinerant electrons should be zero above T∗. As is evi-\ndent from Fig.1, this is not the result within customary\nspin-wave theory.\n/s48/s44/s48 /s48/s44/s50 /s48/s44/s52 /s48/s44/s54 /s48/s44/s56 /s49/s44/s48 /s49/s44/s50 /s49/s44/s52/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s49/s44/s52\n/s77\n/s77/s108\n/s77/s105/s116/s74/s108\n/s47/s74/s61/s48/s46/s53/s32/s32/s32/s32/s74/s105/s116\n/s47/s74/s61/s48/s46/s49\n/s115/s61/s49/s32/s32/s32/s32/s109/s61/s48/s46/s50\n/s84/s47/s74\n/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s91\n/s66\n/s93\nFIG. 1: (color online)Temperature dependence of the ferro-\nmagnetic moments: M(blue line)-the magnetization of the\nsystem, Ml(green line)-contribution of the localized elec-\ntrons,Mit(red line)-contribution of the itinerant electrons\nfor parameters s= 1, m= 0.2, Jl/J= 0.5 andJit/J= 0.1:\nspin-wave theory\nTo solve the problem, we use the idea on description\nof paramagnetic phase of two-dimensional ferromagnets\n(T >0) by means of modified spin-wave theory [13, 14].\nIn the simplest version, the spin-wave theory is modified\nby introducing a parameter which enforces the magneti-\nzation of the system to be equal to zero in paramagnetic\nphase.\nIn the present case, we have two-spin system and we\nintroduce two parameters λlandλitto enforce the mag-\nnetic moments both of the localized and the itinerant\nelectrons to be equal to zero in paramagnetic phase. To4\nthis end, we add two new terms to the effective Hamilto-\nnian Eq.(12),\nˆheff=heff−/summationdisplay\ni/bracketleftbig\nλlSz\ni+λitMz\ni/bracketrightbig\n.(21)\nIn momentum space, the Hamiltonian adopts the form\nEq.(13) with new dispersions ˆ εa\nk=εa\nk+λland ˆεb\nk=\nεb\nk+λit, where the old dispersions are given by equalities\n(14). We utilize the same transformation Eq.(15) with\ncoefficients cos ˆθkand sinˆθkwhich depend on the new\ndispersions in the same way as the old ones depend on\nthe old dispersions Eq.(16). In terms of the αkandβk\nbosons, the Hamiltonian ˆheffadopts the form Eq.(17)\nwith dispersions ˆEα\nkandˆEβ\nk, which can be written in the\nform Eq.(18) replacing εa\nkandεb\nkwith ˆεa\nkand ˆεb\nk.\nWe haveto dosomeassumptionsforparameters λland\nλitto ensure correct definition of the two-boson theory.\nForthat purpose, it is convenientto representthe param-\netersλlandλitin the form λl=mJµl−mJ,andλit=\nsJµit−sJ. In terms of the parameters µlandµit, the\ndispersion reads ˆ εa\nk= 2sJlεk+mJµl,ˆεb\nk= 2mJitεk+\nsJµitThe conventional spin-wave theory is reproduced\nwhenµl=µit= 1(λl=λit= 0). We assume µlandµit\nto be positive ( µl>0, µit>0). Then, ˆ εa\nk>0, ˆεb\nk>0,\nandˆEα\nk>0forallvaluesofthewave-vector k. Toexplore\nthe dispersion ˆEβ\nk=1\n2/bracketleftbigg\nˆεa\nk+ ˆεb\nk−/radicalBig\n(ˆεa\nk−ˆεb\nk)2+4γ2/bracketrightbigg\n,\nwe use the identity (ˆ εa\nk−ˆεb\nk)2+ 4γ2= (ˆεa\nk+ ˆεb\nk)2−\n4(ˆεa\nkˆεb\nk−γ2). It showsthat ˆEβ\nk≥0 if ˆεa\nkˆεb\nk−γ2≥0. Since\nˆεa\nkˆεb\nk≥ˆεa\n0ˆεb\n0=smJ2µlµitforallvaluesofthewavevector\nk, theβkdispersion is non-negative, ˆEβ\nk≥0 ifµlµit≥1.\nIn the particular case, µlµit= 1,ˆEβ\n0= 0, and, near the\nzero wave vector, ˆEβ\nk≈ˆρk2, with spin-stiffness constant\nequals ˆρ= (s2Jlµit+m2Jitµl)/(sµit+mµl). Hence, in\nthis case, βkboson is the long-rangeexcitation (magnon)\nin the system. In the case µlµit>1, bothαkboson and\nβkboson are gapped excitations.\nWe introduced the parameters λlandλit(µl,µit) to\nenforce the magnetic order of localized and itinerant\nelectrons to be equal to zero. We find out the pa-\nrameters µlandµitsolving the system of two equa-\ntionsMl=Mit= 0, where the ordered moments have\nthe same representation as Eq.(19) but with coefficients\ncosˆθk,sinˆθk, and dispersions ˆEα\nk,ˆEβ\nkin the expressions\nfor the Bose functions. The numerical calculations show\nthat for high enough temperature, µit>1, 1> µl>0,\nandµit·µl>1. Hence, αkandβkexcitationsaregapped.\nWhenthetemperaturedecreases, µitdecreasesremaining\nlarger than one, µldecreases too remaining positive, and\nthe product µlµitdecreases remaining larger than one.\nAt temperature TC/J= 2.812, one obtains µit= 5.0427,\nµl= 0.1983, and therefore µlµit= 1. Hence, at TC,\nlong-range excitation (magnon) emerges in the spectrum\nwhich means that TCis the Curie temperature.\nBelow the Curie temperature, the spectrum contains\nmagnon excitations, thereupon µlµit= 1. It is conve-nient to represent the parameters in the following way:\nµit=µ, µl= 1/µ. (22)\nIn ferromagnetic phase, magnon excitations are origin of\nthe suppression of magnetization. Near the zero temper-\nature, their contribution is small, and at zero, tempera-\ntureMit=mandMl=s. Increasing the temperature,\nmagnonfluctuationssuppressthe magnetization. Forthe\nchosen parameters they first suppress the magnetization\nof the itinerant electrons at T∗(Ml(T∗)>0). Once\nsuppressed, the magnetic moment of itinerant electrons\ncannot be restored increasing the temperature above T∗.\nTo formulate this mathematically, we modify the spin-\nwave theory introducing the parameter µEq.(22). Be-\nlowT∗,µ= 1, or in terms of λparameters λl=λit= 0,\nwhich reproduces the customary spin-wave theory. In-\ncreasing the temperature above T∗, the magnetic mo-\nment of the itinerant electron should be zero. This is\nwhy we impose the condition Mit(T) = 0 ifT > T∗. For\ntemperatures above T∗, the parameter µis a solution\nof this equation. We utilize the obtained function µ(T)\nto calculate the magnetization of the localized electrons\nMlas a function of the temperature. Above T∗,Ml\nis equal to the magnetization of the system. The mag-\nnetic moments of the localized and itinerant electrons as\nwell as the magnetization of the system as a function\nof the temperature are depicted in Fig.2 for parameters\ns= 1, m= 0.2, Jl/J= 0.5,andJit/J= 0.1.\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48 /s49/s44/s53 /s50/s44/s48 /s50/s44/s53 /s51/s44/s48/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s49/s44/s50/s77\n/s77/s108\n/s77/s105/s116/s74/s108\n/s47/s74/s61/s48/s46/s53/s32/s32/s32/s32/s74/s105/s116\n/s47/s74/s61/s48/s46/s49\n/s115/s61/s49/s32/s32/s32/s32/s109/s61/s48/s46/s50\n/s84/s47/s74/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s91 /s93\nFIG. 2: (color online) Temperature dependence of the fer-\nromagnetic moments: M(blue line)-the magnetization of\nthe system, Ml(green line)-contribution of the localized\nelectrons, Mit(red line)-contribution of the itinerant elec-\ntrons for parameters s= 1, m= 0.2, Jl/J= 0.5, Jit/J=\n0.1:modified spin-wave theory.\nThe figure shows an anomalous increasing of the mag-\nnetization MbelowT∗which is in a very good agre-\nment with the experiment (see Fig.1 [6]). The present\ntheory enables us to gain insight into the nature of the\ntwo phases. In the low temperature phase (0 ,T∗), the\nlocalized and itinerant electrons contribute to the mag-\nnetization of the system, while in the high temperature5\nphase (T∗,TC), only localized electrons form ferromag-\nnetic moment. At first sight, it seems to be counterintu-\nitive because the local moments build an effective mag-\nnetic field, which, due to spin-Fermion interaction, leads\nto finite itinerant electron spin polarization. This is true\nin the classical limit. In the quantum case, the spin-wave\nfluctuations suppress the magnetic orders of the itiner-\nant and localized electrons at different temperatures T∗\nandTCas a result of different interactions of the magnon\nwith localized and itinerant electrons. the spin fermion\ninteraction increases the alignment of the local moments,\nand magnetic order of itinerant electrons is very strong\nandT∗approaches TC.\nIt is well known that the onset of magnetism in the\nitinerant systems is accompanied with strong anomaly in\nresistivity [15]. This phenomena is experimentally ob-\nserved at T∗in the case of UGe2[4]. This is anothersupport for the theoretical interpretation of T∗as a tem-\nperature at which the itinerant electrons form ferromag-\nnetic order.\nTo conclude, we note that to do more precise fitting\nwith experimental values of the Curie temperature, one\nhastoaccountforthemagnon-magnoninteraction. How-\never, even the approximate calculations in the present\nBrief Report capture the main feature of the two-spin\nferromagnetic systems and the existence of two phases.\nThe next step of our investigation is to understand the\nmechanismofdecreasingthe phasetemperature T∗. This\nwill help us to understand the origin of the superconduc-\ntivity in these materials.\nThis work was financially supported by the Grant-in-\nAid for Scientific Research No19340099 from the JSPS.\nThe author is grateful to Prof. Miyake for the kind hos-\npitality and useful discussions.\n[1] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche,\nR.K.W. Haselwimmer, M.J.Steiner, E.Pugh, I.R.Walker,\nS. R. Julian, P. Monthoux, G. G. Lonzarich, A. Hux-\nley, I. Sheikin, D. Braithwaite, and J. Flouquet, Nature\n(London) 406, 587 (2000).\n[2] A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois, D.\nBraithwaite, R. Calemczuk, and J. Flouquet, Phys. Rev.\nB63, 144519 (2001).\n[3] N. Tateiwa, T. Kobayashi, K. Hanazono, K. Amaya, Y.\nHaga, R. Settai, and Y.Onuki, J. Phys. Condens. Matter\n13, L17 (2001).\n[4] G. Oomi, K. Nishimura, Y. Onuki, and S.W. Yun, Phys-\nicaB186-188 , 758 (1993).\n[5] N. Tateiwa, K. Hanazono, T. C. Kobayashi, K. Amaya,\nT. Inoue, K. Kindo, Y. Koike, N. Metoki, Y. Haga, R.\nSettai, and Y. Onuki, J. Phys. Soc. Jpn 70, 2876 (2001).\n[6] C. Pfleiderer and A. D. Huxley, Phys. Rev. Lett., 89,\n147005 (2002).[7] G. Motoyama, S. Nakamura, H. Kadoya, T. Nishioka,\nand N. K. Sato,Phys. Rev. B 65, 020510 (2001).\n[8] N. Tateiwa, T. C. Kobayashi, K. Amaya, Y. Haga, R.\nSetta, and Y. Onuki,Phys. Rev. B 69, 180513(R) (2004).\n[9] S. Watanabe and K. Miyake, J. Phys. Society of Japan\n71, 2489 (2002).\n[10] K. G. Sandeman, G. G. Lonzarich, and A. J. Schofield,\nPhys. Rev. Lett., 90, 167005 (2003).\n[11] S. Q. Shen, International Journal of Physics B 12, 709\n(1998).\n[12] D. Schmeltzer, Phys. Rev. B 43, 8650 (1991).\n[13] M.Takahashi, Prog. Theor. Physics Supplement 87, 233\n(1986).\n[14] M.Takahashi, Phys. Rev. Lett. 58, 168 (1987).\n[15] P. P. Craig, W. I. Goldburg, T. A. Kitchens, and J. I.\nBudnick, Phys. Rev. Lett., 19, 1334 (1967)."    },    {        "title": "1810.12564v4.Effect_of_Coulomb_interaction_on_the_two_dimensional_electronic_structure_of_the_van_der_Waals_ferromagnet_Cr__2_Ge__2_Te__6_.pdf",        "content": "1  Effect of Coulomb interaction on the two-dimensional electronic structure of the van der Waals ferromagnet Cr2Ge2Te6  M. Suzuki,1∗ B. Gao,2 K. Koshiishi,1 S. Nakata,1 K. Hagiwara,1 C. Lin,1  Y. X. Wan,1 H. Kumigashira,3 K. Ono,3 Sungmo Kang,4 Seungjin Kang,4  J. Yu,4 M. Kobayashi,5,6 S-W. Cheong,2 and A. Fujimori1 1Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 2Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA 3Photon Factory, High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan 4 Center for Theoretical Physics, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea 5Center for Spintronics Research Network, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan 6Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan  In order to investigate the electronic properties of the semiconducting van der Waals ferromagnet Cr2Ge2Te6 (CGT), where ferromagnetic layers are bonded through van der Waals forces, we have performed angle-resolved photoemission spectroscopy (ARPES) measurements and density-functional-theory (DFT+U) calculations. The valence-band maximum at the Γ point is located ~ 0.2 eV below the Fermi level, consistent with the semiconducting property of CGT. Comparison of the experimental density of states with the DFT calculation has suggested that Coulomb interaction between the Cr 3d electrons Ueff ~ 1.1 eV.\t\r The DFT+U calculation indicates that magnetic coupling between Cr atoms within the layer is ferromagnetic if Coulomb Ueff is smaller than 3.0 eV and that the inter-layer coupling is ferromagnetic below Ueff ~ 1.0 eV. We therefore conclude that, for Ueff deduced by the experiment, the intra-layer Cr-Cr coupling is ferromagnetic and the inter-layer coupling is near the boundary between ferromagnetic and antiferromagnetic, which means experimentally deduced Ueff is consistent with theoretical ferromagnetic condition. 2  I. INTRODUCTION Since the discovery of graphene, there has been tremendous interest in the development of new two-dimensional (2D) materials and their functionalities [1,2]. The materials have layered crystal structures and the layers are bonded to each other through van der Waals (vdW) forces. Due to the reduction of dimensionality in the electronic states, 2D materials show fascinating quantum properties which would lead to various potential applications for new devices [3,4]. In addition to graphene and transition-metal dichalcogenides, vdW ferromagnets have recently attracted much attention as candidate materials for new types of spintronic devices. Cr2Ge2Te6 (CGT) is a representative vdW ferromagnet and, along with other vdW ferromagnets, e.g., CrI3 and Cr2Si2Te6, its electronic and magnetic properties have been studied theoretically and experimentally [5 - 12]. The Curie temperature of CGT has been reported to be of 61 K [5 - 7] and the magnetic moment of 2.2 - 3.0 µB/Cr atom with the easy magnetization axis along the c-axis (out-of-plane direction) [5,8]. As for the electric properties, resistivity measurements on CGT have shown semiconducting behaviors and its resistivity prominently increases at low temperatures [5,8]. In their bulk forms, CGT shows physical properties similar to CrI3. The magnetism of CGT nanosheets shows soft behaviors and is well described by the 2D Heisenberg model [9]. In contrast to the Heisenberg description of the CGT nanosheet, ferromagnetism in monolayer of CrI3 is well described by the 2D Ising model [10]. The difference in the magnetic behaviors between CrI3 and CGT could may stem the different strengths of inter-layer magnetic coupling and the different electronic structures of Cr atoms. To unveil the origin of the ferromagnetism in 2D materials, understanding the physical properties from the electronic structure points of view is important. Indeed, both theoretical and experimental studies on 2D ferromagnetic materials have been made in recent years [3-15]. Band-structure calculations on CGT have shown a semiconducting property, which is consistent with experiment. For both fundamental understanding and potential applications of 2D vdW ferromagnets, the knowledge of the electronic structure of CGT related to the ferromagnetism is indispensable. In the present study, we have investigated the electronic properties of CGT single crystals using angle-resolved photoelectron spectroscopy (ARPES) and compared the experimental result with DFT+U calculation. The results suggest that the Coulomb interaction plays an important role in the magnetic properties of CGT.   3  II. METHODS Single crystals of CGT were synthesized using the flux method. High purity Cr powders, Ge powders, and Te powders were mixed in a molar ratio of 2:6:36; the extra Ge and Te were used as a flux. The mixture was loaded in an alumina crucible and sealed in an evacuated quartz tube. A ball of alumina fiber was placed on top of the crucible. The quartz tube was placed in a furnace, heated to 700 oC, held for 5 hours, then slowly cooled down to 480 oC over a period of 3 days. This was followed by centrifugation to remove the flux.  ARPES measurements were performed at beam line BL-28A of Photon Factory, KEK. The incident light was circularly polarized and the energy range used was 34-80 eV . A SCIENTA SES 2002 electron analyzer (Scienta-Omicron Ltd.) was used. Measurements were performed under the vacuum of 2.0 ×10−8 Pa and the sample was cleaved in the analyzer chamber prior to the measurements to obtain fresh surfaces. The sample temperature during the measurements was set to 150 K, above the transition temperature, in order to avoid charging effects. Binding energies were calibrated by measuring the Fermi level (EF) of an Au foil that is in electrical contact with the sample. The energy resolution was set to ~ 50 meV .  First-principles band-structure calculations were conducted based on DFT using the full-potential augmented-plane-wave method implemented in the WIEN2K and OpenMX packages.  By the WIEN2k package, the density of states (DOS) and the band structures of the spin-polarized state were calculated. The lattice constants of a = b = 6.8275 Å and c = 20.5619 Å were used. Generalized-gradient-approximations (GGA) calculation and GGA + U calculation, which leads to orbital dependent potentials for the Cr 3d orbitals, were conducted for bulk materials. Furthermore, using the OpenMX package, local-density-approximation (LDA) and LDA + U calculations were performed for bulk and single-layer CGT in order to investigate the relationships between inter- and intra-layer magnetic coupling and Coulomb interaction. Calculations by WIEN2k and OpenMX for the same Coulomb U give almost the same results.   III. RESLUTS AND DISCUSSION Figure 1(a) shows a constant energy surface mapping at E = - 0.35 eV, where the intensity has been integrated within the energy window of  ± 0.25 eV. There is a circular constant energy surface around the Γ point.  Figure 1(b) shows ARPES spectra along the cut shown by a red line in Fig. 1(a) 4  around the Γ point. In order to closely look at the dispersion around the valence-band maximum (VBM), ARPES spectra enlarged around the Γ point and the corresponding energy distribution curves (EDC) are shown in Figs.1 (c) and (d), respectively. In Figs. 1 (e) and (f), we compare band dispersions around the VBM taken using three photon energies 34, 56, and 80 eV corresponding to kz ~ 17.46, 23.07, and 27.94 p/c. Here, kz’s are estimated using the inner potential of 10 eV. The almost identical band dispersions for the three photon energies indicate that dispersions along the kz direction are negligibly small in this compound and that the electronic structure of CGT is highly 2D reflecting its layered crystal structure. The result demonstrates that there is a hole-like band dispersion from E = - 1 eV to EF and some dispersive bands around E = - (2 - 5) eV. A clear gap is observed below EF, consistent with the semiconducting property [5,8]. The VBM is located at ~ 0.23 eV below EF. Because the band gap of 0.20 - 0.24 eV has been deduced from the resistivity measurements in the intrinsic conduction region (T > ~ 100 K) [5,8], the EF position in the ARPES spectra is just below the conduction band minimum, which contradicts with the fact that CGT should be an intrinsic semiconductor around 150 K, where the ARPES measurements were made. Reported electronic properties show that CGT is an intrinsic semiconductor unless vacancies do not exist at the Ge sites [5,8]. Probably there would be a band bending which lowers both the valence and conduction bands at the surface and only the bended band has been observed due to a short probing depth of ARPES.   In order to investigate the electronic structure in more detail, we have conducted spin-polarized DFT calculation for the ferromagnetic phase of this compound and compared the calculated results with the ARPES spectra. While the present ARPES measurements have been performed above the Curie temperature TC, we consider that the exchange splitting of the ferromagnetic state persists above TC and that the electronic structure in the paramagnetic phase is more similar to that calculated for the ferromagnetic state rather than that calculated for the paramagnetic state [16-18]. Figure 2 shows the spectra of Fig. 1 (b) integrated with respect to k along the cut shown in Fig. 1(a) compared with the calculated k-integrated spectrum. Here, the photoionization cross-sections of the constituent atoms are taken into account. The cross-section of the Cr 3d orbital is larger than those of the other orbitals in the valence band for the photon energy of 80 eV. A prominent peak appears in the experimental spectrum around ~ 1.9 eV, which is significantly deeper than the peak position of ~ 1.3 eV in the calculated one without Coulomb interaction. In order to resolve this discrepancy, we have taken into account Coulomb interaction U between Cr 5  3d electrons in the DFT calculation. We define the effective Coulomb interaction as Ueff = U - J, where U is the atomic Coulomb integral and J is the exchange parameter. Fig. 2 shows that calculation with Ueff = 1.1 eV well reproduces the experiment.  To examine the effects of Coulomb interaction on the valence-band dispersions, the obtained band dispersions are compared with the calculation. Figures 3(a) and (b) show ARPES spectra around the VBM compared with the calculated band dispersions. The band dispersions from the calculation well reproduce the ARPES spectra except for the heaviest bands near the VBM. This is probably because the cross-sections of the bands are not large enough to be observed for the photon energies employed in the present work.  It should be noted that the calculated band dispersions are nearly independent of the Coulomb interaction Ueff between Cr 3d electrons. According to the calculation, the Te 5p states are dominant around the VBM. Furthermore, the width of the entire valence band is seemingly not renormalized even in the presence of the Coulomb interaction, as shown in Figs. 3. These results indicate that the Coulomb interaction affects the Cr 3d states but hardly the Te 5p- and the Ge 3p-derived valence bands.    In a previous DFT calculation on Cr2O3, it has been reported that the value of Ueff for Cr 3d electrons is about 3.5 eV [19]. The present calculation on CGT on the other hand, indicates that the magnetic coupling within the layer is ferromagnetic for Ueff < 3.0 eV , and becomes antiferromagnetic for Ueff > 3.0 eV [20]. On the other hand, inter-layer coupling is ferromagnetic for Ueff < 1.0 eV and antiferromagnetic for Ueff > 1 eV\t\r [20]. Ueff = 1.1 eV estimated in the present experiment well explains the ferromagnetic intra-layer coupling while the inter-layer magnetic coupling is expected to be near the boundary between ferromagnetic and antiferromagnetic. Thus, the present results indicate the importance of Coulomb interaction on the magnetic and electronic structures of CGT.   In the present study, we have confirmed that the valence-band structure of CGT is 2D and the Coulomb interaction affects the Cr 3d states but not the Te 5p and Ge 3p states. In CrI3, it has been found that ferromagnetism persists even in a monolayer thickness but that the inter-layer coupling becomes antiferromagnetic in bilayers, i.e., the inter-layer interaction becomes ferromagnetic or antiferromagnetic depending on the layer number of CrI3 [10]. CGT maintains the soft ferromagnetic properties even in ultrathin films of bilayers [9] and sign change of the inter-layer magnetic coupling has not been observed. This difference probably comes from different environment around the Cr atom. Comparison of the experimental results with the calculation on CGT suggests that the presence 6  of Coulomb interaction makes the energy position of the Cr 3d states deeper, i.e., Coulomb interaction localizes the Cr 3d states and plays a key role in the magnetic properties of CGT. To understand highly 2D vdW ferromagnets from the electronic structure point of view, further studies are desirable. For example, comparison of the electronic structure, especially from the viewpoint of Coulomb interaction, between CGT and CrI3 will shed light on the difference of magnetic properties between CGT and CrI3 in the 2D limit.    IV. CONCLUSION We have performed ARPES measurements and DFT calculation on the vdW ferromagnetic semiconductor CGT in order to investigate its electronic structure. From the ARPES spectra, it is found that the VBM does not reach the EF, consistent with the semiconducting behavior of this compound. The DFT+U calculation well explains the ARPES results. By comparing the experimental results with calculation, it is suggested that the estimated value of Coulomb interaction Ueff favors intra-layer ferromagnetism and is near the boundary between inter-layer ferromagnetic and antiferromagnetic coupling. This suggests the importance of Coulomb interaction between the Cr 3d electrons for understanding both the electronic structure and magnetic properties of CGT. It is also found that the top of the valence band is mainly composed of non-magnetic orbitals, Te 5p, and is not strongly affected by the Coulomb interaction between Cr 3d electrons. Our study has revealed that Coulomb interaction plays an important role not only in its electronic structure but also in its magnetic properties.    Acknowledgments This work was supported by a Grant-in-Aid for Scientific Research from JSPS (grant No. 15H02109) from MEXT and by Center for Spintronics Research Network (CSRN), the University of Tokyo, under Spintronics Research Network of Japan (Spin-RNJ). AF is an adjunct member of CSRN. The synchrotron experiment was done under the approval of the Program Advisory Committee (Proposal Nos. 2015S2-003, 2016G096, and 2018G049). The work at Rutgers University was supported by the NSF under Grant No. DMR-1629059.    7    [1] K. S. Novoselov, A. K. Geim, S. V . Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). [2] R. Mas-Balleste, C. Gomez-Navvaro, J. Gomez-Herrero, and F. Zamora, Nanoscale. 3 20 (2011). [3] N. Samarth, Nature 546, 216 (2017). [4] S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R. Gutierrez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E. Johnston-Halperin, M. Kuno, V. V . Plashnitsa, R. D. Robinson, R. S. Ruo, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer, M. Terrones, W. Windl, and J. E. Goldberger, ACS Nano 7, 2898 (2013). [5] X. Zhang, Y. Zhao, Q. Song, S. Jia, J. Shi, and W. Han, Jpn. J. Appl. Phys. 55, 033001 (2016). [6] V. Carteaux, D. Brunet, G. Ouvard, and G. Andre, J. Phys. Condens. Matter 7, 69 (1995). [7] G. T. Lin, H. L. Zhuang, X. Luo, B. J. Liu, F. C. Chen, J. Yan, Y. Sun, J. Zhou, W. J. Lu, P. Tong, Z. G. 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[14] B. Siberchicot, S. Jobic, V. Carteaux, P. Gressier, and G. Ouvrard, J. Phys. Chem. 100, 5863. (1996). [15] X. Li and J. Yang, J. Mater. Chem. C, 2, 7071 (2014). [16] L. E. Klebanoff, S. W. Robey, G. Liu, and D. A. Shirley, Phys. Rev. B 30, 8  1048 (1984). [17] R. Raue, H. Hopster, and R. Clauberg, Phys. Rev. Lett. 50, 1623 (1983). [18] B. Kim, A. B. Andrews, J. L. Erskine, K. J. Kim, and B. N. Harmon, Phys. Rev. Lett. 68, 1931 (1992). [19] L. Wang, T. Maxisch, and G. Ceder, Phys. Rev. B 73, 195107 (2006). [20] For details of the DFT+U calculation, see S. Kang, S. Kang, & J. Yu, Journal of Elec Materi (2018).   9   \n FIG. 1: ARPES spectra of Cr2Ge2Te6. (a) Constant energy surface mapping at E = - 0.35 eV. The intensity has been integrated within the energy window of ∆E = +/-0.25 eV. The used photon energy was 80 eV.  The hexagon indicates the Brillouin zone. (b) E-k image along the cut though the G point shown by a red line in (a). (c) Enlarged E-k image around the valence-band maximum (VBM) at the Γ point. The hole-like band is clearly observed and the VBM is about 0.23 eV below the Fermi level (EF),\u0001reflecting the semiconducting property of this compound. (d) Energy distribution curves of the data shown in panel (c). (e)(f) ARPES spectra taken with the photon energies of 34 eV and 56 eV . Red dots indicate the peak positions of the top of the valence band in the ARPES spectra taken with the photon energy of 80 eV . \n10     \n  FIG. 2: k-integrated photoemission spectrum (black curve) taken with the photon energy of 80 eV compared with calculated spectra. The k-integration has been made along the cut shown in Fig. 1(a). Calculated result without Coulomb interaction and that with effective Coulomb interaction Ueff = 1.1 eV are compared with experiment. The calculation takes into account the cross sections of atomic orbitals for the photon energy of 80 eV. 11  \n&  \n  FIG.  3:  ARPES spectra around the VBM at the Γ point compared with the band-structure calculation (a) without and (b) with Ueff =1.1 eV. Curves indicate spin-up and spin-down bands. Yellow and green dots are the peak positions obtained from the second derivatives of the EDCs and MDCs, respectively.      \n"    },    {        "title": "1109.2730v2.Large_Magnetoresistance_Effects_in__Ln_CoAsO_with_a_Ferromagnetic_Antiferromagnetic_Transition.pdf",        "content": "arXiv:1109.2730v2  [cond-mat.str-el]  7 Dec 2011Large Magnetoresistance Effects in LnCoAsO with a\nFerromagnetic-Antiferromagnetic Transition\nHiroto Ohta,∗Chishiro Michioka, and Kazuyoshi Yoshimura†\nDepartment of Chemistry, Graduate School of Science, Kyoto Univeristy, Kyoto 606-8502, Japan\n(Dated: December 18, 2018)\nA large magnetoresistance (MR) effect was observed in the lay ered compounds NdCoAsO and\nSmCoAsO, in which ferromagnetically ordered itinerant-el ectrons of Co are sandwiched by localized\n4f-electrons of Ln3+, below ferromagnetic-antiferromagnetic transition (FAF T) temperature TNas\nobserved in other FAFT compounds. In SmCoAsO, the large MR eff ect is also observed up to the\nCurie temperature TC, and it is found to be originating in the presence of another a ntiferromagnetic\nphase in the low-magnetic field region of the ferromagnetic p hase.\nPACS numbers: 75.30.Cr, 75.47.-m, 75.50.Cc, 75.60.Ej\nI. INTRODUCTION\nOne break-through is always going to induce much in-\nfluence onthe vicinityofit. The discoveryofhigh- Tciron\npnictides1,2is such a break-through. Among the several\ntypes ofironbasedsuperconductors,3–8the 1111-type(or\nZrCuSiAs type) compounds, including LnFeAs[O 1−xFx]\n(Lnequal lanthanoids), have many derivatives with in-\nteresting physical properties. The LaCo PnO (Pn=\nP and As) compound is one of such derivatives with\nitinerant-electron ferromagnetism.9,10Magnetic prop-\nerty of LaCo PnO can be well understood within the\nspin-fluctuation theory for the weakly itinerant-electron\nferromagnets.11,12In the 1111-type compounds, two-\ndimensional square lattices of transition metal atoms\nare well separated from each other by LaO layers as\nseen in the inset (a) of Fig. 1, and thus LaCo PnO are\nthoughttobetheferromagnetswithhighly2-dimensional\nanisotropic magnetic nature.12,13Substituting La for\nother lanthanoids, one can introduce magnetic moments\nof 4f-electrons into the in-between layers of Co Pnlay-\ners. The magnetic moments of Ln3+with localized\ncharacter interact with the ferromagnetically ordered\nitinerant-electrons of Co through the Ruderman-Kittel-\nKasuya-Yoshida (RKKY) interaction.14–19Especially, in\nthe cases of Ln= Nd, Sm, and Gd, the ferromagnetic-\nantiferromagnetic transitions (FAFT) occur at TN= 14\nK, 42 K, and about 70 K, respectively, below the Curie\ntemperature ( TC∼70 K),15,17–19whereTNandTCare\nthe FAFT temperature and the Curie temperature at H\n= 0, respectively, in this report.\nRecently, McGuire etal. reported temperature( T) de-\npendence ofelectricresistivity( ρ)ofNdCoAsOunderthe\nmagnetic field ( H) up to 5 T.18In their report, ρshowed\nan abrupt increase at TNwith decreasing Tas observed\nin other compounds with FAFT, especially in FeRh and\nCoMnSi.20–22In these FAFT compounds, large magne-\ntoresistance(MR)ratiohasbeenobservedbelow TN. The\ngiant MR (GMR) effects are of great importance in the\nfield of device technology because of their application to\nthelarge-volumestorageandalsoofinterestinthefieldof\nfundamental physics since the detailed mechanism of theGMR has still remained unsolved. Since the roles of lo-\ncalized magnetic moments and itinerant-electron’s mag-\nnetic moments are well separated in LnCoAsO, it should\nbe less complicate in the mechanism of the MR effect\nin comparison to above two compounds. Furthermore,\nLnCoAsO can be seen as the complete multilayer sys-\ntem with the thickness of one atom size of each magnetic\nlayer. Therefore, it is of much importance both funda-\nmentally and in application to study the electric resistiv-\nity of the newly discoveredFAFT compounds, NdCoAsO\nand SmCoAsO, from the viewpoint of the MR effect.\nIn this paper, we showed the results of electric resistiv-\nitymeasurementsonNdCoAsOandSmCoAsOatvarious\nH. We observed the large MR effect below TNin both\ncompounds, and also observed it between TNandTCin\nSmCoAsO due to the anomalous behavior of ρatH= 0.\nWe successfully showed this anomaly in ρof SmCoAsO\noriginatingin the antiferromagneticphase which is newly\n4\n3\n2\n1\n0 (  m cm )\n300 250 200 150 100 50 0\nT  ( K )TN \nTN TC Nd Ln  = Sm \nTC \nd/dT (arb. unit)\n806040200\nT  ( K )TN \nTN TC Ω(a) \n(b) Ln\nCo As O \nFIG. 1. (color on-line) Tdependence of ρof NdCoAsO and\nSmCoAsOat H=0. Inset: (a)Crystal structureof LnCoAsO\nwith the space group P4/nmm. (b)Tdependence of dρ/dT\nof both compounds.2\n4567891 ( 10-3  cm )\n80 60 40 20 0\nT  ( K ) 0 T\n 0.1 T \n 0.2 T\n 0.5 T\n 1 T\n 2 T\n 4 T\n 6 T\n 8 T\n 10 T\n 14 T \nTN TC \nH  \n78912 ( 10-3  cm )\n80 60 40 20 0\nT  ( K ) 0 T   \n 0.1 T\n 0.2 T \n 0.5 T  \n 1 T\n 2 T\n 4 T\n 6 T\n 8 T\n 10 T\n 14 T H  TC \nTN  Ω Ω(a) Nd (b) Sm\nFIG. 2. (color on-line) Tdependence of ρof (a) NdCoAsO\nand (b) SmCoAsO at various Hfrom 0 to 14 T. Arrows indi-\ncateTCandTN.\nfound in the low- Hregion between TNandTC.\nII. EXPERIMENTS\nFor the synthesis of polycrystalline samples of\nLnCoAsO ( Ln= Nd and Sm), we used powders of Ln\n(purity: 99.9%), As(99.99%)andCoO(99.99%)asstart-\ning materials. The detailed synthesis methods are shown\nin our previous reports.15,23ρof the samples was mea-\nsured with increasing TunderHup to 14 T by a con-\nventionalDC four probesmethod. Forthe measurements\nwe used the samples sintered in the shape of a rectangle\nin size of about 2 ×2×0.5 mm3. Magnetization ( M) of\nboth compounds was measured using the superconduct-\ning quantum interference device (SQUID) magnetometer\nin Research Center for Low Temperature and Materials\nSciences, Kyoto University.\nIII. RESULTS AND DISCUSSION\nFigure 1 shows Tdependence of ρatH= 0. Both\ncompounds show metallic conduction in all temperature\nregions, being consistent to LaCo PnO.9In NdCoAsO,\nthere have been seen a slight decrease at TCand an\nabrupt increase, or a “jump”, at TNwith decreasing T\ninρas reported by McGuire et al.18This behavior can\nalso be observed in dρ/dTas shown in the inset (b) of\nFig. 1 more clearly. The same behavior was reported\nin FeRh and CoMnSi,20–22indicating these characters to\nbe common nature of the FAFT compounds. On the\nother hand, the opposite behavior, i.e., a slight increase\natTCand a slight decrease at TNwas observed in ρof\nSmCoAsO. This “bump”-like behavior of ρindicates the\ndifferent electronic state is realizing between TNandTC-30-20-100/s68/s114/s32/ /s114/s32(0) ( % )\n14121086420\nH  ( T )5 K \n8 K \n10 K 15 K \n20 K  30 K 43 K 60 K (b)  Sm -30-20-100/s68/s114/s32/ /s114/s32(0) ( % )\n5 K \n8 K \n10 K 15 K 20 K \n60 K (a)  Nd \nFIG. 3. (color on-line) Hdependence of ∆ ρ/ρ(0) of (a)\nNdCoAsO and (b) SmCoAsO at various T.\nin SmCoAsO.\nTdependence of ρat various Hfrom 0 to 14 T is\nshown in Fig. 2. In the case of Ln= Nd, the “jump” of\nρobserved at TNshifts to the low- Tdirection. This is\nconsistentwith the previousreport18and with ourreport\nof magnetization.15AroundTC,ρis reduced by H. Such\na reduction of ρcan be understood as that ferromagnetic\nfluctuations are reduced by H. In the case of Ln= Sm,\nexcept for H= 0, the behavior of ρunderHwas simi-\nlar to that of NdCoAsO: the “jump” of ρwas observed\nand its temperature decreased with increasing H, andρ\nwas reduced by HaroundTC. Due to the “bump” of\nρbetween TNandTC,ρof SmCoAsO shows strong H\ndependence in the low- Hregion of this Tregion.\nToseethe detailedchangeof ρagainstH, wemeasured\nHdependence of ρup to 14 T at various T. Figure 3\nshowsHdependence of MR ratio (∆ ρ/ρ(0)) at various T\nfrom 5 K to 60 K, where ∆ ρ=ρ(H)−ρ(0) andρ(H) =ρ\natH. In the case of Ln= Nd, large values of MR ratio\n(∼-30 %) were observed below TN, and just above TN\nthe value of about -20 % was achieved. In addition, such\nlarge values were easily achieved below 1 T in this T\nregion. Ontheotherhand, above20K,wherethereisnot\ninfluenceofFAFTbut thatofferromagnetic-fluctuations,3\n12\n10\n8\n6\n4\n2\n0M  ( 102 emu / mol )\n12010080604020\nT  ( K )(a) Nd 100 mT  \n10 mT\n1 mT (× 2)\nH 12\n10\n8\n6\n4\n2\n0M  ( 102 emu / mol )\n12010080604020\nT  ( K )(b) Sm 100 mT\n10 mT (× 5)\n1 mT (× 40)\nH  \n12\n10\n8\n6\n4\n2\n0M  ( 102 emu / mol )\n12080400\nH  ( mT )dM/dH  ( arb. unit )(d) Sm 12\n10\n8\n6\n4\n2\n0M  ( 102 emu / mol )\n1208040\nH  ( mT )dM/dH  ( arb. unit )(c) Nd \nFIG. 4. (color on-line) Tdependence of Mof (a) NdCoAsO\nand (b) SmCoAsO at H= 1, 10 and 100 mT. The data at 1\nand 10 mT of SmCoAsO and those at 1 mT of NdCoAsO are\nmagnified for a good view. Hdependence of ManddM/dT\nof (c) NdCoAsO at 40 K and (d) SmCoAsO at 43 K. Dotted\nline in the panel (d) shows the Mof NdCoAsO.\nthe valuesofMR ratiowerenot solargeandshowedweak\ndependence on H. Thus it needs high Hto obtain the\nhigh values of MR ratio in this Tregion. In the case of\nLn= Sm, somewhat different behavior was observed as\nseen in Fig. 3 (b). Below TN, large values of MR ratio ( ∼\n-28 %) were obtained as in the case of Ln= Nd although\nrather high Hwas needed compared with the Nd case.\nAbove 43 K, the MR ratio still remained in a high value,\nand the value over -10 % can be achieved below 1 T in\nthisTregion. This difference from the Nd case is clearly\noriginating in the “bump” observed in Tdependence of\nρ(0)between TNandTC, sinceHdependenceofMRratio\nin thisTregion showed the same behavior in the case of\nNd above H= 1 T. Due to this anomalous “bump” of ρ\nSmCoAsO comes to show large MR ratio in much wider\nTregion compared with NdCoAsO.\nTo clarify what happens in the low- Hregion between\nTNandTCin SmCoAsO, we measured magnetization M\nin the low- Hregion presicely. Figures 4 (a) and 4 (b)\nshowTdependence of Mof NdCoAsO and SmCoAsO,\nrespectively, at H= 1, 10, and 100 mT. Since Mof\nSmCoAsO at H= 1 and 10 mT and that of NdCoAsO\nat 1 mT are too small to show in the same scale with\nother data, we magnified these data as indicated in the\nfigures. In the case of Ln= Nd, a ferromagnetic feature,\ni.e., a convex curvature with abrupt changes at TNand30\n25\n20\n15\n10\n5\n0H  ( T )\n50403020100\nT  ( K ) FM  \n AFM-I AFM-II 80\n60\n40\n20\n0H  ( mT )\n806040\nT  ( K )FM \nPM \nAFM-II IM AFM-I\n16\n12\n8\n4\n0H  ( T )\n2520151050\nT  ( K )(a) \n(b) \nAFM FM AFM FM PMAFM-IFMPMAFM-II\nTLn = Sm\nLn = Nd42 K\n15 K72 K\n68 KH\nT\nH\nFIG. 5. (color on-line) (a) Magnetic phase diagram of Sm-\nCoAsO. Circles and triangles show the temperatures where\ndρ/dTtakes zero and the minimum, respectively. Small\n“+” and “ ×” symbols stand for HwheredM/dH shows\nanomaly.23Dotted lines are guides for the eyes. AFM-I and\nFM denote the antiferromagnetically and ferromagneticall y\nordered phases. AFM-II denotes the newly discovered anti-\nferromagnetic phase at the low- Hregion. Inset: the low- H\nregionofphasediagram. Openandclosedcircles showtheval -\nues estimated by the same manner in the case of SmCoAsO.\nIM and PM denote intermediate state during metamagnetic\ntransition and paramagnetic phase. (b) Magnetic phase dia-\ngram of NdCoAsO. Circles and triangles show the tempera-\ntures where dρ/dTtakes zero and the minimum, respectively.\nAFM denotes the antiferromagnetically ordered phase. Inse t:\nschematic phase diagrams of both compounds in the low- H\nregion.\nTC, was observed down to 1 mT between TNandTC,\nwhile in the case of Ln= Sm, an obvious anomaly, i.e.,\na concave curvature, was observed in the corresponding\nTregion below 10 mT. This indicates that ferromagnet-\nically ordered magnetic moments at TCare immediately\nforced to change their arrangement of moments to an-\ntiferromagnetic one. The same anomalous behavior of\nMin the low- Hregion was also reported in Ref. [19],4\nshowing that this anomaly comes from the intrinsic na-\nture of SmCoAsO, not from impurities. Fig. 4 (c) and\n4 (d) show isothermal magnetic curves at 40 K for the\nNd case and at 43 K for the Sm case, where both tem-\nperatures correspond to the “ferromagnetic” phase. In\ncontrast to the case with Ln= Nd, in which Mshows\nferromagnetic H-dependence in all the H-region, in the\ncase ofLn= Sm, a concave curvature, which reminds\nus of a metamagnetic transition, was observed up to 80\nmT, showingtheexistenceofanantiferromagneticphase.\nThis antiferromagneticphaseis easyto breakunderweak\nH, and above 80 mT the itinerant-electronic system of\nSmCoAsO behaves as ferromagneticone in this T-region.\nIn Fig. 5 (a), we summarized the results of mea-\nsurements as the magnetic phase diagram of SmCoAsO.\nClosed circles and triangles respectively show the tem-\nperatures where dρ/dTcomes to zero and the minimum\nat various H. Small “+” and “ ×” symbols show the val-\nues ofHat which dM/dH shows a discontinuity.23Good\nconsistencybetweentheestimatedvaluesfromtheresults\nof electric resistivity and magnetization shows that the\n“jump” of ρcorresponds to the FAFT, which was con-\nfirmed through magnetization measurements. The ob-\ntained phase diagram resembles that of CoMnSi,24in-\ndicating the mechanism of FAFT being similar to each\nother. Inset of Fig. 5 (a) shows the low- Hregion of\nthe phase diagram of SmCoAsO. Open and closed cir-\ncles show the values of Hcorresponding to the upper\nand lower peaks of dM/dH, respectively. It is interest-\ning that both FM and AFM-II phases are on the border\nwith AFM-I phase at TN= 42 K. The magnetic phase di-\nagram of NdCoAsO is quite similar to that of SmCoAsO\nas shown in Fig. 5 (b), except for the absence of AFM-\nII phase. The phase diagrams of both compounds are\nschematically shown in the inset of Fig. 5 (b) for a good\nunderstanding of the similarity and difference between\ntwo compounds. It should be noted that another mag-\nnetically ordered phase is reported in the low- Tregion of\nthe AFM phase.17–19However, we omitted it from Fig.\n5 for simplicity.\nAs observed in Fig. 2 (b), ρshowed an increase with\ndecreasing Tbelow 5 K in SmCoAsO. Recently, an-\nother antiferromagnetic phase was supposed to exist be-\nlow about 5 K.19Therefore, the increase of ρbelow 5\nK may be closely related to the low- Tantiferromagnetic\nphase. It is an interesting and challenging issue althoughwe did not have much information of the low- Tantifer-\nromagnetic phase at this stage.\nFromthe viewpoint ofcrystalstructure, LnCoAsOcan\nbe seen as a “natural” magnetic thin-film system: each\n“film” precisely has a thickness of one atom size, which\nis in a sequence of Co- Ln-Ln-Co-···, where magnetic\nmoments lie within each plane. This is the situation sim-\nilar to the model discussed in the review for artificial\nmultilayer systems by Camley and Stamp,25and the be-\nhavior of ρinLnCoAsO seems to be explained by the\nspin-dependent scattering, i.e., the “jump” of ρwith de-\ncreasing Tis explained to be caused by the change of ar-\nrangement of magnetic spins from ferromagnetic to anti-\nferromagnetic. If this scenario is correct, the component\nofρresponsible to FAFT can be ascribed to the com-\nponent along the c-axis. In LnCoAsO, however, electric\nconductivity along the c-axis is thought to be bad be-\ncause of two-dimensional band structures, and thus re-\nsistivity along the c-axis should be masked by that in the\nab-plane in our measurement condition. For further de-\ntailed study to clarify the mechanism of large MR effects\ninLnCoAsO, single crystalline samples are needed. For-\ntunately, the knowledge obtained through the study of\niron-pnictide superconductors must be available for the\nsynthesis of the single crystal.26\nIV. CONCLUSION\nIn summary, we measured temperature dependence of\nelectric resistivity of NdCoAsO and SmCoAsO at vari-\nous magnetic field up to 14 T. In addition to the abrupt\nincrease in electric resistivity at the FAFT temperature,\nSmCoAsO shows an anomalous behavior, which is clar-\nified to be originating in the newly antiferromagnetic\nphase in the low magnetic field region. Due to this anti-\nferromagnetic phase, large magnetoresistance ratio was\nrealized even above the FAFT temperature up to the\nCurie temperature in SmCoAsO.\nACKNOWLEDGMENTS\nThis work is supported by Grants-in-Aid for Scientific\nResearch from the Japan Society for Promotion of Sci-\nence (Grants No. 19350030 and No. 22350029).\n∗Present address: Institute of Engineering, Division of Ad-\nvanced Applied Physics, Tokyo University of Agriculture\nand Technology; shioshio@kuchem.kyoto-u.ac.jp;\n†kyhv@kuchem.kyoto-u.ac.jp\n1Y. Kamihara, H. Hiramatsu, M. Hirano, R. Kawamura,\nH. Yanagi, T. Kamiya, and H. Hosono, J. Am. Chem. Soc.\n128, 10012 (2006).\n2Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J.\nAm. Chem. Soc. 130, 3296 (2008).3Z. A. Ren, W. 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Tomioka, Physica C 469, 901 (2009)."    },    {        "title": "1706.07904v3.Random_field_induced_disordering_mechanism_in_a_disordered_ferromagnet__Between_the_Imry_Ma_and_the_standard_disordering_mechanism.pdf",        "content": "arXiv:1706.07904v3  [cond-mat.dis-nn]  11 Dec 2017Random-field-induced disordering mechanism in a disordere d ferromagnet:\nBetween the Imry-Ma and the standard disordering mechanism\nJuan Carlos Andresen,1, 2Helmut G. Katzgraber,3, 4, 5and Moshe Schechter2\n1Department of Theoretical Physics, KTH Stockholm, 10691 St ockholm, Sweden\n2Department of Physics, Ben Gurion University of the Negev, B eer Sheva 84105, Israel\n3Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843-4242, USA\n41QB Information Technologies (1QBit), 458-550 Burrard Str eet, Vancouver, British Columbia V6C 2B5, Canada\n5Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501\n(Dated: April 20, 2022)\nRandom fields disorder Ising ferromagnets by aligning singl e spins in the direction of the random field in three\nspace dimensions, or by flipping large ferromagnetic domain s at dimensions two and below. While the former\nrequires random fields of typical magnitude similar to the in teraction strength, the latter Imry-Ma mechanism\nonly requires infinitesimal random fields. Recently, it has b een shown that for dilute anisotropic dipolar systems\na third mechanism exists, where the ferromagnetic phase is d isordered by finite-size glassy domains at a random\nfield of finite magnitude that is considerably smaller than th e typical interaction strength. Using large-scale\nMonte Carlo simulations and zero-temperature numerical ap proaches, we show that this mechanism applies to\ndisordered ferromagnets with competing short-range ferro magnetic and antiferromagnetic interactions, suggest-\ning its generality in ferromagnetic systems with competing interactions and an underlying spin-glass phase. A\nfinite-size-scaling analysis of the magnetization distrib ution suggests that the transition might be first order.\nPACS numbers:\nI. INTRODUCTION\nSpin glasses, where frustration and disorder are introduce d\nthrough random competing ferromagnetic and antiferromag-\nnetic interactions [1], and random-field ferromagnets, whe re\ndisorder is introduced through an effective longitudinal fi eld\n[2], are two archetypal models for the study of disordered\nmagnetic systems [3, 4]. While usually studied indepen-\ndently from each other, random interactions and random field s\nare generic in many nonmagnetic systems [5], and dominate\nthermodynamic and dynamic properties in e.g. orientationa l\nglasses[6] and relaxor ferroelectrics[7, 8]. In magnetic s ys-\ntems, random interactions are abundant, yet the presence of\nan effective longitudinal random field is nontrivial. Appli ed\nmagnetic fields cannot be locally randomized and nonmag-\nnetic disorder cannot produce an effective random magnetic\nfield, because it violates time-reversal symmetry.\nAnisotropic dipolar magnets, and specifically the\nLiHoxY1−xF4compound, are an intriguing exception.\nThe interplay of an applied field in the direction transverse\nto the easy axis of the magnetic holmium ions and the\noff-diagonal elements of the dipolar interaction gives ris e to\nan effective longitudinal field [9–11]. This field is locally\nrandom, transforming spatial disorder coming from the\ndilution of the Ho ions by the nonmagnetic yttrium ions into\na disorder in the effective longitudinal field. Furthermore ,\nas a function of holmium concentration, LiHoxY1−xF4has\nboth a ferromagnetic phase at x/greaterorsimilar0.3and a spin-glass\nphase at 0< x/lessorsimilar0.3including the extreme dilute limit\n[12–14]. The LiHoxY1−xF4system is therefore ideal for the\nstudy of the interplay of competing interactions and effect ive\nlongitudinal random fields, and the effect of this interplay on\nthe thermodynamic and dynamical properties of the system.\nRecently, we have shown [13] that disordered anisotropic\ndipolar magnets in their ferromagnetic phase, are driven in toa quasi-spin-glass phase upon the introduction of a finite effec-\ntive random magnetic field which is considerably smaller than\nthe typical nearest-neighbor interaction . This occurs also in\nthree space dimensions, where it displays a novel disorderi ng\nmechanism, intermediate between the standard disordering of\na ferromagnet and the Imry-Ma [2] mechanism at dimensions\ntwo and below. The disordering field is neither infinitesimal ,\nnor of the order of the interaction. The smallness of the ran-\ndom field at the transition is dictated by the proximity of the\nsystem to the zero-field transition between the ferromagnet\nand the spin-glass phase. Moreover, the disordered phase ne ar\nthe transition consists of neither single spins pointing in the\ndirection of the random field, nor of large ferromagnetic do-\nmains, but of finite-size glassy domains, reminiscent of the\ncompeting spin-glass phase. We denote this phase “quasi-\nspin-glass” (QSG) in the regime of low temperatures, where i t\nis frozen, and “paramagnetic QSG” (pQSG) in the high tem-\nperature regime. The size of the glassy domains is a func-\ntion of the magnitude of the disordering field and temperatur e\n[9, 13, 15].\nHere we consider a more general model of an Ising magnet\nwhere the interactions are short ranged, taken from a Gaussi an\ndistribution with a mean ferromagnetic value J0and a stan-\ndard deviation J, and with longitudinal random fields taken\nfrom a Gaussian distribution of mean zero and standard devi-\nationHr. Using jaded extremal optimization [16] and large-\nscale parallel tempering Monte Carlo simulations [17] we an -\nalyze the thermodynamic properties of the system at zero and\nfinite temperature. For zero random field we establish the\nphase diagram of the system, consisting of low-temperature\nferromagnetic (FM) and spin-glass (SG) phases and a high-\ntemperature paramagnetic (PM) phase. The zero-temperatur e\ntransition between the FM and SG phases occurs at a ratio of\nJ/J0≈1.65. We then analyze the disordering of the FM\nphase atJ/J0<1.65with increasing temperature and ran-2\ndom field. For zero temperature we find that the disordering\nof the FM phase occurs at finite random field, which is much\nsmaller than the typical interaction, 0< Hr≪J0, the value\nofHrdepending on the proximity of the system at zero field\nto the SG phase. At finite temperature we find that the critical\ntemperature of the FM-to-pQSG transition increases linear ly\nas a function of decreasing random field, down to a rather\nsmall value of Hr. Our results here are in agreement with\nour previous results for the dipolar-interacting LiHoxY1−xF4\nsystem [13], as well as with experimental data for this ma-\nterial [18]. This suggests that the disordering mechanism b y\nfinite-size glassy domains is a generic feature of disordered\nferromagnets with competing interactions in the presence o f\nrandom fields. By analyzing the distribution of magnetizati on\nvalues near the random-field-driven transition, we find evi-\ndence for a first-order transition between the FM and pQSG\nphases in the regime where the FM phase is in proximity to\nthe SG phase. No evidence for a first-order transition is foun d\nin the regime where the interactions are strongly ferromag-\nnetic dominated, i.e., J/J0≪1.\nThe paper is structured as follows. In Sec. II we introduce\nthe model. In Sec. III we introduce the methods used for zero-\ntemperature and for finite-temperature calculations. Our r e-\nsults are presented in Sec. IV, and discussion and conclusio ns\nare given in Sec. V. An Appendix lists all parameters of the\ndifferent simulations.\nII. MODEL\nThe model we simulate is given by the Hamiltonian\nH=−/summationdisplay\n/angbracketlefti,j/angbracketrightJijSiSj−/summationdisplay\nihiSi, (1)\nwhere the sum is over nearest neighbors, the spin couplings\nJijare chosen from a Gaussian distribution with standard de-\nviationJand mean J0, and the Siare Ising spins on the\nvertices of a hypercube of dimension d= 3 and linear size\nL. Throughout this work we fix the mean of the distribution\nJ0= 1 and use the standard deviation of the Gaussian distri-\nbutionJas a means of tuning the interaction disorder in the\nmodel. The second term describes the coupling of the Ising\nspins to a site-dependent random field hi. The random fields\nare taken from a Gaussian distribution with zero mean and\nstandard deviation Hr.\nDifferent parameters of the model lead to different phase\ndiagrams. If Jis small compared to |J0|andHr= 0, as the\ntemperature is reduced, the system undergoes a phase transi -\ntion from a PM phase to a FM phase for J0>0. ForJ0<0\nthe system undergoes a phase transition from a PM phase to\nan antiferromagnetic (AFM) phase. By increasing the value o f\nthe parameter Jit is possible to introduce disorder and frus-\ntration into the system. These two ingredients are known to\nbe essential for the emergence of a SG phase. The ratio J0/J\nquantifies the amount of interaction disorder in the system.\nBelow a critical ratio J0/Jc, the system has a PM phase at\nhigh temperatures and a SG phase at low temperatures (keep-\ningHr= 0). Extreme cases occur when J= 0,J0∝negationslash= 002468\n0 0 .5 1 1 .5 202468\n0 0 .5 1 1 .5 2PMTc/J\nJ0/JTc/J\nJ0/JFM\nSG\n1FIG. 1: Dimensionless temperature T/J vs mean of the disor-\nder distribution J0/Jphase diagram for the model given in Eq. (1)\nwith fixed J0= 1 andHr= 0. AtJ/lessorsimilar1.65(J0/J/greaterorsimilar0.606)\nand low temperatures the system is in a FM phase. For J/greaterorsimilar1.65\n(J0/J/lessorsimilar0.606) and low temperatures the system is in a SG phase.\nAt high temperatures the system is in the PM phase independen tly\nof the value of J0/J. Note that the model reduces to the three-\ndimensional Ising model for J= 0 and to the EA spin-glass for\nJ0= 0.\n(J0/J→ ∞ ) where the model reduces to the well-known\nIsing model if Hr= 0, and to the random-field Ising model\n[2] ifHr>0; and when J∝negationslash= 0,J0= 0 (J0/J= 0) where\nthe model reduces to the Edwards-Anderson (EA) spin-glass\nmodel [1]. In Fig. 1 we plot the phase diagram of the system\nas a function of temperature and interaction disorder J(keep-\ningHr= 0) presenting a phase transition at zero temperature\nbetween a FM phase at small disorder and a SG phase at large\ndisorder, as well as a PM phase at high temperatures.\nThe random-field term in Eq. (1) introduces a third axis to\nFig. 1. For both limits of zero Jand finite J0(pure ferromag-\nnet) and zero J0and finite J(pure EA spin glass) the effect\nof finite random field has been thoroughly studied [4, 19–29].\nFor the latter limit two different pictures to describe finit e-\ndimensional spin glasses have been proposed: the replica\nsymmetry breaking (RSB) picture based on the solution of the\nSherrington-Kirkpatrick model [30–34], which predicts th e\nexistence of a spin-glass phase at finite fields and the drople t\npicture [27, 28, 35, 36], which contrary to the RSB predicts t he\ninstability of the spin-glass phase for any infinitesimal sm all\nfield. Which of these pictures correctly describes the three -\ndimensional spin glass is still an open question [29, 37–43] .\nIn this paper, our theoretical considerations are based on t he\ndroplet picture, but we do not undermine the possibility to d e-\nrive an analogous prediction within the RSB picture. Specif -\nically, within the droplet picture of spin glasses, it is arg ued\n[27] that the spin-glass phase is unstable to infinitesimal r an-\ndom fields, as finite-size glassy domains are created, destro y-\ning long-range glass order. Here we study the effect of the\nrandom field in the range, where the system is a ferromagnet,\nalbeit with competing interactions ( 0< J/J 0<1.65). At\nT= 0 we obtain the phase diagram as a function of Jand\nHr. At finite temperature we obtain the phase diagram for\ndifferent values of Jin the ferromagnetic regime, in proxim-3\nity to the spin-glass phase and deep inside the ferromagneti c\nregime. We find the dependence of the critical temperature\nTcon random-field strength and study the nature of the phase\ntransition.\nIII. METHODS\nA. Zero-temperature simulations\nFor the zero-temperature simulations we use the jaded ex-\ntremal optimization heuristic [16, 44]. We set the algorith m\nparameter τ= 1.6,1.8,1.9with an aging parameter Γ =\n0.0001 for at least 224simulation steps. Each disorder real-\nization undergoes at least 512independent runs. We moni-\ntor how many times the lowest energy configuration is found.\nWhen the success rate is more than ∼5%, we assume to have\nfound the ground-state configuration. Ground states are fou nd\nwith high confidence for L/lessorequalslant10forHr= 0 andL/lessorequalslant8for\nHr∝negationslash= 0.\nThe FM to SG (PM) phase boundary is identified through\nthe Binder ratio [45]\ng=1\n2/parenleftBigg\n3−/bracketleftbig\nm4/bracketrightbig\nav\n[m2]2\nav/parenrightBigg\n, (2)\nwherem= 1/N/summationtext\niSiis the magnetization of the system,\nN=L3is the number of spins and [···]avrepresents a disor-\nder average. The Binder ratio gis a dimensionless observable\nthat scales as g∼˜G/bracketleftbig\nL1/ν(J−Jc)/bracketrightbig\n. The argument vanishes\nifJ=Jcfor all linear system sizes L. Therefore, the cross-\ning of the curves for different system sizes gives an estimat e\nof the transition value Jcup to finite-size effects. Simulation\nparameters are shown in table IV.\nThe critical disorder Jcand the standard deviation for each\nrandom field Hrare estimated using a Levenberg-Marquardt\nminimization combined with a bootstrap analysis, where we\nassume that the universal function gis well approximated by\na third-order polynomial. We show in Figure 2 how the esti-\nmatedJcagrees with the crossing of the Binder ratio curves\nwhenH= 0.00. The dashed lines are fit functions to the\ndata used only for visualization purposes and the vertical l ine\nis the average of the intersections of all samples, in this ca se\nJc= 1.63(1) .\nB. Finite-temperature simulations\nThe simulations at finite temperature are done using a com-\nbination of single-spin-flip Monte Carlo with parallel temp er-\ning Monte Carlo [17]. To determine the finite-temperature\ntransitions for a fixed random field strength Hrand disorder\nJwe measure the ferromagnetic and spin-glass two-point cor-\nrelation length [46]\nξsg,fm\nL=1\n2sin(kmin/2)/radicalBigg\nχsg,fm(0)\nχsg,fm(kmin)−1, (3)0.50.60.70.80.91\n1.4 1 .5 1 .6 1 .7 1 .8Jc\nHr= 0.00g\nJL= 4\nL= 6\nL= 8\nL= 10\n1FIG. 2: Binder ratio gas given by Eq. (2) for linear system sizes\nL= 4,6,8, and10at a random field strength Hr= 0 andT= 0.\nThe crossing of the curves for different sizes Lgives an estimate of\nthe critical disorder Jcdenoting the transition between the FM phase\natJ < J cand the SG phase at J > J c. The dashed lines are guides\nto the eye.\nwherekmin= (2π/L,0,0),χsg(k)is the spin-glass wave-\ndependent susceptibility\nχsg(k) =1\nN/bracketleftBig\n∝angbracketleft/summationdisplay\ni,jqiqjeikrij∝angbracketright/bracketrightBig\nav, (4)\nqi=Sα\niSβ\niis the two-replica spin overlap, and χfm(k)is the\nferromagnetic wave-dependent susceptibility\nχfm(k) =1\nN/bracketleftBig\n∝angbracketleft/summationdisplay\ni,jSiSjeikrij∝angbracketright/bracketrightBig\nav. (5)\nHere,∝angbracketleft···∝angbracketright denotes a thermal average and [···]ava disordered\naverage.\nNear the transition the dimensionless ratio of the two-poin t\ncorrelation functions ξsg,fm\nL and the linear system size Lscales\nasξsg,fm\nL/L∼˜X/bracketleftbig\nL1/ν(T−Tc)/bracketrightbig\n. At the critical temper-\nature the argument vanishes and the dimensionless quantity\nbecomes size independent (up to scaling corrections), henc e\nwe expect lines of different system sizes to cross at Tc. If,\nhowever, the lines do not cross, no transition takes place at the\nstudied temperature range. Some example cases are shown\nin Fig. 3. In Fig. 3(a) the FM two-point correlation length\nforJ= 1.60(J0/J= 0.625) andHr= 0.00is depicted.\nClearly, the curves cross at a putative point which indicate s\na transition at a temperature Tc. In Fig. 3(b) we plot the\nsame quantity, but for a higher disorder value J= 1.80\n(J0/J= 0.5¯5). In this case, clearly the curves do not in-\ntersect at any studied temperature, showing a lack of a fer-\nromagnetic phase transition in this temperature range. Mor e-\nover, it suggests a possible lack of ferromagnetic transiti on at\nanyT >0. In Fig. 3(c) we show the spin-glass two-point\ncorrelation length for the same disorder value J= 1.80as in\nFig. 3(b). The curves do cross in the simulated temperature\nrange, signaling a PM to SG phase transition.4\nFor small random field strengths and interactions distribu-\ntion widths the equilibration time is relatively short, mak ing it\npossible to simulate large system sizes. With increasing ra n-\ndom fields and interaction distribution widths equilibrati on\ntimes become longer. To determine the critical temperature\nwe approximate the scaling function by a third-order polyno -\nmial and perform a fit with six free parameters. To estimate\nthe error bars we use a bootstrap analysis as described by\nKatzgraber et al. [47]. Simulation parameters for the finite-\ntemperature simulations are shown in Tables I, II, and III.\nIV . RESULTS\nThe phase diagram of the model presented in Sec. II is\nshown in Fig. 1. It has a high temperature PM phase and low\ntemperature FM and SG phases at small and strong disorder\nJ, respectively. The boundary between the FM and the SG\nphase is at J≈1.65(J0/J≈0.606) with a weak reentrance\natT= 0 [Jc= 1.63(1) ]. The obtained phase diagram qual-\nitatively agrees with previous results[48–50] for the mode l in\nEq. (1), and for the closely related diluted bimodal Ising SG\nmodel[51]. In comparison to the real space rescaling method\nby Southern and Young [50], we find larger values for Tcand\nforJc, in accordance with the typical behavior of real-space\nrescaling method [50]. Note the linear dependence of the PM\nto SG transition on the mean interaction value. The same is\ntrue for the PM to FM phase transition in the regime J/lessorsimilar1.2\n(J0/J/greaterorsimilar0.83¯3).\nWe now study the phase diagram with the inclusion of finite\nrandom fields and consider first the case where T= 0, see\nFig. 4. The FM phase at low disorder Jand low random-field\nstrengthHris disordered at large Jinto a QSG phase, con-\nsisting of finite size glassy domains, but no long range glass\norder except at Hr= 0. The data point at Hr= 0corresponds\nto the FM to SG transition point at T= 0 plotted also in the\nT/J vsJ0/Jphase diagram shown in Fig. 1. At small Jand\nlargeHrthe FM phase disorders into a PM phase. The data\npoint atJ= 0 corresponds to the critical random field value\nHr≈2.16of the random-field Ising model. For low disor-\nderJwe expect the critical random field to be of the order of\nthe mean interaction J0, because in the PM phase single spins\norder along their local effective field.\nThe situation for J/lessorsimilarJcis, however, different. At Hr= 0\nthe FM ground state and the lowest-energy SG state differ in\nenergy, the difference being linear in J−Jc[13]. At finite Hr,\nthe low-energy SG state changes profoundly: long-range or-\nder is destroyed, finite-size glassy domains appear, and the en-\nergy of the resulting QSG state is reduced accordingly [9, 13 ].\nThis leads to a reduction of the energy of the QSG state below\nthat of the ferromagnetic state, and thus to a zero-temperat ure\nphase transition at finite Hr, which is much smaller than J0.\nThe predicted functional form for the boundary between the\nFM and the PM phases at T= 0is given by [13]\nHr/parenleftbig\nJ−1/parenrightbig\n∝/parenleftbig\nJ−1−J−1\nc/parenrightbig(3/2−θ)\n(3−θ). (6)\nHere,θ≈0.19is the stiffness exponent [52]. Indeed, our\nnumerical results are in agreement with this functional for m,00.20.40.60.811.21.4\n2 2.5 3 3 .5 4 4 .5 5(a)\nJ= 1.60\nHr= 0.00ξfm/L\nTL= 6\nL= 8\nL= 10\nL= 12\nL= 16\nL= 20\nL= 24\nL= 28\nL= 32\n100.10.20.30.40.5\n1 1.5 2 2 .5 3 3 .5(b)\nJ= 1.80\nHr= 0.00ξfm/L\nTL= 6\nL= 8\nL= 10\nL= 12\nL= 14\n100.20.40.60.811.21.41.6\n1 1.5 2 2 .5 3 3 .5(c)\nJ= 1.80\nHr= 0.00ξsg/L\nTL= 6\nL= 8\nL= 10\nL= 12\nL= 14\n1FIG. 3: Ferromagnetic and spin-glass two-point correlatio n function\ndivided by the linear system size LforJ= 1.60[panel (a)] and 1.80\n[panels (b) and (c)], respectively. (a) The crossing of the c urves of\nthe dimensionless quantity ξfm\nL/Lfor linear system sizes L= 6 –\n20indicates that a PM to FM transition occurs at Tc∼3.09(4) .\n(b) The dimensionless quantity ξfm\nL/Ldoes not cross in the studied\ntemperature range, showing the lack of a FM transition. (c) C urves\ncorresponding to different system sizes of the dimensionle ss quantity\nξsg\nL/Lcross atTc∼1.90(14) indicating that a PM to SG transition\noccurs in this temperature range.\nas can be seen by fitting the data for Hr<1, fixing the power\nto its theoretically obtained value [13] β= (3/2−θ)/(3−\nθ) = 0.466, and using the above estimated critical disorder\nJc≈1.63for the fit. This leaves the proportionality prefactor\nαas the single free fitting parameter. The inset of Fig. 4 shows\nthe fitting result. We note that our data are limited to Hr≤\n0.5. It was not possible to determine the crossing point of5\n00.511.522.5\n0 0.5 1 1.5 2 2.5 3 3.5 400.20.4\n0.5 0.7J\nHrQSGSG\nFM PM\nHr\nJ−1QSG\nSGFM\n1FIG. 4:J–Hrphase diagram at T= 0. AtHr= 0 there is a FM-\nSG transition at Jc(0) = 1.63. At finite but small Hrthe disordered\nparamagnetic phase has finite size glassy domains, denoted h ere as a\n“quasi-SG” (QSG) phase. The crossover between the QSG phase and\nthe PM phase with short range spin-spin correlations is spec ulative.\nIn the inset, the solid black line is a fit Hr/parenleftbig\nJ−1/parenrightbig\n=α(J−1−J−1\nc)β\nto the data at low random field strength. Here, Jcandβare fixed\nparameters; we use the above estimated value Jc(0) = 1.63, and\nthe analytical result of Ref. [13] β= 0.466.α= 1.52(17) is a free\nfitting parameter.\nthe Binder ratio curves for different system sizes for rando m\nfields with Hr>0.5because of the proximity of the crossing\npoint tog= 1.\nAt finite temperature and for J/lessorsimilarJc, similar considera-\ntions to the ones mentioned above manifest themselves in the\ndependence of Tcon the random field. The underlying glassy\nstate at finite temperature consists of paramagnetic SG do-\nmains of typical size ξ. We denote this phase as “pQSG”.\nFrom moderate Hr, which does not affect the typical domain\nsize, the reduction of the energy per spin of the pQSG state is\n∝Hr/ξ3/2. As a result, only for Hr> h∗∝(Jc−J)ξ3/2\nthe disordering of the FM phase is by the pQSG phase. In this\nregime theory predicts a linear dependence of TconHr[13].\nAtHr< h∗the disordering is to the standard PM phase, with\nthe known weak dependence of TconHr. All the above con-\nsiderations do not apply to J≪Jc, where the FM is far from\nthe SG phase, and the disordering to the PM phase is standard\nfor all strengths of random field.\nIn Fig. 5 we present the TvsHrphase diagrams for dis-\nordersJ= 0.50,J= 1.50, andJ= 1.60. In Fig. 5(a)\nJ≪Jc. The disordering is into the standard PM phase, with\nquadratic dependence T(Hr)−Tc(0)∝H2\nrfor small Hr\nand a small reentrance at low temperatures. For J= 1.50\n[panel (b)] the system is close to the SG phase. Indeed, for\nHr< h∗≈0.1the dependence of TconHris weak, as\nis expected for disordering to the standard PM phase. For\nHr> h∗we find the dependence of TconHrto be linear,\nin agreement with disordering to the pQSG phase. We note\nthat at0.5< Hr<0.6there is a sharp decrease of Tcto zero\nwithin the finite-temperature simulations, and we find a valu e\nofHr= 0.39for the zero temperature transition (inferred\nfrom the fitting function in the inset of Fig. 4). This is in agr ee-\nment with the disordering field by the QSG phase being muchsmaller than the interaction strength also at low temperatu res.\nWe note that as the QSG phase is frozen, susceptibility mea-\nsurements are expected to depict only the crossover between\nthe QSG and the PM phase at large random fields. This is in\nagreement with Ref. [18], where a sharp feature in the sus-\nceptibility is observed at higher temperatures, depicting the\ntransition to the pQSG phase, but a smooth crossover is ob-\nserved for the lower temperatures. The observed peak value\nat the crossover occurs at larger Hrwith diminishing peak\nvalue as temperature is reduced [18], in agreement with the\nscenario of the QSG having smaller glassy domains as a func-\ntion of increasing Hr[9, 13]. In panel (c) of Fig. 5 we present\nthe results for J= 1.60, closer to the zero-field FM-to-SG\ntransition. Results are similar to the case of J= 1.50, with\na weaker linear dependence, and with a smaller random field,\n0.3< Hr<0.4, which disorders the FM at all temperatures.\nLet us now consider the order of the FM to PM/pQSG tran-\nsition. The order of the phase transition of the RFIM is a\nlong standing question. Whereas analytic arguments suppor t a\nsecond-order transition controlled by a zero-temperature fixed\npoint [27, 53, 54], some numerical results support a first-or der\ntransition [19, 20] while others support a second-order tra n-\nsition [21, 22, 55–59]. For bimodal disorder distributions a\nmean-field solution suggests a first-order transition for la rge\nenough values of the random field [60]. Recent numeri-\ncal work supports universality in the RFIM [58], suggesting\nthat the nature of the phase transition is independent of the\nrandom-field distribution. Moreover, high-accuracy estim ates\nfor the magnetic exponent ratio β/ν [59] found the value to\nbe very small, but clearly finite. In the present work, for sma ll\nexchange disorder ( J= 0.50) we do not find any signature\nof a first-order phase transition. This suggests that the rec ent\nphase transition behavior found for the RFIM [58, 59] also\napplies for models with small exchange disorder.\nWe now consider the order of the transition in the pres-\nence of both random fields and strong random interactions.\nIntriguingly, our results suggest, for strong interaction disor-\nder, a continuous transition for Hr< h∗where the FM dis-\norders into a “standard” PM phase, and a first-order transiti on\nforHr> h∗where the disordering is into the pQSG phase.\nIn the latter, we find a discontinuous jump in the magnetiza-\ntion across the transition, but from our microcanonical sim u-\nlations, based on a recently proposed method [61–63], we did\nnot detect any latent heat, which is either zero, or too small to\nbe detected in the current accessible system sizes, similar to\nRefs. [20, 21].\nSpecifically, we study the distribution of the magnetizatio n\nabove and below TcforJ= 1.50at different field strengths\nHr, as shown in Fig. 6. A bimodal distribution close to\nTcis a sign of a first-order phase transition, whereas a nor-\nmal distribution with mean µ0= [∝angbracketleft|m|∝angbracketright]avwould be the\nexpected distribution for a continuous phase transition. F or\nsmallHr= 0.04< h∗[panel (a) of Fig. 6] we note that the\nsingle-Gaussian fit and the double-Gaussian fit agree, sugge st-\ning that the magnetization distribution is likely normal cl ose to\nTcand therefore the phase transition is continuous. However,\nwe find that for large Hr= 0.40> h∗the double-Gaussian\nfit does differ considerably from the single-Gaussian fit, mo re-6\n012345\n0 1 2 3 4012345\n0 1 2 3 4Tc\nHrJ= 0.50\n(a)PM\nFMTc\nHrJ= 0.50\n(a)PM\nFM\n1012345\n0 0 .2 0 .4 0 .6 0 .8012345\n0 0 .2 0 .4 0 .6 0 .8Tc\nHrJ= 1.50\n(b)FMpQSG\nQSGPMTc\nHrJ= 1.50\n(b)FMpQSG\nQSGPM\n1012345\n0 0 .2 0 .4 0 .6 0 .8012345\n0 0 .2 0 .4 0 .6 0 .80.50.7\n1 2 3Tc\nHr012345\n0 0 .2 0 .4 0 .6 0 .8J= 1.60\n(c)FMPM\npQSG\nQSGTc\nHr012345\n0 0 .2 0 .4 0 .6 0 .8J= 1.60\n(c)FMPM\npQSG\nQSG\nξfm/L\nT0.50.7\n1 2 3\n1FIG. 5: T–Hrphase diagram for J= 0.50[panel (a)], J= 1.50[panel (b)] and J= 1.60[panel (c)]. For the slightly-disordered\nsystem with J= 0.50in panel (a) the relation expected from mean-field theory T(Hr)−Tc∼H2\nrholds, except for a weak reentrance\nat the lowest temperatures. The data are consistent with the disordered phase being a standard PM. For the highly-disord ered systems with\nJ= 1.50[panel (b)] and J= 1.60[panel (c)] where the FM phase is close to the SG phase at Hr= 0(see Figure 4) we find a linear relation\nT(Hr)−Tc(h∗)∝HrforHr> h∗consistent with the disordered phase being a pQSG. Furtherm ore, we find strong reentrance at low\ntemperatures with critical Hr≪J0, in agreement with the disordered phase being a QSG. The inse t in panel (c) shows ξfm\nLforL= 8,10,\nand12forHr= 0.40where the curves do not cross, indicating the lack of a phase t ransition to the FM phase for the studied temperature\nrange. The dashed lines in all panels between the pQSG and PM p hases and between the QSG and pQSG phases denote smooth cross overs,\ntheir functional form should be considered as a guide to the e ye.\n05101520\n0 0.1 0.2 0.3 0.4 0.5 0.6J= 1.50\nHr= 0.04\nT= 3.22\nTc≈3.44\n(a)Density\n|m|\n10246810\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9J0= 1.50\nhr= 0.40\nT= 2.00\nTc≈2.86Density\n|m|\n10246810\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9J0= 1.50\nhr= 0.40\nT= 2.89\nTc≈2.86Density\n|m|\n1FIG. 6: Magnetization histograms for J= 1.50with a random field Hr= 0.04andL= 40 (below the critical temperature) [panel (a)],\nHr= 0.40andL= 20 (below the critical temperature) [panel (b)], and Hr= 0.40andL= 20 (above the critical temperature) [panel\n(c)]. The solid red lines are double-Gaussian fits, which are composed of the sum of the two modes (thin dashed black lines) , the blue dashed\nlines are single Gaussian fits. For Hr= 0.04, the double Gaussian fit does not differ much from the single G aussian fit, i.e., the distribution\nis normally distributed as expected for all temperatures in a continuous phase transition. For Hr= 0.40a hump emerges for T < T c,\nthe single Gaussian fit (blue dashed line) cannot capture the hump, but the double Gaussian fit (solid red line) does. Close to the critical\ntemperature the single Gaussian fails to fit the distributio n and a bimodal double-Gaussian structure becomes evident s uggesting a finite jump\nin the magnetization washed out by the disorder at Tc, as expected for a first-order phase transition.\nover, the double Gaussian fit reproduces better the magnetiz a-\ntion histogram below and above Tc. This suggests that the\nmagnetization jumps at Tc, and thus the existence of a first-\norder phase transition.\nTo strengthen the claim of a bimodally-distributed magneti -\nzation for large random fields close to Tc, we perform a boot-\nstrap analysis of the double-Gaussian fit for J= 1.50and\nHr= 0.30,0.40, and0.50, and show the normalized differ-\nence between the mean of the two modes as a function of the\ntemperature T(see Fig. 7). We observe that the maximum\nvalues of the curves are close to Tc. The fact that the relative\ndifference between the modes is maximal close to the criti-\ncal temperature gives further evidence for a first-order pha se\ntransition. These distributions show large finite-size cor rec-tions and a clean determination of the critical temperature is\ndifficult.\nV . DISCUSSION\nDisordered nonmagnetic ferroic systems naturally show\nrandomness in both their inter-particle interactions and i n the\neffective field presented as a bias between the two degenerat e\nsingle-particle states. Recently, it was shown that this in ter-\nplay of randomness of interactions and fields is present also\nin the ferromagnetic phase of the LiHoxY1−xF4compound.\nThis has allowed for new experiments [18, 64] investigating\nthis interplay of randomness, as well as for new insights int o7\n00.511.5\n2.5 3 3 .5 4 TcHr= 0.30(a)|µ10−µ20|/µ10\nTL= 12\nL= 16\nL= 20\nL= 24\n100.511.5\n2 2 .5 3 3 .5 Tc(b)\nHr= 0.40|µ10−µ20|/µ10\nTL= 8\nL= 12\nL= 16\nL= 20\n100.511.5\n2 2.5 3 3 .5 4 TcHr= 0.50(c)|µ10−µ20|/µ10\nTL= 8\nL= 10\nL= 12\nL= 14\n1FIG. 7: Relative difference of the bootstrapped mean value o f the two modes (with mean values µ1\n0andµ2\n0) of the bimodal Gaussian fitted\nfunction as a function of temperature for (a) Hr= 0.30, (b)Hr= 0.40, and (c) Hr= 0.50. The gray vertical line corresponds to the\nestimated critical temperature. The relative difference b etween the modes increases close to Tc, the discrepancy between the estimated value\nforTcand the maximal values of the curves can be attributed to the s trong finite-size correction of this observable.\nolder experiments, e.g., demonstrating different final sta tes\nobtained by in-field and zero-field annealing of the disorder ed\nferromagnetic LiHoxY1−xF4system [65].\nThe effective random field in the LiHoxY1−xF4system is\na consequence of the interplay between the off-diagonal ter ms\nof the dipolar interactions and the applied transverse field . The\napplied field also induces transitions between the spin up |↑∝angbracketright\nand spin down |↓∝angbracketrightstates in Ho, thus giving rise to an effec-\ntive transverse field term in the Ising Hamiltonian. However ,\nat low field, the effective random field dominates because it\nis linear in the applied transverse field [9, 11], whereas the\neffective transverse field is negligible at small fields [66, 67].\nIn a previous study [13] we analyzed the LiHoxY1−xF4\nsystem in the regime where the system is ferromagnetic, al-\nbeit with disorder in both the interaction and the effective lon-\ngitudinal field. We have suggested a novel disordering mech-\nanism where at a finite random field that is much smaller than\nthe typical interaction, finite-size glassy domains disord er the\nFM phase into a PM phase. This mechanism explains vari-\nous experimental features, such as the linear dependence of\nthe critical temperature with increased random field, and th e\ndiminishing and rounding of the susceptibility peak with de -\ncreasing temperature [18].\nThe scaling theory within which the novel disordering\nmechanism was obtained [13] is not particular to dipolar sys -\ntems. In the present work we have shown that the same mech-\nanism applies to a generic short-range random-field Ising\nmodel with competing interactions having a ferromagnetic\nmean. Our results thus support the generality of the disor-\ndering mechanism to random-field ferroic systems with com-\npeting interactions, and it would be of interest to check the ir\napplicability, e.g., to relaxor ferroelectrics[7, 8, 68, 6 9].\nWe find an excellent agreement, quantitative and qualita-\ntive, between our numerical results and the predictions of t he\nscaling theory based on the picture of the disordering of a FM\nwith competing interactions by a QSG (pQSG) phase at low\n(high) temperature. It would be of much interest to further\ncorroborate our results here with a direct microscopic anal y-\nsis of the domain structure in the disordered phase.We further analyze here the nature of the FM to PM transi-\ntion and show evidence suggesting that once the disordering is\ninduced via the above-mentioned mechanism, where the FM\nphase is disordered by finite size glassy domains (i.e. to the\npQSG phase), then the transition is first order. This differs\nfrom the RFIM with no competing interaction. In the latter, i t\nis believed that the thermodynamic phase transition is seco nd\norder[2], albeit experimentally hard to be observed becaus e\nof slow relaxation. It would be of much interest to further\nstudy this transition between the FM and pQSG phases, and\nthe dynamics of the system near the transition, as hysteresi s\nand slow relaxation are expected. With regard to the latter,\nnonequilibrium dynamics of the Ising model were recently\nstudied in the presence of random bonds[70] or in the pres-\nence of random fields[71]. For both systems it was shown\nthat the equilibrium disorder-driven transition shows up w hen\nmeasuring the nonequilibrium aging properties. It would be\nof interest to study whether the quasi-SG driven transition dis-\ncussed here has distinct characteristics in the non equilib rium\naging properties of the system.\nAcknowledgments\nWe would like to thank Amnon Aharony, Michael Moore\nand David Sherrington for useful discussions and valuable\ninput. J.C.A. acknowleges support by the G¨ oran Gustafs-\nson Foundation. M.S. acknowledges support from the Israel\nScience Foundation (Grant No. 821/14). H.G.K. acknowl-\nedges support from the National Science Foundation (Grant\nNo. DMR-1151387) and would like to thank Quantum Keller-\nbier for inspiration. The research of H.G.K. is based upon\nwork supported in part by the Office of the Director of Na-\ntional Intelligence (ODNI), Intelligence Advanced Resear ch\nProjects Activity (IARPA), via MIT Lincoln Laboratory Air\nForce Contract No. FA8721-05-C-0002. The views and con-\nclusions contained herein are those of the authors and shoul d\nnot be interpreted as necessarily representing the official poli-\ncies or endorsements, either expressed or implied, of ODNI,8\nIARPA, or the U.S. Government. The U.S. Government is au-\nthorized to reproduce and distribute reprints for Governme n-\ntal purpose notwithstanding any copyright annotation ther eon.\nWe thank Texas A&M University for access to their Ada and\nCurie clusters, the Swedish National Infrastructure for Co m-\nputing for access to Beskow and Triolith clusters, the KTH\nDepartment of Theoretical Physics for access to the Octopus\ncluster, Mikael Twenst¨ om for access to the Termina cluster ,\nand ETH Zurich for access to the Brutus cluster.VI. APPENDIX\nThe simulation parameters for Fig. 5 are shown in Tab. I,\nTab. II and Tab. III for ˜J= 0.50,˜J= 1.50, and˜J= 1.60, re-\nspectively. The simulation parameters of the zero-tempera ture\nsimulations are shown in Tab. IV.\n[1] S. F. Edwards and P. W. Anderson, Theory of spin glasses , J.\nPhys. F: Met. Phys. 5, 965 (1975).\n[2] Y . Imry and S.-K. Ma, Random-Field Instability of the Or-\ndered State of Continuous Symmetry , Phys. Rev. Lett. 35, 1399\n(1975).\n[3] K. 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Aharony, Absence of ferromagnetic long range order in ran-\ndom isotropic dipolar magnets and in similar systems , Solid\nState Commun. 28, 667 (1978).10\nTABLE I: Parameters of the simulations for ˜J=J/J0= 0.5whereJ0= 1.Nsais the number of samples, Nswis the total number of\nMonte Carlo sweeps used for equilibration, Tminis the lowest temperature simulated, Tmaxis the highest temperature simulated, and NTis\nthe number of temperatures used in the parallel tempering me thod for each system size L.\n˜J H r L N sa Nsw Tmin Tmax NT\n0.50 0.00 16,20,24 512 1024 4 .200 5 .000 100\n0.50 0.00 28,32 512 2048 4 .200 5 .000 100\n0.50 0.00 40 512 4096 4 .200 5 .000 100\n0.50 0.00 48 512 8192 4 .200 5 .000 40\n0.50 0.20 16,20,24 512 2048 4 .100 5 .000 30\n0.50 0.20 28,32,36,40 512 2048 4 .100 5 .000 50\n0.50 0.30 16,20,24 1024 2048 4 .100 5 .000 30\n0.50 0.30 28,32 512 2048 4 .100 5 .000 50\n0.50 0.30 40 512 4096 4 .100 5 .000 50\n0.50 0.50 16 1024 4096 4 .100 5 .000 30\n0.50 0.50 20 1024 8192 4 .100 5 .000 30\n0.50 0.50 24 1024 16384 4 .100 5 .000 30\n0.50 0.50 28,32,40 1024 16384 4 .100 5 .000 50\n0.50 0.70 16 1024 2048 3 .950 4 .850 30\n0.50 0.70 20 1024 8196 3 .950 4 .850 30\n0.50 0.70 24 1024 16384 3 .950 4 .850 30\n0.50 0.70 28,32,40 1024 16384 3 .950 4 .850 50\n0.50 0.90 12 1024 1024 3 .700 4 .800 30\n0.50 0.90 16 1024 4096 3 .700 4 .800 30\n0.50 0.90 20 1024 8192 3 .700 4 .800 30\n0.50 0.90 24 1024 16384 3 .700 4 .800 30\n0.50 0.90 28,32 1024 65536 3 .700 4 .800 40\n0.50 1.30 12 1024 512 2 .200 5 .000 40\n0.50 1.30 16 1900 32768 2 .200 5 .000 50\n0.50 1.30 20 1424 262144 2 .200 5 .000 50\n0.50 1.30 24 1424 1048576 2 .200 5 .000 50\n0.50 1.60 8 2300 8192 1 .500 5 .000 50\n0.50 1.60 10 2300 65536 1 .500 5 .000 50\n0.50 1.60 12 2073 4194304 1 .500 5 .000 50\n0.50 1.60 14 2116 4194304 1 .500 5 .000 50\n0.50 1.80 6 1274 32768 1 .400 5 .000 50\n0.50 1.80 8 2230 65536 1 .400 5 .000 50\n0.50 1.80 10 2392 2097152 1 .400 5 .000 50\n0.50 1.80 12 2752 16777216 1 .400 5 .000 50\n0.50 2.00 4,6 3000 16384 0 .600 4 .500 50\n0.50 2.00 8 4500 1048576 0 .600 4 .500 50\n0.50 2.00 10 1200 33554432 0 .600 4 .500 5011\nTABLE II: Parameters of the simulations for ˜J=J/J0= 1.50, whereJ0= 1.Nsais the number of samples, Nswis the total number of\nMonte Carlo sweeps used for equilibration, Tminis the lowest temperature simulated, Tmaxis the highest temperature simulated, and NTis\nthe number of temperatures used in the parallel tempering me thod for each system size L.\n˜J H r L N sa Nsw Tmin Tmax NT\n1.50 0.00 16,20 1024 16384 3 .30 3 .80 100\n1.50 0.00 24,28 1024 65536 3 .30 3 .80 100\n1.50 0.00 32,40 1024 262144 3 .30 3 .80 30\n1.50 0.02 16 1024 65536 3 .22 3 .983 18\n1.50 0.02 20,24 1024 65536 3 .22 3 .983 30\n1.50 0.02 28 1024 131072 3 .22 3 .983 30\n1.50 0.02 32,40 1024 131072 3 .22 3 .983 40\n1.50 0.04 16 1024 65536 3 .22 3 .983 18\n1.50 0.04 20 1024 65536 3 .22 3 .983 40\n1.50 0.04 24,28,32,40 1024 131072 3 .22 3 .983 40\n1.50 0.06 16 1024 32768 3 .22 3 .983 18\n1.50 0.06 20,24 1024 65536 3 .22 3 .983 40\n1.50 0.06 28,32 1024 262144 3 .22 3 .983 40\n1.50 0.06 40 1200 524288 3 .22 3 .983 40\n1.50 0.08 16,20 1024 16384 3 .25 4 .00 20\n1.50 0.08 24 1024 65536 3 .22 4 .00 18\n1.50 0.08 28 1024 262144 3 .22 4 .00 40\n1.50 0.08 32,40 1024 524288 3 .22 4 .00 40\n1.50 0.10 16 2048 16384 3 .20 4 .00 16\n1.50 0.10 20 2048 32768 3 .20 4 .00 16\n1.50 0.10 24 1024 65536 3 .20 4 .00 16\n1.50 0.10 28 1024 262144 3 .20 4 .00 40\n1.50 0.10 32,40 1024 524288 3 .20 4 .00 40\n1.50 0.20 12 2048 16384 3 .00 4 .00 16\n1.50 0.20 16 1024 32768 3 .00 4 .00 16\n1.50 0.20 20,24 1024 131072 3 .00 4 .00 16\n1.50 0.20 28,32 1024 524288 3 .00 4 .00 30\n1.50 0.30 12,14 2048 32768 2 .78 4 .00 20\n1.50 0.30 16 1024 65536 2 .78 4 .00 20\n1.50 0.30 20 2043 131072 2 .78 4 .00 20\n1.50 0.30 24 3285 1048576 2 .78 4 .00 30\n1.50 0.40 8,10,12 2048 32768 2 .00 4 .50 60\n1.50 0.40 14,16 2048 65536 2 .00 4 .50 60\n1.50 0.40 20 1024 1048576 2 .00 4 .50 60\n1.50 0.50 4 4096 32768 1 .00 5 .00 50\n1.50 0.50 6 3637 65536 1 .00 5 .00 50\n1.50 0.50 8 2048 131072 1 .00 5 .00 50\n1.50 0.50 10 2048 262144 1 .00 5 .00 50\n1.50 0.50 12 1441 1048576 1 .00 5 .00 50\n1.50 0.50 14 2645 2097152 1 .00 5 .00 50\n1.50 0.60 4,6 2048 16384 1 .00 5 .00 50\n1.50 0.60 8 1387 65536 1 .00 5 .00 50\n1.50 0.60 10 2048 262144 1 .00 5 .00 50\n1.50 0.60 12 5275 524288 1 .00 5 .00 50\n1.50 0.60 14 1199 4194304 1 .00 5 .00 50\n1.50 0.60 16 1024 33554432 1 .00 5 .00 5012\nTABLE III: Parameters of the simulations for ˜J=J/J0= 1.60, whereJ0= 1.Nsais the number of samples, Nswis the total number of\nMonte Carlo sweeps used for equilibration, Tminis the lowest temperature simulated, Tmaxis the highest temperature simulated, and NTis\nthe number of temperatures used in the parallel tempering me thod for each system size L.\n˜J H r L N sa Nsw Tmin Tmax NT\n1.60 0.00 12 1024 8192 2 .75 5 .00 100\n1.60 0.00 16 1024 32768 2 .75 5 .00 100\n1.60 0.00 20 1024 65536 2 .75 5 .00 100\n1.60 0.00 24 1024 262144 2 .75 5 .00 30\n1.60 0.00 28 1024 524288 2 .75 5 .00 30\n1.60 0.00 32 1024 1048576 2 .75 5 .00 30\n1.60 0.10 12 1024 32768 1 .75 5 .00 37\n1.60 0.10 16,20 1024 131072 1 .75 5 .00 37\n1.60 0.10 24 1024 262144 2 .75 5 .00 30\n1.60 0.10 28 1024 524288 2 .75 5 .00 30\n1.60 0.10 32 1024 1048576 2 .75 5 .00 30\n1.60 0.20 8 2048 8192 1 .75 5 .00 37\n1.60 0.20 10 2840 16384 1 .75 5 .00 37\n1.60 0.20 12 1500 65536 1 .75 5 .00 37\n1.60 0.20 14 2048 1048576 1 .75 5 .00 40\n1.60 0.20 16 2048 4194304 1 .75 5 .00 40\n1.60 0.30 4 3024 8192 0 .5 4 .00 71\n1.60 0.30 6 3024 32768 0 .5 4 .00 71\n1.60 0.30 8 4096 131072 0 .5 4 .00 71\n1.60 0.30 10 2440 1048576 0 .5 4 .00 71\n1.60 0.30 12 1782 8388608 0 .5 4 .00 71\n1.60 0.40 4 4000 8192 0 .50 4 .00 71\n1.60 0.40 6 2048 32768 0 .50 4 .00 71\n1.60 0.40 8 5800 131072 0 .50 4 .00 71\n1.60 0.40 10 3500 524288 0 .50 4 .00 71\n1.60 0.40 12 2100 4194304 0 .50 4 .00 71\nTABLE IV: Parameters of the zero-temperature simulations w ith˜J=J/J0, whereJ0= 1.Hris the random field strength, Nsais the\nnumber of samples, Neois the total number of simulation steps for each system size L.\n˜J H r L N sa Neo\n0.00 1 .40,1.50,1.55,1.60,1.70,1.80 4 1024 10000000\n0.00 1 .40,1.50,1.55,1.60,1.65,1.70,1.75,1.80 6 1024 25000000\n0.00 1 .40,1.50,1.55,1.60,1.65,1.68,1.70,1.75 8 1024 35000000\n0.00 1 .50,1.55,1.60,1.70 10 1024 268435456\n0.15 1 .50,1.60,1.70 4 1024 15000000\n0.15 1 .50,1.55,1.60,1.65,1.70 6 1024 25000000\n0.15 1 .50,1.55,1.60,1.65,1.70 8 1024 35000000\n0.25 1 .50,1.55,1.60,1.70 4 1024 25000000\n0.25 1 .40,1.50,1.55,1.60,1.65,1.70 6 1024 25000000\n0.25 1 .40,1.50,1.55,1.60,1.65 8 1024 35000000\n0.50 1 .40,1.50,1.55,1.60,1.65 4 1024 25000000\n0.50 1 .40,1.45,1.50,1.55,1.60 6 1024 35000000\n0.50 1 .40,1.45,1.50,1.55,1.60 8 1024 35000000"    },    {        "title": "1810.09053v1.Long_Range_Intrinsic_Ferromagnetism_in_Two_Dimensional_Materials_and_Dissipationless_Future_Technologies.pdf",        "content": "1 \n Long Range Intrinsic Ferromagnetism in Two Dimensional Materials and \nDissipationless Future Technologies \n \nBabar Shabbir1,2†a), Muhammad Nadeem3,4†, Zhigao Dai2, Michael Fuhrer5, Qi-Kun Xue6, Xiaolin Wang3, \nQiaoliang Bao2b) \n \n1Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong \nProvince, Shenzhen University, Shenzhen 518060, P. R. China \n \n2Department of Materials Science and Engineering, and ARC Centre of Excellence in Future Low-Energy \nElectronics Technologies (FLEET) Monash University, Clayton, VIC 3800, Australia  \n3Institute for Superconducting and Electronic Materials, and ARC Centre of Excellence in Future \nLow-Energy Electronics Technologies (FLEET), University of Wollongong, Australia, NSW 2500, \nAustralia \n4School of Electrical Engineering and Computer Science, National University of Sciences and Technology \n(NUST), H-12 Islamabad, Pakistan \n5School of Physics and Astronomy, and Monash Centre for Atomically Thin Materials and ARC Centre \nof Excellence in Future Low-Energy Electronics Technologies (FLEET), Monash University, Monash, \nVIC 3800, Australia \n6State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, \nBeijing 100084, China. \nCorresponding authors: a) Babar Shabbir (Email babar.shabbir@szu.edu.cn) and b) Qiaoliang Bao \n(Email qiaoliang.bao@monash.edu) \n†These authors contributed equally to this work. \nABSTRACT \nThe inherent susceptibility of low-dimensional materials to thermal fluctuations has long \nbeen expected to poses a major challenge to achieving intrinsic long-range ferromagnetic \norder in two-dimensional materials. The recent explosion of interest in atomically thin \nmaterials and their assembly into van der Waals heterostructures has renewed interest in \ntwo-dimensional ferromagnetism, which is interesting from a fundamental scientific point of \nview and also offers a missing ingredient necessary for the realization of spintronic \nfunctionality in van der Waals heterostructures. Recently several atomically thin materials \nhave been shown to be robust ferromagnets. Such ferromagnetism is thought to be enabled by \nmagneto crystalline anisotropy which suppresses thermal fluctuations. In this article, we \nreview recent progress in two-dimensional ferromagnetism in detail and predict new possible \ntwo-dimensional ferromagnetic materials. We also discuss the prospects for applications of \natomically thin ferromagnets in novel dissipationless electronics, spintronics, and other \nconventional magnetic technologies. Particularly atomically thin ferromagnets are promising \nto realize time reversal symmetry breaking in two-dimensional topological systems, 2 \n providing a platform for electronic devices based on the quantum anomalous Hall Effect \nshowing dissipationless transport. Our proposed directions will assist the scientific \ncommunity to explore novel two-dimensional ferromagnetic families which can spawn new \ntechnologies and further improve the fundamental understanding of this fascinating area.  3 \n TABLE OF CONTENTS  \nI. INTRODUCTION  ................................................................................................... 4 \nII. ROLE OF MAGNETIC ANISOTROPY IN 2D FERROMAGNETIC MATERIALS  .. 6 \nIII. PROPOSED 2D FERROMAGNETIC MATERIALS AND THEIR EXPERIMENTAL \nREALIZATION  ...................................................................................................... 8 \nIV. ROLE OF FERROMAGNETISM IN TOPOLIGICAL PHASES  ............................. 15 \nV. TOPOLOGICAL INSULATING PHASE  ............................................................... 17 \nVI. ORIGIN OF FERROMAGNETISM IN TOPOLOGICAL INSULATORS  ................ 21 \nVII. IMPORTANCE OF FERROMAGNETISM IN REALIZATION OF “QAH”  PHASES\n 22 \nVIII. FUTURE DIRECTIONS  ....................................................................................... 26 \n 4 \n I. INTRODUCTION \nFerromagnetism (FM) is a physical phenomenon realized in certain materials that retain a \npermanent magnetic moment in the absence of any external magnetic field as long as the \ntemperature is below a critical value, generally known as the Curie temperature “ Tc”. \nFerromagnetic materials have long been recognized for their technological importance, \nbeginning with the compass, and are still essential today in electric motors and generators and \nnon-volatile information storage. While ferromagnetism in three dimensions (3D) can be \nachieved with Tc much greater than room temperature, the spontaneous rotational symmetry \nbreaking associated with isotropic ferromagnetism is strictly forbidden in two dimensions \n(2D). While certain anisotropic models are known to exhibit stable ferromagnetism in 2D, it \nis generally recognized that the increased instability of 2D materials to thermal fluctuations \nposes challenges to realizing ferromagnetism with high Tc.  \nThe past one and a half decades have seen an explosion of interest in atomically thin \nmaterials since the isolation of graphene, an atom-thick plane of carbon1. This class of \nmaterials is also known as van der Waals materials due to the weak bonding between layers, \nmaking these materials compatible with a variety of substrates and also opening a vast new \nfield of “van der Waals heterostructures” which combine different atomically thin materials to \nform new structures with new properties2. Atomically thin materials have been discovered \nwith a wide range of properties: metals, semiconductors, insulators, superconductors, \ncharge-density-wave materials, and ferroelectrics. Ferromagnetism was notably absent from \nthis list until recent discoveries of atomically thin ferromagnets Cr 2Ge2Te63\n, CrI34, and VSe 25.  \nThe discovery of intrinsic ferromagnetism in 2D materials is vital for understanding spin \nbehavior in low dimensions and could enable novel spintronics applications ranging from \nmolecular quantum devices and sensing to high-density data storage ultrathin devices6-9. 2D \nferromagnets may also be essential components in proposed heterostructures which can \nexhibit the quantum anomalous Hall effect at high temperatures, highly desirable for \napplications in dissipationless electronics and spintronics.   \nSince the discovery of graphene10, several techniques have been proposed to extrinsically \ninduce long-range magnetic order into 2D materials such as defect engineering11, absorption \nof magnetic ions12 or proximity effect13. In defect engineering, the simplest strategy for \nrealizing magnetism in non-magnetic materials is to create unpaired electrons by modifying \nits electronic structure via vacancies14, 15, adatoms12, grain boundaries16 or edges17. In 5 \n absorption technique, magnetic ions are absorbed inside 2D vdW materials where magnetic \norder emerges when the magnetic ions experience exchange coupling. On the other hand, in \nmagnetic proximity effect, 2D vdW materials experience exchange coupling when placed in \ncontact with ferromagnetic insulating substrates. \nDefect engineering and absorption induced ferromagnetic systems usually come along \nwith a change in their crystalline structure. The obtained small magnetic moments in such \nkind of disordered systems, the need to control the doping/adsorption precisely, and/or the \nsacrifice of high carrier mobilities due to scattering are not suitable for hosting robust \nexchange interaction between extrinsically introduced local magnetic moments. Moreover, \napart from the debated experimental evidence for defect engineering and absorption \ntechniques, proposed theoretical models for inducing low density ferromagnetism by tuning \nthe band structure18-20 have yet to be achieved experimentally.  \nOn the other hand, proximity induced ferromagnetic systems have more advantages and \npromise preservation of the original crystalline structure, better controllability of the \nelectronic states, uniform magnetization, and uniformly distributed band gap in space, etc. \nHowever, proximity induced magnetic responses are limited in 2D materials. These potential \nscarce in externally induced ferromagnetism motivates to find ferromagnetism originating \nintrinsically from the parent 2D lattice. \nHowever, little attention has been drawn to investigate the intrinsic ferromagnetism in \ntwo-dimensional “2D” materials because the long-range magnetic order in 2D systems could \nbe destroyed by thermal fluctuations, for example the Mermin–Wagner theorem indicates that \nlong-range magnetic order is impossible at any finite temperature in an isotropic 2D spin \nsystem21-23. To suppress the thermal fluctuations and achieve long range intrinsic \nferromagnetism in 2D materials, magnetic anisotropy is necessary. Based on various forms of \nmagnetic anisotropies, recent groundbreaking experimental realizations of intrinsic \nferromagnetism in Cr 2Ge2Te6, CrI3, and especially VSe 2 at room temperature have been \nachieved. Surprisingly, monolayer VSe 2 has been revealed to be ferromagnetic while 3D \nVSe2 is paramagnetic, pointing out that mono-layers or two-dimensional (2D) structures can \nreveal distinctive properties in contrast to their multi-layers or three dimensional (3D) \nstructures.  \nThis review focuses on intrinsic ferromagnetism in 2D materials. By “intrinsic \nferromagnetism”, we intend to limit the discussion to ferromagnetism occurring in a \nstoichiometric compound or heterostructure of stoichiometric compounds, as opposed to \ndefect or dopant induced (“dilute”) ferromagnetism. By “2D material” we intend those 6 \n materials which can be stably isolated in few atom thick layers; typically these materials \nconsist of covalently bonded layers which interact with each other through weaker van der \nWaals bonding.   \nThis paper also explores the role of 2D ferromagnetism in various topological phases of \nquantum matter. The quantum anomalous Hall effect, which enables robust dissipationless \nelectronic transport over macroscopic distances, is entangled with long-range intrinsic \nferromagnetism in 2D materials24, 25. The realization of quantum anomalous Hall effect \nthrough magnetically doped topological insulators will be discussed within the Landau \nframework of phase transitions, especially emphasizing time reversal symmetry breaking and \nthe associated ferromagnetic order.  \nFinally, this timely review will help the community to discover novel magnetic materials \nthat can lead to ultrathin 2D devices which will be more efficient, compact, and could be \ngoverned by using precise and reliable systems. Due to many available reviews about \nferromagnetism in 3D and/or their other associated phenomena in literature9, 23, 26-28, this \nreview focuses solely on intrinsic ferromagnetism in 2D materials, and especially draws \nattention to proposed materials, i.e. which are mainly predicted by density functional theory \n(DFT) calculations, as well as experimentally realized 2D materials and future directions.  \nII. ROLE OF MAGNETIC ANISOTROPY IN 2D FERROMAGNETIC \nMATERIALS \nMagnetic anisotropy is the dependence of the magnetic properties on the relative \norientation of the applied magnetic field with respect to the crystal lattice. Magnetic \nanisotropy is an essential property of magnetic materials, having significance ranging from \nfundamental to technological, for example, magnetic anisotropy is essential for aligning the \nmagnetic moments in magnetic storage media. Magnetic anisotropy is also a prerequisite to \nobserve ferromagnetism in 2D materials29. There are different types of magnetic anisotropy \nsuch as shape anisotropy, exchange anisotropy, magnetoelastic anisotropy and most \nimportantly magnetocrystalline anisotropy29. Magnetocrystalline anisotropy is caused by the \nspin-orbit coupling which couples the spin orientation to the orientation of the electron \norbitals30. The structure becomes important when the average magnetization of a specimen as \na function of an applied external field is considered since the magnetization depends on the \norientation of the magnetic field with respect to the crystal axes. Additionally, external 7 \n factors such as pressure which is commonly used to tune magnetic properties31-35 can also \nmodify the magnetic anisotropy36. \nWhile magnetically isotropic materials can exhibit long range ferromagnetic order in 3D, \nsuch ordering is prohibited in 2D due to the thermal fluctuations. To achieve long-range \nferromagnetism in 2D systems, magnetic anisotropies need to be incorporated with exchange \ninteraction: \n2\n,1( )2z z\nij i j i i B i\ni j i iH J A S g B S      S S    (1) \nwhere the first term represents Heisenberg isotropic model, second term represents \nmagnetocrystalline anisotropy whereas the last term is the interaction of local moments with \nan external magnetic field B. Here Si is the spin operator on site i, Jij is the exchange coupling \nbetween spins on site i and j whose positive and negative values represent ferromagnetic and \nantiferromagnetic ordering, A is the single-ion anisotropy, whereas g and B are landau \ng-factor and Bohr magneton respectively.  \nCheng Gong et al.,3 showed recently that long-range ferromagnetic order persists in a \nfew layers of a Cr 2Ge2Te6 if magnetocrystalline anisotropy and Zeeman effect are considered \nalong with Heisenberg exchange interaction. Similarly, Bevin Huang et al.,4 also showed that \nferromagnetic order persists even in a monolayer of chromium tri-iodide CrI 3 if \nmagnetocrystalline anisotropy and Zeeman effect are incorporated with Ising exchange \ninteraction. Recently J. L. Lado et al. 37 showed that magnetic anisotropy in CrI 3 mainly \noriginates from the superexchange interaction across the Cr-I-Cr bonds, due to the especially \nlarge spin orbit coupling of iodine, with a small additional contribution due to the single-ion \nanisotropy of Cr. The large magnetic anisotropy due to superexchange can stabilize the \nmagnetic ordering in 2D CrI 3.  \nSimilarly, Bonilla et al., showed that M–H loop VSM measurements of the VSe 2 \nmonolayer /HOPG or VSe 2 monolayer /MoS 2 sample with 1T metallic phase indicate strong \nin-plane anisotropy.  This result is consistent with the predictions made through first \nprinciple simulations that both semiconducting 2H and metallic 1T phases of monolayer \ntransition-metal dichalcogenides exhibit ferromagnetic ordering with in-plane magnetization \ndirection38-40. This strong magnetic anisotropy resulting in-plane spin polarization in \ntransition-metal dichalcogenide monolayer mainly originates from both through-bond and \nthrough-space interactions between transition metal anions and chalcogenide cations39.   8 \n III. PROPOSED 2D FERROMAGNETIC MATERIALS AND THEIR \nEXPERIMENTAL REALIZATION \nFigure 1(A) shows a graph for various proposed 2D ferromagnetic materials with their \nrespective Tc, which were mainly predicted via DFT calculations. The 2D ferromagnetism is \nmostly predicted in transition metal chalcogenides and transition metal halides, therefore, \ntheir general crystal structures including top and side views are presented in Fig 1(B). \nTransition metal trichalcogenides (TMT) are compounds of the formula ABX 3 where A is a \n3d metal (Cr), B can be metalloid or metal (such as Si, Ge, Sn) and X is a chalcogen (S, Se, \nTe)41-45. Generally, the transition metal atoms form a honeycomb lattice structure within each \nlayer, and each of them is octahedrally coordinated by six chalcogenide atoms from its three \nneighboring (B 2X6) ligands, with the centers of the hexagons occupied by the B 2 groups41. \nMagnetic order is predicted in the single-layer limit of these van der Waals materials, and the \nconfiguration of the magnetic phase is sensitive to the transition metal/chalcogen element \ncombinations. Likewise, transition metal dichalcogenides (TMD) are 2D materials of the \nformula AX 2, with A a transition metal (e.g., V, Cr, and Mn) and X a chalcogen (e.g. S and \nSe)46-48. Ferromagnetism is predicted in T phase TMDs, where each transition metal adopts an \noctahedrally coordinated structure with respect to the six chalcogenide atoms around it and \neach chalcogenide atom is bonded to three nearby transition metal atoms46. The localized \nmagnetic moments on the transition metal atoms are coupled ferromagnetically by the \nsuper-exchange interactions mediated by the chalcogenide atoms. Trihalides (TH) are \ncompounds of formula AZ 3 where A is a transition metal ( e.g. Ru, Fe) and Z a halogen ( e.g. I, \nBr)49-52. The crystal structure of TH consists of three flat atomic layers i.e. halide-transition \nmetal-halide from top to bottom layers, with transition metal atom arranged in a honeycomb \nlattice, each coordinated to six halide atoms52. The strong spin exchange interaction and huge \nmagnetic anisotropy play a vital role to observe FM in 2D systems. Dihalides (DH) are \ncompounds of formula AZ 2 with A a transition metal and Z a halogen53, 54. Generally, a layer of \nA atoms arranged in a triangular lattice is sandwiched between two layers of Z atoms in triangular \nlattices, similar to the TMDs, with the 1T crystal structure (C 3v symmetry) preferred in metal \ndihalide families. Each metal atom is surrounded by six neighboring halogen atoms forming an \noctahedral [AX 6]-4 unit. The magnetic states in DH evolve from the competition between \nanti-ferromagnetic direct nearest-neighbor d–d exchange and ferromagnetic \nsuper-exchange via halogen p states53. MXenes  are 2D inorganic compounds, composed of few \natom thick layers of transition metal carbides, nitrides, or carbonitrides. Several combinations of 9 \n MXenes; i.e., M 2AX2 where M = (Mn, Cr, Ti), A= (C, N), X= (O, OH, F, Cl); and A n+1Xn with A= \nCr, X= (C, N) ; are ferromagnetic in nature55-59. Here, the superexchange mechanism contributes to \nthe magnetism. Furthermore, ferromagnetism has also been predicted in several other materials \n(denoted by OM in Table 1) including materials such as MnO 2, FeC2 etc60-71. It is also important \nto mention here that Mounet and co-workers have also theoretically identified a group of \nferromagnetic metals and semiconductors using data-mining algorithms and advanced, \nhigh-throughput electronic structure theory72. Table 1 shows a list of predicted materials with \ntheir curie temperatures (T c). In theoretical studies, T c values are mostly determined by using \na mean field theory (MFT) approach or statistical Monte Carlo simulations (MC). The T c \nvalues predicted by MFT are generally over estimated73, 74. On the other hand, more accurate \nTc values can be computed using statistical Monte Carlo (MC) simulations based on the Ising \nmodel43. \n \n  10 \n (A) \n \n(B) \n \n \n \nFigure 1(A):  Proposed 2D ferromagnetic materials with their respective Curie Temperatures “T c”, determined \nvia DFT calculations (T c details are given in Table 1). TMT (Transition Metal Trichalcogenides)41-45, TMD \n(Transition Metal Dichalcogenides)46-48, 75, TH (Tri-Halides)49-52, 76, DH (Di-Halides)53, MXenes55-59, OM (Other \nmaterials)  60-71. (B): Crystal structures of TMT (Transition Metal Trichalcogenides), T Phase-TMD (Transition \nMetal Dichalcogenides), TH (Transition Metal Trihalides) and DH (Transition Metal Dihalides).  \n11 \n Table 1: Materials and their corresponding Curie temperatures T c(K). Materials listed in black box are \nexperimentally realized ferromagnets. TMT (Transition Metal Trichalcogenides), TMD (Transition Metal \nDichalcogenides), TH (Tri-Halides), DH (Di-Halides), MX (MXenes), OM (Other materials), SCs \n(Semiconductors) and HMs (Half Metals)    \nMaterial Tc(K)  Class Nature Reference  Material Tc(K) Class Nature Reference  \nCrGeTe 3 106 TMT SCs 41 VCl3 80 TH HMs  49 \nCrSiTe 3 80 TMT SCs 42 VI3 98 TH HMs  49 \nCrSnTe 3 170 TMT SCs 43 NiCl3 400 TH Insulator  50 \nCrSiTe 3 36 TMT SCs 44 CrF3 41 TH SCs 51 \nCrGeTe 3 57 TMT SCs 44 CrCl3 49 TH SCs 51 \nCrGaTe 3 71 TMT SCs 45 CrBr3 73 TH SCs 51 \nMnS2 225 TMD SCs 46 CrI3 95 TH SCs 51 \nMnSe 2 250 TMD SCs 46 RuI3 360 TH Insulator 52 \nCrS2 -- TMD SCs 48 MnCl 3 ~680 TH HMs 76 \nVS2 90 TMD SCs 47 MnBr 3 ~700 TH HMs 76 \nCr3C2 886 MX HMs 56 MnF3 450 TH HMs 76 \nMn2NF2 1877 MX HMs 58 MnI3 720 TH HMs 76 \nMn2NO2 1379 MX HMs  58 FeCl2 109 DH HMs 53 \nMn2N(OH) 2 1743 MX HMs 58 FeBr2 81 DH HMs 53 \nCr2NO2 566 MX HMs  58 FeI2 42 DH HMs 53 \nMn2CF2 520 MX HMs 59 CoCl2 85 DH -- 53 \nCr2C -- MX HMs  55 CoBr2 23 DH -- 53 \nCr2CF2 -- MX Metal 57 NiCl2 138 DH SCs 53 \nCr2C(OH) 2 -- MX Metal 57 NiBr2 132 DH -- 53 \nCr2NF2 -- MX Metal 57 NiI2 129 DH -- 53 \nCr2N(OH) 2 -- MX Metal 57 MnO2 140 OM SCs 62 \nCr2NO2 -- MX Metal 57 FeC2 245 OM HMs 67 \nMn2CO2 110 MX HMs 59 Fe3GeTe2 216 OM Metal 64-66 \nMn2C(OH) 2 460 MX HMs  59 Li2N 161 OM HMs 60 \nMn2CCl2 380 MX HMs  59 Fe2Si 780 OM HMs 61 \nMn2CH2 120 MX HMs  59 Cu2MoS4 350 OM SCs 68 \nFe2C 861 MX Metal 77 K2CuF4 8 OM Insulator 63 \nCr2Ge2Te6 30 TMT SCs 3 CrN 675 OM HMs 70 \nCrI3 45 TH SCs 4 Co2S2 404 OM Metal 69 \nVSe2 300 TMD Metal 5 Mn3O4 45 OM -- 71 \n \n  12 \n We now turn to the status of experimental observations of ferromagnetism in 2D \nmaterials. Various magnetic materials have recently been explored to see whether their \nmagnetism can be retained in their reduced dimensions, especially in their monolayers42, 51, 63. \nA few early reports indicated FM in few-layer Cr 2Ge2Te6 and anti-FM in monolayer FePS 378, \n79. However, the first clear observations of long-range intrinsic ferromagnetism in 2D systems \nwere recently reported by two different groups revealing the ferromagnetic order in CrI 3 \nmonolayer and Cr 2Ge2Te6 bilayer flakes at low temperatures3, 4.  \nHuang and colleagues demonstrated the intrinsic FM in a mono layer of CrI 3 with a \nsuppressed Tc of 45 K4, which they explain based on Ising exchange interaction. Interestingly \nthey also observe a strong dependence of the ferromagnetism in CrI 3 on the number of layers \nas can be seen in figure 2. The observed remnant magnetization in a single layer is absent in a \nbilayer, indicating antiferromagnetism i.e. the two layers having oppositely oriented spins. \nInterestingly, the net magnetization is again recovered in a trilayer. This indicates the strong \nlayer-dependent interplay between different mechanisms that stabilize magnetic ordering in \nthe atomically thin limit.  \n 13 \n  \nFigure 2:  Layer-dependent magnetic ordering in atomically-thin CrI 3 4\n. (a) MOKE signal on a monolayer (1L) \nCrI3 flake, showing hysteresis in the Kerr rotation as a function of applied magnetic field, indicative of \nferromagnetic behavior. (b) MOKE signal from a bilayer CrI 3 showing vanishing Kerr rotation for applied fields \n+ 0.65 T, suggesting antiferromagnetic behavior. Insets depict bilayer (2L) magnetic ground states for different \napplied fields. (c) MOKE signal on a trilayer (3L) flake, showing a return to ferromagnetic behavior. Reprinted \nwith permission from B. Huang et al., Nature 546 (7657), 270-273 (2017). Copyright 2018 Springer Nature \nLimited. \nGong and colleagues studied ultrathin layers of Cr 2Ge2Te6, demonstrating long range \nFMO also with a low T c ~ 30K in a bilayer3. The magnetic behavior in Cr 2Ge2Te6 can be \nexplained by the Heisenberg model with anisotropic coupling, in which spins themselves are \n14 \n not direction dependent. The suppression of the material’s Curie temperature with decreasing \nnumber of layers is closely linked to the thermal excitation of spin waves. Another relevant \nstudy about the anomalous paramagnetic to ferromagnetic transition in CrGeTe 3 is also \ncarried out80.  \nThe intrinsic 2D ferromagnetism in CrI 3 and Cr 2Ge2Te6 is observed only far below room \ntemperature (i.e. <293 K) and a challenge remained to realize room temperature intrinsic 2D \nferromagnetism until the very recent remarkable discovery of the strong ferromagnetic \nordering in monolayer VSe 25. Notably, the FM in VSe 2 monolayer with a large magnetic \nmoment persists to above room temperature, which is a major breakthrough in this field and \nthat could lead to potential spintronic devices. Magnetic properties of VSe 2 are illustrated in \nfigure 3. \n \nFigure 3: Magnetic properties of VSe 2 films on MoS 2 substrates5. (a) M–H loops took at 100 K and 300 K for \nmonolayer VSe 2. (b) The strong temperature dependences of H c and M s. (c) Variations of M s and H c with the \nnumber of layers of VSe 2 film. The inset shows the M–H loops for the mono-, bi- and multilayer samples. (d) \nAnomalous Hall-effect measurements. Magnetic field dependences of resistance (R) and voltage (V) taken at \n100 K and 200 K show clear hysteresis with larger loops at a lower temperature (100 K versus 200 K), \nconsistent with the temperature dependence of M–H loops observed by VSM. Reprinted with permission from \nM. Bonilla et al., Nature Nanotechnology 13, 289–293 (2018). Copyright 2018 Springer Nature Limited. \n15 \n Contrary to 2D ferromagnetic materials Cr 2Ge2Te6 and CrI 3, long rang ferromagnetic \nordering persists in monolayer VSe 2 with numerous unique features: (1) While ultrathin \nlayers of Cr 2Ge2Te6 and CrI 3 are ferromagnetic insulators, monolayer of VSe 2 exists in 1T \nmetallic phase with a trademark of CDW transitions. (2) While ultrathin layers of Cr 2Ge2Te6 \nand CrI 3 are cleaved from bulk ferromagnetic layered materials, monolayer VSe 2 exhibits \nferromagnetic ordering even though the 3D bulk is paramagnetic which commonly condenses \nin the 1T structure. (3) Finally, the subject which is under heated discussion currently, \nferromagnetic ordering in VSe 2 monolayer on graphite (highly oriented pyrolytic graphite \n(HOPG) or MoS 2 is possibly arising from the substrate effects.  \n \nThe experimental demonstrations by Bonilla et al., showed that (i) the monolayer of \nVSe2 cleaved from a VSe 2 single crystal has no ferromagnetic ordering. (ii) Measurements on \nHOPG and MoS 2 substrate without a VSe 2 film show a diamagnetic and weak defect-induced \nferromagnetic background respectively. (iii) Both diamagnetic and weak ferromagnetic \nsubstrate’s background cannot explain the large in-plane magnetization value, approximately \n15 µB per unit cell, of VSe 2 monolayer. (iv) M s and H c for VSe 2 monolayer are strongly \ntemperature dependent as shown in figure 3(b). These results suggest that ferromagnetic \nordering is intrinsic property of VSe 2 monolayer and give clear indication that the \nferromagnetic ordering in VSe 2 monolayer /HOPG or VSe 2 monolayer /MoS 2 sample is \npossibly stabilized from the substrate effects. \nIV. ROLE OF FERROMAGNETISM IN TOPOLOGICAL PHASES \nAlthough Landau theory of symmetry or symmetry breaking cannot describe all the \nphases of matter – different phases of fractional quantum Hall effect possess same \nsymmetries and their existence do not require breaking of any symmetries81-88 – the \nsymmetries of the underlying Hamiltonian and the breaking of certain symmetries still play \nan important role in describing and predicting exotic topological phases of quantum matter. \nFor example, topological insulators (TI)89-106 and Dirac semimetals107-120 are protected \nthrough symmetries while the quantum integer/anomalous Hall (QIH/QAH) states25, 121-126 \nand Weyl semimetals109, 127-141 exist only if certain symmetries are broken.  \nThe inter-winding among symmetries of the Hamiltonian and the topology of the ground \nstate of the system – to probe band topology100 – has already been generalized for all 230 \nspace groups142. We are focused here on the role of time-reversal symmetry (TRS) and the 16 \n long-range ferromagnetic order associated with TRS breaking in two-dimensions. Several \ntopological phase transitions, i.e., from Wallace (Semenoff) topologically trivial \nsemi-metallic (insulating) phase143, 144 to Haldane topological Chern insulating phase 25, from \nDirac to Weyl semi-metal phase128-130, and from two-dimensional topological insulating \nphase89-106, 145 to QAH phase 122-126, require breaking of TRS. Hence ferromagnetic materials, \nor heterostructures containing them, are natural candidates for realizing these phases in the \nabsence of an applied magnetic field.  \nSuch a phase transition led by magnetic ordering associated with the breaking of TRS is \nnot limited to topological insulators and semi-metals. For example, the interfacing among \ntopological states with other conventional ordered states such as superconducting order has \nled to the prediction of many new exotic phases of quantum matter 100, 124, 146-164. The \npresence of chiral edge states in both topological Chern insulator, as well as conventional \nspinless px + ipy superconductors, are associated with breaking of TRS. Fu and Kane showed \nthat a two-dimensional state – the Fu-Kane superconductor146 – that resembles spinless px + \nipy superconductor which host both Majorana bound states associated with vortices165-168 as \nwell as chiral edge states 165, 169, 170 – can be obtained through proximity effect between an \nordinary s-wave superconductor and the surface states of strong TI but without TRS breaking. \nHowever, to have chiral Majorana edge states in Fu and Kane model, TRS needs to be broken \nby introducing insulating FMO at the topological insulator-superconductor interface. \nMeng and Balents have also shown a similarity between two-dimensional px + ipy \nsuperconductors and a Weyl superconductor – a superconducting phase emerged from the \nnormal insulator, topological insulator, and superconductor heterostructure171, 172. The \nconsequence of TRS and its breaking are similar to what one finds in the phase transition \nfrom a trivial superconducting phase to a Fu-Kane superconducting phase: A time reversal \nsymmetric (non-centrosymmetric-inversion broken) Weyl superconductor phase is trivial \nwhile a centrosymmetric (time reversal breaking) Weyl superconductor phase is topologically \nnon-trivial superconductor with chiral Majorana surface modes. \nIn short, many phase transitions from topologically trivial semi metallic, insulating, or \nsuperconducting phases to topologically non-trivial semi metallic, insulating, or \nsuperconducting phases require breaking of TRS. These daughter topological non-trivial \nphases allow possible future dissipationless spintronic and electronic devices by achieving \nlong-range ferromagnetic order in parent topologically trivial/non-trivial phases.  \nSince it is not possible to cover all the phase transitions based on TRS breaking in this \nfocused review, we highlight here the possible realization of quantum anomalous Hall effect 17 \n through breaking of TRS in TIs as the future of dissipationless transport is inherited in the \nexperimental realization of quantum anomalous Hall effect. To grasp the idea, we review the \nbasic models for topological insulating phases, the origin of ferromagnetic order in TIs, and \nfinally the realization of QAH phase with the emergence of long range ferromagnetic order in \nTIs. Surface Dirac cones and the bulk band structure of three-dimensional topological \ninsulator doped with TR invariant and TR breaking impurities are shown in figure (4). \n \nFigure 4:  (A and B) A topological insulator doped with TR-invariant impurities hosts a Dirac point similar to \nthe one in the undoped case. ( C and D) Band structure along the K-Γ-K direction of undoped Bi 2Se3 and \nTl-doped (Bi 0.9Tl0.1)2Se3 respectively. The Dirac point remains gapless in both cases. ( E and F) A topological \ninsulator doped with TR-breaking impurities hosts a broken Dirac point. ( G and H) Band structure of two \nFe-doped TIs (Bi 0.88Fe0.12)2Se3.7 and (Bi 0.84Fe0.16)2Se3.7 respectively. A gap formation occurs at the Dirac point in \nboth cases145. Reprinted with permission from Y. L. Chen et al., Science 329 (5992), 659-662 (2010). Copyrights \n2018 American Association for the Advancement of Science. \nV. TOPOLOGICAL INSULATING PHASE \nTo show how TRS and its breaking are associated with the TI phase and the QAH phase \nrespectively, we discuss here a four-band model which serves as a guiding principle for \ngenerating QAH phase in magnetically doped TIs122-126. In a suitably chosen basis, the time \nreversal invariant four-band mode model written as \n\n\n\n\n)(00)(\nkk\nhhH    (2) \nwhere the two-band component blocks )(kh and )(kh are TR partners. Such a model \n18 \n describes both the TI phase found in semiconducting quantum wells (QW) as well as the TI \nphase found in two-dimensional thin films of tetradymite semiconducting family Bi 2Se3, \nBi2Te3, and Sb 2Te3. For example, in the basis  1 1 1 1 ,,, HEHE  where 1E \nand 1H are the degenerate first electron sub-bands and heavy-hole sub-bands \nrespectively, this four-band model represents TI phase in semiconducting QW where )(kh \ntakes the form \nzyx yyxx yx TID kkBMkkAkkDC hh   ))(() ()()( )(22 22\n2   k k    (3) \nThe material parameters A, B, C, D, and M depend on the QW geometry. This four-band \nmodel describes TI phase found in both type-I HgTe and type-II InAs/GaSb semiconducting \nquantum wells; hybridization gap generated through coupling A between electron and hole \nsub-bands in type-II QW is relatively smaller than that in type-I QW. \nSimilarly, this four-band model also represents the two-dimensional limit of 3D TI in \nthe basis  ,,,  where 2/) (  bt , \n2/) (  bt and t (and b) represent the top (and bottom) surface states while ↑, \n↓ represent the spin up and down states respectively. Here, )(kh takes the form \nysurf\nTIDm hh )()( )(3 kk k      (4) \nwhere )( )(22\nyxkkBMm k  represents the tunneling effect between the top and bottom \nsurface states at the surface of 3D TI with normal vector zˆ  \n) ()()(3 yxxyfsurf\nTID kkv h     kk    (5) \nHere fv is the Fermi velocity. Bulk energy bands of the type-I semiconducting quantum \nwell are shown in figure (5) while the local density approximation for the band structure of \nthree-dimensional bulk and the ARPES measurements on the two-dimensional surface (111) \nof Bi2Se3 is shown in figure (6). 19 \n  \nFigure 5:  (A) Bulk energy bands of HgTe and CdTe near the Γ point. ( B) The CdTe-HgTe-CdTe quantum well \nin the normal regime ( d < dc) and in the inverted regime ( d > dc)91. Reprinted with permission from B. A. \nBernevig et al., Science 314 (5806), 1757-1761 (2006). Copyrights 2006 American Association for the \nAdvancement of Science.  \n20 \n  \nFigure 6: (a, b)  Electronic band dispersion of Bi 2Se3 through  ARPES measurements on the surface (111) \nwith an incident photon energy of 22eV near the -point along the M and K momentum-space \ncuts respectively. (c) The corresponding momentum distribution curves. (d) A schematic diagram of the \nthree-dimensional bulk and the two-dimensional surface (111) BZ of Bi 2Se3. (e) When the chemical potential \nlies inside the bulk bandgap, a single ring centered at  represents the Fermi surface of the surface states on \nthe two-dimensional surface (111) along the K M  momentum-space cut. The red dots at  and the \nthree M points represent TRIMs on the surface (111) BZ. f) Local density approximation of the Bulk \n(represented by the shaded areas) and the surface (represented by the red dotted lines) band structure. In the \npresence of strong spin–orbit coupling, the existence of a π Berry phase on the surface is guaranteed since \nthe surface band crosses the Fermi level only once between the TRIMs  and M105. Reprinted with \npermission from Y. Xia et al., Nature Physics 5, 398 (2009). Copyright 2009 Springer Nature Limited. \n21 \n VI. ORIGIN OF FERROMAGNETISM IN TOPOLOGICAL \nINSULATORS \nAt present, no intrinsically ferromagnetic topological 2D material is known. \nFerromagnetism has successfully been introduced into topological materials through \nsubstitutional doping. In analogous dilute magnetic semiconductors, the magnetic order can \nbe established in both metallic and insulating phases through the \nRuderman-Kittel-Kasuya-Yosida (RKKY) mechanism and Van Vleck paramagnetism, \nrespectively. In the conducting phase, exchange coupling between itinerant electron spin \ndensity and spin of magnetic impurity leads to the RKKY interaction among magnetic \nimpurities )(1rS and )(2rS \nint 1 2\n, , ,( ) ( ) ( )RKKY\njk i j\ni j x y zH J\n   r r S r S r    (6) \nwhere )(rrjkJ is the coupling constant between two magnetic impurities separated by \nrr. In the insulating phase, a non-vanishing second order perturbation term in the ground \nstate energy leads to Van Vleck paramagnetic spin susceptibility for band insulator \n\n\nnm nmkmzknB VV\nzEEusu\nV2\n, ,2\n02    (7) \nwhere nmEE. Here kmu, and kmu, are the Bloch functions in valance and conduction \nband,0is the vacuum permeability,Bis Bohr magneton, and zSis the electron spin \noperator. \nIn magnetically doped 2D TIs found in type-II semiconducting quantum wells \nInAs/GaSb95, as well as thin films of 3D TIs, found in magnetic tetradymite semiconducting \nfamily Bi 2Se3173, both RKKY interaction as well as Van Vleck paramagnetic electron \nsusceptibility can contribute in ferromagnetic order. However, due to large hybridization gap \nin type-I semiconducting quantum wells HgTe, the ferromagnetic order is hard to establish. \nAlthough magnetically doped 2D TIs in type-II semiconducting quantum wells has not been \nrealized experimentally, the presence of both RKKY ferromagnetic order and finite Van \nVleck paramagnetic contribution has led to the experimental realization of long range \nferromagnetic order in 2D thin films of dilute tetradymite semiconductors family Bi 2Se3, 22 \n Bi2Te3 , and Sb 2Te3 when doped with transition metal elements such as Fe174, Cr175, and V176. \nThe Van Vleck type spin susceptibility of Bi 2Se3 bulk as a function of the SOC strength is \nshown in figure (7) \n \nFigure 7: The Van Vleck paramagnetic electron spin susceptibility of Bi 2Se3 bulk as a function of the SOC \nstrength. Here λ 0 represents the actual SOC strength125. Reprinted with permission from R. Yu et al., Science 329 \n(5987), 61-64 (2010). Copyrights 2010 American Association for the Advancement of Science.  \n \nVII. IMPORTANCE OF FERROMAGNETISM IN REALIZATION OF \n“QAH” PHASES \nThe QAH insulating phase can be realized in systems with spontaneous magnetization \nand/or strong spin-orbit-interaction. The presence of both collectively gives rise to a \ntopologically non-trivial band structure characterized by non-zero Chern number. Since the \n23 \n observation of the TI phase in 2D semiconducting QW and 3D tetradymite semiconductors is \nthe consequence of strong SOC, the key ingredient to observe QAH phase is to have long \nrange ferromagnetic order in TIs. \nThe role of a ferromagnetic order associated with TR symmetry breaking can be \nformulated in terms of TR breaking four-band mode model as \n\n\n\n\n\nFMOFMO\nh hhhH)( 00 )(~\nkk   (8) \nwhere FMOh represents the conventional ferromagnetic order. To generate QAH insulating \nphase with chiral edge states, FMOh must fulfill the following two constraints such that the \nTR partner 2-band component blocks have (i) different mass terms to assure that TR \nsymmetry in broken and (ii) different band structure; inverted band structure for one block \nwhile normal/non-inverted band structure in the other block. \nFor experimental realization, such ferromagnetic order can be achieved by introducing \nmagnetic exchange coupling either through magnetic proximity effect – by combining \ntopological trivial/non-trivial parent material with a topological trivial magnetic material such \nthat the spin moments of band/iterant electrons in parent material are locally aligned out of \nplane – or through inducing magnetic impurities in the parent material. With a suitable choice \nof Zeeman or magnetic exchange coupling term in TR breaking the four-band model H~, TI \nphase found in both semiconducting QWs as well as two-dimensional thin films of \ntetradymite semiconducting family could lead to QAH phase. \nOriginally, Liu et al. proposed the QAH insulating phase in Mn doped 2D TI found in \ntype-I semiconducting quantum wells such as Hg1-xMnxTe QW where FMO takes the form \nz FMO GIGh      (9) \nHere 2/)(21GGG and G1 and G2 induce spin splitting in electron sub-bands ,1E and \nheavy hole sub-bands ,1Hrespectively. To fulfill the above-mentioned constraints on \nFMOh and achieve the QAH phase in Hg 1-xMnxTe QW, G1 and G2 must be such that 21GG \nand 021GG. \nHowever, QAH phase in Hg 1-xMnxTe QW system could not be observed because Mn \nmoments do not order spontaneously and hence cannot generate required spin splitting energy. 24 \n As a result, QAH phase is hard to realize experimentally without applying an external \nmagnetic field in a 2D magnetic TI like Hg 1-xMnxTe QW. To overcome this problem, Yu et \nal125 predicted that QAH phase could be realizable in two-dimensional thin films of \nmagnetically doped 3D TI found in tetradymite semiconductors Bi 2Te3, Bi2Se3, and Sb 2Te3. In \nthis case, the ferromagnetic order takes the form  \nz FMOgmh    (10) \nwhere the effective g factor and magnetization along the z-axis m take the same value for top \nand bottom surfaces. The constraint Mm fulfills both the conditions on FMOh and \nhence the QAH phase is guaranteed. This prediction led to the experimental realization of \nQAH phase in thin films of Cr-doped (Bi,Sb) 2Te3177. Magnetic field dependence of ρ yx and ρxx \nfor Cr0.15(Bi0.1Sb0.9)1.85Te3 film at different temperatures and the different gate voltage is shown \nin figure  (8,9).  \n  \n \nFigure 8: (A) A schematic representation for the realization of the QAH effect in a two-dimensionally thin film of \nmagnetically doped TI. The chemical potential of the film can be controlled by applying gate voltage to a \ndielectric substrate. ( B) A schematic drawing showing the expected dependence of zero field conductivities σ xx(0) \n25 \n and σxy(0) on chemical potential. ( C) An optical image of a Hall bar device where the light gray areas are the \nCr0.15(Bi0.1Sb0.9)1.85Te3 film, dark gray areas are bare substrate from where the film is removed, the red arrow \nindicates the direction of current flow during the measurements, and the black areas are the attached indium \nelectrodes. ( D) Magnetic field dependence of ρ yx for Cr 0.15(Bi0.1Sb0.9)1.85Te3 film measured at different \ntemperatures and the temperature dependence of zero field ρ yx(0) shown in the ansatz which indicates a Curie \ntemperature of 15 K177. Reprinted with permission from C.-Z. Chang et al., Science 340 (6129), 167-170 (2013). \nCopyrights 2013 American Association for the Advancement of Science.  \n \n \n \nFigure 9: (A and C) Magnetic field dependence of ρ yx and ρ xx at different gate voltage applied to dielectric \nsubstrate Vg. (B and D) Vg dependence of zero field resistivities ρ yx(0), ρxx(0) and zero field conductivities σ xy(0) \nand σxx(0)177. Reprinted with permission from C.-Z. Chang et al., Science 340 (6129), 167-170 (2013). \nCopyrights 2013 American Association for the Advancement of Science.  \n  \nIdeally, the QAHE would manifest in exactly zero ρ xx at zero magnetic field, \ncorresponding to the dissipationless transport of charge along the QAH edges. However, the \npresence of spin scattering centers due to non-uniform magnetization and parallel dissipative \nchannels at the edge of Cr-doped (Bi,Sb) 2Te3177 likely induces several KΩ longitudinal \n26 \n resistance in the absence of an external magnetic field. Hence the ideal QAH phase with \ndissipationless edge conduction in Cr-doped (Bi,Sb) 2Te3177 still requires an external magnetic \nfield of several teslas. Later, both of these mechanisms responsible for dissipation are \naddressed in V-doped178 and Cr-doped179 tetradymite family (Bi,Sb) 2Te3.   \n Moreover, difficulty in achieving both a high T c and a large band gap opening \nsimultaneously in magnetically doped TIs limits the applications of QAHE at room \ntemperature. Room-temperature ferromagnetism is the key requirement for the realization of \nQAHE and its utilization in related spintronics and low-energy electronics technologies. In \nthe QAH phase, quantized transverse resistance and extremely low longitudinal resistance \nallow possible future dissipationless spintronic and electronic devices. However, the highest \ntemperature at which QAH phase is observed so far is less than 100 mK. For practical \napplications, the prime goal is to achieve dissipationless chiral edge states at room \ntemperature. To achieve this, the key factor is intrinsic ferromagnetic order sustainable \nagainst thermal fluctuations at high temperature in the two-dimensional limit. In short, the \ndiscovery of intrinsic ferromagnetic order with high T C in atomically thin materials with \ntopological non-trivial band structure would offer a much better platform to realize the \nQAHE without random magnetic doping. \nVIII. FUTURE DIRECTIONS \nThe discovery of intrinsic 2D ferromagnetism could revolutionize modern day \ntechnologies in many ways from both comprehending the underlying physics of electronic \nand spin processes, and applications ranging from sensing to information, and \ncommunication technologies180. If 2D magnetic semiconductors are the building blocks for \nmodern sensing and communication technologies, ferromagnetic metals with reduced \ndimensionality on the other hand could help us to develop fast and non volatile \nstorage/memory devices purely based on the fundamentals of physics where not only the \ncharge transport is controlled by the magnetization orientation but the magnetization \norientation can be electrically modulated181-183.  \nIntrinsic ferromagnetism in 2D materials also plays an essential role in realizing new \nultra-low energy electronics in several ways. Firstly, strong spin-orbit coupling in thin film \nmaterials along with surface and interface effects are important in many spin-based \nphenomena and low-energy spintronic applications. Secondly, topologically protected \nspin-polarized surface states of topological insulators provide efficient conversion from spin 27 \n to charge transport and vice versa. This spin-charge conversion is greatly enhanced with the \nreduction of dimension. Finally, the intrinsic ferromagnetic order in two-dimensionally thin \nfilms of topological insulators opens the possibility of dissipationless charge transport along \nthe edges of the quantum anomalous Hall phase. \nThe integration of semiconductor spintronic devices with ferromagnetism could also be \nuseful in quantum information applications that are based on manipulation of spin states in \nsemiconductors184-186. Recent discoveries of intrinsic 2D ferromagnetism could potentially \nlead us to discover new spin-gapless semiconductors (SGS)187 which link semiconductors and \nhalf-metals. The idea of the SGS was first coined by Wang et al. in 2008188 based on the \nnovel designs of band structures with either linear or parabolic energy dispersion. The SGSs \nwith linear energy dispersion are promising materials for realizing massless and \ndissipationless states.  \nIn summary, the discovery of intrinsic ferromagnetism in two-dimensionally thin \nmaterials with signatures of dissipationless spintronic and electronic transport urges the \ndiscovery of new spin-and-charge based phenomena where robustness against disorder and \nthermal fluctuations promises spintronic applications at room temperature. Intrinsic \nferromagnetism in combination with recent thin film growth techniques provides novel \ndirections to synthesize and fabricate possible candidates to observe high temperature QAHE \nphenomena for electronics and related spintronics applications. \n  28 \n Additional Note: The experimental demonstration of intrinsic ferromagnetism in VSe 2 \nmonolayer /HOPG or VSe 2 monolayer /MoS 2 sample in 1T metallic phase and CDW \ntransitions triggered a heated debate on substrate-induced ferromagnetic ordering and CDW \ntransitions. Numerous articles have appeared on long range ferromagnetism and CDW phase \nmaintainable on substrates since drafting this manuscript and its consideration. For example, \ntransition metal compound monolayers such as 1T-MnSe 2189, 190, Fe3GeTe2 191, and transition \nmetal dihydride MH 2 (M=Sc, Co)192 host magnetic states and the van der Waals substrate \nstabilizes the former and leads toward 2D itinerant ferromagnetism. 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Furtherm ore, our experiments reveal that not only \nnanoparticles of metal oxides but also  metal nitrides such as NbN (T c = 6 - 12 K) and δ-\nMoN (T c~ 6 K) exhibit room-temperature ferromagnetism. \n \n \nPACS:  75.30.Pd; 71.24.+q; 74.72.Bk  \nKeywords:  A. ferromagnetism; A. nanoparticles; A. superconductors \n *Electronic address: sundaresan@jncasr.ac.in\n \n*Electronic address: cnrrao@jncasr.ac.in  \n   \n \n  2 \n 1. Introduction  \nIt is well known that the materials with partially filled d or f shells exhibit \ninteresting magnetic properties. In the r ecent years, ferromagnetism above room \ntemperature has been observed in se veral materials containing no unpaired d or f \nelectrons. High-temperature ferromagnetis m has been reported in alkaline-earth \nhexaboride.\n1 Ferromagnetic hysteresis and high Curie temperature was observed in \nproton irradiated graphite.2 Nanoparticles of gold capped with thiol were shown to \nexhibit ferromagnetism at room temperature.3 Thin films of many oxides with no \nunpaired electrons are also  reported to be ferromagnetic at room temperature.4,5 It was \nsuggested that defects or oxygen vacancies  could be responsible for the observed \nferromagnetism.  Ab initio  electronic structure calcula tions using density functional \ntheory in HfO 2 have shown that isolated halfnium vacancies lead to ferromagnetism.6 \nMeanwhile, there is a conflicting report attributing the ferromagnetism in HfO 2 to \npossible iron contamination while using stai nless-steel tweezers in handling thin films.7 \nVery recently, we have reported room-temperature ferromagnetism in \nnanoparticles of nonmagnetic oxides such as CeO 2 and Al 2O3.8 The origin of \nferromagnetism in these nanoparticles has been suggested to be due to magnetic moments \narising from oxygen vacancies at the surfaces of the nanoparticles. We also suggested \nthat all oxide nanopar ticles would exhibit ferromagnetis m at room temperature. These \nresults motivated us to study whether the ferromagnetism occurs in nanoparticles of \nsuperconductors in the normal as well as the superconducting states, since the phenomena  3of ferromagnetism and superconductivity are an tagonistic. For this pur pose, we selected \nthe well known oxide high temp erature superconductor, YBa 2Cu3O7-δ (YBCO). We also \nexamined whether the ferromagnetism exists in the nanoparticles of metal nitrides such as \nNbN and δ-MoN which are also known to be superconductors with low T c. Interestingly, \nwe found room-temperature ferromagnetism not only in the nanoparticles of oxide \nsuperconductor but also in the metal nitrides. \n2. Experimental \nNanoparticles of YBCO were prepared by the citrate-gel met hod described in the \nliterature except that the sample was cooled from 790o C to room temperature at a faster \nrate (5o C/min) than the reported one (1o C/min).9 The nanoparticles of NbN were \nprepared by heating NbCl 5 with urea in 1:6 molar ratio at ~ 1000o C.10 The nanoparticles \nof δ-MoN were prepared by heating γ-Mo 2N nanoparticles which in turn were prepared \nby the urea method described elsewhere.11 Powder x-ray diffraction (XRD) was used to \nidentify the phase and its purity. The partic le size and morphology were studied by Field \nEmission Scanning Electron Microscopy  (FESEM) and Transmission Electron \nMicroscopy (TEM). Magnetization measurements were carried out with  vibrating sample \nmagnetometer in physical property measuri ng system (PPMS, Quantum Design, USA). \n3. Results and discussion \nThe XRD pattern of nanopart icles of YBCO showed the orthorhombic structure \nwith a = 3.829 Å,   b = 3.874 Å and c = 11.670 Å. The XRD pattern of NbN \nnanoparticles was consistent with the cubic lattice with a = 4.44 Å.  The structure of δ-\nMoN was confirmed to be hexagonal with a = 5.68 Å and c = 5.56 Å. The particle size of  4YBCO ranges from 100 - 200 nm as seen from the image of FESEM (Fig. 1). The \nparticles size of δ-MoN and NbN were 20 and 30 nm, respectively.10 \nFig. 2 shows zero-field cooled (ZFC) and fiel d cooled (FC) susceptibility data as a \nfunction of temperature. It can be seen  that nanoparticles of YBCO exhibit \nsuperconducting diamagnetism with a critical temperature, T c = 91 K similar to that of \nbulk YBCO despite the faster cooling rate adopted in this  work. In contrast to bulk \nYBCO, the reversible region in  the nanoparticles extends down  to 40 K. This is due the \nincreased threshold for flux penetration in  the nanoparticles due to Bean-Livingston \nsurface barrier.12 The magnetization versus field curve measured at 5 K is shown in the \ninset of the Fig. 2. The field decreasing pa rt of the magnetization remains negative, \nconsistent with the granular nature of th e superconductor where the flux pinning is also \nweak compared to the bu lk or single crystals. \nIntriguingly, the room-temperature ma gnetization of YBCO nanoparticles show \nhysteresis, with a coercivity of ~ 200 Oe, t ypical of ferromagnetic behavior as shown in \nFig. 3. This observation is consistent with our prediction that a ll oxide nanoparticles \nwould exhibit ferromagnetism.8 The origin of ferromagnetism is likely to be due to \nmagnetic moments arising from the oxygen vacanc ies at the surfaces of the nanoparticles.  \nA bulk YBCO was prepared by pressing th e nanoparticles into re ctangular bars and \nannealed at high temperatures. The magnetiz ation measurement at room temperature \nrevealed paramagnetic behavior (Fig. 3) wh ich is typical of bulk YBCO. This supports \nthe suggestion that the ferromagnetism is confin ed to the surface of the nanoparticles. It \nwould be interesting to explore whether such surface ferromagnetism exists in the \nsuperconducting state of the nanoparticles. Fo r this purpose we measured M(H) across  5the superconducting transition. At T c (= 91 K), the ferromagnetic hysteresis still remains \nwith increased coercivity (300 Oe). At 90 K, th e M(H) data is shown in the inset of Fig. \n3. It can be seen that the M(H) behavior is that of a superconductor rather than that of \nferromagnetism. In fact, it would be difficu lt to infer the presence of ferromagnetism \nbelow T c from the magnetization measurement.  \nNbN nanoparticles showed T c in the range 6 - 12 K de pending on the preparative \nconditions. The diamagnetic susceptibility an d M(H) curve in the superconducting state \nare shown in the inset of Fig. 4. More importa ntly, they also exhibit room-temperature \nferromagnetism with coercivity ~ 500 Oe as seen in Fig. 4. In order to compare this result with bulk or bigger particles of NbN, the nanopa rticles were pressed into bars and heated \nat high temperature (920\noC) for a prolonged time in amm onia. This treatment gave an \naverage particle size of ~ 200 nm. The results  of the magnetic measurement are shown in \nFig. 5.  It can be seen that the magnetic hysteresis measur ed in the superconducting state \n(T=3.3 K) is that of a bulk sample. Furt her, the magnetization measured at room-\ntemperature shows essentially a linear behavior but for a small hysteresis and step at very \nlow fields which arises from the tape used to fix the sample with the quartz sample holder.  However, it is remarkable to note the M(H) behavior of smaller (Fig. 4) and \nbigger (Fig. 5)  particles of NbN which demonstrate that the ferromagnetism is associated \nwith the smaller particles. Further, as th ere was no oxide impurity  phase in the x-ray \ndiffraction, we suggest that the observed ferro magnetism is intrinsic to the nanoparticles \nof NbN. This suggests that there can be vacancies of n itrogen at the surfaces of the \nnanoparticles that can also l ead to creation of magnetic moment. In fact, we have \nobserved room-temperature ferromagnetism in  CdS and CdSe nanoparticles. Similarly,  6room-temperature ferromagnetism and supe rconducting diamagnetism are observed in \nnanoparticles of δ-MoN as shown in Fig. 6. These results demonstrate that \nferromagnetism may be a universal featur e of nanoparticles of most compounds. \n \n4. Conclusions \nRoom-temperature ferromagnetism is observed in nanoparticles of \nsuperconducting YBCO as well as metal nitrides. It requires further st udies to explore the \npresence or absence of ferromagnetism in the superconducting state. The observation of \nroom-temperature ferromagnetism in a variety of nanoparticles such as oxides, nitrides \nand chalcogenides suggests that ferromagnetism may be a universal feature of \nnanoparticles of most compounds. \n \n5. Acknowledgement \nWe thank C. Madhu for his assistance in  preparing some of the samples.  \n \n              7References \n \n[1] D. P. Young, D. Hall, M. E. Torelli, Z. Fisk, J. L. Sarrao, J. D. Thompson, H. R. \nOtt, S.  B. Oseroff, R. G. Goodrich, and R. Zysler, Natu re (London) 397 (1999) \n412. \n[2] P. Esquinazi, A. Setzer, R. Höhne, C.  Semmelhack, Y. Kopelevich, D. Spemann, \nT. Butz, B. Kohlstrunk, and M. Lösche, Phys. Rev. B 66 (2002) 024429. \n[3] P. Crespo, R. Litrán, T. C. Rojas, M. Multig ner, J. M. de la Fuente, J. C. Sánchez- \nLópez, M. A. García, A. Hernando, S. Pe nadés, and A. Fernández, Phys. Rev. \nLett. 93 (2004) 087204. \n[4] M. Venkatesan, and C. B. Fitzgerals, J. M. D. Coey, Nature 430 (2004) 630. \n[5] N. H. Hong, J. Sakai, N. Poirot, V. Brizé, Phys. Rev. B 73 (2006) 132404 . [6] C. D. Pemmaraju and S. Sanv ito, Phy. Rev. Lett. 94 (2005) 217205  \n[7] D. W. Abraham, M. M. Frank, and S. Guha, Appl. Phys. Lett. 87 (2005) 252502 \n[8] A. Sundaresan, R. Bhargavi, N. Rangara jan, U. Siddesh, and C. N. R. Rao, Phys. \n            Rev.B 74 (2006) 161306(R) [9] E. Blinov, V. G. Fleisher, H. Huhtinen, R. Laiho, E. Lähderanta, P. Paturi, Yu. P. \n            Stepanov, and L. Vl asenko, Supercond. Sci. Technol. 10 (1997) 818  \n[10] A. Gomathi, and C. N. R. Ra o, Mater. Res. Bull. 41 (2006) 941  \n[11] A. Gomathi, A. Sundaresan, and C. N.  R. Rao, J. Solid State Chem. 180 (2007) \n291.              [12] K. I. Kugel and A. L. Rakhmanov, Physica C 196 (1992) 17.  \nFigure Captions  81. FESEM image of YBCO nanoparticles. Particle size ranges from 100 – 200 nm. \n2. Volume susceptibility as a function of temperature showing the superconducting \ntransition around 91 K. Note the absence of irreversibility between ZFC and FC   \ncurves  down to 40 K. Inset shows the M(H) curve at 5 K. \n3. M(H) data of YBCO nanoparticles at 300 K and 91 K showing the ferromagnetic  \nbehavior. M(H) data of bulk YBCO is al so shown for comparison. Inset shows the \nhysteresis at 90 K which is typical of a superconductor. \n4. M(H) data at 300 K for NbN nanoparticle s (30 nm) showing ferromagnetic behavior. \nInset at the bottom shows the superconducting diamagnetic onset around 6 K and the top \ninset shows the hysteresis in  the superconducting state. \n5. M(H) data at 300 K for NbN particles with  an average size of ~ 200 nm showing a \nparamagnetic behavior. Hysteresis at 3.3 K s hown in the top inset is typical of a bulk \nsuperconductor. The diamagnetic transition is shown in the bottom inset. \n6. M(H) data at 300 K for δ-MoN. Inset shows the diamagnetic response under the  \n    applied field of 5 Oe. \n \n        9\n \nFig1 \n \nFig2  10\n \nFig3 \n \nFig4 \n  11\n \nFig5 \n \nFig6 "    },    {        "title": "0803.0174v2.Ferromagnetic_Pairing_States_on_Two_Coupled_Chains.pdf",        "content": "arXiv:0803.0174v2  [cond-mat.supr-con]  28 Jul 2008Ferromagnetic Pairing States on Two-Coupled Chains\nAkinori Tanaka1\nDepartment of General Education, Ariake National College o f Technology, Omuta\n836-8585, Japan\n(March 3, 2008)\nAbstract\nWe propose a concrete model which exhibits ferromagnetism a nd electron-pair con-\ndensation simultaneously. The model is defined on two chains and consists of the\nelectron hopping term, the on-site Coulomb repulsion, and a ferromagnetic interaction\nwhich describesferromagnetic couplingbetween two electr ons, one on abondin achain\nand the other on a site in the other chain. It is rigorously sho wn that the model has\nfullypolarized ferromagnetic pairing ground states. The h igher dimensional version of\nthe model is also presented.\n1akinori@ariake-nct.ac.jp\n11 Introduction\nRecently, UGe 2[1], URhGe [2], and UCoGe [3] were discovered to exhibit ferromagnetic\nsuperconductivity. Experimental results suggest that the same electrons are contributing\nto both ferromagnetism and superconductivity, and thus, in the s uperconducting phase of\nthese materials, the electrons are considered to condensate into a spin-triplet pair state\nunlike usual low temperature superconductors, in which electrons are forming non-magnetic\nspin-singlet pairs. Microscopic explanation of this phenomenon is a ch allenge in condensed\nmatter physics, but the problem is rather subtle and difficult since we have to treat spin-\nrotation symmetry breaking and electron-pair condensation simult aneously. In fact, the\nmechanism for ferromagnetism alone in itinerant electron systems h as not yet been fully\nunderstood, although there have been some rigorous developmen ts [4, 5, 6, 7, 8, 9] in the\nHubbard model, which is one of the simplest models of itinerant electro n systems. As for\nelectron-pair condensation, some models have been shown to have pairing ground states [10,\n11, 12, 13, 14, 15], but little is known about spin-triplet pairing groun d states.\nThe Coulomb repulsion between electrons combined with the Pauli exc lusion principle\ncan generate ferromagnetism in itinerant electron systems [4, 5, 6, 7, 8, 9], but it is unclear\nat the present time that the same can also lead to ferromagnetic su perconductivity. Since\nthe electrons form spin-triplet pairs, it is expected that there app ears a kind of effective\nferromagnetic interaction between electrons in the ferromagnet ic superconductors. Thus\na model containing ferromagnetic interactions as well as the Coulom b repulsion between\nelectronswill besuitableforaninvestigation asafirst steptowardu nderstanding microscopic\nmechanisms for ferromagnetic superconductivity.\nIn this paper we propose a model which has fullypolarized ferromagn etic pairing ground\nstates. The model consists of the electron hopping term, the on- site Coulomb repulsion and\na short-range ferromagnetic interaction term. It is possible to co nsider the model in any\ndimension, but for simplicity we mainly concentrate on a one-dimension al version here. In\nthe one-dimensional case, our model is defined on two chains. The e lectrons can hop along\nthe chain direction, feeling on-site repulsion, and furthermore the two electrons, one on a\nbondinachainandtheotheronasiteintheotherchain, feelaferro magneticinteraction(see\nFig.1). Itisshowninourmodelthat,owingtotheferromagneticint eractionintroducedhere,\nthe electrons form spin-triplet pairs, and then the fullypolarized fe rromagnetic pairing state\nis selected by the on-site Coulomb repulsion as the unique ground sta te (up to degeneracy\ndue to spin-rotation symmetry). It is also shown that the ground s tate exhibits off-diagonal\nlong-range order associated with local spin-triplet pairs.\nThis paper is organized as follows. In the next section we set up the m odel and state the\nmain results. In section 3 we prove the main results. In section 4 we m ake some remarks\non our model and ferromagnetic pairing states. We also present mo dels with anisotropic\nspin-interactions and show that the models exhibit other kinds of sp in-1 electron-pair con-\n2Figure 1: The image of the ferromagnetic interaction considered he re. The solid circles\nrepresent lattice sites, and thick arrows represent electron spin s. The electron on a bond in\na chain favors to couple ferromagnetically with the one on a site in the other chain.\ndensation. In the final section we briefly comment on the higher dime nsional version of the\nmodel.\n2 Definition and Main Results\nWe start by defining a lattice of our model. Let Λ 1and Λ 2be linear chains with Lsites. We\nlabel sites in Λ 1and Λ 2by integers, and half integers, respectively, as Λ 1={0,1,...,L−1}\nand Λ 2={1/2,3/2,...,L−1/2}. It is assumed that Lis a positive odd integer, and periodic\nboundary conditions are imposed. Then let Λ = Λ 1∪Λ2, on which our Hamiltonian will be\ndefined.\nWe next introduce fermion operators. As usual, we denote by cx,σandc†\nx,σthe annihi-\nlation and creation operators, respectively, of an electron with sp inσ=↑,↓at sitex∈Λ.\nThey satisfy the canonical anticommutation relations,\n{cx,σ, cy,τ}={c†\nx,σ, c†\ny,τ}= 0, (2.1)\nand\n{c†\nx,σ, cy,τ}=δx,yδσ,τ (2.2)\nforanyx,y∈Λandσ,τ=↑,↓. Wedenote by Netheelectron number andby Φ 0astatewith-\nout electrons. An Ne-electron state can be constructed by operating Necreation operators\nc†\nx,σon Φ0.\nWe also introduce other fermion operators which play an essential r ole in our model. For\neachx∈Λ, letϕx= (ϕx(y))y∈Λbe a vector whose components are given by [16]\nϕx(y) =/braceleftigg\n1 if|y−x|= 1/2,\n0 otherwise, (2.3)\n(see Fig. 2) and define\nbx,σ=/summationdisplay\ny∈Λϕx(y)cy,σ=cx−1\n2,σ+cx+1\n2,σ. (2.4)\n3PSfrag replacements\nx\n1 1ϕx\n−1PSfrag replacements\nx\n1\n21\n21\n2˜ϕx\n−1\n2−1\n2\nFigure 2: The components of the vectors ϕxand˜ϕx.\nThis operator corresponds to a single-electron state localized at a bond of nearest neighbour\nsites in a chain. Since the set of all vectors ϕxis not orthonormal, the b-operators do not\nsatisfy the canonical anticommutation relations (in fact, it is easy t o see that {b†\nx,σ, by,σ}= 1\nif|x−y|= 1), so that we introduce dual operators ˜bx,σas follows. First recall that we adopt\nthe periodic boundary conditions, and thus, for each x∈Λ1, sitesy∈Λ2can be written as\ny=x+1/2+n(x,y) wheren(x,y) is an integer in 0 ≤n(x,y)≤L−1. Then, for x∈Λ1,\nlet˜ϕx= (˜ϕx(y))y∈Λbe a vector whose components are given by\n˜ϕx(y) =\n\n1/2 ify∈Λ2andn(x,y) is even\n−1/2 ify∈Λ2andn(x,y) is odd\n0 ify∈Λ1,(2.5)\n(see Fig. 2) and, for x∈Λ2, let˜ϕxbe a vector obtained by ˜ ϕx(y) = ˜ϕx−1/2(y−1/2). It is\neasy to see that the inner product of ˜ϕxandϕysatisfies\n∝an}bracketle{t˜ϕx,ϕy∝an}bracketri}ht=/summationdisplay\nz∈Λ˜ϕ∗\nx(z)ϕy(z) =δx,y. (2.6)\nBy using ˜ϕx, we define\n˜bx,σ=/summationdisplay\ny∈Λ˜ϕx(y)cy,σ. (2.7)\nThe˜b-operators are dual to the b-operators in the sense\n{˜b†\nx,σ,by,τ}=∝an}bracketle{t˜ϕx,ϕy∝an}bracketri}htδσ,τ=δx,yδσ,τ. (2.8)\nThe anticommutation relation (2.8) implies that the set {b†\nx,σΦ0}x∈Λis linearly indepen-\ndent and so is the set {˜b†\nx,σΦ0}x∈Λ. Furthermore, (2.8) implies that the c-operators can be\nrepresented in terms of the ˜b-operators as\ncx,σ=/summationdisplay\ny∈Λ{b†\ny,σ, cx,σ}˜by,σ=/summationdisplay\ny∈Λϕ∗\ny(x)˜by,σ\n=/summationdisplay\ny∈Λϕx(y)˜by,σ=˜bx−1\n2,σ+˜bx+1\n2,σ. (2.9)\n4Let us define the number operators nx,σwithσ=↑,↓and the spin operators Sx,αwith\nα= 1,2,3 for the c-operators by\nnx,σ=c†\nx,σcx,σ, (2.10)\nSx,1=1\n2/parenleftig\nc†\nx,↑cx,↓+c†\nx,↓cx,↑/parenrightig\n, (2.11)\nSx,2=1\n2i/parenleftig\nc†\nx,↑cx,↓−c†\nx,↓cx,↑/parenrightig\n, (2.12)\nSx,3=1\n2/parenleftig\nc†\nx,↑cx,↑−c†\nx,↓cx,↓/parenrightig\n. (2.13)\nWe also define\nnx=nx,↑+nx,↓. (2.14)\nThenumber operators nb\nx,σandnb\nx, andthespinoperators Sb\nx,αfortheb-operatorsaredefined\nsimilarly by using bx,σin place of cx,σin (2.10) – (2.14).\nBy using the operators introduced as above, we define the Hamilton ianHas follows:\nHhop=t/summationdisplay\nx∈Λ/summationdisplay\nσ=↑↓b†\nx,σbx,σ, (2.15)\nHint,J=−J/summationdisplay\nx∈Λ/parenleftiggnb\nxnx\n4+Sb\nx·Sx/parenrightigg\n, (2.16)\nHint,U=U/summationdisplay\nx∈Λnx,↑nx,↓, (2.17)\nH=Hhop+Hint,J+Hint,U, (2.18)\nwheret,J >0 andU≥0. By using the c-operators, Hhopis rewritten as\nHhop=t/summationdisplay\nx,y∈Λ\n|x−y|=1/summationdisplay\nσ=↑,↓c†\nx,σcy,σ+2t/summationdisplay\nx∈Λnx, (2.19)\nwhichrepresentsthemotionofelectronsinthetwochains. TheHam iltonianHint,Urepresents\nthe repulsive interaction between two electrons with the opposite s pins at the same site\nandHhop+Hint,Ureduces to the usual Hubbard Hamiltonian. In our model, owing to\nthe Hamiltonian Hint,J, an electron at a bond in one chain and an electron at a site in\nthe other chain feel the attractive and ferromagnetic interactio n. We stress that all the\ninteraction terms considered here are of short-range. It is also n oted that the Hamiltonian\nHconserves the electron number and possesses spin-rotation sym metry. The occurrence of\nferromagnetism and electron-pair condensation in our model is thu s not trivial at all.\n5Let [Gx,y]x,y∈Λbe an antisymmetric matrix whose elements are given by\nGx,y=\n\n−1/2 ifx∈Λ1,y∈Λ2and|x−y|= 1/2\n1/2 ifx∈Λ2,y∈Λ1and|x−y|= 1/2\n0 otherwise ,(2.20)\nand define ˜ζστandζστ, corresponding to pair states with spin-1, by\n˜ζστ=/summationdisplay\nx,y∈ΛGx,y˜bx,σ˜by,τ (2.21)\nand\nζστ=/summationdisplay\nx,y∈ΛGx,ycx,σcy,τ, (2.22)\nrespectively.\nOur main results are summarized as follows:\nProposition 2.1 Assume that the electron number Neis an even integer in 2≤Ne≤\n|Λ|, and consider the Hamiltonian Hwith2t=JandU >0. Then the fullypolarized\nferromagnetic pairing state,\nΦG=/parenleftig˜ζ†\n↑↑/parenrightigNe\n2Φ0, (2.23)\nis the ground state of H. Furthermore the ground state is unique up to degeneracy due to\nspin-rotation symmetry.\nProposition 2.2 Consider the Hamiltonian Hwith2t=JandU >0, and take a sequence\nof the ground states ΦGofHfor evenNeandΛsuch that the electron filling factor Ne/(2|Λ|)\nconverges to νin0< ν≤1/2. Let∆ =ζ↑↑/Landgk= 2cos(k/2). Then we have\nµ(ν) = lim\n|Λ|,Ne→∞∝an}bracketle{tΦG,∆∆†ΦG∝an}bracketri}ht\n∝an}bracketle{tΦG,ΦG∝an}bracketri}ht≥1\n2(1−2ν)I(ν) (2.24)\nwith\nI(ν) = 2/parenleftigg1\n2π/integraldisplay\n|k|≤πχ[|gk|2≤ǫ(ν)]|gk|2dk/parenrightigg\n, (2.25)\nwhereχ[X]equals1ifXis true and zero otherwise, and ǫ(ν)is determined by\nν=1\n2/parenleftigg1\n2π/integraldisplay\n|k|≤πχ[|gk|2≤ǫ(ν)]dk/parenrightigg\n. (2.26)\n63 Proof\nProof of Proposition 2.1 . In the proof we fix the electron number Neto an even integer\nin 2≤Ne≤ |Λ|.\nBy some straightforward calculations, we can rewrite Hint,Jas\nHint,J=−J\n2/summationdisplay\nx∈Λ(b†\nx↑bx,↑c†\nx,↑cx,↑+b†\nx↓bx,↓c†\nx,↓cx,↓+b†\nx↑bx,↓c†\nx,↓cx,↑+b†\nx↓bx,↑c†\nx,↑cx,↓).(3.1)\nNotingtheassumption2 t=Jandtheanticommutationrelations(2.1),(2.2)and {b†\nx,σ,cx,σ}=\n0, we obtain\nH=¯Hint,J+Hint,U (3.2)\nwith\n¯Hint,J=J\n2/summationdisplay\nx∈Λ(b†\nx,↑cx,↑+b†\nx,↓cx,↓)(c†\nx,↑bx,↑+c†\nx,↓bx,↓). (3.3)\nSince¯Hint,JandHint,Uare sums of positive semi-definite operators\nJ\n2(b†\nx,↑cx,↑+b†\nx,↓cx,↓)(c†\nx,↑bx,↑+c†\nx,↓bx,↓) (3.4)\nand\nUnx,↑nx,↓=Uc†\nx,↑c†\nx,↓cx,↓cx,↑, (3.5)\nrespectively, the energy eigenvalues of ¯Hint,J+Hint,Uare greater than or equal to zero.\nTherefore, one can conclude that a zero-energy state Φ of both of these Hamiltonians, which\nsatisfies\n(c†\nx,↑bx,↑+c†\nx,↓bx,↓)Φ = 0 (3.6)\nand\ncx,↓cx,↑Φ = 0 (3.7)\nfor allx∈Λ (if it exists), is a ground state. We shall show that the state Φ Gis indeed a\nzero-energy state of both ¯Hint,JandHint,U, and thus is a ground state of H.\nSince there is no electron with down-spin in Φ G, the operation of c†\nx,↑bx,↑+c†\nx,↓bx,↓on ΦG\nreduces to c†\nx,↑bx,↑ΦG. Here, by using (2.8) and (2.9), we have\nc†\nx,↑bx,↑˜ζ†\n↑↑=c†\nx,↑/braceleftig\n(−1)l+1c†\nx,↑+˜ζ†\n↑↑bx,↑/bracerightig\n=˜ζ†\n↑↑c†\nx,↑bx,↑ (3.8)\nforx∈Λlwithl= 1,2. Therefore, we have c†\nx,↑bx,↑ΦG= 0, from which ¯Hint,JΦG= 0 follows.\nFurthermore, noting again that there is no electron with down-spin in ΦG, one immediately\nfindsHint,UΦG= 0. We conclude that Φ Gis a zero-energy state of ¯Hint,J+Hint,U.\n7In what follows, we shall show that any zero-energy state of ¯Hint,J+Hint,Umust be\nexpanded in terms of Φ Gand itsSU(2) rotations.\nAssume that Φ is a zero-energy state of Hand thus satisfies (3.6) and (3.7) for all x∈Λ.\nWe firstly show that Φ can be written as [17]\nΦ =/summationdisplay\nC⊂Λ1\n|C|=Ne/2/summationdisplay\nσ,τf(C;σ,τ)/parenleftigg/productdisplay\nx∈C˜b†\nx,σx/parenrightigg/parenleftigg/productdisplay\nx∈Cc†\nx,τx/parenrightigg\nΦ0, (3.9)\nwith coefficients f, whereσandτareshort hands of spin configurations ( σx)x∈Cand (τx)x∈C\nwithσx,τx=↑,↓, respectively, and the sum/summationtext\nσ,τruns over all possible spin configurations.\nTo prove the above claim, we begin with preparing basis states for Ne-electron states.\nNoting the linear independence of ˜b-operators and the relation (2.9), we find that all states\nof the form\nΦ2(B↑,B↓;C↑,C↓) =\n/productdisplay\nx∈B↑˜b†\nx,↑\n\n/productdisplay\nx∈B↓˜b†\nx,↓\n\n/productdisplay\nx∈C↑c†\nx,↑\n\n/productdisplay\nx∈C↓c†\nx,↓\nΦ0 (3.10)\nwith subsets Bσ,Cσof Λ2such that/summationtext\nσ=↑,↓(|Bσ|+|Cσ|) =Neform complete basis states for\ntheNe-electron Hilbert space. By using these basis states we expand Φ as\nΦ =/summationdisplay\nB↑,B↓⊂Λ2/summationdisplay\nC↑,C↓⊂Λ2f2(B↑,B↓;C↑,C↓)Φ2(B↑,B↓;C↑,C↓), (3.11)\nwithcoefficients f2, wherethesumrunsoverallpossiblesubsetsofΛ 2suchthat/summationtext\nσ=↑,↓(|Bσ|+\n|Cσ|) =Ne. (For notational simplicity, here and in the rest of the proof we do n ot explicitly\nwrite this restriction in the expressions.)\nLet us derive the condition on f2which follows from (3.6) and (3.7). We choose a site\nz∈Λ2andB′\nσ,Cσ⊂Λ2withσ=↑,↓satisfying z /∈B′\nσand/summationtext\nσ(|Bσ|+|Cσ|) =Ne−2. Then,\nfrom (3.6), we have that\n\n/productdisplay\nx∈B′\n↓bx,↓\n\n/productdisplay\nx∈B′\n↑bx,↑\n\n/productdisplay\nx∈C↓cx,↓\n\n/productdisplay\nx∈C↑cx,↑\nbz,−σcz,σ(c†\nz,↑bz,↑+c†\nz,↓bz,↓)Φ = 0,(3.12)\nwhich leads to f2(B′\n↑∪ {z},B′\n↓∪ {z};C↑,C↓) = 0 for z /∈Cσ. Therefore, if z /∈C↑∩C↓,\nthenf2(B′\n↑∪ {z},B′\n↓∪ {z};C↑,C↓) = 0 or equivalently f2(B↑,B↓;C↑,C↓) = 0 for B↑and\nB↓such that z∈B↑∩B↓. On the other hand, by examining (3.7) for sites in Λ 2, we also\nobtainf2(B↑,B↓;C↑,C↓) = 0 for C↑andC↓withC↑∩C↓∝ne}ationslash=∅. As a result we find that\nf2(B↑,B↓;C↑,C↓) = 0 unless B↑∩B↓=∅as well as C↑∩C↓=∅is satisfied.\n8We furthermore examine (3.6). Fix C↑andC↓withC↑∪C↓∝ne}ationslash= Λ2and choose a site\nz∈Λ2which is not included in C↑∪C↓. Using the condition (3.6) for z, we have that\n\n/productdisplay\nx∈B′\n↓bx,↓\n\n/productdisplay\nx∈B′\n↑bx,↑\n\n/productdisplay\nx∈C↓cx,↓\n\n/productdisplay\nx∈C↑cx,↑\ncz,σ(c†\nz,↑bz,↑+c†\nz,↓bz,↓)Φ = 0 (3.13)\nfor arbitrary B′\n↑,B′\n↓⊂Λ2such that/summationtext\nσ(|B′\nσ|+|Cσ|) =Ne−1. From this we obtain\nf2(B′\n↑∪ {z},B′\n↓;C↑,C↓) = 0 and f2(B′\n↑,B′\n↓∪ {z};C↑,C↓) = 0 for z /∈C↑∪C↓, i.e., that\nf2(B↑,B↓;C↑,C↓) = 0 if there exists a site zsuch that z∈B↑∪B↓andz /∈C↑∪C↓.\nTherefore, we conclude that f2(B↑,B↓;C↑,C↓) = 0 unless all the conditions C↑∩C↓=∅,\nB↑∩B↓=∅and (B↑∪B↓)⊂(C↑∪C↓) are satisfied.\nTaking into account the above result and noting (2.7) and (2.9), we e xpand Φ as\nΦ =/summationdisplay\nB↑,B↓⊂Λ1/summationdisplay\nC↑,C↓⊂Λ1χ/bracketleftigg/summationdisplay\nσ|Bσ| ≥/summationdisplay\nσ|Cσ|/bracketrightigg\nf1(B↑,B↓;C↑,C↓)Φ1(B↑,B↓;C↑,C↓),(3.14)\nwith coefficients f1, where we define Φ 1as in (3.10), replacing subsets CσandBσof\nΛ2with those of Λ 1. Then, repeating the same argument as above, we conclude that\nf1(B↑,B↓;C↑,C↓) = 0 unless C↑∩C↓=∅,B↑∩B↓=∅and (B↑∪B↓) = (C↑∪C↓).\nWe thus reach the desired expression (3.9) of a zero-energy stat e Φ.\nWe shall next show that the coefficients fin (3.9) satisfy\nf(C;σ,τ) =f(C′;σ′,τ′) (3.15)\nif/summationtext\nx∈Λ1(σx+τx) =/summationtext\nx∈Λ1(σ′\nx+τ′\nx).\nForz∈Λ2andC⊂Λ1a straightforward calculation yields\ncz,↓cz,↑/parenleftigg/productdisplay\nx∈C˜b†\nx,σx/parenrightigg/parenleftigg/productdisplay\nx∈Cc†\nx,τx/parenrightigg\nΦ0\n=/summationdisplay\ny,y′∈C˜ϕy(z)˜ϕy′(z)sgn[y,y′;C](χ[σy=↑,σy′=↓]−χ[σy=↓,σy′=↑])\n×\n/productdisplay\nx∈C\\{y,y′}˜b†\nx,σx\n/parenleftigg/productdisplay\nx∈Cc†\nx,τx/parenrightigg\nΦ0. (3.16)\nHeresgn[···] is a sign factor arising from exchanges of the fermion operators. By definition,\nfor anyy,y′inC⊂Λ1, ˜ϕy(z)˜ϕy′(z) withz∈Λ2is non-vanishing. We thus have from (3.7)\nand (3.16) that f(C;σ,τ) =f(C;σ[y↔y′],τ), where σ[y↔y′] is a spin configuration\nobtained by switching σywithσy′.\n9It is also easy to see that\n(c†\nz,↑bz,↑+c†\nz,↓bz,↓)/parenleftigg/productdisplay\nx∈C˜b†\nx,σx/parenrightigg/parenleftigg/productdisplay\nx∈Cc†\nx,τx/parenrightigg\nΦ0\n=sgn[z;C](χ[σz=↑,τz=↓]−χ[σz=↓,τz=↑])\n×c†\nz,↑c†\nz,↓\n/productdisplay\nx∈C\\{z}˜b†\nx,σx\n\n/productdisplay\nx∈C\\{z}c†\nx,τx\nΦ0 (3.17)\nforz∈C⊂Λ1. We thus have from (3.6) and (3.17) that f(C;σ,τ) =f(C;σ[z],τ[z]) where\nσ[z] andτ[z] are obtained by switching σzandτzinσandτ.\nSo far we have shown (3.15) with C=C′. We shall complete the proof of the claim by\nexamining (3.6) with z∈Λ2.\nBy using (2.4) and (2.9), (3.6) is rewritten as\n/summationdisplay\ny,y′∈Λ1ϕz(y)ϕz(y′)(˜b†\ny,↑cy′,↑+˜b†\ny,↓cy′,↓)Φ = 0. (3.18)\nLety,y′be a pair of sites with |y−y′|= 1 in Λ 1. For this pair, there exists a site zin\nΛ2such that ϕz(y)ϕz(y′)∝ne}ationslash= 0. Let Dbe an arbitrary subset of Λ 1such that y,y′/∈Dand\n|D|=Ne/2−1. Then, noting (3.18), we obtain from\n/parenleftigg/productdisplay\nx∈Dcx,υx/parenrightigg\n/productdisplay\nx∈D∪{y,y′}˜bx,ιx\n(c†\nz,↑bz,↑+c†\nz,↓bz,↓)Φ (3.19)\n=ϕz(y)ϕz(y′)/parenleftigg/productdisplay\nx∈Dcx,υx/parenrightigg\n/productdisplay\nx∈D∪{y,y′}˜bx,ιx\n(˜b†\ny,ιycy′,ιy+˜b†\ny′,ιy′cy,ιy′)Φ = 0 (3.20)\nthat\nf(D∪{y};σ,τ) =f(D∪{y′};σ′,τ′) (3.21)\nwhereσy=ιy′,τy=ιy,σy′=ιy,τy′=ιy′andσx=σ′\nx=υx,τx=τ′\nx=ιxforx∈D. LetC\nbe an arbitrary subset of Λ 1with|C|=Ne/2 and let Cy→y′be a subset which is obtained by\nremoving yfrom and adding y′toC(we assume y∈Candy′/∈C). Then the above result\nimplies that (3.15) holds for C′=Cy→y′ifyandy′satisfy|y−y′|= 1. It is easy to check\nthat, for arbitrary subsets CandC′of Λ1with|C|=|C′|, there exist pairs ( {yl,y′\nl})n\nl=1with\n|yl−y′\nl|= 1 such that C1=Cy1→y′\n1,C2=C1\ny2→y′\n2,···,C′=Cn−1\nyn→y′n. This completes the\nproof of the desired relation (3.15).\nWe write f(M) forf(C;σ,τ) with/summationtext\nx∈Λ1(σx+τx) =M. By using (3.15), (3.9) is\nrewritten as\nΦ =/summationdisplay\nC⊂Λ1\n|C|=Ne/2Ne/2/summationdisplay\nM=−Ne/2/summationdisplay\nσ,τf(M)χ[/summationdisplay\nx∈Λ1(σx+τx) =M]/parenleftigg/productdisplay\nx∈C˜b†\nx,σx/parenrightigg/parenleftigg/productdisplay\nx∈Cc†\nx,τx/parenrightigg\nΦ0\n10=/summationdisplay\nC⊂Λ1\n|C|=Ne/2Ne/2/summationdisplay\nM=−Ne/2f′(M)/parenleftig\nS−\ntot/parenrightigNe\n2−M/parenleftigg/productdisplay\nx∈C˜b†\nx,↑/parenrightigg/parenleftigg/productdisplay\nx∈Cc†\nx,↑/parenrightigg\nΦ0\n=Ne/2/summationdisplay\nM=−Ne/2f′′(M)/parenleftig\nS−\ntot/parenrightigNe\n2−MΦG (3.22)\nwhere\nf′(M) =(Ne/2+M)!(Ne/2−M)!\nNe!f(M), (3.23)\nf′′(M) = (−1)Ne\n4(Ne\n2−1)1\n(Ne/2)!f′(M), (3.24)\nand the total spin lowering operator S−\ntotis defined as S−\ntot=/summationtext\nx∈Λc†\nx,↓cx,↑. To get the final\nexpression we used\n˜ζ↑↑=/summationdisplay\nx∈Λ1,y∈Λ2Gx,y˜bx,↑˜by,↑+/summationdisplay\nx∈Λ2,y∈Λ1Gx,y˜bx,↑˜by,↑=/summationdisplay\nx∈Λ1cx,↑˜bx,↑. (3.25)\nThis completes the proof of Proposition 2.1.\nProof of Proposition 2.2 . As in the case of ˜ζ↑↑, we have\nζ↑↑=/summationdisplay\nx∈Λ1bx,↑cx,↑. (3.26)\nThe momentum representation of ζ↑↑is\nζ↑↑=/summationdisplay\nk∈Kgkˆc(2,k),↑ˆc(1,−k),↑, (3.27)\nwhere\nˆc(l,k),σ=1√\nL/summationdisplay\nx∈Λle−ikxcx,σ, (3.28)\n(3.29)\nwithl= 1,2 and\nK=/braceleftbigg\nk=2π\nLn/vextendsingle/vextendsingle/vextendsingle−L−1\n2≤n≤L−1\n2/bracerightbigg\n. (3.30)\n11By using (3.25) and (3.26), we have\nΦG∝/summationdisplay\nC⊂Λ1\n|C|=Ne/2/parenleftigg/productdisplay\nx∈C˜b†\nx,↑c†\nx,↑/parenrightigg\nΦ0∝/summationdisplay\nC⊂Λ1\n|C|=Ne/2/parenleftigg/productdisplay\nx∈C˜b†\nx,↑c†\nx,↑/parenrightigg\n/productdisplay\nx∈Λ1bx,↑cx,↑\nΦF\n∝/summationdisplay\nC⊂Λ1\n|C|=Ne/2\n/productdisplay\nx∈Λ1\\Cbx,↑cx,↑\nΦF∝(ζ↑↑)Nh\n2ΦF (3.31)\nwhere Φ F=/parenleftig/producttext\nx∈Λc†\nx,↑/parenrightig\nΦ0andNh=|Λ|−Ne, which is the number of holes counted from\nhalf-filling of electrons. We use the above representation of Φ Gwithζ↑↑in (3.27) to estimate\na lower bound on µ(ν).\nIt is easy to check the anticommutation relation {ˆc†\n(l,k),σ,ˆc(l′,k′),σ′}=δl,l′δk,k′δσ,σ′, and by\nusing this relation we have\nζ†\n↑↑ζ↑↑=/summationdisplay\nk∈K|gk|2−/summationdisplay\nk∈K/summationdisplay\nl=1,2|gk|2ˆc(l,k),↑ˆc†\n(l,k),↑+ζ↑↑ζ†\n↑↑. (3.32)\nWe furthermore have\nˆc(1,k),↑ˆc†\n(1,k),↑ζ↑↑=g−kˆc(2,−k),↑ˆc(1,k),↑+ζ↑↑ˆc(1,k),↑ˆc†\n(1,k),↑, (3.33)\nwhich leads to\nˆc(1,k),↑ˆc†\n(1,k),↑(ζ↑↑)mΦF=mg−kˆc(2,−k),↑ˆc(1,k),↑(ζ↑↑)m−1ΦF (3.34)\nfor integers mand thus\n(ζ↑↑)Nh\n2−mˆc(1,k),↑ˆc†\n(1,k),↑(ζ↑↑)mΦF=2m\nNhˆc(1,k),↑ˆc†\n(1,k),↑(ζ↑↑)Nh\n2ΦF. (3.35)\nThe same relation holds for ˆ c(2,k),↑ˆc†\n(2,k),↑. We thus obtain\n∝an}bracketle{tΦG,∆∆†ΦG∝an}bracketri}ht\n∝an}bracketle{tΦG,ΦG∝an}bracketri}ht=Nh\n21\nL2/summationdisplay\nk∈K|gk|2\n−1\n2/parenleftbiggNh\n2−1/parenrightbigg1\nL2/summationdisplay\nk∈K/summationdisplay\nl=1,2|gk|2∝an}bracketle{tΦG,ˆc(l,k),↑ˆc†\n(l,k),↑ΦG∝an}bracketri}ht\n∝an}bracketle{tΦG,ΦG∝an}bracketri}ht\n≥Nh\n41\nL2/summationdisplay\nk∈K/summationdisplay\nl=1,2|gk|2∝an}bracketle{tΦG,ˆc†\n(l,k),↑ˆc(l,k),↑ΦG∝an}bracketri}ht\n∝an}bracketle{tΦG,ΦG∝an}bracketri}ht\n≥|Λ|−Ne\n2|Λ|2\nLNe/2/summationdisplay\nm=1|gk(m)|2, (3.36)\nwhere we arranged the elements in Kask(1),k(2),...,k(L) so that |gk(m)|2≤ |gk(m′)|2if\nm≤m′. Taking the limit |Λ|,Ne→ ∞withNe/(2|Λ|) converging to νcompletes the proof.\n124 Remarks\nIn this section we remark some aspects of our model and its ground states.\nThe state ( S−\ntot)Ne\n2−MΦGis the representative of the ground state in the subspace where\nthe third component of the total spin is M. If we use the expression (3.31) of Φ G, this\nbecomes ( ζ↑↑)Nh\n2ΦMwhere Φ M= (S−\ntot)Ne\n2−MΦF. This implies that the ground state can be\nregarded as a hole-condensation state in which all holes form the sp in-triplet pair in the back\nground of the fullypolarized ferromagnetic state.\nBy using the so-called d vector d= (d1,d2,d3), one can obtain a useful representation of\na pair operator [18]. In our case, we define\n˜ζ†\nd=/summationdisplay\nx,y∈Λ1/summationdisplay\nσ,τ=↑↓Fστ\nx,y(d)˜b†\nx,σ˜b†\ny,τ (4.1)\nwith /parenleftigg\nF↑↑\nx,y(d)F↑↓\nx,y(d)\nF↓↑\nx,y(d)F↓↓\nx,y(d)/parenrightigg\n=Gx,y/parenleftigg\n−d1+idyd3\nd3d1+idy/parenrightigg\n. (4.2)\nThe pair operator ˜ζ†\n↑↑corresponds to the case of the complex d vector d= (−1/2,−i/2,0),\nand thus the ground state Φ Gis a nonunitary spin-triplet pairing state similar to the one\ndescribing the A1-phase of superfluid3He in the magnetic field [18]. It is noted that, unlike\nthe case of A1-phase realized in the magnetic field, the states\n/parenleftig˜ζ†\nd/parenrightigNe\n2Φ0 (4.3)\nwith any dobtained by rotating ( −1/2,−i/2,0) are the ground states of H, since our Hamil-\ntonianHhas spin-rotation symmetry. The direction of the magnetic moment is given by\nthe cross product i d×d∗where∗means the complex conjugation.\nLetBbe the collection of pairs {x,y}of sites such that x∈Λ1,y∈Λ2and|x−y|= 1/2.\nThen, we have from (3.26) that\n∆†=1\nL/summationdisplay\n{x,y}∈Bc†\nx,↑c†\ny,↑. (4.4)\nTherefore, the order parameter µ(ν) measures the long-range correlation between local spin-\n1 electron pairs, and Proposition 2.2 implies that there exists this long -range correlation in\nthe ground state of Hin the thermodynamic limit.\nWith respect to the ground state Φ Gexpectation values of operators constituted of either\nannihilation or creation operators, such as ∆ and ˆ c(2,k),↑ˆc(1,−k),↑, are always zero since there\nare exactly Neelectrons in Φ G. In order to obtain a particle number symmetry breaking\nground state, which is usually discussed in mean field approximations, we need to form a\n13linear combination of Φ Gwith different electron numbers. Here let us consider the Bardeen-\nCooper-Schrieffer type state\nΦ′\nG=/productdisplay\nk∈K/parenleftig\n1+gkˆc(2,k),↑ˆc(1,−k),↑/parenrightig\nΦF. (4.5)\nIt is noted that the projection of Φ′\nGonto the Hilbert space with the fixed electron number\nis proportional to Φ Gand thus Φ′\nGattains the ground state energy of H. Then, for Φ′\nGthe\nexpectation values of pair annihilation and creation operators are c alculated as\n∝an}bracketle{tΦ′\nG,ˆc(2,k),↑ˆc(1,−k),↑Φ′\nG∝an}bracketri}ht\n∝an}bracketle{tΦ′\nG,Φ′\nG∝an}bracketri}ht=∝an}bracketle{tΦ′\nG,ˆc†\n(1,−k),↑ˆc†\n(2,k),↑Φ′\nG∝an}bracketri}ht\n∝an}bracketle{tΦ′\nG,Φ′\nG∝an}bracketri}ht=gk\n1+g2\nk, (4.6)\nwhich is finite for −π < k < π .\nWhen we set U= 0, the model with 2 t=Jhas degenerate ground states and does not\nexhibit ferromagnetism. In fact, states of the form\n/parenleftig˜ζ†\n↓↓/parenrightigN↓↓/parenleftig˜ζ†\n↑↓/parenrightigN↑↓/parenleftig˜ζ†\n↑↑/parenrightigN↑↑Φ0 (4.7)\nwith non-negative integers N↓↓,N↑↓andN↑↑are zero-energy states of ¯Hint,J, since we have\n(c†\nx,↑bx,↑+c†\nx,↓bx,↓)˜ζ†\nστ=˜ζ†\nστ(c†\nx,↑bx,↑+c†\nx,↓bx,↓) (4.8)\nfor anyx∈Λ andσ,τ=↑,↓, as in (3.8). The on-site repulsion removes this degeneracy and\ngenerates the unique ferromagnetic pairing ground state.\nWe can extent the present model, which has spin-rotation symmetr y, to anisotropic spin-\ninteraction cases. Consider the Hamiltonian\nHα,β=Hhop+Hα,β\nint,J+Hint,U (4.9)\nwhereHα,β\nint,Jis given by\nHα,β\nint,J=−J/summationdisplay\nx∈Λ/braceleftiggnxnb\nx\n4+βSb\nx,1Sx,1+βSb\nx,2Sx,2+(α−αβ+β)Sb\nx,3Sx,3/bracerightigg\n(4.10)\nwithα=±1 andβin 0< β≤1. It is noted that H±1,1becomes H.\nThe following results are obtained for the Hamiltonian Hα,β. In the case of α= 1,0<\nβ <1,2t=J, U >0, for even Nein 2≤Ne≤ |Λ|, there exist exactly two ground states of\nHα,β, which are given by\nΦσσ\nG=/parenleftig˜ζ†\nσσ/parenrightigNe\n2Φ0 (4.11)\n14withσ=↑,↓. In this case, therefore, the ground states of Hα,βexhibit ferromagnetism and\ncondensation of the spin-1 electron pairs whose spins point in the sa me direction parallel to\nthe third component axis. In the case of α=−1,0< β <1,2t=J, U= 0, the ground\nstate ofHα,βis unique for even Nein 2≤Ne≤2|Λ|, and is given by\nΦ↑↓\nG=/parenleftig˜ζ†\n↑↓/parenrightigNe\n2Φ0. (4.12)\nIn this case the ground state exhibits condensation of the spin-1 e lectron pairs whose spins\npoint in the plane perpendicular to the third component axis, but doe s not exhibit ferro-\nmagnetism. One can prove these results in the same way as in section 3, by noting the\nrelation\nHα,β=β¯Hint,J+(1−β)¯Hα\nint,J+Hint,U (4.13)\nwith\n¯Hα\nint,J=J\n2/summationdisplay\nx∈Λ/summationdisplay\nσ=↑,↓b†\nx,σcx,ασc†\nx,ασbx,σ. (4.14)\nIn this paper, we treat only the case of 2 t=Jwhere the exact spin-1 pairing ground\nstates can be constructed. Apart from 2 t=Jwe have the following exact results. When we\nsetJ= 0, the model becomes the decoupled two Hubbard chains, and thu s the ground state\nofHis non-magnetic. In another limit of U→ ∞, where double occupancies at sites are\nforbidden, it can be proved, by using the Perron-Frobenius theor em [19], that the ground\nstate ofHfor 2t < J,0< Ne≤ |Λ|with open boundary conditions is ferromagnetic. So we\nexpect that the ground state phase diagram of our model has a ric h structure depending on\nJ,U, and the electron filling factor ν(and also the anisotropy parameters αandβin the\nanisotropic cases). It is desirable to describe the phase diagram in d etail, but it is beyond\nof our scope at the present time. We leave this problem for future s tudy.\n5 Higher Dimensional Cases\nLetLbe a positive odd integer and define Λ 1= [0,L−1]d∩Zd. Letabe a vector whose\nall components are 1 /2, and define Λ 2={x+a|x∈Λ1}and Λ = Λ 1∪Λ2. We impose\nperiodic boundary conditions in all directions on Λ. For each x∈Λ, letϕx= (ϕx(y))y∈Λ\nbe a vector whose components are given by\nϕx(y) =/braceleftigg\n1 if|y−x|=|a|=/radicalig\nd/2d\n0 otherwise .(5.1)\nRecalling that the periodic boundary conditions are adopted, one no tices that for each\nx∈Λ1, sitesy∈Λ2can be written as y=a+x+/summationtextd\nl=1nl(x,y)δlwhereδlis the unit\n15vector along the l-axis, and nl(x,y) withl= 1,...,dare integers in 0 ≤nl(x,y)≤L−1.\nThen, for x∈Λ1, let˜ϕx= (˜ϕx(y))y∈Λbe a vector whose components are given by\n˜ϕx(y) =\n\n1/2dify∈Λ2and/summationtextd\nl=1nl(x,y) is even\n−1/2dify∈Λ2and/summationtextd\nl=1nl(x,y) is odd\n0 ify∈Λ1,(5.2)\nand forx∈Λ2, let˜ϕxbe a vector obtained by ˜ ϕx(y) = ˜ϕx−a(y−a). Let [Gx,y]x,y∈Λbe an\nantisymmetric matrix whose elements are given by\nGx,y=\n\n−1/2 ifx∈Λ1,y∈Λ2and|x−y|=|a|\n1/2 ifx∈Λ2,y∈Λ1and|x−y|=|a|\n0 otherwise .(5.3)\nBy using ϕx,˜ϕxand [Gx,y]x,y∈Λintroduced as above, we define b- and˜b-operators and\npair operators ˜ζστandζστ, and then define the Hamiltonian H, as in section 2. For this\nd-dimensional Hamiltonian we can obtain the same results as in Proposit ion 2.1 and 2.2,\nwhereI(ν) is replaced with\nId(ν) = 2/parenleftigg1\n(2π)d/integraldisplay\n|k1|≤π···/integraldisplay\n|kd|≤πχ[|gk|2≤ǫ(ν)]|gk|2dk1···dkd/parenrightigg\n. (5.4)\nHeregk= 2d/producttextd\nl=1cos(kl/2) andǫ(ν) is determined by\nν=1\n2/parenleftigg1\n(2π)d/integraldisplay\n|k1|≤π···/integraldisplay\n|kd|≤πχ[|gk|2≤ǫ(ν)] dk1···dkd/parenrightigg\n. (5.5)\nWe can also obtain the d-dimensional anisotropic spin-interaction model as in section 4.\nAcknowledgements\nI would like to thank Masanori Yamanaka for useful discussions in th e early stage of the\nstudy. 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Theor. 40, 10415 (2007).\n[16] Here |·|means the absolute value. The same symbol |X|is used to denote the number\nof elements in a set X.\n[17] We assume that the product is ordered according to the labels a ssociated with the sites.\n[18] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).\n[19] In Ref. [6] one can find the detailed explanation for applications o f the Perron-Frobenius\ntheorem to ferromagneitsm in one-dimension.\n17"    },    {        "title": "1211.2701v1.Spin_polarization_and_magnetoresistance_through_a_ferromagnetic_barrier_in_bilayer_graphene.pdf",        "content": "Spin polarization and magnetoresistance through a ferromagnetic barrier in bilayer\ngraphene\nHosein Cheraghchi\u0003and Fatemeh Adinehvand\nSchool of Physics, Damghan University, 6715- 364, Damghan, Iran\n(Dated: June 30, 2021)\nWe study spin dependent transport through a magnetic bilayer graphene nanojunction con\fgured\nas two dimensional normal/ferromagnetic/normal structure where the gate-voltage is applied on the\nlayers of ferromagnetic graphene. Based on the fourband Hamiltonian, conductance is calculated by\nusing Landauer Butikker formula at zero temperature. For parallel con\fguration of the ferromag-\nnetic layers of bilayer graphene, the energy band structure is metallic and spin polarization reaches\nto its maximum value close to the resonant states, while for antiparallel con\fguration, the nanojunc-\ntion behaves as a semiconductor and there is no spin \fltering. As a result, a huge magnetoresistance\nis achievable by altering the con\fgurations of ferromagnetic graphene especially around the band\ngap.\nPACS numbers: 73.23.-b,73.63.-b\nI. INTRODUCTION\nSince spin-orbit coupling in graphene1is very weak2\nand also there is no nuclear spin3, spin \rip length is so\nlong about 1 \u0016min dirty samples and room temperature4.\nClean samples are expected to have longer spin co-\nherency. This is good opportunity for spintronic ap-\nplications based on candidatory of graphene. On the\nother hand, graphene has not intrinsically ferromagnetic\n(FM) properties, however, it is possible to induce ferro-\nmagnetism externally by doping and defects5, Coulomb\ninteractions6or by applying an external electric \feld\nin the transverse direction in nanoribbons7. Recently,\nHaugen8proposed the FM correlations due to strong\nproximity of magnetic states close to graphene. The over-\nlap between the wave functions of the localized magnetic\nstates in the magnetic insulator and the itinerant elec-\ntrons in graphene induces an exchange \feld on itiner-\nant electrons in graphene giving rise spin splitting of the\ntransport. The exchange splitting which is induced by\nFM insulator Euo in graphene was estimated to be of\norder of 5meV . This splitting which is e\u000bectively simi-\nlar to a Zeeman interaction has so large magnitude that\ncan have important e\u000bects. Such spin splitting can be\ndirectly evaluated from the transmission resonances or\nmagnetoresistance of FM graphene junction. The ferro-\nmagnetism leads to a spin splitting e\u000bectively similar to\na Zeeman interaction but of much larger magnitude. The\ninduced exchange \feld is tunable by an in-plane external\nelectric \feld9. The possibility of controlling spin conduc-\ntance in FM monolayer graphene insulator has also been\nstudied by Yokoyama10. It was found that the spin con-\nductance has an oscillatory behavior in terms of chemical\npotential and the gate voltage.\nBilayer graphene on the other hand has shown to have\ninteresting properties for application in nanoelectronic\ndevices such as transistors based on graphene substrate.\nThe new type of integer quantum Hall e\u000bects11and also\nthe electronic band gap controllable by vertically applied\nelectric \feld are of its unusual properties in compared\nFIG. 1: Schematic view of normal/ferromagnetic/normal bi-\nlayer graphene. Two gate electrodes can be coated on top of\nthe magnetic insulator strips which are located on the upper\nand lower layers.\nto monolayer graphene12{15. Moreover, the parabolic\nband structure close to the Dirac points transforms to a\nMexican-hat like dispersion when an electric \feld is ap-\nplied on graphene. Optical measurements and theoretical\npredictions propose a 200 meV gap in bilayer graphene.\nThis controllable gap makes bilayer graphene as an ap-\npropriate candidate for spintronic devices. An e\u000bective\ntwo-band Hamiltonian can describe the low energy exci-\ntations of a graphene bilayer in the regime of low barrier\nheights16. However, the four-band Hamiltonian is known\nto give a better agreement with both experimental data\nand theoretical tight-binding calculations12,17. Very re-\ncently, spin splitting of conductance in bilayer graphene\nhas been investigated by using an e\u000bective two-band\nHamiltonian emerging from low energy approximation18.\nThis approximation is valid when energy of electrons hit-\nting on the potential barrier is about the barrier height.\nApplication of a potential di\u000berence between upper and\nlower layers intensi\fes the failure of this approxima-\ntion. On the other hand, magnetoresistance in bilayer\ngraphene has been studied in Ref.19by using 8\u00028 Hamil-arXiv:1211.2701v1  [cond-mat.mes-hall]  12 Nov 20122\ntonian when the induced exchange \felds are laid in plane\nof each layer with a rotation in their orientations aginst\neach other. By using Kobo-Greenwood formula, they\nhave investigated the dependence of conductivity and\nmagnetoresistance on temperature and induced exchange\n\feld.\nMotivated by these studies, based on the four-band\nHamiltonian and close to the Dirac points, we study\nspin current through a magnetic barrier creating by use\nof the proximity of a ferromagnetic insulator on bilayer\ngraphene. Conductance is calculated by use of Landauer-\nButtiker formula at zero temperature. The parameters\nof the barrier, energy and angle of incident electrons can\na\u000bect transport through a magnetic barrier classifying in\nthe propagating or evanescent modes. The dependnce\nof resonant peaks in transmission on system parameters\nis proposed to follow a resonance condtion. We have\nfound in some resonant energies and barrier parameters\nand also around the gapped region that a remarkable spin\npolarization and also magnetoresistance can be achieved.\nModel : We consider a normal/ferromagnetic/normal\nbilayer graphene nanojunction. The model we have used\nfor a bilayer graphene sandwiched between two ferromag-\nnetic insulator is shown schematically in Fig. 1. The\nexchange \felds induced by the ferromagnetic insulators\nis supposed to be perpendicular to the graphene plane.\nTherefore, Hamiltonian of the spin up detaches from the\nspin down. Two gate electrodes can be attached to the\nferromagnetic graphene from the upper and lower layers\nwhich control the barrier height in each of layers. This\nset-up is di\u000berent from the systems studied by Refs.19,20.\nThe exchange \feld splits this potential depending on the\nspin parallel (+) or antiparallel ( \u0000) to the exchange \feld.\nSo in the ferromagnetic part of bilayer graphene, we have\nV\u0006=V0\u0007\u0001 where \u0001 and V0are the exchange \feld and\nthe potential barrier made by the gate voltage, respec-\ntively. So the two spins are scattered from the barriers\nwith di\u000berent heights. It means that energy shift of the\ntop of the valance band in the barrier is di\u000berent for par-\nallel and antiparallel spins to the exchange \feld. This\nspin splitting causes to shift conductance as a function\nof energy for each spin resulting in magnetoresistance.\nTo investigate spin polarization and also magnetoresis-\ntance, we have considered two di\u000berent con\fgurations so\nthat the exchange \feld inducing by the magnetic insula-\ntors on each layer are parallel or antiparallel with respect\nto each other. The con\fguration prepared for observa-\ntion of magnetoresistance di\u000bers from the con\fguration\nconsidered by Ref.19,20. The parallel con\fguration has\na metallic behavior while the antiparallel con\fguration\ninduces a potential di\u000berence between upper and lower\nlayers concluding that the system has a semiconductor\nbehavior with the band gap of 2\u0001.\nThis paper is organized as follows: we brie\ry explain\nthe formalism which is used for calculating of transmis-\nsion based on the four-band Hamiltonian. Before present-\ning our results, it is so important to have a short review in\nsection III on transport through a barrier deposited on bi-\nFIG. 2: Schematic view of the potential barrier with height\nV0and width wand the wave number at normal incidence\ndirected in the x-axis for three regions.\nlayer graphene and its dependence on the system param-\neters such as energy of incident quasi-particles and their\nangle hitting into the barrier and also barrier parame-\nters. The method presented in section II is a detailed\nanalysis accompanied with some small corrections on the\nmethod used by Ref.17. We will present spin polarization\nin the parallel con\fguration in section IV. Magnetoresis-\ntance and its dependence on energy of incident particles\nand also induced magnetic \feld will be investigated in\nsection V. Finally, the last section concludes our results.\nII. FORMALISM\nIn the unit cell of bilayer graphene, we suppose that\ntwo independent sublattices A and B related to each\nmonolayer graphene are connected to each other in the\nBernal stacking. Close to the Dirac points and in the\nnearest neighbor tight binding approximation, the four\nband Hamiltonian and also its eigenfunction is written\nas the following:\nH=0\nBB@V1\u0019 t?0\n\u0019yV10 0\nt?0V2\u0019y\n0 0\u0019 V 21\nCCA;\t =0\nB@ A\n B\n B0\n A01\nCA (1)\nwhere\n\u0019= (px+ipy)vf=\u0000i~vf(@x\u0000ky)\nand in the above formula ky=ksin\u0012where\u0012andkare\nthe incident angle and wave number of quasi-particles\nhitting on a barrier which is created by applying a bias\nto a metallic strip deposited on bilayer graphene. More-\nover,V1andV2are the gate potentials applied on the\nupper and lower layers of bilayer graphene. Such a gate\npotential can be manipulated by applying a perpendic-\nular electric \feld on graphene sheet. Here, the barrier\nis approximated by a square potential of barrier with\nsharp variation. By solving the eigenvalue equation of3\nH\t =E\t, the four band spectrum can be concluded as\nthe following:\n(\"0)2=k2+\u000e2+(t0)2\n2\u0006t0r\n4k2\u000e2=(t0)2+ (t0\n2)2+k2(2)\nwhere the above parameters are de\fned as the follow-\ning:\n\"0= (E\u0000V0)=~vf=\"\u0000\u001d0; V 0= (V1+V2)=2\n\u000e= (V1\u0000V2)=2~vf; t0=t=~vf:(3)\nIn the case of \u000e= 0 andk << t0, in low energy limit,\nenergy spectrum behaves as E\u0000V0=\u0006~2k2=2mwhere\nm=t=(2v2\nf) is an e\u000bective mass. This approximation\nwhich results in a e\u000bective two-band Hamiltonian is valid\nwhen energy of incident electrons is close to the barrier\nheight. On the other word, in the case of zero poten-\ntial di\u000berence between two layers, the absolute value of\n(E\u0000V0) should be much smaller than the interlayer cou-\npling strength (0.4eV). For \u000e6= 0, this approximation\nmay fail for large potential di\u000berences. So one should\ncare to choose valid energy and potential ranges. How-\never, in spite of Ref.18, in this paper, we use four-band\nHamiltonian in which the only approximation is the Dirac\ncone.\nIf we assume plane wave solution for the Schroedinger\nequation, the wave function in each region with a con-\nstant potential is written as the following matrix prod-\nuct.\n\t =GM0\nB@a\nb\nc\nd1\nCA (4)\nwhere matrix elements of the matrices GandMare de-\n\fned as the following:\nG=0\nBB@1 1 1 1\nf+\n+f+\n\u0000f\u0000\n+f\u0000\n\u0000\nh+h+h\u0000h\u0000\ng+\n+h+g+\n\u0000h+g\u0000\n+h\u0000g\u0000\n\u0000h\u00001\nCCA\nM(x) =0\nBB@ei\u000b+x0 0 0\n0e\u0000i\u000b+x0 0\n0 0 ei\u000b\u0000x0\n0 0 0 e\u0000i\u000b\u0000x1\nCCA(5)\nf+\n\u0006=\u0006\u000b+\u0000iky\n\"0\u0000\u000e; f\u0000\n\u0006=\u0006\u000b\u0000\u0000iky\n\"0\u0000\u000e\ng+\n\u0006=\u0006\u000b++iky\n\"0+\u000e; g\u0000\n\u0006=\u0006\u000b\u0000+iky\n\"0+\u000e\nh\u0006=(\"0\u0000\u000e)2\u0000\u000b2\n\u0006\u0000k2\ny\nt0(\"0\u0000\u000e)(6)\nFIG. 3: Transmission at normal incidence \u0012= 0 as a function\nof energy di\u000berence between energy of incident particles and\nthe barrier height ( E\u0000V0)=~vFfor potential di\u000berence of\nupper layer against lower layer \u000e= 0 and 10meV .\nHere\u000bis the wave number in the x direction and is de-\n\fned as:\n\u000b2\n\u0006= [\u000e2+(\"0)2\u0000k2\ny\u0006p\n4(\"0)2\u000e2+ (t0)2((\"0)2\u0000\u000e2)] (7)\nIn the special case of normal incident angle and zero gate\npotential where ky=\u000e= 0, the wave number \u000b+is real\nin the energy range of 0 < \"0< t0and\u000b\u0000is real if\nthe energy of incident particles is in the range of \"0<\n0;\"0>\u0000t0. In this paper, studied system contains a\nmagnetic or electrostatic barrier of potential as shown in\nFig.2. The barrier width is w. The electrostatic potential\nwhich plays the role of the gate voltage is set to be V0\nin the barrier part and zero in the \frst and last regions.\nWe suppose that the energy range of incident particles\nis limited to the range of 0 < \"0< t0. Consequently\nin the barrier part we have \u0000V0< \"0\n2< t0\u0000V0. The\nwave numbers behind and in front of the barrier \u000b(1)\n+and\n\u000b(3)\n+are real while \u000b(1)\n\u0000and\u000b(3)\n\u0000are imaginary. In the\nbarrier part, for \"0\n2<0, the wave numbers \u000b(2)\n+and\u000b(2)\n\u0000\nare imaginary and real, respectively, while for \"0\n2>0\nthey behave vice versa. A schematic view of the barrier\nat normal incidence and wave numbers in each part are\nshown in Fig.2.\nBy applying continuity of the wave functions on the\nboundaries of the barrier, one can connect coe\u000ecient ma-\ntrix of the wave function for the last region A3to the\ncoe\u000ecient matrix for the \frst region A1.\nA1=NA 3\nN=M\u00001\n1(0)G\u00001\n1G2M2(0)M\u00001\n2(w)G\u00001\n2G3M3(w)(8)\nwhere N is called as transfer matrix. Since \u000b(1)\n\u0000and\u000b(3)\n\u0000\nare imaginary in the interested energy range, that part of4\nFIG. 4: Contour plot of transmission in plane of the incident\nangle and energy di\u000berence of \"0\n2= (E\u0000V0)=~vFaccompanied\nwith the band structure for potential di\u000berence of the upper\nlayer against the lower layer to be as a) \u000e= 0 and b) 10 meV .\nwave function which are associated by such wave numbers\nare exponentially a growing or decaying function. So we\nhave to set the coe\u000ecient of plane wave ei\u000b(1)\n\u0000x(cin Eq. 4)\nto be zero for the \frst region, because this part of wave\nfunction grows exponentially when x!\u00001 . Therefore,\ncoe\u000ecients matrix in the \frst region is supposed to be as\nA1= [1;r;0;eg]T, where the superscript Trefers to the\ntranspose of a matrix and egis the coe\u000ecient of growing\nevanescent state and ris the coe\u000ecient of the re\rected\npart of the wave function. In the last region, we have\nto set the coe\u000ecient ( din Eq. 4) of e\u0000i\u000b(3)\n\u0000xto be zero\nbecause this part of the wave function increases exponen-\ntially when x!1 . Therefore, the coe\u000ecients matrix in\nthe last regions is supposed to be as A3= [t;0;ed;0]T\nwheretis the coe\u000ecient of the transmitted part of wave\nfunction and edis the coe\u000ecient of decaying evanescent\nstate. In this region, there is no re\rected wave. How-\never, in equation 8 of Ref.17, matricesA1andA3have\nbeen considered to be completely displaced which leas to\ndi\u000berent results.\nBy rearrangement of the transfer matrix elements of\nEq. 8, the coe\u000ecient of transmitted part of wave function\nis derived in terms of transfer matrix elements as the\nfollowing;\nt= [N11\u0000N13N31=N33]\u00001: (9)\nSince the \frst and last regions possess similar wave num-\nbers, transmission probability is given as T=jtj2. Be-\nfore presenting our results, in the next section, we will\nshortly review transport properties through a potential\nbarrier by using the mentioned formalism.III. TRANSMISSION THROUGH A BARRIER\nON BILAYER GRAPHENE\nThe Klein tunneling in monolayer graphene results\nin a complete transmission through a barrier poten-\ntial in normal incident. However, in contrast to mono-\nlayer graphene, as a result of chiral symmetry in bilayer\ngraphene, transmission is zero for quasiparticles with en-\nergies lower than the barrier height. In the special case\nof\u000e=ky= 0, transmission through a potential barrier\ncan be analytically calculated in normal incident and for\ntwo ranges of energy \"0<0 and\"0>0.\nt=ei\u000b(1)w[cos(\u000b(2)w)\u0000iQsin(\u000b(2)w)]\u00001 (10)\nwhere\nQ=1\n2(\"0\n1\u000b(2)\n\"0\n2\u000b(1)+\"0\n2\u000b(1)\n\"0\n1\u000b(2))\nwhere in the above formula, parameters are de\fned as\n\"0\n1=\"=~vf;\"0\n2= (\"\u0000V0)=~vf. So the real part of the\nwave numbers inside and outside of the barrier part are\nde\fned as \u000b(1)= [(\"0\n1)2+\"0\n1t0]1=2and\u000b(2)= [(\"0\n2)2+\n\"0\n2t0]1=2, respectively. The energy of incident particle is\nsupposed to be always \"0\n1>0. So\u000b(1)is always real.\nFor the energy range of E <V 0, the wave number in-\nside the barrier part \u000b(2)and consequently Qare imag-\ninary so that \u000b(2)=i\u0014andQ=iqwhere\u0014andqare\nreal. As a result, transmission tends to zero as a function\nof the system parameters such as wand\"0\n2.\nT(\u0012= 0;\"0\n2<0) =tt\u0003= [cosh2(\u0014w) +q2sinh2(\u0014w)]\u00001\n(11)\nThis behavior is the trace of chiral symmetry in bilayer\ngraphene. However, if the incident angle is nonzero,\nsome resonant peaks appear in the transmission curve\n(see Fig. 4). For the energy range of E >V 0, all param-\neters such as \u000b(2)andQare real. Thus transmission has\nan oscillatory behavior as a function of \"0\n2as the following\nform,\nT(\u0012= 0;\"0\n2>0) = [cos2(\u000b(2)\n+w) +Q2sin2(\u000b(2)\n+w)]\u00001\n(12)\nIn the high energies limit E\u001dV0, we haveQ\u0000!1\nand so transmission is complete ( T\u0000!1). By apply-\ning a vertically electric \feld in the barrier part, a band\ngap is opened in the band structure of bilayer graphene\nwhich is proportional to the potential di\u000berence between\npotentials of each layers. In this case, chiral symme-\ntry is failed and therefore transmission in normal inci-\ndence is nonzero for energies lower than the barrier height\n(E < V 0). Transmission at normal incidence is repre-\nsented in Fig. 3 as a function of \"0\n2for\u000e= 0 and 10meV .\nApplication of a vertically electric \feld causes to emerge\nsome resonant tunneling states for energies of E <V 0. In5\nFIG. 5: Contour plot of transmission in plane of the incident\nangle and the barrier width for the barrier height 50 meV , the\nenergy of incident angle 17 meV and for the case of \u000e= 0.\nFIG. 6: Energy band structure for parallel and antiparallel\ncon\fgurations. In the parallel con\fguration, the direction of\nthe exchange \felds inducing in each layers are parallel with\nrespect to each other. a) for spin parallel and antiparallel to\nthe exchange \feld direction in the parallel con\fguration. b)\nfor both spins up and down in the antiparallel con\fguration.\nthis energy range, resonant states originates from inter-\nference of the incident and scattered waves. For all cases\nsuch as nonzero incident angles and \u000e6= 0, the resonant\npeaks are interpreted by the resonance condition relation\nproposed as\n\u000bbw=n\u0019 (13)\nwhere\u000bbis the x-component of the wave number inside\nthe barrier which can be calculated by Eq. 7.\nTo have more complete view, we prepare a contour\nplot of transmission in plane of the incident angle and \"0\n2\nwhich is shown in Fig. 4 for a \fxed width of the barrier.\nFor the normal incidence ( \u0012= 0), transmission behavior\nis compatible with the results shown in Fig. 3.\nFor energies higher than the barrier height \"0\n2>0,\ntransmitting channels are opened over all ranges of en-\nergies. However, transmitting window for the incident\nFIG. 7: a) Conductance and b) spin polarization as a func-\ntion of\"0\n2for di\u000berent induced exchange \felds \u0001 in the par-\nallel con\fguration. Here, the barrier height and width are\nconsidered to be as 50 meV and 40 nm, respectively.\nangles is limited with the condition that \u000b(2)\n+(in Eq. 7)\nis real. In the case of \u000e= 0, the range of incident an-\ngle in which transmission is high can be extracted as\n\u0000sin\u00001p\n(\"0\n2)2+\"0\n2t0\nk\u0014\u0012\u0014sin\u00001p\n(\"0\n2)2+\"0\n2t0\nk. Therefore,\nby increasing \"0\n2, the range of angles with high trans-\nmission becomes more extended. In the energy range of\n\"0\n2<0, independent of the value of \u000e, resonant peaks\nemerge for nonzero incident angles ( \u00126= 0) which obey\nthe resonance condition \u000bbw=n\u0019. So additional to some\nresonant energy states, we have some resonant widths in\nwhich transmission is high. Fig. 5 shows transmission\nin plane of the incident angle and the barrier width for\n\"0<0 and\u000e= 0. It is shown that based on the reso-\nnance condition (Eqs. 13,7), in large incident angles, \u000bb\nreduces and so in a \fxed resonance order ( n), the reso-\nnance condition is satis\fed for wide barriers. Therefore,\nthe resonance strips with complete transmission shown\nin Fig. 5, depend strongly on the incident angle in the\nrange of wide barriers.\nBy applying a vertically electric \feld in the barrier\npart, a band gap is opened around \"0\n2= 0. This band\ngap also has a trace in transmission as a transport gap\nshown in Fig. 4b.\nIV. RESULTS\nBy application of an averaged gate voltage V0, band\nstructure in the barrier part is shifted by V0value. Fig. 6\nshows band structure of parallel and antiparallel con\fgu-\nration magnetic insulators when a gate voltage is applied\non the barrier part. In case the exchange \felds inducing\nin each layers of bilayer graphene are parallel, particles\nwith spin parallel (spin up) and antiparallel (spin down)\nto the exchange \felds are scattered from barriers with dif-6\nFIG. 8: a) Conductance and b) spin polarization as a func-\ntion of barrier width for di\u000berent induced exchange \felds \u0001\nin the parallel con\fguration. Here, the barrier height and in-\ncident energy are considered to be as 50 meV and 17 meV,\nrespectively.\nferent heights. In the parallel con\fguration, spin splitting\nof the barrier potential in the ferromagnetic graphene is\nwritten asV\u0000\u0000V+= 2\u0001. Such spin splitting is also seen\nin the band structure that is shown in Fig. 6a. It is seen\nthat the top of valance band are shifted to lower/higher\nenergies for spins up/down. However, in the antiparallel\ncon\fguration, the band structure shown in Fig. 6b is the\nsame for both up and down spins. A band gap which is\nproportional to 2\u0001 appears in the band structure of the\nantiparallel con\fguration.\nA. Spin Polarization\nHere, there is a correspondence between the band\nstructure and transmission. According to the band struc-\nture, we expect to emerge spin polarization just for par-\nallel con\fguration because energy bands for up and down\nspins are shifted by 2\u0001 with respect to each other. How-\never, since the band structure for antiparallel con\fgu-\nration is the same for both spins, it is not expected to\nhave spin polarization for this con\fguration. The spin\npolarization is de\fned as:\nP=Gup\u0000Gdown\nGup+Gdown(14)\nwhereGupandGdown are conductance for up and down\nspins. The conductance is calculated by using Landauer\nformalism in the linear regime. Therefore, conductance\nis proportional to angularly averaged transmission pro-\njected along the current direction.\nG=Z\u0019=2\n\u0000\u0019=2T(E;cos (\u0012))cos\u0012d\u0012It is clear that additional to the transmission curves\n(Fig.4), resonance peaks also appears in conductance.\nSince up and down spins in the parallel con\fguration see\nbarriers with di\u000berent heights, resonance peaks in con-\nductance as a function of Fermi energy Eare shifted to\nhigher energies as \u0001 for spin down and to lower energies\nas\u0000\u0001 for spin up. This mismatching of conductance\npeaks for two spins causes to a large spin polarization at\nresonance states. Fig. 7 displays conductance and spin\npolarization as a function of \"0\n2for the parallel con\fgu-\nration. It is shown that conductance peaks and conse-\nquently spin polarization appears in the energy range of\n\"0\n2<0. It is seen that by inducing an exchange \feld, con-\nductance peaks in Fig. 7a split into two peaks which are\nrelated to each spin. This spin splitting is about 2\u0001. Spin\npolarization shown in Fig. 7b has an oscillatory behavior\nwith energy of incident particles for energies lower than\nthe barrier height \"0\n2<0. The amplitude of spin polar-\nization increases with the induced exchange \feld \u0001 and\nreaches to its maximum value. However, spin polariza-\ntion tends to zero for energies greater than the potential\nheight\"0\n2>0 except at E\u0018V0.\nIn the parallel con\fguration and for \"0\n2<0, Fig. 8a\nshows that conductance in the resonance widths has a\npeak. These peaks which are also seen in the transmission\ncurves of Fig. 5 are explained by the resonance condition\nof Eq. 13. It is shown that spin splitting of conductance\npeaks also appears in the resonance widths which is orig-\ninated from di\u000berent barrier heights for two spins up and\ndown. It should be noted that the conductance at res-\nonance widths decreases for wide barriers. In the wide\nrange of widths, the angularly window for transmitting\nchannels shown in Fig. 5 decreases with the widths.\nFig. 8b shows spin polarization as a function of the\nbarrier width. Again, spin polarization has an oscilla-\ntory behavior with the barrier width. The amplitude of\nspin polarization strongly increases by an increase of the\ninduced exchange \feld. Therefore, to manifest this spin\npolarization, we should manufacture the ferromagnetic\ngraphene part with the special widths in which spin po-\nlarization reaches to the value of unity.\nB. Magnetoresistance\nIn this section, we will show that by switching between\nparallel and antiparallel con\fgurations, one can obtain\nlarge magnetoresistance. Magnetoresistance is de\fned as\nthe following:\nMR =Gp\u0000Gap\nGp+Gap(15)\nwhereGp=Gp\nup+Gp\ndownandGap=Gap\nup+Gap\ndownare\nconductance for parallel and antiparallel con\fgurations.\nFig. 9 displays conductance in the parallel and antipar-\nallel con\fgurations and also magnetoresistance as a func-\ntion of\"0\n2and the barrier width for a \fxed exchange \feld7\nFIG. 9: Conductance in the parallel and antiparallel con\fg-\nurations as a function of a) \"0\n2= (E\u0000V0)=~vFfor a barrier\nwith the width of 40 nm and, c) the barrier width for a barrier\nwith the height of 50 meV. Magnetoresistance as a function\nof b)\"0\n2for a barrier with the width of 40 nm, d) the barrier\nwidth for a barrier with the height of 50 meV. The induced\nexchange \feld is considered to be as \u0001 = 5meV.\n\u0001 = 5meV . As we before expressed, spin splitting at\nthe resonance states (for \"0\n2<0) emerges in conductance\npeaks in the case of the parallel con\fguration. This be-\nhavior is clear in Fig. 9a and 9c. However, this splitting\nwill not occur for the case of antiparallel con\fguration.\nTherefore, large magnetoresistance appears around the\nconductance resonance peaks. In the parallel con\fgura-\ntion, a band gap appears around the barrier edge in the\nintervalV0\u0000\u0001<E <V 0+\u0001. This band gap has a trace\nin transmission and consequently conductance. Zero con-\nductance region around the barrier edge \"0\n2\u00180 which is\nseen in Fig.9a, is a result of the band gap. Since there is\nno such a band gap in the parallel con\fguration, magne-\ntoresistance as shown in Fig.9b reaches to its maximum\nvalue in the energy band gap. In the energy range greater\nthan the barrier height \"0\n2>0, there is no spin splitting\nand therefore, magnetoresistance tends to zero.\nAs we showed before, conductance has peak at reso-\nnant widths. Similar to the previous case, spin splitting\noccurs just for the parallel con\fguration. So magnetore-\nsistance increases around the resonance widths. The os-\nFIG. 10: a) Conductance in the parallel and antiparallel con-\n\fguration and b) magnetoresistance as a function of the in-\nduced exchange \feld \u0001 for a barrier with the height of 50\nmeV and energy of incident particles as 40 meV. Here the\nbarrier width is 20 nm.\ncillatory behavior of magnetoresistance as a function of\nthe barrier width is represented in Fig.9d.\nAs we showed, there is a large magnetoresistance\naround the barrier edge E\u0019V0. In this range of ener-\ngies, we investigate the dependence of magnetoresistance\nto the induced exchange \feld. This exchange \feld of\ngraphene can be controlled by an in-plane external elec-\ntric \feld9. Fig. 10b represents that magnetoresistance\nincreases monotonically by increasing the exchange \feld\n\u0001. It is interesting by increasing the exchange \feld up\nto 10meV , magnetoresistance reaches to its maximum\nvalue.\nTo explain this behavior, we investigate the depen-\ndence of conductance on the exchange \feld in the par-\nallel and antiparallel con\fgurations. In the antiparal-\nlel con\fguration, the band gap which is limited in the\ninterval of V0\u0000\u0001< E < V 0+ \u0001, enhances by in-\ncreasing the exchange \feld. Therefore, conductance in\nthe antiparallel con\fguration goes to zero when the ex-\nchange \feld is increased. Suppression of the conductance\nwith the exchange \feld in the antiparallel con\fguration\nis shown in Fig. 10a. However, in the parallel con\fg-\nuration, conductance increases by enhancement of the\nexchange \feld. The reason of this enhancement comes\nback to have larger angularly transmitting windows for\nlarger\"0\n2(see Fig. 4). In fact, e\u000bective potential for spins\nupV+=V0\u0000\u0001 is decreased by an increase in \u0001. So\n\"0\n2= (E\u0000V0)=~vFfor a \fxed energy is increased and\nconsequently Gupand soGpis increased by \u0001. As a\nconclusion, for the exchange \felds up to 10 meV, sup-\npression of Gapand an increase of Gpresults in a large\nmagnetoresistance which is so useful for designing spin\nmemory devices.8\nV. CONCLUSION\nWe have studied spin polarization and magnetoresis-\ntance of a normal/ferromagnetic/normal junction of bi-\nlayer graphene by using transfer matrix method and\nbased on the four-band Hamiltonian. Transport prop-\nerties simultaneously is controlled by two gate electrodes\n(V0), which are applied on the ferromagnetic graphene.\nTwo con\fgurations of the exchange \feld is considered\nperpendicular to the graphene sheet. This exchange \feld\nis induced by the proximity of a localized magnetic or-\nbital in a magnetic insulator coating on top of each layers\nof bilayer graphene in the barrier part. In the parallel\ncon\fguration which graphene has a metallic behavior, a\nspin splitting 2\u0001 occurs for the conductance at the reso-nant states just for energies lower than the barrier height\nE < V 0. However, there is no spin splitting in the an-\ntiparallel con\fguration. A band gap of 2\u0001 is opened\nin the antiparallel con\fguration which makes it a semi-\nconductor. As a result of spin splitting in the parallel\ncon\fguration, an oscillating spin polarization emerges for\nenergies lower than the barrier height. Furthermore, an\noscillatory of magnetoresistance with large amplitude is\nachievable for E <V 0when we are able to switch between\ntwo con\fgurations. There is also a large magnetoresis-\ntance in the energy range around the barrier edge origi-\nnating from the band gap which is openned by a vertically\nelectric \feld. In this range of energy, magnetoresistance\nreaches to its maximum value when the exchange \feld is\nincreased by an in-plane external electric \feld.\n\u0003Electronic address: cheraghchi@du.ac.ir\n1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.\nZhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov,\nScience 306, 666 (2004).\n2D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B. 74, 155426 (2006); Hongki Min, J. E. Hill, N. A.\nSinitsyn, B. R. Sahu, L. Kleinman, and A. H. MacDonald,\nPhys. Rev. B. 74, 165310 (2006); Y. Yao, F. Ye, X. L.\nQi, S. C. Zhang, and Z. Fang, Phys. Rev. B. 75, 041401,\n(2007).\n3B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nat.\nPhys. 3, 192 (2007).\n4N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, B.\nJ. van. Wees, Nature, 448, 571 (2007).\n5O. V. Yazyev and L. Helm, Phys. Rev. B. 75, 125408\n(2007); T. O. Wehling, K. S. Novoselov, S. V. Morozov,\nE. E. Vdovin, M. I. Katsnelson, A. K. Geim and A. 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Phys.\n109, 073717 (2011)."    },    {        "title": "1710.10895v2.RbEu_Fe___1_x__Ni__x____4_As__4___From_a_ferromagnetic_superconductor_to_a_superconducting_ferromagnet.pdf",        "content": "arXiv:1710.10895v2  [cond-mat.supr-con]  8 Jan 2018RbEu(Fe 1−xNix)4As4: From a ferromagnetic superconductor to a superconducting\nferromagnet\nYi Liu,1,∗Ya-Bin Liu,1,∗Ya-Long Yu,1Qian Tao,1Chun-Mu Feng,1and Guang-Han Cao1,2,3,†\n1Department of Physics, Zhejiang University, Hangzhou 31002 7, China\n2State Key Lab of Silicon Materials, Zhejiang University, Han gzhou 310027, China\n3Collaborative Innovation Centre of Advanced Microstructu res, Nanjing 210093, China\n(Dated: January 9, 2018)\nThe intrinsically hole-doped RbEuFe 4As4exhibits bulk superconductivity at Tsc= 36.5 K and\nferromagnetic ordering in the Eu sublattice at Tm= 15 K. Here we present a hole-compensation\nstudy by introducing extra itinerant electrons via a Ni subs titution in the ferromagnetic\nsuperconductor RbEuFe 4As4withTsc> Tm. With the Ni doping, Tscdecreases rapidly, and the Eu-\nspin ferromagnetism and its Tmremain unchanged. Consequently, the system RbEu(Fe 1−xNix)4As4\ntransforms into a superconducting ferromagnet with Tm> Tscfor 0.07≤x≤0.08. The\noccurrence of superconducting ferromagnets is attributed to the decoupling between Eu2+spins and\nsuperconducting Cooper pairs. The superconducting and mag netic phase diagram is established,\nwhich additionally includes a recovered yet suppressed spi n-density-wave state.\nPACS numbers: 74.70.Xa, 74.25.Ha, 74.62.Dh, 75.30.-m\nINTRODUCTION\nSuperconductivity (SC) and ferromagnetism (FM) are\nbasically antagonistic and incompatible [1, 2], and only\nin a very few cases can they coexist simultaneously in\na single material [1, 3]. The relative robustness of SC\nand FM can be reflected by the superconducting criti-\ncal temperature Tscand the (ferro)magnetic transition\ntemperature Tm. Materials with Tsc> Tmwere earlier\ncalled “ferromagnetic superconductors” (FSCs) [1], and\nthose with Tm> Tscwere then termed “superconducting\nferromagnets” (SFMs) [4–6]. Generally, a ferromagnetic\nexchange field prevails over the intrinsic superconducting\nupper critical field H∗\nc2forTm≤Tsc, hence SFMs\nare particularly rare. So far, examples of SFMs only\ninclude U-based germanides with spin-triplet SC [7]\nand ruthenocuprates with spin-singlet high-temperature\nSC[8], thelatterofwhichactuallyexhibitthecoexistence\nwith a canted antiferromagnetism. Note that the\nclassification into FSCs and SFMs is meaningful for\nstudying the way of coexistence of the two antagonistic\nphenomena [5, 9].\nIn recent years, Eu-containing 122-type iron arsenides\nhave earned a lot of interest owing to the intriguing\ninterplay between SC and FM [3, 10]. The crystal\nstructure allows the magnetic Eu-atomic planes away\nfrom the superconductively active Fe-atom sheets. In\nnon-doped EuFe 2As2, the Eu sublattice is of an A-\ntype antiferromagnetism below ∼19 K while the Fe\nsublattice exhibits a spin-density wave (SDW) order\nbelow∼190 K [11–14]. SC at Tsc= 20-30 K\ncan be induced by the chemical doping with P [15],\nCo [16], Ru [17], Ir [18, 19], etc. Simultaneously,\nthe Eu2+local spins become ferromagnetically ordered\natTm∼18 K [20–24]. It has been concluded\nthat SC appears only when Tsc> Tmin systems ofEuFe2(As1−xPx)2[25, 26], Eu(Fe 1−xCox)2As2[27], and\nSr1−yEuy(Fe0.88Co0.12)2As2[28]. Theconclusionalsofits\nwith the absence of SC in Eu(Fe 1−xNix)2As2[29], since\nthe Eu-free analogous system Sr(Fe 1−xNix)2As2shows a\nmaximum Tscof 9.8 K, which is significantly lower than\nthe expected Tm[30].\nVeryrecently,avariantofEuFe 2As2, i.e. the1144-type\nAEuFe4As4(A=Rb andCs), weresynthesizedandchar-\nacterized [31–33]. The twin compounds adopt a crystal\nstructureidenticaltothatpreviouslydesigned[34], which\nwas first realized in AeAFe4As4(Ae= Ca, Sr; A= K,\nRb, Cs) [35]. In RbEuFe 4As4, the Rb+and Eu2+planes,\nsandwiched by FeAs layers, stack alternately along the c\naxis. The structure can also be viewed as an intergrowth\nbetween non-doped EuFe 2As2and heavily over-doped\nRbFe2As2. As a result, RbEuFe 4As4is intrinsically hole-\ndoped, exhibiting SC at Tsc= 36.5 K without any SDW\nordering [31, 32]. Additionally, evidence of FM of the\nEu2+spins below Tm= 15 K is given by magnetization\nmeasurements [32]. Compared with 122-type FSCs, the\nTscvalue is significantly higher, and the Tmvalue is\nslightly lower. Important to be noted is that the Tsc\nvalue of RbEuFe 4As4are almost the same as, or even\nlarger than, those of the non-magnetic analogues (e.g.\nTsc= 35.1 K in RbSrFe 4As4[35]), indicating that the\nEu2+spins hardly break superconducting Cooper pairs.\nIn this context, SC may survive easily in the presence of\nEu-spin order, and therefore, it is of interest to seek for\nan SFM in the 1144-type system.\nNow that RbEuFe 4As4bears an intrinsic hole doping\n(0.25 holes per Fe atom), it is natural to tune the\nTscvalue by hole depletion via electron doping. Our\npreliminary trial with a Ba-for-Rb substitution was\ndemonstrated unsuccessful, since a 122-type phase, in-\nsteadoftheexpected1144-typephase, becamestabilized.\nThen, we turned to a substitution at the Fe site. To2\ncompensate the doped holes more effectively, we chose Ni\nas the dopant, because Ni2+(3d8) has twomore itinerant\nelectrons than Fe2+(3d6) does, and more importantly,\nprevious Ni-doping studies indeed show such an effect of\nelectron doping [30, 36, 37].\nIn this paper, we report a systematic investigation\non the magnetic and superconducting properties in\nRbEu(Fe 1−xNix)4As4. As expected, Tscdecreases\nrapidly with the Ni doping. On the other hand, the\nEu-spin FM and its Tmvalue remain unchanged. This\nleads to a discovery of SFMs in RbEu(Fe 1−xNix)4As4\nwith 0.07≤x≤0.08, in which SC survives when Tsc\nbecomes lower than Tm. The SFMs are found to show\nno Meissner state with a broadened resistive transition,\nbecausetheyarealwaysundertheinternalfieldgenerated\nby the FM of Eu2+spins. The superconducting and\nmagnetic phase diagram has been established, and the\nreason for the existence of SFMs is discussed.\nEXPERIMENTAL METHODS\nPolycrystalline samples of RbEu(Fe 1−xNix)4As4with\n0≤x≤0.125 were prepared by solid-state reactions\nin evacuated quartz ampoules sealed, similar to our\nprevious report [32]. The source materials were the\nconstituent elements: Rb (99.75%), As (99.999%), Eu\n(99.9%), Fe (99.998%), and Ni (99.99%), all from Alfa\nAesar. Firstly, precursors of EuAs, FeAs, NiAs, and\nRbFe2As2(with 5% excess of Rb) were prepared by\nsolid-state reactions in evacuated quartz tubes at 873-\n1023 K for 24 hours. These precursors and additional Fe\npowders were then mixed together in the nominal com-\nposition ofRbEu(Fe 1−xNix)4As4, followedby thoroughly\nhomogenizing with ball milling in an Ar-filled glove box.\nSecondly, the mixtures were pressed into pellets which\nwere loaded in an alumina container jacketed with a\ndouble-layer protector (a sealed Ta tube inside and a\nquartz ampoule outside). Finally, the samples were\nrapidly heated to 1133 K, holding for 20 hours, ended\nwith quenching in cool water. To improve samples’\npurity, the synthesis was repeated once or twice, with\nan intermediate grinding.\nPowder x-ray diffraction (XRD) was performed on a\nPANalytical x-ray diffractometer with a monochromatic\nCu-Kα1radiation at room temperature. The lattice\nparameters were obtained by a least-squares fit of 15-\n25 reflections in the range of 5◦≤2θ≤80◦. The\nsample’s chemical composition was checked by energy-\ndispersive x-ray spectroscopy (EDS). The resistivity\nand specific-heat measurements were conducted on a\nQuantum Design Physical Property Measurement Sys-\ntem (PPMS-9). The dc magnetization was measured\non a Quantum Design Magnetic Property Measurement\nSystem (MPMS-XL5)./s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s51/s46/s56/s56/s51/s46/s56/s57/s51/s46/s57/s48\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s49/s51/s46/s50/s52/s49/s51/s46/s50/s54/s49/s51/s46/s50/s56/s49/s51/s46/s51/s48/s42\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s50/s49/s51/s49/s49/s54/s49/s48/s53\n/s50/s48/s48/s49/s49/s52/s49/s49/s51/s49/s49/s50/s49/s49/s48/s49/s48/s51/s48/s48/s52/s48/s48/s50/s120 /s32/s61/s32/s48/s46/s49\n/s120 /s32/s61/s32/s48/s46/s48/s57\n/s120 /s32/s61/s32/s48/s46/s48/s56\n/s120 /s32/s61/s32/s48/s46/s48/s55\n/s120 /s32/s61/s32/s48/s46/s48/s54\n/s120 /s32/s61/s32/s48/s46/s48/s53\n/s120 /s32/s61/s32/s48/s46/s48/s50/s53\n/s120 /s32/s61/s32/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s97/s41/s48/s48/s49/s82/s98/s69/s117/s40/s70/s101\n/s49/s45 /s120/s78/s105\n/s120/s41\n/s52/s65/s115\n/s52\n/s42/s42/s32/s70/s101/s65/s115\n/s40/s98/s41\n/s120/s97 /s32/s40 /s197 /s41/s40/s99/s41\n/s120/s99 /s32/s40 /s197 /s41\nFIG. 1. (a) Powder X-ray diffraction patterns of\nRbEu(Fe 1−xNix)4As4(0≤x≤0.1) at room temperature.\n(b) and (c) Lattice parameters aandcas functions of the\nnominal Ni concentration x. The dashed line in (c) gives the\nlinear fit.\nRESULTS AND DISCUSSION\nX-ray diffraction\nFigure 1(a) shows XRD patterns of the series samples\nof RbEu(Fe 1−xNix)4As4(0≤x≤0.1), which can be\nwell indexed by a tetragonal unit cell with a≈3.89˚A\nandc≈13.3˚A. The variations in relative intensity is\nmainly due to (00l) preferred orientations. One sees that\nall the samples from x= 0 to 0.1 are nearly single phase\n(only small amount of impurities such as FeAs appear\nin some of the samples). In the case of x= 0.125, the\nXRDpattern(notshown)indicatesformationof122-type\nphase. Therefore, the present study is limited to samples\nwith 0≤x≤0.1.\nThe lattice constants were determined by a least-\nsquares fit, the results of which are displayed in Table I.\nFigs. 1(b,c) plot the fitted lattice parameters aandc,\nrespectively, as a function of the nominal Ni content.\nAs is shown, while the aaxis of the unit cell basically\nremains unchanged, the caxis decreases significantly\nwith the Ni doping. The result is quite similar to\nthose in Eu(Fe 1−xNix)2As2[29], LaFe 1−xNixAsO [36],\nand Ba(Fe 1−xNix)2As2[37] systems. The linear decrease\ninc, which obeys the Vegard’s law, confirms that the\ndopant Ni indeed substitutes for Fe. The EDS on the3\nTABLE I. Room-temperature lattice constants and physical- property parameters of RbEu(Fe 1−xNix)4As4(0≤x≤0.1).\nThe digits in parentheses give twice of the standard deviati on of the least-squares fit. Tρ\nscis the superconducting midpoint\ntransition temperature in the resistivity measurement. TSDWdenotes the spin-density-wave transition temperature. Tmand Θ\nare the magnetic-transition and Curie-Weiss temperatures , respectively. µeffandMsatare the effective magnetic moment in\nthe paramagnetic state and the ordered moment in the ferroma gnetic state, respectively. Hcoerefers to the apparent coercive\nfield.\nLattice Constants Physical-Property Parameters\nx a (˚A) c(˚A) Tρ\nsc(K)TSDW(K)Tm(K) Θ (K) µeff(µB/Eu) Msat(µB/Eu) Hcoe(Oe)\n0 3.8891(4) 13.304(17) 36.4 - 15.0 23.6 7.95 6.5 360\n0.025 3.8900(6) 13.284(19) 30.3 - 15.0 24.3 7.88 6.5 258\n0.05 3.8909(3) 13.274(12) 23.0 28.9 15.0 24.5 8.00 6.4 88\n0.06 3.8878(6) 13.268(27) 18.1 31.1 15.0 24.3 7.65 5.9 67\n0.07 3.8889(7) 13.263(37) 11.2 35.0 15.1 24.2 7.46 5.9 44\n0.08 3.8877(11) 13.261(48) 2.1 33.6 14.7 24.4 7.85 6.5 21\n0.09 3.8892(8) 13.256(40) - 31.3 14.7 24.8 7.79 6.0 20\n0.1 3.8877(4) 13.250(17) - 29.4 14.8 24.4 7.74 6.3 24\nsample of x= 0.1 shows that the Ni content is 0.089(8),\nwhich remains unchanged throughout the sample. This\nconfirms that the sample is homogeneous for the Ni\ndoping, and the actual Ni-doping level is close to the\nnominal one within the measurement uncertainty.\nMagnetic properties\nWe first address the Eu-spin state by focus-\ning on the high-temperature magnetic properties in\nRbEu(Fe 1−xNix)4As4. As shown in Fig. 2, the magnetic\nsusceptibility ( χ) at high temperatures is of Curie-Weiss-\ntype paramagnetism. The χ(T) data in the temperature\nrange of 50 K ≤T≤300 K can be well fitted with an\nextended Curie-Weiss law, χ=χ0+C/(T−Θ), where\nχ0is the temperature-independent term, Cgives the\nCurie constant from which the effective local moment is\nderived, and Θ represents the Curie-Weiss temperature.\nThe derived effective moment µeff(see Table I) ranges\nbetween 7.46 and 8.00 µB/f.u. (f.u. refers to formula\nunit), independent of the Ni substitution. These µeff\nvalues are close to that (7.9 µB) expected for a free Eu2+\nion, indicating a spin state with total spins of S= 7/2for\nthe Eu2+ions. The Θ values fitted (from 23.6 to 24.8 K)\nis also independent of the Ni doping. The positive sign of\nΘ reflects dominant ferromagnetic interactions between\nEu2+spins.\nAt low temperatures, the system undergoes super-\nconducting and/or magnetic transitions. To determine\ntheir transition temperatures, TscandTm, we performed\nmeasurement of temperature-dependent magnetization,\nM(T), under a low field of 10 Oe. In general, a\nsuperconducting transition can be easily recognized by\nthe strong diamagnetic signal owing to Meissner effect.\nHowever, in cases of the coexistence between SC and FM\nwithTsc≤Tm, the diamagnetic signal may be covered\nup by FM. For the ferromagnetic transition, on the other/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s101/s102/s102/s32/s61/s32/s56/s46/s48/s48/s32\n/s66/s47/s69/s117\n/s32/s61/s32/s50/s52/s46/s53/s32/s75/s82/s98/s69/s117/s40/s70/s101\n/s49/s45 /s120/s78/s105\n/s120/s41\n/s52/s65/s115\n/s52/s32/s40 /s120 /s32/s61/s32/s48/s46/s48/s53/s41\n/s72 /s32/s61/s32/s49/s32/s107/s79/s101/s32/s40/s101/s109/s117/s47/s109/s111/s108/s41\n/s84 /s32/s40/s75/s41/s48/s49/s48/s50/s48/s51/s48\n/s49/s47 /s32/s40/s109/s111/s108/s47/s101/s109/s117/s41\nFIG. 2. Temperature dependence of dc magnetic susceptibil-\nity (χ=M/H) for a typical sample of RbEu(Fe 1−xNix)4As4\nwithx= 0.05 at a magnetic field of H= 1 kOe. The\nright-hand axis plots 1 /χ, indicating dominant Curie-Weiss\nparamagnetism above 25 K. The data in the range of 50 K\n≤T≤300 K are fitted with Curie-Weiss law (displayed with\nsolid lines), from which the effective local moment µeffand\nthe Curie-Weiss temperature Θ are extracted as shown. See\nthe text for details.\nhand, the Curie temperature is traditionally determined\nby Arrot approach [38]. However, the presence of SC\nmakes the method invalid. Here we take advantage\nof the magnetic hysteresis arising from the appearance\nof magnetic domains. Namely, Tmis defined by the\nbifurcation temperature between field-cooling (FC) and\nzero-field-cooling (ZFC) M(T) data measured under\nmagnetic fields lower than the coercive field. In fact, the\nbifurcation point basically coincides with the kink (peak)\nin the FC (ZFC) curves. Furthermore, the resultant\nTmvalue is precisely consistent with the heat-capacity\nmeasurement [32]. Note that a type-II SC may also give\nrise to a bifurcation between FC and ZFC curves owing\nto magnetic flux-pinning effect.\nFigure 3(a-h) shows the low-field M(T) data for4\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s50/s49 /s50/s50 /s50/s51 /s50/s52/s49/s53 /s49/s54\n/s49/s53 /s49/s54/s49/s52 /s49/s53 /s49/s54/s49/s52 /s49/s53 /s49/s54/s49/s52 /s49/s53 /s49/s54/s84\n/s115/s99/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s120 /s32/s61/s32/s48/s40/s97/s41\n/s84\n/s109\n/s32/s70/s67\n/s32/s90/s70/s67/s120 /s32/s61/s32/s48/s46/s48/s50/s53\n/s32/s70/s67\n/s32/s90/s70/s67/s84\n/s109/s84\n/s115/s99/s40/s98/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s120 /s32/s61/s32/s48/s46/s48/s53/s32/s70/s67\n/s32/s90/s70/s67/s84\n/s109\n/s84\n/s115/s99/s40/s99/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s120 /s32/s61/s32/s48/s46/s48/s54/s32/s70/s67\n/s32/s90/s70/s67/s84\n/s109/s40/s100/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41\n/s32/s70/s67\n/s32/s90/s70/s67/s120 /s32/s61/s32/s48/s46/s48/s55/s40/s101/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s32/s70/s67\n/s32/s90/s70/s67/s120 /s32/s61/s32/s48/s46/s48/s56/s40/s102/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s32/s70/s67\n/s32/s90/s70/s67/s120 /s32/s61/s32/s48/s46/s48/s57/s40/s103/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s32/s70/s67\n/s32/s90/s70/s67/s120 /s32/s61/s32/s48/s46/s49/s40/s104/s41\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s84 /s32/s40/s75/s41/s84\n/s115/s99/s77\n/s84 /s32/s40/s75/s41/s84\n/s109\n/s84\n/s115/s99/s77\n/s84 /s32/s40/s75/s41\n/s77\n/s70/s67/s45/s77\n/s90/s70/s67\n/s48\n/s84\n/s109/s77\n/s84 /s32/s40/s75/s41/s84\n/s109/s77\n/s84 /s32/s40/s75/s41/s84\n/s109/s77\n/s84 /s32/s40/s75/s41/s84\n/s109/s77\n/s84 /s32/s40/s75/s41\nFIG. 3. Temperature dependence of magnetization under a mag netic field of 10 Oe for RbEu(Fe 1−xNix)4As4(0≤x≤\n0.1) samples. FC (solid symbols) and ZFC (open symbols) denote field cooling and zero-field cooling, respectively, in the\nmagnetic measurements. TscandTmmarked with arrows represent the superconducting and magne tic transition temperatures,\nrespectively. The insets show close-ups near the supercond ucting/magnetic transitions. The inset in panel (d) also pl ots the\ndifference between the FC and ZFC data, using the righ-hand ax is.\nsamples of the RbEu(Fe 1−xNix)4As4series. The non-\ndoped (x= 0) compound shows SC at Tsc= 36.5 K\nand FM below Tm= 15 K [32]. For x= 0.025, the\nsuperconducting transition keeps robust, yet with an\nobviously reduced Tscof 30.5 K. At x= 0.05, in which\nTscis further decreased to 22 K, the superconducting\ntransition becomes much less remarkable. For x= 0.06,\nonly weak signature of SC at 15.7 K can be traced\nfrom the slight difference between the FC and ZFC data,\nthe latter of which comes from flux-pinning effect. The\nexistence of SC is also evidenced from the diamagnetism\nin the ZFCdataat lowertemperaturesand, in particular,\nfrom the zero resistance at 16 K in the resistivity\nmeasurement shown below. In the cases of x≥0.07,\nno magnetic signal for SC can be detected, although the\nresistivity measurement clearly shows superconducting\ntransitions at lower temperatures for 0 .07≤x≤0.08.\nIn contrast with the monotonic suppression in Tsc\nwith increasing Ni doping, the magnetic transition\ntemperature Tmalmost keeps unchanged. Note that the\nCurie-Weiss temperature Θ, which is remarkably higher\nthanTm, does not depend on the Ni doping either.\nThe lower-than-expected Tmvalue is probably related to\nthe quasi-two-dimensional magnetism caused by a much\nweaker magnetic coupling along the caxis [32]. We will\ndiscuss on the magnetic interactions later on.\nOne may also note that, for lower Ni doping with x≤\n0.05, the FC magnetization decreases with decreasing\ntemperature just below Tm, showing a peak-like anomaly\n(PLA) at Tm, which casts doubt on the nature of the/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s52/s54\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s52/s54\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s52/s54/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s52/s54\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s52/s54\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s52/s54/s120 /s32/s61/s32/s48/s46/s48/s53/s32/s32/s32/s32 /s72 /s32/s61\n/s32/s48/s46/s49/s32/s107/s79/s101\n/s32/s48/s46/s53/s32/s107/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s50/s32/s107/s79/s101\n/s32/s53/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s40/s98/s41\n/s40/s102/s41\n/s32/s32/s32/s32 /s72 /s32/s61\n/s32/s48/s46/s49/s32/s107/s79/s101\n/s32/s48/s46/s53/s32/s107/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s50/s32/s107/s79/s101\n/s32/s53/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s120 /s32/s61/s32/s48/s46/s49/s40/s101/s41\n/s120 /s32/s61/s32/s48/s46/s48/s56/s32/s32/s32/s32 /s72 /s32/s61\n/s32/s48/s46/s49/s32/s107/s79/s101\n/s32/s48/s46/s53/s32/s107/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s50/s32/s107/s79/s101\n/s32/s53/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s120 /s32/s61/s32/s48/s46/s48/s50/s53/s32/s32/s32/s32 /s72 /s32/s61\n/s32/s48/s46/s49/s32/s107/s79/s101\n/s32/s48/s46/s53/s32/s107/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s50/s32/s107/s79/s101\n/s32/s53/s32/s107/s79/s101\n/s32/s49/s48/s32/s107/s79/s101/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s32/s70/s67\n/s32/s90/s70/s67\n/s32/s32/s32 /s72 /s32/s61\n/s32/s48/s46/s49/s32/s107/s79/s101\n/s32/s48/s46/s53/s32/s107/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s50/s32/s107/s79/s101/s40/s99/s41\n/s120 /s32/s61/s32/s48/s46/s48/s54/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41/s32/s32/s32 /s72 /s32/s61\n/s32/s48/s46/s49/s32/s107/s79/s101\n/s32/s48/s46/s53/s32/s107/s79/s101\n/s32/s49/s32/s107/s79/s101\n/s32/s50/s32/s107/s79/s101/s40/s100/s41\n/s120 /s32/s61/s32/s48/s46/s48/s55/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s84 /s32/s40/s75/s41\nFIG. 4. Temperature dependence of field-cooling (FC,\nsolid symbols) and zero-field-cooling (ZFC, open symbols)\nmagnetization under different magnetic fields for x= 0.025\n(a), 0.05 (b), 0.06 (c), 0.07 (d), 0.08 (e), and 0.1 (f) in\nRbEu(Fe 1−xNix)4As4. The magnetization is conversed into\nBohr magnetons per formula unit.5\nmagnetic transition. To address this issue, we measured\ntheM(T) data at elevated magnetic fields shown in\nFig.4. OneseesthatthePLAdisappearsunderrelatively\nlow fields ( ∼0.5 kOe). At higher fields with H≥\n5 kOe, all the M(T) curves become almost identical\nwithout bifurcations in the FC and ZFC data, suggesting\ncommonality ofFM in the system. In fact, a similar PLA\nbehavior is also seen in 122-type FSCs [16, 39] where\nan Eu-spin FM is unambiguously demonstrated [20–22].\nNote that the PLA here happens only for Tsc> Tm. This\nsuggests that the PLA is probably in relation with the\npresence of SC. As is pointed out theoretically, the state\nof FM may be modified into crypto-ferromagnetism [40]\nor dense-domain structure [41] owing to the presence of\nSC. The expected fine ferromagnetic domains that are\nantiferromagnetically aligned could give rise to a PLA\nin the FC M(T) curve. Future studies with single-\ncrystalline samples by magnetic-force microscopy [42]\nseem promising to clarify this issue.\nThe Eu-spin FM in RbEu(Fe 1−xNix)4As4is further\nconfirmed by the isothermal magnetization, M(H),\nshown in Fig. 5(a-f). At high temperatures (e.g., 40\nK), theM(H) data are essentially linear, consistent with\nthe paramagnetic state of Eu2+spins. At temperatures\nbelowTm, by contrast, the M(H) curves are of an S\nshape, characteristic of an FM. The saturation magne-\ntization Msat, defined here as the magnetization value\nat 1 T and 2 K, scatters from 5.9 to 6.5 µB/f.u. (see\nTable I). Samples containing more FeAs impurity tend to\nhave relatively low Msat(andµeffalso). Other samples\nshow an Msatvalue that is close to the expected one\n(7.0µBper Eu2+) [43], indicating that Eu2+spins order\nferromagnetically.\nNote that for samples of x≤0.05, which have higher\nTscvalues, the magnetic hysteresis is extended to high\nfields where the magnetization is saturated. This clearly\nindicates the existence of type-II SC that commonly\nexhibits flux-pinning effect, the latter of which also gives\nrise to an enhancement of the apparent coercive field\nHcoe. As shown in Table I, the intrinsic Hcoe, given\nby non-superconducting samples of 0 .08< x≤0.1, is\nactually around 20 Oe. The sample of x= 0.07 shows an\nenhanced Hcoeof 44 Oe, implying the existence of SC.\nIndeed, the resistivity measurement below demonstrates\na superconducting transition at Tsc= 11 K, 4 K lower\nthanTm. That is to say, the sample is actually an SFM.\nNotably, no superconducting diamagnetism is detected\nby theM(T) data [Fig. 3(e)] and the virgin M(H)\ncurve [Fig. 5(c)]. This observation is consistent with\nthe absence of Meissner state, as expected from the\ninternalfield( ∼4.5kOe[44])thatismuchhigherthanthe\nintrinsic lower critical field. It is of great interest for the\nfuture to look into the anisotropic magnetic properties\nwith using the single-crystalline samples./s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54/s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54/s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52/s45/s48/s46/s51/s48/s46/s48/s48/s46/s51\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s54/s45/s51/s48/s51/s54\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s32/s50/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75/s120 /s32/s61/s32/s48/s46/s48/s53/s40/s98/s41\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41\n/s32/s50/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75/s120 /s32/s61/s32/s48/s46/s49/s40/s102/s41\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41\n/s32\n/s72 /s32/s40/s107/s79/s101/s41\n/s32/s32\n/s72 /s32/s40/s107/s79/s101/s41/s32/s50/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41/s120 /s32/s61/s32/s48/s46/s48/s50/s53/s40/s97/s41\n/s32/s50/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s50/s48/s32/s75\n/s32/s52/s48/s32/s75/s120 /s32/s61/s32/s48/s46/s48/s56/s40/s101/s41\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41\n/s32\n/s72 /s32/s40/s107/s79/s101/s41\n/s32\n/s72 /s32/s40/s107/s79/s101/s41/s32/s50/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s50/s48/s32/s75/s120 /s32/s61/s32/s48/s46/s48/s54/s40/s99/s41\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41/s32/s50/s32/s75\n/s32/s49/s48/s32/s75\n/s32/s50/s48/s32/s75/s120 /s32/s61/s32/s48/s46/s48/s55/s40/s100/s41\n/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s107/s79/s101/s41/s32\n/s72 /s32/s40/s107/s79/s101/s41\n/s32\n/s72 /s32/s40/s107/s79/s101/s41\nFIG. 5. Isothermal magnetization at low temperatures in\nthe RbEu(Fe 1−xNix)4As4series. The insets show close-ups\nof the magnetic hysteresis at 2 K, from which the virgin\nmagnetization as well as the coercive field can be seen clearl y.\nElectrical resistivity\nFigure 6 shows the electrical resistivity ( ρ) for\nRbEu(Fe 1−xNix)4As4polycrystalline samples. To high-\nlight the evolution of the temperature dependence, and\nalso to present the superconducting/SDW transitions\nclearly, we normalize the ρ(T) data relative to the\nresistivity values at 200 and 50 K, respectively. First\nof all, the slope d ρ/dTin the normal state decreases\nmonotonically with the Ni doping, giving rise to an\nincrease in the residual resistivity at low temperatures.\nThis is consistent with the increase of Fe-site disorder\nwithNidoping. Secondly, thesuperconductingtransition\ntemperature Tρ\nscdecreases with the Ni doping, and SC\nis completely suppressed at x= 0.09 (the slight drop\naround 5 K is probably due to sample’s inhomogeneity).\nThe result is basically consistent with the magnetic\nmeasurement for x≥0.06 with Tsc> Tm. In the\ncase ofTsc≤Tm, nevertheless, SC cannot be directly\ndetected in the magnetic measurement above, and\nthe resistive transitions become remarkably broadened.\nThe broadened resistive transition is similar to the\nobservation in Eu(Fe 0.81Co0.19)2As2[45] also with Tsc<\nTm, which can be explained in terms of the dissipative\nflow of spontaneous vortices [9].\nAnother interestingly point is that the samples with6\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s55/s48/s46/s56/s48/s46/s57/s49/s46/s48/s120 /s32/s61/s32/s48/s46/s49/s82/s98/s69/s117/s40/s70/s101\n/s49/s45 /s120/s78/s105\n/s120/s41\n/s52/s65/s115\n/s52\n/s32/s120 /s32/s61/s32/s48\n/s32/s120 /s32/s61/s32/s48/s46/s48/s50/s53\n/s32/s120 /s32/s61/s32/s48/s46/s48/s53\n/s32/s120 /s32/s61/s32/s48/s46/s48/s54\n/s32/s120 /s32/s61/s32/s48/s46/s48/s55\n/s32/s120 /s32/s61/s32/s48/s46/s48/s55/s53\n/s32/s120 /s32/s61/s32/s48/s46/s48/s56\n/s32/s120 /s32/s61/s32/s48/s46/s48/s57\n/s32/s120 /s32/s61/s32/s48/s46/s49/s47\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s120 /s32/s61/s32/s48\n/s84\n/s83/s68/s87 /s40/s98/s41\n/s32/s32/s32/s32/s32 /s120 /s32/s61\n/s32/s48/s46/s48/s50/s53\n/s32/s48/s46/s48/s53\n/s32/s48/s46/s48/s54\n/s32/s48/s46/s48/s55\n/s32/s48/s46/s48/s55/s53\n/s32/s48/s46/s48/s56\n/s32/s48/s46/s48/s57\n/s32/s48/s46/s49/s47\n/s84 /s32/s40/s75/s41/s84\n/s109\nFIG. 6. Temperature dependence of resistivity for\nRbEu(Fe 1−xNix)4As4polycrystalline samples at zero field.\nThe top and bottom panels adopt a normalized scale relative\nto the resistivity values at 200 and 50 K, respectively. In\nthe bottom panel, the arrows labeled with TSDWpoint to the\nresistivity upturn that probably comes from a spin-density\nwave (SDW) transition. The left dashed line labeled with\nTmmarks the tiny kinks at which the Eu2+spins order\nferromagnetically.\nx≥0.05 show a resistivity upturn above Tsc, which is\nprobably due to a spin-density-wave (SDW) ordering in\nthe Fe sublattice. At x= 0.05, the hole concentration is\nreduced to nh= 0.15 holes per Fe atom, if assuming that\neverydopedNiatomdepletestwoholes. Intheprototype\nBa1−xKxFe2As2system, SDW order remains at nh=\n0.15 [46]. Note that the SDW transition temperature\nTSDWhere is much lower than expected. Furthermore,\nTSDWdecreases with xin the high doping regime. These\nresults can be ascribed to the Fe-site disorder mentioned\nabove. Similarly, a recovery of SDW by charge com-\npensation was reported in Ba 1−xKxFe1.86Co0.14As2[47]\nandEu 0.5K0.5(Fe1−xNix)2As2[48] systems. Herewenote\nthat, according to a recent report on Ni- and Co-doped\nCaKFe 4As4[49], the recovered SDW phase may have\nstrikingly different magnetic order.\nIn addition to the SDW-like anomaly above, Fig. 6(b)\nalsoshowsa veryslight(yet observable)kink at Tm. This/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s50/s48/s52/s48/s54/s48\n/s50 /s52 /s54/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52\n/s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s49/s53 /s50/s48 /s50/s53/s49/s46/s50/s49/s46/s51/s49/s46/s52\n/s40/s98/s41\n/s67 /s32/s40/s74/s32/s75/s45/s49\n/s32/s109/s111/s108/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s32/s66/s32/s40 /s120 /s32/s61/s32/s48/s46/s48/s55/s53/s41\n/s32/s65/s32/s40 /s120 /s32/s61/s32/s48/s46/s49/s41/s82/s98/s69/s117/s40/s70/s101\n/s49/s45 /s120/s78/s105\n/s120/s41\n/s52/s65/s115\n/s52/s68/s47/s84 /s32/s40/s74/s32/s75/s45/s50\n/s32/s109/s111/s108/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s68 /s32/s61/s32 /s67\n/s66/s32/s45/s32 /s67\n/s65\n/s68 /s32/s61/s32 /s67\n/s66/s32/s45/s32 /s67\n/s65\n/s84\n/s83/s68/s87/s68/s47/s84 /s32/s40/s74/s32/s75/s45/s50\n/s32/s109/s111/s108/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s42/s67 /s32/s40/s74/s32/s75/s45/s49\n/s32/s109/s111/s108/s45/s49\n/s41\n/s84 /s32/s40/s75/s41/s32/s120 /s32/s61/s32/s48\n/s32/s120 /s32/s61/s32/s48/s46/s48/s53\n/s32/s120 /s32/s61/s32/s48/s46/s48/s54/s82/s98/s69/s117/s40/s70/s101\n/s49/s45 /s120/s78/s105\n/s120/s41\n/s52/s65/s115\n/s52/s40/s97/s41\n/s67/s47/s84 /s32/s40/s74/s32/s75/s45/s50\n/s32/s109/s111/s108/s45/s49\n/s41\n/s84 /s32/s40/s75/s41\nFIG. 7. Temperature dependenceofspecific heat capacity( C)\nof the ferromagnetic superconductors (a) and the supercon-\nducting ferromagnet (b) in RbEu(Fe 1−xNix)4As4under zero\nmagnetic field. The superconducting transitions are marked\nby arrows. Note that the insets in top and bottom panels plot\nC/TandD/T, respectively, where Dstands for the specific-\nheat difference between samples of x= 0.075 and 0.1.\nis due to the reduction of magnetic scattering on the\ncharge carriers when Eu2+spins become ordered, akin to\nthe case in EuFe 2As2[11]. The tiny resistivity change\natTmseems to reflect weak interactions between Eu2+\nspins and the charge carriers.\nHeat capacity\nFigure 7 shows the temperature dependence of specific\nheat of RbEu(Fe 1−xNix)4As4under zero field, tracking\nfor superconducting transitions in the FSCs and SFMs.\nForx= 0, the specific-heat jump is clearly seen with\n∆C/Tsc= 208 mJ K−2mol−1at the superconducting\ntransition [32]. With increasing the Ni doping, the\nspecific-heat jump becomes unapparent. At x= 0.05\nand 0.06, for example, the ∆ C/Tscvalues are estimated\nto be∼50 and ∼20 mJ K−2mol−1, respectively.\nSimilar dramatic reduction in ∆ Cwas also observed\nin underdoped Ba 1−xKxFe2As2[50]. As a comparison,7\nthe underdoped Ba 0.77K0.23Fe2As2sample ( Tsc= 23\nK) shows a ∆ C/Tscvalue of ∼40 mJ K−2mol−1[50],\ncomparable to that of the x= 0.05 sample (with a\nsimilarTsc) in the present system. Note that the\nstrength of superconducting coupling varies with the\ndoping level [50], the former of which is reflected by\nthe dimensionless parameter, p= ∆C/γnTsc, whereγn\ndenotes the Sommerfeld coefficient in the normal state.\nThe drastic reduction in ∆ Cfor the underdoped samples\nis actually a consequence of the concurrent decrease in\nall the three factors: γn,Tscandp.\nIn the case of SFMs with Tsc< Tm, no specific-\nheat anomaly at Tsccan be directly seen from the\nraw data. Nevertheless, by subtraction of the specific\nheat between the superconducting ( x= 0.075) and\nnon-superconducting ( x= 0.1) samples, the specific-\nheat jump is still observable at 4.1 K (at which the\nresistivity drop to 6%), as shown in the upper-left\ninset of Fig. 7(b) (note that the kink at ∼5 K is\nprobably due to an antiferromagnetic transition from\nthe very small amount of Eu 3O4impurity [51]). Since\nthe Sommerfeld coefficient is greatly reduced in the\nunderdoped regime [50], the ∆ C/Tscvalue (∼10 mJ\nK−2mol−1) is actually appreciable, which supports the\nbulk nature of superconductivity. Additionally, from the\nspecific-heat difference shown in the lower inset, one can\nsee another anomaly at ∼36 K (albeit no anomaly is\nobservable again in the raw data), in accordance with\nthe resistivity upturn. This anomaly, if being intrinsic,\nshould be related to the recovered SDW transition.\nPhase diagram\nThe results above allow us to construct the phase\ndiagram in RbEu(Fe 1−xNix)4As4, which is displayed\nin Fig. 8. As usual, the bottom axis employs the\ndirect control parameter, i.e. the Ni content x. Since\nsubstitution of Fe2+(3d6) with Ni2+(3d8) doubly\ncompensates the self-doped holes, the expected hole\nconcentration is nh= 0.25−2x, which is also shown\nin the middle horizontal axis. With the hole depletion\nby Ni doping, Tscdecreases monotonically. Note that Tsc\ndecreases more rapidly for x≥0.05 where SDW order\nappears, suggesting competing nature between SC and\nSDW. SC disappears at x >0.08 or at nh≤0.09. One\nnotes that TSDWgoes down for x≥0.075.\nTo understand the possible role of disorder,\nwe compare with the electronic phase diagram of\nBa1−yKyFe2As2[52] which is disorder-free at the Fe site.\nOne sees that the Tscvalues in RbEu(Fe 1−xNix)4As4are\noverall lower in the doping area, and the critical hole\nconcentration (for appearance of SC) is significantly\nhigher. On the other hand, the TSDWvalues are even\nmuch lower, especially in the high doping regime. Both\nresults strongly suggest that the disorder by Ni doping\n/s70/s83/s67\n/s83/s70/s77/s32/s32/s32/s32/s82/s98/s69/s117/s40/s70/s101\n/s49/s45 /s120/s78/s105\n/s120/s41\n/s52/s65/s115\n/s52\n/s32/s84\n/s115/s99/s32/s32/s32 /s32/s84\n/s115/s99\n/s32/s84\n/s109/s69/s117\n/s32/s32 /s32/s84\n/s83/s68/s87\n/s70/s83/s67/s84 /s32/s40/s75/s41\n/s78/s105/s32/s67/s111/s110/s116/s101/s110/s116/s44/s32 /s120/s83/s67\n/s70/s77/s83/s68/s87/s48/s46/s53 /s48/s46/s52 /s48/s46/s51 /s48/s46/s50 /s48/s46/s49\n/s66/s97\n/s49/s45 /s121/s75\n/s121/s70/s101\n/s50/s65/s115\n/s50/s32/s91/s52/s52/s93 /s32\n/s32/s84\n/s115/s99\n/s32/s84\n/s83/s68/s87/s75/s32/s67/s111/s110/s116/s101/s110/s116/s44 /s32/s121\n/s48/s46/s50/s48 /s48/s46/s49/s53 /s48/s46/s49/s48/s72/s111/s108/s101/s45/s68/s111/s112/s105/s110/s103/s32/s76/s101/s118/s101/s108/s44/s32 /s110\n/s104\nFIG. 8. Superconducting and magnetic phase diagram in\nRbEu(Fe 1−xNix)4As4containing various electronic phases\nhighlighted with colors. Tρ\nscis the midpoint temperature\nof superconducting resistive transitions, and the error ba rs\ndenote the transition widths. Tχ\nscis the onset temperature\nof superconducting diamagnetic transitions. Other abbre-\nviations include: SC, superconductivity; SDW, spin-densi ty\nwave; FM, ferromagnet; FSC, ferromagnetic superconductor ;\nSFM, superconducting ferromagnet. For comparison, the\nphase lines of Ba 1−yKyFe2As2[52] are plotted using the top\naxis. Both horizontal axes share the same hole-doping level ,\nnh= 0.25−2x=y/2, as shown in the middle axis.\nplays an important role in suppressing SC as well as\nSDW.\nThen, how much do the Eu2+spins influence the\nTsc? As we emphasize in the Introduction, first of all,\nSC in RbEuFe 4As4is not suppressed at all. Secondly,\ntheTscvalues in RbEu(Fe 1−xNix)4As4are on average\n∼10 K lower than those in Ba 1−yKyFe2As2[52]. The\namount of Tscreduction is very close to that of Eu-\nfree Ba 1−yKyFe1.86Co0.14As2(compared in the same\nway with Ba 1−yKyFe2As2) [47], suggesting that disorder\nplays the dominant role for suppression of Tsc. In other\nwords, the Eu2+spins playarelativelyminor(if notnone\nat all) role in suppressing SC. Thirdly, the Tscvalues\nof RbEu(Fe 1−xNix)4As4are even higher than those in\nEu-diluted Eu 0.5K0.5(Fe1−xNix)2As2[48], the latter of\nwhich shows an antiferromagnetism (e.g., TN= 8.5 K\nforx= 0.08) for Eu2+spins which implies relatively less\ninfluence on SC. This comparison further corroborates\nthat Eu2+spins hardly suppress Tscin the 1144-type\nsystem.\nIn contrast to the dramatic changes in the states\nassociatedwith the Fe sublattice, the ferromagneticstate\nin the Eu sublattice remains unchanged. In particular,\nTmdoes not depend on the Ni doping, resulting in a8\ncrossing at x∼0.065 between the data lines of Tscand\nTm. According to the classification for materials with\ncoexistence of SC and FM [5, 9], the system changes\nfrom FSC to SFM at the crossing point. For x >0.08,\nthe system shows coexistence of Eu-spin FM and Fe-site\nSDW below 15 K. Although the sample with x= 0.125\n(correspondingto nh= 0)couldnotbesynthesizedowing\nto the solubility limit, one may expect by extrapolation\nthat this completely hole-compensated material would\nshow a similar behavior to that of the x= 0.1 sample.\nDiscussion\nIn the following, we discuss why both FSCs and\nSFMs exist in RbEu(Fe 1−xNix)4As4system. First of\nall, coexistence of SC and FM in Eu-containing 122-\ntype iron pnictides was tentatively explained in terms\nof Fe-3d multi-orbitals and robustness of SC [3]. On\nthe one hand, multi-3d-orbitals in the valence band\nallow both SC mainly from 3d yz/zxelectrons and Eu-\nspin FM via Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteractions through 3d x2−y2and 3d z2electrons. On the\nother hand, the intrinsic upper critical field H∗\nc2of the\nsuperconductors alike (e.g., an Eu-free analogue) could\nbe high enough to overcome the exchange field between\nEu2+spins and Cooper pairs.\nIn this context, the absence of SC for Tsc< Tm\nin EuFe 2(As1−xPx)2[25, 26], Eu(Fe 1−xNix)2As2[29],\nand Eu(Fe 1−xCox)2As2[27] can be attributed to the\nrelatively low H∗\nc2in relation with the lower Tsc. Here\nwe note that the Eu(Fe 0.81Co0.19)2As2single crystals\ngrown from Sn flux were reported to show SC with\nTsc< Tm[45]. The possible reason is that the Eu-\nspin exchange field could be reduced due to existence\nof defects in the Eu sublattice. One notes that the\nSn-flux-grown “Eu(Fe 0.82Co0.18)2As2” samples showed\nSDW order [53], indicating that they were actually in an\nunderdoped regime. The underdoped status with heavy\nCo-doping levels suggests the possibility of significant\nEu deficiencies. Besides, the flux Sn could also be\nincorporated into the Eu site [54]. Both factors lead to\ndilution in the Eu sublattice, such that the exchangefield\nthat breaks Cooper pairs may be reduced, which helps\nthe survival of SC.\nIn the 1144-type system of RbEu(Fe 1−xNix)4As4,\nthe Eu2+spins hardly suppress SC. Therefore, not\nonly do FSCs exist, but also SFMs occur in\nRbEu(Fe 1−xNix)4As4. Note that the internal field\ngenerated from the Eu-spin FM is about 4.5 kOe, being\nhigh enough to induce spontaneous vortices, yet not high\nenough to destroy SC.\nFinally, we comment on the magnetic interactions\nbetween Eu2+spins in RbEu(Fe 1−xNix)4As4. The\nexperimental fact is that neither Tmnor Θ change\nwith the Ni doping. This result contrasts to thechange in Tmfrom 20 to 16 K, accompanying with\nanantiferromagnetic-to-ferromagnetictransition,onlyby\n3% Ni doping in Eu(Fe 1−xNix)2As2[29]. The sensitivity\nto Ni doping dictates an indirect RKKY interaction\nwhose strength is proportional to cos(2 kFr)/r3, where\nkFis the Fermi vector and, ris the distance between\nEu2+spins. Conversely, the invariance of Tmand Θ\nagainstelectrondopingin RbEu(Fe 1−xNix)4As4suggests\nthat the RKKY interaction may not be the dominant\nmagnetic exchange interaction. This reminds us of the\nferromagnetic europium chalcogenides, EuO ( Tm= 69.2\nK) and EuS ( Tm= 16.6 K) [55], where there are no\nitinerant electrons for an indirect RKKY interaction.\nSo, the effective ferromagnetic couplings between Eu2+\nspins in RbEu(Fe 1−xNix)4As4may be due to the so-\ncalled d−f [56] and/or As −Eu−As superexchange inter-\nactions. Such exchange interactions naturally explain\nthe decoupling between Eu-4f and Fe-3d orbitals, which\nconversely sheds light on the mechanism of iron-based\nsuperconductivity. Future theoretical analyses and\ncalculations may help to clarify this issue.\nCONCLUSION\nIn summary, we have systematically studied\nthe magnetic and superconducting properties in\nRbEu(Fe 1−xNix)4As4(0≤x≤0.1). With the Ni\ndoping that introduces extra itinerant electrons, the self-\ndoped holes are gradually compensated. Resultantly, the\nsuperconducting transition temperature Tscdecreases\nrapidly, and superconductivity disappears at x∼0.08.\nThe hole depletion also brings the recovery of SDW\norder for x≥0.05. For the Eu sublattice, the Eu-\nspin ferromagnetism in RbEuFe 4As4remains, and its\nCurie temperature keeps unchanged. 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Dev. 14, 214 (1970)."    },    {        "title": "0801.3350v1.Quantum_Oscillations_of_Tunnel_Magnetoresistance_Induced_by_Spin_Wave_Excitations_in_Ferromagnet_Ferromagnet_Ferromagnet_Double_Barrier_Tunnel_Junctions.pdf",        "content": "arXiv:0801.3350v1  [cond-mat.mes-hall]  22 Jan 2008Quantum Oscillations of Tunnel Magnetoresistance Induced by\nSpin-Wave Excitations in Ferromagnet-Ferromagnet-Ferro magnet\nDouble Barrier Tunnel Junctions\nXi Chen, Qing-Rong Zheng, and Gang Su∗\nCollege of Physical Sciences, Graduate University of Chine se Academy of Sciences,\nP.O. Box 4588, Beijing 100049, China\nAbstract\nThepossibility of quantum oscillations of thetunnelcondu ctance and magnetoresistance induced\nby spin-wave excitations in a ferromagnet-ferromagnet-fe rromagnet double barrier tunnel junction,\nwhen the magnetizations of the two side ferromagnets are ali gned antiparallel to that of the mid-\ndle ferromagnet, is investigated in a self-consistent mann er by means of Keldysh nonequilibrium\nGreen function method. It has been found that owing to the s-d exchange interactions between\nconduction electrons and the spin density induced by spin ac cumulation in the middle ferromagnet,\nthe differential conductance and the TMR indeed oscillate wit h the increase of bias voltage, being\nconsistent with the phenomenon that is observed recently in experiments. The effects of magnon\nmodes, the energy levels of electrons as well as the molecula r field in the central ferromagnet on\nthe oscillatory transport property of the system are also di scussed.\nPACS numbers: 75.47.m, 73.63.Kv, 75.70.Cn\n∗Author to whom correspondence should be addressed. Email: gsu@ gucas.ac.cn\n1I. INTRODUCTION\nIn past decades, the spin-dependent transport properties in ma gnetic tunnel junctions\n(MTJs) have been extensively investigated both experimentally and theoretically, where a\ngreat progress has been made (see, e.g. Refs. [1, 2, 3, 4, 5] for r eviews). It has been unveiled\nthat owing to the conduction electron scatterings, the tunnel cu rrent through the MTJ is\nmodulated by the relative orientation of magnetizations, giving rise t o the so-called tunnel\nmagnetoresistance (TMR) effect. As the quality of tunnel junctio ns is being improved, a\nlargeTMR,whichisexpectedbypracticalapplications, hasbeenach ieved inseveral systems.\nOn the other hand, a reverse effect of TMR, coined as the spin tran sfer effect [6, 7], has also\nbeen proposed, which predicts that the orientation of magnetizat ion of free ferromagnetic\nlayer can be switched by passing a spin-polarized electrical current , and spin waves could\nalso beexcited. This latter effect has been confirmed experimentally ina number of systems.\nAlthough single barrier MTJs already show abundant characteristic s concerning the spin-\ndependent electrical transport, a double barrier magnetic tunne l junction (DBMTJ), in\nwhich the formation of quantum well states and the resonant tunn eling phenomenon are\ntheoretically anticipated, has also attracted much attention in rec ent years [8, 9, 10, 11, 12,\n13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In orde r to observe the coherent\ntunneling thru the DBMTJ, people have attempted to improve the ju nction quality to elim-\ninate the influences from the interface roughness and impurity sca ttering, and remarkable\nadvances have been achieved on this aspect.\nRecently, an unusual magnetotransport phenomenon in the ferr omagnet-ferromagnet-\nferromagnet (FM-FM-FM) DBMTJs was reported by Zeng et al. [27]. They observed that,\nwhen the magnetization of center (free) magnetic layer was antipa rallel (AP) to the mag-\nnetization of the two outer (pinned) magnetic layers, the conduct ance and TMR oscillate\ndistinctly with the applied bias voltage, while for the parallel (P) situat ion, no such os-\ncillation was seen. Unlike the previous oscillatory tunnel magnetores istance, this unusual\nphenomenon can neither be explained by Coulomb blockade effect sinc e the middle FM layer\nis continuous, the charge effect should be equal in P and AP configur ations and the charging\nenergy is negligibly small, nor be attributed to the resonant tunneling , because the observed\nperiod of oscillation is too small to account for the energy level spac ing of the quantum well\nstates. Considering that the conductance oscillation is asymmetric al for P and AP configu-\n2rations, and the energy level of the unusual phenomenon is the sa me as the typical energy of\na magnon, one may speculate that the unusual oscillation behavior c ould be induced by the\nmagnon-assisted tunneling [27]. This is because in the AP state, the n onequilibrium spin\ndensity, which is proportional to the applied bias, could be accumulat ed near the interfaces\ninthemiddle regiontoemit spin waves, andthemagnon-assisted tunn eling wouldcontribute\nto the conductance, while in the P state the spin wave emission is forb idden due to the spin\nangular momentum conservation, as discussed previously [28].\nAs there is no previous theoretical study devoting to the investiga tion on the possible\nquantum oscillations induced by spin wave excitations, in this paper, b y using the nonequi-\nlibrium Green function method, we shall examine theoretically the abo ve-mentioned idea\nby studying the possibility of magnon-assisted tunneling in the FM-FM -FM DBMTJ, and\nexplore whether the magnon-assisted tunneling could really cause t he oscillations of the\ndifferential conductance and TMR with the applied bias voltage.\nThe rest of this paper is organized as follows: In Sec. II, a model is p roposed. The\ntunnel current and relevant Green functions are obtained in term s of the nonequilibrium\nGreen function technique in Sec. III. In Sec. IV, the transport p roperties of the system are\nnumerically investigated, and some discussions are presented. Fina lly, a brief summary is\ngiven in Sec. V.\nII. MODEL\nLet us consider a FM-FM-FM DBMTJ with three FM layers separated b y two thin\ninsulating films. Suppose that the left (L) and right (R) FM electrode s with magnetizations\naligned parallel are applied by bias voltages −V/2 andV/2, respectively. The magnetization\nof the middle FM layer is presumed to be antiparallel to those of the L a nd R electrodes so\nthat spin waves can be emitted in the middle FM layer because of spin ac cumulation. The\nschematic layout of this system is depicted in the inset of Fig. 1(a). T he Hamiltonian of\nthe system reads\nH=HL+HR+HC+HLC+HCR, (1)\n3with\nHα=/summationdisplay\nkασεkασa+\nkασakασ,(α=L,R) (2)\nHC=/summationdisplay\nkσεkσc+\nkσckσ+/summationdisplay\nq/planckover2pi1ωqb+\nqbq, (3)\nHLC=/summationdisplay\nkLkσTd\nkLk(a+\nkLσckσ+h.c.)\n+1√\nN/summationdisplay\nkLkqTJ\nkLkqS(q)(a+\nkL↑ck↑−a+\nkL↓ck↓+c+\nk↑akL↑−c+\nk↓akL↓)\n+1√\nN/summationdisplay\nkLkqTJ\nkLkq√\n2S(a+\nkL↑ck↓b+\nq+c+\nk↑akL↓b+\nq+a+\nkL↓ck↑bq+c+\nk↓akL↑bq),(4)\nHCR=/summationdisplay\nkRkσTd\nkRk(a+\nkRσckσ+h.c.)\n+1√\nN/summationdisplay\nkRkqTJ\nkRkqS(q)(a+\nkR↑ck↑−a+\nkR↓ck↓+c+\nk↑akR↑−c+\nk↓akR↓)\n+1√\nN/summationdisplay\nkRkqTJ\nkRkq√\n2S(a+\nkR↑ck↓b+\nq+c+\nk↑akR↓b+\nq+a+\nkR↓ck↑bq+c+\nk↓akR↑bq),(5)\nwhereakασandckσare annihilation operators of electrons with momentum kand spin σin\ntheαelectrode and in the middle FM layer, respectively, εkασ=εkα−σMα−eVαwithεkα\nthe single-electron energy and Mαthe molecular field in the αelectrode, εkσ=εk−σM\nwithεkthe single-electron energy and Mthe molecular field in the middle FM layer, bq\nis the annihilation operator of magnon with momentum qin the middle region, /planckover2pi1ωqis the\nmagnon energy, N=/summationtext\nq/an}bracketle{tns\nq/an}bracketri}htwithns\nq=b+\nqbqis the number of magnons, S(q) =S−ns\nqwhere\nS= 1/2 is the spin of electron, Td\nkαkare tunneling matrix elements of electrons between the\nαelectrode and middle FM layer, TJ\nkαkqare coupling matrix elements between the electrons\ninαelectrode and magnons in the middle FM region.\nIt is noting that HLC(HCR) describes the coupling between electrons in the L (R) elec-\ntrode and electrons as well as magnons in the central FM region, wh ere the terms containing\nTJ\nkαkqin Eqs. (4) and (5) are due to the s-dexchange interactions[28]. Without loss of gen-\nerality, we further assume TJ\nkLkq=γTd\nkLkin the following discussions.\n4III. TUNNEL CURRENT AND GREEN FUNCTIONS\nA. Tunnel Current\nStarting from Eq. (1), after some cubersome but straightforwa rd calculations, one may\nobtain the tunnel electrical current I\nI=IL↑+IL↓, (6)\nIL↑(t) =2e\n/planckover2pi1ℜe[/summationdisplay\nkLkTd\nkLkG<\nk↑kL↑(t,t)+1√\nN/summationdisplay\nkLkqTJ\nkLkq(S−/an}bracketle{tns\nq/an}bracketri}ht)G<\nk↑kL↑(t,t)\n+/radicalbigg\n2S\nN/summationdisplay\nkLkqTJ\nkLkqGq<\nk↓kL↑(t,t)],\nIL↓(t) =2e\n/planckover2pi1ℜe[/summationdisplay\nkLkTd\nkLkG<\nk↓kL↓(t,t)−1√\nN/summationdisplay\nkLkqTJ\nkLkq(S−/an}bracketle{tns\nq/an}bracketri}ht)G<\nk↓kL↓(t,t)\n+/radicalbigg\n2S\nN/summationdisplay\nkLkqTJ\nkLkqGq<\nk↑kL↓(t,t)],\nwhere the lesser Green functions are defined as\nG<\nkσkLσ(t,t′) =i/an}bracketle{ta+\nkLσ(t′)ckσ(t)/an}bracketri}ht, (7)\nGq<\nk↓kL↑(t,t′) =i/an}bracketle{ta+\nkL↑(t′)b+\nq(t′)ck↓(t)/an}bracketri}ht, (8)\nGq<\nk↑kL↓(t,t′) =i/an}bracketle{ta+\nkL↓(t′)bq(t′)ck↑(t)/an}bracketri}ht. (9)\nIt should be remarked that in the above derivations, we have made d ecoupling approxima-\ntions for the terms containing ns\nqto simplify the calculations. From these above equations,\none may see that to get the tunnel electrical current, the lesser Green functions must be ob-\ntained. In the following, we shall employ Keldysh’s nonequilibrium Green function method\nto get all self-consistent equations to determine the lesser Green functions. As the lesser\nGreen function is closely related to the retarded and advanced Gre en functions according\n5to Keldysh formalism, the relevant retarded and advanced Green f unctions of electrons and\nmagnons should be first calculated.\nAccordingly, the differential tunnel conductance ( G) is obtained by G(V) =dI(V)/dV,\nand the TMR can be calculated by TMR= (1−G↑↓/G↑↑) whereG↑↓(G↑↑) is the differen-\ntial conductance when the magnetizations of the middle FM and the s ide FM are aligned\nantiparallel (parallel).\nB. Green Functions of Electrons\nLet us define useful retarded Green functions for electrons as\nGr\nkσk′σ(t,t′) =−iθ(t−t′)/an}bracketle{t{ckσ(t),c+\nk′σ(t′)}/an}bracketri}ht, (10)\nGr\nkσkασ(t,t′) =−iθ(t−t′)/an}bracketle{t{ckσ(t),a+\nkασ(t′)}/an}bracketri}ht, (11)\nGr(q)\nk↑kα↓(t,t′) =−iθ(t−t′)/an}bracketle{t{ck↑(t),a+\nkα↓(t′)bq(t′)}/an}bracketri}ht, (12)\nGr(q)\nk↑k′↓(t,t′) =−iθ(t−t′)/an}bracketle{t{ck↑(t),c+\nk′↓(t′)bq(t′)}/an}bracketri}ht, (13)\nGr(q)\nk↓kα↑(t,t′) =−iθ(t−t′)/an}bracketle{t{ck↓(t),a+\nkα↑(t′)b+\nq(t′)}/an}bracketri}ht, (14)\nGr(q)\nk↓k′↑(t,t′) =−iθ(t−t′)/an}bracketle{t{ck↓(t),c+\nk′↑(t′)b+\nq(t′)}/an}bracketri}ht. (15)\nIn terms of the equation of motion, after a tedious calculation, up t o the third-order of\nGreen functions, we get the following equations\n(ε−εkα↑)Gr\nk↑kα↑(ε) = [Td+TJ\n√\nN/summationdisplay\nq(S−/an}bracketle{tns\nq/an}bracketri}ht)]/summationdisplay\nk′Gr\nk↑k′↑(ε)\n+/radicalbigg\n2S\nNTJ/summationdisplay\nk′qGr(q)\nk↑k′↓(ε),\n6(ε−εk′↑)Gr\nk↑k′↑(ε) = [Td+TJ\n√\nN/summationdisplay\nq(S−/an}bracketle{tns\nq/an}bracketri}ht)]/summationdisplay\nα/summationdisplay\nkαGr\nk↑kα↑(ε)\n+/radicalbigg\n2S\nNTJ/summationdisplay\nα/summationdisplay\nkαqGr(q)\nk↑kα↓(ε),\n(ε−εkα↓+/planckover2pi1ωq)Gr(q)\nk↑kα↓(ε) = [Td−TJ\n√\nN/summationdisplay\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)+TJ\n√\nN/an}bracketle{tnkα↓/an}bracketri}ht]/summationdisplay\nk′Gr(q)\nk↑k′↓(ε)\n+/radicalbigg\n2S\nNTJ(/an}bracketle{tns\nq/an}bracketri}ht+/an}bracketle{tnkα↓/an}bracketri}ht)/summationdisplay\nk′Gr\nk↑k′↑(ε),\n(ε−εk′↓+/planckover2pi1ωq)Gr(q)\nk↑k′↓(ε) =/braceleftBigg\nTd−TJ\n√\nN[/summationdisplay\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)−/summationdisplay\nk′′/an}bracketle{tc+\nk′↓ck′′↓/an}bracketri}ht]/bracerightBigg/summationdisplay\nα/summationdisplay\nkαGr(q)\nk↑kα↓(ε)\n+/radicalbigg\n2S\nNTJ(/an}bracketle{tns\nq/an}bracketri}ht+/summationdisplay\nk′′/an}bracketle{tc+\nk′↓ck′′↓/an}bracketri}ht)/summationdisplay\nα/summationdisplay\nkαGr\nk↑kα↑(ε),\n(ε−εkα↓)Gr\nk↓kα↓(ε) = [Td−TJ\n√\nN/summationdisplay\nq(S−/an}bracketle{tns\nq/an}bracketri}ht)]/summationdisplay\nk′Gr\nk↓k′↓(ε)\n+/radicalbigg\n2S\nNTJ/summationdisplay\nk′qGr(q)\nk↓k′↑(ε),\n(ε−εk′↓)Gr\nk↓k′↓(ε) = [Td−TJ\n√\nN/summationdisplay\nq(S−/an}bracketle{tns\nq/an}bracketri}ht)]/summationdisplay\nα/summationdisplay\nkαGr\nk↓kα↓(ε)\n+/radicalbigg\n2S\nNTJ/summationdisplay\nα/summationdisplay\nkαqGr(q)\nk↓kα↑(ε),\n(ε−εkα↑−/planckover2pi1ωq)Gr(q)\nk↓kα↑(ε) = [Td+TJ\n√\nN/summationdisplay\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)+TJ\n√\nN/an}bracketle{tnkα↑/an}bracketri}ht]/summationdisplay\nk′Gr(q)\nk↓k′↑(ε)\n+/radicalbigg\n2S\nNTJ(/an}bracketle{tns\nq/an}bracketri}ht+1−/an}bracketle{tnkα↑/an}bracketri}ht)/summationdisplay\nk′Gr\nk↓k′↓(ε),\n(ε−εk′↑−/planckover2pi1ωq)Gr\nk↓k′↑(ε) =/braceleftBigg\nTd+TJ\n√\nN[/summationdisplay\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)+/summationdisplay\nk′′/an}bracketle{tc+\nk′↑ck′′↑/an}bracketri}ht]/bracerightBigg/summationdisplay\nα/summationdisplay\nkαGr(q)\nk↓kα↑(ε)\n+/radicalbigg\n2S\nNTJ(/an}bracketle{tns\nq/an}bracketri}ht+1−/summationdisplay\nk′′/an}bracketle{tc+\nk′↑ck′′↑/an}bracketri}ht)/summationdisplay\nα/summationdisplay\nkαGr\nk↑kα↑(ε),\n7where we have presumed, for simplicity, the coupling matrix elements Td\nkLkandTJ\nkLkqin-\ndependent of momentum by considering that only those electrons n ear the Fermi surface\nparticipate in the transport process, and nkασ=a+\nkασakασ.\nFrom these equations, the required Green functions can be obtain ed self-consistently.\nOn the other hand, the lesser self-energy Σ<can be approximated by Ng’s ansatz [29]:\nΣ<= Σ<\n0(Σr\n0−Σa\n0)−1(Σr−Σa), where Σr−Σa=Ga−1−Gr−1, Σr\n0and Σ<\n0are given by the\nfollowing equations\n\nΣr\n0↑↑(ε) Σr\n0↓↑(ε)\nΣr\n0↑↓(ε) Σr\n0↓↓(ε)\n=\n−iΓL↑\n2−iΓR↑\n20\n0−iΓL↓\n2−iΓR↓\n2\n, (16)\n\nΣ<\n0↑↑(ε) Σ<\n0↓↑(ε)\nΣ<\n0↑↓(ε) Σ<\n0↓↓(ε)\n=\niΓL↑f(ε−eV\n2)+iΓR↑f(ε+eV\n2) 0\n0 iΓL↓f(ε−eV\n2)+iΓR↓f(ε+eV\n2)\n,\nwhere Γ ασ(ε) is the linewidth function defined by Γ ασ(ε) = 2π/summationtext\nkαρσ(kα)/vextendsingle/vextendsingleTd\nkLk/vextendsingle/vextendsingle2withρσ(kα)\nthe density of states of electrons with momentum kαand spin σin theαth FM electrode,\nandf(ε) is the Fermi distribution function. By means of G<=GrΣ<Ga, the lesser Green\nfunctions can be procured.\nC. Green Functions of Magnons\nAs the number of magnons, N, enters into the formalism, we need to obtain the Green\nfunctions of magnons to determine Nself-consistently. Define the retarded Green function\nof magnons as\nGr\nqq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),b+\nq(t′)]/an}bracketri}ht. (17)\nBy using the equation of motion, we have\n(ε−/planckover2pi1ωq)Gr\nqq(ε) = 1+/radicalbigg\n2S\nN/summationdisplay\nα/summationdisplay\nkαkTJ\nkαkq[Gr(1)\nkq(ε)+Gr(2)\nkq(ε)]\n+1√\nN/summationdisplay\nα/summationdisplay\nkαkTJ\nkαkq[−Gr(3)\nkq(ε)+Gr(5)\nkq(ε)−Gr(4)\nkq(ε)+Gr(6)\nkq(ε)],(18)\n8whereGr(i)\nkq(ε) (i= 1,···,5) are the Fourier transforms of the Green functions defined as\nbelow\nGr(1)\nkq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),a+\nkα↓(t′)ck↑(t′)]/an}bracketri}ht,\nGr(2)\nkq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),c+\nk↓(t′)akα↑(t′)]/an}bracketri}ht,\nGr(3)\nkq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),b+\nq(t′)a+\nkα↑(t′)ck↑(t′)]/an}bracketri}ht,\nGr(4)\nkq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),b+\nq(t′)c+\nk↑(t′)akα↑(t′)]/an}bracketri}ht,\nGr(5)\nkq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),b+\nq(t′)a+\nkα↓(t′)ck↓(t′)]/an}bracketri}ht,\nGr(6)\nkq(t,t′) =−iθ(t−t′)/an}bracketle{t[bq(t),b+\nq(t′)c+\nk↓(t′)akα↓(t′)]/an}bracketri}ht.\nByusing repeatedly theequation ofmotion, andmaking appropriate cut-offapproximations,\nup to the second order, we have\nGr(1)\nkq(ε) =−TJ/radicalbigg\n2S\nN/an}bracketle{tnkα↓/an}bracketri}ht−/summationtext\nk′/an}bracketle{tc+\nk′↑ck↑/an}bracketri}ht\nε−εkα↓+εk↑Gr\nqq(ε), (19)\nGr(2)\nkq(ε) =TJ/radicalbigg\n2S\nN/an}bracketle{tnkα↑/an}bracketri}ht−/summationtext\nk′/an}bracketle{tc+\nk↓ck′↓/an}bracketri}ht\nε+εkα↑−εk↓Gr\nqq(ε), (20)\nGr(3)\nkq(ε) =−{TJ√\nN(1−/an}bracketle{tnkα↑/an}bracketri}ht)/summationtext\nk′/an}bracketle{tc+\nk′↑ck↑/an}bracketri}ht−[Td+TJ√\nN/summationtext\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)](/summationtext\nk′/an}bracketle{tc+\nk′↑ck↑/an}bracketri}ht−/an}bracketle{tnkα↑/an}bracketri}ht)\nε−/planckover2pi1ωq−εkα↑+εk↑}Gr\nqq(ε),\n(21)\nGr(4)\nkq(ε) =−{TJ√\nN(1−/summationtext\nk′/an}bracketle{tc+\nk↑ck′↑/an}bracketri}ht)/an}bracketle{tnkα↑/an}bracketri}ht+[Td+TJ√\nN/summationtext\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)](/summationtext\nk′/an}bracketle{tc+\nk↑ck′↑/an}bracketri}ht−/an}bracketle{tnkα↑/an}bracketri}ht)\nε−/planckover2pi1ωq+εkα↑−εk↑}Gr\nqq(ε),\n(22)\nGr(5)\nkq(ε) ={TJ√\nN(1−/an}bracketle{tnkα↓/an}bracketri}ht)/summationtext\nk′/an}bracketle{tc+\nk′↓ck↓/an}bracketri}ht+[Td−TJ√\nN/summationtext\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)](/summationtext\nk′/an}bracketle{tc+\nk′↓ck↓/an}bracketri}ht−/an}bracketle{tnkα↓/an}bracketri}ht)\nε−/planckover2pi1ωq−εkα↓+εk↓}Gr\nqq(ε),\n(23)\nGr(6)\nkq(ε) ={TJ√\nN(1−/summationtext\nk′/an}bracketle{tc+\nk↓ck′↓/an}bracketri}ht)/an}bracketle{tnkα↓/an}bracketri}ht−[Td−TJ√\nN/summationtext\nq′(S−/an}bracketle{tns\nq′/an}bracketri}ht)](/summationtext\nk′/an}bracketle{tc+\nk↓ck′↓/an}bracketri}ht−/an}bracketle{tnkα↓/an}bracketri}ht)\nε−/planckover2pi1ωq+εkα↓−εk↓}Gr\nqq(ε).\n(24)\n9The number of magnons can thus be obtained by the spectral theo rem\nN=/summationdisplay\nq/an}bracketle{tns\nq/an}bracketri}ht=/summationdisplay\nqℑm/integraldisplaydε\n2πG<\nqq(ε) =/summationdisplay\nq/integraldisplaydε\n2πfs(ε)[Gr\nqq(ε)−Ga\nqq(ε)],(25)\nwherefs(ε) is the Bose distribution function.\nTogetphysical quantitiesunder interest, allaboveequations sho uldbenumerically solved\nin a self-consistent manner.\nIV. RESULTS AND DISCUSSIONS\nTo proceed the numerical calculations, we need to make some assum ptions. Since the\nnumber of the above self-consistent equations nonlinearly increas es with increasing the num-\nber of wave vectors of electrons and the number of spin-wave mod es in the middle FM\nregion, which makes the calculations too complicated to perform, fo r the sake of simplic-\nity but without losing the generality, we shall only consider the situat ions where both the\nnumbers of kandqtaken in the following calculations are not so large that the numeri-\ncal calculations can be readily proceeded. This is plausible, because t he magnon-assisted\ntransport property mainly depends on the low-lying quantum well st ates of electrons in the\nmiddle FM, and only the lower modes of spin waves are easy to emit[6], lea ding to small\nenergy levels of magnons[27]. Besides, considering that only those e lectrons near the Fermi\nsurface participate in the tunneling process, we may take εkσ≈εkF−σM, denoted by\nε↑andε↓for spin up and down electrons, respectively. In addition, we suppo se that the\ntwo side FM electrodes are made of the same materials, i.e., ML=MR,PL=PR=P,\nwherePL(R)= [ΓL(R)↑−ΓL(R)↓]/[ΓL(R)↑+ ΓL(R)↓] is the polarization of the left (right) FM\nlayer. Then, the linewidth function can be written as Γ L↑,↓= ΓR↑,↓= Γ0(1±P), where\nΓ0= ΓL(R)↑(P= 0) = Γ L(R)↓(P= 0) will be taken as an energy scale. In the following, we\nwill take P= 0.7,kBT= 0.04Γ0,I0=eΓ0\n/planckover2pi1andG0=e2\n/planckover2pi1will be taken as scales for the tunnel\ncurrent and the differential conductance, respectively.\nA. Effect of Magnon-Assisted Tunneling\nIn order to study whether the quantum oscillations of the conduct ance and TMR ob-\nserved in the FM-FM-FM tunnel junction are induced by spin-wave e xcitations owing to\n10spin accumulation, let us first examine the bias dependence of the tr ansport properties by\nconsidering the effect of magnon-assisted tunneling in the AP state .\nFor given energy levels of the electrons and magnons in the middle FM r egion, the bias\ndependent tunnel current ( I/I0), thedifferential conductance ( G/G0) andTMRfordifferent\nγ(=TJ/Td)areshowninFig. 1. Itisobservedthatthetunnelcurrentincrea sesnonliniearly\nwith increasing the bias voltage. The twisted behavior of I/I0forγ/ne}ationslash= 0, presented in Fig.\n1(a), comes from the magnon-assisted tunneling, as no such beha viors are found for γ= 0 in\neither AP or P state. This can be clearly seen from the bias dependen ce of the differential\nconductance, shown in Fig. 1(b), where the peaks and dips appear for appropriate γ.\nCorrespondingly, the TMR shows oscillating behavior with increasing t he bias voltage, as\ndemonstrated in Fig. 1(c). This observation manifests that the qu antum oscillations of\nthe conductance and TMR in the FM-FM-FM tunnel junction can be c aused by spin-wave\nexcitations, because when we turn off the effect of spin-wave excit ations, the oscillating\nbehaviors of GandTMRdisappear. Note that the peak in G/G0and one dip and one\npeak inTMRare from the quantum resonant tunneling of electrons. A larger TM R can be\nobtained for large γ, and due to the s-d exchange interactions that could lead to spin-fl ip\nscatterings, the TMR can be negative, as presented in Fig. 1(c).\nIt should be remarked that the above oscillating behaviors of GandTMRappear only\nwhenγis in a suitable range, say, when γis too small, no oscillations can be observed, while\nγis too large, the self-consistent equations have no solutions.\nThe reason for the appearance of oscillations is that, when the app lied bias voltage\nexceeds a certain value, the non-equilibrium spin density can be accu mulated in the middle\nFM region, and spin waves can be excited. When polarized electrons f rom the left FM layer\ntunnel into the central FM layer, they are subject to scattering s from not only the polarized\nelectrons in the central region but also the spin waves from the acc umulation owing to s-d\nexchange interactions. It is possible that the electrons may turn t heir spin directions by\nemission and absorption of magnons, leading to that the tunnel con ductance and TMR\noscillate under a combination of effects of magnon-assisted tunnelin g as well as quantum\nresonant tunneling[23] through the quantum well states in the cen tral FM region.\nApparently, the polarized electrical current can excite magnons, while these magnons\nparticipate in the tunneling process and in turn influence the tunnel current. The number of\nmagnonsmustbeestimatedself-consistently. Asanexample, inFig. 2, thebiasdependent of\n11/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s50 /s51 /s52 /s53 /s54 /s55 /s56/s45/s49/s48/s49/s50/s84/s77/s82/s32/s32/s71/s47/s71\n/s48\n/s101/s86/s47\n/s48\n/s32/s32\n/s40/s98/s41/s32\n/s32\n/s32 /s61/s48/s46/s49/s53/s32/s40/s65/s80/s41\n/s32/s32/s48/s46/s49/s55/s53/s32/s40/s65/s80/s41\n/s32/s32/s48/s46/s50/s32/s40/s65/s80/s41\n/s32/s32/s48/s46/s48/s32/s40/s65/s80/s41\n/s32/s32/s48/s46/s48/s32/s40/s80/s41/s73/s47/s73\n/s48\n/s32/s32\n/s40/s97/s41\n/s70/s77 /s70/s77 /s70/s77/s45/s86/s47/s50/s86/s47/s50/s76 /s67 /s82/s32\n/s32\n/s32/s32\n/s40/s99/s41\nFIG. 1: (Color Online) Bias dependenceof (a) the tunnel curr ent I, (b) the differential conductance\nG, and (c) TMR for the different γ, where/planckover2pi1ωq= 0.5Γ0,ε↑= 3.0Γ0andε↓= 2.0Γ0.\nthe number of magnons Nfor some parameters is presented. We may see that the number of\nmagnons oscillates with the bias voltage, which could be the main reaso n for the oscillatory\ntransport property of the system.\nB. Effect of Magnon Modes\nThere are a number of factors including magnon energy /planckover2pi1ωqthat can affect the transport\nbehavior of the FM-FM-FM tunnel junction. The bias dependence o f the current, the\ndifferent conductance and the TMR for different magnon energies is shown in Fig. 3. It can\nbe found that with increasing /planckover2pi1ωq, apart from some quantitative changes of peak positions\n12/s50 /s51 /s52 /s53 /s54 /s55 /s56/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s32\n/s32/s78\n/s32/s32\n/s101/s86/s47\n/s48/s61/s48/s46/s50/s44 /s61/s51/s46/s48\n/s48/s44 /s61/s50/s46/s53\n/s48\n/s119\n/s113/s61/s48/s46/s53\n/s48\nFIG. 2: (Color Online) Bias dependence of the number of magno ns N in the AP state.\nand amplitudes, there are not much qualitative changes of the curr ent, conductance and\nTMR. Therefore, for a given magnon mode, the magnon energy doe s not affect qualitatively\nthe transport oscillating behavior of the system. It is noted that, while some positions of\nthe peaks and dips of GandTMRare influenced by the magnon energy, the others are not.\nThis observation indicates that the magnon energy is only one of fac tors determining the\npositions and amplitudes of the peaks.\nFor the sake of simplicity, in the aforementioned analysis we have ado pted a single spin-\nwave mode. Whether are the transport properties of the system much affected qualitatively\nwhen we take more spin-wave modes into account? The answer is pre sented in Fig. 4,\nwhere we have taken two and three spin-wave modes to get the tun nel current, differential\nconductance and TMR. For a comparison, we have also included the c ase with single mode.\nOne may see that at low biases, the magnon modes do not have so muc h effect on the\nbehaviors of I/I0,G/G0andTMR, but at higher voltages the magnitudes of the current,\ndifferential conductance as well as TMR change somewhat remarka bly. This is because\nat low biases the spin accumulation effect is small, and the interaction b etween tunneling\nelectrons and spin-wave modes is weak, leading to the transport pr operties less influenced;\nat large biases the spin accumulation effect becomes more pronounc ed, and the interactions\nbetween electrons and magnons are strong, the transport beha viors of the system are thus\naltered quantitatively. Note that the round peaks and dips in the cu rves of the tunnel\ncurrent shown in Fig. 4 may come from combinations of the magnon-a ssisted tunneling as\nwell as the quantum resonant tunneling.\n13/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s50 /s51 /s52 /s53 /s54 /s55 /s56/s45/s49/s48/s49/s50/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s84/s77/s82/s71/s47/s71\n/s48/s73/s47/s73\n/s48/s32\n/s32\n/s101/s86/s47\n/s48\n/s32/s32\n/s32 /s119\n/s113/s61/s48/s46/s52\n/s48/s32/s32/s40/s65/s80/s41\n/s32/s32/s48/s46/s53\n/s48/s32/s32/s40/s65/s80/s41\n/s32/s32/s48/s46/s54\n/s48/s32/s32/s40/s65/s80/s41/s40/s97/s41/s32\n/s32\n/s32/s32\n/s40/s99/s41/s32\n/s32\n/s32/s32\n/s40/s98/s41\nFIG. 3: (Color Online) Bias dependenceof (a) the tunnel curr ent I, (b) the differential conductance\nG and (c) TMR for the different /planckover2pi1ωq, whereγ= 0.2,ε↑= 3.0Γ0andε↓= 2.0Γ0.\nC. Effect of Electron Level in the Middle FM Layer\nThe bias dependence of the current, the differential conductanc e and the TMR for dif-\nferent energy levels ( εkF≡ε) of the electrons in the middle FM region is given in Fig. 5. It\ncan be observed that with lifting the energy levels of electrons in the central FM layer, the\npeak and dip positions of I/I0,G/G0andTMRchange dramatically with increasing the\nbias voltage, while the shapes of the curves retain quite similar for diff erent energy levels\nof electrons in the central region. This signifies that the energy lev els of electrons in the\nmiddle FM region do not affect the oscillating behavior itself of the tran sport properties,\n14/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55\n/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s50 /s51 /s52 /s53 /s54 /s55 /s56/s45/s49/s48/s49/s50/s51/s32/s32/s32/s32 /s119\n/s113/s61/s48/s46/s52Γ\n/s48/s32/s40/s65/s80/s41\n/s32 /s119\n/s113/s49/s61/s48/s46/s52Γ\n/s48/s44/s32 /s119\n/s113/s50/s61/s48/s46/s53Γ\n/s48/s32/s40/s65/s80/s41\n/s32 /s119\n/s113/s49/s61/s48/s46/s52Γ\n/s48/s44/s32 /s119\n/s113/s51/s61/s48/s46/s53Γ\n/s48/s44\n/s119\n/s113/s50/s61/s48/s46/s54Γ\n/s48/s32/s40/s65/s80/s41/s73/s47/s73\n/s48\n/s32/s32\n/s40/s97/s41\n/s101/s86/s47Γ\n/s48/s32/s32 /s84/s77/s82/s71/s47/s71\n/s48\n/s32/s40/s98/s41\n/s32/s32 /s32/s32\n/s40/s99/s41\nFIG. 4: (Color Online) Bias dependenceof (a) the tunnel curr ent I, (b) the differential conductance\nG, and (c) TMR for different spin-wave modes, where γ= 0.2,ε↑= 3.0Γ0andε↓= 2.0Γ0.\nbut affect the positions of oscillating peaks and dips. This can be unde rstandable, because\nthe transport behavior of electrons are mainly determined by the s catterings from polarized\nelectrons and magnons to that the electrons are subject, when t he energy levels of electrons\nin the middle region are promoted, the resonant energies in magnon- assisted and quantum\nresonant tunneling processes become different, resulting in the be haviors shown in Fig. 5.\nD. Effect of Molecular Field in the Middle FM Layer\nThe molecular fields of the middle FM layer could also have effect on the t ransport\nproperties of the system. The bias dependence of the current, d ifferential conductance\nand TMR for different molecular fields of the central FM region is show n in Fig. 6. It\n15/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57/s45/s49/s48/s49/s50/s32/s32/s71/s47/s71\n/s48\n/s32/s32/s40/s98/s41/s32/s32/s73/s47/s73\n/s48\n/s101/s86/s47\n/s48\n/s32/s32\n/s40/s97/s41/s32\n/s32/s84/s77/s82/s32 /s61/s50/s46/s53\n/s48/s32\n/s32 /s61/s51/s46/s48\n/s48/s32/s32\n/s32 /s61/s51/s46/s53\n/s48/s32\n/s32/s32\n/s40/s99/s41\nFIG. 5: (Color Online) Bias dependenceof (a) the tunnel curr ent I, (b) the differential conductance\nG, and (c) TMR in the AP state for different energy levels of elec trons in the middle FM region,\nwhereγ= 0.2,/planckover2pi1ωq= 0.5Γ0andM= 1.0Γ0.\nis seen that as the molecular field increases, the magnitude of the tu nnel current becomes\nsmaller, and the oscillations of the differential conductance as well a s TMR become more\napparent, where not only the number but also the positions of the o scillating peaks change\nwith increasing the molecular fields. This fact suggests that the leve l spacing between the\nmajorty and minority subbands of electrons in the central region p lays an important role in\nthe oscillating behaviors of the differential conductance and TMR in t he present system.\n16/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57/s45/s49/s48/s49/s50/s32/s32/s71/s47/s71\n/s48\n/s32/s32/s40/s98/s41/s32/s32/s73/s47/s73\n/s48\n/s101/s86/s47\n/s48\n/s32/s32\n/s40/s97/s41/s32\n/s32/s84/s77/s82/s32/s32/s77/s61/s48/s46/s50/s53\n/s48/s32\n/s32/s32/s77/s61/s48/s46/s53\n/s48/s32\n/s32/s32/s77/s61/s48/s46/s55/s53\n/s48/s32\n/s32/s32\n/s40/s99/s41\nFIG. 6: (Color Online) Bias dependenceof (a) the tunnel curr ent I, (b) the differential conductance\nG, and (c) TMR for different molecular fields of the middle FM lay er, where γ= 0.2,ε= 2.5Γ0\nand/planckover2pi1ωq= 0.5Γ0.\nV. SUMMARY\nInsummary, wehave probedthepossibility ofquantum oscillations of thedifferential con-\nductance and TMR in the FM-FM-FM tunnel junction in the AP state. By self-consistently\ntaking the s-d exchange interactions between conduction electro ns and the nonequilibrium\nspin density induced by spin accumulation in the middle FM layer into acco unt, we have\nfound that the differential conductance and TMR indeed oscillate wit h increasing the bias\nvoltage, thereby theoretically confirming qualitatively the inferrer and experimental results\n17presented in Ref. [27]. It has been unveiled that the average numbe r of magnons oscillates\nwith the bias, which could be the main reason for the oscillations of the conductance and\nTMR of the system. When we turn off the s-d exchange interactions , i.e., taking γ= 0,\nno oscillations of the conductance and TMR were observed, showing that the oscilations\nare indeed caused by the spin-wave excitations induced by spin accu mulations. In the P\nstate, owing to the absence of spin accumulation[21, 22], no oscillatio ns of the conductance\nand TMR with the bias can be found. We have also investigated the effe cts of the magnon\nmodes, the energy levels of electrons as well as the molecular field in t he middle FM region,\nand found that in spite of changes of the positions and amplitudes of the oscillating peaks\nand dips, the oscillatory behavior of the transport properties is no t qualitatively affected.\nWe anticipate that our findings could offer clues for better underst anding the experimental\nobservation presented in Ref. [27].\nFinally, we would like to remark that our preceding discussions could be applicable to\nthe system with a magnetic quantum dot coupled to two ferromagne tic electrodes, where\nthe oscillatory behavior of the transport properties with the bias w ould be expected if the\nspin accumulation effect is not neglected. The work toward this direc tion is under progress.\nAcknowledgments\nWe are grateful to S. S. Gong, B. Gu, X. F. Han, H. F. Mu, Z. C. Wan g and Q. B.\nYan for helpful discussions. This work is supported in part by the Na tional Science Fund\nfor Distinguished Young Scholars of China (Grant No. 10625419), t he National Science\nFoundation of China (Grant Nos. 90403036 and 20490210), and by the MOST of China\n(Grant No. 2006CB601102).\n[1] G. A. Prinz, Science 282, 1660 (1998).\n[2] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln´ ar, M. L. Roukes,\nA. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001)\n[3] I.ˇZuti´ c, J. Fabian and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).\n[4] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, B.I. Halperin , Rev. Mod. Phys. 77, 1375 (2005).\n18[5] G. Su, in Progress in Ferromagnetism Research , edited by V. N. Murray (Nova Science Pub-\nlishers, Inc., New York, 2006), pp. 85-123.\n[6] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[7] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[8] X. Zhang, B. Z. Li, G. Sun, and F. C. Pu, Phys. Rev. B 56, 5484 (1997).\n[9] F. Montaigne, J. Nassar, A. Vaur´ es, F. Nguyen Van Dau, F. Petroff, A. Schuhl, and A. Fert,\nAppl. Phys. Lett 73, 2829 (1997).\n[10] X. Zhang, B. Z. Li, G. Sun, and F. C. Pu, Phys. Lett. A 245, 133 (1998).\n[11] L. Sheng, Y. Chen, H. Y. Teng, and C. S. Ting, Phys. Rev. B 59, 480 (1999).\n[12] S. Stein, R. Schmitz, and H. Kohlstedt, Solid State Comm um117, 599 (2001).\n[13] Z. G. Zhu, G. Su, B. Jin and Q. R. Zheng, Inter. J. Mod. Phys .B 16, 2857 (2002).\n[14] B. Jin, G. Su, Q. R. Zheng and M. Suzuki, Phys. Rev. B 68, 144504 (2003).\n[15] Z.G. Zhu, G. Su, Q. R. Zheng and B. Jin, Phys. Rev. B 68, 224413 (2003).\n[16] S. Colis, G. Gieres, L. Bar, and J. Wecker, Appl. Phys. Le tt.83, 948 (2003).\n[17] X. F. Han, S. F. Zhao, F. F. Li, T. Daibou, H. Kubota, Y. And o, and T. Miyazaki, J. Magn.\nMagn. Mater. 282, 225 (2004).\n[18] B. Jin, G. Su and Q. R. Zheng, J. Appl. Phys. 96, 5654 (2004).\n[19] Z. G. Zhu, G. Su, Q. R. Zheng and B. Jin, Phys. Rev. B 70, 174403 (2004).\n[20] H. F. Mu, G. Su, Q. R. Zheng and B. Jin, Phys. Rev. B 71, 064412 (2005).\n[21] B. Jin, G. Su and Q. R. Zheng, Phys. Rev. B 71, 144514 (2005).\n[22] B. Jin, G. Su and Q. R. Zheng, Phys. Rev. B 73, 064518 (2006).\n[23] H. F. Mu, G. Su, and Q. R. Zheng, Phys. Rev. B 73, 054414 (2006).\n[24] Z. P. Niu, Z. B. Feng, J. Yang, and D. Y. Xing, Phys. Rev. B 73, 014432 (2006).\n[25] T. Nozaki, N. Tezuka, and K. Inomata Phys. Rev. Lett. 96, 027208 (2006).\n[26] Y. Wang, Z. Y. Lu, X. -G. Zhang, and X. F. Han, Phys. Rev. Le tt.97, 087210 (2006).\n[27] Z. M. Zeng, X. F. Han, W. S. Zhan, Y. Wang, Z. Zhang, and Shu feng Zhang, Phys. Rev. B\n72, 054419 (2005).\n[28] S. Zhang, and P. M. Levy , Phys. Rev. Lett. 79, 3744 (1997).\n[29] T. K. Ng, Phys. Rev. Lett. 89, 286803 (2002).\n[30] H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of Semiconductors\n(Springer, Berlin, 1998), p.166.\n19"    },    {        "title": "0708.2143v2.Spin_transfer_torques_in_anti_ferromagnetic_metals_from_first_principles.pdf",        "content": "arXiv:0708.2143v2  [cond-mat.mtrl-sci]  4 Jun 2008Spin-transfer torques in anti-ferromagnetic metals from fi rst principles\nYuan Xu, Shuai Wang, and Ke Xia\nState Key Laboratory for Surface Physics, Institute of Phys ics,\nChinese Academy of Sciences, P.O. Box 603, Beijing 100080, C hina\n(Dated: November 16, 2018)\nIn spite of the absence of a macroscopic magnetic moment, an a nti-ferromagnet is spin-polarized\non an atomic scale. The electric current passing through a co nducting anti-ferromagnet is polarized\nas well, leading to spin-transfer torques when the order par ameter is textured, such as in anti-\nferromagnetic non-collinear spin valves and domain walls. We report a first principles study on\nthe electronic transport properties of anti-ferromagneti c systems. The current-induced spin torques\nacting on the magnetic moments are comparable with those in c onventional ferromagnetic materials,\nleading to measurable angular resistances and current-ind uced magnetization dynamics. In contrast\nto ferromagnets, spin torques in anti-ferromagnets are ver y nonlocal. The torques acting far away\nfrom the center of an anti-ferromagnetic domain wall should facilitate current-induced domain wall\nmotion.\nPACS numbers: 72.25.Ba,75.50.Ee,75.47.De\nAnti-ferromagnet metals (AFMs) are widely used to\npin and exchange-bias ferromagnets in devices such as\nmagnetic spin valve read heads [1, 2]. The absence of\nmagnetic stray fields makes them very useful as probe\ntips in spin-polarized scanning tunnelling microscopes\n[3]. AFMs are materials with spontaneous magnetiza-\ntion below a critical N´ eel temperature at which the mag-\nnetic moments of two (or more) sub-lattices point into\nopposite directions, such that the net magnetic moment\nof AFM vanishes. External magnetic fields cannot be\nused to manipulate the strongly coupled magnetic mo-\nments of AFMs. An alternative for magnetic fields to\nexcite ferromagnets (FMs) is the current-induced spin-\ntransfer torque (STT) [4]. Since the electric current is\nspin-polarized on an atomic scale, the STT phenomenon\nmay excite the anti-ferromagnetic order parameter. It is\nnot obvious whether the STT is strong enough and how\ndifferent sub-lattices are affected.\nRecent theoretical studies [5, 6, 7] predicted that in\nspin valves with uncompensated anti-ferromagnetsangu-\nlar resistance (AR) effect can be expected. Reversal of\ntheanti-ferromagneticorderparameterisequivalentwith\na phase shift by a half period. Furthermore, STT is not\nan interface effect as in ferrromagnets, but effective over\nthe whole AFM. Very recently, strong experimental evi-\ndencesofSTTbetween AFM andFM havebeen reported\n[8]. In addition, the resistance of anti-ferromagnetic do-\nmain walls (AFDWs) in chromium has been measured[9].\nIn contrast to FMs, AFMs have much smaller shape\nanisotropies which makes manipulation of its magnetic\nmoments easier — provided one can apply a sizablespin\ntorque on the moments. What is the magnitude of the\nspin torque for an atomic scale spin polarization? Can\nan atomic scale spin polarization gives rise to observable\nAR effect? These arethe questions we attempt to answer\nin this work.\nγ-FeMn is an AFM used as pinning layer in spin\nvalves, and we demonstrate that an electric current caninduce angular momentum transfer between magnetic\nmoments in AFMs. By a detailed analysis of the cur-\nrent and non-equilibrium charge densities, we show that\nan atomic scale spin polarization exists even for com-\npensated AFMs. A sizable AR is predicted in anti-\nferromagnetic spin valves (AFSVs). In an AFDW the\nspin torque is effective not only around the domain wall\ncenter but throughout the entire AFM, which can be un-\nderstood by the suppressed spin precession.\nIn our calculations, the atomic potential was deter-\nmined in the framework of the tight-binding (TB) linear\nmuffin-tin-orbital (MTO) method based on density func-\ntional theory in the local density approximation [10] and\nan exchange-correlation potential parameterized by von\nBarth and Hedin [11]. The self-consistent crystal po-\ntentials were used as input to a TB-MTO wave-function-\nmatchingcalculationandthescatteringwavefunctionsof\nthe whole system were obtained explicitly [12]. To model\nnon-collinear magnetic configurations in AFDWs, a rigid\npotential approximation is applied, which is a good ap-\nproximation for sufficiently wide domain walls [13].\nFIG. 1: (color online) (a) Magnetic structure of γ-FeMn. (b)\nfcc(111) plane of γ-FeMn, where the dashed lines connect the\natoms in a unit cell.\nThe spin current and charge density can be calculated\nfrom the scattering wave functions. The torques act-\ning on an atomic plane are defined as the difference be-\ntween incoming and outgoing spin current [14]. Finally,\nwe focus on transport along the direction perpendicu-\nlar to the plane fcc(111) as illustrated in Fig.1(b), where2\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s53/s48 /s54/s48 /s55/s48 /s56/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s53/s48 /s54/s48 /s55/s48 /s56/s48/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s32/s80\n/s83\n/s76/s97/s121/s101/s114/s32/s105/s110/s100/s101/s120/s58/s32 /s73/s32/s70/s101/s49\n/s32/s70/s101/s50\n/s32/s77/s110/s50\n/s32/s77/s110/s49\n/s32\n/s32/s32/s32\n/s40/s97/s41/s120/s49/s48/s48\n/s32\n/s32/s32/s32\n/s40/s98/s41/s120/s49/s48/s48\nFIG. 2: (color online) Layer resolved spin-polarization of non-\nequilibrium charge density, Pα\nS, on the four inequivalent sub-\nlattices of the FeMn |Cu system, where FeMn occupies 1st-\n40th layers and Cu 41th-80th layers. The electron current\nflows from FeMn to Cu. Inset (a) gives the PSon the Cu\nside for the inequivalent sub-lattices corresponding to Fe 2 and\nMn1, Inset(b)gives the PSon theCu side for theinequivalent\nsub-lattices corresponding to Fe1 and Mn2.\nfouratomsconnectedbythe dashedlinesformaunit cell.\nfcc(111) is a compensated plane with zero net magnetic\nmoment in each layer.\nThemagneticstructureof γ-FeMnisshowninFig.1(a).\nThis configuration has been observed experimentally [15]\nand confirmed to be energetically favorable by ab.initio\ncalculation [16]. There are four inequivalent atoms in a\nunit cell. The magnetic moments of Fe2 and Mn1 and\nthose of Fe1 and Mn2 point into opposite directions. Our\ncalculation [17] yields the magnetic moments as mFe=\n1.4µBandmMn= 1.9µB.\nWhen a current passes through an anti-ferromagnetic\nmaterial, the current-induced non-equilibrium charge\ndensity integrated over the atomic sphere of an atom is\nspin-polarized. We define the polarization of atom αas\nPα\nS= (ρα\n↑−ρα\n↓)/(ρα\n↑+ρα\n↓), whereρα\n↑(↓)is the spin resolved\nnon-equilibrium charge density. ↑(↓) indicates the elec-\ntron spin parallel or antiparallel to the spin quantization\naxis.\nIn Fig.2 we plot Pα\nSfor a single FeMn |Cu interface.\nWe group the system into four inequivalent sub-lattices\naccording to the inequivalent atoms in the unit cell of\nγ-FeMn. A nonzero polarization is found even at the Cu\nside. The sum of the spin polarizations of the four sub-\nlattices vanishes in each layer. These results show that a\ncompensated anti-ferromagnetinjects a spin polarization\ninto a normal metal that oscillates on an atomic scale.\nLet us consider now an AFSV denoted as\nFeMn1(θ)|Cu|FeMn2|Cu, where an infinitely thick\nFeMn1 serves as the fixed layer and FeMn2 as the free\nlayer. The zaxis of spin space is set along the magne-\ntization of the free layer and θgives the relative angle\nbetween both order parameters. In our calculations, the/s48 /s51 /s54 /s57 /s49/s50 /s49/s53 /s49/s56 /s50/s49 /s50/s52 /s50/s55/s45/s48/s46/s48/s51/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s67/s117/s67/s117/s70/s101/s77/s110/s50\n/s32\n/s32/s76/s97/s121/s101/s114/s32/s105/s110/s100/s101/s120/s58/s32 /s73/s32/s32/s73\n/s83/s90\n/s32/s40/s101/s86\n/s98/s41/s70/s101/s77/s110/s49\nFIG. 3: (color online) Top panel: cartoon of the parallel con -\nfiguration, θ= 0o, of an AFSV. The red and blue arrows\ndenote the two sub-lattices A and B, respectively. Bottom\npanel: the layer resolved zcomponent of spin-polarized cur-\nrent, Iz\ns⊥, where the subscript ⊥denotes the current perpen-\ndicular to the fcc(111) plane and the red circles (blue squar es)\nrepresent the spin-polarized current on the sub-lattice A ( B).\nthicknesses of the Cu spacer and the FeMn2 free layer\nare chosen to be both 10 monolayers (ML) [18].\nThe top panel of Fig.3 illustrates the magnetic struc-\nture of the so-called parallel configuration ( θ= 0o). Here\nand in the following we group the four sub-lattices into\ntwo according to the direction of their magnetic mo-\nments. When the magnetization of FeMn1 is rotated\nfromθ= 0otoθ= 180o(the antiparallel configuration),\nthe electric resistance changes.\nThe bottom panelofFig.3givesthe spin-polarizedcur-\nrent at the two sub-lattices. When the magnetizations of\nthe two AFM layers are collinear, there is no net cur-\nrent spin polarization. However, the spatial distribution\nof spin-polarized current is not trivial. In fact, the spin-\npolarized currents on the two sub-lattices do not vanish\neven in Cu. They are identical in magnitude but oppo-\nsitely spin-polarized. Another finding is that the spin-\npolarized current oscillates between the two sub-lattices\nand is not conserved within one sub-lattice.\nThe AR ratio in terms of conductance as [ G(0o)−\nG(180o)]/G(180o) amounts to a measurable 5%. The re-\nmainingquestionistherobustnessofthisAR effectinthe\npresence of interfacial disorder. We modelled disorder by\nan interfacial alloy with a lateral super-cell method [12].\nFor a 1ML (FeMn) 1−xCuxinterfacial alloy, the AR de-\npendence on interfacial alloy concentration xis shown\nin Fig.4, which indicates that the AR is somewhat sup-\npressed in the presence of disorder, but should remain to\nbe observable.\nIt is interesting to investigate the STT due to the\natomic scale spin dependent scattering. Before going\ninto the numerical results, we analyze the symmetry of\nthe STT. For one sub-lattice, similar to that of FM, the3\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53\n/s32/s32/s65/s82/s32/s40/s37/s41\n/s67/s117/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s120/s32/s40/s37/s41\nFIG. 4: Cu concentration xdependence of AR in the system\nFeMn1|Cu(10ML) |FeMn2(10ML) |Cu.\ntorque on a local magnetic moment mis proportional to\nm×M×m(in-plane) and m×M(out-of-plane), where\nMdenotes the source of spin current. For another sub-\nlattice, both mandMare reversed. The in-plane torque\nis also reversed whereas the out-of-plane torque remains\nunchanged. Because the in-plane torque T/bardblon two sub-\nlattices are identical in magnitude and opposite in di-\nrection and the magnetization of two sub-lattices points\ninto opposite directions, the net effect of STT tends to\nrotate those moments together. In contrast to T/bardbl, the\nout-of-plane torque T⊥acts identically on the two mag-\nnetic sub-lattices. As the exchange coupling in the AFM\nis strong, out-of-plane torques will not result in any sig-\nnificant effect on the dynamics of the magnetic moments.\nAs shown in Fig.5(a), the spin torque in the AFM\nlayer oscillates deep into material which differs from\nconventional FM spin valve [19, 20]. The angular de-\npendence of the total in-plane torque acting on sub-\nlattice (Fe1&Mn2) in FeMn2 is shown in Fig.5(b). For\nθ= 150o, the torque (4 .1×10−4eVb/µB) acting on the\nsurface atoms is much smaller than that in FM spin valve\n(∼10−3eVb/µB) [21]. However, due to slow decay of the\nspin torque in AFMs, the integrated torque on per atom\nof AFMs (2 .4×10−4eVb/µB) is comparable to that in\nFMs.\nSimilar spin transfer effect should occur in AFDWs.\nAs domain wall physics in AFMs has not been fully\nunderstood [22], we simply consider a Bloch-like do-\nmain wall which could exist in the experiment [23], as\nshown in Fig.6(a). The configuration is described by\nθ(y) =π\n2+ arcsin[tanh(y−yo\nλDW)], where λDWis the char-\nacteristic length which is selected to be 4ML and yothe\ncenter of the wall. The spin direction in each monolayer\nalong the wall is shown in Fig.6(b). The change of the\ntotal conductance due to the formation of a wall is 2.8%\nfora4ML thickdomain wall, which iscomparabletothat\nin a FM with the same width.\nFig.6(c) shows the in-plane torque exerted on one sub-/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s49/s52 /s49/s53 /s49/s54 /s49/s55 /s49/s56 /s49/s57 /s50/s48 /s50/s49 /s50/s50 /s50/s51 /s50/s52 /s50/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s32/s40/s100/s101/s103/s41\n/s32/s32/s84\n/s124/s124/s32/s40 /s49/s48/s45/s50\n/s101/s86\n/s98/s41\n/s40/s98/s41/s32\n/s32/s32/s84\n/s124/s124/s32/s40 /s49/s48/s45/s50\n/s101/s86\n/s98/s41\n/s32/s32/s76/s97/s121/s101/s114/s32/s105/s110/s100/s101/s120/s58/s32 /s73/s70/s101/s77/s110/s50/s70/s101/s77/s110/s49/s67/s117/s67/s117/s40/s97/s41\nFIG. 5: (color online) (a) The layer resolved in-plane\ntorqueT/bardblon one sub-lattice (Fe1&Mn2) of FeMn2 in\nFeMn1(θ)|Cu(10ML) |FeMn2(10ML) |Cu, as shown by the car-\ntoon. Here θ= 150o, electrons flow from FeMn1 to FeMn2\nand FeMn2 is located at 15 ≤I≤24. (b) Angular depen-\ndence of the in-plane torque T/bardblon the sub-lattice (Fe1&Mn2)\nof FeMn2 in the same system.\nlattice(Fe1&Mn2)asafunctionofpositioninthedomain\nwall. There is a substantial STT in the region far away\nfrom the domain wall center. This is attributed to the\nspin degeneracy of the electronic band structure in the\nAFM. The injected spin can penetrate and precess co-\nherently until deep into the AFM. As a result, STT in\nAFM is very non-local, at least in the absence of disorder\nscattering.\nWe investigate the dynamics of the AFDW with the\nLandau-Lifshitz-Gilbert (LLG) equation [24] using our\ncalculated STT. An AFM with two sub-lattices is de-\nscribed by two sets of coupled LLG equations. Here we\ndecouple the equations by assuming that the magnetiza-\ntions of two sub-lattices in the same layerare antiparallel\nto each other at any time. This should be reasonable due\nto a strong anti-ferromagnetic coupling between the two\nsub-lattices.\nToinvestigatedynamicsofthe domainwallinthe pres-\nence of a current, we follow the method of Ohe and\nKramer[25] who studied FM domain walls. This method\nconsists of several steps: (i) Given a domain wall config-\nuration, we calculate the STT acting on each site under\na certain current density; (ii) the obtained spin torque at\neach site is substituted into the LLG equation to obtain\na set of equations for each site in the domain wall at a\ngiven time t; (iii) by integrating the LLG equation, we4\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48 /s49/s50/s48/s45/s48/s46/s48/s48/s51/s48/s46/s48/s48/s48/s48/s46/s48/s48/s51/s48/s46/s48/s48/s54/s48/s52/s53/s57/s48/s49/s51/s53/s49/s56/s48\n/s32/s32/s84/s111/s114/s113/s117/s101/s32/s40/s101/s86\n/s98/s41\n/s76/s97/s121/s101/s114/s32/s105/s110/s100/s101/s120/s58/s32 /s73 /s32/s84\n/s88\n/s32/s84\n/s90\n/s32/s124/s84\n/s124/s124/s124\n/s40/s99/s41\n/s32/s105/s110/s105/s116/s105/s97/s108/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110\n/s32\n/s32/s32/s32/s40/s32/s100/s101/s103/s32/s41\n/s40/s98/s41/s40/s97/s41\nFIG. 6: (color online) (a) Schematic view of the AFDW, in\nwhich two sub-lattices are indicated by the green and blue\narrows, the magnetization is rotated in the plane by an angle\nθand the electrons are injected from the left side. The in-\nplane torque directions are indicated by the red arrows. (b)\nθas the function of position. (c) Spin torques acting on the\nsub-lattice (Fe1&Mn2) in each monolayer with initial config -\nuration, where TX,TZdenote the in-plane components and\n|T|||is the absolute value of in-plane torques.\nobtain the domain wall configuration after an interval at\nt′=t+δt; (iv) once a new domain wall configuration is\nobtained, we can calculate the STT again and the pro-\ncess is repeated. The interval δtbetween two consecutive\nsteps is chosen small enough to guarantee numerical con-\nvergence of the solution.\nIn particular, a magnetic anisotropy constant of 1 .35×\n105erg/cm3, exchange stiffness constant of 0 .94×10−9\nerg/cm, magnetization of sub-lattice of 650 Gs and\nGilbert damping constant of 0 .1 are used in our calcu-\nlation.\nUnder a current density of 5 ×107A/cm2and start-\ning from the initial domain wall configuration mentioned\nabove, we obtained results for 10 time steps of δt= 2\nps. The domain wall moves into the −Ydirection and\nthe estimated velocity is 4 m/s. In contrast to the FM\ndomain wall, the velocity is maintained due to the van-\nishing de-magnetization field. For the same reason, when\nno pinning field is taken into account, the critical current\nthat sets the AFDW into motion is also expected to be\nvery small as compared to a FM domain wall.\nIn summary, we find an atomic scale spin polariza-\ntion in completely compensated AFMs. An STT can be\ninduced by an electric bias. In AFSVs, a sizable AR ispredicted and the STT is found to be comparable to that\nin conventional FM spin valves. In AFDWs, the torques\nturn out to be nonlocal. The spin dynamics of AFDWs\nis studied by including the STT into the LLG equation\nfor AFMs.\nWe acknowledge G. E. W. Bauer and H. Guo for\ntheir critical reading of the manuscript. We also ac-\nknowledge M. D. Stiles for his suggestion to define an\natomic scale spin polarization. This work is supported\nby the NSF(10634070) and MOST(2006CB933000,\n2006AA03Z402) of China.\n[1] W. H. Meiklejohn and C. P. Bean, Phys. Rev. 102, 1413\n(1956).\n[2] J. Bass, A. Sharma, Z. Wei, M. Tsoi, arXiv:0804.3358\n[3] A. Kubetzka, et al., Phys. Rev. Lett. 88, 057201 (2002).\n[4] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\nL. Berger, Phys. Rev. B 54, 9353 (1996). J.A. Katine, et\nal., Phys. Rev. Lett. 84, 3149 (2000); X. Waintal, et al.,\nPhys.Rev.B 62, 12317 (2000). J.Z.Sun, J. Magn. Magn.\nMater.202, 157 (1999). M. Tsoi et al., Phys. Rev. Lett.\n80, 4281 (1998).\n[5] A. S. N´ u˜ nez, et al., Phys. Rev. B 73, 214426 (2006).\n[6] R. A. Duine, et al., Phys. Rev. B 75, 014433 (2007).\n[7] P. Haney, et al., Phys. Rev. B 75, 174428 (2007).\n[8] Z. Wei, et al., Phys. Rev. Lett. 98, 116603 (2007).\n[9] R. Jaramillo, et al., Phys. Rev. Lett. 98, 117206 (2007).\n[10] I. Turek, et al. Electronic Structure of Disordered Al-\nloys, Surfaces and Interfaces (Kluwer, Boston-London-\nDordrecht, 1997).\n[11] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972).\n[12] K. Xia et al., Phys. Rev. B 73, 064420 (2006).\n[13] Selfconsistent calculations show that when the relati ve\nangle between neighboring magnetic atoms is less than\n20o, magnetic moments change less than 2%.\n[14] S. Wang, Y. Xu, and K. Xia, arXiv:0801.3135.\n[15] J. S. Kouvel and J. S. Kasper, J. Phys. Chem. Solids 24,\n529 (1963), P. Bisanti, G. Mazzone and F. Sacchetti, J.\nPhys. F: Met. Phys 17, 1425 (1987).\n[16] D. Spiˇ s´ ak and J. Hafner, Phys. Rev. B 61, 11569 (2000).\n[17] The Wigner-Seitz radii of the two atoms satisfy\nrFe/rMn= 0.98. We use a grid of 57600 k/bardblpoints in\nthe full 2DBZ for the transport calculations.\n[18] Several relative configurations between the two FeMn\n(111) planes facing each other across the Cu spacer can\nbe considered. In this work, we take the most convenient\none byreplacing 10 ML of the bulk FeMn with Cu atoms.\n[19] M. D. Stiles, et al., Phys. Rev. B 66, 014407 (2002).\n[20] M. Zwierzycki, et.al., Phys. Rev. B 71, 064420 (2005).\n[21] D. M. Edwards, et.al., Phys. Rev. B 71, 054407 (2005).\n[22] M. Ando and S. Hosoya, Phys. Rev. Lett. 29, 281 (1972).\nF. Nolting et.al., Nature (London) 405, 767 (2000). C.\nL. Chien, et.al.,Phys. Rev. B 68, 014418 (2003).\n[23] F. Radu and H. Zabel, arXiv:0705.2055.\n[24] Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004).\n[25] J. Ohe andB. Kramer, Phys. Rev.Lett. 96, 027204(2006)"    },    {        "title": "1002.2499v1.Quantum_Spin_Tomography_in_Ferromagnet_Normal_Conductors.pdf",        "content": "arXiv:1002.2499v1  [cond-mat.mes-hall]  12 Feb 2010Quantum Spin Tomography in Ferromagnet-Normal Conductors\nP. Samuelsson1and Arne Brataas2\n1Division of Mathematical Physics, Lund University, Box 118 , S-221 00 Lund, Sweden and\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nWe present a theory for a complete reconstruction of non-loc al spin correlations in ferromagnet-\nnormal conductors. This quantum spin tomography is based on cross correlation measurements of\nelectric currentsintoferromagnetic terminals with contr ollable magnetization directions. For normal\ninjectors, non-local spin correlations are universal and s trong. The correlations are suppressed by\nspin-flip scattering and, for ferromagnetic injectors, by i ncreasing injector polarization.\nPACS numbers: 72.25.Mk,73.23.-b\nSpintronics utilizes the electron spin in electronics ap-\nplications and is an importantsubfield ofcondensed mat-\nter physics. It is possible to create metallic or semicon-\nducting hybrid ferromagnet-normal conductor systems\nsmaller than the spin-flip length [1, 2], yet semiclassically\nlarge. Topics of current interest such as spin injection,\nprecession, and relaxation [1–3], spin Hall effects [4], cur-\nrent induced magnetization excitations [5], the reciprocal\nmagnetization dynamics induced spin-pumping [6], spin\nbased transistors [7], and ferromagnet-superconductor\nheterostructures [8] focus on the averagenon-equilibrium\nspin accumulation and dynamics.\nThe correlations between injected spins in\nferromagnet-normal conductor systems have received\nmuch less attention. In two-terminal junctions, current\ncorrelations have been investigated in few-level quantum\ndots [9] as well as semiclassically large systems [10, 11].\nThe prime targets have been noise due to spin-flip\nscattering and the super or sub poissionian nature of the\nauto correlations.\nIn multiterminal junctions, current cross correlations\nallow investigations of non-local spin transport proper-\nties. Of main interest has been the sign of the cross\ncorrelations, studied in quantum dots [12], diffusive [13]\nand superconducting [14] systems and chaotic cavities\n[15]. Moreover, in the context of entanglement of itin-\nerant spins, works on few-mode [16] and recently also\nsemiclassical [17, 18] conductors considered non-local de-\ntection schemes with cross correlations between currents\nin non-collinear ferromagnetic terminals.\nA fundamental and important question which has not\nbeen addressed is if known non-local spin injection and\ndetection schemes [1, 3] can be extended to identify non-\nlocal spin-correlations. Imagine spins injected into a nor-\nmal conductor and detected at two different spatial loca-\ntions byferromagneticterminals. What arethe non-local\nspatial correlations between the spins? Is it possible to\ncompletely characterize the correlations by experimen-\ntally accessible electrical current correlations? We pro-\nvide answers to these questions for semiclassical systems:\ni) non-local spin correlations are strong, and for normal\ninjectors, universaland ii) spin-correlationscanbe recon-structed by a sequence of measurements of correlations\nof currents at ferromagnetic detectors with controllable\nmagnetization directions, a quantum spin tomography.\nWe consider a semiclassically large, normal (metal or\nsemi-) conductor connected to a normal or ferromagnetic\ninjector, biased at a voltage V, and two spatially sepa-\nrated detectors, AandB, see Fig 1. Detector A(B)\nconsists of a normal node coupled to grounded ferromag-\nnetic terminals A1 andA2 (B1 andB2) via tunnel con-\ntacts with conductances GA1andGA2(GB1andGB2).\nThroughout,conductancesaredimensionlessandinunits\nof the conductance quantum 2 e2/h. The detectors A and\nB probe non-invasively the non-local spin correlations.\nFIG. 1: a) A normal conductor is connected to an injector bi-\nased at voltage Vand two detector nodes AandB. The node\nA(B) is coupled to grounded ferromagnetic detector termi-\nnalsA1 andA2 (B1 andB2). b) Node Ais connected to the\nnormal conductor, as well as nodes A1 andA2 via tunnel con-\nductances GA,GA1, andGA2, respectively. The polarizations\nPA1andPA2of the contacts to the ferromagnetic terminals\nare in opposite directions.\nLet us first summarize and explain our main results i)\nand ii) for the non-local correlated spin transport prop-\nerties in the device in Fig. 1: First, combining scatter-\ning theory and a Boltzmann-Langevin approach we de-\nrive an expression for the current correlations SAiBj=\n(2e2/h)/integraltexteV\n0dEsAiBj(E) with\nsAiBj= 4GAiGBj∝an}bracketle{t(δfc\nA+PAi·δfA)(δfc\nB+PBj·δfB)∝an}bracketri}htf,\n(1)\nwherePAi(PBj) is the polarization of the tunnel con-\ntact to terminal Ai(Bj),δˆfA/B=δfc\nA/Bˆ1 +δfA/B·ˆσ2\nis the fluctuating part of the 2 ×2 spin distribution ma-\ntrix atA/B,ˆσ= [ˆσx,ˆσy,ˆσz] is a vector of Pauli ma-\ntrices, and ∝an}bracketle{t..∝an}bracketri}htfdenotes the average over fluctuations.\nThe matrix δˆfAB, with elements δfpq\nAB=∝an}bracketle{tδfp\nAδfq\nB∝an}bracketri}htf,\np,q∈ {c,x,y,z}, is the spin correlation matrix, describ-\ning the irreducible, or exchange, correlations between\nspins at A and B.\nWethenshowourresultii): δˆfABcanbe reconstructed\nby a sequence of measurements of e.g.SA1B1with differ-\nent settings of PA1andPB1. Importantly, this quantum\nspin tomography can be performed for arbitrary (finite)\nmagnitudes of the polarizations |PA1|and|PB1|and\nspin-flip scattering in the conductor. Moreover, global\nspin symmetries limit the number of finite elements of\nδˆfAB, allowing for a simplified quantum spin tomogra-\nphy with fewer cross correlation measurements.\nFor a normal injector we derive a generic expression\nforδˆfAB, with nonzero elements\nδfcc\nAB=δf0\nAB/2, δfxx\nAB=δfyy\nAB=δfzz\nAB=γδf0\nAB/2,\n(2)\nwhereδf0\nABis the equal-spin correlator and γquantifies\nthe spin coherence in the conductor. γ= 1 for a coherent\nsystem,i.e.no spin-flip scattering, and γ= 0 for a sys-\ntem with strong spin-flip relaxation. For a ferromagnetic\ninjector, the correlations depend on the properties of the\nconductor, as shown below.\nInserting Eq. (2) into (1) gives a cross correlator\nsAiBj= 2GAiGBjδf0\nAB[1+γPAi·PBj],(3)\ndepending on the relative orientation of the polarizations\nPAiandPBj. This together with Eq. (2) demonstrate\nour counter-intuitive result i): any conductor with a nor-\nmal injector displays strong and universal non-local spin-\ncorrelations . We note that for the current cross correla-\ntor, similar results have been obtained in particular ge-\nometries [16–18] with no spin-flip scattering, γ= 1.\nWe now describe the quantum spin tomography, start-\ning for clarity with the known properties [19] of the\naverage spin distribution matrix in node A,ˆfA=\nfc\nAˆ1 +fA·ˆσ, where the real polarization vector fA=\n[fx\nA,fy\nA,fz\nA] with|fA| ≤1. The average current is IA1=\n(e/h)/integraltexteV\n0dEiA1(E) with [19]\niA1= 2GA1[fc\nA+PA1·fA]. (4)\nFor the quantum spin tomography, we transform the or-\nbital scheme developed in Ref. [21] to the spin degree\nof freedom and extend it to account for arbitrary de-\ntector polarization. Formally, to determine fc\nA,fAfour\nindependent measurements of the current are needed.\nThe theoretically most convenient set {I(k)\nA1},k= 1−4\nhas the polarizations P(1)\nA1/PA1= [0,0,1],P(2)\nA1/PA1=\n[0,0,−1],P(3)\nA1/PA1= [1,0,0] andP(4)\nA1/PA1= [0,1,0],\nwherePA1=|PA1|, but other settings are also fea-\nsible. The expression in Eq. (4) then allows writing({k}={c,x,y,z})\nfk\nA=/summationtext4\nl=1Qkl\nA1I(j)\nA1\n4GA1e2V/h,QA1\nPA1=\nP−1\nA1P−1\nA10 0\n−1−1 2 0\n−1−1 0 2\n1−1 0 0\n(5)\nKnowing the polarization PA1and the conductance GA1\nfrom independent measurements, the spin-distribution\nmatrixˆfAis fully reconstructed by current measure-\nments. Importantly, for a normal injector, only fc\nAis\nnon-zero. For a ferromagnetic injector, when the spin\nquantizationaxisalongthedirection ofpolarization, only\nfc\nAandfz\nAare non-zero.\nWe then turntothe spin correlationmatrix δˆfAB, with\nthe 16 real elements δfpq\nAB. This implies that we need\n16 independent cross correlator measurements to deter-\nmine all elements δfpq\nABand reconstruct δˆfAB. From Eq.\n(1) we obtain the formal relation between the coefficients\nδfpq\nABand the cross correlators\nδfpq\nAB=1\n8GA1GB1Ve3/h4/summationdisplay\nk,l=1Qpk\nA1Qql\nB1S(k,l)\nA1B1,(6)\nwhereS(k,l)\nA1B1is the cross correlatorwith the detector ter-\nminal setting katA1 andlatB1. HereQB1is obtained\nfromQA1by changing PA1toPB1.\nFor a normal injector, the requirement [22] of invari-\nance ofδˆfABunder any global spin rotation means that\nthere is only four non-zero elements δfcc\nABandδfxx\nAB=\nδfyy\nAB=δfzz\nAB. For a ferromagnetic injector (defining\nthe spin quantization axis) invariance of δˆfABunder\nthe global rotation {|↑∝an}bracketri}ht,|↓∝an}bracketri}ht} → { eiφ|↑∝an}bracketri}ht,e−iφ|↓∝an}bracketri}ht}yields\n[23] six non-zero elements δfcc\nAB,δfcz\nAB,δfzc\nAB,δfzz\nABand\nδfxx\nAB=δfyy\nAB.\nFrom Eqs. (5) and (6) the detector polarization set-\ntings necessary to determine the non-zero components\nofˆfAandδˆfABare found: For a normal injector, only\ncollinear polarizationsat A and B are needed for both ˆfA\nandδˆfAB. For a ferromagnetic injector, for ˆfAthe de-\ntector polarizations in addition have to be collinear with\nthe injector one. However, for δˆfABnon-collinear polar-\nizations at A and B are necessary, e.g.both along the x\nandzaxis, since δfzz\nAB∝ne}ationslash=δfxx\nAB=δfyy\nAB. Importantly, for\nan unknown direction of the injector polarization or two\n(or more) non-collinear ferromagnetic injectors, the full\ntomographicscheme with detector polarizationsalongall\nthree axes x,yandzare required.\nWe will now detail our calculations, assumptions, and\napproximations. In addition to the information given\nabove, the normal conductor in Fig. 1 is connected to\ndetector nodes AandBvia tunnel barriers with conduc-\ntancesGAandGB. The two ferromagnetic terminals A1\nandA2 (B1andB2)haveoppositedirectionsofpolariza-\ntion. We assume the limit of low temperature kT≪eV.\nAll conductances are much larger than unity.3\nIt is assumed that the normal conductor consists of\ndiffusive and/or chaotic parts, allowing a semiclassical\ntreatment of the orbital properties. In contrast, spin is\ntreated fully quantum mechanically. Furthermore, scat-\ntering is elastic. Following the magnetoelectronic circuit\ntheory of Ref. [19], we discretize the system into nodes\nconnected via tunnel barriers, see Fig. 1. Each node ν,\nspatially much smaller than the spin-flip length, is char-\nacterized by a 2 ×2 distribution matrix with an average,\nˆfν, andafluctuating, δˆfν, part. Toensurethatthedetec-\ntorsdonotinfluencethespin-propertiesofthesystem, we\nrequirei) GA≪GA1+GA2andGB≪GB1+GB2sothat\nan electron entering e.g.nodeAfrom the conductor is\nemitted into A1orA2anddo notreturntothe conductor\nand ii)GA1PA1=−GA2PA2andGB1PB1=−GB2PB2,\nwhich ensures that no spin polarization is induced into\nthe conductor from the ferromagnetic terminals, i.e.the\nmeasured spin signal arises from the conductor exclu-\nsively and not from the detector circuits.\nDeriving Eqs. (4) and (1), we first review [19] the\nspin information present in the average spectral current\niA1(E). In the scatteringapproach[20], with no particles\nincident from terminal A1 in the bias window (0 ≤E≤\neV), the spectral current is\niA1=/summationdisplay\nnσ∝an}bracketle{tnσ\nA1,n∝an}bracketri}ht, nσ\nA1,n=bσ†\nA1,nbσ\nA1,n(7)\nwherebσ†\nA1,ncreates an electron on the ferromagnetic side\nin the contact between A1 andA, in conduction mode n\npropagating into A1 and the energy-dependence is sup-\npressed. The spin quantization axis σ=↑,↓is along the\ndirection of PA1. The creation operators bσ†\nA1,nare re-\nlated to the operators bτ†\nAmfor electrons on the normal\nconductor side, emitted from node Atowards A1, via\nthe spin-dependent transmission matrix of the normal-\nferromagnetic interface tA1with elements tστ\nA1,nm. Fol-\nlowing Ref. [19], we make the semiclassical approxima-\ntion that the spin distribution matrix in node Ais in-\ndependent on mode index, i.e.∝an}bracketle{tbσ†\nAnbσ′\nAm∝an}bracketri}ht=ˆfσσ′\nAδnm,\ngiving\niA1=/summationdisplay\nστˆTτσ\nA1ˆfστ\nA=GA1tr/braceleftig\n(ˆ1+PA1·ˆσ)ˆfA/bracerightig\n.(8)\nHere [19] ˆTA1=/summationtext\nnm(ˆtA1,nm)T(ˆtA1,nm)∗=GA1(ˆ1 +\nPA1·ˆσ) where the elements of the 2 ×2 matrix are\n(ˆtA1,nm)στ=tστ\nA1,nm. Eq. (8) directly gives Eq. (4).\nSimilar relations hold for the average currents into A2,\nB1 andB2.\nWethenturntothelowfrequencycorrelationsbetween\nelectrical currents in e.g.terminals A1 andB1,SA1B1=/integraltext\ndt∝an}bracketle{t∆IA1(0)∆IB1(t)∝an}bracketri}ht. Scattering theory [20] gives\nsA1B1=/summationdisplay\nnm,στ/bracketleftbig\n∝an}bracketle{tnσ\nA1,nnτ\nB1,m∝an}bracketri}ht−∝an}bracketle{tnσ\nA1,n∝an}bracketri}ht∝an}bracketle{tnτ\nB1,m∝an}bracketri}ht/bracketrightbig\n(9)wherenτ\nB1,m=bτ†\nB1,mbτ\nB1,mandbτ†\nB1,mcreates an out-\ngoing electron on the ferromagnetic side, in conduction\nmodeminB1 with spin quantization axis along the di-\nrection of the magnetization nB. Disregarding terms of\nsecondorderin GA/(GA1+GA2)orGB/(GB1+GB2), the\noperators bσ†\nA1,nandbτ†\nB1,mare expressed in terms of the\noperators bσ′†\nAkandbτ′†\nBland the scattering amplitudes of\nthe respective normal-ferromagnetic interfaces. Making\nthe semiclassical approximation that the non-local irre-\nduciblecorrelator ∝an}bracketle{tbσ†\nAnbσ′\nAmbτ†\nBkbτ′\nBl∝an}bracketri}ht−∝an}bracketle{tbσ†\nAnbσ′\nAm∝an}bracketri}ht∝an}bracketle{tbτ†\nBkbτ′\nBl∝an}bracketri}ht ≡\nδˆfσσ′,ττ′\nABδnmδkl, we arrive at\nsA1B1=/summationdisplay\nσσ′ττ′ˆTσ′σ\nA1ˆTτ′τ\nB1ˆfσσ′,ττ′\nAB=GA1GB1\n×tr/braceleftig/bracketleftbig/parenleftbigˆ1+PA1·ˆσ/parenrightbig\n⊗/parenleftbigˆ1+PB1·ˆσ/parenrightbig/bracketrightbig\nδˆfAB/bracerightig\n(10)\nwhereˆTB1is obtained by changing all indices AtoBin\nˆTA1and⊗is the tensor product. Here we work in the\nbasis{|στ∝an}bracketri}ht}= [|↑↑∝an}bracketri}ht,|↑↓∝an}bracketri}ht,|↓↑∝an}bracketri}ht,|↓↓∝an}bracketri}ht],i.e.the matrix el-\nements ( δˆfAB)|στ/angbracketright,|σ′τ′/angbracketright=δˆfσσ′,ττ′\nAB. As is shown below,\nthe 4×4 spin correlation matrix δˆfAB=∝an}bracketle{tδˆfA⊗δˆfB∝an}bracketri}htf,\nprovides a semiclassical interpretation of δˆfAB. This\nmeans that Eq. (10) directly gives Eq. (1). Moreover,\nthe expressions for the other correlators SAiBjcan sim-\nilarly be given in terms of δˆfAB. This shows that δˆfAB\ncontains all information about non-local spin-correlatio ns\nthat can be obtained from cross correlations .\nTo further investigate the properties of ˆfA,ˆfBand\nδˆfABwe now turn to the spin-dependent Boltzmann-\nLangevin approach of Ref. [11]. The average part of\nthe distribution matrix ˆfνat nodeνis determined from\nthe condition of conservation of matrix currents into the\nnode,/summationtext\nµˆiνµ= 0,iνµ=−iµν. The 2×2 matrix current\nbetween a normal node νand a ferromagnetic or normal\nnodeµisˆiνµ= (Gνµ/2)/braceleftig\n(ˆ1+Pµ·ˆσ),(ˆfν−ˆfµ)/bracerightig\nwith\n{.,.}the anti-commutator, Gνµthe tunnel conductance\nbetweenthenodesand Pµthepolarizationvectorofnode\nµ(Pµ= 0 for a normal node). The distribution matrices\nfor normal and ferromagnetic terminal nodes are ˆ1 for\nbiased terminals and 0 for grounded. This allow us to\ncalculate the distribution matrices of all nodes.\nFor the fluctuating part of the distribution matrix, we\nfirst note that the total fluctuations ofthe matrix current\n∆ˆiνµflowing between two nodes νandµis a sum of the\nbare fluctuations δˆiνµandδˆIνµdue to the fluctuating\ndistribution matrices. For νnormal and µnormal or fer-\nromagnetic δˆIνµ= (Gνµ/2)/braceleftig\n(ˆ1+Pµ·ˆσ),(δˆfν−δˆfµ)/bracerightig\n.\nThe requirement of matrix current fluctuation conser-\nvation/summationtext\nµ∆ˆiνµ= 0 then gives δˆfνin terms of δˆiνµ.\nThe bare fluctuations δˆiνµat different contacts are\nuncorrelated while for ν,µnormal∝an}bracketle{tδˆiνµ⊗δˆiνµ∝an}bracketri}htf=\n(Gνµ/2)/bracketleftig\nˆfν⊗(ˆ1−ˆfµ)+ˆfµ⊗(ˆ1−ˆfν)/bracketrightig\nˆW+h.c.where4\nh.c.denotehermitianconjugateandthe permutationma-\ntrixˆWhas nonzero elements W11=W23=W32=\nW44= 1. For νnormal and µferromagnetic we have\n∝an}bracketle{tδˆiνµ⊗δˆiνµ∝an}bracketri}htf= (Gνµ/2){([ˆ1+Pµ·ˆσ]fµ)⊗(ˆ1−ˆfν) +\n([ˆ1+Pµ·ˆσ][1−fµ])⊗ˆfν}ˆW+h.c.. Here we used that\nferromagnetic ( i.e. terminal) distributions do not fluctu-\nate. From theserelationsanyelectricalcurrentcorrelator\n∝an}bracketle{t∆iν∆iµ∝an}bracketri}htf, with ∆iν= tr[∆ˆiν], can be obtained.\nSpin flip scattering is taken into account on the level\nof the relaxation time approximation. This amounts to\ncoupling each node nto a spin-flip node ϕνwith a tun-\nnel contact with conductance Gϕν∝1/τϕν, withτϕνthe\nspin-flip time of the node, and requiring conservation of\nelectrical current and current fluctuations into the spin-\nflip node. Here we give the universal results of the cal-\nculation, i.e.we consider an arbitrary normal conductor\nwith any amount of (spatially dependent) spin-flip scat-\ntering, the details of the calculations are given elsewhere.\nFirst, by comparing the obtained expressionfor the spec-\ntralcrosscorrelators sAiBj=∝an}bracketle{t∆iA1∆iB1∝an}bracketri}htfwithEq. (10)\nwe conclude that δˆfAB=∝an}bracketle{tδˆfA⊗δˆfB∝an}bracketri}htf, discussed above.\nSecond, for normal injectors, we find the generic form\nδˆfAB= (δf0\nAB/2)/bracketleftig\n(1−γ)ˆ1⊗ˆ1+2γˆW/bracketrightig\n(11)\nThis is just the result in Eq. (2).\nFurther insight is obtained by calculating the prop-\nerties of the simplest possible conductor, a single node\n[2]. For a normal injector we find the distribution func-\ntion at e.g.A asˆfA=GA/(GA1+GA2)fˆ1, with\nf=G/(G+GA+GB) the distribution function of the\nconductor node and Gthe injector-conductor node con-\nductance. This is independent on spin-flip scattering.\nFor the spin-correlation matrix we get the result in Eq.\n(11) with δf0\nAB=f3GAGB/[G(GA1+GA2)(GB1+GB2)]\nandγ= [1+τ/τϕ]−1withτ/τϕ=Gϕ/(GA+GB+G)\nthe ratio of spin-flip and dwell times in the central node.\nFor a ferromagnetic injector with polarization PIthe\nspin distribution matrix at e.g.A has two non-zero com-\nponents ( ˆfA)↑↑≡f↑\nAand (ˆfA)↓↓≡f↓\nAwithf↑,↓\nA=\nf[(1±PI)(1∓PIf) +τϕ/τ]/[(1−P2\nIf2) +τϕ/τ] with\nPI=|PI|. For the spin-correlation matrix, the full\nexpression, including spin-flip scattering, becomes very\nlengthy and we only present the result for γ= 1. This\nisδfcc\nAB=δfzz\nAB=δf0\nAB(c++c−)/2,δfcz\nAB=δfzc\nAB=\nδδf0\nAB(c+−c−)/2 andδfxx\nAB=δfyy\nAB=δf0\nABc0/2 with\nc±= (1±PI)2/(1±PIf)3andc0= (1−P2\nI)/[1−(PIf)2].\nInserting this into Eq. (10) we get the cross correlator\nSA1B1= 2GA1GA1δf0\nAB(cI+c0[1+PA1·PB1]) (12)\nwithcI= (1 + Pz\nA1Pz\nB1)(c++c−−2c0)/2 + (Pz\nA1+\nPz\nB1)(c+−c−)/2. This clearly demonstrates that while\na ferromagnetic injector leads to a polarization of the\nconductor, it suppresses the spin-correlations.\nIn conclusion we have presented a scheme for quan-\ntum state tomography of non-local spin correlations innormal-ferromagneticconductors. Non-local correlations\nare genericallystrong but suppressed by spin-flip scatter-\ning and ferromagnetic injectors.\nWe acknowledge discussions with Daniel Huertas Her-\nnando. This work was supported by the Swedish VR and\nthe Research Council of Norway, Grant No 162742/V00.\n[1] Y. Ohno et al, Nature 402790 (1999); R. Fiederling et\nal, Nature 402787 (1999); F. J. Jedema, A. T. Filip, B.\nJ. van Wees, Nature 410345, (2001); F. J. Jedema et al,\nNature416713 (2002).\n[2] M. Zaffalon, and B. J. van Wees, Phys. Rev. Lett, 91\n186601, (2003).\n[3] M. Johnson, R. H. Silsbee, Phys. Rev. B 375312 (1988);\nPhys. Rev. B 375326 (1988).\n[4] H.-A. Engel, E. I. Rashba, and B. I. Halperin,\narXiv:0603306.\n[5] D. C. 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B, 73, 041305(R)\n(2006).\n[22] B. Michaelis, and C.W.J. Beenakker, Phys. Rev. B, 73,5\n115329 (2006).\n[23] X. Wang, and P. Zanardi, Phys. Lett. A 3011 (2002)."    },    {        "title": "0801.1552v2.Controllable_spin_transport_in_ferromagnetic_graphene_junctions.pdf",        "content": "arXiv:0801.1552v2  [cond-mat.mes-hall]  14 Jan 2008Controllable spin transport in ferromagnetic graphene jun ctions\nTakehito Yokoyama\nDepartment of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan\n(Dated: November 12, 2018)\nWe study spin transport in normal/ferromagnetic/normal gr aphene junctions where a gate elec-\ntrode is attached to the ferromagnetic graphene. We find that due to the exchange field of the\nferromagnetic graphene, spin current through the junction s has an oscillatory behavior with respect\nto the chemical potential in the ferromagnetic graphene, wh ich can be tuned by the gate voltage.\nEspecially, we obtain a controllable spin current reversal by the gate voltage. Our prediction of\nhigh controllability of spin transport in ferromagnetic gr aphene junction may contribute to the\ndevelopment of the spintronics.\nThere is a rapidly growing attention to graphene be-\ncause it has a rich potential from fundamental and ap-\nplied physics point of view.1,2,3Graphene has a two-\ndimensional honeycomb network of carbon atoms and,\nas a result, electrons in graphene are governed by Dirac\nequation, which provides a bridge between condensed\nmatter physics and quantum electrodynamics, thus fasci-\nnating theoriticians toward the ultimate goal of the ‘uni-\nfied’ theory.4,5The progress of practical fabrication tech-\nniques for single graphene sheets has allowed experimen-\ntal study of this system, which has attracted a tremen-\ndous interest from the scientific community.6,7,8There\nhave been intensive studies on graphene to this date, for\ninstance, half integer and unconventional quantum Hall\neffect7,9,10, observation of minimum conductivity8and\nbipolar supercurrent11.\nGraphene has many important natures for applica-\ntions: it exhibits gate-voltage-controlled carrier conduc-\ntion, high field-effect mobilities and a small spin-orbit\ninteraction.12,13Thus, it is extremely promising for fu-\nture technologies, especially spintronics. Motivated by\nthis, some studies on graphene have been implemented.\nSpin injection into a graphene thin film has been success-\nfully demonstrated by using non-local magnetoresistance\nmeasurements.14,15,16The possibility of inducing ferro-\nmagnetic correlations in graphene due to the proximity\neffect by magnetic gates in close proximity to graphene is\nalso discussed.17In these works, induced ferromagnetism\nin graphene is ‘extrinsic’.\nOn the other hand, there is another attempt to induce\nferromagnetism ‘intrinsically’ in graphene. It has re-\ncently been predicted that zigzagedge graphene nanorib-\nbon become half-metallic by an external transverse elec-\ntric field due to the different chemical potential shift at\nthe edges,18,19,20which indicates the high controllability\nof ferromagnetism in graphene and hence opens the pos-\nsibility of the spintronics application for graphene. In\nview of this, the study of spin transport in ferromagnetic\ngrapheneisverytimelyanddesirableforthedevelopment\nof the spintronics.\nStimulated by this, in this paper, we study spin trans-\nport in normal/ferromagnetic/normal graphene junc-\ntions where a gate electrode is attached to the ferro-\nmagnetic graphene. We find that due to the exchange\nfield in the ferromagnetic graphene, spin current throughthese systems has an oscillatory behavior with respect\nto the chemical potential in the ferromagnetic graphene,\nwhich can be tuned by the gate electrode.7,8,9Especially,\nwe find a controllable spin current reversal by the gate\nvoltage. Since the exchange field of graphene is also\ntunable by the inplane external electric field and even\nthe half-metallicity can be induced18,19,20, our prediction\nof high controllability of spin transport in ferromagnetic\ngraphene junction may facilitate the development of the\nspintronics. Note that the model itself in this paper is\nthe same as that in Ref.17and here we mainly focus on\nthe case with the exchange field comparable to the Fermi\nenergy in the model of Ref.17.\nNow, let us explain the formulation. The fermions\naround Fermi level in graphene obey a massless relativis-\ntic Dirac equation. The Hamiltonian is given by\nH±=vF(σxkx±σyky) (1)\nwith Pauli matrices σxandσy, and the velocity vF≈106\nm/s in graphene. This is roughly 100 times larger than\nin a normal metal, and thus it is safe to neglect the\nCoulomb interaction compared to the kinetic energy in\ngraphene.21,22The Pauli matrices operate on the two\ntriangular sublattice space of the honeycomb structure.\nThe±signreferstothe twoso-calledvalleysof KandK′\npoints in the Brillouinzone. Also, thereis a valley degen-\neracy, which allows one to consider one of the H±set.23\nThe linear dispersion relation is valid for Fermi levels as\nhighas1eV,24suchthatthefermionsingraphenebehave\nlike massless Dirac fermions in the low-energy regime.\nWe consider a two dimensional nor-\nmal/ferromagnetic/normal graphene junction where\na gate electrode is attached to the ferromagnetic\ngraphene. This junction may be realized by applying an\nexternal transverse electric field to a part of graphene\nnanoribbon to make it ferromagnetic partially. See Fig-\nure 1 for the schematic of the model with corresponding\ndispersion relations. The interfaces are parallel to the\ny-axis and located at x= 0 andx=L. Since there\nis a valley degeneracy, we focus on the Hamiltionian\nH+withH+=vF(σxkx+σyky)−V(x),V(x) =EF\nin the normal graphenes and V(x) =EF+U±Hin\nthe ferromagnetic graphene. Here, EF=vFkFis the\nFermi energy, Uis the chemical potential shift tunable2\n\u0001\n\u0000\n\u0002 \u0003 \u0004 \u0005 \u0006 \u0007\b \t \n \u000b\f \r \u000e \u000f\n\u0010 \u0011 \u0012 \u0013 \u0014 \u0015\u0016 \u0017 \u0018 \u0019\u001a \u001b \u001c \u001d\u001e \u001f  ! \" # $ % & ' ( ) *\n+ , - . / 0 1 2 3 4 5 6 78 9 : ; < = > ?\nFIG. 1: (color online) Schematic of the model of nor-\nmal/ferromagnetic/normal graphene junction with corre-\nsponding Dirac cones. In the ferromagnetic graphene, dif-\nferent Dirac cones correspond to different chemical potenti als\nof majority and minority spins.\nby the gate voltage, and His the exchange field. ±signs correspond to majority and minority spins. The\nwavefunctions are given by\nψ1=/parenleftbigg1\neiθ/parenrightbigg\neipcosθx+ipyy\n+a±/parenleftbigg1\n−e−iθ/parenrightbigg\ne−ipcosθx+ipyy, (2)\nψ2=b±/parenleftbigg1\neiθ′/parenrightbigg\neip′\n±cosθ′x+ipyy\n+c±/parenleftbigg1\n−e−iθ′/parenrightbigg\ne−ip′\n±cosθ′x+ipyy, (3)\nψ3=d±/parenleftbigg1\neiθ/parenrightbigg\neipcosθx+ipyy(4)\nwith angles of incidence θandθ′,p= (E+EF)/vFand\np′\n±= (E+EF+U±H)/vF. Here,ψ1andψ3are wave-\nfunctions in the left and right normal graphenes, respec-\ntively, while ψ2is a wavefunction in the ferromagnetic\ngraphene. Because of the translational symmetry in the\ny-direction, the momentum parallel to the y-axis is con-\nserved:py=psinθ=p′sinθ′.\nBymatchingthewavefunctionsattheinterfaces( ψ1=\nψ2atx= 0 andψ2=ψ3atx=L), we obtain the coef-\nficients in the wavefunctions. Note that these conditions\nare equivalent to ˆ vxψ1= ˆvxψ2atx= 0 and ˆvxψ2= ˆvxψ3\natx=Lwith velocity operator ˆ vx=∂H+/∂kx=vFσx\nand hence the current is conserved at the interfaces. The\ntransmittion coefficient has the form\nd±=cosθcosθ′e−ipLcosθ\ncos(p′\n±Lcosθ′)cosθcosθ′−isin(p′\n±Lcosθ′)(1−sinθsinθ′). (5)\nThen, the dimensionless spin-resolved conductances\nG↑,↓are given by\nG↑,↓=1\n2/integraldisplayπ/2\n−π/2dθcosθT↑,↓(θ) (6)\nwithT↑,↓(θ) =|d±(θ)|2. Finally, the spin conductance\nGsis defined as Gs=G↑−G↓. Below, we focus on the\nzero voltage conductances, namely we set E= 0.\nNow, let us explain the underlying mechanism of\nspin manipulation by the gate voltage. In the limit of\n|U±H| ≫EF, we haveθ′→0 and hence the transmit-\ntion coefficient of the form\nd±→cosθe−ipLcosθ\ncosχ±cosθ−isinχ±(7)withχ±=χ±χH,χ=UL/vF,andχH=HL/vF. The\nresulting transmittion probability is represented as25\nT↑,↓(θ)→cos2θ\n1−sin2θcos2χ±. (8)\nFrom this expression, we see the π-periodicity with re-\nspect toχ±orχ.17,25,26,27We also find that G↑,↓has a\nmaximum (minimum) value of 1 (2/3) at χ±= 0 (π/2).\nThe phase difference between G↑andG↓is given by\nχ+−χ−= 2χH= 2HL/vF. If this is equal to the\nhalf period π/2 (namely, H/EF=πkFL/4), we expect\na large spin current which oscillates with χ, namely the\ngate voltage, because when one of G↑andG↓has a maxi-\nmumatacertain χ, theotherhasaminimum atthesame\nχ. In this case, the value of Gsoscillates between −1/33\nand 1/3. Note that the electrical conductance G↑+G↓\nin the junctions is always positive and therefore spin cur-\nrent reversal in our model is not accompanied with the\ncurrent reversal.@ A B\nC D E F\nG\nH I J\nK L M\nN O P\nQ R S\nT\nU V WX Y Z\n[ \\ ]^ _ `\nFIG. 2: (color online) Plots as a function of χtunable by the\ngate voltage, in the limit of U→ ∞. (a) spin resolved con-\nductances where the phases of G↑(solid line) and G↓(dotted\nline) are shifted by half period, χ↑−χ↓=π/2. (b) Spin con-\nductance Gswhich oscillates with the period πwith respect\ntoχbut is never damped.\nThe results in this limiting case are shown in Fig. 2.\nFigure 2 (a) shows spin resolved conductances as a func-\ntion ofχwhich is tunable by the gate voltage. Here,\nthe phases of G↑andG↓are shifted by half period,\nχ+−χ−=π/2. Then, we have a finite spin current as\nshown in Fig. 2 (b). Remarkably, it oscillates with the\nperiodπwith respect to χbut is never damped. As is\nseen, wecanreversethespincurrentbychangingthegate\nvoltage. Here, we focus on the limit of |U±H| ≫EF.\nNext, we consider more general cases without taking this\nlimit.\nFigure 3 (a) exhibits the spin conductance Gsas a\nfunction of χfor several values of kFL. For a ferro-\nmagnetic graphene with kFL= 0.1, theπ-periodicity\nis seen. With increasing kFL, however, the π-periodicity\ngetsbrokenasseeninFig. 3(a). Thisisbecause U≫EF\nis no more satisfied for small kFLsinceχ=kFLU/E F.\nThus, by choosing large χfor largekFL, we again get\nπ-periodicity of Gswith respect to χas shown in Fig. 3\n(b). From this figure, we find that U/EF>1 is required\nto obtain controllable spin current reversal.\na b c\nd ef g\nh ij k\nl\nmn o\npq r\nst\nu v wx y z{\n| } ~ \n    \n    \n     \n  \n\n \n \n¡ ¢ £¤¥\n¦ § ¨©\nFIG. 3: (color online). Spin conductance Gsas a function of\nχfor several values of kFLwithχ+−χ−=π/2.\nFigure 4 displays Gsas a function of χH, which is\nproportional to the exchange field, for several values of\nkFL. Here, we fix χ=π/4 to obtain finite spin current.\nFor a short ferromagnetic graphene with kFL= 0.1, the\nπ-periodicity is satisfied. With increasing kFL, theπ-\nperiodicity requires larger magnitude of χHas shown in\nthis figure, similar to Fig. 3.\nOur prediction is applicable as long as the continuum\nDirac equation is valid, which requires wide graphene\nnanoribbon. For the realization of our prediction of con-\ntrollable spin current by the gate voltage, U/EF>1\nis required. If we choose ferromagnetic graphene with\nkFL= 1 andEF≃1 meV as an example, we need\nthe length around 1 µm. Also, χ∼U/EF,χH∼\nH/EFand hence U,H∼1-10 meV is required. These\nvalues can be achieved by the present experimental\ntechnique.7,8,9,18,19,20\nIn summary, we studied spin transport in nor-\nmal/ferromagnetic/normal graphene junctions where a\ngate electrode is attached to the ferromagnetic graphene.\nWe found that because of the exchange field in the fer-\nromagnetic graphene, spin current through the junctions\nhas an oscillatory behavior with respect to the chemical\npotential in the ferromagnetic graphene, which can be\ntuned by the gate electrode.7,8,9Especially, spin current\nreversal by the gate voltage, which is not accompanied4ª « ¬\n® ¯ °\n± ² ³ ´\nµ\n¶ · ¸\n¹ º »\n¼ ½ ¾\n¿À\nÁ Â Ã Ä Å ÆÇ\nÈ É Ê Ë Ì ÍÎ\nÏ Ð ÑÒ Ó Ô Õ Ö × Ø\nÙ Ú Û Ü\nÝ\nÞ ß à\ná â ã\nä å æ çè\né ê ëìí\nî ï ð\nñ ò óô õ ö\nFIG. 4: (color online) Gsas a function of χH, which is propor-\ntional to the exchange field, for several values of kFL. Here,\nwe choose χ=π/4.with the current reversal, is obtained. The exchangefield\nof graphene is also known to be tunable by the inplane\nexternal electric field.18,19,20Therefore, our prediction of\nhigh controllability of spin transport in ferromagnetic\ngraphene junctions will contribute to the development\nof the spintronics.\nThe author acknowledges support by the JSPS.\n1T. Ando, J. Phys. Soc. Jpn. 74, 777 (2005).\n2M. I. Katsnelson and K. S. Novoselov, Solid State Com-\nmun.143, 3 (2007).\n3A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov and A. K. Geim, arXiv:0709.1163v1.\n4G. E. Volovik, The Universe in a Helium Droplet (Claren-\ndon, Oxford, 2003).\n5A. M. Tsvelik, Quantum Field Theory in Condensed Mat-\nter Physics (Cambridge Univ Pr, 2007).\n6K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y.\nZhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov,\nScience306, 666 (2004).\n7Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, Nature\n438, 201 (2005).\n8K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M.\nI. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A.\nFirsov, Nature 438, 197 (2005).\n9K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko,\nM. I. Kastenelson, U. Zeitler, D. Jiang, F. Schedin, and A.\nK. Geim, Nat. Phys. 2, 177 (2006).\n10K. Yang, Solid State Commun. 143, 27 (2007).\n11H. B. Heersche, P. Jarillo-Herrero, J. B. Oostinga, L. M. K.\nVandersypen, and A. F. Morpurgo, Nature 446, 56 (2007).\n12C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801\n(2005).\n13D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys.\nRev. B74, 155426 (2006).14N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and\nB. J. van Wees, Nature (London) 448, 571 (2007).\n15M. Ohishi, M. Shiraishi, R. Nouchi, T. Nozaki, T. Shinjo\nand Y. Suzuki, Jap. J. Appl. Phys. 46, L605 (2007).\n16S. Cho, Yung-Fu Chen, and M. S. Fuhrer, Appl. Phys.\nLett.91, 123105 (2007).\n17H. Haugen, Daniel Huertas-Hernando, and Arne Brataas,\narXiv:0707.3976v1.\n18Y.-W. Son, M. Cohen, and S. G. Louie, Nature (London)\n444, 347 (2006).\n19Er-Jun Kan, Zhenyu Li, Jinlong Yang, and J. G. Hou,\narXiv:0708.1213v1.\n20S. Dutta and S. K. Pati, arXiv:0712.4130v1.\n21J. Gonz´ alez, F. Guinea, M. A. H. Vozmediano, Phys. Rev.\nB59, R2474 (1999).\n22C. L. Kane and E. J. Mele, Phys. Rev. Lett. 93, 197402\n(2004).\n23A. F. Morpurgo and F. Guinea, Phys. Rev. Lett. 97,\n196804 (2006).\n24P. R. Wallace, Phys. Rev. 71, 622 (1947).\n25M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nature\nPhys.2, 620 (2006).\n26S. Bhattacharjee and K. Sengupta, Phys. Rev. Lett. 97,\n217001 (2006).\n27J. Linder and A. Sudbø, Phys. Rev. Lett. 99, 147001\n(2007)."    },    {        "title": "2111.15260v1.Majorana_Zero_Modes_in_Ferromagnetic_Wires_without_Spin_Orbit_Coupling.pdf",        "content": "Majorana Zero Modes in Ferromagnetic Wires without Spin-Orbit\nCoupling\nGiorgos Livanas , Nikolaos Vanas and Georgios Varelogiannis\nDepartment of Applied Mathematical and Physical Sciences, National Technical University of Athens,\n15780 Athens, Greece; n.vanas@protonmail.com (N.V .); varelogi@mail.ntua.gr (G.V .)\n*Correspondence: livanasg@mail.ntua.gr;\nAbstract: We present a novel controllable platform for engineering Majorana zero modes. The\nplatform consists of a ferromagnetic metallic wire placed among conventional superconductors, which\nare in proximity to ferromagnetic insulators. We demonstrate that Majorana zero modes emerge\nlocalised at the edges of the ferromagnetic wire, due to the interplay of the applied supercurrents and\nthe induced by proximity exchange fields with conventional superconductivity. Our mechanism does\nnot rely on the pairing of helical fermions by combining conventional superconductivity with spin-\norbit coupling, but rather exploits the misalignment between the magnetization of the ferromagnetic\ninsulators and that of the ferromagnetic wire.\nKeywords: conventional superconductors; ferromagnetic wire; ferromagnetic insulators; supercur-\nrents; Majorana zero modes\n1. Introduction\nThe development of quantum computers promises a new technological revolution [ 1].\nHowever, a fundamental obstacle hindering the development of quantum computation is\nquantum decoherence, the loss of quantum mechanical phase coherence and, therefore, of\ninformation encoded in qubits. Attempts to deal with this problem through quantum error\ncorrection algorithms lead to elaborate schemes consuming a highly disproportional amount\nof qubits. On the contrary, topological quantum computers suppress quantum decoherence at\nthe hardware level. Qubits in these devices are realized from topologically protected entities,\nwhich are immune to environmental noise and, therefore, decoherence [2–4].\nSuch topological protected entities are Majorana zero modes—massless neutral par-\nticles that constitute their own antiparticles. Characterised by their particle–antiparticle\nsymmetry, MZMs emerge as quasiparticles bound to defects or boundaries of topological\nsuperconductors [ 5]. The simplest topological superconductor is the effectively spinless\nsuperconducting phase of p-wave symmetry [ 6,7]. Although a topological p-wave state\nis proposed for the superconducting phase of Sr 2RuO 4[8], experimental results are yet\ninconclusive [9].\nInstead, several proposals have been put forward for engineering topological p-wave\nsuperconductivity out of more trivial materials. Among them, we distinguish conventional\nsuperconductor/topological insulator heterostructures [ 10], semiconductors in proximity to\nconventional superconductors [ 11], non-centrosymmetric superconductors [ 12] and planar\nJosephson junctions [ 13]. All these proposals effectuate the pairing of helical fermions by\ncombining conventional superconductivity with spin orbit coupling in the presence of a\nspin-splitting field, in order to realize an effectively spinless superconducting state.\n2. Majorana Zero Modes in Superconductor/Ferromagnet Heterostructures\nAs aforementioned, realising MZMs using conventional superconductors requires\nthe presence of a spin-splitting field. The simplest way to satisfy this requirement is byarXiv:2111.15260v1  [cond-mat.supr-con]  30 Nov 20212 of 14\napplying an external magnetic field. However, magnetic fields are detrimental to con-\nventional superconductivity; thereby, many proposals for realising MZMs are based on\nheterostructures among superconductors with ferromagnetic materials. Superconductor-\nferromagnet heterostructures do not require the application of an external magnetic field\nand, therefore, are advantageous with respect to other platforms. Following this line of ar-\ngument, topological superconductivity and MZMs have been demonstrated to emerge in\nhalf-metal/superconductor [ 14] or ferromagnet/unconventional superconductor [ 15] het-\nerostructures, in ferromagnetic wires proximised to conventional superconductors [16–18]\nand in ferromagnetically aligned chains of magnetic impurities embedded in conven-\ntional superconductors [ 19–26]. Spin-orbit coupling is an essential component in all of\nthese proposals.\n2.1. A Novel Platform\nHere, we put forward a novel platform for engineering topological superconductivity\nand Majorana zero modes, which does not require spin orbit coupling. The platform, as\npresented in Figure 1, consists of a ferromagnetic metallic wire, which is placed between\ntwo conventional superconducting layers where supercurrents flow in opposite directions.\nThe superconducting layers are deposited on ferromagnetic insulators. The magnetisations\nof the two ferromagnetic insulators point in opposite directions and both are perpendicular\nto the magnetisation of the ferromagnetic metallic wire. Majorana zero modes emerge\ny\nzxSC\nFISC\nFM wire\nFIFM wire\nFigure 1. (Left) Novel platform for engineering Majorana zero modes. A ferromagnetic metallic wire\nplaced among two conventional superconductors where supercurrents flow in opposite directions\n(blue arrows). The superconducting layers are deposited upon ferromagnetic insulators with opposite\nmagnetisations (black arrows), which are perpendicular to the magnetisation of the ferromagnetic\nwire (white arrow). ( Right ) The energy bands, Es,\u0006where srefers to spin components and \u0006to hole\nand particles, respectively, of a conventional superconductor when a supercurrent and an exchange\nfield are applied. Supercurrent breaks inversion symmetry ( k,\u0000kmomenta are not equivalent)\nand the exchange field breaks spin symmetry ( sands0are not equivalent) leading to the emergence\nofp-wave superconducting correlations. Arrows indicate the pairing components of conventional\nsuperconductivity.\nlocalised at the edges of the FM, due to the synergy between the externally applied super-\ncurrents with the emergent in the SCs exchange fields [ 27]. We remark that several works\nhave demonstrated both theoretically [ 28,29] and experimentally [ 30–32] the emergence of\nexchange fields in conventional SCs proximised by FIs. Moreover, developments in the\nfabrication of SC-FI heterostructures [ 33] and ferromagnetic metallic nanonwires embedded\nin conventional SCs [ 34–36], enhance the experimental feasibility of the proposed platform.3 of 14\n2.2. The Underlying Mechanism\nAccording to our mechanism, a topological superconducting state is stabilized over the\nFM wire, when charge supercurrents are applied in conventional SCs where finite exchange\nfields are present. The supercurrent and the exchange field break inversion and spin\nsymmetries, respectively, and, therefore, partially convert conventional superconductivity\nto ap-wave superconducting field. This emergent p-wave superconducting field still pairs\nelectrons of different energy bands of the superconductor, while MZMs require intraband\nsuperconductivity.\nHowever, in an FM wire with magnetisation perpendicular to the magnetisation of the\nferromagnetic insulators, the induced p-wave superconducting field pairs electrons of the\nsame energy band of the FM. An intraband superconducting component emerges when\nthed-vector of p-wave superconductivity [ 37] is misaligned to the exchange field of the\nFM wire [ 38]. Thus, when this p-wave superconducting field mediates into the FM wire,\ndue to the proximity effect, an effectively spinless superconducting state is stabilised, and\nMZMs emerge. In the next section, we present analytically how the triplet superconducting\ncomponent emerges, due to the coexistence of the exchange field and the supercurrents\nwith conventional superconductivity.\n3. Induced p-Wave Superconductivity\nIn order to demonstrate how triplet p-wave superconductivity is induced, we consider\na superconducting wire with order parameter D, where a supercurrent Jis applied in the\npresence of an exchange field hz. We consider the following Hamiltonian\nH=Z\ndxY†(x)\u0012\n[¶2\nx\n2m+m]tz+hztzsz+Dtysy\u0013\nY(x) (1)\nwhere Y†= (y†\n\",y†\n#,y\",y#)the extended Nambu spinor and tandsare the Pauli ma-\ntrices acting on particle-hole and spin space, respectively, y(†)\ns(x)the operator destroy-\ning(creating) an electron with spin s at coordinate x, hzthe exchange field and D=jDjeiJxtz\nthe singlet superconducting field. Under the gauge transformation\nY(x)!e\u0000iJ/2xtzY(x) (2)\nkinetic term¶2x\n2mtransforms to¶2x\n2m+iJ\n2m¶x\u0000J2\n8mtzwhere iJ\n2m¶xis a current term and \u0000J2\n8mtz\ncan be absorbed in the chemical potential m. Thus, Hamiltonian Equation (1) takes the form\nH=Z\ndxY†(x)\u0012\n[¶2\nx\n2m\u0000m]tz+iJ\n2m¶x+hztzsz+Dtysy\u0013\nY(x) (3)\nConsidering a translationally invariant wire Hamiltonian Equation (3) takes the following\nform upon the Fourier transformation\nY†(x) =Z\ne\u0000ikxY(k)†dk (4)\nto momenta kspace, whereR\nY†(x)Y(x) =R\nY†(k)Y(k) =1.\nH=Z\ndkY†(k)ˆH(k)Y(k) =Z\ndkY†(k)\u0012\n[k2\n2m\u0000m]tz+J\n2mk+hztzsz+Dtysy\u0013\nY(k)(5)4 of 14\nThe energy bands of the wire\nEs,\u0006(k) =\u0000shz+J\n2mk\u0006s\u0012k2\n2m\u0000m\u00132\n+D2 (6)\nwhere s=\u0006corresponds to#and\"bands respectively, derive from the transformation\nU(k) =p\n2\n2\u0014\u0002\nA\u0000(k)(tz+t0) +A+(k)(tx+ity)\u0003\neip\n4sy+ (tx\u0000ity) + ( t0\u0000tz)\u0015\n(7)\nand\nU(k)\u00001=p\n2\n2\u0014\nA\u0000(k)(t0\u0000tz) +A+(k)(tx\u0000ity) +B(k)(tx+ity+t0+tz)e\u0000ip\n4sy\u0015\n(8)\ndiagonalising Hamiltonian matrix ˆH(k), where\nA\u0006(k) =k2\n2m\u0000m\u0006q\n(k2\n2m\u0000m)2+D2\nD(9)\nand\nB(k) =Dq\n(k2\n2m\u0000m)2+D2(10)\nThe induced p-wave correlations derive from the equation\n<Dp>µå\nkå\nmnF(Em)[U(k)†ktxd\u0001˜sisyU(k)]mm (11)\nwhere nF(Em)the Fermi distribution and ˜s= (sx,tzsy,sz). For d=dz= (0, 0, 1 ), we find\n<Dpz>µå\nkå\nmknF(Em)[U(k)†txsxU(k)]mm (12)\nwhere\n[U(k)†txsxU(k)] =Dq\n(k2\n2m\u0000m)2+D2tzsz+k2\n2m\u0000m\nDtxsz+(k2\n2m\u0000m)2\nDq\n(k2\n2m\u0000m)2+D2itysz(13)\nTherefore,\n<Dpz>µå\nkkDq\n(k2\n2m\u0000m)2+D2(nF(E+,\u0000)\u0000nF(E+,+)\u0000[nF(E\u0000,\u0000)\u0000nF(E\u0000,+)])\n<Dpz>µå\nkkDq\n(k2\n2m\u0000m)2+D2å\ns,\u0006\u0007snF(Es,\u0006) (14)\nand apparently the <Dpz>correlations emerge from the imbalance between singlet pairing\nin the k\",\u0000k#channel <y†\nk,\"y†\n\u0000k,#+y\u0000k,#yk,\">µnF(E+,\u0000)\u0000nF(E+,+)and in the\nk#,\u0000k\"channel <y†\n\u0000k,\"y†\nk,#+yk,#y\u0000k,\">µnF(E\u0000,\u0000)\u0000nF(E\u0000,+). From Equation (14) it\nis straightforward that, for h=0,<Dpz>=0since, in this case, nF(E\u0000,+) =nF(E+,+)and\nnF(E\u0000,\u0000) =nF(E+,\u0000), and therefore ås,\u0006\u0007snF(Es,\u0006) =0. Moreover, for J=0expression5 of 14\nås,\u0006\u0007snF(Es,\u0006)is even in momenta, and therefore <Dpz>=0, due to the multiplication\nwith momenta k. Thus, p- wave correlations are induced only when both h6=0 and J6=0.\nFor the other two components of the dvector, dy= (0, 1, 0 )anddx= (1, 0, 0 ), there are\nnop-wave correlations induced. Particularly, for dy= (0, 1, 0 )we get\n<Dpy>µå\nkå\nmknF(Em)[U(k)†tys0U(k)]mm (15)\nwhere\n[U(k)†tys0U(k)] =Dq\n(k2\n2m\u0000m)2+D2tzsy+k2\n2m\u0000m\nDtxsy+(k2\n2m\u0000m)2\nDq\n(k2\n2m\u0000m)2+D2itysy(16)\nTherefore,\n<Dpy>=0 (17)\nsince matrix [U(k)†tys0U(k)]has no diagonal term. Similarly, for dx= (1, 0, 0 )we obtain\n<Dpx>µå\nkå\nmknF(Em)[U(k)†txszU(k)]mm (18)\nwhere\n[U(k)†txsxU(k)] =Dq\n(k2\n2m\u0000m)2+D2tzsx+k2\n2m\u0000m\nDtxsx+(k2\n2m\u0000m)2\nDq\n(k2\n2m\u0000m)2+D2itysx(19)\nTherefore,\n<Dpx>=0 (20)\nsince, again, matrix [U(k)†txszU(k)]has no diagonal terms. Thus, we conclude that only\np-wave correlations with <dkh>emerge, due to the imbalance among the two spin\nconfigurations of interband conventional pairing.\n4. Engineering Majorana Zero Modes without Spin-Orbit Coupling\n4.1. Intraband p-Wave Superconductivity in the Ferromagnetic Wire\nAs demonstrated in the previous section, triplet p-wave correlations are induced in\na conventional superconductor when a supercurrent and an exchange or Zeeman field\nare applied. However, topological superconductivity and Majorana zero modes cannot be\nengineered in a superconducting wire simply by applying a supercurrent and a magnetic\nfield. The triplet p-wave correlations induced in this setup still connect different energy\nbands of the wire, while MZMs require intraband or effectively spinless superconductivity.\nNevertheless, when these correlations mediate to a material with magnetisation perpendic-\nular to the exchange field induced in the superconductors, an intraband superconducting\nstate that hosts MZMs is realised.\nConsidering the particular setup presented in Figure 1, the induced Dpcorrelations\nin the SCs correspond to Cooper pairs of electrons with opposite spin, i.e., pz\u0011txsx\ncorrelations. These correlations, for which d=dz, mediate in the ferromagnet with6 of 14\npolarisation along the x-axis, i.e., perpendicular to the dvector of the induced correlations,\ndescribed by the following Hamiltonian\nHFM=Z\ndxY†(x)\u0012\n[¶2\nx\n2mFM\u0000mFM]tz+hFMtzsx+Dp¶xtxsx\u0013\nY(x). (21)\nThe energy bands of the FM wire derive from the following transformation\nY0=UY, U=0\nBB@0 0 1 1\n0 0\u00001 1\n\u00001 1 0 0\n1 1 0 01\nCCA(22)\nwhere Uis the matrix that diagonalises the Hamiltonian HFM. In this eigenbasis, the\ninduced pztriplet correlations are expressed as\nDpµtxs0 (23)\nFrom Equation (23), it becomes apparent that, in the eigenbasis of the FM, the induced\nby proximity p-wave field Dppairs electrons from the same energy band, i.e., we obtain\nintraband pairing. The Hamiltonian Equation (21) upon the Utransformation acquires\nthe form\nH0\nFM=Z\ndxY0†(x)\u0012\n[¶2\nx\n2mFM\u0000mFM]tz+hFMtzsz+Dp¶xtxs0\u0013\nY0(x). (24)\nwhich can be reduced into two copies of the Kitaev’s Hamiltonian\nH0\nFM,s=Z\ndxY0†\ns(x)\u0012\n[¶2\nx\n2mFM\u0000mFM,s]tz+Dp¶xtx\u0013\nY0\ns(x). (25)\nwhere Y0†\ns= (y†\ns,ys),s=\u0006the band index and mFM,s=mFM+shFM.\nBased on the above analysis, it becomes clear that the optimal configuration for the\nemergence of MZMs in the particular platform, is that of anti-parallel supercurrents in\nthe SCs and anti-parallel magnetisations in the FIs. This configuration guarantees that no\nsupercurrent or exchange fields parallel to the magnetisation of the FIs are induced in the\nFM wire. Thus, the d-vector of the induced p-wave superconducting correlations remains\nperpendicular to the magnetisation of the FM wire and an intraband superconducting\nstate is realised. However, as we will elaborate in a future work, MZMs are robust against\nsignificant deviations from this optimal configuration for the supercurrents and the FIs\nmagnetisations.\n4.2. Topological Criteria for the Ferromagnetic Wire\nBased on Hamiltonian Equation (24) describing the FM wire, we derive, in this section,\nthe criteria regarding the exchange field hFMand the chemical potential mFMfor which\nMZMs emerge localised at the edges of the FM. To this aim, we consider periodic boundary\nconditions and transform Hamiltonian Equation (24) in momenta kspace. Introducing the\nspin-dependent Nambu spinor Yk= (yk,s,yk,s0,y†\n\u0000k,s0,y†\n\u0000k,s)T, where y(†)\nk,sare the operators7 of 14\ndestroying (creating) an electron with momenta k and spin s and the Pauli matrices tfor\nthe particle-hole and sfor the spin space, the Hamiltonian of the FM acquires the form\nH=å\nkY†\nk\u0002(2tcosk+mFM)tz+hFMtzsz+Dtysy+Dpsinktxs0\u0003\nY†\nk, (26)\nwhere we also considered an induced by proximity conventional superconducting field\nD. We remark that Hamiltonian Equation (26) is expressed in lattice instead of continuum\nspace, in order to connect with the numerical results presented in the next section. In the\nparticular case, the reality conditions, QH(k)Q\u00001=H(\u0000k)andXH(k)X\u00001=\u0000H(\u0000k)\nwhere QandXanti-unitary operators [ 39], are satisfied for X=txKwithX2=Iand\nQ=itzszKwithQ2=I. In addition, a unitary chiral symmetry operator S=Q\u0001X=tysz\nanticommutes with the Hamiltonian.\nThus, the system belongs to the BDI symmetry class, characterised by an integer Z\ntopological invariant, and Hamiltonian Equation (26) acquires a block off-diagonal form\nH0(k) =\u00120A(k)\nA(k)†0\u0013\nin the eigenbasis of the chiral operator S. Therefore, by\napplying the transformation Us=i[tx+ity\u0000tz]sy+ [tx\u0000ity+tz]sx, we obtainA(k) =\nhFMsz\u0000[2tcosk+mFM+iDpsink] +iDsx. The topological invariant of the Hamiltonian\nis defined as the winding number N=\u0000i\n2pRk=2p\nk=0dz(k)\nz(k)[40] , where z(k)the unimodular\ncomplex number defined as z(k) = Det(A(k))/jDet(A(k))j. For the particular system,\nwe find\n<z(k) =h\n(2tcosk+mFM)2\u0000(hFM)2+D2+D2\npsin2ki\n=z(k) =2[(2tcosk +mFM)px]sink. (27)\nSince within each topological phase, the q(k)is a continuous function of momentum k,\nthe maximum possible value of the winding number relates to the number n of roots\n=Det(A(k))within the interval [0, 2p),maxjNj=n/2. Notice that because =Det(A(k)) =\n\u0000=Det((A(\u0000k)), number n is always even. Next, we distinguish the two following cases.\nThe first case holds for jmj>2t. In this case,=Det(A(k)) = 0only for k=0,p,\nand therefore the maximum absolute value of the winding number is maxjNj=1. The\ntopologicalN=1 phase is realised when z(k=0)z(k=p) =\u0002\nD2+ (2t+mFM)2\u0000h2\nFM\u0003\n\u0002\nD2+ (2t\u0000mFM)2\u0000h2\nFM\u0003<0, which results in the following condition for the chemi-\ncal potential\nj2t\u0000q\nh2\nFM\u0000D2j<jmFMj<j2t+q\nh2\nFM\u0000D2j (28)\nor equivalently for the exchange field\nq\nD2+ (2t\u0000mFM)2<jhFMj<q\nD2+ (2t+mFM)2. (29)\nThe second case holds for jmFMj<2t. When this condition holds, =Det(A(k)) =\n0is realised also for another pair of momenta k\u0003=\u0006cos\u00001\u0000\f\fmFM\n2t\f\f\u0001\n. However, since\nforp\nD2+ (2t\u0000mFM)2<jhFMj<p\nD2+ (2t+mFM)2,z(k=0)z(k=p)<0, it is\nstraightforward that jNj=1within these limits, even in this case. In general, considering\nDp<<D, ajNj=2 topological phase can be realised for\nD.jhFMj<q\nD2+ (2t\u0000mFM)2 (30)8 of 14\nfor which z(k=0)z(k=p)>0 and z(k=k\u0003)z(k=p)<0.\n4.3. Numerical Calculations for the FI-SC-FM-SC-FI Heterostructure\nIn order to verify the above arguments, we employ the following lattice Bogoliubov\nde Gennes equation,\nH= [Hl,SC+Hl,tc+HFM+Hr,tc+Hr,SC] (31)\nwhich describes the FI-SC-FM-SC-FI heterostructure presented in Figure 1, with\nHa,SC=å\ni,jY†\na,i\u0002(t fi,j+mSC)tz+ha,SCtzsz+Dtysy+Ja\u0001gi,j\u0003\nY†\na,j (32)\nthe Hamiltonian of the superconductor, where a=l,rstand for left and right SC with\nhl,SC=\u0000hr,SC,Jl=\u0000Jrand\nHFM=å\ni,jF†\ni\u0002(t fi,j+mFM)tz+hFMtzsx\u0003\nF†\nj (33)\nthe Hamiltonian of the FM, where Y†\na,i,SC= ( y†\na,\",i,y†\na,#,i,ya,\",i,ya,#,i)andF†\ni,FM\n= (f†\n\",i,f†\n#,i,f\",i,f#,i)with y(†)\na,s,iand f(†)\ns,ithe operators destroying (creating) an electron\nwith spin s of the a=l,rSC and FM, respectively, at lattice site i,tand sthe Pauli\nmatrices acting on particle-hole and spin space respectively, fi,j=dj,i\u0006ˆx+dj,i\u0006ˆyand\ngi,j= (\u0006idj,i\u0006ˆx,\u0006idj,i\u0006ˆy)the even and odd in spatial inversion, respectively, functions\nconnecting nearest neighbours lattice points. Moreover, tis the hopping integral and Jthe\nsupercurrent term. The proximity of the SCs to the FIs is modelled by the introduction of\nan exchange field term in the HSCEquation (32). The SCs are connected with the FM wire\nthrough the following Hamiltonian\nHa,tc=å\ni,j,stcy†\na,i,sfj,s+c.c. . (34)\nIn general, we consider tc=tfor simplicity. MZMs are anticipated to emerge localised at\nthe edges of the wire for j2t\u0000p\n(hFM)2\u0000D02j<jmFMj<j2t+p\n(hFM)2\u0000D02jwhere D0\nis the induced by proximity conventional superconducting field over the FM. In Figure 2a\nwe present the phase diagram for the setup of Figure 1 considering D/2t=1,mSC/2t=0,\nJ/2t=0.2and hSC/2t=0.2. Based on this diagram, we observe that for mFM<2t, the\ninduced singlet superconducting field over the FM is D0/2t'0.25 while the hopping term\nalong the wire is normalised to t0'0.86t, due to the coupling of the wire to the SCs [41].\nFormFM>2t, again, the hopping term along the wire is normalised to t0'0.86t\nand the black lines hFM/2t'mFM/2t\u00002t0/2tandhFM/2t'mFM/2t+2t0/2tdefine the\nregion of parameters for which N=1and a pair of MZMs emerge at the edges of the FM\nwire. Notice that the particular phase diagram is in accordance with the topological criteria\nfor theN=1 andN=2 topological phases presented in the previous section. However,\nwe remark that the particular results correspond to a fixed conventional order parameter,\nwhich has not been self-consistently determined over the SC regions.\nTherefore, the effect of finite momentum Cooper pairs, which, in principle, can emerge\ndue to the presence of the supercurrent and the exchange field in the conventional super-\nconductor, on the phase diagram (Figure 2), has not been investigated and will be examined\nin a future work.9 of 14\nN=1N=0\nN=0 N=2a)\n0 0.2 0.4 0.6 0.8 1\nhFM/2t-0.15-0.1-0.0500.050.10.15E/∆\nb)0.3\n0.2\n0.1\n0\n-0.1\n-0.2\n-0.3\n0 0.5 1 1.5\nhFM/2t-0.15-0.1-0.0500.050.10.15E/∆\nc)\n0.20.3\n0.1\n0\n-0.1\n-0.2\n-0.3\n0 0.5 1 1.5 2\nhFM/2t-0.4-0.200.20.4E/∆\nd)0.8\n0.4\n0\n-0.4\n-0.8\nFigure 2. ForD/2t=1,mSC/2t=0,J/D=0.2andhSC/D=0.2, (a) topological phase diagram with\nrespect to the chemical potential mFMand the exchange field hFMof the ferromagnet. The low energy\nspectra with respect to hFMfor (b)mFM/2t=0, (c)mFM/2t=0.4and ( d) for mFM/2t=1.2denoted\nby red dashed lines in figure ( a). The numerical results verify the topological criteria presented in the\nprevious section.\n4.4. Local Density of States\nThe pursuit of functional platforms for topological quantum computation, requires\nthe experimental verification of the theoretical proposals for engineering topological su-\nperconductivity, by detecting the emergent MZMs. A characteristic signature of MZMs in\nconductance spectroscopy measurements is the quantized zero-bias tunnelling conductance.\nEmanating from an Andreev reflection resonant at zero energy, the zero bias conductance\nin the presence of a single MZMs pair equals2e2\nh, i.e., double the conductance quantum,\nirrespective of the tunnelling barrier [42,43].\nSeveral tunnelling experiments have demonstrated such zero bias conductance peaks\nand therefore provided supporting evidence for the emergence of MZMs in the semicon-\nductor nanowires [ 44], the topological insulator/conventional superconductor heterostruc-\ntures [ 45,46], the magnetic adatoms embedded in conventional superconductor [ 47], the\nheavy metal surfaces [ 48], the quantum anomalous Hall insulator [ 49], the iron-based su-\nperconductors [ 50] and the planar Josephson junctions [ 51] platforms. However, the results\nare far from being conclusive, since in each case several deviations from the corresponding\ntheoretical predictions are observed.\nSince differential conductance measurements in scanning tunnelling spectroscopy\nexperiments are proportional to the local density of states of the sample, when the scanning\ntunnelling junction can be approximated by a point contact and the tunnelling amplitude\nis small, we calculate the local density of states at characteristic positions in the FM wire10 of 14\nin order to link our results with relevant experiments. To this end, we solve the following\neigenvalue equation\nˆHi,j0\nBB@un,\"(j)\nun,#(j)\nvn,\"(j)\nvn,#(j)1\nCCA=En0\nBB@un,\"(i)\nun,#(i)\nvn,\"(i)\nvn,#(i)1\nCCA(35)\nwhere ˆHi,jthe Hamiltonian matrix of the FI-SC-FM-SC-FI heterostructure (Equation (31))\nand\u0000\nun,\"(j)un,#(j)vn,\"(j)vn,#(j)\u0001Tthe eigenstate corresponding to eigenenergy\nEn. Based on the eigenstates and eigenvalues of the Hamiltonian we calculate the local\ndensity of states for spin up and down components according to the following equations\nN(i,w)\"=\u00001\npImˆGR\ni,i,w,\"\"=å\nnh\njun,\"(i)j2d(w\u0000En)i\n(36)\nN(i,w)#=\u00001\npImˆGR\ni,i,w,##=å\nnh\njun,#(i)j2d(w\u0000En)i\n(37)\nwhere ˆGR\ni,j,w=1\nw+ih\u0000ˆHi,jthe retarded Green’s function. In order to account for the presence\nof impurities in the system, we consider a scattering time t, which corresponds to a mean\nfree path l=uFt, where uFthe Fermi velocity. Including the scattering from impurities,\nthe density of states derives from equation\nN(i,w) =1\npå\nn\u0014\n(jun,\"(i)j2+jun,#(i)j2)g\n(w\u0000En)2+g2\u0015\n(38)\nwhere the Dirac function d(w\u0000En)has been substituted by a Cauchy–Lorentz function\n1\npg\n(w\u0000En)2+g2, with g=1/t. Taking the clean superconductor limit, we assume that l/x'\n1000, where x=uF\npDthe superconducting coherence length, and thereforeg\nD=px\nl'0.003.\nIn Figure 3, we present the local density of states N(w)for three values of the exchange\nfield hFMof the wire considering mFM/2t=0.4, which corresponds to diagram (c) of\nFigure 2 . From Figure 3a we deduce that the singlet pairing field induced in the wire, due\nto proximity with the conventional SC, is D0/D'0.25. Since in this case hFM<D0the\nFM is in a non-topological phase with N=0. When hFM>D0the wire is in a topological\nN=2(Figure 3b) orN=1(Figure 3c) phase, and a peak for w=0emerges at the local\ndensity of states spectrum only at the edges of the wire, signifying the presence of localised\nzero energy states, the MZMs.\nForhFM/2t=0.36 whereN=2the two MZMs localised at the beginning of the wire,\nacquire opposite spin polarisation. The difference in the two zero energy peaks in Figure 3 b\nindicates the different localization of the two MZMs, due to the difference in the Fermi\nvelocity, uF=2tsin(cos\u00001(\u0000(\u0006hFM+mFM)\n2t), among the two spin bands. The localization\nlength lMof the MZMs equals lM=uF\nDp. For hFM/2t=0.72, a single pair of MZMs with\nparticular spin polarisation emerges localised at the FM wire as indicated from the zero\nenergy peak, which is significant for the spin down component. However, the zero energy\ndensity of states is also finite for the other spin component, due to the finite hSC.11 of 14\n-0.5 -0.25 0 0.25 0.5\nE/∆05101520N(ω)/N0a)\n-3 -2 -1 0 1 2 3\nω/Eg0204060N(ω)/N0b)\n-3 -2 -1 0 1 2 3\nω/Eg0102030N(ω)/N0c)\nFigure 3. The local density of states N(w)at the edge and the middle of the FM wire for ( a)hFM/2t=\n0.18 for which the wire is in a non-topological phase where the induced singlet pairing D0prevails,\n(b) for hFM/2t=0.36 for which a topological N=2phase is realised and ( c) for hFM/2t=0.72 for\nwhich the wire is in a topological N=1phase. Solid and dashed lines correspond to the edge, mean\nvalue of the density at the first five lattice sites of the wire, and the middle of the wire, respectively. The\nblue line corresponds to the spin up, while the green line corresponds to the spin down component of\nthe local density of states. The particular diagrams are relevant to spin-polarised scanning tunnelling\nmicroscopy (SP\u0000STM) measurements tips polarised (anti \u0000)parallel to the magnetisation of the FM\nwire. In ( b,c) diagramms energy is normalised with respect to the triplet superconducting energy gap\nEginduced in the wire. N0is the density of states in the normal phase.\n4.5. Robustness of Emergent Majorana Zero Modes against Disorder\nIn realistic experimental setups, the presence of disorder is unavoidable. On this\naccount, we demonstrate, in this section, the robustness of the emergent MZMs against\ndisorder in the FM wire. In particular, we consider a random on-site potential dmFM(x)tz\nthat derives from a normal distribution with mean value 0 and variance s2corresponding\nto correlator <dmFM(x)dmFM(x0)>=s2d(x\u0000x0), i.e., dmFM(x) =D(0,s).\nThe mean free path lalong the wire is related to the variance through the following re-\nlation l=uFt=u2\nf/s2=uF/[ps2N(m)]where uFthe Fermi velocity and N(m) =1/puF\nthe density of states at the Fermi level. In the lattice model described by Hamiltonian\nEquation (33), uFin the FM wire equals uF(m) =2tsin[cos\u00001(\u0000m/2t)]considering a com-\nmon Fermi velocity for the two spin components. We define the dimensionless parameter\nk=l\nx=Dp\ns2N(m), where x=uF\npDpis the coherence length of the p-wave superconductivity\ninduced in the FM wire. In Figure 4, we present the low energy spectrum of the wire for\nmFM/2t=0.4 and hFM/2t=0.36 for whichN=2 and hFM/2t=0.72 for whichN=1.\nForhFM/2t=0.36, we observe that the energy gap essentially closes for l/x=2,\nhowever, the spin up MZM remains at zero energy until 2l=x, which is the critical value\nof disorder for p-wave superconducting wires [ 52]. We attribute the difference in the critical\nmean free path for the two MZMs pairs in their different localization. For hFM/2t=0.72\nthe energy gap closes at l'x. We remark that in both cases, the energy gap is reduced by\n50% with respect to that of a pristine wire when l'10x, signifying the robustness of the\nemergent MZMs against disorder of the chemical potential of the FM wire.\nFinally, we investigate the robustness of the emergent MZMs against disorder in the\nexchange field hFMof the FM wire. We remark that our mechanism does not depend on\nthe structural characteristics of the FM wire, which can therefore also be considered as an\narray of ferromagnetically aligned magnetic impurities. Magnetic impurities induce in the\nsuperconductor subgap bound states, the Yu–Shiba–Rusinov states [ 53–55]. Depending on\nthe hopping integral among the impurities, these subgap states form an energy band, which\nmay lie within the superconducting gap (Shiba limit), or extent above the superconducting\nenergy gap (wire limit) [18]. Our mechanism is valid in both limits.\nConsidering this possible formation of the FM wire, it is worth examining the ro-\nbustness of the emergent MZMs against disorder in the ferromagnetic alignment of the12 of 14\n-1 -0.5 0 0.5 1\nlog(l/ξ)-1-0.500.51E/Egap,0\n-1 -0.5 0 0.5 1\nlog(l/ξ)-1-0.500.51E/Egap,0\n-1 -0.5 0 0.5 1\nlog(lm/ξm)-1-0.500.51E/Egap,0\nFigure 4. Low energy spectrum of the FM wire with respect to the chemical potential disorder\nstrength l/xformFM/2t=0.4and ( left)hFM/2t=0.36, (middle )hFM/2t=0.72. The dashed black\nline denotes critical disorder 2l=x, for which a topological phase transition occurs in p\u0000wave\nsuperconducting wires. In both cases, the energy gap is reduced by 50% with respect to that of a\npristine wire, for l'10x. (right ) For hFM/2t=0.72 the low energy spectrum with respect to magnetic\ndisorder lm/xm. Although the energy gap is again reduced by approximately 50% forlm=10xm,\nthe MZMs appear to acquire finite energy only for lm<3xm. Green and blue colours for the zero\nenergy modes correspond to spin components as in Figure 3. Egap,0is the energy gap in the pristine\nferromagnetic wire.\nmagnetic impurities and, therefore, in the corresponding orientation of the local exchange\nfield, which otherwise is considered to have a fixed magnitude. In complete analogy to\nthe previous case, we consider two normal distributions Dq= (0,sm)andDf= (0,sm)for\nthe angle qof the local magnetisation with respect to the x-axis and for the angle fin the\ny\u0000zplane.\nWe define a magnetic mean free path lm=uF\ns2mN(m)as a measure of disorder of the local\nexchange field and the magnetic coherence length xm=uF\nphFM. In Figure 4, we present\nthe low energy spectrum of the FM wire with respect to disorder strength lm/xmonly for\nhFM/2t=0.72. We note that for hFM/2t=0.36 the two pairs of MZMs interact with each\nother and acquire finite energy even for infinitesimal magnetic disorder, since magnetic\ndisorder breaks the spin symmetry that prevents the two MZMs from interacting with\neach other.\n5. Conclusions\nWe present a novel platform for realising Majorana zero modes based on\nsuperconductor–ferromagnet heterostructures. The platform consists of a ferromagnetic\nwire placed among conventional superconductors, which are proximised by ferromagnetic\ninsulators. Instead of spin-orbit coupling, the essential element of the physical mechanism\nunderlying the proposed platform is the misalignment between the magnetisation of the\nferromagnetic wire and that of the ferromagnetic insulators.\nThe robustness of the emergent MZMs against impurities disorder resembles that\nof a psuperconducting wire, although the emergence of two pairs of MZMs and the\nobservation of a zero bias conductance peak in the corresponding phase is hindered even\nfor infinitesimal disorder in the magnetisation of the FM wire. 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Here we have developed the Landau theory description of the phase diagram and established\nthe speci\fc ordering arising at each type of transition.\nThe phase transitions to the ferromagnetic superconducting state are inevitably accompanied\nby the emergency of screening currents. The corresponding magnetostatics considerations allow\nto establish the signi\fcant di\u000berence between the transition from the ferromagnetic to the ferro-\nmagnetic superconducting state and the transition from the superconducting to the ferromagnetic\nsuperconducting state.\nPACS numbers: 74.20.De, 74.25.Dw, 74.25.Ha, 74.20.Rp, 74.70.Tx\nI. INTRODUCTION\nThe superconductivity in the uranium ferromagnetic\ncompounds UGe 2and URhGe discovered more than\ndecade ago [1, 2] and more recently in the related com-\npound UCoGe [3] is still the subject of quite active in-\nvestigations (see recent experimental [4] and theoretical\n[5] reviews and references therein). The existence of su-\nperconducting state at temperatures far below the Curie\ntemperature and very high upper critical \feld in these\nmaterials do not leave doubts that here we deal with the\ntriplet superconductivity like it is in the super\ruid3He.\nThis is also con\frmed by the measurements of the Knight\nshift on nucleus of59Co which is proved unchanged in the\nsuperconducting state [6].\nOne of many peculiar properties of UCoGe is that the\nferromagnetism in this compound is suppressed by pres-\nsure whereas the superconductivity arising at small pres-\nsures inside of the ferromagnetic state continues to exist\nat high pressures in the paramagnetic state.The pressure-\ntemperature phase diagram shown in Fig.1, has been es-\ntablished \frst in the paper [7] and then con\frmed in\nmany subsequent studies (see f.i. the last one [8]). The\nphase transition from the paramagnetic to the ferromag-\nnetic state and the following to it the phase transition to\nthe ferromagnetic superconducting state at low pressures\nand the phase transition from the normal to the super-\nconducting state at high pressures are \frmly registered.\nWhile the phase transition from the superconducting to\nthe ferromagnetic superconducting state shown on the\nFig.1 by the dashed line is still not con\frmed experimen-\ntally.\nA theoretical phase diagram description has been pro-\nposed recently by Cheung and Raghu [9]. Making use\nthe numerical calculations applied to the minimal Lan-\ndau model of neutral ferromagnetic super\ruid state with\none component order parameter for each spin up-up and\nspin down-down Cooper pair states they were able to\nreproduce the general structure of the UCoGe phase di-agram and to predict a \frst order phase transition near\nthe boundary between the normal phase and the ferro-\nmagnetic superconducting phase.\nHere I reconsider the same problem making use the\nanalytical calculations applied to the same minimal\nmodel for neutral ferromagnetic or nonmagnetic super-\n\ruid states. The results of Ref.9 were con\frmed.\nA phase transition of normal metallic to the super-\nconducting state has its own speci\fc properties di\u000berent\nfrom the properties of a phase transition in the neutral\nFermi liquid to the super\ruid state. So, in the last part of\nthe paper I will discuss the signi\fcant di\u000berence between\nthe two transitions, namely, between the phase transition\nfrom the ferromagnetic normal state to the ferromagnetic\nsuperconducting state and the transition from the super-\nconducting state to the ferromagnetic superconducting\nstate. This di\u000berence arises due to the essentially di\u000ber-\nent screening of magnetic moment at these two transi-\ntions. In the latter case the screening is complete and\ninstead of a bulk phase transition there is the gradual\nformation of the Meissner state as it occurs in a super-\nconductor of the second kind under external magnetic\n\feld smaller than Hc1.\nII. MODEL\nThe triplet-pairing superconducting state order pa-\nrameter is given by the complex spin-vector [10]\nd(k;r) = (1)\n1\n2\u0002\n\u0000\u0001\"(k;r)(^x+i^y) + \u0001#(k;r)(^x\u0000i^y)\u0003\n+ \u00010(k;r)^z;\nwhere \u0001 \"(k;r), \u0001 #(r;k;r), \u0001 0(k;r) are the amplitudes\nof spin-up, spin-down and zero-spin of superconducting\norder parameter depending on the Cooper pair centre\nof gravity coordinate rand the momentum kof pair-\ning electrons. In the tetragonal ferromagnets with easy\naxis along ^zdirection there are only two superconductingarXiv:1611.08440v4  [cond-mat.supr-con]  8 Feb 20172\nstates A and B with di\u000berent critical temperature [11].\nThe general form of the order parameter for the A-state\nin a two-band spin-up, spin-down superconducting ferro-\nmagnet\n\u0001\"\nA(k;r) =^kx\u0011\"\nx(r) +i^ky\u0011\"\ny(r);\n\u0001#\nA(k;r) =^kx\u0011#\nx(r) +i^ky\u0011#\ny(r); (2)\n\u00010\nA(k;r) =^kz\u00110\nz(r)\ndepends from the \fve complex amplitudes\n\u0011\"\nx; \u0011\"\ny; \u0011#\nx; \u0011#\ny; \u00110\nz, which obey to coupled Ginzburg-\nLandau equations, derived in the linear approximation\nin the papers [5, 12]. ^ki=ki=jkj; i =x;y;z are the\nprojections of the unit ^kvector on the coordinate axis.\nThe order parameter of the paramagnetic supercon-\nducting state [5, 13] in a orthorhombic metal looks like\nthe order parameter of super\ruid3He-B phase [10]\n\u0001\"(k;r) =\u0000^kx\u0011x(r) +i^ky\u0011y(r);\n\u0001#(k;r) =^kx\u0011x(r) +i^ky\u0011y(r); (3)\n\u00010\nA(k;r) =^kz\u0011z(r):To avoid excessive di\u000eculties the authors of [9] con-\nsidered the minimal model for the superconducting fer-\nromagnetic state with the order parameter\n\u0001\"(k;r) =^kx\u0011\"(r);\n\u0001#(k;r) =^kx\u0011#(r): (4)\nCorresponding simplest order parameter for the param-\nagnetic superconducting state looks like the order param-\neter for the discovered recently polar state of super\ruid\n3He [14]\n\u0001\"(k;r) =\u0000^kx\u0011(r);\n\u0001#(k;r) =^kx\u0011(r): (5)\nIn neglect of interactions of electron charges with mag-\nnetic \feld created by the magnetization one can write\nfollowing Ref.9 the gradient independent Landau free en-\nergy density as\nF=\u000bM2+\fM4+\u000b1(j\u0011\"j2+j\u0011#j2) +\r1M(j\u0011\"j2\u0000j\u0011#j2) +\r2(\u0011\"\u0011?\n#+\u0011?\n\"\u0011#) +B(j\u0011\"j2+j\u0011#j2)2+C(j\u0011\"j2\u0000j\u0011#j2)2;(6)\nwhereMis the density of magnetic moment component\nalong the easy axis,\n\u000b=\u000b0(T\u0000Tc); \u000b 1=\u000b10(T\u0000Tsc0); (7)\nTc(P) is the pressure dependent Curie temperature and\nTsc0(P) is the formal critical temperature of supercon-\nducting transition in the single band (say just spin-up)\ncase. The phenomenological treatment does not allow\nto \fx the pressure dependences of these critical tempera-\ntures and the other coe\u000ecients in Eq.(6). In what follows\nwe shall assume that the pressure dependences Tc(P) and\nTsc0(P) qualitatively correspond to the phase diagram\nwith the intersection of the phase transition lines shown\nin Fig.1.\nOne can note that the symmetry allows also the fol-\nlowing interaction i\r3M(\u0011\"\u0011?\n#\u0000\u0011?\n\"\u0011#) between the super-conducting and magnetic order parameters [13], but the\ngeneral enough microscopic calculations [5, 12] do not\ncon\frm the existence of this term.\nIn general, the free energy fourth order terms actually\nhave the form di\u000berent from B(j\u0011\"j2+j\u0011#j2)2+C(j\u0011\"j2\u0000\nj\u0011#j2)2used in Ref.9. If the normal state Green functions\nare diagonal in the band indices (in our case they are\nspin-up and spin-down indices ) the Wick's decoupling\ndoes not produce any mixing terms between the band or-\nder parameters (see f.i. [15]). This case the fourth order\nterms in respect of the superconducting order parameters\nare\n\f1(j\u0011\"j4+j\u0011#j4) +~\f1M(j\u0011\"j4\u0000j\u0011#j4): (8)\nIf in the normal state there is the band mixing interaction\nthis leads to emergence of the additional terms\n\f2j\u0011\"j2j\u0011#j2+\f3[(\u0011\"\u0011?\n#)2+ (\u0011?\n\"\u0011#)2] +\f4(j\u0011\"j2+j\u0011#j2)(\u0011\"\u0011?\n#+\u0011?\n\"\u0011#) +~\f4M(j\u0011\"j2\u0000j\u0011#j2)(\u0011\"\u0011?\n#+\u0011?\n\"\u0011#): (9)\nOne can show that the additional fourth order terms do\nnot introduce a qualitative modi\fcation in the phase dia-gram. So, we will work with the same free energy density\nas in Ref.93\nF=\u000bM2+\fM4+\u000b1(j\u0011\"j2+j\u0011#j2) +\r1M(j\u0011\"j2\u0000j\u0011#j2) +\r2(\u0011\"\u0011?\n#+\u0011?\n\"\u0011#) +\f1(j\u0011\"j4+j\u0011#j4) +\f2j\u0011\"j2j\u0011#j2:(10)\nIII. PHASE TRANSITIONS IN NEUTRAL\nFERMI LIQUID\nAt low pressures the system \frst passes from the para-\nmagnetic to the ferromagnetic state and then from the\nferromagnetic state to the ferromagnetic superconduct-\ning state. We begin with consideration of these phase\ntransitions and then discuss the high pressures transi-\ntions from the normal to the superconducting state and\nfrom the superconducting state to the ferromagnetic su-\nperconducting state, and also as well as the transition\nfrom the normal to the ferromagnetic superconducting\nstate.\nA. Phase transition from the paramagnetic to the\nferromagnetic state\nThe second order transition from paramagnetic to fer-\nromagnetic state occurs at T=TCurie (P). Below this\ntemperature the magnetic moment acquires the \fnite\nvalue and a superconducting ordering is absent\nM2= (M0(T))2=\u0000\u000b0(T\u0000Tc(P)\n2\f; \u0011 \"=\u0011#= 0:(11)B. Phase transition from the ferromagnetic state\nto the superconducting ferromagnetic state\nAt the subsequent phase transition the superconduct-\ning order parameter amplitudes \u0011\";\u0011#appear and the\nmagnetic moment acquires a magnitude M=M0+m.\nAccepting for certainty that coe\u000ecient \r2=\u0000j\r2jis neg-\native we see from Eq. (10) that the phase di\u000berence be-\ntween the superconducting order parameters is absent\n\u0011\"=\u00111ei'; \u0011 #=\u00112ei': (12)\nHere,\u00111and\u00112are the modules of the superconducting\norder parameters. Thus, one can rewrite the free energy\ndensity (10) as\nF=\u000bM2+\fM4+\u000b1(\u00112\n1+\u00112\n2) +\r1M(\u00112\n1\u0000\u00112\n2)\n\u00002j\r2j\u00111\u00112+\f1(\u00114\n1+\u00114\n2) +\f2\u00112\n1\u00112\n2: (13)\nThe minimization of the free energy density (13) in\nrespect\u00111;\u00112andmyields the equations\n\u000b1\u00111+\r1(M0+m)\u00111\u0000j\r2j\u00112+ 2\f1\u00113\n1+\f2\u00111\u00112\n2= 0; (14)\n\u000b1\u00112\u0000\r1(M0+m)\u00112\u0000j\r2j\u00111+ 2\f1\u00113\n2+\f2\u00112\n1\u00112= 0; (15)\n2\u000bm+ 12\fM2\n0m+ 12\fM 0m2+ 4\fm3+\r1(\u00112\n1\u0000\u00112\n2) = 0: (16)\nHere, we have taken into account that M0is the minimum\nof free energy at \u00111=\u00112= 0 and omitted the fourth order\nterms. The corresponding linear equations for \u00111,\u00112\n(\u000b1+\r1M0)\u00111\u0000j\r2j\u00112= 0; (17)\n\u0000j\r2j\u00111+ (\u000b1\u0000\r1M0)\u00112= 0 (18)\nare not coupled with linear equation for m. Equating\nthe determinant of this system to zero and taking into\naccount Eq.(11) we obtain the equation\nTsc=Tsc0+p\n(\r1(M0(Tsc))2+\r2\n2\n\u000b10(19)\nfor the temperature Tscof transition to the superconduct-\ning ferromagnetic state. We shall not write the explicit\nformula for Tscin view of its cumbersome shape. Let\nus only note that according to this equation the pres-\nsure decrease of the Curie temperature Tc(P) causes theincrease of the superconducting transition temperature\nTsc(P), although this is not the only reason for the Tsc(P)\npressure dependence.\nThe linear equation in respect of mgives\nm\u0018=\u0000\r1(\u00112\n1\u0000\u00112\n2)\n8\fM2\n0: (20)\nSo,mis proved to be of the next order of smallness\nin comparison with \u00111/\u00112/pTsc\u0000T. Substitution\nEq.(20) to the Eqs. (14) and (15) gives the equations of\nthe third order in respect to the amplitudes \u00111;\u00112. Ana-\nlytic solution of this system is possible only at negligibly\nsmall coe\u000ecientj\r2j. This case at \r1>0 we obtain\n\u00112\n2\u0018=\u0000\u000b10\n2\f1\u0000\r2\n1\n8\fM2\n0\u0012\nT\u0000Tsc0\u0000\r1M0\n\u000b10\u0013\n; (21)4\n\u00111\u0018=j\r2j\n\u000b1+\r1M0\u00112: (22)\nThis description of the second order phase transition\nfrom ferromagnetic to the ferromagnetic superconduct-\ning state is valid at the assumption m<<M 0. However,\nat pressure enhancement the Curie temperature and the\ncritical temperature of superconducting transition (see\nFig.1) approach each other, the value of M0gets smaller\nand according to Eq.(20) the value of mincreases. One\ncan expect the turning of the second order transition into\nthe \frst order transition such that the order parameters\n\u00111;\u00112; m undergo \fnite jumps from zero to the \fnite\nvalues at temperature larger than the critical tempera-\nture given by Eq.(19). Indeed, the this type of behavior\nwas established in Ref.9 by the numerical solution of non-\nlinear equations for the order parameter components at\nclose enough values TCurie andTsc.\nC. Phase transitions from the normal to the\nferromagnetic superconducting state\nTo establish the whole phase diagram one must con-\nsider the phase transition from the normal nonmagnetic\nstate to the superconducting state. The free energy den-\nsity (13) minimization in respect \u00111; \u00112; M yields\n\u000b1\u00111+\r1M\u0011 1\u0000j\r2j\u00112+ 2\f1\u00113\n1+\f2\u00111\u00112\n2= 0;(23)\n\u000b1\u00112\u0000\r1M\u0011 2\u0000j\r2j\u00111+ 2\f1\u00113\n2+\f2\u00112\n1\u00112= 0;(24)\n2\u000bM+ 4\fM3+\r1(\u00112\n1\u0000\u00112\n2) = 0:(25)\nAt\u000b > 0 there are two type solutions of these equa-\ntions such that\n\u00111=\u00112; M = 0; (26)\nand\n\u001116=\u00112; M6= 0: (27)In the \frst case the transition to the ferromagnetic su-\nperconducting state occurs by means of two consecutive\nphase transitions: the phase transition from the normal\nstate to the nonmagnetic superconducting state followed\nat lower temperature by the transition to the ferromag-\nnetic superconducting state. In the second case the phase\ntransition to the ferromagnetic superconducting state oc-\ncurs directly from the normal state. We consider these\nsituations separately.\n1. Two consecutive phase transitions from the normal to\nthe ferromagnetic superconducting state\nThe solution (26) is realized at large enough positive \u000b\nwhen formation of a ferromagnetic state is not energeti-\ncally pro\ftable. This case the common magnitude of the\nsuperconducting amplitudes is\n\u00112=\u0000\u000b1\u0000j\r2j\n2\f1+\f2: (28)\nAt positive sum 2 \f1+\f2>0 this phase transition is of\nthe second order and occurs at\nTsc=Tsco+j\r2j\n\u000b10; (29)\nthat coincides with Eq.(19) at M0= 0.\nTo pass in the ferromagnetic superconducting state the\nsystem must undergo one more phase transition. At this\ntransition the magnetization Mspontaneously appears\nand the superconducting order parameter amplitudes ac-\nquire the deviations from the value given by Eq. (28)\n\u00111=\u0011+\u000e1; \u0011 2=\u0011+\u000e2: (30)\nThe free energy acquires the following form\nF=\u000bM2+\fM4+\u000b1(\u000e2\n1+\u000e2\n2) +\r1M\u0002\n2\u0011(\u000e1\u0000\u000e2) +\u000e2\n1\u0000\u000e2\n2\u0003\n\u00002j\r2j\u000e1\u000e2\n+\f1\u0002\n6\u00112(\u000e2\n1+\u000e2\n2) + 4\u0011(\u000e3\n1+\u000e3\n2) +\u000e4\n1+\u000e4\n2\u0003\n+\f2\u0002\n\u00112(\u000e2\n1+\u000e2\n2+ 4\u000e1\u000e2) + 2\u0011\u000e1\u000e2(\u000e1+\u000e2) +\u000e2\n1\u000e2\n2\u0003\n(31)\nHere we have taken into account that \u0011is the minimum\nof free energy at M=\u000e1=\u000e2= 0 and omitted the zero\norder terms in respect of M;\u000e 1;\u000e2. The order parameters\nare determined from the conditions of the free energy\nminimum\n@F\n@\u000e1= 0;@F\n@\u000e2= 0;@F\n@M= 0: (32)\nOne can easily check that in linear approximation theequations for ( \u000e1\u0000\u000e2) andM\n\u0002\n\u000b1+j\r2j+ (6\f1\u0000\f2)\u00112\u0003\n(\u000e1\u0000\u000e2) + 2\r1\u0011M= 0;\n\r1\u0011(\u000e1\u0000\u000e2) +\u000bM= 0 (33)\nare decoupled from the equation for ( \u000e1+\u000e2). Hence,\nthe latter combination is of the next order of smallness\nin comparison with\nM/(\u000e1\u0000\u000e2)/p\nTscM\u0000T: (34)5\nHereTscMis the critical temperature of transition from\nthe superconducting to the superconducting ferromag-\nnetic state which is determined from the equation given\nby the equality to zero of the determinant of the system\n(33)\n\u0002\n\u000b1+j\r2j+ (6\f1\u0000\f2)\u00112\u0003\n\u000b\u00002 [\r1\u0011]2= 0: (35)\n2. Direct phase transition from the normal to the\nferromagnetic superconducting state\nThe second type solution (27) is realized at small\nenough positive \u000b. The analytical treatment is possible\nin neglect of third order term 4 \fM3in Eq.(25), then\nM\u0018=\u0000\r1(\u00112\n1\u0000\u00112\n2)\n2\u000b: (36)\nPassing to the sum and the di\u000berence of Eqs.(23) and\n(24) and using the Eq. (36) we come to the equations\n\u0011\u0014\n\u000b1\u0000j\r2j\u00002\r2\n1\n\u000b\u000e2+ 2\f1(\u00112+ 3\u000e2) +\f2(\u00112\u0000\u000e2)\u0015\n= 0;(37)\n\u000e\u0014\n\u000b1+j\r2j\u00002\r2\n1\n\u000b\u00112+ 2\f1(3\u00112+\u000e2)\u0000\f2(\u00112\u0000\u000e2)\u0015\n= 0;(38)\nwhere\n\u0011=1\n2(\u00111+\u00112); \u000e =1\n2(\u00111\u0000\u00112): (39)\nThe solution of these equations at \u00116= 0; \u000e6= 0 is\n\u00112=1\n2\u000b1\u000b\n\r2\n1\u00004\f1\u000b+1\n2j\r2j\u000b\n\r2\n1\u0000(2\f1+\f2)\u000b; (40)\n\u000e2=1\n2\u000b1\u000b\n\r2\n1\u00004\f1\u000b\u00001\n2j\r2j\u000b\n\r2\n1\u0000(2\f1+\f2)\u000b: (41)\nThus, at direct transition from the normal to the fer-\nromagnetic superconducting state the order parameter\ncomponents M;\u0011 1;\u00112undergo the \fnite jumps. This is\nthe phase transition of the \frst order.\nAt a phase transition the free energy is not changed,\nthat gives the equation for the phase transition temper-\nature\nF=\u000bM2+\fM4+\u000b1(\u00112\n1+\u00112\n2) +\r1M(\u00112\n1\u0000\u00112\n2)\n\u00002j\r2j\u00111\u00112+\f1(\u00114\n1+\u00114\n2) +\f2\u00112\n1\u00112\n2= 0: (42)\nWe solve this equation in the assumption that the co-\ne\u000ecientj\r2jis negligibly small. Then at small enough\nvalues of\u000bone can use the approximate expressions\n\u00112\u0019\u000e2\u0019\u000b\u000b1\n2\r2\n1; M\u0019\u00002\r1\n\u000b\u00112: (43)\nSubstituting them to the Eq.(42) we obtain\n\u00114\u0014\n\u000b1\u0000\r2\n1\n\u000b+ 4\f\u0015\n= 0 (44)So, at small enough positive \u000bthe phase transition tem-\nperature is given by\nTsc\u0019Tsco+\r2\n1\n\u000b\u000b10; (45)\nthat exceeds the critical temperature of the second order\ntransition given by Eq. (29).\nD. Phase diagram\nThe analytic derivation made at several not strongly\nrestrictive assumptions leads to the conclusion that the\ndirect phase transition from the normal to the ferromag-\nnetic superconducting state is of the \frst order. This\ncon\frms the statement numerically established in Ref.9.\nOn the other hand, as we have pointed out in the section\nIIIB. , when the temperatures of phase transitions to the\nnormal ferromagnetic and to the superconducting ferro-\nmagnetic state are close to each other, the transition from\nthe ferromagnetic to the ferromagnetic superconducting\nstate is of the \frst order.\nThus, the simple phase diagram with intersection of\nferromagnetic and superconducting phase transition lines\nlike it is drown in Fig.1 cannot be realized at least in\nframe of the model under consideration. Near the in-\ntersection of the critical lines of the ferromagnetic and\nthe superconducting transitions there is a piece of the\ntransition of the \frst order as this is shown in Fig.2 by\nthe thick line. The pressure interval where this transi-\ntion takes place can be quite small. The presence of the\n\frst order transition seems to be in correspondence with\nthe sharp drop of resistivity at superconducting phase\ntransition in this pressure interval found in the paper [8].\nIV. MAGNETOSTATICS\nThe authors of Ref.9 discussed the phase transition\nbetween the SC state and FM+SC state in the neutral\nFermi liquid. The situation is changed in a charged Fermi\nliquid because the magnetization in the superconduct-\ning state is inevitably accompanied by the screening cur-\nrents. The emergency of superconducting state in the\nferromagnet state of UCoGe takes place at \fnite mag-\nnetization. Whereas the arising of ferromagnetic state\nin the superconducting state of UCoGe accompanied by\nthe smooth increasing of magnetization from zero to the\n\fnite value. This determines the di\u000berence between the\ntransition from the ferromagnetic to the superconducting\nferromagnetic state and the transition from the super-\nconducting to the superconducting ferromagnetic state\nwhich we discuss here.\nLet us consider a cylindrical sample of radius Rwith\naxis parallel to the easy magnetization axis. A phase\ntransition to the superconducting ferromagnetic state is\naccompanied by the appearance of super-currents. The6\ncorresponding London equation for magnetic induction\nis\ncurlB=4\u0019j\nc=\u0000A\n\u000e2+ 4\u0019 curl M; (46)\nwhere\u000eis the London penetration depth. The contri-\nbution to the current due to the term c curl Mis non-\nvanishing only in the surface layer with thickness of the\norder of coherence length \u0018 <<\u000e <<R [10], hence, the\nsmall enough magnetic \feld has to decay in the sample\nvolume:\nB(r) =B(R) exp\u0012\n\u0000R\u0000r\n\u000e\u0013\n; (47)\nwhere the surface magnetic \feld is determined by the\nmagnetic moment created by the super-currents \rowing\nin the surface layer\nB(R) = 4\u0019M: (48)\nIn UCoGe according to the phase diagram drown in Fig.2\nthe formation of ferromagnetic superconducting state oc-\ncurs (i) from the normal ferromagnetic state, (ii) from the\nnonmagnetic normal state either through the two consec-\nutive phase transitions of the second order N!SCand\nthenSC!FM +SCor directly by means of the \frst\norder transition.\nA. Magnetostatics below transition from the\nferromagnetic to the ferromagnetic superconducting\nstate\nJust below the temperature of the phase transition\nfrom the ferromagnetic to the ferromagnetic supercon-\nducting state discussed in the section III.B the \feld at\nsurface is\nB(R) = 4\u0019\u0012\nM0\u0000\r1(\u00112\n1\u0000\u00112\n2)\n8\fM2\n0\u0013\n: (49)\nAccording to the experimental results reported in [16{\n18] this \feld at ambient pressure is larger than the lower\ncritical \feld \feld Hc1in UCoGe. So, the complete \feld\nscreening is not realised, and the phase transition occurs\ndirectly to the superconducting mixed state. The vortex\ncores occupy the small part of the sample volume, and\nalmost whole volume is in the superconducting state with\nthe order parameter given by Eqs.(21), (22). The speci\fc\nheat jump at phase transition to the superconducting\nstate at the ambient pressure has the \fnite value\n\u0001C\u0018=\u000b10Tsc\n2\f1\u0000\r2\n1\n8\fM2\n0: (50)\nHere, we have neglected the temperature dependence of\nM0(T), just taking it as the constant M0=M0(Tsc). The\nspeci\fc heat jump at this transition has been registered\n[4].B. Magnetostatics below transition from the\nsuperconducting to the ferromagnetic\nsuperconducting state\nAnother situation takes place at the transition from\nthe superconducting to the ferromagnetic superconduct-\ning state. This case discussed in the section III.C the\nsurface \feld is determined by the small magnetic moment\narising at transition in magnetic superconducting state\nB(R)\u0018=4\u0019M/p\nTscM\u0000T: (51)\nThis \feld is certainly smaller than the lower critical \feld\nin well developed superconducting state with supercon-\nducting density ns/\u00112/(Tsc\u0000T)\nHc1= 2\u0019\u0016bnsln\u000e\n\u0018/(Tsc\u0000T): (52)\nThe transition to the ferromagnetic superconducting\nstate is characterized by the emergency of magnetic mo-\nmentMand the magnetic part of superconducting or-\ndering\u0018(\u000e1\u0000\u000e2). But the super-currents completely\nscreen this magnetism in the bulk of material. The grad-\nual increase of magnetization from zero to some \fnite\nvalue is not accompanied by a bulk phase transition as it\nis in the process of the Meissner state formation in a su-\nperconductor of the second kind under external magnetic\n\feld smaller than Hc1.\nThe pressure decrease stimulates ferromagnetism.\nHence, at low temperatures and low pressures the mag-\nnetization will exceed the lower critical \feld and a sample\npasses to the ferromagnetic superconducting mixed state.\nSo, instead a phase transition between the superconduct-\ning and the ferromagnetic superconducting state one can\nexpect just the transition between the Meissner and the\nmixed superconducting states.\nThus, there is no bulk phase transition at all. The fer-\nromagnetic superconducting Meissner state exists in the\nregion between two dashed lines shown in Fig.3. The ac-\ntual position of Hc1line is subject to experimental dede-\ntermination. It can be in principle much more to the\nleft, than it is drawn in Fig.3, and also as well as more\nto the right such that near the \frst order transition the\ntwo dashed lines can be merged to one line.\nV. CONCLUSION\nThe Landau theory allows to establish speci\fc proper-\nties of phase transformations in anisotropic ferromagnetic\nsuperconducting material UCoGe. There was found that\nthe phase transition from the ferromagnetic to the fer-\nromagnetic superconducting state at ambient pressure is\ncharacterized by the appearance of superconducting part\nof the order parameter whereas the ferromagnetic com-\nponent does undergo insigni\fcant changes. However, at\nhigher pressures this transition can turn to the transition\nof the \frst order.7\nThere was proven that the direct phase transition from\nthe nonmagnetic normal state to the ferromagnetic su-\nperconducting state is of the \frst order and exists in\nsmall pressure interval. Out of this interval the tran-\nsition to the ferromagnetic superconducting state occurs\nby means of two consecutive phase transitions of the sec-\nond order from normal to nonmagnetic superconducting\nstate and then from the nonmagnetic superconducting\nstate to ferromagnetic superconducting state. The model\nphase diagram in neutral Fermi liquid acquires the shape\nshown in Fig.2.\nThe phase diagram modi\fcation introduced by screen-ing currents in charged Fermi liquid is shown in Fig.3.\nThe magnetic moment at phase transition from the fer-\nromagnetic to the ferromagnetic superconducting state\nis just partially screened by the superconducting cur-\nrents. Whereas at phase transition from the supercon-\nducting state to the ferromagnetic superconducting state\nthis screening is complete what shades the manifestations\nof a bulk phase transition. The ferromagnetic mixed su-\nperconducting state occurs only at lower pressures where\nthe spontaneous magnetic moment exceeds the lower crit-\nical \feld. The position of Hc1line is subject to experi-\nmental determination.\n[1] S. S. Saxena, P. Agarval, K. Ahilan, F. M. Grosche, R. K.\nW. Hasselwimmer, M. J. Steiner, E. Pugh, I. R. Walker,\nS. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley,\nI. Sheikin, D. Braithwaite and J. Flouquet, Nature 406,\n587 (2000).\n[2] D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J.\nFlouquet, J.-P. Brison, E. Lhotel and C. Paulsen, Nature\n413, 613 (2001).\n[3] N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C.\nP. Klaasse, T. Gortenmulder, A. de Visser, A. Hamann,\nT. Gorlach, and H. v. Lohneysen, Phys. Rev. Lett. 99,\n067006, (2007).\n[4] D. Aoki and J. Flouquet, J. Phys. Soc. Jpn. 83, 061011\n(2014).\n[5] V. P. Mineev, Phys. Usp. 60, (2017)\nhttps://doi.org/10.3367/UFNe.2016.04.037771 ; V.\nP. Mineev, Usp. Fizich. Nauk 187, 129 (2017)\nhttps://doi.org/10.3367/UFNr.2016.04.037771 and\narXiv:1605.07319v1 [cond-mat.supr-con].\n[6] T.Hattory, K.Karube, Y.Ihara, K.Ishida, K.Deguchi and\nN.K.Sato, Phys. Rev. B 88, 085127 (2013).\n[7] E. Slooten, T. Haka, A. Gasparini, Y. K. Huang, and A.\nde Visser, Phys. Rev. Lett. 103, 097003, (2009).[8] G. Bastien, D. Braithwaite, D. Aoki, G. Knebel, and J.\n.Flouquet, Phys. Rev. B 94, 125110 (2016).\n[9] A. K.C. Cheung and S. Raghu, Phys. Rev. B 93, 134516\n(2016).\n[10] V. P. Mineev and K. V. Samokhin, \"Intoduction to\nUnconventional Superconductivity\", Gordon and Breach\nScience Publishers, 1999.\n[11] V. P. Mineev, Phys. Rev. B 66, 134504 (2002).\n[12] V. P. Mineev, Phys. Rev. B 90, 064506 (2014).\n[13] V. P. Mineev AIP Conference Proceedings 1134 68\n(2009).\n[14] V. V. Dmitriev, A. A. Senin, A. A. Soldatov, and A. N.\nYudin, Phys. Rev .Lett. 115, 165304 (2015).\n[15] M. E. Zhitomirsky and V.-H. Dao, Phys. Rev. B 69,\n054508 (2004).\n[16] K. Deguchi, E. Osaki, S. Ban, N. Tamura, Y. Simura, T.\nSakakibara, I. Satoh, and N. K. Sato,\nJ. Phys. Soc. Jpn. 79, 083708 (2010).\n[17] C. Paulsen, D. J. Hykel, K. Hasselbach, and D. Aoki,\nPhys. Rev. Lett. 109237001 (2012).\n[18] D. J. Hykel, C. Paulsen, D. Aoki, J. R. Kirtley, and K.\nHasselbach, Phys. Rev. B 90, 184501 (2014).8\n \n \nFIG. 1: (Color online) Temperature-pressure phase diagram\nof UCoGe. Notations FM, SC and PM used for ferromag-\nnetic, superconducting and paramagnetic phases correspond-\ningly [4].9\n  \nFM PM SC FM+SC P T   \nFIG. 2: Schematic temperature-pressure phase diagram of\nUCoGe in frame of neutral Fermi liquid model. Notations\nFM, SC and PM used for ferromagnetic, superconducting\nand paramagnetic phases correspondingly. The thin and thick\nlines are the lines of the second and the \frst order transitions\ncorrespondingly.10\n     FM PM SC   P T   \n FM PM SC FM+SC P T   \nM Mixed \nFIG. 3: Schematic temperature-pressure phase diagram of\nUCoGe taking into account the e\u000bect of super-currents screen-\ning. Notations FM, SC and PM used for ferromagnetic, super-\nconducting and paramagnetic phases correspondingly. The\nthin and thick lines are the lines of the second and the \frst\norder transitions correspondingly. The right dashed line is\nthe imaginary line of the transition between the nonmagnetic\nand ferromagnetic Meissner superconducting states which is\nnot a phase transition in the bulk of sample. The left dashed\nline is the line of Hc1dividing the Meissner and the mixed\nferromagnetic superconducting states."    },    {        "title": "0810.3206v2.Ferromagnetic_Spin_Coupling_as_the_Origin_of_0_7_Anomaly_in_Quantum_Point_Contacts.pdf",        "content": "arXiv:0810.3206v2  [cond-mat.mes-hall]  6 Feb 2009Ferromagnetic Spin Coupling as the Origin of 0.7Anomaly in Quantum Point Contacts\nK. Aryanpour and J. E. Han1\n1Department of Physics, SUNY at Buffalo, Buffalo, NY 14260\n(Dated: October 29, 2018)\nWe studyone-dimensional itinerant electron models with fe rromagnetic couplingtoinvestigate the\norigin of 0 .7 anomaly in quantum point contacts. Linear conductance cal culations from the quan-\ntum Monte Carlo technique for spin interactions of different spatial range suggest that 0 .7(2e2/h)\nanomaly results from a strong interaction of low-density co nduction electrons to ferromagnetic fluc-\ntuations formed across the potential barrier. The conducta nce plateau appears due to the strong\nincoherent scattering at high temperature when the electro n traversal time matches the time scale\nof dynamic ferromagnetic excitations.\nQuantum point contacts (QPC) are narrow constric-\ntions inside two-dimensional electron gas. They con-\nstruct one of the building blocks of submicrometer de-\nvices such as quantum dots and qubits [1, 2]. The dc\nconductance through a QPC is quantized in steps of\nG0= 2e2/h[3, 4]. However, experiments also reveal\nthe appearance of an additional shoulder in the conduc-\ntance measurement near 0 .7G0widely referred to as the\n0.7 anomaly [2]. The origin of 0 .7 anomaly in QPC has\nremained a puzzle over almost a decade. The evolution\nofthe 0.7G0plateau to 0 .5G0with magnetic field and the\nenhancement of the gfactor [2] have strongly suggested\nthat the origin of the anomaly is the electron spin.\nAnumberofscenarioshavebeenproposed,suchasspin\npolarizationoftheitinerantelectrons[5,6,7,8,9, 10,11],\nferromagnetic correlation [12, 13, 14], formation of a\nspin 1/2 magnetic moment in the conductance chan-\nnel [15, 16, 17, 18] and Kondo effect [16, 17, 19], Hub-\nbard chain [20, 21, 22, 23, 24, 25, 26], Wigner crystal-\nlization and antiferromagnetism [27]. These approaches\nhave produced a wide range of different phenomenolo-\ngies, sometimes inconsistent with experiments, and there\nis no widely accepted microscopic theory to date. The\nproblem is partly due to the approximate methods used\nin the strongly interacting limit and therefore it becomes\nessentialtoperformexactcalculationstotestmicroscopic\nmodels against experiments. Here we use numerically\naccurate quantum Monte Carlo technique to study the\nstrong correlation effects in QPC devices.\nWe find that the 0.7 anomaly at high temperature\narises from the incoherent electron scattering from itin-\nerant ferromagnetic fluctuations near the Stoner insta-\nbility [28], in the strong correlation limit of low electron\ndensity created by spatially inhomogeneous gate poten-\ntial. We show, through a comparison with a model with\non-site interactions, that the relevant electron scattering\nis due to the spin fluctuations which are spatially coher-\nent acrossthe potential barrier. With decreasingtemper-\nature, the magnetic excitation becomes slower than the\nitinerant electrons. The current is then carried by the\nquasiparticles and the 0.7 plateau gradually disappears.\nWith the Zeeman magnetic field, the 0.7 plateau evolvesApplied biasg\nQPC (x = 0)\nInteracting block−VV 2\n2Gate voltage ~ V  /cosh (x /L  )2g\nFIG. 1: (color online) Profile of the 1D chain with the inter-\nacting block (including seven sites in this figure). The gate\nvoltage potential V(x) acts as an adiabatic potential barrier\nthrough the QPC region. A bias with a ramp passing through\nthe QPC is applied across the chain to compute the dc con-\nductance in the linear response regime.\nto a robust 0.5 plateau in agreement with experiments.\nWe model our system using a one-dimensional (1D)\nelectron gas with spin-spin interaction among itinerant\nelectrons as depicted in Fig.1 with the assumption that\nthe higher 1D subbands play indirect roles for the first\nconductance plateau. The Hamiltonian reads\nH=/integraldisplay\ndx ψ†(x)/bracketleftbigg\n−¯h2\n2m∂2\n∂x2+V(x)−µ+1\n2/vector σ·/vectorH/bracketrightbigg\nψ(x)\n+/integraldisplay\ndx/braceleftbigg\nK1(x)[/vector s(x)]2+1\n2K2(x)[∂x/vector s(x)]2/bracerightbigg\n,(1)\nwithψ(x) = [ψ↑(x),ψ↓(x)]Tthe field operator vector,\nµthe chemical potential, Hthe Zeeman magnetic field,\n/vector σthe Pauli matrices and V(x) the external gate volt-\nage barrier in order to pinch off the electron current\nthrough the QPC. V(x) is defined in our model as\nV(x) =Vg/cosh2(x/Lg) withx= 0 corresponding to\nthe center of the chain, Lga characteristic length and\nVgthe gate voltage. The operator /vector s(x) =ψ†(x)/vector σ\n2ψ(x)\nrepresents the spin density of itinerant electrons along\nthe chain. Spins interact locally with the coupling con-\nstantK1(x) =α(x)K1(K1<0 for repulsive on-site\nCoulomb interaction) with α(x) = 1/cosh(x/Ls) an at-\ntenuating function with characteristic length Ls. We set\nthe spin coupling to adiabatically fall off to take into ac-\ncount the screening effects in the leads and to reduce the\nbackscattering due to interaction away from the QPC2\nconstriction. K2(x) =α(x)K2(K2>0) is the coeffi-\ncient of the gradient term accounting for the ferromag-\nnetic Heisenberg interaction. We discretize the contin-\nuum model Hamiltonian to a tight-binding chain of lat-\ntice constant ∆ xwith the nearest-neighbor hopping t.\nDefiningt=¯h2\n2m∆x2,µ=µ−¯h2\nm∆x2,√\n∆x ψ(xi) =ci,\nJ0=−(K1\n∆x+K2\n∆x3),J1=K2\n∆x3and/vector sp=c†\np/vector σ\n2cpthe dis-\ncretized Hamiltonian reads\nH=−t/summationdisplay\n<ij>,σc†\niσcjσ−/summationdisplay\niσ/parenleftbigg\nµ−Vi+1\n2σH/parenrightbigg\nc†\niσciσ\n−/summationdisplay\np∈block(J0α2\np/vector sp·/vector sp+J1αpαp+1/vector sp·/vector sp+1),(2)\nwithHtaken along the zdirection and σ=±1 the\nspin index. The microscopic parameters K1andK2\nare unknown and we treat J0,J1>0 (for ferromagnetic\ncoupling) as the model parameters throughout this let-\nter. The index pruns only within the interacting block\nin Fig.1 near the QPC saddle point. The discretiza-\ntion is valid since we are in the low-density limit with\n/angbracketleftc†\niσciσ/angbracketright<1 inside the interacting block. Near the pinch-\noff gate voltage, /angbracketleftc†\niσciσ/angbracketrightat the top of the potential bar-\nrierrapidlyapproachedzeroin the followingcalculations.\nUsingm≈0.067mefor GaAs and ∆ x≈20 nm (exper-\nimental QPC length is about 200 nm, roughly 10∆ xfor\nLg= 4),t= 1.4 meV. Due to CPU limitations, we\nrestrict the interacting block to about seven sites.\nIn our calculations, we modify Eq.(2) by allowing the\nnonlocal part of the interaction term to extend beyond\nnearest neighboring sites. We define the block spin oper-\nator/vectorS=/summationtext\np∈blockαp/vector spand rewrite a new Hamiltonian\nH=−t/summationdisplay\n<ij>,σc†\niσcjσ−/summationdisplay\niσ/parenleftbigg\nµ−Vi+1\n2σH/parenrightbigg\nc†\niσciσ\n−/summationdisplay\np∈block/parenleftbigg\nJ0−J1\n2/parenrightbigg\nα2\np/vector sp·/vector sp−J1\n2/vectorS·/vectorS.(3)\nCompared to Eq.(2), Eq.(3) incorporates stronger spin\ninteraction among all the spins within the interacting\nblock. This modification makes the decoupling scheme\nin quantum Monte Carlo more efficient. Interactions\nbeyond nearest neighbors can be thought of a coarse-\ngrained effective Hamiltonian on the discretized lattice\nin the low wave-vectorlimit. The effective interaction re-\nsults from virtual fluctuations to high momentum states\nwhich are excluded in the discretized model and it takes\na form similar to the RKKY interaction. Since we are in-\nterestedin the low-densitylimit nearthe pinch-offregime\nwith the effective Fermi wave-vector kF,effinside the con-\nstrictionapproachingzero,the kF,effRifactorforposition\nRiinside the constriction also goes to zero and the effec-\ntive spin interaction over the interacting block becomes\npredominantly ferromagnetic.\nWe use acontinuousHubbard-Stratonovichdecoupling\nfor the/vectorS·/vectorSterm in Eq.(3) and discrete decoupling forH = 0\nH = 0.06t\nH = 0.1t\nH = 0.2t\nJ0 = 0.5t, J1 = 0.5tµ = 0.4t, T = 0.025t\n(b)Lg = 4, Ls = 2.5\n0 0.2 0.4 0.6 0.8 11.2\nVgH = 0, T = 0.025t\nH = 0, T = 0.014t\nH = 0.1, T = 0.025t\nµ = 0.4t, J1 = 0\n(d) Lg = 4.5, Ls = 3, J0 = 0.9t00.20.40.60.81G(ω = 0) (in units of 2e2/h) J0 = 0, J1 = 0\n J0 = 0.35t, J1 = 0.36t \n J0 = 0.5t, J1 = 0.5t \n J0 = 0.58t, J1 = 0.5t \n00.20.40.60.81G(ω = 0) (in units of 2e2/h)\nLg = 4\nLg = 6, Ls = 4.5\nLg = 4, Ls = 2.5\nLg = 4, Ls = 2.5µ = 0.4t, T = 0.025t\n(a)\n0 0.2 0.4 0.6 0.8 1\nVg00.20.40.60.81G(ω = 0) (in units of 2e2/h)T = 0.014t\nT = 0.025t\nT = 0.033t\nJ0 = 0.44t, J1 = 0.44t (c)µ = 0.4t, Lg = 4, Ls = 2.5\nFIG. 2: (color online) (a) dc conductance as a function of\ngate voltage Vgat different values for Lg,Ls,J0andJ1at\nT= 0.025t. Conductance plateaus form at 0 .6−0.7 times\nofG0= 2e2/h. (b) The evolution of the plateau to 0 .5G0\nwith the Zeeman magnetic field. (c) The gradual increase of\nconductance with decreasing temperature. (d) Purely local\nmodel (J1= 0) as a function of temperature and the Zee-\nman magnetic field. Local interaction produces qualitative ly\ndifferent transport mechanism from the nonlocal model.\nlocal/vector sm·/vector smterm. We calculate the dc conductance\nusing the Kubo formula in the linear response regime,\nGdc(ω= 0) = lim ω→0Re i/integraltext∞\n0eiωt′/angbracketleftbig/bracketleftbig\nj(t′),Hsd/bracketrightbig/angbracketrightbig\ndt′,\nwherej(t′) =iet/summationtext\nσ[c†\n1,σ(t′)c0,σ(t′)−c†\n0,σ(t′)c1,σ(t′)] is\nthe current operator evaluated at the center of the QPC\nandHsd=−e/summationtext\nm,σV(xm)nm,σistheexternalperturba-\ntion across the chain with V(xm) the normalized source-\ndrainbiaswith maximum(minimum) voltage1 /2(−1/2)\non the left (right) hand side as depicted in Fig.1. The\nrange of summation in Hsd,|m|<∼100, produced well-\nconverged conductance. Conductance by the Kubo for-\nmula is obtained in terms of the bosonic Matsubara fre-\nquencies (iνn=i2πn\nβwhereβ= 1/kBTwithkBthe\nBoltzmann constant and positive integer n) and it needs\nto be analytically continued to real frequency to take the\ndc limitGdc(ω= 0). This task is done by fitting the con-\nductance defined on the Matsubara frequency into the\nLehman representation G(iνn) =i/integraltext∞\n−∞dω′ρ(ω′)\niνn−ω′with\nthe spectral function ρ(ω) as the fitting parameter. This\nmethod has been extensivelytested to an excellent agree-\nment in comparison with the rational function fit [26].\nAfter taking the analytic continuation iνn→ω+iη, we\nobtain the conductance Gdc(ω= 0) =ρ(0).\nFig.2(a) plots the dc conductance as a function of\nthe gate voltage Vgfor different values of LgandLs\nat severalJ0andJ1values and fixed chemical poten-3\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8\nVg00.050.10.150.20.25C(+ −)\nFM(ν0 = 0)T = 0.014t\nT = 0.025t\nT = 0.033t\nT = 0.014t, J0 = J1 = 0\nT = 0.025t, J0 = J1 = 0µ = 0.4t, Lg = 4, Ls = 2.5\nJ0 = 0.44t, J1 = 0.44t\n(a)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1 1.11.2\nω00.0010.0020.0030.0040.005ρFM(ω)Vg = 0\nVg = 0.2t\nVg = 0.4t\nVg = 0.45t\nVg = 0.5t\nVg = 0.6t\nVg = 0.8tµ = 0.4t, Lg = 4, Ls = 2.5J0 = 0.44t, J1 = 0.44tT = 0.025t\n(b)ωpeak ~ 0.1135t\nFIG. 3: (color online) (a) Static ferromagnetic spin correl a-\ntion function versus the gate voltage at different tempera-\ntures. The susceptibility has a strong many-body enhance-\nment (compared to the noninteracting model). (b) Spectral\nfunctionfor dynamicferromagnetic spinsusceptibility ρFM(ω)\nat different values of gate voltage at T= 0.025t. The dashed\nline indicates the excitation frequency.\ntialµ= 0.4twhenH= 0. With increasing interac-\ntion strength, the plateau evolves from near 0 .5G0[25]\nto higher values 0 .6−0.7G0in the strongly interacting\nlimit withJ0,J1>µ. Fig.2(b) exhibits the gradual evo-\nlution of the plateau near 0 .7G0to 0.5G0as the Zee-\nman magnetic field is applied. The significantly wide\ngate voltage interval for the plateau region (∆ Vg≈0.4t\nat 0.5G0forH= 0.2t) compared to the bare Zeeman\nsplitting (∆ Vg=H= 0.2t) is a clear indication for the\nenhancedgfactor as seen in the experiment [2]. Fig.2(c)\nshows that, with decreasing temperature, the conduc-\ntance plateau consistently moves upward with decreas-\ning width. Despite the phase problem of the quantum\nMonte Carlo method for temperature T <0.014t, we are\nable to capture the correct trend as seen in the experi-\nment for 0.014t≤T≤0.033t.T= 0.025tcorresponds\ntoT≈0.41K, falling within the experimental range [16].\nFeatures in Fig.2(a)-(c) correctly reproduce the exper-\nimental results on the temperature and magnetic field\ndependence.\nFig.2(d) plots the conductance in the purely local limit\n(J1= 0). The local spin model is equivalent to the re-\npulsive Hubbard model [20, 21, 22, 26] with the on-site\nCoulomb parameter Ugiven asU= 3J0/4 by redefining\nthe gate potential Vito absorb the one-body terms. Al-\nthough the local limit produces a well-defined 0.7 feature\nat zero field, it is inconsistent with the 0.7 phenomenol-0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1\nω00.0010.0020.0030.0040.0050.0060.007ρFM(ω)T = 0.014t\nT = 0.025t\nT = 0.033t\nµ = 0.4t, Lg = 4, Ls = 2.5\nJ0 = 0.44t, J1 = 0.44t\nVg = 0.5t(a)ωpeak(T = 0.014t)\nωpeak(T = 0.025t)\nωpeak(T = 0.033t)\n0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8\nVg0.050.0750.10.1250.150.1750.20.2250.250.275ωpeakT = 0.014t\nT = 0.025t\nT = 0.033t\nReciprocal traversal time Lg = 4, Ls = 2.5\nµ = 0.44t, J0 = 0.44t, J1 = 0.4t\n(b)\nFIG. 4: (color online) (a) ρFM(ω) atVg= 0.5tfor different\ntemperatures. The peak frequency ωpeakcorresponds to fer-\nromagnetic excitation energy. (b) The excitation energy as a\nfunction of gate voltage Vgat different temperatures. When\nthe traversal time τtr(dashed line) is long inside the constric-\ntion,i.e.ωpeakτtr>1, the electron scattering becomes strong\nand the conductance plateau results.\nogy. First, the 0.7 feature becomes more pronounced\nas temperature is lowered. Second, more interestingly,\nat finite magnetic field two plateaus appear with the\n0.7 feature shifted to higher conductance and another\nplateau emerging near G∼0.3G0. It is very interest-\ning that these results are consistent with the scenario of\nspin singlet-triplet formation discussed in Refs.[18, 29].\nThe main difference here is that the spin is self-generated\nfrom itinerant electrons in our model, not as an exter-\nnal spin [18] or from a quasibound state [29]. Near the\npinch-off,electronandspindensitiesarelowandthespin-\nsinglet does not form and the conductance plateau does\nnot appear at H= 0. However, at finite H, the spin\nmoment becomes enhanced enough to produce the spin-\nsinglet conductance plateau.\nIn the presence of nonlocal interactions, an itinerant\nelectron interacts with many neighboring sites and the\nresulting spin multiplets are not necessarily S= 0 or\nS= 1. With the finite interacting range, a spatially\ncoherent ferromagneticstate extends over the interacting\nblock at finite Hand becomes harder to be flipped by\nelectron scattering. Therefore, the nonlocal interaction\nblocks the minority-spin band and the 0.5 plateau results\nfrom spin splitting, instead of the 0 .3 plateau through\nthe singlet formation. It has been shown previously that\nferromagneticcouplingbeyondlocalinteractionstabilizes\nthe ferromagnetic phase in a uniform 1D chain [14].\nDue to the low electron density within the constriction\nat the pinch-off, the ferromagnetic spin correlations are\nconsiderably enhanced at low T. We compute the ferro-\nmagnetic spin correlation function (FSCF), C(+−)\nFM(νn) =\n1\nN2/summationtext\np,m∈block/integraltextβ\n0eiνnτ/angbracketleftbig\nS+\np(τ)S−\nm(0)/angbracketrightbig\ndτ,withνn=2πn\nβ\n(n≥0) the bosonic Matsubara frequency, τthe imag-\ninary time and Nthe number of sites inside the inter-\nacting block in Fig.1. Fig.3(a) plots the static FSCF\n(ν0= 0) as a function of Vgat different Tvalues corre-\nsponding to the conductance curves in Fig.2(c). In com-4\nparison with the noninteracting system, the static FSCF\nis significantly enhanced as Tis lowered when there is\ninteraction. Enhancement of the static FSCF is effec-\ntive only around the pinch-off due to the singular na-\nture of the 1D density of states. The gate voltage range\nfor the maximum enhancement of static FSCF also co-\nincides with that of the plateaus in Fig.2(a)-(c), indi-\ncating the effect of strong ferromagnetic correlations on\nthe appearance of the plateau. The dynamic ferromag-\nnetic spin susceptibility can be obtained by analytically\ncontinuing the FSCF using the same method employed\nfor the conductance. Throughthe fluctuation-dissipation\ntheorem,ρFM(ω), the spectral function for magnetic ex-\ncitations can be obtained from the analytically contin-\nuedC(+−)\nFM(iνn→ω+iη) from positive νn. In Fig.3(b),\nρFM(ω) has been plotted for different values of Vgat\nT= 0.025tcorresponding to Fig.3(a). The height of the\npeakgrowsas Vgincreasestowardsthepinchoff, reaching\nits maximum for Vg≈0.4t−0.5t, aroundωpeak≈0.1135t\nas indicated by the dashed line. As Vgis further in-\ncreased, the height of the peak decreases in agreement to\nthedecreaseinthestaticFSCFinFig.3(a). However,the\nlocation of the peak continues to stay at ωpeak≈0.1135t\nas the characteristicexcitation energy at T= 0.025up to\nVg≈0.7twhere ferromagnetic correlations have almost\nbeen obliterated as shown in Fig.3(a).\nTokura and Khaetskii [11] addressed the effect of scat-\ntering for electrons in the current due to the local param-\nagnons as the origin of the 0 .7 anomaly in the QPC using\nthe second order perturbation approach with local inter-\nactions. Following B¨ uttiker and Landauer in Ref.[30], we\narguethatmagnons,perceivedasspinwaveswithcharac-\nteristic frequency ωpeak, can strongly interfere with elec-\ntrons while transmitting through the gate voltage barrier\nat frequencies ωpeak∼1/τtrwhereτtris the traversal\ntime for tunneling through the barrier in the absence of\nmagnons defined as/integraltextx2\nx1dx/radicalBig\nm\n2[V(x)−E], withx1andx2\nthe classical turning points, E≈µthe incident energy\nandV(x) =Vg/cosh2(x/Lg) the gate voltage barrier.\nFig.4(a) presents ρFM(ω) forVg= 0.5t(maximum\nFSCF in Fig.3(a)) at three different Tvalues. The char-\nacteristicfrequency ωpeakdecreasesas the temperature is\nlowered and therefore the spin waves soften at the onset\nof Stoner instability in the system as explained in detail\ninRef.[28]. Fig.4(b)plots ωpeakasfunctionof Vgforthese\nthree different temperature values along with the recip-\nrocal traversal time 1 /τtr(nearly constant as a function\nofVg). Whileωpeak∼1/τtr, strong interference between\nthe spin fluctuations and electrons tunneling through the\ngate voltage barrier leads to the suppression of the con-\nductance for the gate voltage values in the vicinity of the\npinch off. The manifestation of this suppression is the\nappearance of a plateau in the conductance as a function\nof the gate voltage near the pinch off. With decreas-\ning temperature and therefore ωpeak≪1/τtr, as seen inFig.4(a) and (b) for T= 0.014t, the itinerant electrons\nbegin to see an effectively static ferromagneticmean field\nand scattering of electrons off magnetic excitations be-\ncomes coherent. As a result, the conductance increases\nand the plateau disappears.\nWe have demonstrated the incoherent electron scatter-\ningduetoferromagneticspin-spincorrelationsinaquasi-\n1D chain as the underlying physics behind 0 .7 anomaly\nphenomenon in the QPC. We thank Jonathan Bird and\nIgorˇZuti´ c for helpful discussions. This project was sup-\nported by NSF DMR-0426826 and we acknowledge the\nCCR at the SUNY Buffalo for computational resources.\n[1] J. Lantz et al., Physica C, 368, 315 (2002).\n[2] K. J. Thomas et al., Phys. Rev. Lett. 77, 135 (1996).\n[3] B. J. van Wees, et al., Phys. Rev. Lett, 60, 848 (1988).\n[4] D. A. Wharam et al., J. Phys. C, 21, L209 (1988).\n[5] Chuan-Kui Wang and K.-F. Berggren, Phys. Rev. B. 54,\nR14257 (1996); Phys. Rev. B. 57, 4552 (1998)\n[6] L. Calmels and A. Gold, Solid State Commun. 106, 139\n(1998)\n[7] N. Zabala, M. J. Puska, and R. M. Nieminen, Phys. Rev.\nLett.80, 3336 (1998).\n[8] H. Bruus, V. V. Cheianov, and K. Flensberg, Physica E,\n(Amsterdam) 10, 97 (2001).\n[9] D. J. Reilly, Phys. Rev. Lett. 89, 246801 (2002); Phys.\nRev. B.72, 033309 (2005)\n[10] A. A. Starikov, I. I. Yakimenko, and K.-F. Berggren,\nPhys. Rev. B. 67, 235319 (2003)\n[11] Y. Tokura, and Khaetskii, Physica E, 12, 711 (2002).\n[12] B. Spivak, and F. Zhou, Phys. Rev. B. 61, 16730 (2000)\n[13] L. Bartosch, M. Kollar, and P. Kopietz, Phys. Rev. B.\n67, 092403 (2003)\n[14] K. Yang, Phys. Rev. Lett. 93, 066401 (2004).\n[15] T. Rejec and Y. Meir, Nature 442, 900 (2006).\n[16] S. M. Cronenwett et al., Phys. Rev. Lett. 88, 226805\n(2002).\n[17] K. Hirose, Y. Meir, and N. S. Wingreen, Phys. Rev. Lett.\n90, 026804 (2003).\n[18] V I Puller et al., J. Phys.: Condens. Matter 17(2005)\n5269-5284\n[19] Y. Meir, K. Hirose, and N. S. Wingreen, Phys. Rev. Lett.\n89, 196802 (2002); J. Phys. : Condens. Matter 20(2008)\n164208\n[20] S. Kirchner, H. G. Everts, and W. Hanke Phys. Rev. B.\n59, 1825 (1999).\n[21] J. V. Alvarez, and C. Gros, Phys. Rev. Lett. 88, 077203\n(2002); Phys. Rev. B 66, 094403 (2002).\n[22] K. Louis, and C. Gros, Phys. Rev. B 68, 184424 (2003).\n[23] D. Schmeltzer et al., Phys. Rev. B. 71, 045429 (2005).\n[24] A. M. Lunde et al., arXiv:cond-mat/0707.1989v1.\n[25] C. Sloggett, A. I. Milstein, and O. P. Sushkov,\narXiv:cond-mat/0606649v2.\n[26] O. F. Sylju rasen, Phys. Rev. Lett. 98, 166401 (2007).\n[27] K.A.Matveev, Phys.Rev.Lett. 92, 106801(2004); Phys.\nRev. B.70, 245319 (2004).\n[28] S. Doniach, Proc. Phys. Soc. (London) 91, 86 (1967).\n[29] T. Rejec, A. Ram˘ sak, and J. H. Jefferson, Phys. Rev. B\n62, 12985 (2000).5\n[30] M. B¨ uttiker and R. Landauer, Phys. Rev. Lett. 49, 1739\n(1982)."    },    {        "title": "1411.5339v2.Dynamical_spin_injection_at_a_quasi_one_dimensional_ferromagnet_graphene_interface.pdf",        "content": "1 \n Dynamical  spin injection at a quasi -one-dimensional ferromagnet -graphene \ninterface  \nS. Singh1, A. Ahmadi1, C.T. Cherian2, 4, E. R. Mucciolo1, E. del Barco1, and \nB. Özyilmaz2,3,4,5  \n1Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542  \n3NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n4Graphene Research Center, National University of Singapore, Singapore 117542  \n5NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of \nSingapore, Singapore 117456  \nAbstract:  We present a s tudy of dynamical spin injection  from a three -dimensional ferromagnet \ninto two-dimensional single -layer graphene.  Comparative f erromagnetic resonance (F MR) \nstudies of f erromagnet/g raphene strips buried underneath the central line of a coplanar \nwaveguide show that t he FMR line width  broadening is the largest  when the graphene layer \nprotrudes laterally  away from  the ferromagnet ic strip, indicating that the spin current is injected \ninto the graphene areas away from the area directly underneath the ferromagnet being excited. \nOur results confirm that the observed damping is indeed a signature of dynamical spin injec tion, \nwherein a pure spin current  is pumped into the single -layer graphene from the precessing \nmagnetization of the ferromagnet . The observed spin pumping efficiency is difficult to reconcile \nwith the expected backflow of spins according to  the standard sp in pumping theory and the \ncharacteristics of graphene, and constitutes a n enigma for spin pumping in two-dimensional \nstructures.  2 \n The efficient generation  of pure spin currents holds a great deal of prom ise for spintronics \napplications , with several existin g methods  already demonstrated, such as  the spin H all effect  1, \nelectrical spin injection  2, voltage -based spin pumping3 dynamical spin pumping  4,5, and optical \ngenerat ion of  spin packets6. Particularly,  the dynamical generation of spin current s carries special  \ninterest because no net charge current is involved in the process . In this method, spin a ngular \nmomentum is transferred from the precessional magnetization in a ferromagnet  (FM)  to an \nadjacent  non-magnetic  (NM)  system. This approach has already been experimentally \ndemonstrated in several FM/NM interfaces, including NM systems such as metals7, \nsemiconductors  5,8 or organic based materials9. A few advantages of this method are  that it does \nnot suffer from the impedance mismatch problem, it is scalable to large samples, it provides a \nhigh spin injection efficiency , and it is not based on the spin -orbit coupling to function, allowing \nits employment in sy stems without this interaction .  \nWithin the exciting cur rent trend to explore novel low -dimensional systems, the possibility to \ninject pure spin currents in graphene and other two -dimensional (2D) crystals has attracted \nconsiderable  attention in the past f ew years . In particular, graphene rises as  a prototype system to \nexplore this physics  due to its crystalline nature, excellent  electronic  properties10, tunable spin -\norbit co upling (i.e., via adatom engineering )11,12 and long spin relaxation lengths13. In addition, \ngraphene can act as a  high-fidelity channel for spin information  transfer  (due to its small intrinsic \nspin-orbit coupling and absence  of nuclear spins) , as well as provide the platform  for electrical \nmanipulat ion of the  spin polarization  (i.e., on-demand enhancement of spin -orbit coupling11,12). \nDynamical spin injection in FM/g raphene (FM/Gr) interfaces has been recently demonstrated  \n14,15. In the original work, we associated  the observed enhan cement in dynamical damping of \nextended FM/Gr films  to the generation of pure spin current in graphene  result ing from loss es of 3 \n spin angular mome ntum  in the ferro magnet  (i.e., spin pumping14). Subsequently, the dynamical \ninjection of spin currents in graphene was demonstrated by spin -charge conversio n \nmeasurements in a P d strip  placed laterally and in close proximity to a FM/Gr interface  \nundergoing FMR15. Although providing evidence for spin injection, none of these w orks shed \nlight into the real nature of spin pumping at the FM/Gr interface.  Estimates of the spin -mixing \nconductance obtained from the broadening of the FMR peaks resulted in surprisingly high values \n(e.g., \ng 5.261019 m-2 from Py/Gr  = Py/Gr - Py = 110-2), comparable to systems with high \nspin-orbit coupling7 (such as heavy metals  Pt and Pd). In addition, the direct deposition of the \nferromagnetic Permalloy (Py) film on top of the graphene layer could cause magneto -structural16 \nchanges in the Py surface  and a subsequent increase in straight fields altering the spin dynamics \nand accumulation in the semiconductor17, responsible for the observed change in damping when \ncomparing with films without graphene, i.e., deposited on the bare wafer.  A direct measuremen t \nof the spin Hall angle in FM/Gr interfaces has not been reported yet. A  recent work by R. \nOhshima et al.  18 published during the preparation of this manuscript , claims the observation of \nthe ISHE signal in single -layer graphene upon conversion of a spin current pumped from an \nextended insulating ferromagnet ( i.e., YIG) into a charge current. According to th at report, the \nspin current is injected perpendicularly to the YIG/G r interface , and the ISHE electric field \nmeasured within the graphene plane, as is the norm in other ferromagnet/conductor \nheterostructures. However, we find this interpretation rather questionable, since  conduction \nperpendicular to a single -layer graphene in not possible  simply due to the lack of a third \ndimension. Since conduction perpendicular to the graphene sheet is not possible, the geometry \nproposed by the authors, where the ISHE is measured along the graphene  plane, is impracticable.  4 \n The observations in that work  could be explained in terms of alternative physics, such as an \ninverse Rashba -Edelstein effect19 or similar, but never in terms of the ISHE.  \nIn this letter we present FMR experiments performed on different FM/Gr interfaces designed \nto systematically identify and eliminate  damping enhancement arising  from process es other than  \nspin pumping. In particular, a substantial enhancement of the Gilbert  damping observed in Py/Gr \nstrips when the graphene layer protrudes a few micrometers away from the edges of a narrow Py \nstrip is univocally a ssociated to spin pumping at the quasi -one-dimensional  interface between the \nPy edge and graphene , which shows lower spin -mixing conductance values (\ng 6.891018 \nm-2) than in extended films but still comparable to those obtained in Py/Pt interfaces  (e.g., \ng\n~1-41019 m-2). We also provide a theoretical analysis which  shows  the observed  spin \ninjection efficiency to lie well beyond that expected from the spin conduction channels provided \nby single -layer graphene, open ing fundamental  question s about the nature of spin injection in to \nthis 2D crystal . \nThe graphene  layers used in our experiments are  grown by the standard chemical vapor \ndeposition ( CVD ) method on thin Cu foil s20. Graphene is subsequently transferred onto the \nsubstrate  using a wet chemistry  process  and characterized by Raman spectroscopy . We use \n14 nm-thick films of Ni20Fe80 Permalloy  (deposited by e -beam evaporation in high vacuum \nconditions) as the ferromagnet for  all our studies. For FMR on extended films, we place  the \nsample  upside -down  on the central part of our broad -band -CPW  FMR sensor , which is  coated \nwith an insulating polymer  to prevent electrical contact with the sample14. For FMR on patterned \nfilms  in the shape of long and narrow  Py/Gr strips , the sample is buried directly underneath the \ncentral line of the CPW , isolated  from it by a thick  (~100 nm) insulating layer of oxide.    5 \n Before getting into the detailed discussion of the main results of this work , we want to briefly \ndiscuss a set of experiments designed to address the effect of magneto -structural changes in the \nsurface of the Py due to the immediate presence of graphene underneath, which  could  cause a \nnon-dynamical  broadening of the FMR 21. In the first experiment, the stacking order  of the Py \nand graphene layers has been reversed with respect to the original experiments14, where the Py \nwas deposited directly on top of the graphene layer  (i.e., a Py/Gr stacking) . In the present case, \nthe Py film is depo sited on a bare Si wafer coated with 300nm of thermally grown SiO 2 and \ngraphene transferred on top afterwards  (i.e., a Gr/Py stacking) , with the objective of maintaining \nthe Py film unaltered by the presence of graphene. A clear enhancement of the Gilbert damping \nis obtained when graphene is present  (i.e.,  Gr/Py   = Gr/Py  - Py = 3.410-3), resulting in a spin -\nmixing conductance of \ng 1.951019 m-2, i.e., three times lower than in previous Py/Gr \nsamples. In the second experiment, a 20  nm-thick Cu spacing l ayer was inserted in between the \nPy and graphene ( i.e., a Py/Cu/Gr stacking), and FMR results compared to those in Py and Py/Cu \nsamples. The rationale is to use the Cu layer as a structural spacer between the Py and the single -\nlayer graphene in order to ma intain the Py film unchanged. Note that a thin Cu film does not \ncontribute to the absorption/diffusion of the spin pumped away from the Py film, since Cu has a \nsubstantially larger spin diffusion length than the Cu  layer  thickness used in these experiments22. \nAgain, a clear enhancement of the G ilbert damping is observed when graphene is present ( i.e., \nPy/Cu/Gr   =  Py/Cu/Gr  -  Py/Cu = 4.210-3), resulting in a spin -mixing conductance of \ng\n2.381019 m-2, i.e., comparable to the values obtained in Gr/Py samples.   6 \n \n \nFigure 1: Sketches illustrating the strips used in the experiments. The Py strips are all the same \ndimensions, with a length of 3 mm and a width w  = 25m. a) Py strip. b) Py/Gr strip. c)  Py/Gr -prt strip, \nwith graphene protruding from the sides of the Py strip by d  = 20 m. \n \nThe set of experiments described above eliminate structu ral changes in the Py as a possible \ncause for the observed damping enhancement. However, the presence of the ferromagnet i n close \nproximity to the single -layer graphene , even in areas away from the FMR excitation , may \ninfluence the diffusion of the spins p umped away from the Py film, which can  still act as a spin \nsink since electrons can flow back into it . To avoid this situation , we patterned the Py/Gr film \ninto long  (l = 3mm)  and narrow  (w = 25 m) strips that  are placed directly underneath the central \nline of the -CPW, as shown in Fig.  1 and in the inset s to Fig. 2. Essentially, we prepare d three \ndifferent samples for this study : a) a Py strip  (Fig.  1a); b) a Py/Gr  strip (Fig.  1b); and,  c) a Py/Gr \nstrip with  the single -layer  graphene protruding  away  from  the Py strip  on both sides , which we \nshall call Py/Gr -prt henceforth (Fig.1c) . The upper inset to Fig. 2 shows a  scanning e lectron  \nmicroscope  image of a Py/Gr -prt strip, where  one can clearly see the cont inuous sheet of \ngraphene extending  away from the central ferromagnet  strip. Note that t he length by which \ngraphene protrudes  on each side of the ferromagnet strip , i.e., d ~12 m, is larger than the  spin \ndiffusion length of CVD graphene ( s ~2 m)23, in order to allow for a total relaxation of the spin \npumped away from the ferromagnet.  The devices are prepared by  transferr ing a single -layer 7 \n graphene  onto a GaAs (undoped) /SiO 2(100nm)  substrate , after which  unwanted graphene areas  \nare etch ed away using a photo resist mask  and standard optical lithography . Following etching  \nand e-beam  evaporation  of the Py strip , a 100  nm-thick layer of silicon oxide is grown atop to \ninsulate the device from the  central line of the -CPW , which ultimately cov ers the sample  (as \ndepicted in  the lower inset to Fig. 2). This geometry guarantees a homogeneous FMR excitation \nof the whole Py strip .  \n \nFigure 2: Field -derivative of the  FMR response for the three strips measured  (Py: Permalloy Gr: \ngraphene and Gr -prt: graphene protruding from the sides of the Py strip).  The upper inset shows an \nelectronic microscope image of the Py/Gr -prt stripe before being placed underneath the central line of \nthe -CPW sensor  (lower inset) . \n \nStandard  broadban d FMR measurements are performed on the samples described above to \nextract the FMR line width s. The corresponding field-derivative s, dM/dH, obtained at  an \nirradiation frequency of  12 GHz with the dc magnetic field applied in the plane of the Py strips \nare shown in F ig. 2. The FMR linewidth, defined as the peak -to-peak distance in the dM/dH data, \nis the largest for the sample with graphene protruding away from the Py strip ( i.e., the Py/Gr -prt 8 \n strip) and the smallest for the sample with Py only (i.e., the Py strip).  The frequency dependence \nof the in-plane excited FMR linewidth for the se samples is shown in Fig. 3a. The observed linear \nfrequency dependence of the linewidth can be explained by  means of the dynamical Gilbert \ndamping model , using the following ex pression:  \n                  \nf H H\n\n34\n0  ,                                (1)                                                                                        \nwhere \nh gB/  is the gyromagnetic ratio and α is the damping parameter, which is related to \nthe Gilbert damp ing through the expression \nSM G , with \nSM  being the saturation \nmagnetization . The two different contributions to the damping in Eqn. 1 are: a) a sample \ndependent i nhomogeneous part  (first term in the equation) , which can be calculated from the \nintercept at zero  extrapolated frequency and it does not depend on frequency ; and, b) dynamical \ndamping  (second term in the equation) , which scales  linearly with frequency  and from whose \nslope the Gilbert damping can be calculated . These extracted damping parameters  for in -plane \nFMR excitation (Fig.  3a) are as follows:   Py = 9.110-3 & GPy = 0.239  GHz; Py/Gr = 11.310-3 \n& GPy/Gr = 0.299  GHz ; Py/Gr-prt = 13.010-3 & GPy/Gr-prt = 0.333 GHz . There is a considerable \nenhancement in damping when going  from the Py -only strip t o the Py/Gr strip, where graphene \nis only present underneath the ferromagnet . This damping can not be attributed to spin pumping \ngiven the 2D nature of graphene, which  is located only underneath the Py strip and  does not \nprovide any conduction channel perpe ndicular to the interface.  Indeed , it has been shown that \ngraphene can act as an effective tunnel barrier for electrical spin injection into silicon due to the \nvery large resistivity of carriers across the graphene sheet24. Most likely, t he observed FMR \nbroadening is due to changes in the magnetic response of the Py due to surface changes induced \nby the graphene, such as an enhancement  of two-magnon scattering  processes . It has been 9 \n recently shown that the deposition of Co films  on graphene results  in magnetic variations a nd \nenhanced ma gnetic coercivity16. In our case, for example, we observe a slight  change (<10%) in \nthe magnetization saturation when graphene is present  (not shown here) . Out-of-plane excited \nFMR measurements on these strips seem to support this hypothesis . Two-magnon processes are \nsubstantially weaker when the precession of the magnet ization is excited wit h the dc field out of \nthe plane, which would explain the similar  H vs. f slopes for Py and Py/Gr strips in the out -of-\nplane excited FMR data of Fig.  3b (black and red data) .  \n \nFigure 3: a) In -plane and b) ou t-of-plane frequency dependences of the FMR linewidth of the three strips \nmeasured  (Py: Permalloy Gr: graphene and Gr -prt: graphene protruding from the sides of the Py strip) . \nThe insets show the FMR field excitation situations and the corresponding direct ions of propagation and \npolarization of the pumped spin currents  (Js) into the graphene area protruding away from the edges of \nthe Py strip.  \n \nThe central result of this work is the clear enhancement of the dynamical damping observed \nin Py/Gr -prt strips und er both in -plane and out -of-plane excited FMR (blue data in Figs.  3a and \n3b). Importantly, t his additional damping can only result from the relaxation of spins  in the area \nof graphene away from the Py strip, where an enhanced relaxation due to proximity ef fects as 10 \n discussed in the introduction for extended films is not an option . Comparing the FMR \nbroadening in the Py/Gr and Py/Gr -prt strips, and using Eqn.  (1) we extract a change in the \ndamping parameter Py/Gr -prt  = Py/Gr -prt - Py/Gr = 1.310-3, resul ting in a spin -mixing \nconductance of \ng 6.891018 m-2. These are a factor 2 -3 smaller than the values found in \nexperiments performed on extended films but still comparable to those found in Py/Pt or Py/Pd \nsamples.  \nWe now discuss the fundamental implications of the experiments described above. The \nobserved additional damping enhancement  provided by the protruding graphene  support s our \nassertion  that spin  pumping must occur across the quasi -one-dimensional  Py/Gr -prt interface at \nthe very edge of the Py strip. The picture we propose is the following. The proximity  of Py to \ngraphene induces a wea k equilibrium ferromagnetization in the latter25. The precessing \nmagnetization in the Py film pumps a spin current into the graphene layer  underneath the Py , \nthus creating an additional non-equilibrium spin accumulation in that layer. Part of the excess \nspin polarization is rel axed by local defects and impurities present on graphene (through  local -\nmoment scattering or spin -orbit coupling). When the graphene layer does not protrude away \nfrom the Py, t he remainder non -equilibrium spin accumulation creates a coherent backflow spin \ncurrent into the Py.  Thus, in steady state, there is no net spin current and the enhanced damping \nof the FMR is mainly due to spin relaxation in the graphene layer underneath the Py. However, \nwhen the graphene layer extends beyond the Py, the non -equilibri um spin accumulation causes a \nspin current to flow into the protruded graphene regions, reducing the amount of coherent \nbackflow into the Py and thus increasing the damping of the FMR due to losses of angular \nmomentum . In this case, it is standard to obtai n the spin -mixing conductance associated to the \npumping of spin into the extended graphene regions through  the expression26 11 \n  \n .                                                      (2) \nYet, this is only justified if the spin current relaxes much faster than the charge diffuses ( i.e., \nwhen the electronic motion in the extended graphene regions is ballistic or when very strong spin \nscattering is present, which are unlikely in our samples). We rather expect the charge to diffus e \nwith a relaxation time \n , where \n  is the spin relaxation time. In this case, a non -\nequilibrium spin population  builds up  on the protruding graphene  near the Py edge. This causes \nthe spin current  to partially diffuse  back into the graphene underneath the Py.  Thus, t he resulting  \nspin-mixing conductance is smaller than that obtained from  measuring the excess damping26 \n                                                               (3) \nwhere \n  is the backflow (dimensionless) parameter, with \n  denot ing the \ndensity of states of graphene at the Fermi level , \n being the length of the Py strip,  A being the \narea of the Py/graphene interface, and \n  representing the spin diffusion length (notice \nthat \n , where \n  = 20 m is the length of the protruding  graphene region). Assuming \ns, \n m2/s, and \n , with \n  meV and \nm/s (see Ref.13 ) we find \n  m2, which i s a much large r value (by several \norders of magnitude) than  the experimental value \n  m2. This implies  a 12 \n negative  value for \n  and therefore indicates that Eqn. (3) may not be directly applicable to \nour setup . We note that this analysi s may change in experiments  performed  in devices using h -\nBN substrates, for which the spin relaxation parameters would drastically change. H -BN has also \nbeen shown to decrease the conductance mismatch in electrical spin injection  and therefore may \naffect s pin reflection and relaxation at FM/graphene interfaces  27. We believe that the main \nproblem  in our analysis  is not in the est imate of the backflow parameter  β, since this follows \nstraightforwardly from reasonable estimates for the graphene parameters l, D, τs, and NF. Instead, \nwe believe that the problem lies on the assumption that spin currents pumped by the Py are fully \ninjected into the protruding graphene sheets. It may be possible that very close to the edge where \nthe protruding graphene meets the Py, there is a strongly enhanced spin relaxation, effectively \nmaking λs a much shorter length scale, rendering the backflow negligible. This relaxation could \nbe due to the Py -Gr edge acting approximately as a semiconducting p -n junction, regaining the \nconducta nce mismatch that were supposed to be eliminated by the spin pumping method, with \ngraphene underneath the Py being heavily p -doped, while protruding areas  are almost undoped . \nHowever, this interpretation requires further experimental verification and a mor e detailed \ntheoretical modeling.   \nIn conclusion, we have presented experimental evidence of a substantial increase in damping \nin Py/Gr strips when graphene is left to protrude from the sides of the ferromagnet. The \nsurprisingly high spin mixing conductance  obtained from the observations raises question s about \nthe physics of dynamical spin injection into two -dimensional structures such as graphene.  Our \nimmediate future objective is the direct measurement of the ISHE voltage generated in the \nprotruding graph ene region as a result of the  pumped spins from the ferromagnet, for which \nelectrodes will be placed at the opposite ends of the protruding graphene lines. However, YIG-13 \n based insulating ferromagnetic strips will be used for this purpose, since the low -resistivity Py \nstrip in the present configuration acts as an electrical shunt and prevents the observation of the \neffect.   \n \nAknowledgements  \nWe thank A. Brataas for valuable discussions. S.S. E.M. and E.d.B. acknowledge support from \nthe National Science Foundat ion (ECCS #1402990).  A.A., C.T.C. and B.O. acknowledge support \nfrom the Singapore National Research Foundation (Grant Nos. NRFRF2008 -07 and NUS -YIA \nR144 -000-283-101). \n \n \n \n \n \n \n \n \n \n 14 \n Supplemental Material  \nDynamical spin injection at a quasi -one-dimensional ferrom agnet -graphene \ninterface  \nS. Singh1, A. Ahmadi1, C.T. Cherian2, 4, E. R. Mucciolo1, E. del Barco1, and \nB. Özyilmaz2,3,4,5  \n1Department of Physics, University of Central Florida, Orlando, Florida USA, 32816  \n2Department of Physics, National University of Singa pore, 2 Science Drive 3, Singapore 117542  \n3NanoCore, 4 Engineering Drive 3, National University of Singapore, Singapore 117576  \n4Graphene Research Center, National University of Singapore, Singapore 117542  \n5NUS Graduate School for Integrative Sciences and E ngineering (NGS), National University of \nSingapore, Singapore 117456  \nIn this document we present a set of experiments designed to study and prevent changes of the \nmagnetic properties of the Py films due to structural changes associate to the adjacent singl e-\nlayer graphene and their possible effect in broadening the FMR linewidth of the ferromagnet.  \n1. Reversing the FM/Gr stacking order  \nWe first turn to experiments in which the graphene layer is transferred onto pre -deposited Py \nfilms, in contrast with our prev ious studies14, where the graphene was transferred onto the \nsubstrate ( i.e., GaAs) and subsequently Py was deposited atop. Since graphene is not atomically \nflat when deposited on most su bstrates28 (with exception of boron nitride), the growth of the \nferromagnet on top of the graphene surface can result in a rough interface and damping 15 \n contributions due to processes other than the spin pumping, such as two magnon scattering. In \naddition, the saturation magnetization of the film can also be modified. We have performed \nnanoscale tunneling microscopy on our graph ene films on SiO 2 substrates and observed the \nnanometer sized ripples of our graphene surfaces. To eliminate an enhancement of damping due \nto structural changes at the interface, we transfer single -layer graphene onto of Py films which \nhave been previously  deposited on SiO 2 wafers, so that the Py is unaffected, or weakly affected \n(see Fig.  SM1a).  \n \nFigure SM4: a) Sketch illustrating a Py film in where graphene is wet transferred on the top of \nthe film. b) Frequency dependence of th e FMR linewidth of Py and Gr/Py films.  \n \nWe performed FMR measurements in both Py and Gr/Py by applying the dc magnetic field \nin the plane of the films. The width of the resonance, in units of external dc field, is extracted by \ntaking the peak -to-peak dista nce of the FMR field -derivatives. The frequency dependences of \nFMR linewidth for both samples are shown in figure SM1b . The frequency -dependent width of a \nFMR resonance represents the effective dynamical damping of the precessing magnetization of \nthe ferro magnet. Any loss of spin angular momentum  (in this case due to the presence of 16 \n graphene)  will correspondingly result in an increase in linewidth. Since pure spin currents carry \naway spin angular momentum, we can theoretically associate t he damping enhancem ent to be a \ndirect consequence of dynamical spin pumping from the Py into the graphene layer .  \nThe damping parameters extracted for these samples are: Py = 9.610-3 & GPy = 0.257  GHz; \nPy/Gr = 13.010-3 & GPy/Gr = 0.339  GHz. This corresponds to a change i n the damping \nparameter ( Δα) of 35%, as compared to 88% change observed in our previous studies of films \nwith the Py deposited onto of the graphene layer14. The difference in the relative damping change \njust by reversing the stacking order of the Py and graphene layers is indicative that a good \nfraction of the FMR broadening in Py/Gr samples was likely due to structural changes in the \nsurface of the Py film, which has been deposited on top of the rough graphene s urface. It is \npossible that in our experiment the wet transfer of graphene into only one of the Py films causes \nsome degree of contamination ( e.g., oxidation of the Py surface) altering the magnetic properties \nof the ferromagnetic film or the spin polariza tion at the interface. Recent experiments have \nshown how contamination during the wet transfer process of two -dimensional crystals can \ndecrease spin polarization at the interface29. We do not observe any appreciable change in the \nsaturation magnetization, so such contamination is not drastically changing the \nmagnetostructural properties of the Py film.  Nevertheless, the substantial amount of damping \nobserved in the Gr/Py samples o f Fig.  SM1 indicates that spin pumping at the Gr/Py interface is \nlikely the main cause of FMR broadening.  \n2. Inserting a Cu spacing layer in between the Py film and graphene  \nIt was brought to our attention that while graphene is transferred onto evaporated Py  films \nfrom an aqueous solution there is a possibility that the Py is oxidized, causing an enhancement of 17 \n damping of structural origin. Although we find this possibility unlikely to explain the observed \nresults, we made controlled extended films by inserti ng a thin layer of Copper (Cu) in between \nthe Py film and graphene, and compared the FMR results with Py/Cu films without graphene \nunderneath. The actual samples that we developed are: a) bare Py (14nm); b) Py(14nm)/Cu \n(20nm); and, Py(14nm)/Cu(20nm)/Gr. Th e cartoons representing samples are shown in the inset \nto Fig. SM2a. We use 20nm -thick Cu because it is substantially less than the spin diffusion \nlength and one expects to see negligible spin relaxation and hence no enhanced damping due to \nthe presence of  Cu itself in the FMR experiments22. In addition, the effect of a graphene layer \n20nm away from the Py sample will be negligible and the Py/Cu interface will remain unaltered, \nwhether or not graphene is underneath the heterostructures.  \n \nFigure SM5: a) Illustrations of the Py, Cu and Gr heterostructures. b) Frequency dependence of \nthe FMR linewidth of Py , Py/Cu and Py/Cu/Gr  films . \n \nThe in -plane excited FMR linewidths for the above mentioned samples are presented as a \nfunction of frequency in Fig.  SM2b. As exp ected, there is no change in damping due to presence \nof the Cu film underneath the Py, since the pumped spins reflect coherently back into the 18 \n ferromagnet (no diffusion in the Cu film). One would need to increase the thickness of the Cu \nfilm beyond the spi n diffusion length for pumping -induced damping to be observable22. Now, in \nthe case of the Py/Cu/Gr stacking we do observe a clear change in the damping (i.e. the slope of \nthe frequency dependence of FMR linewidth changes). The extracted damping parameters for \nthese samples are: Py = 1210-3 & GPy = 0.315  GHz; Cu/Py = 11.910-3 & GCu/Py = 0.312  GHz ; \nand, Gr/Cu/Py = 16.210-3 & GGr/Cu/Py = 0.422  GHz . This is about 35 % change in damping due to \nthe presence of graphene underneath the Cu film, which is similar to what we observe in \nexperiments with graphene transferred atop of the Py films.  \n3. Comparison between different experiments  \nA first inclination is to attribute the observed damping enhancements to spin pumping into \nthe whole graphene area underneath the ferromagnet. However, one  needs to keep in mind that \ngraphene is a two dimensional material with no conduction channels perpendicular to the plane. \nFor standard 3D FM/NM heterostructures where spin pumping has been studied7, the spin \ngradient is perpendicular to the FM/NM interface, i.e. the spin polarization decays \nperpendicularly to the plane. In the case of graphene, the only d irection spins can decay is along \nthe plane of the graphene sheet, and this needs to be away from the microwave excitation area. In \nour above explained experiments on extended thin films only a small part (~20  um  1000  µm) \nof the total film (~5  mm  5 mm)  is excited by to the central line of the µ -CPW. Therefore, the \nspin density pumped from the excited part of the ferromagnet into the graphene directly \nunderneath can only decay in the horizontal graphene plane which is not under microwave \nexcitation. Unfo rtunately, in experiments with extended films (as those presented in this \nSupplemental Information document), the areas of graphene away from excitation are in close 19 \n proximity to the ferromagnet, which is covering the whole film. Given that graphene is hig hly \nsensitive to its environment30, the presence of the ferromagnet is expected to change the intrinsic \nelectronic and spin relaxation properties and result in an enhanced diffusion of the spin currents. \nThis asks for a device geometry in which the damping assoc iated with spin pumping can be \nsystematically differentiated, which is the purpose of the experiments discussed in the main text \nof this article.  \nFor the sake of clarity, we have compared the change in damping observed for different \nconfiguration samples i n Table SM1, including the results in Py/Gr -prt strips discussed in the \nmain text, where the enhanced spin diffusion due to the proximity of the ferromagnet in areas of \ngraphene away from the excitation has been eliminated. Although being the smallest, t he 11% \ndamping enhancement observed in the latter case can be univocally associated to spin pumping \ninto single -layer graphene and still stands comparable to values observed in heavy metals. It has  \nbeen shown that a small density of  adatoms ( e.g., atomic hyd rogen) on the graphene surface can \ninduce local deformations resulting in huge enhancement s of spin orbit coupling11. Recently, a \nlarge spin Hall Effect  has also been reported for CVD grown graphene (similar to what is used \nfor this work) du e to presence of residual Cu adatoms at the graphene surface. Although Cu itself \nhas low spin-orbit  interaction , it can induce local deformation in the graphene lattice which \nresults in an enhance ment of the spin -orbit interaction .  \n \n \n \n 20 \n  \nTable  SM1 : Comparison between damping enhancements for all the samples discussed in this \narticle.  \n \nReferences:  \n1 A. K. Patra, S. Singh, B. Barin, Y. Lee, J. -H. 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V enkata Kamalakar, André Dankert, Johan Bergsten, Tommy Ive, and Saroj P. Dash,  Sci. Rep. \n4 (2014).  \n28 Masa Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, and E. D. Williams,  Nano Letters 7 (6), \n1643 (2007).  \n29 André Dankert, M. Venkata Kamalakar, Abdu l Wajid, R. S. Patel, and SarojP Dash,  Nano Res., 1 \n(2014).  \n30 F. Schedin, A. K. Geim, S. V. Morozov, E. W. Hill, P. Blake, M. I. Katsnelson, and K. S. Novoselov,  \nNat Mater 6 (9), 652 (2007).  \n "    },    {        "title": "2110.05624v1.Magnetoelastic_Coupling_in_Hole_doped_Two_dimensional__b_eta__PbO.pdf",        "content": " \nMagnetoelastic Coupling in Hole-doped Two-dimensional  β-PbO \nYan Liang, Xingshuai Lv*, Thomas Frauenheim* \nYan Liang  \nBremen Center for Computational Materials Science, University of Bremen, 28359 Bremen, \nGermany . \nXingshuai Lv  \nShenzhen JL Computational Science and Applied Research Institute, 518109 Shenzhen, P.R. China.  \nProf. Thomas Frauenheim  \nBremen Center for Computational Materials Science, University of Bremen, 28359 Bremen ; \nGerman; Shenzhen JL Computational Science and Applied Research Institute, 518109 Shenzhen ; \nP.R. China; Beijing Computational Science Research Center, 100193 Beijing, P.R. China . \nE-mail: lvxingshuai@csar.ac.cn  (X.L);  thomas.frauenheim@bccms.uni -bremen.de  (T.F.)  \n \n \nAbstract : \nThe realization of i ntertwined  ferroelasticity and ferromagnetism in two -dimensional (2D) lattices is \nof great interest for b road nanoscale applications, but still remains a remarkable challenge. Here, we \npropose an alternative approach to realize the strongly coupled ferromagnetism and ferroelasticity by \ncarrier doping. We demonstrate that prototypical 2D β-PbO is dynamically, thermally and \nmechanically stable. Under hole doping, 2D  β-PbO possesses ferromagnetism and ferroelasticity \nsimultaneously. Moreover, the robustness of ferromagneti c and ferroelastic orders are doping tunable. \nIn particular, 2D β-PbO featu res in -plane easy magnetization axis that is coupled with lattice direction, \nenabling ferroelastic manipulation of spin direction. Our work highlights a new direction for 2D \nmagnetoelastic research and enables the possibility for multifunctional devices.  \n \n \n \n \n \n \n \n  \nI. Introduction  \nMultiferroics refer to an important quantum matter that simultaneously possess more than one ferroic \norder parameters, namely (anti) ferromagnetism ( AFM/ FM), ferroelasticity (FEL) and ferroelectricity  \n(FE), has brought great opportunitie s for applications in spintronics, transistors and data storage [1-4]. \nIn this material class, both multiple ferroic  orders and strong coupling between them are extraordinary \ndesirable for nonvolatile  modulat ion one or two ferroic  orders  by using another . In accompany with \nthe continuing quest for miniaturiz ed electronic  devices,  the upsurge of two -dimension al (2D) \nmultiferroics  is promot ed [5-8]. Among all the multiferroics, magnetism based multiferroics stand at \nthe paramount place  due to its best aspects of low -cost reading combined with efficient \nmechanical/electrical writing for nonvolatile memory [9]. However, FE -AFM/FM multiferroics are \nrarely reported in single layers because of the inherent mutual exclusion between magnetism and FE, \ndubbed  as “d0 rule” [10]. FEL, in contrast, shows better compatibility with AFM/FM because neither \nsymmetry nor partial filled orbitals are specially required, which serves as a fantastic playground for \nexploring emergent physics at 2D limit. Nevertheless, AFM/FM -FEL systems has b een scarcely \nreported, arising from the fact that 2D magnets are rather rare itself [11-14]. To date, several 2D systems \nwith FEL and AFM/FM are proposed  [9,15-20], but strong magnetoelastic effect only exist in a few \ncases, including FeOOH  [21], MnNX  [22], CrSX [23], VF 4 [24], and VTe 2 [17].  \nThe model of carrier doping -induced Stoner ferromagnetism has been well established in many \narchetypal 2D systems, such as GaSe [25], α-SnO [16] and phosphorene  [26], which provide a valuable \nalternative to obtain both ferroelastic and FM orders in 2D single layers. Recently, carrier doping -\ninduced multiferroicity has been predicted in several 2D systems, but none of this category exhibits \ncoupled FEL and FM  so far [15,16,27-29]. For example, doping α-SnO and PbO  single layers indeed \nsucceed in realizing multiferroic phases present FM and FEL simultaneously, but two ferroic order \nparameters in all of them just coexist due to the robust out -of-plane easy magnetization , excluding \nany magnetoelastic coupling [15,16]. To this end, realizing strong magnetoelastic coupling in 2D limit,  \nusing doping as a start point, is highly desirable for both fundamental scientific interest and \ntechnological applications.  \nIn this paper, based on first -principles, we report the discover y of strong magnetoelastic effect in \n2D β-PbO by hole doping. It is demonstrated that semiconducting 2D  β-PbO prototype  can be easily \naccessed from the layered bulk counterpart. Moreover, 2D β-PbO features strain -driven 90o variant \nswitching with moderate barrier and strong transition signal, suggesting intrinsic FEL. Furthermore, \nthe non -dispersive nature of the valence b ands leads to the electronic instabilities, exhibiting both FM \nand ferroelastic orders simultaneously under hole doping. Most strikingly, hole -doped 2D β-PbO \nshows robust in -plane easy magnetization with tunable magnetic anisotropy that is closely linked w ith \nFEL, enabling strong magnetoelastic effect, namely, doping -induced in -plane FM can be preciously  \nmanipulated via reversible ferroelastic strain.  \nII. Results and  Discussions  \n \nFIG. 1. (a) Top and side views of 2D β-PbO. (b) Variation of total  energy with respect to d-d0 for 2D \nβ-PbO,  where d is the distance between the top β-PbO layer and the underlying layers.  d0 is the \nequilibrium distance. The inset shows the exfoliation process.  (c) 2D Brillouin zone. (d) Phonon \nspectrum  for 2D β-PbO. (e) Top and s ide views and total energy of  β-PbO supercell  captured from \nthe end of AI MD simulations at 300 K  and the corresponding fluctuation of total energy and \ntemperature . (f) ELF for 2D β-PbO.  \nThe crystal structure of bulk β-PbO is shown in Figure S1 in the Supplemental Material , which is \ncrystallized in yellow orthorhombic form with van der Waals layered structure and has been \nsynthesized several decades ago  [30-32]. The atomic layer has a tetragonal symmetry and consists four \nPb and four O atoms per unit cell, in which Pb (O) is four -coordinated with adjacent O (Pb) atoms as \npresented in Fig. 1(a) for top and side views of 2D β-PbO. The lattice constants of 2D β-PbO are \nfound to be a = 5.80  Å and b = 4.76 Å, which is quite close to its bulk phase [32]. The weak interlayer \nvan der Waals forces in bulk β-PbO suggests that monolayer β-PbO could be exfoliated from its \nlayered bulk.  We evaluate the possibility of mechanical or liquid isolation by calculating the cleavage \nenergy. The resulting cleavage energy as a function of separation d-d0 is shown in Fig.  1(b), the total \nenergy increases gradually with the separation distance and fi nally converges to a constant value  of \n~ 0.31 J  m-2. This value is smaller than that of graphene ( 0.37 ± 0.01Jm-2) [33], Nb2ATe 4 (0.42/0.43 \nJm-2) [34] and GeS (0.52 Jm-2) [35], which firmly indicates that the experimentally exfoliation of 2D \n \nβ-PbO from its bulk is highly feasible.  \nTo examine the stability of 2D β-PbO, the phonon spectrum is first calculated as depicted in Fig. \n1(d). No imaginary phonon  frequencies  are observed in the entire Brillouin zone, confirming the \ndynamical stability  of 2D β-PbO. The thermal stability is further checked by performing AIMD \nsimulations. After heating at 300 K for 6 ps, the total energy of 2D β-PbO only fluctuate slightly and \nno broken bonds or structure  reconstruction appear  [Fig. 1(e)], demonstrating that 2D β-PbO is rather \nstable to keep their structural integrity at room temperature. Additionally, the four independent elastic \nconstants are calculated to be C11 = 20.03  N/m, C22 = 59.98  N/m, C12 = 16.18  N/m and C66 = 20. 74 \nN/m, respectively, which meet the Born criteria for mechanical stable 2D lattice: C11C22 – C122 >0 \nand C66>0 [36]. We also characterize the bonding types in 2D β-PbO by calculating its electron \nlocalization function (ELF), which is plotted in Fig. 1(f). Substantial concentrations of electrons are \ndistributed at the Pb and O sites as well as the centers between them, reflecting a clear covalent \nbonding feature.  \n \nFIG. 2. (a) Schematic plot of ferroelastic switching path between two ferroelastic states F and F′ for \n2D β-PbO. (b) Energy barriers of ferroelastic switching under hole doping concentration of 0 × \n1014/cm2, 0.14 × 1014/cm2 and 7.23 × 1014/cm2. The direction -dependent (c)Young’s modulus and (d) \nPossion’s ratio of 2D β-PbO.  \n \nFEL is defined by the coexistence of multiple orientation variant s of spontaneous lattice strain, \nwhich can be switched from one variant to another energetically equivalent state under external \nmechanical stimuli. Hence, FEL is expected in 2D β-PbO taking the crystal symmetry of 2D β-PbO \ninto account. The two orientation states with the same energy (states F and F′) are presented in Fig. \n2(a). For ferroelastic state F, lattice parameter a is larger than b. Under the external compressive strain \nalong a axis, the larger lattice switches to b axis. That is, ferroelastic state F transforms to another \nferroelastic state F′, in which |a′| = |b| and |b′| = |a|. As a result, t he lattice transformation trace between \nF and F′ can be regarded as the result of 90o rotation operation on F or F′ state. The intermediate state, \nalso refer to paraelastic state P, that connect the transition pathway is a centrosymmetric square lattice \nwith lattice constants |a| = |b| = 5.29 Å. As displayed in Figure S2 , the phonon spectrum of paraelastic \nstate P shows imaginary modes aro und Γ and M points, suggesting it is unstabl e. Thus, the relaxation \nof paraelastic state P is spontaneous, giving rise to ferroelastic states F or F′. \nTo get deeper understanding on the FEL of 2D β-PbO, the transformation strain matrices η for the \nferroelastic  transition is studied based on the ferroelastic states F, F′ and paraelastic state P. \nTransformation matrix J relates to F, F′ and P, and the transformation strain matrix η can be obtained  \nby the definition of Green -Lagrange strain tensor [37]: \n11 11( ) ([ ] )22TT\nP I I P  − −= − = −J J I H H H H I\n \nwhere  superscript T is the matrix transpose,  I is a 2 × 2 identity matrix. HP = |5.290\n05.29| and HI = \n|5.800\n04.76| are the lattice constant matrices for the paraelastic state P and ferroelastic states F, \nrespectively. Then, the strain matrix η = |0.120\n0−0.10|  is obtained, implying that there is 10% \ncompressive strain along b direction and 12% tensile strain along a direction for ferroelastic state F, \nin comparation with paraelastic state.  \nIn order to judge the excellence (feasibility and robustness) of FEL in 2D β-PbO,  the energy \nbarrier for ferroelastic transition is first calculated using NEB method. As shown in Fig. 2(b), in view \nof structural symmetry, the revolution paths of P → F and P → F′ are identical. The intrinsic energy \nbarrier for 2D β-PbO is obtained to be 34.32 meV/atom. This value is higher than those of SnS ( 4.2 \nmeV/atom ) [7] and ScP (2.53 meV /atom)  [39] but much lower than those of GaTeCl (0.16 eV/atom)  [39] \nand CrSX (X = Cl, Br, I)  (116 - 178 meV/atom)  [23]. This moderate energy berrier not only facilitates \nthe ferroelastic orientation switching under mechanical stimuli, but also highlights t he robustness of  \nferroelastic states against external perturbation. Another criterion that determines the ferroelastic \nperformance is the reversible ferroelastic strain, which is defined as [(|a/b| - 1) × 100%]. The high \nferroelastic strain  for 2D β-PbO reaches 22%, comparable or larger than those of frequently reported \nferroelastics  [9,22,34,40], ensuring the distinct  ferroelastic signal intensity of 2D β-PbO that in favor of \nshape memory applications.  \nActually, structural rigidity and transverse synthe tic strain  of the system against corresponding \naxial strain  play decisive  roles in FEL. High flexibility and positive Posson’s ratio along ferroelactic \nvariant orientations are two indispensable  preconditions.  Thus, Young’s modulus Y(θ) and Poisson’s \nratio υ(θ) of 2D β-PbO are investigated  based on the elastic constants above. Young’s modulus Y(θ) \nand Poisson’s ratio υ(θ) can be obtained by the expression as follows [41]: \n2\n11 22 12\n4 2 2 4\n22 11-( ) = cos + cos sin + sinC C CYθCθCθθCθ\n， \n4 2 2 4\n12 12\n4 2 2 4\n22 11cos - cos sin + sin( ) = cos + cos sin + sinCθBθθCθθCθAθθCθ\n. \nHere, A = (C11C22 - C122)/C66 - 2C12 and B = C11 + C22 - (C11C22 - C122)/C66. θ = 0o corresponds to \na axis. The θ-dependent values of Y(θ) and υ(θ) are presented in Fig. 2(c) and 2(d). Young’s modulus \nof 2D β-PbO exhibits strong anisotropy due to the anisotropic structural m orphology . Besides,  values \nof Young’s modulus for 2D β-PbO range from 15.66 - 54.17 N/m, which is comparable with AgF 2 \n(32 N/m)  [9], 1S′-MSSe  (M = Mo, W)  (70-72 N/m)  [42] and GaTeCl (42 N/m)  [39] and much smaller \nthan those of α-MPI (141-152 N/m)  [43] and graphene (340 N/m)  [44], indicative of the superior \nmechanical flexibility  and thus it is easier to apply strain in the in -plane directions for switching the \nFEL. As for Poisson’s ratio, it also shows anisotropic behavior. Poisson’s ratio is obtained to be 0.27 \n(0.81) along a (b) axis, suggesting sensitive structural deformation  under external stress . And \ninterestingly, negative Poisson’s ratio of -0.025 is found along diagonal direction in 2D β-PbO, as \ndepicted in F igure S 3.  \n \nFIG. 3. (a) Orbital  resolved band structure and (b) density of states of 2D β-PbO. (c) Variation of the \nmagnetic energy and magnetic moment (per carrier) as a function of doping concentration. (d) Density \nof states under the hole doping concentration of 3.61 × 1014/cm2. (e) Doping -dependent in out -of-\nplane and in -plane  MAEs. (f) Schematic representation of ferroelastic hysteresis loop that is coupled \nwith doping -induced magnetism in 2D β-PbO. Two magnetic states M1 and M2 possess orthogonal \neasy magnetization axes . \nWe next focus on the electronic properties of 2D β-PbO. The orbital -resolved band structure along \nhigh-symmetric points is shown in Fig. 3(a). Clearly, 2D β-PbO is an indirect band gap semiconductor, \nwith the valence band maximum (VBM) locat es along  Γ-X line and conduction band minimum (CBM) \nsits between X -M. The band gap is calculated to be 2.61 eV at the PBE functional, whereas HSE06 \ncorrection to the PBE gap is about 1.10 eV . In addition, CBM of 2D β-PbO is dominated primarily \nby py and pz prbitals  of Pb, while VBM is mainly formed by py and pz orbitals of O. Another interesting \npoint of the band structure worth noting is that valence bands are distinctive by very flat dispersion \nin the momentum space. This is reasonable as valence bands nea r the Fermi level is O -dominated, \nwhile the bonding environment results in the rather weak p-orbital coupling between the nearest \nneighboring O atoms and is responsible for the less disperse d valence bands. As illustrated in Fig. \n3(b), The flat valence ban ds create multiple saddle points in the vicinity of X, M and Y points, leading \nto a remarkably extended van Hove singularity in the density of states  (DOS)  that is first peaked at \n \nthe energies of 0.17 eV below the VBM. According to band -picture model,  spontaneous \nferromagnetic ground state emerges  when  relative exchange interaction is larger than the loss in \nkinetic energy, that is, Stoner criterion N(EF)I > 1 [45]. Therefore, Stoner ferromagnetism is expected \nin 2D β-PbO, once  doping pushes  Fermi level approaching the  divergent point s. \nTo verify the electronic instability of 2D β-PbO driven by dole doping, Fig. 3(c) shows calculated \nconcentration -dependent magnetic energy per carrier, ΔEmag (i.e., total energy difference between \nnon-magnetic  phase and ferromagnetic phases), and magnetic moment (i.e., \n| | / |n hole n holen n n n−   kkk m k k k\n , where m and | nk> denote spin magnetic moment \noperator and the Bloch states ). Ferromagnetic state begins to be more stable when the hole \nconcentration is larger than 0.72 × 1014/cm2 (0.2 hole per unit cell). Then, larger spin magnetic \nmoment develops as concentration increase, and saturates at 1 μB/hole upon hole density between \n1.45 - 10.12 × 1014/cm2 (0.4 - 2.8 hole per unit cell). Further increment of doping concentration \nreduces the magnetic moment. In contrast  to magnetic moment, magnetic energy ΔEmag increases \nmonotonically as a function of doping concentration. In Fig . 3(d), we plot the spin -polarized DOS of \n2D β-PbO under the  doping level of 3.61 × 1014/cm2 (1 hole per unit cell) . Compared with pristine \n2D β-PbO, doping causes a considerable energy splitting of 0.49 eV between spin -up and spin -down \nchannels with Fermi level traverse  only spin -down bands, indicating its fascinating prospect for \ncontrollable half -metallic spintronics.  Given that carriers doping  concentration  of ~1015/cm2 has \nalready been realized experimentally [46,47], high efficiency carrier  injection  and ferromagnetism in \nmetal shrouded  2D β-PbO is readily to be achieved.  \nIt is worth noting that obvious structural phase transition from 2D to one -dimensional -like \nframework of 2D β-PbO is observed in the doping process when the doping density surpass 9.40 × \n1014/cm2 (2.6 hole per unit cell), which is also the critical point f or sudden surge of magnetic energy \nand is similar to previous report  [48]. To show this more clearly, the evolution of lattice constants and \nfinal structure are shown in Fi gure S4 (a). Besides, such new metastable phase will experience \nspontaneous collapsing to the original structure after doped carriers are removed. Moreover , FEL \nmaintains  under hole doping  concentration smaller than 9.40 × 1014/cm2, whereas both the stability \nof ferroelastic state and ferroelastic signal intensity of 2D β-PbO could be significantly enhanced as \nshown in Fig. 2(b) and Figure S4(b).  \nTo get deeper understanding on the doping -induced ferromagnetism, the MAE of 2D β-PbO under  \ndoping is further investigat ed by involving SOC. Two in -plane ([100] and [010]) and one out -of-plane \ndirections ([001]) are considered. Fig. 3(e) depicts the MAE as a function of doping density, from \nwhich we can clearly find the minimum energy spin orientation is [100]. Therefore, it favors in -plane  \neasy magnetization over out -of-plane, and easy magnetization axis for hole doped 2D β-PbO lies  \nrobustly along the [100] direction throughout the hole doping process. More interestingly, it is found \nthat MAEs in both in -plane and out -of-plane are boosted by doping. For example, the MAE of [001] \n([010]) direction is 150 (23) μeV per unit cell larger than [100] direction at low doping level of 1.45 \n× 1014/cm2 but becomes as high as 3824 (1104) μeV per unit cell at doping concentration of 8.68 × \n1014/cm2, which is sufficient to stabilize the in -plane ferromagnetic ordering.  \nAs discussed above, 2D β-PbO hold FEL and ferromagnetism simultaneously upon hole doping. \nAlthough the FEL and ferromagnetism have different origins, the mutual coupling between them can \nbe expected. The most favorable in -plane easy magnetization axis is s trongly coupled to the lattice \ndirection (i.e., along the longer lattice vector), allowing for precise direction tailor of  spin orientation \nby FEL. As schematically plotted in Fig. 3(f), ferroelastic  switching rotate the lattice orientation of \n2D β-PbO by 90o, thus the easy magnetization axis will also undergo a 90o rotation. That’s to say, \nferroelastic transition switches easy magnetization axis from a to b axis, achieving strong \nmagnetoelastic effec t. Different from doped α-SnO [16] and PbO [15], as shown in Figure S5, strong \nmagnetoelastic effect maintains even beyond monolayer. To our best knowledge, this is the first \nexample that systematically demonstrate s strong magnetoelastic coupling in 2D limit by carrier \ndoping , and we believe this work will ignite more efforts to forge  2D functionalities via carrier doping.  \nIII. Conclusion  \nIn summary, we show  that stable 2D β-PbO harbors ferroelastic  and ferromagnetic orders \nsimultaneously, depending on the concentration of hole doping. Meanwhile, the robustness of FEL \nand ferromagnetism are doping tunable. Remarkable ly, strong magnetoelastic effect is observed in \n2D β-PbO. Under hole doping, easy mag netization axis along the longer lattice vector is closely linked \nwith FEL, rendering ferroelastic control of MAE. These findings imply the possibility of achieving \nstrong magnetoelastic effect in 2D lattices by doping, which are useful for both fundamenta l research \nand technological applications.  \nExperimental Section  \nFirst-principles calculations  are performed on the basis of density functional theory  as implemented \nin the Vienna Ab Initio  Simulation Package (V ASP) [49]. The Per dew–Burke –Ernzerhof  (PBE)  \nparametrization of  the generalized  gradient approximation (GGA) is  used to account for  electron \nexchange -correlation  [50]. Monkhorst –Pack k-point meshes  [51] of 9 × 9 × 1 and 11  × 11 × 1 are used \nfor geometry optimization and self -consistent electron ic structure calculations . The convergence \ncriteria of energy and Hellmann -Feynman force are 1.0×10-5 eV per atom and 0.01 eV  Å-1, \nrespectively, at the cutoff energy of 500 eV .  The thickness of  vacuum space is set to be larger than  18 \nÅ in the direction normal  to the basal plane to avoid  the interaction between its periodic adjacent \nlayers . Spin-orbital coupling (SOC) is considered  in magnetocrystalline anisotropy energy (MAE)  \ncalculations . Phonon spectrum  is obtained using the  finite di splacement method as implemented in \nthe Phonopy code [52]. Ab Initio  molecular dynamic  (AIMD ) simulation is performed  with the NVT \nensemble  and t he temperature is controlled by using the Nosé –Hoover thermostat  [53]. 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We rigorously prove the emergence of chiral\nground states and we compute, by performing the \u0000-limits of proper renormalizations and scalings,\nthe energy for a chirality transition.\n1 Introduction\nLow-dimensional magnets have attracted the attention of the scienti\fc community in the last years\n(see [12] and the references therein). Among them, edge-sharing chains of cuprates provide a natural\nexample of frustrated lattice systems, the frustration resulting from the competition between ferromag-\nnetic (F) nearest-neighbor (NN) and antiferromagnetic (AF) next-nearest-neighbor (NNN) interactions.\nIn this paper we study some of the multiple scale properties of these systems, focusing on a classical spin\nmodel as a \frst step towards the understanding of its quantum analogue (see [13] for a discussion about\nthe relation between classical and quantum models in chains of cuprates).\nWe consider a minimal energy model describing the magnetic properties of one dimensional frustrated\nmagnetic systems: the so called F-AF spin chain model. On the one-dimensional torus [0 ;1], the state\nof the system is described by the values of a vectorial spin variable parameterized over the points of the\nlattice Z. The energy of a given state of the system u:i2Z7!ui2S1is\nE(u) =\u0000J1X\ni2Z(ui;ui+1) +J2X\ni2Z(ui;ui+2); (1.1)\nwhereJ1;J2>0 are the NN and the NNN interaction parameters, respectively. While the \frst term of\nthe energy is ferromagnetic and favors the alignment of neighboring spins, the second, being antiferro-\nmagnetic, frustrates it as it favors antipodal next-to-nearest neighboring spins. As a result the frustration\nof the system depends on the relative strength of the ferromagnetic/antiferromagnetic constants. A more\nre\fned analysis shows that the frustration can be actually measured in terms of \u000b=J1=J2. More specif-\nically (see Proposition 3.2 and Remark 3.3), for \u000b\u00154 the ground state of the system is ferromagnetic,\nwhile for 0 < \u000b\u00144 it is helimagnetic (see \fgure 1). The description of the ground states of the F-AF\nsystem for a choice of the parameters such that \u000b'4 is the main aim of our analysis. In this case\nthe system is said to be close to the ferromagnet/helimagnet transition point (examples of edge-sharing\ncuprates in the vicinity of the ferromagnetic/helimagnetic transition point can be found in [14], while an\nanalysis of the thermodynamic properties of such spin chains can be found in [13], [15] and [16]).\nIn order to study F-AF chains close to the ferromagnet/helimagnet transition point we need to\nperform a multiple scale analysis of the energy in (1.1). We start by \frst scaling the functional in (1.1)\n\u0003Zentrum Mathematik - M7, Technische Universit at M unchen, Boltzmannstrasse 3, 85748 Garching, Germany. Email:\ncicalese@ma.tum.de\nyZentrum Mathematik - M7, Technische Universit at M unchen, Boltzmannstrasse 3, 85748 Garching, Germany. Email:\nfrancesco.solombrino@ma.tum.de\n1arXiv:1312.5573v3  [math-ph]  27 Jan 2014Figure 1: Ground states of the spin system for 0 <J1<4 for clockwise and counter-clockwise chirality\nby a small parameter \u0015n(\u0015n!0 asn!1 ). Further setting J2= 1 (so that now the frustration\nparameter is now \u000b=J1) and Zn=fi2Z:\u0015ni2[0;1]gwe de\fneEn:fu:Zn7!ui2S1g!Ras\nEn(u) =\u0000J1X\ni2Zn\u0015n(ui;ui+1) +X\ni2Zn\u0015n(ui;ui+2): (1.2)\nIt turns out that the ground states of Encan be completely characterized (see Proposition 3.2 and Re-\nmark 3.3). Neighboring spins are aligned if J1\u00154 (ferromagnetic order), while they form a constant\nangle'=\u0006arccos(J1=4) if 0<J 1<4 (helimagnetic order). In this last case the two possible choices\nfor'correspond to either clockwise or counter-clockwise spin rotations, or in other words to a positive\nor a negative chirality (see Fig 1). Such a degeneracy is known in literature as chirality symmetry . The\nenergy necessary to break this symmetry is, to the best of our knowledge, an open problem. In this\npaper we provide a solution to this problem in the case of a system close to the ferromagnet/helimagnet\ntransition point, that is to say that we are able to \fnd the correct scaling to detect the symmetry breaking\nand to compute the asymptotic behavior of the scaled energy describing this phenomenon as J1is close\nto 4. Before coming to the description of our analysis, it is worth noticing that if instead of a vector\nspin parameter with continuous symmetry we consider a scalar one, i.e. u2f\u0006 1g, then the helicity\nsymmetry translates into the periodicity of the ground states. In this case in [8] it has been proved that\nthe asymptotic analysis of these systems can be performed without any restriction on the values of J1.\nTo set up our problem we let the ferromagnetic interaction parameter J1depend on nand be close\nto 4 from below, that is in (1.2) we substitute J1byJ1;n= 4(1\u0000\u000en) for some vanishing sequence\n\u000en>0. For such energies in Theorem 2.1 we prove that, as a consequence of an abstract result proven\nin [3], their \u0000-limit (with respect to the weak- ?convergence in L1) asn!1 is a constant functional\nwhose value can be approached by weakly vanishing sequences unthat may mix on a mesoscopic scale\ncon\fgurations having opposite chirality. Such a poor description suggests that, in order to get further\ninformations on the ground states of the system we need to consider higher order \u0000-limits (see [7] and [10]\nfor more details as well as for the general theory of development by \u0000-convergence). Note that the choice\nof the right energy scaling which may capture the phenomena we are interested in is not straightforward.\nIn fact the continuous symmetry of the order parameter u2S1adds a new di\u000eculty: it allows for very\nslow variations in the angle between neighboring spins which results in the emergence of very low energy\nphase changes. This implies that, even if we expect to \fnd a limit energy accounting for 0-dimensional\ndiscontinuities of some parameter related to the chirality, the continuous symmetry of the spins makes the\ncorrect scaling not a 'surface'-type scaling. Note that this would not be the case if the spin \feld u2f\u0006 1g\nas it is shown in [1] (see also [8]) where the degeneracy of the ground states is solved by a surface scaling.\nSimilar problems regarding the continuos symmetry of the order parameter arise already in [2] for NN\nsystems in the context of XY spin models (see also [4], [5], [6] and [9] for related Ginzburg-Landau-type\nmodels). In [2, example 1] it is explicitly proved that the system does not undergo any phase separation\nthat may be detected by a surface scaling. Such an example can be straightforwardly exported in the\n2context of frustrated spin chains and, as a consequence, we are led to renormalize the energy of the system\nand study the asymptotic behavior of a new functional Hnde\fned as\nHn(u) =En(u)\u0000minEn\n\u0016n\u0015n(1.3)\nfor some\u0016n!0 to be found. In terms of Hn, \fnding the energy the system spends in a transition\nbetween two states with di\u000berent chirality translates into the following problem: depending on the scale\n\u000en\n(i) \fnd a scaling \u0016nand an order parameter znsuch that if supnHn(un)\u0014Cthen, asn!1 ,\nznconverges to some zdescribing a system whose chirality may have at most a \fnite number of\ndiscontinuities,\n(ii) for such a choice of \u0016ncompute the \u0000-limit of Hn(with respect to the convergence zn!zin\nthe previous step) and interpret the limit functional as the energy the system spends on the scale\n\u0015n\u0016nfor a \fnite number of chirality transitions.\nThe main result of this paper is contained in Theorem 4.2 which states that the right scale to consider in\norder to keep track of energy concentration is \u0015n\u000e3=2\nn(corresponding to the choice \u0016n=\u000e3=2\nn). We prove\nthat, within this scaling, several regimes are possible. Roughly speaking, for nlarge enough, we show\nthat the spin system has a chirality transition on a scale of order \u0015n=p\u000en. As a result, depending on the\nvalue of lim n\u0015n=p\u000en:=l2[0;+1] di\u000berent scenarios are possible (see Fig 2 for a schematic picture\nof the transition). If l= +1chirality transitions are forbidden (equivalently we \fnd that the energy\nfor a transition is in\fnite). If l >0 the spin system may have di\u000buse and regular macroscopic (on an\norder one scale) chirality transitions whose limit energy is \fnite on W1;2(I) (provided some boundary\nconditions are taken into account). When l= 0 transitions on a mesoscopic scale are allowed. In this\ncase the continuum limit energy is \fnite on BV(I) and counts the number of jumps of the chirality of\nthe system.\nWe think it is worth noticing that, to the best of our knowledge, this paper shows for the \frst time\nthe presence of multiple scale regimes in a chirality transition. It is our opinion that this phenomenon is\nquite general and suggests that the analysis of frustrated discrete systems should take advantage from a\nrigorous variational method any time the parameters describing frustration and scaling may compete. As\na \fnal technical remark we would like to point out that, although our analysis is presently con\fned to the\n1-dimensional case, it can be easily extended to an n-dimensional systems for which NNN interactions\nare present only along the coordinate directions. Indeed, in such a case the \u0000-limit of the energy of such\nsystems can be straightforwardly obtained by a slicing procedure starting from our 1-dimensional result.\n2 Notation and Preliminaries\nDenoted by J\u001aRan open interval and by \u0015na vanishing sequence of positive numbers, we de\fne\nZn(J) as the set of those points i2Zsuch that\u0015ni2J. Givenx2R, we denote by [ x] the integer part\nofx. The symbol S1stands as usual for the unit sphere of R2. Given two vectors a;b2R2we will denote\nby (a;b) their scalar product. We will denote by Un(J) the space of functions u:i2Zn(J)7!ui2S1\nand byUn(J) the subspace of those usuch that\n(uimin+1;uimin) = (uimax;uimax\u00001); (2.1)\nwhereiminandimaxare the minimum and the maximum of Zn(J), respectively. We analogously denote\nbyU(J) the space of functions u:Z!S1such that (2.1) holds with iminandimaxthe minimum and\nthe maximum of Z\\J, respectively. Given K2Rmwe denote by co(K) the convex hull of K. We set\nQh= (0;h). Givenv= (v1;v2);w= (w1;w2)2S1we de\fne the function \u001f[v;w] :S1\u0002S1!\u00061 as\n\u001f[v;w] = sign(v1w2\u0000v2w1);\n3Figure 2: The optimal con\fguration of the spins of the F-AF chain in a chirality transition (above).\nThe transition in terms of a scalar parameter related to the chirality of the system (see Section for more\ndetails)clockwise chirality\n\u0015n=p\u000en!lcounter-clockwise chirality\nwith the convention that sign(0) = \u00001.\nWe recall some preliminary results concerning the general theory of spin-type discrete systems in the\nbulk scaling. The following theorem has been proved in [3]. We state it here in a version which best \fts\nour setting. Let K\u001aRmbe a bounded set. For all \u00182Zletf\u0018:R2m!Rbe a function such that\n(H1)f\u0018(u;v) =f\u0000\u0018(v;u),\n(H2) for all \u0018,f\u0018(u;v) = +1if (u;v)62K2,\n(H3) for all \u0018, there exists C\u0018\u00150 such thatjf\u0018(u;v)j\u0014C\u0018for all (u;v)2K2, andP\n\u0018C\u0018<1.\nLet us de\fne the set of functions\nDn(J;Rm) =fu:R!Rm:uconstant on \u0015n(i+ [0;1)) for any i2Zn(J)g\nand the family of functionals Fn:L1(J;Rm)!(\u00001;+1]\nFn(u) =8\n><\n>:X\n\u00182ZX\ni2R\u0018\nn(J)\u0015nf\u0018(ui;ui+\u0018) ifu2Dn(J;Rm)\n+1 otherwise;(2.2)\nwhereR\u0018\nn(J) :=fi2Zn(J) :i+\u00182Zn(J)g. Givenv:Z!RmandA\u001aRopen and bounded, we\nde\fne the discrete average of vinAas\nhvi1;A=1\n#(Z\\A)X\ni2Z\\Avi:\nTheorem 2.1. Letff\u0018g\u0018satisfy hypotheses (H1)-(H3). Then Fn\u0000(w\u0003\u0000L1)-converges to\nF(u) =Z\nJfhom(u(x))dx\nfor allu2L1(J;co(K)), wherefhom is given by the following homogenization formula\nfhom(z) = lim\n\u001a!0lim\nh!+11\nhinf8\n<\n:X\n\u00182ZX\n\f2R\u0018\n1(Qh)f\u0018(v(\f);v(\f+\u0018));hvi1;Qh2B(z;\u001a)9\n=\n;; (2.3)\n4whereR\u0018\n1(J) :=fi2Z\\J:i+\u00182Z\\Jg.\nWe now state (with minor variations) a result proved in [11] regarding the discrete approximations\nof Modica-Mortola type energies. We say that a function W:R![0;+1) is a double-well potential if\nit is locally Lipschitz and satis\fes the following properties:\n(1)W(z) = 0 if and only if z2f\u0006 1g,\n(2) limz!\u00061W(z) = +1,\n(3) there exists C0>0 such thatfz:W(z)\u0014C0g=I1[I2withI1;I2intervals on which Wis\nconvex.\nLet\u000bn;\fnbe two sequences of positive numbers such that lim n\u000bn= 0, limn\u000bn=\fn= 1 and lim n\u0015n=\u000bn=\n0 and letGn:L1(J)![0;+1] be de\fned as\nGn(u) =8\n><\n>:\u000bnX\ni\u0015n\u0012ui+1\u0000ui\n\u0015n\u00132\n+1\n\fnX\ni\u0015nW(ui) ifu2Dn(J;R)\n+1 otherwise;(2.4)\nwithWa double-well potential. The following \u0000-convergence result holds.\nTheorem 2.2. LetGn:L1(J)![0;+1]be as in (2.4) , then, with respect to the L1(J)convergence,\n\u0000-lim\nnGn(u) =(\nCW#(S(u)\\J)ifu2BV(J;f\u00061g)\n+1 otherwise in L1(J);(2.5)\nwhereCW:= 2R+1\n\u00001p\nW(s)ds.\nProof. The proof follows by Theorem 2 :1 in [11] once we observe that for all unsuch that supnGn(un)\u0014\nCwe have \f\f\f\f1\n\fn\u00001\n\u000bn\f\f\f\fX\ni\u0015nW(ui)\u0014Cj\fn\u0000\u000bnj\n\u000bn;\nso that\nGn(un) =\u000bnX\ni\u0015n\u0012ui+1\u0000ui\n\u0015n\u00132\n+1\n\u000bnX\ni\u0015nW(ui) +o(1):\nRemark 2.3. In the explicit case W(s) = (1\u0000s2)2the constant cW=8\n3.\n3 The energy model: the bulk scaling\nIn this section we introduce the F-AF model of a frustrated ferromagnetic spin chain and prove a\n\frst result concerning the \u0000-limit of its bulk scaling.\nLetI= (0;1) and let us consider a pairwise-interacting discrete system on the lattice Zn(I) whose\nstate variable is denoted by u:Zn(I)!S1. Such a system is driven by an energy En:Un(I)!\n(\u00001;+1) given by\nEn(u) =\u0000J1[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+1) +J2[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+2);\nfor some non negative constants J1;J2. Without loss of generality we will set J2= 1, thus considering\nthe family of energies\nEn(u) =\u0000J1[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+1) +[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+2): (3.1)\n5Moreover we will consider the case J12(0;4], the case J1>4 will be shortly discussed in Remark 3.3.\nSince we are not interested to the possible formation of boundary layers, we \fx periodic boundary\nconditions on the system:\n(u1;u0) = (u[1=\u0015n];u[1=\u0015n]\u00001) (3.2)\nor equivalently u2Un(I).\nRemark 3.1. The periodic boundary conditions in (3.2) are an alternative to the computation of the\n\u0000-limit ofEnwith respect to a local convergence.\nAs usual in the analysis of discrete systems we may embed the family of functionals on a common\nfunctional space, extending Ento some Lebesgue space. To this end we associate to any u2Un(I) a\npiecewise-constant interpolation belonging to the class\nCn(I;S1) :=fu2Un(I) :u(x) =u(\"i) ifx2\u0015n(i+ [0;1)); i2Zn(I)g: (3.3)\nAs a consequence we may see the family of energies Enas de\fned on a subset of L1(I;S1) and consider\ntheir extension on L1(I;S1). With an abuse of notation we do not relabel these functionals and set\nEn:L1(I;S1)!(\u00001;+1] as\nEn(u) =8\n>><\n>>:\u0000J1[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+1) +[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+2)ifu2Cn(I;S1)\n+1 otherwise.(3.4)\nWe now de\fne the functional Hn:L1(I;S1)!(\u00001;+1] as\nHn(u) =8\n>><\n>>:1\n2[1=\u0015n]\u00002X\ni=0\u0015n\f\f\f\fui+2\u0000J1\n2ui+1+ui\f\f\f\f2\nifu2Cn(I;S1)\n+1 otherwise.(3.5)\nSincejuij= 1 for all i2Zn, thanks to (3.2), the energy in (3.4) can be rewritten, in terms of Hnas\nEn(u) =Hn(u)\u0000\u0012\n1 +J2\n1\n8\u0013\n(1\u0000cn\u0015n); (3.6)\nforcn=1\n\u0015n\u0000h\n1\n\u0015ni\n+ 12[1;2), so that\n[1=\u0015n]\u00002X\ni=0\u0015n= (1\u0000cn\u0015n): (3.7)\nEquality (3.6) suggests that in order to study the asymptotic properties of Enwe can equivalently study\nthe non negative functional Hn.\n3.1 Ground states of Hn\nIn this section we characterize the global minimizers of En, we give upper and lower bounds on\nits \u0000-limit as n!1 for di\u000berent values of J1. As a corollary we show that in the case J1= 4, the\ncontinuum limit is indeed trivial.\nProposition 3.2. LetEn:L1(I;S1)!(\u00001;+1]be the functional in (3.1) and0\u0014J1\u00144. Then\nmin\nu2L1(I;S1)En(u) =\u0000\u0012\n1 +J2\n1\n8\u0013\n(1\u0000cn\u0015n): (3.8)\nFurthermore, a minimizer unofEnoverL1(I;S1)satis\fes\n(ui\nn;ui+1\nn) =J1\n4and (ui\nn;ui+2\nn) =J2\n1\n8\u00001 (3.9)\nfor alli= 0;:::; [1=\u0015n]\u00002.\n6Proof. LetHnbe de\fned as in (3.5). Since Hn\u00150, by (3.6) we deduce En(u)\u0015\u0000\u0010\n1 +J2\n1\n8\u0011\n(1\u0000cn\u0015n)\nfor allu2L1(I;S1). Now, \fx '2[\u0000\u0019\n2;\u0019\n2] so that cos( ') =J1\n4. Then, we construct un2Cn(I;S1)\nby setting, for all i= 0;:::; [1=\u0015n],\nui\nn= (cos('i);sin('i)):\nBy the prosthaphaeresis formulas we get\nui\nn+ui+2\nn= 2 cos(')ui+1\nn=J1\n2ui+1\nn\nfor alli= 0;:::; [1=\u0015n]\u00002. This implies Hn(un) = 0, thus En(un) =\u0000\u0010\n1 +J2\n1\n8\u0011\n(1\u0000cn\u0015n) and (3.8)\nfollows.\nConsider now a minimizer unofEnoverL1(I;S1). By de\fnition of En, we have that un2\nCn(I;S1). By (3.8) it must be Hn(un) = 0, which in turn implies\nui+1\nn=2\nJ1(ui\nn+ui+2\nn) (3.10)\nfor alli= 0;:::; [1=\u0015n]\u00002. Sinceuntakes values on the unit sphere, by taking the modulus squared in\n(3.10) we further get that\n1 =4\nJ2\n1jui\nn+ui+2\nnj2=8\nJ2\n1(1 + (ui\nn;ui+2\nn));\nso that\n(ui\nn;ui+2\nn) =J2\n1\n8\u00001:\nBy this and (3.10) we also get\n(ui\nn;ui+1\nn) =2\nJ1(ui\nn;ui\nn+ui+2\nn) =2\nJ1(1 + (ui\nn;ui+2\nn)) =J1\n4;\nas required.\nRemark 3.3. Note that the case J1>4 is trivial. In fact the ground states are all ferromagnetic, that\nisui\nn= \u0016ufor alli= 0;:::; [1=\u0015n] and for some \u0016 u2S1. Indeed in this case, set E(J1=4)\nn the energy in\n(3.4) forJ1= 4, we have that, for all u2Un(I)\nEn(u) =\u0000J1[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+1) +[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+2) (3.11)\n=E(J1=4)\nn (u)\u0000(J1\u00004)[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+1): (3.12)\nBy the previous proposition E(J1=4)\nn is minimized on uniform states, which trivially also holds true for\nthe second term in the above sum. In particular the minimal value can be straightforwardly computed:\nminEn(u) =\u0000(J1\u00001) (1\u0000cn\u0015n):\n3.2 Zero order estimates\nThe following theorem is the main result of this section.\nTheorem 3.4. LetEn:L1(I;S1)!(\u00001;+1]be the functional in (3.1) . Then \u0000-limnEn(u)with\nrespect to the weak- \u0003convergence in L1(I)is given by\nE(u) :=(R\nIfhom(u(x))dx ifjuj\u00141;\n+1 otherwise in L1(I;R2);(3.13)\n7where the convex function fhom:B1!Ris given by the following asymptotic homogenization formula:\nfhom(z) := lim\n\u001a!0lim\nk!11\nkinf\nu2U(Qh)f\u0000J1k\u00002X\ni=0(ui;ui+1) +k\u00002X\ni=0(ui;ui+2); <u> 1;Qh2B(z;\u001a)g: (3.14)\nFurthermore\n(i) ifJ1\u00154thenfhom(z) =\u0000(J1\u00001),\n(ii) if 0<J1\u00144then the following estimate hold:\n(J1\u00004)2\n8jzj2\u0014fhom(z) + (1 +J2\n1\n8)\u0014(J1\u00004)2\n8jzj: (3.15)\nMoreover there exists h: [0;1]!Rconvex and monotone non-decreasing such that fhom(z) =\nh(jzj).\n(iii) if 0<J1\u00144we have that minE(u) =E(0) =\u0000(1\u0000J2\n1\n8).\nProof. The formula in (3.14) follows applying Theorem 2.1 in the special case\nf\u0018(u;v) =8\n><\n>:\u0000J1\n2(u;v) ifj\u0018j= 1;\n1\n2(u;v) ifj\u0018j= 2;\n0 otherwise(3.16)\nandK=S1. To prove (i) we notice that, as observed in Remark 3.3, Enis minimized by constant\nS1-valued functions and its minimum is \u0000(J1\u00001)(1\u0000cn\u0015n). SinceEn\u0000-converges to Egiven by\n(3.13) we have that\nfhom(z)\u0015\u0000(J1\u00001);8z2B1; (3.17)\nfhom(z) =\u0000(J1\u00001);8z2S1: (3.18)\nBy the convexity of fhom, (i) follows.\nWe divide the proof of (ii) into the lower bound and the upper bound estimates.\nLower bound: letun2Cn(I;S1) be such that un\u0003*u inL1(I;R2), by (3.6) it is left to prove that\nlim inf\nnHn(un)\u0015(J1\u00004)2\n8Z\nIju(x)j2dx: (3.19)\nWe de\fne the functions wnto be piece-wise constant on the cells of the lattice and such that\nwi\nn=(\nui\nn+ui+2\nn\n2ifi= 0;:::; [1=\u0015n]\u00002;\n0 ifi= [1=\u0015n]\u00001:(3.20)\nLet us show that wn\u0003*u inL1(I;R2). Since supnkwnk1\u00141 andun\u0003*u inL1(I;R2), it su\u000eces to\nshow that, for all ( a;b)\u001a\u001aIit holds\nZb\na(un(x)\u0000wn(x))dx!0: (3.21)\nThe above limit follows on observing that\n\f\f\f\f\fZb\na(un(x)\u0000wn(x))dx\f\f\f\f\f\u0014\f\f\f\f\f\f[b=\u0015n]X\ni=[a=\u0015n]\u0015n(ui\nn\u0000wi\nn)\f\f\f\f\f\f+o(1)\n=1\n2\f\f\f\f\f\f[b=\u0015n]X\ni=[a=\u0015n]\u0015n(ui\nn\u0000ui+2\nn)\f\f\f\f\f\f+o(1)\u00144kunk1\u0015n+o(1)!0:\n8We also need to de\fne the functions ^ unpiece-wise constant on the cell of the lattice and such that\n^ui\nn:=ui+1\nn. An analogous computation as the one above shows that ^ un\u0003* u inL1(I;R2). We now\nmay write that\nHn(un) = 2[1=\u0015n]\u00002X\ni=0\u0015n\f\f\f\fui\nn+ui+2\nn\n2\u0000J1\n4ui+1\nn\f\f\f\f2\n\u00152Z\nIjwn(x)\u0000J1\n4^un(x)j2dx+o(1): (3.22)\nBy the weak lower semicontinuity of the L2norm we deduce (3.19).\nUpper bound: we \frst prove that fhom(0) =\u0000(1\u0000J2\n1\n8). Using the already proved lower bound in ( ii)\nit is left to show that fhom(0)\u0014\u0000(1\u0000J2\n1\n8). To this end we construct the sequence of piecewise-constant\nfunctionsunon the cells of the lattice such that ui\nn= (cos'i;sin'i). It holds that un\u0003*0 and moreover,\nas shown in Proposition 3.2 En(un) = (1\u0000cn\u0015n)(\u00001\u0000J2\n1\n8). As a result\nfhom(0) =Z\nIfhom(0)dx\u0014lim inf\nnEn(un) =\u00001\u0000J2\n1\n8: (3.23)\nWe now prove the upper bound for z2S1. Let us consider a constant sequence un=z. Using formula\n(3.4) and (3.6), we have that\nEn(un) = (1\u0000cn\u0015n)(\u00001\u0000J2\n1\n8+(J1\u00004)2\n8):\nArguing as before, it follows that, for all z2S1,fhom(z) + (1 +J2\n1\n8)\u0014(J1\u00004)2\n8.\nNow for all z2B1the upper bound follows by the convexity of fhom.\nFinally, by the de\fnition of fhom it follows that, for all z2B1fhom(Rz) =fhom(z) for all\nR2SO(2). As a consequence of this and [19, Corollary 12.3.1 and Example below] we also get that\nfhom(z) =h(jzj) for some h: [0;1]!Rconvex and monotone non-decreasing. Eventually (iii) follows\nby (ii).\nRemark 3.5. We notice that 0 is the unique minimizer of fhom, in all the cases when the \u0000-limit is\nnon trivial, that is for 0 <J1<4.\n4 Renormalization of the energy close to the ferromagnetic state\nand chirality transitions\nIn this section, motivated by the study of spin systems close to the helimagnet/ferromagnet transition\npoint, we let the ferromagnetic interaction parameter J1be scale dependent and approach the transition\nvalue 4 from below. Namely we set J1=J1;n= 4(1\u0000\u000en) for some \u000en>0,\u000en!0. We then perform a\nrenormalization of the energy Enand introduce a new functional whose asymptotic behavior will better\ndescribe the ground states of the system. More precisely we de\fne Ehf\nn:L1(I;R2)!(\u00001;+1] and\nHhf\nn:L1(I;R2)![0;+1]as:\nEhf\nn(u) :=8\n>><\n>>:\u00004(1\u0000\u000en)[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+1) +[1=\u0015n]\u00002X\ni=0\u0015n(ui;ui+2)ifu2Cn(I;S1)\n+1 otherwise.(4.1)\nHhf\nn(u) :=(\n1\n2P\ni\u0015n\f\fui+2\u00002(1\u0000\u000en)ui+1+ui\f\f2ifu2Cn(I;S1)\n+1 otherwise.(4.2)\nNote that by Theorem 3.4 it holds\nHhf\nn(u) =Ehf\nn(u)\u0000minEhf\nn=Ehf\nn(u) + (3\u00004\u000en+ 2\u000e2\nn)(1\u0000cn\u0015n) (4.3)\n9Proposition 4.1. LetEhf\nn:L1(I;S1)!(\u00001;+1]be the functional in (4.1) . Then \u0000-limnEn(u)\nwith respect to the weak- \u0003convergence in L1is given by\nE(u) :=(\n\u00003 ifjuj\u00141;\n+1 otherwise in L1(I;R2):\nProof. Observing that for all u2Cn(I;S1) it holds that\njEhf\nn(u)\u0000EJ1=4\nnj\u00144\u000en; (4.4)\nthe result immediately follows by Theorem 3.4.\nIn what follows we will de\fne a convenient order parameter such that the \u0000-limit of a scaled version Hhf\nn\nis given by a functional penalizing the helimagnetic transition around the ferromagnetic state.\nWe \frst introduce the order parameter. Given un2Cn(I;S1), fori= 0;1;:::; [1=\u0015n]\u00001 we\nassociate to each ui\nn;ui+1\nnthe corresponding oriented central angle \u0012i\nn2[\u0000\u0019;\u0019) given by\n\u0012i\nn:=\u001f[ui\nn;ui+1\nn] arccos((ui\nn;ui+1\nn)): (4.5)\nWe now set\nwi\nn= sin\u0012\u0012i\nn\n2\u0013\n: (4.6)\nWe eventually de\fne the order parameter of our problem as\nzi\nn=r\n2\n\u000enwi\nn: (4.7)\nNote that, the above procedure de\fnes Tn:un7!znwhich associate to any given un2Cn(I;S1) a\npiecewise-constant function zn2~Cn(I;R) where\n~Cn(I;R) :=fz: [0;1)!R:z(x) =zi\nn;ifx2\u0015nfi+ [0;1)g; i= 0;1;:::; [1=\u0015n]\u00001g\nwithznas in (4.7). We observe that if zn=Tn(un) =Tn(vn) thenunandvndi\u000ber by a constant\nrotation (possibly depending on n) so thatHhf\nn(un) =Hhf\nn(vn). Therefore, with a slight abuse of\nnotation, we now regard Hhf\nnas a functional de\fned on z2L1(I;R) by\nHhf\nn(z) =(\nHhf\nn(u);ifz2~Cn(I;R)\n+1 otherwise.(4.8)\nNote that in the de\fnition above, uis any function such that Tnu=z. As a consequence, it will\nbe natural to state the \u0000-convergence theorem considering the convergence with respect to the order\nparameterz.\nTheorem 4.2. LetHhf\nn:L1(I;R)![0;+1]be the functional in (4.8) . Assume that there exists l:=\nlimn\u0015n=(2\u000en)1=2. ThenHhf(z) := \u0000 -limnHhf\nn(z)=(p\n2\u0015n\u000e3=2\nn)with respect to the L1(I)convergence is\ngiven by one of the following formulas:\n(i) ifl= 0\nHhf(z) :=(\n8\n3#(S(z))ifz2BV(I;f\u00061g);\n+1 otherwise.(4.9)\n(ii) ifl2(0;+1)\nHhf(z) :=(\n1\nlR\nI(z2(x)\u00001)2dx+lR\nI(z0(x))2dx ifz2W1;2\njperj(I);\n+1 otherwise,(4.10)\nwhere we have set W1;2\njperj(I) :=fz2W1;2(I); s:t:jz(0)j=jz(1)jg.\n10(iii) ifl= +1\nHhf(z) :=(\n0 ifz=const;\n+1 otherwise.(4.11)\nIn the following proposition we consider an equi-bounded sequence of spins and obtain a \frst bound\non the scalar product between neighbors.\nProposition 4.3. Let\u0016n!0and letunbe such that\nsup\nnHn(un)\u0014C\u0015n\u0016n; (4.12)\nthen for all iwe have that\njJ1\n4\u0000(ui\nn;ui+1\nn)j\u0014p\nC\u00122\nJ1+1\n2\u0013\n\u00161=2\nn (4.13)\nProof. Since for all iwe have that\n\f\f\f\fui+2\u0000J1\n2ui+1+ui\f\f\f\f2\n\u0015\u0012\f\f\f\fui\u0000J1\n2ui+1\f\f\f\f\u00001\u00132\n;\nby (4.12) and the de\fnition of Hnwe have that\nX\ni\u0015n\u0012\f\f\f\fui\u0000J1\n2ui+1\f\f\f\f\u00001\u00132\n\u0014C\u0015n\u0016n\nwhich implies that, for all i,\u0012\f\f\f\fui\u0000J1\n2ui+1\f\f\f\f\u00001\u00132\n\u0014C\u0016n:\nAs a result we have that \f\f\f\fui\u0000J1\n2ui+1\f\f\f\f2\n\u00001!2\n\u0014C\u0012\n2 +J1\n2\u00132\n\u0016n:\nBy an explicit computation we \fnally get (4.13).\nProof of Theorem 4.2. We prove the theorem only in cases ( i) and (ii), since the proof of ( iii) involves\nonly minor changes of the arguments we need in the other two cases.\nLet us consider a sequence zn2~Cn(I;R) such that supnHhf\nn(zn)\n\u0015n\u000e3=2\nn\u0014C < +1. Equivalently there is a\nsequenceun2Cn(I;S1) satisfying supnHhf\nn(un)\n\u0015n\u000e3=2\nn\u0014C < +1. We claim that\nHhf\nn(un)p\n2\u0015n\u000e3=2\nn\u0015p\n2\u000e1=2\nn\n\u0015n[1=\u0015n]\u00002X\ni=0\u0015n\u0000\n(zi\nn)2\u00001\u00012+\u0015np\n2\u000e1=2\nn(1\u0000\rn)[1=\u0015n]\u00002X\ni=0\u0015n\u0012zi+1\nn\u0000zi\nn\n\u0015n\u00132\n(4.14)\nfor some\rn!0. Associating to each ui\nnthe angles \u0012i\nnand the functions wi\nnintroduced in (4.6), by\nmeans of the trigonometric identity 1 \u0000cos(2x) = 2 sin2(x) we can write that\n1\u0000(ui\nn;ui+1\nn) = 1\u0000cos(\u0012i\nn) = 2 sin2\u0012\u0012i\nn\n2\u0013\n= 2(wi\nn)2\n1\u0000(ui\nn;ui+2\nn) = 1\u0000cos(\u0012i+1\nn+\u0012i\nn):\nBy Lemma 4.3 with \u0016n=\u000e3\n2nthere exists a constant C0such that\n1\u0000(ui\nn;ui+1\nn)\u0014C0\u000e3\n4n; (4.15)\n11so that in particular \u0012i\nn!0.\nIntroducing the function wnand the angles \u0012n, by(3.7) and (4.3) we may rewrite Hhf\nn(un) as follows\nHhf\nn(un) = 4(1\u0000\u000en)[1=\u0015n]\u00002X\ni=0\u0015n(1\u0000(ui\nn;ui+1\nn))\u0000[1=\u0015n]\u00002X\ni=0\u0015n(1\u0000(ui\nn;ui+2\nn))\n+2\u000e2\nn(1\u0000cn\u0015n)\n= 8(1\u0000\u000en)[1=\u0015n]\u00002X\ni=0\u0015n(wi\nn)2\u0000[1=\u0015n]\u00002X\ni=0\u0015n(1\u0000cos(\u0012i+1\nn+\u0012i\nn))\n+2\u000e2\nn(1\u0000cn\u0015n):\nWe further point out the following identities:\n4(wi\nn)2\u0000sin2(\u0012i\nn) = 4(wi\nn)4;\n2[1=\u0015n]\u00002X\ni=0\u0015nsin2(\u0012i\nn) =[1=\u0015n]\u00002X\ni=0\u0015n(sin2(\u0012i\nn) + sin2(\u0012i+1\nn)):\nThe \frst one comes from the trigonometric identity 4 sin2(x)\u0000sin2(2x) = 4 sin4(x) while the second\nfollows from the boundary condition (3.2). Moreover the following limit holds true\nlim\n(x;y)!(0;0)\nx6=ysin2(x) + sin2(y)\u0000(1\u0000cos(x+y))\n(sin(x=2)\u0000sin(y=2))2= 2 (4.16)\nupon observing that\nsin2(x) + sin2(y)\u0000(1\u0000cos(x+y)) = (sin(x)\u0000sin(y))2\u0000(1\u0000cos(x\u0000y)):\nWe can therefore continue estimate Hhf\nn(un) as\nHhf\nn(un) =[1=\u0015n]\u00002X\ni=0\u0015n(8(wi\nn)2\u00002 sin2(\u0012i\nn))\u00008\u000en[1=\u0015n]\u00002X\ni=0\u0015n(wi\nn)2\n+2[1=\u0015n]\u00002X\ni=0\u0015nsin2(\u0012i\nn)\u0000[1=\u0015n]\u00002X\ni=0\u0015n(1\u0000cos(\u0012i+1\nn+\u0012i\nn))\n+2\u000e2\nn(1\u0000cn\u0015n)\n= 8[1=\u0015n]\u00002X\ni=0\u0015n\u0012\n(wi\nn)4\u0000\u000en(wi\nn)2+\u000e2\nn\n4\u0013\n+[1=\u0015n]\u00002X\ni=0\u0015n\u0000\nsin2(\u0012i+1\nn) + sin2(\u0012i\nn)\u0001\n\u0000[1=\u0015n]\u00002X\ni=0\u0015n(1\u0000cos(\u0012i+1\nn+\u0012i\nn))\n\u00158[1=\u0015n]\u00002X\ni=0\u0015n\u0012\n(wi\nn)2\u0000\u000en\n2\u00132\n+ 2(1\u0000\rn)[1=\u0015n]\u00002X\ni=0\u0015n(wi+1\nn\u0000wi\nn)2;\nfor some\rn!0. The last inequality is a consequence of (4.16) once we recall that, by (4.15), \u0012n!0\nuniformly. In terms of znthe inequality in (4.16) becomes:\nHhf\nn(un)\u00152\u000e2\nn[1=\u0015n]\u00002X\ni=0\u0015n\u0000\n(zi\nn)2\u00001\u00012+ (1\u0000\rn)\u000en[1=\u0015n]\u00002X\ni=0\u0015n(zi+1\nn\u0000zi\nn)2:\n12Figure 3: In black the function tanh( t). In red the function z\"(t) in the limsup constructionR\"R\"+\"1\n1\u0000\"\nThe claim (4.14) is proved on dividing byp\n2\u0015n\u000e3=2\nn.\nThe claim implies the liminf inequality both in case ( i) and (ii). In case ( i) it is obtained apply-\ning Theorem 2.2 and Remark 2.3. For what concerns ( ii), it su\u000eces to observe that piecewise a\u000ene\ninterpolations of the sequence znassociated to an equibounded unare, in this case, weakly compact in\nW1;2\njperj(I) so that the lower bound follows by standard lower semicontinuity.\nIn order to prove the limsup inequality we separately discuss cases ( i) and (ii).\nCase(i). By the locality of the construction it su\u000eces to exhibit a recovery sequence for z=\u0000\u001f(0;1=2]+\n\u001f(1=2;1). As it is well known, (see for example [18], [17]) zmin(t) = tanh(t) is the solution of the following\nproblem\nmin\u001aZ+1\n\u00001((z0(t))2+ (z(t)2\u00001)2)dt;z2W1;2(R); z(\u00061) =\u00061\u001b\n=:m (4.17)\nand, by a direct computation, the above minimum is m=8\n3. For all\">0 there exists R\u000f>0 such that\nmaxf sup\nt2(\u00001;\u0000R\")jzmin(t) + 1j;sup\nt2(R\";+1)jzmin(t)\u00001jg\u0014\" (4.18)\nZ+R\"\n\u0000R\"((z0\nmin(t))2+ (zmin(t)2\u00001)2)\u0014m+\"\nLet us de\fne z\":R!Ras the odd C1function such that\nz\"(t) :=8\n><\n>:zmin(t)ift2[0;R\"];\np\"(t) ift2(R\";R\"+\");\n1 ift2(R\"+\";+1);\nwherep\"is a suitable third order interpolating polynomial that we may choose such that kp0\n\"k1\u00142.\nNote that, by the de\fnition of z\"and by the properties (4.18) above we have that there exists C > 0\nsuch that Z+1\n\u00001((z0\n\"(t))2+ (z\"(t)2\u00001)2)dt\u0014m+C\": (4.19)\nLetzn;\"2~Cn(I;R) be de\fned as follows\nzi\nn;\"=z\"\u0012p2\u000en\n\u0015n(\u0015ni\u00001\n2)\u0013\n: (4.20)\n13We have that zn;\"!zinL1(I) asn!+1. If we set\ni+=\u00141\n\u0015n\u00121\n2+\u0015np2\u000en(R\"+\")\u0013\u0015\n+ 1 and i\u0000=\u00141\n\u0015n\u00121\n2\u0000\u0015np2\u000en(R\"+\")\u0013\u0015\n;\nthenjzi\nn;\"j= 1 for all i\u0015i+ori\u0014i\u0000. We now put wi\nn;\"=q\n\u000en\n2zi\nn;\", so that in particular for all i\njwi\nn;\"j\u0014q\n\u000en\n2. We can therefore de\fne the angles\n'i\nn;\"= 2iX\nj=0arcsin(wj\nn;\"):\nLet us observe that sign( 'i+1\nn;\"\u0000'i\nn;\") = sign(wi\nn) and that '1\nn;\"\u0000'0\nn;\"='[1=\u0015n]\nn;\"\u0000'[1=\u0015n]\u00001\nn;\" . As a\nconsequence, upon de\fning ui\nn;\"= (cos('n;\")i;sin('i\nn;\")), we have that un2Un(I) and that Tn(un;\") =\nzn;\". Using again the limit (4.16) and repeating the computation as in the proof of the lower bound, we\nobtain the existence of a sequence \u0011n!0 such that\nHhf\nn(un;\")p\n2\u0015n\u000e3=2\nn\u0014p\n2\u000e1=2\nn\n\u0015n[1=\u0015n]\u00002X\ni=0\u0015n\u0000\n(zi\nn;\")2\u00001\u00012+\u0015np\n2\u000e1=2\nn(1 +\u0011n)[1=\u0015n]\u00002X\ni=0\u0015n \nzi+1\nn;\"\u0000zi\nn;\"\n\u0015n!2\n:(4.21)\nDe\fne now the piecewise constant functions z1\n\";n(s) via\nz1\n\";n(s) :=8\n<\n:\u0010zi+1\nn;\"\u0000zi\nn;\"p\n2\u000e1=2\nn\u0011\nifs2hp\n2\u000e1=2\nn\n\u0015n(\u0015ni\u00001\n2);p\n2\u000e1=2\nn\n\u0015n(\u0015n(i+ 1)\u00001\n2)\u0011\n;i= 0;:::; [1=\u0015n]\u00002\n0 otherwise.\nNotice that by constuction z1\n\";n(s) = 0 whenjsj\u0015R\"+\"+ 2\u0015n.\nSince each of the intervalshp\n2\u000e1=2\nn\n\u0015n(\u0015ni\u00001\n2);p\n2\u000e1=2\nn\n\u0015n(\u0015n(i+ 1)\u00001\n2)\u0011\nhas lengthp\n2\u000e1=2\nn!0, and since\nz0\n\"is uniformly continuous in R, we get thatjz1\n\";n(s)\u0000z0\n\"(s)j! 0 uniformly with respect to s2R.\nBeingz1\n\";n(s) = 0 outside a compact set independent of n, this implies\nlim\nn!+1Z+1\n\u00001(z1\n\";n(s))2ds=Z+1\n\u00001(z0\n\"(s))2ds:\nOn the other hand, by a direct computation, we get that\n\u0015np\n2\u000e1=2\nn(1 +\u0011n)[1=\u0015n]\u00002X\ni=0\u0015n \nzi+1\nn;\"\u0000zi\nn;\"\n\u0015n!2\n= (1 +\u0011n)[1=\u0015n]\u00002X\ni=0p\n2\u000e1=2\nn \nzi+1\nn;\"\u0000zi\nn;\"p\n2\u000e1=2\nn!2\n\u0014(1 +\u0011n)Zp\n2\u000e1=2\nn\n2\u0015n\n\u0000p\n2\u000e1=2\nn\n2\u0015n(z1\n\";n(s))2ds\u0014(1 +\u0011n)Z+1\n\u00001(z1\n\";n(s))2ds\nso that\nlim sup\nn!+1\u0015np\n2\u000e1=2\nn(1 +\u0011n)[1=\u0015n]\u00002X\ni=0\u0015n \nzi+1\nn;\"\u0000zi\nn;\"\n\u0015n!2\n\u0014Z+1\n\u00001(z0\n\"(s))2ds: (4.22)\nTo estimate the other term we proceed in a similar way. We de\fne the piecewise constant functions\n^z\";n(s) via\n^z\";n(s) :=(\nzi\nn;\" ifs2hp\n2\u000e1=2\nn\n\u0015n(\u0015ni\u00001\n2);p\n2\u000e1=2\nn\n\u0015n(\u0015n(i+ 1)\u00001\n2)\u0011\n;i= 0;:::; [1=\u0015n]\u00002\n0 otherwise.\nNotice that by constuction ^ z\";n(s)2= 1 whenjsj\u0015R\"+\"+ 2\u0015n.\nSince each of the intervalshp\n2\u000e1=2\nn\n\u0015n(\u0015ni\u00001\n2);p\n2\u000e1=2\nn\n\u0015n(\u0015n(i+ 1)\u00001\n2)\u0011\nhas lengthp\n2\u000e1=2\nn!0, and since\n14z\"is uniformly continuous in R, we get thatj^z\";n(s)\u0000z\"(s)j! 0 uniformly with respect to s2R.\nBeing ^z\";n(s)2= 1 outside a compact set independent of n, this implies\nlim\nn!+1Z+1\n\u00001(^z\";n(s)2\u00001)2ds=Z+1\n\u00001(z\"(s)2\u00001)2ds:\nOn the other hand, since by construction\n^z\";n p\n2\u000e1=2\nn\n\u0015n(t\u00001\n2)!\n=zi\nn;\"()t2[\u0015ni;\u0015n(i+ 1))\nvia the change of variables t\u00001\n2=\u0015np\n2\u000e1=2\nnswe have\nZ+1\n\u00001(^z\";n(s)2\u00001)2ds\u0015Zp\n2\u000e1=2\nn\n2\u0015n\n\u0000p\n2\u000e1=2\nn\n2\u0015n(^z\";n(s)2\u00001)2ds\n=p\n2\u000e1=2\nn\n\u0015nZ1\n00\n@^z\";n p\n2\u000e1=2\nn\n\u0015n(t\u00001\n2)!2\n\u000011\nA2\ndt\u0015p\n2\u000e1=2\nn\n\u0015n[1=\u0015n]\u00002X\ni=0\u0015n\u0000\n(zi\nn;\")2\u00001\u00012\nso that\nlim sup\nn!+1p\n2\u000e1=2\nn\n\u0015n[1=\u0015n]\u00002X\ni=0\u0015n\u0000\n(zi\nn;\")2\u00001\u00012\u0014Z+1\n\u00001(z\"(s)2\u00001)2ds: (4.23)\nCombining (4.21), (4.22), and (4.23) we obtain\nlim sup\nn!+1Hhf\nn(un;\")p\n2\u0015n\u000e3=2\nn\u0014Z+1\n\u00001((z0\n\"(t))2+ (z\"(t)2\u00001)2)dt\u0014m+C\":\nThis gives the required upper bound by arbitrariness of \".\nCase (ii). We argue by density. Let us consider z2W1;2\njperj(I)\\C1(I). We de\fne\nzi\nn=8\n<\n:z(\u0015ni) ifi= 1;2;:::;h\n1\n\u0015ni\n\u00001;\nz(1) ifi=h\n1\n\u0015ni\n:(4.24)\nNote that, by taking the piecewise a\u000ene interpolation of such a znwe have that\nlim\nn0\n@p\n2\u000e1=2\nn\n\u0015n[1=\u0015n]\u00002X\ni=0\u0015n\u0000\n(zi\nn)2\u00001\u00012+\u0015np\n2\u000e1=2\nn(1\u0000\rn)[1=\u0015n]\u00002X\ni=0\u0015n\u0012zi+1\nn\u0000zi\nn\n\u0015n\u001321\nA=Hhf(z):\nTo conclude, we construct unas in the proof of ( i) and observe that (4.21) still holds true.\nReferences\n[1] Roberto Alicandro, Andrea Braides, and Marco Cicalese. Phase and anti-phase boundaries in binary\ndiscrete systems: a variational viewpoint. Netw. Heterog. Media , 1(1):85{107, 2006.\n[2] Roberto Alicandro and Marco Cicalese. Variational analysis of the asymptotics of the XY model.\nArch. Rat. Mech. Anal. , 192(3):501{36, 2009.\n[3] Roberto Alicandro, Marco Cicalese, and Antoine Gloria. Integral representation of the bulk limit of\na general class of energies for bounded and unbounded spin systems. Nonlinearity , 21:1881{1910,\n2008.\n15[4] Roberto Alicandro, Marco Cicalese, and Marcello Ponsiglione. Variational equivalence between\nGinzburg-Landau, XY spin systems and screw dislocation energies. Indiana Univ. Math. J. ,\n60(1):171{208, 2011.\n[5] Roberto Alicandro, Lucia De Luca, Adriana Garroni, and Marcello Ponsiglione. Metastability and\ndynamics of discrete topological singularities in two dimensions: a \u0000-convergence approach. preprint ,\n2013.\n[6] Roberto Alicandro and Marcello Ponsiglione. Ginzburg-Landau functionals and renormalized energy:\nA revised \u0000-convergence approach. preprint , 2011.\n[7] Andrea Braides. \u0000 -convergence for beginners , volume 22 of Oxford Lecture Series in Mathematics\nand its Applications . Oxford University Press, Oxford, 2002.\n[8] Andrea Braides and Marco Cicalese. Spatially-modulated phases in discrete systems. forthcoming .\n[9] Andrea Braides, Marco Cicalese, and Francesco Solombrino. Q-tensor continuum energies as limits\nof head-to-tail-symmetric spin systems. arXiv preprint arXiv:1310.4084 , 2013.\n[10] Andrea Braides and Lev Truskinovsky. Asymptotic expansions by \u0000-convergence. Contin. Mech.\nThermodyn. , 20(1):21{62, 2008.\n[11] Andrea Braides and Nung Kwan Yip. A quantitative description of mesh dependence for the dis-\ncretization of singularly perturbed nonconvex problems. SIAM Journal on Numerical Analysis ,\n50(4):1883{1898, 2012.\n[12] HT Diep. Frustrated spin systems . World Scienti\fc, 2005.\n[13] D. V. Dmitriev and V. Ya Krivnov. Universal low-temperature properties of frustrated classical\nspin chain near the ferromagnet-helimagnet transition point. The European Physical Journal B ,\n82(2):123{131, 2011.\n[14] S-L Drechsler, O Volkova, AN Vasiliev, N Tristan, J Richter, M Schmitt, H Rosner, J M\u0013 alek,\nR Klingeler, AA Zvyagin, et al. Frustrated cuprate route from antiferromagnetic to ferromagnetic\nspin-1/2 heisenberg chains: Li f2gzrcuof4gas a missing link near the quantum critical point.\nPhysical review letters , 98(7):077202, 2007.\n[15] I Harada and HJ Mikeska. One dimensional classical planar rotor model with competing interactions.\nZ. Phys. B , 72:391, 1988.\n[16] Isao Harada. One-dimensional classical planar model with competing interactions. J. Phys. Soc.\nJapan , 53:1643{51, 1984.\n[17] Luciano Modica. The gradient theory of phase transitions and the minimal interface criterion.\nArchive for Rational Mechanics and Analysis , 98(2):123{142, 1987.\n[18] Luciano Modica and Stefano Mortola. Un esempio di \u0000\u0000-convergenza. Boll. Un. Mat. Ital. B (5) ,\n14(1):285{299, 1977.\n[19] Ralph Tyrrell Rockafellar. Convex Analysis . Princeton University Press, 1970.\n16"    },    {        "title": "1310.3354v2.Spin_diffusion_and_magnetoresistance_in_ferromagnet_topological_insulator_junctions.pdf",        "content": "arXiv:1310.3354v2  [cond-mat.mes-hall]  14 Dec 2013Spin diffusion and magnetoresistance in ferromagnet/topol ogical-insulator junctions\nTakehito Yokoyama1and Yaroslav Tserkovnyak2\n1Department of Physics, Tokyo Institute of Technology, Toky o 152-8551, Japan\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n(Dated: July 2, 2021)\nWe study spin and charge diffusion in metallic-ferromagnet/ topological-insulator junctions. The\ndiffusive theory is constructed for the coupled transport of the spin-dependent electron densities in\nthe ferromagnet and the charge density in the topological in sulator. The diffusion equations for the\ncoupled transport are derived perturbatively with respect to the strength of the interlayer tunneling.\nWe analytically calculate spin accumulation in the ferroma gnet and junction magnetoresistance\nassociated with a current bias along the interface. It is fou nd that due to the helical spin texture\nof the surface Dirac fermion, the spin accumulation and the j unction magnetoresistance depend on\nthe angle between the magnetization and the current induced spin polarization on the surface of the\ntopological insulator.\nI. INTRODUCTION\nRecent discovery of three-dimensional (3D) topologi-\ncal insulators (TI’s) offers a new state of matter that is\ntopologically distinct from the conventional band insu-\nlators.1–6Surface states of the strong TI are topologi-\ncally protected and immune to small perturbations that\nrespect time-reversal symmetry. At low energies, these\nstates are described by Dirac fermions and hence exhibit\nstrong spin-orbit interactions and intriguing Berry-phase\nphenomena.\nSince the electron’s spin and momentum on the sur-\nface of a TI are essentially interlocked, TI’s offer a\npromising arena for developing spintronics. In par-\nticular, there have been an intense interest in cou-\npling TI’s with ferromagnets (F’s). In previous works,\nthe focus has been on the surface properties of the\nTI’s that are standalone or exchange coupled to the\ninsulating F’s,7–9such as magnetoelectric effect,4,10,11\nspin torque and magnetization dynamics,12–18mag-\nnetic domain walls,19–21charge pumping,22,23magnetic\nheat transport,24spin rotation,25,26spin and charge\ndynamics,27–29magnetotransport,30–37and Majorana\nfermions (in superconducting heterostructures).38–45\nThesephenomenaareaffectedoreffectedbytheexchange\nfield of the F: The in-plane exchange field acts like a vec-\ntor potential while the out-of-plane exchange field makes\nthe Dirac fermions massive. On the other hand, the cou-\npled spin and charge transport in metallic F’s interfaces\nwith TI’s remains largely unexplored.46We study this\ntransporthere, in the weak-tunneling regimethrough the\nF/TI junction.\nMotivated by theoretical predictions, there has\nalso been intense interest on the magnetic proxim-\nity effect47–53and observation of Dirac fermions cou-\npled with exchange field. Recently, signatures of Dirac\nfermions coupled to magnetization have been observed\nby doping Fe, Mn, Cr, Gd, or ferrocene into TI’s,54–66\ndepositing Fe or Co on the surface of TI’s,67–70making\njunctionofaTIwithFe, GdNorEuS,71–74orintergrowth\nwith Fe 7Se8.75\nIn this paper, we construct the diffusive theory for thecoupled transport of the spin-dependent electron densi-\nties in the F and the charge density in the TI. We derive\nthecoupledspinandchargediffusionequationsinlayered\nmetallic F/TI junctions, under current bias along the TI\nsurface, which are perturbed by the tunneling across the\ninterface. We analytically solve the diffusion equations\nto calculate the current-inducedspin accumulationin the\nferromagnet and junction magnetoresistance due to the\nF/TI interlayer tunneling. It is found that due to the\nhelical spin texture of the surface Dirac fermion, the spin\naccumulationandthejunctionmagnetoresistancedepend\non the angle between the magnetization and the current\ninduced spin polarization on the surface of the TI.\nII. MODEL\nWe consider an F/TI junction sketched in Fig. 1(a).\nThe Hamiltonian ofthe systemreads H=HF+HT+H′,\nwhere\nHF=/summationdisplay\nkˆa†\nk/parenleftbigg¯h2k2\n2m+h·ˆσ−εF/parenrightbigg\nˆak(1)\nis the Hamiltonian of the ferromagnet. Here, kdenotes\n3D wave vector, h=h(sinθcosϕ,sinθsinϕ,cosθ) is the\nexchange field, ˆσ= (ˆσx,ˆσy,ˆσz) is a vector of Pauli ma-\ntrices, and ˆ a†\nk= (a†\nk↑,a†\nk↓) is the spin-1/2creation opera-\ntor. We suppose that the magnetization is uniform. The\nHamiltonian on the unperturbed surface of the topolog-\nical insulator reads\nHT=/summationdisplay\nkˆb†\nk[¯hv(kyˆσx−kxˆσy)−εT]ˆbk,(2)\nwhere the 2D wave vector kis parallel to the interface.\nThe dispersions of the TI and F are shown in Fig. 1(b),\nwhere the horizontal line denotes their common equilib-\nrium Fermi level.\nThe F and TI are coupled by the tunneling Hamilto-\nnian\nH′=/radicalbigg\n¯hγ\nπ/summationdisplay\nk,k′ˆa†\nkˆbk′+H.c., (3)2\nµ\nεTεF\nh\nmv\nFIG. 1: (Color online) (a) Schematic picture of the model.\n(b) Dispersions of the TI and F. The horizontal line denotes\nthe Fermi level.\nwhere/radicalBig\n¯hγ\nπdenotes the tunneling amplitude. It is as-\nsumed that the interface between F and TI is rough, such\nthat, upon transmission through the interface, the mo-\nmentum of the electron is scrambled while the spin and\nthe energy are conserved. This spin-conserving form of\ntunneling would generally need to be extended to ac-\ncount for details of atomistic matrix elements for the\nstates composing the Dirac electron band, in order to\nconstruct a more quantitative theory. We believe, how-\never, thatoursimple treatmentgivesrisetotheappropri-\nate symmetry-based phenomenological structure of the\nfinal diffusion equations. We will be assuming the am-\nbient temperature Tto be low, kBT≪εF,εT>0, and\nwill focus on small deviations from an equilibrium state.\nSpin-resolved electron-number density (measured rela-\ntive to the equilibrium state) in the F, nσwithσ=↑and\n↓for minority and majority spins, respectively, obeys thediffusion equation:76–79\n∂nσ\n∂t=Dσ∇2nσ−/parenleftbiggnσ\nτσ−n¯σ\nτ¯σ/parenrightbigg\n−i′\nσ,(4)\nwherei′\nσis the spin- σparticle flux from the F into the\nTI across the F/TI interface and ¯ σ=−σ. According to\nequilibrium considerations, the spin-flip rates are related\nbyρ↑/τ↑=ρ↓/τ↓≡Γ, where ρσis the Fermi-level spin- σ\ndensity of states in the F. Electron density on the surface\nof the TI, n, in general obeys the continuity equation:\n∂n\n∂t=−∇·j+(i′\n↑+i′\n↓). (5)\nDue to the strong spin-orbit coupling in the TI, we do\nnot separate diffusion into two spin components. In lieu\nof the diffusive relations for n↑andn↓in the F, we have\nthose for nandj= (jx,jy) in the TI. The nonequilib-\nrium spin density, as a hydrodynamic quantity, is thus\nreplaced in the TI by the current density (which is phys-\nically motivated by the helicity of electron transport).\nTo the lowest order in tunneling, the spin-dependent\ninterlayer currents can be calculated using the Fermi’s\ngolden rule:\ni′\nσ=2π\n¯h/summationdisplay\nk,k′,σ|∝angb∇acketleftˆbk′H′a†\nkσ∝angb∇acket∇ight0|2(fF,kσ−fT,k′)δ(εkσ−εk′).\n(6)\nHere,fTandfFaretheassociateddistributionfunctions;\nεk= ¯hvk−εTandεkσ= (¯hk)2/2m+σh−εFare the\ncorresponding energy eigenvalues; and the expectation\nvalue∝angb∇acketleft...∝angb∇acket∇ight0istakenwithrespecttotheabsolutevacuum.\nIn the state of global thermal equilibrium, the electron\ndistributions are given by the Fermi-Dirac distribution\nfunction: f0(ε) = [eε/kBT+ 1]−1. The nonequilibrium\nparticle densities in the F and TI can be accounted for\nby simply shifting their chemical potentials by µ=n/ρ\nandµσ=nσ/ρσ, respectively, where ρis the Fermi-level\ndensity of states in the TI.\nInthe dilute limitofscatteringimpurities, theparticle-\ncurrent density jin thexyplane on the surface of the TI\nresults in a shift δkof the electron distribution function\nin the momentum space:\nj=vk\n4πδk, fT=f0−∂f0\n∂k·δk. (7)\nHere,kis the Fermi wave number in the TI.3\nIII. RESULTS\nA. Diffusion equations\nThe tunneling matrix elements entering in Eq. (6) are determined by t he spinor eigenfunctions\n|k∝angb∇acket∇ight=1√\n2/parenleftbiggi\neiφ/parenrightbigg\n,|↑∝angb∇acket∇ight=/parenleftbigg\ncosθ\n2\neiϕsinθ\n2/parenrightbigg\nand|↓∝angb∇acket∇ight=/parenleftbigg\n−sinθ\n2\neiϕcosθ\n2/parenrightbigg\n, (8)\nin the TI and F, respectively, where φ= tan−1(ky/kx) is the polar angle of kin the TI. We find (absorbing the\nferromagnet’s volume into γ)\ni′\nσ=γ/integraldisplayk2dk\n2π2k′dk′dφ\n(2π)2[1+σsinθsin(φ−ϕ)](fF−fT)δ(εkσ−εk′) =γ/bracketleftbiggσρσsinθ\nveϕ·j+ρnσ−ρσn/bracketrightbigg\n,(9)\nwhereeϕ= (sinϕ,−cosϕ).\nSubstituting spin-dependent tunneling current density (9) into Eq . (4) complements the diffusion equation in the\nF with the tunneling leakage current, which is itself expressed in term s of the consistent hydrodynamics quantities.\nIn order to complete our diffusion theory, we, however, still need t o relate the current on the surface of the TI, j, to\nthe out-of-equilibrium densities, nandnσ, which we proceed to do in the following.\nDriving the TI electrons by a spatially homogeneous in-plane electric fi eldE, in the presence of the interlayer\ntunneling, the current jon the surface of the TI obeys the following scattering relation:\n∂j\n∂t=−j−gE/e\nτ+∂j\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt, (10)\nwhereτis the transport mean free time, e <0 is the electron charge, and gis the electrical conductivity, which is\ngiven, according to the Einstein relation, by g=e2Dρ, in terms of the diffusion coefficient D=v2τ/2. The last term\non the right hand side represents the tunneling contribution to the time derivative of the current. At frequencies\nω≪1/τ, this reduces to\nj=eDρE+τ∂j\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt≡jd+jt, (11)\nwhere the second term stands for the tunneling contribution to th e planar TI current. In general, −eEshould be\nreplaced with the electrochemical potential gradient, −eE→∇µ, which also captures the diffusive contribution to\nthe current: jd=−D∇n. The tunneling contribution to the current is also evaluated using th e Fermi’s golden rule,\ngiving\n∂j\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nt=2π\n¯h/summationdisplay\nk,k′,σ∝angb∇acketleftˆbk′vˆb†\nk′∝angb∇acket∇ight0|∝angb∇acketleftbk′H′a†\nkσ∝angb∇acket∇ight0|2(fF,kσ−fT,k′)δ(εkσ−εk′) =−γ/bracketleftbigg\nρ+j+vsinθ\n2(ρn−−ρ−n)eϕ/bracketrightbigg\n.(12)\nwherevi=∂HT\n¯h∂ki(i=x,y),ρ±≡ρ↑±ρ↓andn±≡n↑±n↓. (We will later also benefit from the definition\nD±≡D↑±D↓.) The first term gives a correction of order O(γτ) to the diffusion coefficient D, which can be absorbed\nby a redefinition of D. Note that in the absence of spin-orbit interactions in the F, ∂jσ/∂t|t= 0 there, which allowed\nus to use simple diffusive relations in Eq. (4).\nSubstituting the current Eq. (11) into Eq. (5), we obtain the diffus ion equation in the TI (to the lowest order in\ntunneling):\n∂n\n∂t=−∇·jd−∇·jt+γ/bracketleftbiggρ−sinθ\nveϕ·jd+ρn+−ρ+n/bracketrightbigg\n, (13)\nwhich becomes, upon substituting the above explicit expressions fo rjdandjt:\n∂n\n∂t=D∇2n+γ/bracketleftbiggρDsinθ\nveϕ·∇n−−2ρ−Dsinθ\nveϕ·∇n+ρn+−ρ+n/bracketrightbigg\n. (14)\nIn the most general case, our final diffusion equations for the TI a nd F are written in terms of the full electrochemical\npotentials (including electrostatic contributions), µandµσ:\n∂n\n∂t=Dρ∇2µ+γρ/bracketleftbiggDsinθ\nveϕ·(ρ↑∇µ↑−ρ↓∇µ↓)−2ρ−Dsinθ\nveϕ·∇µ+ρ↑µ↑+ρ↓µ↓−ρ+µ/bracketrightbigg\n(15)4\nand\n∂nσ\n∂t=Dσρσ∇2µσ−Γ(µσ−µ¯σ)+γρρσ/bracketleftbiggσDsinθ\nveϕ·∇µ+µ−µσ/bracketrightbigg\n. (16)\nThese equations constitute the central result of this paper.\nB. Static 1D case\nAs an example of F/TI junctions, consider the static 1D case. The d iffusion equations in the F then become\n0 =Dσρσ∂2\nxµσ−σΓ(µ↑−µ↓)+γρρσ/bracketleftbiggσDsinθsinϕ\nv∂xµ+µ−µσ/bracketrightbigg\n, (17)\nand that on the surface of the TI is\n0 =D∂2\nxµ+γ/bracketleftbiggDsinθsinϕ\nv(ρ↑∂xµ↑−ρ↓∂xµ↓)−2ρ−Dsinθsinϕ\nv∂xµ+ρ↑µ↑+ρ↓µ↓−ρ+µ/bracketrightbigg\n.(18)\nThe diffusion equations in the F can be combined to obtain the equation for the spin accumulation µs≡µ↑−µ↓:\n0 =∂2\nxµs−µs\nλ2+γρ/bracketleftbigg/parenleftbigg1\nD↑+1\nD↓/parenrightbiggDsinθsinϕ\nv∂xµ+/parenleftbigg1\nD↑−1\nD↓/parenrightbigg\nµ−µ↑\nD↑+µ↓\nD↓/bracketrightbigg\n, (19)\nwhereλis the spin-diffusion length given by\n1\nλ2= Γ/parenleftbigg1\nρ↑D↑+1\nρ↓D↓/parenrightbigg\n. (20)\nFor the internal consistency of our quasi-1D treatment of the diff usion in the F, we need the F layer to be thinner\nthanλ. The charge transport in the F is governed by the following relation:\n0 = (D↑ρ↑∂2\nxµ↑+D↓ρ↓∂2\nxµ↓)+γρ/bracketleftbiggDρ−sinθsinϕ\nv∂xµ+ρ+µ−ρ↑µ↑−ρ↓µ↓/bracketrightbigg\n. (21)\nWe will in the following solve the static 1D equations,\nEqs.(18), (19), and(21), iteratively,tofirstorderin γ. In\nthe absenceoftunneling, i.e., γ= 0, the generalsolutions\nare\nµ0=Ax+B,\nµ0\ns=aex/λ+be−x/λ,\nµ0\nc=µ0+cx+d−Pµ0\ns\n2, (22)\nwhereP≡(D↑ρ↑−D↓ρ↓)/(D↑ρ↑+D↓ρ↓)istheFconduc-tivity polarization (according to the Einstein’s relation)\nandµc≡(µ↑+µ↓)/2isthespin-averagedchemicalpoten-\ntial. Here, a,b,c,d,A , andBare yet unknown constants,\nwhich should be determined by the boundary conditions.\nThe solutions to the first order in γare then obtained\nby inserting these zeroth-order solutions into the terms\nproportional to γin the diffusion equations, which pro-\nducesO(γ) source terms. After some algebra, we thus\nfind solutions to the first order in γ:5\nµs=aex/λ+be−x/λ+γρλ2\nD↑D↓\n×/bracketleftbigg\nfex/λ+ge−x/λ+DD+sinθsinϕ\nvA+D−(cx+d)+D++PD−\n4λx/parenleftBig\naex/λ−be−x/λ/parenrightBig/bracketrightbigg\n,\nµc=µ0+cx+d−γρ\n2(D↑ρ↑+D↓ρ↓)\n×/bracketleftbigg\nux+v+Dρ−sinθsinϕ\nvAx2−ρ+/parenleftBigc\n3x3+dx2/parenrightBig\n+(Pρ+−ρ−)λ2/parenleftBig\naex/λ+be−x/λ/parenrightBig/bracketrightbigg\n−Pµs\n2,\nµ=Ax+B+γ\n2D\n×/bracketleftbigg\nCx+E−Dsinθsinϕ\nv/braceleftBig\n(ρ+−Pρ−)λ/parenleftBig\naex/λ−be−x/λ/parenrightBig\n+ρ−(c−A)x2/bracerightBig\n−ρ+/parenleftBigc\n3x3+dx2/parenrightBig\n+(Pρ+−ρ−)λ2/parenleftBig\naex/λ+be−x/λ/parenrightBig/bracketrightbigg\n. (23)\nHere,f,g,u,v,C , andEare to be determined by the boundary conditions (at order γ).\nIn order to impose specific boundary conditions, sup-\npose now that a ferromagnetic layer is attached to the TI\nalong−W/2< x < W/ 2. Since the ferromagnet is ter-\nminated at x=−W/2 andW/2, we require vanishing of\nthe diffusive spin-dependent planar fluxes normal to the\nboundaries, such that ∂xµc=∂xµs= 0 atx=±W/2.\nApplying a uniform 2Dparticleflux jto the TI surfacein\nthexdirection, we have j=−Dρ∂xµ, in the absence of\ntunneling, which determines A=−j/Dρ. We can, fur-\nthermore, set B= 0 without loss of generality(absorbing\nit by a gauge potential shift, if necessary). Solving the\nfirst two of Eqs. (23) then gives the spin accumulation in\nthe F of the form:\nµs=−4γλ2j/braceleftbiggsinθsinϕ\nv(D+−PD−)−D−\nD(D2\n+−D2\n−)\n×/bracketleftbigg\nx−λsinh(x/λ)\ncosh(W/2λ)/bracketrightbigg/bracerightbigg\n.(24)\nThe spin accumulation, which is a nonequilibrium quan-\ntity, is proportional to the current j, as expected. The\nfirst term in the curly brackets is proportional to the\nyprojection of the F exchange field. This can be inter-\npretedtoresultdirectlyfromthecurrent-inducedspinac-\ncumulationintheTI,whichpointsalongthe −ydirection\n(when the current is flowing along the xaxis). Since the\nTIcurrentisspinpolarizedhelicallyevenwithout thefer-\nromagnet, this contribution is position-independent and\npersistsevenwhen D−→0. Theothercontributionto µs\nis independent of the direction of the exchange field, and\nrequires D−∝negationslash= 0, which means that it is due to nonequi-\nlibrium transport that is spin polarized by the ferromag-\nnet. This contribution is position dependent, vanishing\nwhenW≪λand growing linearly with xwhenW≫λ.\nIt should be remarked that dis given by d=\n−j(ρ−/vρρ+)sinθsinϕ, to the zeroth order in tunneling,\nreflecting the fact that the average chemical potential inthe F can be shifted by spin injection. The tangential\nparticle flux in the F is obtained as\n/summationdisplay\nσjσ=−∂x(D↑ρ↑µ↑+D↓ρ↓µ↓) =γρ+\n2Dj/parenleftbiggW2\n4−x2/parenrightbigg\n.\n(25)\nThe current in the F flows in the same direction as that\non the surface of the TI and reaches its maximum value\nat the center of the junction.\nThechemicalpotentialonthesurfaceoftheTI(setting\nE= 0) can be expressed as\nµ\nµ0= 1−γ\n2/bracketleftbiggρC\nj−ρ−sinθsinϕ\nvx−ρ+\n3Dx2/bracketrightbigg\n,(26)\nwhereµ0=−jx/Dρ. The continuity of the TI current\n(11) at junction boundaries, x=±W/2, at first order in\nγ, gives\nρC\nj=ρ+W2\n4D−4Dρ+\nv2−2Dρ2\n−sin2θsin2ϕ\nv2ρ+.(27)\nOnly when ρ−is nonzero, the chemical potential on the\nsurfaceoftheTIdependsonthedirectionoftheexchange\nfield. Finally, the additional resistance δR≡R−R0of\nthe TI/F junction (normalized to the bare TI resistance\nR0) is found to be\nδR\nR0=δµ(−W/2)−δµ(W/2)\njW/Dρ\n=γ/bracketleftBigg\n−ρ+W2\n12D+2Dρ+\nv2−Dρ2\n−sin2θsin2ϕ\nv2ρ+/bracketrightBigg\n,(28)\nwhereδµ≡µ−µ0is the chemical-potential shift due to\ntunneling. The magnetoresistance ∝ −h2\nyreaches max-\nimum for h⊥yand minimum for h∝ba∇dbly. Note that6\nthis resistance respects the Onsager reciprocity for two-\nterminal systems. Namely, δR∝const−h2\nyis invariant\nunder time reversal: h→ −h. The normalized magne-\ntoresistance is given by\nR(h⊥y)−R(h∝ba∇dbly)\nR0=γρ2\n−τ\n2ρ+. (29)\nForρ−/ρ+∼1, ¯hγρ−∼10−3eV, and τ∼10−13s, we\nestimate γρ2\n−τ/(2ρ+)∼0.1.\nIV. CONCLUSION\nIn summary, we studied spin and charge diffusion in\nferromagnet/topological-insulator junctions under cur-\nrent bias on the surface of the topological insulator. The\nferromagnet is assumed to be metallic. The diffusive the-\nory is constructed for the coupled transport of the spin-\ndependent electron densities in the ferromagnet and the\ncharge density in the topological insulator. The diffusion\nequations which include the interlayer coupling and the\ncurrent bias are derived based on the perturbative ap-proach. By solving the diffusion equations analytically,\nwe obtained spin accumulation in the ferromagnet in-\nduced by the current and magnetoresistance due to the\ninteraction between the magnetization and the current\nflowing on the surface of the topological insulator. 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B 56, 3296 (1997)."    },    {        "title": "0712.3082v1.Dynamic_Phase_Transitions_for_Ferromagnetic_Systems.pdf",        "content": "DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC\nSYSTEMS\nTIAN MA AND SHOUHONG WANG\nAbstract. This article presents a phenomenological dynamic phase transition\ntheory for ferromagnetism, leading to a precise description of the dynamic tran-\nsitions, and to a physical predication on the spontaneous magnetization. The\nanalysis also suggests asymmetry of \ructuations in both the ferromagnetism\nand the PVT systems.\n1.Introduction\nClassically, phase transitions are classi\fed by the Ehrenfest classi\fcation scheme,\nbased on the lowest derivative of the free energy that is discontinuous at the transi-\ntion. For ferromagnetic systems, it has been observed that the magnetization, which\nis the \frst derivative of the free energy with the applied magnetic \feld strength,\nincreases continuously from zero as the temperature is lowered below the Curie tem-\nperature, and the magnetic susceptibility, the second derivative of the free energy\nwith the \feld, changes discontinuously. Hence, the ferromagnetic phase transition\nin materials such as iron is regarded as a second order phase transition. However,\na theoretical understanding of the transition is still lacking. The main objective\nof this article is to provide theoretical approach to dynamic phase transitions for\nferromagnetic systems.\nFor the classical GL free energy, although both the steady state and time-\ndependent models provide some results in agreement with experiments, there are\nobvious discrepancies on both susceptibility and spontaneous magnetization. Hence\na revised GL free energy is proposed and analyzed, leading to a precise description\nof the dynamic transitions, and to a physical predication on the spontaneous mag-\nnetization.\nThe analysis is based on the recently developed dynamic transition theory by\nthe authors, together with a new dynamic classi\fcation scheme, which classi\fes\nphase transitions into three categories: Type-I, Type-II and Type-III, correspond-\ning mathematically to continuous, jump, and mixed transitions, respectively; see\nSection 2 and the Appendix as well as two recent books by the authors [1, 3] for\ndetails.\nWe remark also that the analysis leads naturally to a physical conjecture on\nasymmetry of \ructuations, which appears in both the ferromagnetic system studied\nin this article and in PVT systems studied in [2].\nKey words and phrases. Ferromagnetism, Curie point, time-dependent Ginzburg-Landau\nmodel, dynamic transition theory, dynamic classi\fcation scheme of phase transitions, asymmetry\nof \ructuations.\nThe work was supported in part by the O\u000ece of Naval Research and by the National Science\nFoundation.\nhttp://www.indiana.edu/ \u0018fluid .\n1arXiv:0712.3082v1  [cond-mat.mtrl-sci]  19 Dec 20072 MA AND WANG\nThis article is organized as follows. In Section 2, we review the dynamic classi\f-\ncation scheme and the new time-dependent Ginzburg-Landau model for equilibrium\nphase transitions. Section 3 deals with the dynamic transition based on the classi-\ncal Ginzburg-Landau energy, and Section 4 addresses the dynamic transition theory\nusing a revised Ginzburg-Landau energy. Physical conclusions are given in Section\n5, and dynamic transition theory is recapitulated in the Appendix.\n2.General Principles of Phase Transition Dynamics\nIn this section, we recapitulate the new dynamic phase transition classi\fcation\nscheme to classify phase transitions into three categories: Type-I, Type-II and\nType-III, corresponding mathematically to continuous, jump, and mixed transi-\ntions, respectively.\nThen we recall a new time-dependent Ginzburg-Landau theory for modeling\nequilibrium phase transitions in statistical physics, derived based on the le Ch^ atelier\nprinciple and some mathematical insights on pseudo-gradient systems.\nBoth the classi\fcation scheme and the Ginzburg-Landau theory was developed\nrecently by the authors, and we refer interested readers to [1, 3, 2] for details.\n2.1.Dynamic classi\fcation scheme. In sciences, nonlinear dissipative systems\nare generally governed by di\u000berential equations, which can be expressed in the\nfollowing abstract form\n(2.1)du\ndt=L\u0015u+G(u;\u0015);\nu(0) =';\nwhereu: [0;1)!Xis the unknown function, \u00152RN(N\u00151) is the control\nparameter, XandX1are two Banach spaces with X1\u001aXbeing a dense and\ncompact inclusion, L\u0015=\u0000A+B\u0015andG(\u0001;\u0015) :X1!XareCr(r\u00150) mappings\ndepending continuously on \u0015,L\u0015:X1!Xis a sectorial operator, and\n(2.2)A:X1!X a linear homeomorphism ;\nB\u0015:X1!Xa linear compact operator :\nIn following, we introduce some basic and universal concepts in nonlinear sci-\nences.\nFirst, a state of the system (2.1) at \u0015is usually referred to as a compact invariant\nset \u0006\u0015. In many applications, \u0006 \u0015is a singular point or a periodic orbit. A state\n\u0006\u0015of (2.1) is stable if \u0006 \u0015is an attractor, otherwise \u0006 \u0015is called unstable.\nSecond, we say that the system (2.1) has a phase transition from a state \u0006 \u0015at\n\u0015=\u00150if \u0006\u0015is stable on \u0015<\u0015 0(or on\u0015>\u0015 0) and is unstable on \u0015>\u0015 0(or on\n\u0015 < \u0015 0). The critical parameter \u00150is called a critical point. In other words, the\nphase transition corresponds to an exchange of stable states.\nThe concept of phase transition originates from the statistical physics and ther-\nmodynamics. In physics and chemistry, \"phase\" means the homogeneous part in\na heterogeneous system. However, here the so called phase means the stable state\nin the systems of nonlinear sciences including physics, chemistry, biology, ecology,\neconomics, \ruid dynamics and geophysical \ruid dynamics, etc. Hence, here the\ncontent of phase transition has been endowed with more general signi\fcance. In\nfact, the phase transition dynamics introduced here can be applied to a wide varietyDYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 3\nof topics involving the universal critical phenomena of state changes in nature in a\nuni\fed mathematical viewpoint and manner.\nThird, if the system (2.1) possesses the gradient-type structure, then the phase\ntransitions are called equilibrium phase transition; otherwise they are called the\nnon-equilibrium phase transitions.\nFourth, classically, there are several ways to classify phase transitions. The one\nmost used is the Ehrenfest classi\fcation scheme, which groups phase transitions\nbased on the degree of non-analyticity involved. First order phase transitions are\nalso called discontinuous, and higher order phase transitions ( n > 1) are called\ncontinuous.\nHere we introduce the following notion of dynamic classi\fcation scheme:\nDe\fnition 2.1. Let\u001502RNbe a critical point of (2.1), and (2.1) undergo a\ntransition from state \u00061\n\u0015to\u00062\n\u0015. There are three types of phase transitions for (2.1)\nat\u0015=\u00150, depending on their dynamic properties: continuous, jump, and mixed as\ngiven in Theorem A.1, which are called Type-I, Type-II and Type-III respectively.\nThe main characteristics of Type-II phase transitions is that there is a gap be-\ntween \u00061\n\u0015and \u00062\n\u0015at the critical point \u00150. In thermodynamics, the metastable states\ncorrespond in general to the super-heated or super-cooled states, which have been\nfound in many physical phenomena. In particular, Type-II phase transitions are\nalways accompanied with the latent heat to occur.\nIn a Type-I phase transition, two states \u00061\n\u0015and \u00062\n\u0015meet at\u00150, i.e., the system\nundergoes a continuous transition from \u00061\n\u0015to \u00062\n\u0015.\nIn a Type-III phase transition, there are at least two di\u000berent stable states \u0006\u0015\n2\nand \u0006\u0015\n3at\u00150, and system undergoes a continuous transition to \u0006\u0015\n2or a jump\ntransition to \u0006\u0015\n3, depending on the \ructuations.\nIt is clear that a Type-II phase transition of gradient-type systems must be\ndiscontinuous or the zero order because there is a gap between \u00061\n\u0015and \u0006j\n\u0015(2\u0014j\u0014\nK). For a Type-I phase transition, the energy is continuous, and consequently, it\nis ann-th order transition in the Ehrenfest sense for some n\u00152. A Type-III phase\ntransition is inde\fnite, for the transition from \u00061\n\u0015to \u00062\n\u0015it may be continuous, i.e.,\ndF\u0000=d\u0015=dF+\n2=d\u0015, and for the transitions from \u00061\n\u0015to \u00063\n\u0015it may be discontinuous:\ndF\u0000=d\u00156=dF+\n3=d\u0015at\u0015=\u00150.\n2.2.Time-dependent Ginzburg-Landau models for equilibrium phase tran-\nsitions. In this subsection, we introduce the time-dependent Ginzburg-Landau\nmodel for equilibrium phase transitions.\nWe start with thermodynamic potentials and the Ginzburg-Landau free energy.\nAs we know, four thermodynamic potentials{ internal energy, the enthalpy, the\nHelmholtz free energy and the Gibbs free energy{are useful in the chemical ther-\nmodynamics of reactions and non-cyclic processes.\nConsider a thermal system, its order parameter uchanges in \n\u001aRn(1\u0014n\u00143).\nIn this situation, the free energy of this system is of the form\n(2.3)H(u;\u0015) =H0+Z\n\nh1\n2mX\ni=1\u0016ijruij2+g(u;ru;\u0015)i\ndx4 MA AND WANG\nwhereN\u00153 is an integer, u= (u1;\u0001\u0001\u0001;um);\u0016i=\u0016i(\u0015)>0, andg(u;ru;\u0015) is a\nCr(r\u00152) function of ( u;ru) with the Taylor expansion\n(2.4) g(u;ru;\u0015) =X\n\u000bijkuiDjuk+NX\njIj=1\u000bIuI+o(jujN)\u0000fX;\nwhereI= (i1;\u0001\u0001\u0001;im);ik\u00150 are integer,jIj=Pm\nk=1ik, the coe\u000ecients \u000bijk\nand\u000bIcontinuously depend on \u0015, which are determined by the concrete physical\nproblem,uI=ui1\n1\u0001\u0001\u0001uim\nmandfXthe generalized work.\nThus, the study of thermal equilibrium phase transition for the static situation\nis referred to the steady state bifurcation of the system of elliptic equations\n\u000e\n\u000euH(u;\u0015) = 0;\n@u\n@nj@\n= 0;(oruj@\n= 0);\nwhere\u000e=\u000eu is the variational derivative.\nA thermal system is controled by some parameter \u0015. When\u0015is for from the\ncritical point \u00150the system lies on a stable equilibrium state \u0006 1, and when \u0015\nreaches or exceeds \u00150the state \u0006 1becomes unstable, and meanwhile the system\nwill undergo a transition from \u0006 1to another stable state \u0006 2. The basic principle\nis that there often exists \ructuations in the system leading to a deviation from the\nequilibrium states, and the phase transition process is a dynamical behavior, which\nshould be described by a time-dependent equation.\nTo derive a general time-dependent model, \frst we recall that the classical le\nCh^ atelier principle amounts to saying that for a stable equilibrium state of a system\n\u0006, when the system deviates from \u0006 by a small perturbation or \ructuation, there\nwill be a resuming force to retore this system to return to the stable state \u0006. Second,\nwe know that a stable equilibrium state of a thermal system must be the minimal\nvalue point of the thermodynamic potential.\nBy the mathematical characterization of gradient systems and the le Ch^ atelier\nprinciple, for a system with thermodynamic potential H(u;\u0015), the governing equa-\ntions are essentially determined by the functional H(u;\u0015). When the order param-\neters (u1;\u0001\u0001\u0001;um) are nonconserved variables, i.e., the integers\nZ\n\nui(x;t)dx=ai(t)6= constant:\nthen the time-dependent equations are given by\n(2.5)@ui\n@t=\u0000\fi\u000e\n\u000euiH(u;\u0015) + \bi(u;ru;\u0015);\n@u\n@nj@\n= 0 (oruj@\n= 0);\nu(x;0) ='(x);\nfor any 1\u0014i\u0014m, where\u000e=\u000euiare the variational derivative, \fi>0 and \bisatisfy\n(2.6)Z\n\nX\ni\bi\u000e\n\u000euiH(u;\u0015)dx= 0:\nThe condition (2.6) is required by the Le Ch^ atelier principle. In the concrete\nproblem, the terms \b ican be determined by physical laws and (2.6).DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 5\nWhen the order parameters are the number density and the system has no ma-\nterial exchange with the external, then uj(1\u0014j\u0014m) are conserved, i.e.,\n(2.7)Z\n\nuj(x;t)dx= constant:\nThis conservation law requires a continuous equation\n(2.8)@uj\n@t=\u0000r\u0001Jj(u;\u0015);\nwhereJj(u;\u0015) is the \rux of component uj. In addition, Jjsatisfy\n(2.9) Jj=\u0000kjr(\u0016j\u0000X\ni6=j\u0016i);\nwhere\u0016lis the chemical potential of component ul,\n(2.10) \u0016j\u0000X\ni6=j\u0016i=\u000e\n\u000eujH(u;\u0015)\u0000\u001ej(u;ru;\u0015);\nand\u001ej(u;\u0015) is a function depending on the other components ui(i6=j). When\nm= 1, i.e., the system consists of two components AandB, this term \u001ej= 0.\nThus, from (2.8)-(2.10) we obtain the dynamical equations as follows\n(2.11)@uj\n@t=\fj\u0001\u0014\u000e\n\u000eujH(u;\u0015)\u0000\u001ej(u;ru;\u0015)\u0015\n;\n@u\n@nj@\n= 0;@\u0001u\n@nj@\n= 0;\nu(x;0) ='(x);\nfor 1\u0014j\u0014m, where\fj>0 are constants, \u001ejsatisfy\n(2.12)Z\n\nX\nj\u0001\u001ej\u0001\u000e\n\u000eujH(u;\u0015)dx= 0:\nIf the order parameters ( u1;\u0001\u0001\u0001;uk) are coupled to the conserved variables\n(uk+1;\u0001\u0001\u0001;um), then the dynamical equations are\n(2.13)@ui\n@t=\u0000\fi\u000e\n\u000euiH(u;\u0015) + \bi(u;ru;\u0015);\n@uj\n@t=\fj\u0001\u0014\u000e\n\u000eujH(u;\u0015)\u0000\u001ej(u;ru;\u0015)\u0015\n;\n@ui\n@nj@\n= 0 (oruij@\n= 0);\n@uj\n@nj@\n= 0;@\u0001uj\n@nj@\n= 0;\nu(x;0) ='(x):\nfor 1\u0014i\u0014kandk+ 1\u0014j\u0014m.\nThe model (2.13) gives a general form of the governing equations to thermody-\nnamic phase transitions. Hence, the dynamics of equilibrium phase transition in\nstatistic physics is based on the new Ginzburg-Landau formulation (2.13).\nPhysically, the initial value condition u(0) ='in (2.13) stands for the \ructuation\nof system or perturbation from the external. Hence, 'is generally small. However,\nwe can not exclude the possibility of a bigger noise '.6 MA AND WANG\nFrom conditions (2.6) and (2.12) it follows that a steady state solution u0of\n(2.13) satis\fes\n(2.14)\bi(u0;ru0;\u0015) = 081\u0014i\u0014k;\n\u0001\u001ej(u0;ru0;\u0015) = 08k+ 1\u0014j\u0014m:\nHence a stable equilibrium state must reach the minimal value of thermodynamic\npotential. In fact, u0ful\flls\n(2.15)\fi\u000e\n\u000euiH(u0;\u0015)\u0000\bi(u0;ru0;\u0015) = 0;\n\fj\u0001\u000e\n\u000eujH(u0;\u0015)\u0000\u0001\u001ej(u0;ru0;\u0015) = 0;\n@ui\n@nj@\n= 0 (oruij@\n= 0);\n@uj\n@nj@\n= 0;@\u0001uj\n@nj@\n= 0;\nfor 1\u0014i\u0014kandk+ 1\u0014j\u0014m. Multiplying \b i(u0;ru0;\u0015) and\u001ej(u0;ru0;\u0015)\non the \frst and the second equations of (2.15) respectively, and integrating them,\nthen we infer from (2.6) and (2.12) that\nZ\n\nX\ni\b2\ni(u0;ru0;\u0015)dx= 0;\nZ\n\nX\njjr\u001ej(u0;ru0;\u0015)j2dx= 0;\nwhich imply that (2.14) holds true.\n3.Classical Theory of Ferromagnetism\nA ferromagnetic material consists of lattices containing particles with a magnetic\nmoment. When no external \feld is present and the temperature is above some\ncritical value, called the Curie temperature, the magnetic moments are oriented at\nrandom and there is no net magnetization. However, as the temperature is lowered,\nmagnetic interaction energy between lattice sites becomes more important than\nthe random thermal energy. Below the Curie temperature, the magnetic moments\nbecome ordered in the space and a spontanous magnetization appears. The phase\ntransition from a paramagnetic to a ferromagnetic system takes place at the Curie\nTemperature.\nThe phase diagrams for magnetic systems are given in Figures 3.1-3.3. In Figure\n3.1, below the Curie temperature, the magnetization occurs spontaneously, and the\nzero magnetic \feld H= 0 separates the two possible orientations of magnetization.\nFigure 3.2 provides a sketch of the isotherms of magnetic system, and Figure 3.3\ngives the magnetization as a function of temperature; see also Reichl [5] and Onuki\n[4] for details.\nBased on the classical Ginzburg-Landau theory, for an isotropic system, the\nHelmholtz free energy can be expressed as\nA(M;T ) =A0(T) +1\n2\u000b2(T)jMj2+1\n4\u000b4(T)jMj4+\u0001\u0001\u0001;DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 7\nH =∞+\nH = 0\nH = ∞\nTTc_C\nFigure 3.1. Below the Curie point the magnetization occurs\nspontaneously; the curve H= 0 separates the two possible ori-\nentations of magnetization.\nMM\nMT = TT > T\nT < T\nT = 0T = 0\nT < T\nT = T\nT > T\nH0\n0_cc\ncc\nc\nc\nFigure 3.2. A sketch of the isotherms for a magnetic system.\nT\nTM\nM\n-Mc\n00\nFigure 3.3. A sketch of the spontaneous magnetization of a mag-\nnetic system.\nwhereA0(T) is a magnetization-independent contribution to the free energy, jMj2=\nM\u0001M, andM= (M1;M2;M3) is the magnetization of the system. When an8 MA AND WANG\nexternal \feld His present, the Gibbs free energy is given by\nG(M;H;T ) =A(M;T )\u0000H\u0001M\n=A0(T)\u0000H\u0001M+1\n2\u000b2(T;H)jMj2+1\n4\u000b4(T;H)jMj4+\u0001\u0001\u0001:\nFor smallH,\u000b2and\u000b4can be considered to be independent of H, and near the\nCurie point Tcwe have\n\u000b2(T) =\u000b0(T\u0000Tc); \u000b 4(T)>0:\nUsually,G(M;H;T ) is called the Ginzburg-Landau free energy. To omit the\nhigher order terms than jMj4, it is known that the equilibrium state Mof the\nferromagnetic system satis\fes\n(3.1)\u000e\n\u000eMG=\u000b4jMj2M+\u000b2M\u0000H= 0:\nThus above the Curie point we obtain from (3.1) that\n(3.2)M'1\n\u000b2H;\n\u001f=@M\n@H=1\n\u000b2(T)=1\n\u000b0(T\u0000Tc);\nwhere\u001fis the isothermal susceptibility, which is a scalar because the system is\nisotropic. Below the critical point, for H= 0, the magnetization Mobeys\n(3.3)jMj=s\n\u000b0(Tc\u0000T)\n\u000b4;@jMj\n@T=\u00001\n2r\u000b0\n\u000b4(Tc\u0000T):\nThe heat capacity at T=Tcis\nC(T <Tc)\u0000C(T >Tc) =\u0000T@2G\n@T2\f\f\f\nT=Tc(3.4)\n=\u0000Tc@2\n@T2\u00121\n2\u000b2jMj2+1\n4\u000b4jMj4\u0013\f\f\f\nT=Tc\n=\u000b2\n0Tc\n2\u000b4:\nWe infer then from (3.2)-(3.4) the following classical conclusions for an isotropic\nmagnetic system:\n(1) When as external magnetic \feld is present, a nonzero magnetization exists\nabove the Curie point Tc, which has the same direction as the applied \feld\nH.\n(2) Near the critical point Tcthe susceptibility \u001ftends to in\fnite with the rate\n(T\u0000Tc)\u00001, i.e., a very small applied \feld at T=Tccan yield a large e\u000bect\non the magnetization.\n(3) In the absence of an external \feld (i.e., H= 0), below the critical point a\nspontaneous magnetization Mappears, which depends continuously on T\nand tends to zero with the rate ( T\u0000Tc)1=2; namely the transition is of the\nsecond order.\n(4) The heat capacity at T=Tchas a jump with the gap \u0001 C=\u000b2\n0Tc\n2\u000b4, and the\njump has the shape of a \u0015, as shown in Figure 3.4.DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 9\nC\nTT\nc\nFigure 3.4.\nQualitatively, part of the above conclusions are in agreement with experimental\nresults.\nHowever these conclusions lead to wrong susceptibility and spontaneous magne-\ntization, whose experimental rates are given by \u001f/(T\u0000Tc)\u0000rwithr= 1:3, and\nbyM/(T\u0000Tc)\fwith\f= 1=3.\nFree energy Gmust be a function of the magnetization M. Hence the errors are\noriginated from the fact that the expression of Gin the Ginzburg-Landau theory\nis an approximation. It is di\u000ecult to derive a precise formula because Gis not\nanalytic onjMj, even the di\u000berentibility of GonjMjis very low.\nIf we study the dynamical properties of ferromagnetic systems by using the\nclassical Ginzburg-Landau free energy, we shall see a more serious error when an\nexternal \feld is present.\nTo see this, the dynamic equation of classical theory is given by\n(3.5)dM\ndt=\u0000\u000b2M\u0000\u000b4jMj2M+H:\nFor simplicity, we take H= (h;0;0) withh>0, it is equivalent that we take the\nx1-axis in the direction of H. Then the equation\n\u000b4jMj2M+\u000b2M\u0000H= 0\nhas a steady state solution M0= (m0;0;0) forT\u00150, which is the magnetization\ninduced by H. Make the transformation\nM=M0+M0:\nThen, the equation (3.5) is rewritten as (drop the primes)\n(3.6)dM1\ndt=\u0000(\u000b2+ 3\u000b4m2\n0)M1\u00002\u000b4m0M2\n1\u0000\u000b4m0jMj2\u0000\u000b4jMj2M1;\ndM2\ndt=\u0000(\u000b2+\u000b4m2\n0)M2\u00002\u000b4m0M1M2\u0000\u000b4jMj2M2;\ndM3\ndt=\u0000(\u000b2+\u000b4m2\n0)M3\u00002\u000b4m0M1M3\u0000\u000b4jMj2M3:\nComparing the two critical parameter curves\n\u000b2+ 3\u000b4m2\n0= 2)T1=Tc\u00003\u000b4m2\n0=\u000b0;\n\u000b2+\u000b4m2\n0= 0)T2=Tc\u0000\u000b4m2\n0=\u000b0;\nwe \fnd that T2> T 1. By Theorem A.1, (3.6) has the \frst transition at T=T2,\nwhere a new magnetization M= (M1;M2;M3), withM26= 0 andM36= 0, appears.10 MA AND WANG\nThis is unrealistic because any magnetization Mof this system must have the same\ndirection as H= (h;0;0); see Figure 3.1.\nIn fact, when a magnetic \feld His applied on an isotropic system the direction\nofHis a favorable one for magnetization. However, in (3.5) this point is not man-\nifested. Therefore, to investigate the phase transition dynamics of ferromagnetic\nsystems we need to revise the GLfree energy.\n4.Dynamic Transitions in Ferromagnetism\n4.1.Revised Ginzburg-Landau free energy. Let the ferromagnetic system be\nisotropic. When a magnetic \feld His present, we introduce a second order sym-\nmetric tensor\nA(T;H) = (aij(T;H)); aij=aji;1\u0014i; j\u00143;\nsuch thatA(T;0) = 0, and A(T;H) has eigenvalues\n\u00151=\u00151(T;H); \u0015 2=\u00153= 0;with\u00151(T;H)>0 asH6= 0;\nandHis the eigenvector of Acorresponding to \u00151:\n(4.1) AH=\u00151H (\u00151>0 asH6= 0; \u0015 1= 0 asH= 0):\nIt is clear that if we take the coordinate system ( x1;x2;x3) withx1-axis in the\nH-direction, then H= (H1;0;0) and\n(4.2) A=0\n@\u001510 0\n0 0 0\n0 0 01\nA:\nPhysically, condition (4.1) means that His a favorable direction of magnetization\nif we add a term\n\u0000MAMT=\u0000X\naijMiMj\nin the free energy. We also need to consider the nonlinear e\u000bect acted by H. To\nthis end we introduce the term \u0000jMj2M\u0001Hin the free energy.\nThus when the applied \feld Hmay vary in \n\u001aRn(n= 2;3), then the GLfree\nenergy is in the form\nG(M;T;H ) =G0+1\n2Z\n\nh\n\u0016jrMj2+\u000b2jMj2+\u000b4\n2jMj4(4.3)\n\u0000X\naijMiMj\u0000f(T;H;M )M\u0001Hi\ndx;\nwhereG0=G0(T) is independent of MandH, andfis a scalar function of T;H\nandM, de\fned by\n(4.4) f(T;H;M ) = 2(1 +\fjMj2); \f =\f(T;H)>0:\nFor\u000b2and\u000b4we assume that\n(4.5)\u000b2=\u000b0(H)(T\u0000T0(H)); T 0(0) =Tc; \u000b 0>0; \u000b 4>0:\nBy the standard model (2.5), we derive from (4.3) and (4.4) the following dy-\nnamical equations:\n@Mi\n@t=\u0016\u0001Mi\u0000\u000b2Mi+3X\nj=1aijMj\u0000\u000b4jMj2Mi (4.6)\n+\fjMj2Hi+ 2\f(M\u0001H)Mi+Hi for 1\u0014i\u00143:DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 11\nThe boundary condition is given by\n(4.7)@M\n@n\f\f\f\f\n@\n= 0:\nObviously, if H= 0, (4.6) coincide with the classical equations. For simplicity,\nhereafter we always take\n(4.8) H= (h;0;0) (h>0 is a constant) :\nWhenHis constant, in the study of phase transitions of magnetic systems, (4.6)\ncan be replaced by a system of ordinary di\u000berential equations as follows:\n(4.9)dM1\ndt= (\u00151\u0000\u000b2)M1+\fhjMj2+ 2\fhM2\n1\u0000\u000b4jMj2M1+h;\ndM2\ndt=\u0000\u000b2M2+ 2\fhM 1M2\u0000\u000b4jMj2M2;\ndM3\ndt=\u0000\u000b2M3+ 2\fhM 1M3\u0000\u000b4jMj2M3:\nEquations (4.9) have a steady state solution induced by H:\nM\u0003= (m0;0;0);with lim\nh!0m0= 0:\nWe see in Figure 3.2 that the magnetization M\u0003has an upper bound. Namely there\nis anM0such that\n(4.10)jM\u0003j<jM0j 8H2R3; T\u00150;\nM\u0003!M0 ifh!1:\nTo satisfy (4.10) it is necessary to assume that the coe\u000ecients \u000b0and\u000b4as in (4.5)\npossess the properties\n(4.11) \u000b0(H)!+1; \u000b 4(T;H)!+1;ash!1:\nBoth conditions (4.5) and (4.11) are physical.\n4.2.Dynamic transitions. In this subsection, to illustrate the main ideas, we\nonly consider the case where His a constant on \n. Therefore we shall study phase\ntransition dynamics of the ferromagnetic systems by using equations (4.9) for h>0.\nAnalysis for more general case can be carried out in the same fashion, and will be\nreported elsewhere.\nTake the transformation in (4.9)\n(4.12) M=M\u0003+M0:\nThen equations (4.9) are rewritten as (drop the primes)\n(4.13)dM1\ndt=\f1M1\u00002a2M2\n1\u0000a2jMj2\u0000\u000b4jMj2M1;\ndM2\ndt=\f2M2\u00002a2M1M2\u0000\u000b4jMj2M2;\ndM3\ndt=\f2M3\u00002a2M1M3\u0000\u000b4jMj2M3;12 MA AND WANG\nwhere\na2=\u000b4m0\u0000\fh;\n\f1=\u00151+ 6\fhm 0\u00003\u000b4m2\n0\u0000\u000b2;\n\f2= 2\fhm 0\u0000\u000b4m2\n0\u0000\u000b2:\nThe critical parameter curves \f1= 0 and\f2= 0 are given by\n\f1= 0)T1=T0(H) +1\n\u000b0(\u00151+ 6\fhm 0\u00003\u000b4m2\n0);\n\f2= 0)T2=T0(H) +1\n\u000b0(2\fhm 0\u0000\u000b4m2\n0):\nIt is clear that T1>T2provided\n(4.14) \u00151>2m0(\u000b4m0\u00002\fh);forh>0:\nTherefore, under condition (4.14), the equations (4.13) have a transition at T=\nT1in the space\n(4.15) E=f(M1;0;0)j \u00001<M 1<1g:\nMore precisely, we have the following transition theorem.\nTheorem 4.1. Assume the condition (4.14) and a26= 0. Then (4.13) has a Type-\nIII (mixed) transition at T=T1, and the transition occurs in the space E. The\nphase diagram is as shown in Figure 4.1. Moreover we have the following assertions:\n(1)There are two stable equilibrium states near T=T1, which are given by\nM+\n1=8\n<\n:0 ifT >T 1;\n1\n2\u000b4[\u00003a2+q\n9a2\n2+ 4\u000b4\f1]ifT <T 1;\nM\u0000\n1=\u00001\n2\u000b4[3a2+q\n9a2\n2+ 4\u000b4\f1]:\n(2)IfT < T 1,M+\n1is stable in the region 0< M 1<1, andM\u0000\n1is stable in\n\u00001<M 1<0.\n(3)IfT > T 1,M+\n1= 0 is stable in\u0000b < M 1<1andM\u0000\n1is stable in\n\u00001<M 1<\u0000b, where\nb=1\n2\u000b4[3a2\u0000q\n9a2\n2+ 4\u000b4\f1]>0 forT >T 1:\nProof. It is clear that if (4.14) holds, then\n\f1(T)8\n><\n>:<0 ifT >T 1;\n= 0 ifT=T1;\n>0 ifT <T 1;\n\f2(T1)<0:\nHence, by Theorem A.1 the system (4.13) has a transition at T=T1. Obviously,\nthe spaceEde\fned by (4.15) is the center manifold of (4.13) near T=T1. Hence,\nthe reduced equation of (4.13) on Eis expressed as\n(4.16)dM1\ndt=\f1M1\u00003a2M2\n1\u0000\u000b4M3\n1:DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 13\nM\nTM\nM+\n_1\n1\nM_\n1M+\n11\nT10= 0\nFigure 4.1.\nAsa26= 0, by Theorem A.2 we infer from (4.16) that this transition is of type-III,\nand the transition solutions satisfy\n\u000b4M2\n1+ 3a2M1\u0000\f1= 0:\nBy a direct compute one obtains Assertions (1) and (2).\nThe proof is complete. \u0003\n5.Physical Conclusions and Remarks\n5.1.Physical predications based on Theorem 4.1. By (4.12), the stable steady\nstates of (4.9) near T=T1are\nM+= (m0+M+\n1;0;0);\nM\u0000= (m0+M\u0000\n1;0;0);\nFrom the physical point of view, it should be\n(5.1) M+\n1\u00150; M\u0000\n1<0; m 0+M\u0000\n1\u00150:\nThe condition (5.1) requires that\n0<3a2<\u000b4m0;\nwhich is equivalent to\n(5.2) \fh<\u000b 4m0<3\n2\fh; (h>0)\nwherem0>0 is a solution of the equation\n(5.3) \u000b4m3\n0\u00003\fhm2\n0+ (\u000b2\u0000\u00151)m0\u0000h= 0;(h>0);\nnearT=T1.\nThus, the stable steady states M+andM\u0000of (4.9) near T=T1are physical\nprovided that the cvoe\fcients \u000b2(T;h);\u000b4(T;h);\f(T;h) and\u00151(T;h) satisfy (5.2)\nand (5.3). In this case the temperature T1is greater than the Curie temperature\nTc:\nT1(H)>Tc=T1(0) forH6= 0:\nThe two states M+andM\u0000are mathematically equal, therefore only by Theo-\nrem 4.1 we can not determine the magnetization behaviors of ferromagnetic systems\nnearT=T1. However, we see that the magnetization M+is stronger than M\u0000.\nPhysically, it implies that M+is favorable in T < T 1, andM\u0000is inT > T 1.14 MA AND WANG\nThus, from Theorem 4.1 there are two possible magnetization behaviors, i.e., two\nmagnetization functions:\n\u00161(T) =M+(T) = (m0(T) +M+\n1(T);0;0);\n\u00162(T) =(\nM+(T) = (m0(T) +M+\n1(T);0;0) ifT <T 1;\nM\u0000(T) = (m0(T) +M\u0000\n1(T);0;0) ifT\u0015T1:\nThe function \u00161(T) is continuous on T, as shown in Figure 5.1(a), and its derivative\nis discontinuous at T=T1:\n\u00160\n1(T\u0000\n1)\u0000\u00160\n1(T+\n1)'\u00121\n3d\ndT\f1(T1)\n\u000b2(T1);0;0\u0013\n:\nThe function \u00162(T) has a jump at T=T1, as shown in Figure 5.1(b).\nM\nTm\nTμ  (  T )\n0\n11\n(a)\nM\nTm\nTμ  (  T )\n0\n12\n(b)\nFigure 5.1. (a) The graph of function \u00161(T); (b) the graph of\nfunction\u00162(T).\nOn the other hand, by direct computation, the free energies of M+andM\u0000are\nshown to be given by\nG(M+) =8\n<\n:G(m0) if T\u0015T1;\nG(m0) +1\n4(M+\n1)2(a2M+\n1\u0000\f1) ifT <T 1;\nG(M\u0000) =G(m0) +1\n4(M\u0000\n1)2(a2M\u0000\n1\u0000\f1);\nwherem0is the magnetization induced by H= (h;0;0) satisfying (5.3), and\n\f1(T1) = 0. It is clear that\n(5.4) G(M+)>G(M\u0000) nearT=T1:\nHence, it follows from (5.4) that the magnetization behavior described by \u00162(T)\nis prohibited in real world because the free energy can not abruptly increase (or\ndecrease) in a temperature decreasing (or increasing) process. Thus, by Theorem\n4.1 and (5.4) we can derive the following physical conclusion:\nPhysical Conclusion 5.1. When an external \feld His present, the magnetization\nMH(T)of an isotropic ferromagnetic system is continuous on the temperature T,\nand there is a T1(H)>Tc(Tcthe Curie temperature) with T1(H)!TcasH!0\nsuch thatMH(T)is not di\u000berentiable at T=T1, whose derivative has a \fnite jump\nM0\nH(T\u0000\n1)\u0000M0\nH(T+\n1) =a>0 (a<1):DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 15\nMoreover, the graph of MH(T) =\u00161(T)as shown in Figure 5.1(a), and MH(T)!\nM0(T)asH!0with\nM0(T) =(\n0 ifT\u0015Tc;\nMs(T)ifT <Tc;\nwhereMs(T)is the spontaneous magnetization (see Figure 3.3).\n5.2.Asymmetry of \ructuations. The above discussions suggest that for ferro-\nmagnetic systems, there are two possible phase transition behaviors near a critical\npoint, and theoretically each of them has some probability to take place, however\nonly one of them can appear in reality. For the ferromagnetic systems we again\nsee this situation. This phenomena is also observed in phase transitions for PVT\nsystems [2].\nOne explanation of such phenomena is that the symmety of \ructuation near a\ncritical point is not generally true in equilibrium phase transitions. To make the\nstatement more clear, we \frst introduce some concepts.\nLetG(u;\u0015) be free energy of a thermodynamic system, u= (u1;\u0001\u0001\u0001;un) be the\norder parameter, and \u0015= (\u00151;\u0001\u0001\u0001;\u0015m) the control parameter ( n;m\u00151). Assume\nthatuis de\fned in the function space L2(\n;Rn) and\u00152Rm. Then the space\nX=f(u;\u0015)ju2L2(\n;Rn);\u00152Rmg\nis called the state space of the system.\nLet (u0;\u00150)2Xbe a stable equilibrium state of the system; namely ( u0;\u00150)\nis a locally minimal state of G(u;\u0015). We say that the system has a \ructuation at\n(u0;\u00150) if it deviates randomly from ( u0;\u00150) to (eu;e\u0015) with\nkeu\u0000u0k+je\u0015\u0000\u00150j>0:\nIn this case, ( eu;e\u0015) is called a state of \ructuation.\nThe so called symmetry of \ructuation means that for given r > 0, all states\n(eu;e\u0015) of \ructuation satisfying\nkeu\u0000u0k+je\u0015\u0000\u00150j=r; (eu;e\u0015)2X;\nhave the same probability to appear in real world. Otherwise, we say that the\n\ructuation is asymmetric.\nThe observations in both the PVT systems and the ferromagnetic systems strongly\nsuggest the following physical conjecture, regarding to the uniqueness of transition\nbehaviors.\nPhysical Conjecture (Asymmetry of Fluctuations). The symmetry of \ructua-\ntions for general thermodynamic systems may not be universally true. In other\nwords, in some systems with multi-equilibrium states, the \ructuations near a criti-\ncal point occur only in one basin of attraction of some equilibrium states, which are\nthe ones that can be physically observed.\nAppendix A.Recapitulation of the Dynamic Transition Theory\nIn this appendix we recall some basic elements of the dynamic transition theory\ndeveloped by the authors [1, 3], which are used to carry out the dynamic transition\nanalysis for the ferromagnetism systems in this article.16 MA AND WANG\nLetXandX1be two Banach spaces, X1\u001aXa compact and dense inclusion.\nIn this chapter, we always consider the following nonlinear evolution equations\n(A.1)du\ndt=L\u0015u+G(u;\u0015);\nu(0) =';\nwhereu: [0;1)!Xis unknown function, and \u00152R1is the system parameter.\nAssume that L\u0015:X1!Xis a parameterized linear completely continuous \feld\ndepending contiguously on \u00152R1, which satis\fes\n(A.2)L\u0015=\u0000A+B\u0015a sectorial operator ;\nA:X1!X a linear homeomorphism ;\nB\u0015:X1!X a linear compact operator :\nIn this case, we can de\fne the fractional order spaces X\u001bfor\u001b2R1. Then we also\nassume that G(\u0001;\u0015) :X\u000b!XisCr(r\u00151) bounded mapping for some 0 \u0014\u000b<1,\ndepending continuously on \u00152R1, and\n(A.3) G(u;\u0015) =o(kukX\u000b);8\u00152R1:\nHereafter we always assume the conditions (A.2) and (A.3), which represent that\nthe system (A.1) has a dissipative structure.\nIn the following we introduce the de\fnition of transitions for (A.1).\nDe\fnition A.1. We say that the system (A.1) has a transition of equilibrium from\n(u;\u0015) = (0;\u00150)on\u0015>\u0015 0(or\u0015<\u0015 0)if the following two conditions are satis\fed:\n(1)when\u0015 < \u0015 0(or\u0015 > \u0015 0);u= 0 is locally asymptotically stable for (A.1);\nand\n(2)when\u0015 > \u0015 0(or\u0015 < \u0015 0), there exists a neighborhood U\u001aXofu= 0\nindependent of \u0015, such that for any '2Un\u0000\u0015the solution u\u0015(t;')of\n(A.1) satis\fes that\nlim sup\nt!1ku\u0015(t;')kX\u0015\u000e(\u0015)>0;\nlim\n\u0015!\u00150\u000e(\u0015)\u00150;\nwhere \u0000\u0015is the stable manifold of u= 0, with codim \u0000\u0015\u00151inXfor\n\u0015>\u0015 0(or\u0015<\u0015 0).\nObviously, the attractor bifurcation of (A.1) is a type of transition. However,\nbifurcation and transition are two di\u000berent, but related concepts. De\fnition A.1\nde\fnes the transition of (A.1) from a stable equilibrium point to other states (not\nnecessary equilibrium state). In general, we can de\fne transitions from one attrac-\ntor to another as follows.\nDe\fnition A.2. Let\u0006\u0015\u001aXbe an invariant set of (A.1). We say that (A.1)\nhas a transition of states from (\u0006\u00150;\u00150)on\u0015 > \u0015 0(or\u0015 < \u0015 0)if the following\nconditions are satis\fed:\n(1)when\u0015<\u0015 0(or\u0015>\u0015 0);\u0006\u0015is a local minimal attractor, and\n(2)when\u0015 > \u0015 0(or\u0015 < \u0015 0), there exists a neighborhood U\u001aXof\u0006\u0015\nindependent of \u0015such that for any '2Un(\u0000\u0015[\u0006\u0015), the solution u(t;')DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 17\nof (A.1) satis\fes that\nlim sup\nt!1dist(u(t;');\u0006\u0015)\u0015\u000e(\u0015)>0;\nlim\n\u0015!\u00150\u000e(\u0015) =\u000e\u00150;\nwhere \u0000\u0015is the stable manifolds of \u0006\u0015with codim \u0000\u0015\u00151.\nLet the eigenvalues (counting multiplicity) of L\u0015be given by\nf\fj(\u0015)2Cjj= 1;2;\u0001\u0001\u0001g\nAssume that\nRe\fi(\u0015)8\n><\n>:<0 if\u0015<\u0015 0;\n= 0 if\u0015=\u00150;\n>0 if\u0015>\u0015 0;81\u0014i\u0014m; (A.4)\nRe\fj(\u00150)<0 8j\u0015m+ 1: (A.5)\nThe following theorem is a basic principle of transitions from equilibrium states,\nwhich provides su\u000ecient conditions and a basic classi\fcation for transitions of non-\nlinear dissipative systems. This theorem is a direct consequence of the center man-\nifold theorems and the stable manifold theorems; we omit the proof.\nTheorem A.1. Let the conditions (A.4) and (A.5) hold true. Then, the system\n(A.1) must have a transition from (u;\u0015) = (0;\u00150), and there is a neighborhood\nU\u001aXofu= 0such that the transition is one of the following three types:\n(1)Continuous Transition : there exists an open and dense set eU\u0015\u001aU\nsuch that for any '2eU\u0015, the solution u\u0015(t;')of (A.1) satis\fes\nlim\n\u0015!\u00150lim sup\nt!1ku\u0015(t;')kX= 0:\nIn particular, the attractor bifurcation of (A.1) at (0;\u00150)is a continuous\ntransition.\n(2)Jump Transition : for any\u00150<\u0015<\u0015 0+\"with some\">0, there is an\nopen and dense set U\u0015\u001aUsuch that for any '2U\u0015,\nlim sup\nt!1ku\u0015(t;')kX\u0015\u000e>0;\nwhere\u000e >0is independent of \u0015. This type of transition is also called the\ndiscontinuous transition.\n(3)Mixed Transition : for any\u00150<\u0015<\u0015 0+\"with some\">0,Ucan be\ndecomposed into two open sets U\u0015\n1andU\u0015\n2(U\u0015\ninot necessarily connected):\n\u0016U=\u0016U\u0015\n1+\u0016U\u0015\n2; U\u0015\n1\\U\u0015\n2=;;\nsuch that\nlim\n\u0015!\u00150lim sup\nt!1ku(t;')kX= 0 8'2U\u0015\n1;\nlim sup\nt!1ku(t;')kX\u0015\u000e>0 8'2U\u0015\n2:\nAn important aspect of the transition theory is to determine which of the three\ntypes of transitions given by Theorem A.1 occurs in a speci\fc problem. We refer\nthe interested readers to [3, 1] for more discussions. Instead, here we consider the18 MA AND WANG\ntransition of (A.1) from a simple critical eigenvalue. Let the eigenvalues \fj(\u0015) of\nL\u0015satisfy\n(A.6)\f1(\u0015)8\n><\n>:<0 if\u0015<\u0015 0;\n= 0 if\u0015=\u00150;\n>0 if\u0015>\u0015 0;\nRe\fj(\u00150)<08j\u00152;\nwhere\f1(\u0015) is a real eigenvalue.\nLete1(\u0015) ande\u0003\n1(\u0015) be the eigenvectors of L\u0015andL\u0003\n\u0015respectively corresponding\nto\f1(\u0015) with\nL\u00150e1= 0; L\u0003\n\u00150e\u0003\n1= 0; <e 1;e\u0003\n1>= 1:\nLet \b(x;\u0015) be the center manifold function of (A.1) near \u0015=\u00150. We assume that\n(A.7) <G(xe1+ \b(x;\u00150);\u00150);e\u0003\n1>=\u000bxk+o(jxjk);\nwherek\u00152 an integer and \u000b6= 0 a real number.\nWe have the following transition theorems.\n(a)v u = 0λ\n(b)u = 0\nu = 0v\nvv1 2λ λ\nλ\n(c)\nFigure A.1. Topological structure of the mixing transition of\n(A.1) when k=even and \u000b6= 0: (a)\u0015 < \u0015 0; (b)\u0015=\u00150; (c)\n\u0015 > \u0015 0. HereU\u0015\n1is the unstable domain, and U\u0015\n2the stable do-\nmain.\nTheorem A.2. Under the conditions (A.6) and (A.7), if k=even and \u000b6= 0, then\nwe have the following assertions:\n(1)(A.1) has a mixed transition from (0;\u00150). More precisely, there exists a\nneighborhood U\u001aXofu= 0 such thatUis separated into two disjoint\nopen setsU\u0015\n1andU\u0015\n2by the stable manifold \u0000\u0015ofu= 0 satisfying the\nfollowing properties:\n(a)U=U\u0015\n1+U\u0015\n2+ \u0000\u0015,\n(b)the transition in U\u0015\n1is jump, and\n(c)the transition in U\u0015\n2is continuous. The local transition structure is as\nshown in Figure A.1.\n(2)(A.1) bifurcates in U\u0015\n2to a unique singular point v\u0015on\u0015 > \u0015 0, which is\nan attractor such that for any '2U\u0015\n2,\nlim\nt!1ku(t;')\u0000v\u0015kX= 0;\nwhereu(t;')is the solution of (A.1).\n(3)(A.1) bifurcates on \u0015<\u0015 0to a unique saddle point v\u0015with the Morse index\none.DYNAMIC PHASE TRANSITIONS FOR FERROMAGNETIC SYSTEMS 19\n(4)The bifurcated singular point v\u0015can be expressed as\nv\u0015=\u0000(\f1(\u0015)=\u000b)1=(k\u00001)e1+o(j\f1j1=(k\u00001)):\nReferences\n[1]T. Ma and S. Wang ,Bifurcation theory and applications , vol. 53 of World Scienti\fc Series on\nNonlinear Science. Series A: Monographs and Treatises, World Scienti\fc Publishing Co. Pte.\nLtd., Hackensack, NJ, 2005.\n[2] ,Dynamic phase transitions in pvt systems , submitted, (2007).\n[3] ,Stability and Bifurcation of Nonlinear Evolution Equations , Science Press, 2007.\n[4]A. Onuki ,Phase transition dynamics , Cambridge University Press, 2007.\n[5]L. E. Reichl ,A modern course in statistical physics , A Wiley-Interscience Publication, John\nWiley & Sons Inc., New York, second ed., 1998.\n(TM) Department of Mathematics, Sichuan University, Chengdu, P. R. China\n(SW) Department of Mathematics, Indiana University, Bloomington, IN 47405\nE-mail address :showang@indiana.edu"    },    {        "title": "1809.08644v1.Ferromagnetic_resonance_in_thin_ferromagnetic_film_with_surface_anisotropy.pdf",        "content": "Ferromagnetic resonance in thin ferromagn etic film with surface anisotropy \n \nN. A. Usov1,2\n \n1National University of Science and Tec hnology «MISiS», 119049, Moscow, Russia  \n2Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, \nRussian Academy of Sciences, IZMIRAN, 108480, Troitsk, Moscow, Russia \n \nAbstract The ferromagnetic resonance frequencies are obtaine d for a thin ferromagnetic film with surface \nanisotropy for the cases when th e external magnetic field is applied perpe ndicularly or parallel to the film surface, \nand for various combinations of boundary conditions on the film surface. It is shown that in the presence of surface \nanisotropy the ferromagnetic resonance frequency essentially depends both on th e film thickness and on the value of \nthe surface anisotropy constant. The results obtained provide  a basis for the correct interpretation of experimental \ndata obtained by means of broadband ferroma gnetic resonance in thin film structures. \n \nPACS:    75.50.Kj, 75.80.+q, 75.60.Ej, 75.60.Jk \nKeywords:   Thin ferromagnetic film, surface magnetic anisotropy, ferromagnetic resonance frequency \n \nI. Introduction \nThin ferromagnetic films with su rface magnetic anisotropy are cu rrently of great technological \ninterest [1-4]. In particular, the ultra-thin CoFe B/MgO bilayers [2-4] exhibit a perpendicular magnetic \nanisotropy. The latter provides a high thermal stability and is highly de sirable for reducing the critical \ncurrent for spin-torque switching [2]. Therefore, thin  CoFeB ferromagnetic films are considered currently \nas promising electrode material in magnetic rando m access memory and spin-transfer torque memory \ndevices. \n From physical considerations, it is obvious that only the surface magnetic anisotropy [5-8] is \ncapable of providing an out-of-plane magnetization in a thin ferromagnetic  film made of a soft magnetic \nmaterial. Actually, if a surface anisotropy constant is negative and sufficiently large in absolute value, the \norientation of the unit magnetization v ector perpendicular to the film pl ane is energetically favorable, in \nspite of a significant increase in the magnetostatic en ergy of the film. Experimentally, a rotation of the \nunit magnetization vector as a function of the film thickness was observed in a number of experiments \nwith thin ferromagnetic films of iron, cobalt, and other ferromagnets [9-12]. \nThe existence of surface magnetic anisotropy may be related to various physical origins, such as \nthe specifics of the spin-orbit interaction on the ferromagnetic surface [13], the difference in atomic periods of various layers of a multilayer sample, or  a film substrate [14], th e presence of non-uniform \nmechanical stresses near the inte rface [15], etc. However, in terms of phenomenological description of \nmagnetic phenomena [5], it is important that the energy density of surface anisotropy is concentrated in a \nvery narrow region near the sample surface. Therefor e, it is a surface contri bution to the total energy \nwhich is proportional to the surface area of the film , and not to its volume. The general variational \napproach [5] shows that the influence of surface ma gnetic anisotropy on the behavior of a ferromagnet is \nmanifested only under special boundary condition. The latt er acts on the sample surface or the interface \nbetween different materials [5,16].  \nSince the theoretical calculation of  the surface anisotropy constant Ks is rather complicated [13-\n15], it is desirable to have reliable methods to dete rmine this phenomenological constant experimentally. \nOne of such methods can be the ferromagnetic res onance (FMR) [17,18], in whic h the film magnetization \nundergoes a rapid precession near a certain equilib rium position of the unit magnetization vector. The \nbroadband FMR [19-23] can provide valuable information on the presence or absence of the energy \ninteractions on the surface of a ferromagnetic film, or on the interfaces of ferromagnetic multilayers. The \ninfluence of surface anisotropy on the ferromagnetic resonance in a thin ferromagnetic film was previously considered in seminal papers by Rado and Weertman [24,25]. Un fortunately, no explicit \n 1formulas for the FMR frequency were actually obt ained. Whereas in the present paper, the FMR \nfrequencies have been derived for ferromagnetic f ilm with surface anisotropy for the cases when the \nexternal magnetic field is applied perpendicularly or parallel to the film surface. Besides, various \ncombinations of boundary conditions on the film surface s have been studied. Thes e results seem helpful \nfor correct interpretation of the experimental data obt ained by means of the broadband FMR in thin film \nstructures.  \n \nII. Out-of-plane external magnetic field \n Consider a thin ferromagnetic film of thickness L parallel to the XY plane and located in the \ndomain 0 < z  < L. The unit magnetization vector of the film ()t r,rrα  satisfies the Landau-Lifshitz-Gilbert \n(LLG) equation \n[]⎥⎦⎤\n⎢⎣⎡\n∂∂× + × − =∂∂\ntHtefαα κ α γαrrrrr\n,       ( 1 )  \nwhere γ is the gyromagnetic ratio, κ being the phenomenological dampi ng constant. The total effective \nmagnetic field of the film efHr takes into account the contributions due to the exchange and dipolar \ninteractions as well as from th e magnetic anisotropy energy. \n \n0H HMw\nMCH\nsa\nsefr r\nrr r+′+∂∂− Δ =αα .       ( 2 )  \nHere C = 2A is the exchange constant, Ms is the saturation magnetization,  is the vector of the \nhomogeneous applied magnetic field, 0Hr\nH′r is the vector of the demagnetiz ing field which is created by the \nvolume and surface magnetic charges distributed in  the volume and on the surface of the film, \nrespectively. Finally, ()αrw  is the energy density of the magne tic anisotropy in the film volume. \n The presence of surface magnetic anisotropy can be described [5-8] by introducing a surface \ninteraction with the ener gy density per unit area  \n \n     ()2n K ws sarrα= .       ( 3 )  \nHere Ks is the phenomenological surface anisotropy constant, and nr is the unit vector of the external \nnormal to the film surface. In th e presence of the surface magnetic  anisotropy, Eq. (3), the boundary \ncondition for the unit magnetiza tion vector is given by [5] \n \n    () ()(α α α)α rrrr rrr\nn n n KnCs− =∂∂2 .      ( 4 )  \nIt works on the upper and lower surfaces of the ferro magnetic film. In the absence of surface anisotropy \non one or both surfaces of the film, Ks = 0, the usual boundary condition, 0= ∂ ∂nαr, acts on the \ncorresponding surface.  \n Suppose that the ferromagnetic film is placed in  a sufficiently strong external magnetic field \nperpendicular to the film surface, ()0 0 , 0 , 0H H=r. In what follows we neglect for simplicity the effect of a \nsmall volume magnetic anisotropy of the film, setting ()0=αrw . In equilibrium, in the absence of \nmagnetization perturbations, the unit magnetization vector is perpe ndicular to the film surface, \n. Indeed, in the case considered the v ector of effective ma gnetic field has only z component, ()(1 , 0 , 00=αr)\n) (s ef M H H H H π4 , 0 , 00) 0 (\n0) 0 (− = ′+ =rrr. Therefore, the equilibrium condition for the unperturbed unit \nmagnetization vector  \n     .  () ()[]00 0= ×efHrrα .      ( 5 )  \n 2as well as the boundary condi tion (4) are satisfied. \n Let us now consider small deviations of th e unit magnetization vector from the equilibrium \nposition. The first-order correction of  the perturbation theory [5] to the unit magnetization vector is given \nby () () ()()0 , ,1 1 1\ny xα α α=r. The components of the first-order co rrection satisfy the equation of motion \n \n   ()\n() ()[]() ()[]()()\n⎥⎦⎤\n⎢⎣⎡\n∂∂× + × − × − =∂∂\ntH Htef ef1\n0 1 0 0 11αα κ α γ α γαrrrrrrr\n   (6) \nSince the magnetization in the film pl ane is homogeneous, it is  reasonable to assume that the perturbation \nof the magnetization depends only on the z coordinate. Then the first order correction to the effective \nmagnetic field vector is given only by the exchange interaction, so that \n \n    () ()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂\n∂∂= 0 , ,21 2\n21 2\n) 1 (\nz MC\nz MCHy\nsx\nsefα α r.      ( 7 )  \nTaking into account Eqs.  (6), (7) and neglec ting the attenuation, κ → 0, one obtains the following \nequations of motion for the component s of the unit magnetization vector \n \n    () ()\n()()1\n21 2 1\ny H my\nsx\nz MC\ntα ω ωαγα− +∂∂=∂∂;     ( 8 a )  \n    () ()\n()()1\n21 2 1\nx H mx\nsy\nz MC\ntα ω ωαγα− −∂∂− =∂∂,     ( 8 b )  \nwhere we denote ωH = γH0 and ωm = 4πγMs. The boundary conditions to Eq. (8) follow from Eq. (4)  \n \n  ()()()()t L Kzt LCx sx, 2,1\n11\nαα− =∂∂;  ()()()(t L Kzt LCy sy, 2,1\n11\nαα− =∂∂);  (9a) \n  ()()()()t KztCx sx, 0 2, 01\n21\nαα=∂∂;  ()()()(t KztCy sy, 0 2, 01\n21\nαα=∂∂)\n).  (9b) \nHere it is assumed that the surface anisotropy constants at the upper, K1s, and lower, K2s, film surfaces can \nbe different.  \n If one of the surface anisotropy constants is zero, for example, K2s = 0, the solution of Eq. (8), (9) \nhas the form of the unit ma gnetization vector precession \n \n  ;  ()() ( )( t kz A t zx ω α cos cos ,1=()()()()t kz B t zy ω α sin cos ,1= .    (10) \nFrom the equations of motion (8) one obtains for the precession frequency \n \n    2kMC\nsm Hγ ω ω ω+ − = ,       ( 1 1 )  \nwhereas the boundary conditions (9) lead to the dispersion equation \n \n     ()CKkL ks12tan= .       ( 1 2 )  \nIn the limit kL << 1 for the lowest quasi-homogeneous pre cession mode one obtains from Eq. (12) \n 3    CLKks12= ;   L MK\nss\nm H12γ ω ω ω+ − = .   (13) \nFor inhomogeneous FMR modes in the limit 11<<C L Ks  one has  \n \n     n CK\nLnks\nnππ 12+ ≈ ;  n = 1,2,  … \n However, if the surface anisotropy is pres ent on both surfaces of the ferromagnetic film, the \ndispersion relation (1 2) takes the form \n \n    ()( )\nCK KkLkCK Kks s s s 2 1\n22 1 2tan4 +=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− .     (14) \nAs a result, in the limit kL << 1 the resonance frequency of the quasi-homogeneous mode equals  \n \n   ( )\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+++ − =C MK K\nL MK K\nss s\nss s\nm H2 1 2 1 4 2γ ω ω ω .    (15) \n Eqs. (11-15) for FMR frequencies are valid in a sufficiently strong perpe ndicular magnetic field, \nunder the condition that the unperturbed film magnetizati on is perpendicular to the film surface. They can \nbe used to determine experimentally the values of the film surface anisotropy constants. \n \nIII. In-plane external magnetic field \nSuppose now that a sufficiently strong external ma gnetic field is applied parallel to the film \nsurface, for definiteness, along the Y axis, so that ()0 , , 00 0 H H=r. Then in equilibrium, in the absence of \nmagnetization perturbation, the unit magnetization vector is parallel to the external magnetic field, \n. Indeed, in the case under considera tion the demagnetizing field is absent, , and the \neffective magnetic field vector is given by ()(0 , 1 , 00=αr) 0) 0 (=′Hr\n()0 , , 00) 0 (\n0) 0 (H H H Hef =′+ =rrr. Therefore, both the equilibrium \nequation (5) and the boundary condition (4) are satisfied.  \nNext, the first-order correction of the unit magnetization vector has the form () ()()()1 1 1, 0 ,z xα α α=r. \nThe first order correction to the eff ective magnetic field vector contains the contributions of exchange and \nmagneto- dipole interactions associated with the devi ation of the unit magnetization vector perpendicular \nto the film surface \n    () ()\n()\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−∂∂\n∂∂=1\n21 2\n21 2\n) 1 (4 , 0 ,z sz\nsx\nsef Mz MC\nz MCH α πα α r.   (16) \nAs a result, neglecting attenuation, the equation of motion (6) takes the form \n \n   () ()\n()()1\n21 2 1\nz H mz\nsx\nz MC\ntα ω ωαγα+ +∂∂− =∂∂;      ( 1 7 a )  \n   () ()\n()1\n21 2 1\nx Hx\nsz\nz MC\ntα ωαγα−∂∂=∂∂.      (17b) \nThe boundary conditions for the components of the unit magnetization vector are given by \n \n   ()()0,1\n=∂∂\nzt Lxα;  ()() ()(t L Kzt LCz sz, 2,1\n11\nαα=∂∂);   (18a) \n 4   ()()0, 01\n=∂∂\nztxα;  ()() ()(t Kzt\nz sz, 0 2, 01\n21\nαα− =∂∂).    (18b) \nIn the absence of the surface magnetic anisotropy, K1s = K2s = 0, the solution of Eqs. (17), (18) is \nsimilar to Eq. (10). For the FMR frequency one obtains the Kittel’s relation [18] \n \n ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+ + =2 2\nn\nsH n\nsm H n kMCkMCγ ω γ ω ω ω ; kn = πn/L;   n = 0,1 … \nFor the homogeneous precession mode, n = 0, it reduces to well known result [17,18] ()m H Hω ω ω ω+ = . \nIf the surface magnetic anisotropy is present on ly on the upper surface of the ferromagnetic film, \nK1s < 0, K2s = 0, the solution of Eqs. (17), (18) is of the form \n \n   ()() ()() () ()t z k A z k A t zx ω α cos cosh cosh ,2 2 1 11+ = ;  \n   ()() ()() ( )()t z k B z k B t zz ω α sin cosh cosh ,2 2 1 11+ = .     (19) \nHere the wave vectors k1,2 and the coefficients B B1,2 are determined by the equations \n \n  ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ ± + =4 22\n2 2\n2 , 1m m\nHs\nCMkωωωωγ;   2 , 12\n2 , 1\n2 , 1 AkMC\nBsH\nωγω−\n= . \nIn the given case, K2s = 0, the boundary conditions (18b) for the so lutions (19) are auto matically satisfied, \nwhereas the boundary conditions (18a) lead to the dispersion equation for the FMR frequency ω \n \n () ()\n22 2\n21 2\n2\n11 2\n11 2\n1coth 2 coth 2\nkL kkMC\nСKkMC\nkL kkMC\nСKkMC\nsHs\ns sHs\ns⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− + =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− +γωγ γωγ.  (20) \nIn the limit k1L, k2L << 1, using the series expansion coth( x) = 1/ x + x/3 + … for the lowest precession \nmode one obtains from Eq. (20) the relation \n \n    ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− − + =CL K\nL MKs\nss\nm H H1 1\n3212γ ω ω ω ω .    (21) \nEq. (21) is valid for sufficiently small film thicknesses L, when dimensionless parameter 11<C L Ks . \nWith increasing film thickness, the value of the FMR frequency ω should be found numerically from Eq. \n(20). \nEq. (21) for the lowest FMR frequency is used in  the literature [19-21,23] by analogy with Kittel's \nformulas [17], but without any ju stification. In fact, Eq. (21) is valid only in the limit k1L, k2L << 1, and \nfor the case when the surface anisotropy is present only on one of the film surfaces.  \n If the surface anisotropy is present both on th e upper and lower surfaces of the ferromagnetic film, \nK1s, K2s < 0, the solution of Eqs. ( 17) has a more general form \n \n ()() ()()()() () t z k A z k A z k A z k A t zx ω α cos sinh cosh sinh cosh ,2 4 2 3 1 2 1 11+ + + = ();  \n ()() () ()()() () ()t z k B z k B z k B z k B t zz ω α sin sinh cosh sinh cosh ,2 4 2 3 1 2 1 11+ + + = ,   (22) \nwhere the coefficients B Bi are expressed in terms of Ai as follows \n 5   2 , 12\n1\n2 , 1 AkMC\nBsH\nωγω−\n= ;   4 , 32\n2\n4 , 3 AkMC\nBsH\nωγω−\n= . \nUsing the solutions (22), it is pos sible to satisfy the general boundary  conditions (18). Then, from the \ncorresponding dispersion equation in the limit k1L, k2L << 1 one can obtain an approximate expression for \nthe FMR frequency \n \n   ( )\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−+− + =C MK K\nL MK K\nss s\nss s\nm H H2 1 2 1 4 2γ ω ω ω ω .   (23) \nAs an example, Fig. 1 shows the FMR frequencies, f = ω/2π, in the thin-film structure [2-4] \nTa/CoFeB/MgO in the case when the external magnetic field H0 is applied parallel to the film plane, and \nunder the assumption that the surface anisotropy c onstant is nonzero only at the CoFeB/MgO interface. \nThe solid curves in Fig. 1 are drawn according to Eq. (21). \n For sufficiently small CoFeB film thicknesses this equation approximates the numerical solution of \nEq. (20) with reasonable accuracy. The values of the saturation magnetization Ms = 1250 emu/cm3, and \nthe exchange constant A = C/2 = 1.5×10-6 erg/cm, used in the calculation were determined experimentally \n[22] for ferromagnetic fi lm of the composition Co 40Fe40B B20.  \n \n02468 1 0 1 2 1 4010203040\n432f  (GHz)\nH0  (kOe)Ks = - 1.0 erg/cm2\nMs = 1250 emu/cm3\nA = 1.5*10-6 erg/cmCo40Fe40B20 film\n1\n(a)\n   0 2 4 6 8 1 01 21 4010203040\n32Co40Fe40B20 filmL = 1.0 nm\nMs = 1250 emu/cm3\nA = 1.5*10-6 erg/cmf  (GHz)\nH0  (kOe) (b)1\n \nFig. 1. a) The frequency of ferromagne tic resonance in a ferromagnetic film Co 40Fe40B B20 as a function of \nin-plane magnetic field for different film thicknesses: 1) L = 1.0 nm; 2) L = 0.8 nm; 3) L = 0.7 nm; 4) L = \n0.6 nm. b) The same for different values of the surface anisotropy constant at fixed thickness L = 1.0 nm: \n1) K 1s = - 1.0 erg/cm , 2) K2\n1s = - 1.25 erg/cm , 3) K2\n1s = - 1.5 erg/cm . 2\n \nIt was experimentally shown [3] that for thin -film structure Ta/CoFeB /MgO the out-of-plane \nmagnetization exists in the film thickness range 0.6 - 1.2 nm. It ca n be shown also [26] that the \nmagnetization vector becomes parallel to the film plane above the saturation field Hs = 2|K1s|/MsL - 4πMs. \nThis value closely coincides with the magnetic fiel d at which the FMR frequencies, Eq. (21), become \nnonzero. \nAs Fig. 1 shows, in the presence of surface an isotropy the FMR frequenc ies essentially depend \nboth on the ferromagnetic film thickness and on the valu e of the surface anisotropy constant. Note that \nEqs. (21), (23) are valid al so at film thicknesses greater than the cr itical value, when in the absence of \napplied magnetic field there is a canted or inhomoge neous micromagnetic state in the film [11,16]. The \n 6only condition is that there is a su fficiently strong in-plane magnetic field in which the unperturbed film \nmagnetization is parallel  to the film surface. \n \nIV. Conclusion \nIt this paper, it is shown that the peculiarit ies of the FMR spectra observed in the ultra-thin \nCoFeB/MgO bilayers [2-4] and similar ferromagnetic structures may be related with the presence of the \nsurface magnetic anisotropy of appreciable value at various surfaces of the f ilm. A proper theoretical \nmethod to describe the influence of the surface magnetic anisotropy on the film behavior is the use of the \nnatural boundary condition that can be  obtained on the basis of the genera l variational approach [5]. Using \nthis technique, the FMR frequencies are obtained in th is paper for a thin ferromagnetic film with surface \nanisotropy for the cases when the external magnetic field is applied perpendicularly or parallel to the film \nplane, and for various combinations of boundary conditio ns on the film surface. These results can be used \nfor determination of the surface anisotropy constant of the film by means of the broadband FMR \nexperiment. \n \nAcknowledgement \n This work was supported by the Ministry of E ducation and Science of the Russian Federation in \nthe framework of Increase Competitiveness Program of NUST «MISiS», contract № K2-2017-008. \n \nReferences \n[1] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, E. E. Fullerton, Nature Mater. 5 (2006) 210. \n[2] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, H. \nOhno, Nature Mater . 9 (2010) 721. \n[3] W. X. Wang, Y. Yang, H. Naganuma, Y. Ando, R. C. Yu, X. F. Han. Appl. Phys. Lett. 99 (2011) 012502. \n[4] Q. L. Ma, S. Iihama, T. Kubota, X. M. Zhang, S. Mi zukami, Y. Ando, T. Miyazaki, Appl. Phys. Lett. 101 (2012) \n122414. \n[5] W. F. Brown, Jr., Micromagnetics, Wiley-Interscience, New York - London, 1963. \n[6] M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broe der, J. J. de Vries, Rep. Prog. Phys. 59 (1996) 1409.  \n[7] H. N. Bertram, D. I. Paul, J. Appl. Phys. 82 (1997) 2439. \n[8] C. A. F. Vaz, J. A. C. Bland, G. Lauhoff, Rep. Prog. Phys. 71 (2008) 056501. \n[9] R. Allenspach, M. Stampanoni, A. Bischof, Phys. Rev. Lett. 65 (1990) 3344. \n[10] H. Fritzsche, J. Kohlepp, H. J. Elmers , U. Gradmann, Phys. Rev. B 49 (1994) 15665.  \n[11] M. Speckmann, H. P. Oepen, H. Ibach, Phys. Rev. Lett. 75 (1995) 2035.  \n[12] M. Hehn, S. Padovani, K. Ounadjela, J. P. Bucher, Phys. Rev. B 54 (1996) 3428. \n[13] H. X. Yang, M. Chshiev, B. Dieny, J. H. Lee, A. Manchon, K. H. Shin, Phys. Rev. B 84 (2011) 054401.  \n[14] K. H. He, S. J. Chen, J. Appl. Phys. 111 (2012) 07C109.  \n[15] J. I. Hong, S. Sankar, A. E. Berkowitz, W. F. Egelhoff Jr., J. Magn. Magn. Mater. 285 (2005) 359. \n[16] N. A. Usov, O. N. Serebryakova, J. Appl. Phys. 121 (2017) 133905. \n[17] C. Kittel,  Phys. Rev. 73 (1948) 155.  \n[18] C. Kittel, Phys. Rev. 110 (1958) 1295. \n[19] J.-M. L. Beaujour, W. Chen, A. D. Kent, J. Z. Sun, J. Appl. Phys. 99 (2006) 08N503. \n[20] J-M. L. Beaujour, J. H. Lee, A. D. Kent, K. Krycka, C-C. Kao, Phys. Rev. B 74 (2006) 214405. \n[21] X. Liu, W. Zhang, M. J. Carter, G. Xiao, J. Appl. Phys. 110 (2011) 033910. \n[22] A. Conca, J. Greser, T. Sebastian, S. Klingler, B. Obry, B. Leven, B. Hillebra nds, J. Appl. Phys. 113 (2013) \n213909. \n[23] A. Conca, A. Niesen, G. Reiss, B. Hillebra nds, J. Phys. D.: Appl. Ph ys. (2018), DOI: 10.1088/1361-\n6463/aab5d8. \n[24] G. T. Rado, J. R. Weertman, J. Phys. Chem. Solids 11 (1959) 315. \n[25] G. T. Rado, Phys. Rev. B 26 (1992) 295. \n[26] N. A. Usov, O. N. Serebryakova, J. Magn. Magn. Mater. 142 (2018) 453. \n 7"    },    {        "title": "1611.02834v1.Theoretical_study_of_the_stripline_ferromagnetic_resonance_response_of_metallic_ferromagnetic_films_based_on_an_analytical_model.pdf",        "content": " 1  Theoretical study of the stripline ferromagnetic resonance  \nresponse of metallic ferromagnetic films based on an \nanalytical model  \n \nR.Hai 1,2  and M. Kostylev 1  \n \n1.School of Physics, University of Western Australia, Crawley, W.A. 6009, Australia  \n2.School of Phy sics, Nanjing University, Nanjing 210093, China  \n \nAbstract:  We develop an advanced analytical model for calculating the broadband stripline \nferromagnetic resonance (FMR) response for metallic ferromagnetic films, taking into account the \nexchange interaction  as well as the exchange boundary conditions at the film surface. This approach \nleads to simple analytical expressions in the Fourier space. As a result, a numerical code which \nimplements inverse Fourier transform of these equations is very quick. This all ows us to explore a wide \nspace of parameters as numerical examples of application of this theory . In particular, we investigate \nthe joint effect of microwave eddy current shielding and magnetisation pinning at the ferromagnetic \nfilm surfaces on the shape o f the stripline FMR response of the film.  \n \n \nI.  INTRODUCTION  \nStripline broadband ferromagnetic resonance (FMR) response has been receiving a significant \nattention in recent years because of its potential for characterisation of magnetic thin films and \nnanostr uctures  [1 - 3]. Also, the stripline geometry is important for various applications of magnetization \ndynamics, such as sensing fields, particles and substances [4 - 6] and in microwave spintronics [7].  \nThe geometry of a stripline ferromagnetic resonance exper iment setup usually contains a \nmicroscopic coplanar (CPW) or microstrip line (MSL) through which a microwave current flows \n(direction z  in Fig. 1). In this work we will be focusing on MSL, and Fig. 1 reflects this geometry . A \nferromagnetic film which is to  be characterized, sits on top of the microstrip line. We will be dealing \nwith metallic films; therefore an insulating spacer separates the film from the microstrip in order to \navoid an electrical contact between them. A static magnetic field H  is applied along the microstrip line. \nPrecession of magnetization in the material is driven by the Oersted field of a microwave current \nflowing through the microstrip line. On resonance, the amplitude of precession increases sharply which \nis seen as an increase in th e microstrip line transmission losses or a decrease in the microstrip line \ntransmission coefficient S21 [1].  \nA number of important peculiarities of the stripline FMR response have been recently discovered. \nFor instance, it has been shown that the linewidth  of the FMR peaks for the samples as measured with \nthis method can be broadened by excitation of travelling spin waves [8] and over - coupling to the \nprobing stripline (“radiation losses”) [9].  \nAlso, it has been demonstrated both experimentally [10 - 13] and theoretically [14 - 16] that the  2  geometry of Fig. 1 breaks the symmetry of the microwave magnetic field incidence on the sample \nsurface. This leads to excitation of microwave eddy currents in a sample under test if it is conducting. \nConsequences of this are especially important for samples with high conductivity of metals -  see Fig. \n21 in [1] for details of the microwave magnetic field configuration in this case.  \nUltimately, the eddy currents lead to strong excitation of standing spin wave modes (SSWM) for  \nt hose ferromagnetic - film samples, for which one would expect much smaller or even vanishing SSWM \namplitudes in the conditions of the conventional cavity FMR [10 - 11].  \nThe first theoretical paper on this subject treated the stripline FMR response only in the limit of a \nvery large microstrip width [15]. Later on, it was found that the width of the microstrip line has a large \neffect on the response [16,17]. In particular, it was shown that the decrease in the microstrip width \ndecreases the FMR peak amplitudes co rresponding to excitation of the standing spin wave modes [16].  \nHowever, the theoretical approaches from those publications have significant drawbacks which did \nnot allow exploring the impact of the finite width of the stripline on the entire space of par ameters for \nthe stripline FMR experiment. The numerical model constructed in [16] is just very slow . The analytical \nsolution from [17] delivers quantitative results almost instantly; however, it was obtained in the \nexchange - free approximation, hence it all ows simulation of only the fundamental FMR peak.  \nIn this work we fill the gap and report an analytical solution for the dipole - exchange case and carry \nout a number of calculations by using the derived model. The obtained results agree well with the fully \nnumerical model from [16]. Furthermore, since a numerical code we developed based on this analytical \nsolution is very fast, we are now able to explore a range of details of the broadband FMR experiment \nwhich have not been studied before. These are the effe cts of the finite width of the stripline and of the \nspacer thickness on the amplitudes of the response of the standing - spin wave modes. Also, we are now \nable to consider radiation losses for the standing wave excitations and the effect of surface anisotrop ies \n(surface spin pinning) on these peaks.  \nThe paper is organized as follows. In Section II we construct the analytical model. In Section III it \nis used to obtain a number of quantitative results which are discussed in detail in that section.  Section \nIV c ontains conclusions.  \n \nII.  Theory  \nWe consider a metallic ferromagnetic film of thickness L  which is homogeneous throughout its volume, \nbut may possess perpendicular surface anisotropy at one or both surfaces. In the model, the \nperpendicular anisotropy is intro duced as surface magnetization pinning [18]. We treat the film as \ninfinitely long along both in - plane axes -  x  and z. The external magnetic field H  is applied in the \npositive z  direction. The film sits on top of a dielectric spacer of a thickness s  separat ing the film from a \nmicrostrip of a width w . We neglect the fact that real microstrips have a small (with respect to w ), but \nfinite thickness and treat the microstrip as infinitely thin in the direction y . The microstrip is supported \nby a dielectric substr ate of thickness d  whose second surface is uniformly metalized. Although the \nmagnetostatic approximation utilized below does not need specifying the dielectric constant for the \nsubstrate, we implicitly assume a specific value for it. It is one which result s in a 50   of characteristic \nimpedance for the microstrip line in the absence of a film on its top.  \nOne standard approach to calculation of the complex impedance for the microstrip lines loaded by \nferromagnetic films is by exploiting the translation invar iance in the z - direction [19]. This corresponds \nto a quasi - static approach for description of microwave transmission lines and results in a \ntwo - dimensional problem which we will be dealing with below . The actual form of the electromagnetic \nfield dependence  on the z  co - ordinate –  in the form of an electromagnetic wave travelling along the \nstripline -  will be accounted for while employing the obtained expression for the complex impedance \nfor calculations of the transmission coefficient S21.   3  In order to descri be the magnetization dynamics in the film we employ the linearized \nLandau - Lifschitz equation (LLE)  \n ( ) eff t        m m H M h ,                               (1)  \nwhere  0 z H  H u ,  0 z M  M u  , and u z  is a u nit vector in the direction z.  In Eq.(1), the dynamic \nmagnetization m has only two components :     , x y m m  m . They are perpendicular to the static \nmagnetization M  whose magnitude 0 M is equal to the s aturation magnetization of the film \n0 S M M   . The dynamic effective field  eff h  consists of two parts: the effective exchange field ex h  is \ngiven by  \n 2 2 \n2 2 ( ) ex x y        h m   ,                               (2)   \nand the dynamic magnetic field h  being solution of  Maxwell equations in the electric - bias free \napproximation  \n    h e  ,                                            ( 3)   \n  h m ,                                          (4)  \n 0 ( ) i     e h m  .                                   (5)   \nHere e  is the microwave electric field,   is electric conductivity,  0  is the permittivi ty of vacuum and   \nis the frequency of the microwave driving field. Dynamic magnetic field h  has two components  \n( , ) x y h h  h  which are perpendicular to the direction along the stripline. Note that the term containing \nelectric permi ttivity is missing on the right hand side of Eq. (3); this is consistent with the standard \nmagnetostatic approximation for ferromagnetic materials [20]. Furthermore, for metals at microwave \nfrequencies, the contribution of the electric bias field to Eq.(3)  is negligible with respect to the \nconductivity one [21,15,17].  \nEven in two dimensions, previous calculations  faced serious difficulties when trying to employ \nreal - space methods [1,14] because of the incompatibility of the length scales for L , w  and d . By  \nexploiting the translational symmetry in the x - direction, that is, by applying the spatial Fourier \ntransformation to both sides of Eqs. (1 - 5) we can significantly simplify the problem. This is because the \nmicrostrip has a infinitely small thickness in the  y - direction and hence can be treated as a boundary \ncondition involving the surface current density [19,22,16,17]. The surface current and its distribution \nalong x  are assumed to be given. Alternatively , it can be calculated self - consistently [17,19], but the \nlatter is out of scope of the present paper.  \nOne more important advantage of the Fourier - space approach is that simple analytical solutions \nexist for the areas above and below the film [16,17]. This allows one to exclude those areas from \nconsideration and consider only the dynamic equations for the film. Exclusion of those areas produces \nspecific boundary conditions for the electromagnetic fields on the film surfaces [17].        \nAccording to the Fourier space approach, we have  \n, exp( ) i t ikx   m h  .                       (6)  \nSubstituting Eq. (6) into Eqs. (3 - 5), we have  \n 0 ( ) z y y ke h m    ,                                    (7)   4   x \ny z h ikh e y      ,                                       (8)  \n y y \nx x h m ikh ikm y y         ,                               (9)    \n 0 (h m ) z \nx x e i y      .                                     (10)  \nIt is easy to verify that, similar to Ref. [21] the ansatz   , qy e h m   solves the sys tem of equations \n(1), (7 - 10). Eqs. (7 - 10) then reduce to  \n   2 2 2 0 x y y kqh iK h i k K m     ,                           (11)  \n 0 x y y x ikh qh qm ikm      ,                               (12)  \nwhere  2 2 \n0 K k i    ,  \nand Eq.( 1) takes the form as follows:  \n   2 2 H \nx x y \nM M h k q m i m                 ,                      (14)  \n   2 2 H \ny y x \nM M h k q m i m                 ,                        (15)  \n where ( ) H H i H      ,  M S M    , and / G H       is the magnetic loss parameter which scales as \nthe Gilbert magnetic damping constant G  .  The magnetic losses have been introduced into Eqs. \n(14,15) phenomenologically, as it is known that the linearized Landau - Lifshitz - Gilb ert Equation \nreduces to the linearised Landau - Lifshitz Equation (1) with a complex value of the applied field given \nby the expression above [20].  Equations (11 - 15) form a homogeneous system of linear algebraic \nequations. On elimination of the h x  and h y  va riables, the system reduces to two equations (A1) for m x  \nand m y  shown in Appendix.  \n Equating the determinant of the system (A1) to zero produces a characteristic equation for the \nproblem.  The above - mentioned approach of first eliminating h x  and h y  from t he system has an \nimportant advantage –  it results in a compact characteristic equation which is bi - cubic with respect to q :  \n3 2 \n0 0 aQ bQ cQ d     ,                            (16)    \nwhere Q = q 2  and the coefficients a , b , c  and d 0  are given in Appendix (E qs.(A2)). The sixth order for the \ncharacteristic equation is in agreement with the earlier treatment of the dipole - exchange spectrum [21] \nand, as we will see below, with the number of available boundary conditions. 1    \nThe cubic equation (16) allows an anal ytic solution.  For completeness, it is also given in the \nappendix, although this is a textbook result. The three roots of (16) are Q 1 , Q 2  and Q 3  (see Eqs.(A3)). \n                                                             \n1 D irect evaluation of the determinant of ( 11 - 14 ) results in an equation of 8 th  order with respect to q , but as the method of elimination of the \nh x  and h y  variab les demonstrates, two of eight roots in total of the 8 th  order equation are meaningless.      5  Accordingly, the 6 roots of the characteristic equation read: 1 1 q Q  , 2 2 q Q  , 3 3 q Q  , \n4 1 q Q  , 5 2 q Q  , 6 3 q Q  . Note that since the coefficients of Eq.(16) are complex numbers, the \nroots are also complex.    \nThe presence of the six roots indicate s that the complete solution for the two components of m  can \nbe expressed in the following form  \n  6 \n1 exp y \ny i i \ni m M q y \n   ,                                    (17)  \n  6 \n1 exp x \nx i i \ni m M q y \n    .                                (18)  \n     In order to determine the co efficients i \nx M  (or i \ny M ), we need six boundary conditions at the film \nsurfaces. Furthermore, in order to obtain a nontrivial solution, at least one of these boundary \nconditions should be inhomogeneous.  \n     The first set of available boundary conditions is the exchange boundary conditions for the \noriginal Landau - Lifshitz equation. They apply to the magnetization vector [18]  \n0 p d y     m m  .                                    (19)  \nWhere d p  is the s urface magnetization pinning constant. The sign in front of p d is nega tive for the \nsurface 0 y  while it is posi tive for the surface y L  . The pinning constant is  non - vanishing if surface \nanisotropies are present at the film surfaces. For instance, in the case of a perpendicular uni - axial \nsurface anisotropy which will be considered in Section III, the m y  component can be pinned but the m x  \ncomponent remains free ( 0, 0 y x \np p d d   ). In the case of an in - plane uni - directional in - plane anisotropy \n(exchange bias), both pinning constants are non - vanishing and depend on the direction of the exchange \nbias field with respect to H [23].   \nAs m  has two components -   x m  and  y m , and this exchange boundary condition applies to both \nsurfaces of the film and to the two m - vector components separately, altogether four independent \nboundary conditions are avail able.  \n     Two more boundary conditions are still needed. These are electromagnetic boundary conditions \nwhich follow from Maxwell Equations. As shown in [17], for the geometry of Fig. 1, they can be cast \nin the form which involves field components inside  the film only . For the surface y L   the respective \nboundary condition reads  \n  0 y y x k h m ih k     .                                      (20)  \nHere we would like to recall that we seek solution of the problem in the Fourier space, ther efore the \nfield and magnetization vector components entering Eq.(20) are actually spatial Fourier components of \nthese quantities.   6  For the surface 0 y  the boundary condition is inhomogeneous because of the presence of a \nmicro wave current I  flowing through the microstrip in the direction z . The linear density of this current \nj ( x ) is assumed to be given. Its spatial Fourier transform reads: 1 ( )exp( ) 2 k j j x ikx dx   \n  .  \nThe inhomogeneous boundary condition for y =0 reads   \n| | sinh(| | ) | | ( )coth(| |( )) sinh(| |( )) y y x k k k d k h m k d s i h i j k k d s k      .                          (21)  \nSubstitution of the solution (17 - 18) together with the relations between the components of m  and h  \n(Eqs. (A4) in Appendix) into the boundary conditions results in a system of 6 algebraic equations \n(Eqs.(A5) in Appendix). The system is non - homogeneous, as one of the equations has a non - vanishing \nright - hand side which follows from the right - hand - side of Eq.(21).  \n   In our work, we solve the system of equations (A5) using numerical methods of linear algebra. \nThe so lution is obtained for the q - values which solve the characteristic equation (16). As the order of \nthe vector - matrix equation (A6) is just 6, the numerical solution is instantaneous, while run on a \npersonal computer.  \nThe obtained numerical solution for m  is  employed to calculate the linear impedance Z r  of the \nmicrostrip loaded by the film, since the latter quantity is an indicator of the microwave magnetic \nabsorption by the film  [15]. The linear impedance can be defined as follows:  \nr U Z I  ,                                              (22)  \nwhere  U  is the linear voltage along the microstip which can be defined as the mean value of the total \nelectric field e z ( x ) induced at the surface of the strip  \n/2 \n/2 1 ( , ) w \nz \nw U e x y s dx w    .                           (23)  \nKeeping in mind that our solution is in the Fourier space it is useful to express U  in terms of the Fourier \ncomponents zk e  of e z ( x ) ( 1 ( )exp( ) 2 zk z e e x ikx dx   \n  ). From the solution of Eqs.(7 - 10) for the area y <0 \n(characterized by   =0, m =0) it follows that  \n \n0 cosh( ) ( 0) ( ) sinh( ) sinh[ ( )] cosh[ ( )] k x \nzk k d j i h y e y s k d k k d s k d s                ,             (24)    \n \nwhere the Fourier transform of the magnetic field at the film surface y =0 reads  \n6 \n1 ( 0) ( ) x \nx hx i i \ni h y C q M \n    .                                (25)  \nAlso, it is convenient to express U  in t erms of the Fourier components of e z ( x ). This expression reads  \nsin 2 \n2 zk kw \nU e dk kw  \n        .                                  (26)    7  This concludes the solution for Z r . Similar to our previous works [16,17], the integral in (26) is \ncomputed numerically in the pr esent work.  \nThus, the complete process of the numerical simulation consists of three steps. The first one (Step 1) \nis to calculate the roots q 1 to q 6  by using the expressions (A3). Because of the analytical character of this \nsolution, the program code imp lementing it delivers those values instantaneously . The next step is to \nsubstitute these q - values into Eqs. (A5) and solve the inhomogeneous system of linear algebraic \nequations numerically . This gives the six amplitudes i \nx M . This comp utation is practically instantaneous, \nas already stated above. Substitution of the obtained i \nx M values into Eq.(25) and the result into Eq.(24) \nconcludes Step 2. Thus, the output of Step 2 is a value of zk e  for some  given value of k .  \nThe two steps have to be repeated numerous times, as we need zk e  values for a large number of \nFourier wave numbers k , in order to implement Step 3 –  carrying out the numerical integration in \nEq.(26).  \nIn our previous  work [16] up to ten hours of computation were needed to produce the final result -  \na Z r  dependence on the applied field H  for a given microwave frequency . The computation was slow \nbecause the model developed in [16] was based on a numerical solution for S tep 1. Half a minute or so \nwas needed to complete this step. As it is repeated numerous times for different k  and H  values, the \ntotal computation time becomes very large.  \nOn the contrary, the present code is very fast. It has been implemented as MatLAB an d MathCAD \nworksheets for a usual personal computer. In both cases a complete Z r ( H ) dependence is obtained \nwithin 10 to 20 seconds. It takes a couple of more seconds to convert the obtained Z r  values into S21 by \nusing Eqs. (18 - 25) from [16].  \n \nIII.  DISCUSSION  \nSev eral numerical tests were run to validate our theory . First we checked the value of Im( Z r ) off \nthe resonance peaks (e.g. for H =0 or a very large H ) and for  =0 (a non - conducting film). We found \nthat Im( Z r ( H =0,  =0)) is in excellent agreement with an analyt ical formula for the linear inductive \nimpedance for a microstrip line not loaded by a ferromagnetic film (see [16,17] for detail).   \nWe also checked the off - resonance distribution of h x  across the film thickness for a highly \nconducting film. For a 50nm - thi ck film, w =1.5mm and s =0, we found that ( 0, 0) ( 0) x h x y j x      \nand ( 0, ) x h x y L    is practically vanishing, as expected. Furthermore, the h x ( y ) dependence is very \nclose to linear between these two points, as also expected. This behavior is cons istent with a strong \nmicrowave shielding effect present for thin metallic films when microwave power is incident on one \nfilm surface only [1,15].  \nAlso, the shape of Z r (H)  dependence as a function of the film thickness was checked. The results \nare shown in Fig. 2.  The first observation from Fig. 2 is that two resonance peaks are present in Panels \n(b) and (c), although no surface pinning was assumed for either of the film surfaces. This is consistent \nwith the calculation in [16] where it was explained as a co nsequence of a perfect microwave shielding \neffect. In the presence of the microwave eddy currents induced in the film by the incident microwave \nmagnetic field of the microstrip line, the magnetization dynamics is driven by the sum of the spatially \n(quasi) - uniform Oersted field of the microwave current in the microstrip and the spatially \nanti - symmetric Oersted field of the eddy current in the film (see Fig. 21 in [1]). The uniform  8  component is responsible for excitation of the fundamental mode of uniform pre cession (the right - hand \npeak in the panels of Fig. 2) and the anti - symmetric component for excitation of the 1 st  standing spin \nwave mode (1 st  SSW , the left - hand peak in the panels). The latter mode is characterized by an \nanti - symmetric distribution of dyna mic magnetization across the film thickness.  \nThe second observation from Fig. 2 is that an increase in L  leads to an upshift in the 1 st  standing \nspin wave mode peak. This behavior and the field positions of both peaks are consistent with Kittel \nEquation fo r the dipole - exchange modes [24]. This is one more confirmation of the validity of our \ntheory . Noteworthy is the absence of the 2 nd  SSW peak in Fig. 2(b), although Kittel Equation predicts \nthat this peak should be located at about 2000 Oe. Hence, the eddy current effect allows one to probe \nthe 1 st  SSW mode only (unless a film possesses asymmetric surface magnetization pinning).  \nNext, we employ the developed numerical code in order to understand the dependence of the \namplitude of the 1 st  SSW peak on the widt h of the microstrip w . In [16] it has been observed that the \namplitude of this peak increases with an increase in w . Now, given the much higher computation speed \nof the present software, we are able to explore this effect in detail.  \nIn order to perform th is study, the calculations are repeated for a number of w  values. The obtained \nZ r ( H ) and S21( H ) traces are fitted with a complex function  \n1 2 \n0 \n1 1 2 2 ( ) ( ) ( ) A A F H A H H i H H H i H           .                (27)  \nHere H 1(2)  is the extracted resonance field for Mode 1 or 2 respectivel y, 1(2) H   is the linewidth of the \nrespective peak. The quantities A 0 , A 1  and A 2  are complex numbers; they are also extracted from the fits. \nIn the following, we use quantities 1 1 / A H   and 2 2 / A H   to chara cterize the resonance peak \nheights.  \nThe fits show that the shape of the Z r ( H ) dependence is in excellent agreement with Eq.(27). Hence, \nthe complex function (27) is the “natural” dependence of Z r  on H . This implies that the shape of the \nS21( H ) dependence may deviate from the one given by Eq.(27), because S21 depends on Z r  in a \ncomplicated and nonlinear way (see Eq.(31) in [17]). Consequently, values of the parameters of Eq.(27) \nextracted from the fits may be different for Z r and S21 traces. This conclusion  is confirmed by our \nnumerical calculations.  \nFig. 3 displays examples of traces obtained for different values of w . One sees that the 1SSW peak \ngrows in amplitude with an increase in w . In [16] this was explained as a more pronounced microwave \nshielding ef fect for wider microstrips. The last panel of Fig. 3 shows the peak amplitude ratio \n2 2 1 1 / / / r A H A H     as a function of w . (Here the index “1” denotes the fundamental mode and “2” \nthe 1 st  SSW .). From this figure one sees that there is a strong depend ence of the amplitude of the 1 st  \nSSW mode on the microstrip width for w  values below 300 micron. For larger w  values the dependence \nsaturates and the effect becomes the same as in the case of normal incidence of a travelling plane \nelectromagnetic wave on t he surface of a ferromagnetic film [25,26]. One also notices that the traces \nare slightly different for Z r  and S21. As discussed above, this is consequence of the fundamental \ndifference in the shapes of Zr ( H ) and S21( H )  dependences. \nSimilar behavior is ob served as a function of the thickness of the dielectric spacer s  (Fig. 4(e).) From \nthe examples of the raw traces in Fig. 4(a - c), one sees that r  increases with an increase in s . This \nhappens because lifting the film with respect to the stripline makes the  microwave Oersted field of the \ncurrent in the stripline more spatially uniform at the position y  where the film is located. The more \nuniform is the field, the more pronounced the microwave shielding is. Hence, the effect of lifting the  9  film is analogous t o increasing w  while keeping s  constant.  \nThis conclusion implies that the r ( s ) dependence should be significant for small w  values only . \nIndeed, the graphs in Fig. 4 were obtained for w =10 microns. For significantly wider striplines ( w >300 \nmicrons) this d ependence is practically vanishing.  \nFrom the raw traces in Fig. 4 one also notices that the increase in s  is accompanied by a decrease in \nthe heights of both peaks. This behavior is caused by a decrease of the strength of coupling of the \nmagnetization dyn amics to the driving magnetic field [9, 17]).   \nThe change in the coupling strength also leads to a dependence of the resonance peak linewidth on \ns . From Fig. 4(e) one sees that the linewidth decreases with a decrease in s . This is consistent with the \neffe ct of radiation losses [9]. Interestingly, the linewidth broadening due to the radiation losses is more \nsignificant for the 1 st  SSW than for the fundamental mode, with broadening being the same for both \nS21 and Z r . This is actually in qualitative agreement  with the experiment in [9], see Fig. 3 (c) in that \npaper. On the contrary, the fundamental - mode linewidth for small values of s  is noticeably larger for \nS21 than for Z r . The latter fact is in agreement with the exchange - free model from [17].  \nLet us now lo ok at the effect of surface magnetization pinning on magnetization dynamics. Fig. 5 \ndisplays a number of S21( H ) traces calculated for different values of the pinning constant. As before, \nwe assume that the films have large conductivity of metals; this lead s to important peculiarities of the \nstripline FMR responses, as we will show below .   \nLet us first discuss the effect of symmetric pinning –  the situation when ( 0) ( ) x x \np p d y d y L    . As seen \nfrom Fig. 5 (a,b,e,f), the main impact of the symmetric pinning  is a shift of the resonance peaks \nupwards or downwards, depending on the sign of the pinning constant. This is in full agreement with \nKittel Equation for spin wave resonance frequencies [24].  No noticeable difference in r  is seen for \nPanels (e) and (f) w ith respect to Panel (a).  \nOn the contrary, asymmetric pinning Fig.5(c,d,g,h) has a strong effect on r . This is because the \nsymmetry is now doubly broken –  on top of the asymmetry of h x ( y ) originating from the single - side \nincidence of the microwave field, t here is also asymmetry in the film’s magnetic parameters in the \ndirection of the film thickness. This makes the value of r  dependent on the film orientation with respect \nto the microstrip line.  \nThis effect was theoretically found in [1] based on considerat ion of a simple model w =  . In that \nwork, it was suggested that it might be useful for experimental determination of the particular film \nsurface (from the two) at which magnetization pinning is present. Fig. 5 confirms this funding with a \nrigorous calculati on for a realistic value of w  and hence the practical importance of this effect.  \n \nIV.  CONCLUSION  \nIn this work, we constructed a new two - dimensional model for calculation of the stripline \nferromagnetic resonance response of metallic ferromagnetic films. Our mo del works much more \nrapidly and is capable of taking exchange interaction and surface magnetisation pinning into account. \nThe acceleration was achieved by analytically solving the initial system of equations describing the \ndynamics.  \nWe also conducted a nu mber of computations with a numerical model which followed from this \ntheory . The numerical code enabled us to explore a large parameter space for the problem. Our \ncomputations confirmed that microwave shielding by eddy currents induced in a ferromagnetic f ilm \nstrongly affects its stripline FMR response if the film has large conductivity . The eddy currents are \nexcited in the film because of the single - side incidence of the microwave field on the film in the \ngeometry of Fig. 1. They lead to excitation of the 1 st  Standing Spin Wave Mode. The amplitude of the \nrespective peak in the raw FMR traces depends on the width of the microstrip – with an increase in the  10  width the microwave magnetic field incident on the film becomes more spatially uniform which leads \nto mo re efficient shielding by the eddy currents.  \nThe developed model also allowed us to investigate radiation losses for the FMR modes. It was \nfound that they are larger for the 1 st  Standing Spin Wave Mode than for the Fundamental one, in \nqualitative agreemen t with an earlier experiment [9].  \nAlso the effect of the surface magnetization pinning on the response was explored in the \nframework of this more rigorous model. It confirmed a conclusion from a previous work that the \nstripline FMR method can be used to e xtract the degree of symmetry of pinning, and if the pinning \nturns to be asymmetric to identify which of film surfaces is characterized by larger (smaller) pinning.  \n \nAppendix  \nBy substituting Eq. (14) and Eq. (15) into Eqs. (11 - 12), we obtain  \n \n  \n  2 2 4 2 2 2 \n4 2 2 2 2 ( ) ( ) 0 \n( ) ( ) ( ) 0 M x M K M k M y \nM K M k M x M y K q kq m i q q K v m \ni q q K m K q kq m      \n                  \n              ,       (A1)  \nw here  2 \n0 i    , 2 \nk H M k       , and 2 \nK k M K      .  \nEquating the determinant of the matrix of this system of equations to zero results in the characteristic \nequation (16). The coefficients of this  equation are as follows:  \nM a   ,  \n2 2 2 ( ) M K M M b k v              ,  \n2 2 2 2 2 2 2 2 2 (3 2 ) 2 ( )( ) ( ) M M M H H H M H c k k v k v k                        ,  \n2 2 2 2 2 2 2 2 2 2 2 \n0 ( ) 2( )( ) ( ) M M M H M H M H d k k v k k v k v                             .        (A2)  \nThe three roots of the cubic equation (16) are as follows.  \n1 Q D R C     \n* \n2 3 3( ) \n2 2 2 R D i R D Q Q B       ,                         (A3)  \nwhere  3 b B a  , 3 c C a   2 C B R D   , and   1/3 3 \n1 /(2 ) 3 / 2 D D B d a BC     ,  and \n1/2 3 2 2 2 \n3 \n1 2 2 3 \n4 2 4 B d B C Bcd d D C a a a            .  Once we have obtained the Q - values, we are able to express m y  \nin terms of m x  with the hel p of Eq.(A1):  \ny my x m C m   ,                        (A4 - 1)  \nw here  4 2 2 \n2 2 ( ) ( ) \n( ) M K M k M \nmy \nM i q q K \nC K q kq    \n                 . A lso, the following relations between the other \nfield components follow from Eqs.(14) - (15):   11  x hx x h C m  ,                       (A4 - 2)  \nwhere  2 2 2 ( )/( ) hx my C K qkC K q    , and  \ny hy x h C m  ,                       (A4 - 3)  \nwhere  ( 1)/ hy my hx C C ik C q    .  \n \nThe system of equations which follows from the boundary value problem is given by:  \n \n6 \n1 ( ) 0 x i \ni p x \ni q d M \n    ,  \n6 \n1 ( ) exp( ) 0 x i \ni p x i \ni q d M q L \n    ,  \n6 \n1 ( ) ( ) 0 y i \ni p my i x \ni q d C q M \n    ,  \n6 \n1 ( ) ( ) exp( ) 0 y i \ni p my i x i \ni q d C q M q L \n    ,                        (A5)  \n  6 \n1 sgn( )sinh( ) ( ) ( ) coth ( ) sgn( ) ( ) \nsinh ( ) i \nmy i hy i hx i x \ni i k k d C q C q k d s i k C q M \nk d s                    ,  \n  6 \n1 ( ) ( ) sgn( ) ( ) exp 0 i \nmy i hy i hx i i x \ni C q C q i k C q q L M \n         .   \nIn this work this system is solved numerically to obtain the six coefficients x \ni M .  \n \nReferences  \n1.  I.S. Maksymov and M. Kostylev, J. Ph ys. E  69 , 253 (2015).  \n2.  J.M. Shaw, H.T . Nembach, T.J. Silva, C.T. Boone, J. Appl. Phys . 114 , 243906 (2013).  \n3.  C. Bilzer, T. Devolder, P . Crozat, C. Chappert, S. Cardoso, and P .P . Freitas, J. Appl. Phys.  101 , \n074505 (2007).  \n4.  M. Inoue, A. Baryshev, H. Takagi, P . B. Lim, K. Hatafuku, J. Noda and K. Togo, Appl. Phys. Lett . \n98 , 132511 (2011).  \n5.  P .J. Metaxas, M. Sushruth, R.A. Begley, J. Ding, R.C. Woodward, I.S. Maksymov, M. Albert, W . \nWang, H. Fangohr, A.O. Adeyeye and M. Kostylev, Appl. Phys. Lett.  106 , 232406 (2015) .  \n6.  C.S. Chang, M. Kostylev, and E. Ivanov, Appl. Phys. Lett.  102 , 142405 (2013).  \n7.  E. Saitoh, M. Ueda, H. Miyajima and G. Tatara, Appl. Phys. Lett . 88 , 182509 (2006).  \n8.  G. Counil, J. - V . Kim, T. Devolder, C. Chappert, K. Shigeto and Y . Otani, J. Appl. Phys.  95 , 5646 \n(2004).  \n9.  M.A. W . Schoen, J.M. Shaw, H.T. Nembach, M. Weiler, and T. J. Silva, Phys. Rev. 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J.  31 , 875 (1988).  \n20.  A.G. Gurevich, and G.A. Melkov. Magnetization oscillation s and waves . CRC P ress, 1996.  \n21.  T. Wolfram and R.E. De Wames, Phys. Rev. B  4 , 3125 (1971).  \n22.  P .R. Emtage, J. Appl. Phys . 49 , 4475 (1978).  \n23.  R. Magaraggia, K. Kennewell, M. Kostylev, R.L. Stamps, M. Ali, D. Greig, B.J. Hickey, C.H. \nMarrows, Phys.Rev. B  83 ,  054405  ( 2011).  \n24.  D.D. Stancil and A. Prabhakar, Spin Waves: Theory and Application , Springer, Berlin, 2009.  \n25.  M. Kostylev, J. Appl. Phys . 112 , 093901 (2012).  \n26.  M. Kostylev, J. Appl. Phys.  113 , 053908 (2013).  \n \n \n \n \n  13   \n \n \n \n \nFigure 1 . Sketch of t he modeled geometry . (1) The ground plane of the microstrip \nline. (2) Substrate of the microstrip line of thickness d . (3) Infinitely thin strip of \nwidth w  carrying a microwave current in the direction z . (4) Ferromagnetic film of \nthickness L  in the direct ion y and of width w  in the direction x. The static \nmagnetic field H  is applied along z.   14   \n \nFigure 2 . Calculated FMR traces for different ferromagnetic film thicknesses L . (a) \nL =30nm; (b) L =50nm; (c) L =80nm. Parameters of calculat ion: width of \nmicrostrip line w =350  m; thickness of the line substrate d=300  m; spacer \nthickness s =1  m; microwave frequency is 20GHz; saturation magnetization \n4  M s =10000G; conductivity of the film  =4.5 10 6 S/m; magnetic loss parameter \nfor the ferromagnetic film:  H =53.017 Oe (Gilbert dampin g constant  G =0.008), \nexchange constant A =0.8 10  6  erg/cm. Unpinned surface spins  ( 0 x y \np p d d   )  are \nassumed for both film surfaces.   15  \n \n \n \n30nm \n             (d) \n0 1 2 3 4 5 Ratio of pek amplitudes r \n(unitless)\n0.0 0.1 0.2 0.3 0.4 \nCol 2 vs S21 \nCol 2 vs Zr 85nm \n                (e) \nMicrostrip width (mm) 0 1 2 3 4 5 Ratio of pek amplitudes r \n(unitless)\n0.0 0.1 0.2 0.3 0.4 \nS21 \nZ r \n \nFig. 3. Panels (a) to (c): FMR traces for different microstrip line widths w =10, 100 \nand 350  m resp ectively . Other parameters are the same as for Fig.2(a). (d) and (e): \ndependence of the ratio of the peak amplitudes on w  for two film thicknesses \nL =30nm (d) and L =85nm (e). Parameters of calculation for (d) and (e): L =85 \nmicron, 4  M s =10000G ,  =4 .5 10 6 S/m,  exchange constant A =1.3 10  6 erg/cm,  \nGilbert damping constant 0.008,  w =0.1 mm, s =10micron, spacer dielectric \npermittivity: 1; microstrip substrate thickness d =0.1mm and permittivity 10. \nLength of the film in the direction along the microstrip is 5 mm. Freq uency is 22 \nGHz.   16  \n \n \n \nSpacer thickness (  m) 0 100 200 300 400 500 600 Peak amplitude ratio\n(unitless)\n0.0 0.1 0.2 0.3 0.4 \nZ r \nS21 \nSpacer thickness (  m) 0 100 200 300 400 500 600 Peak linewdth \n(Oe)\n28 32 36 40 44 48 \nfundamental mode, Z r \nfundamental mode, S21 \n1st exchange mode, S21 \n1st exchange mode, Z r (d) \n(e) \n \nFig. 4. Panels (a) to (c) are plotted for spacers of thickness s =10, 100 and 200 \nmic rons respectively . The microstrip line width is 10 micron and other parameters \nare same as in Fig.2(a). (d): dependence of the peak amplitude rati o on s . (e) Peak \nlinewidth dependence on s . Parameters of calculation for (d) and (e): L =85 nm, \nw =10  m, frequency is 10 GHz. The remainder of parameters is the same as for \nFig. 3(d - e).   17  \n \nFigure 5. Examples of FMR traces for different surface magnetization pinning \nconditions. (a) and (e): magnetisation is unpinned 0 y \np d   at both film surfaces. \n(b) and (f): it is equally pinned at both surfaces. The pinning constant 8 1 10 m y \np d    \nfor (b) and 8 1 10 m y \np d   for (f). (c) and (g): single - sided pinning. 0 y \np d   for y = L  \nand 8 1 10 m y \np d    (c) or 8 1 10 m y \np d    (g) for y =0.  (d) and (h): the same, but \nnow 0 y \np d   for y =0, and 8 1 10 m y \np d    (d) or 8 1 10 m y \np d    (h) for y = L .  0 x \np d   \nfor all panels. Width of the microstrip line w =350  m, thickness of the film \nL =50nm; thickness of the spacer s =0; microwave frequency is 19 GHz; saturation \nmagnetization 4  M s =17900G; conductivity of the film  =1.8 10 7 S/m;  Gilbert \ndamping constant is 0.008 . Note that that (a) and ( e) show the same plot; the plot \nis repeated to facilitate its comparison with the other plots.  \n "    },    {        "title": "2308.02336v1.Anisotropy_of_the_spin_Hall_effect_in_a_Dirac_ferromagnet.pdf",        "content": "Anisotropy of the spin Hall effect in a Dirac ferromagnet\nGuanxiong Qu,1, 2Masamitsu Hayashi,1, 3Masao Ogata,1, 4and Junji Fujimoto1\n1Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n2RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan.\n3Trans-scale Quantum Science Institute (TSQSI),\nThe University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n4Trans-scale quantum science institute (TSQSI), The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan\n(Dated: August 7, 2023)\nWe study the intrinsic spin Hall effect of a Dirac Hamiltonian system with ferromagnetic exchange\ncoupling, a minimal model combining relativistic spin-orbit interaction and ferromagnetism. The\nenergy bands of the Dirac Hamiltonian are split after introducing a Stoner-type ferromagnetic or-\ndering which breaks the spherical symmetry of pristine Dirac model. The totally antisymmetric spin\nHall conductivity (SHC) tensor becomes axially anisotropic along the direction of external electric\nfield. Interestingly, the anisotropy does not vanish in the asymptotic limit of zero magnetization. We\nshow that the ferromagnetic ordering breaks the spin degeneracy of the eigenfunctions and modifies\nthe selection rules of the interband transitions for the intrinsic spin Hall effect. The difference in\nthe selection rule between the pristine and the ferromagnetic Dirac phases causes the anisotropy\nof the SHC, resulting in a discontinuity of the SHC as the magnetization, directed orthogonal to\nthe electric field, is reduced to zero in the ferromagnetic Dirac phase and enters the pristine Dirac\nphase.\nI. INTRODUCTION\nThe spin Hall effect (SHE) allows generation of a spin\ncurrent transverse to the external electric field within ma-\nterials with large spin-orbit coupling (SOC)[1]. It is a\nstandard method of spin current generation in the field\nof spintronics. The effect has been studied with strong\ninterest in theoretical and experimental point of views [2–\n5]. In the early stage of research, substantial effort was\nput in understanding the SHE in paramagnetic materials\nwith large SOC, e.g., heavy metals [5–7]. Recent studies\nhave expanded the system to include ferromagnetic and\nantiferromagnetic materials. First principles calculations\n[8, 9] have shown that the SHE in ferromagnets displays\nthe interplay of ferromagnetic ordering and SOC. How-\never, a general model that can account for such interplay\nhas not been established.\nOne of the simplest models that includes both ferro-\nmagnetism and SOC is the Dirac ferromagnet [10]. The\nSOC, a relativistic effect, naturally exists in the Dirac\nequation [11]. Despite its origin in relativistic particle\nphysics, the Dirac equation is employed as a low energy\neffective model in condensed matter physics, partly be-\ncause of its diverse mathematical forms [12]. For ex-\nample, bismuth [13, 14] and the bulk states of three-\ndimensional topological insulators [15–17] are effectively\ndescribed by the Dirac Hamiltonian based on k·pper-\nturbation theory. Particularly, the Dirac Hamiltonian\nis equivalent with an isotropic Wolff Hamiltonian [13],\nwhich effectively describes the low-energy states of semi-\nconductors and semimetals with strong SOC and small\ngaps. Various transport properties that arise from such\na unique Hamiltonian have been discussed [10, 18–20].\nFerromagnetic ordering can be incorporated into the\nDirac equation through a Stoner-type mean field. Along\nthis line, historical proposals were made: (i) a ferromag-netic order parameter having opposite signs between the\npositive and the negative energy states introduced by\nMacDonald and Vosko [21]; (ii) an order parameter hav-\ning the same sign in both states proposed by Ramana\nand Rajagopal [22]. It is known that ferromagnetic order\ncan be induced in topological insulators by doping mag-\nnetic impurities [15, 23, 24]. A recent study also showed\nthe possibility of introducing ferromagnetism into a Dirac\nsemimetal by a proximity effect [25].\nIt is well known that the two-dimensional Rashba-type\nHamiltonian can be regarded as a minimal model to treat\nthe anomalous Hall effect (AHE) in ferromagnets [26–\n28]. However, in the Rashba Hamiltonian, the form of\nSOC breaks the mirror symmetry and hence it cannot be\nused to study the coupling between ferromagnetism and\nSOC in mirror symmetric materials. In contrast, the\nDirac Hamiltonian inherently includes the SOC without\nbreaking its spherical symmetry, which is broken if fer-\nromagnetic order is introduced in the system. We, thus,\nconsider a system that can be described by a ferromag-\nnetic Dirac Hamiltonian, i.e., a Dirac ferromagnet as a\nminimal model to study the SHE in the ferromagnets.\nIn this paper, we study the SHE of a Dirac ferro-\nmagnet with particular focus on its anisotropy. The\nintrinsic SHC is calculated using the Kubo formula in\nthe clean limit. The SHC tensor is found to be axially\nanisotropic along the axis parallel to the external electric\nfield. The anisotropy of the SHC tensor is found to be\nmaintained even when the ferromagnetic ordering asymp-\ntotically vanishes. Comparing the eigenfunctions of the\nferromagnet and pristine Dirac electrons through degen-\nerate perturbation theory, the anisotropy of the SHC ten-\nsor and its asymptotic limits are explained by the modi-\nfication of selection rules associated with the set of eigen-\nfunctions in Dirac ferromagnet.\nThe paper is organized in the following order. InarXiv:2308.02336v1  [cond-mat.mes-hall]  4 Aug 20232\nSec. II, the theoretical model is defined, and the calcu-\nlation method is presented.The main result is shown in\nSec. III where the anisotropy of the intrinsic SHC ten-\nsor and its asymptotic behavior in zero magnetization\nlimit are discussed. In Sec. IV, we compare the ferro-\nmagnetic and pristine Dirac electron systems to explain\nthe anisotropy and discontinuity of the SHC. A summary\nfollows in Sec. V.\nII. MODEL DESCRIPTION\nThe model Hamiltonian of a Dirac ferromagnet is de-\nscribed by a 4 ×4 Dirac Hamiltonian with magnetization\nM= (M1, M2, M3) representing the ferromagnetic or-\ndering:\nH0=ℏvkiρ1⊗σi+ ∆ρ3⊗σ0+Miρ3⊗σi,(1)\nwhereℏis the reduced Planck’s constant. ρi(i= 1,2,3)\nandσi(i= 1,2,3) are the Pauli matrices spanning the\nelectron-hole space and the spin space, respectively.1ρ0\nandσ0are 2×2 identity matrices. The first two terms\nof the right hand side of Eq. (1) are known to represent\nthe low energy electronic states of systems with large\nSOC (and a small band gap), which is equivalent with\nan isotropic Wolff Hamiltonian [13]. In the context of\nsuch a low-energy effective Hamiltonian, vand ∆ are\nthe Fermi velocity and the band gap around the Dirac\ncone, respectively. See, for example, ref. [18] in which\nthe Hamiltonian is derived for the electronic states near\nthe L point of bismuth. Note that the k·σterm in Eq. (1)\ncontains the SOC, which originates from the off-diagonal\ncomponents of velocity matrix in the k·pexpansion.\nIn Eq. (1), we express the magnetization Musing the\nspin magnetic moment operator ρ3⊗σi[14, 29], where\nthe field Macts oppositely on electron (positive energy)\nand hole (negative energy) states [21]. Miρ3⊗σiis the\nphysical description of spontaneous magnetization since\ngenerators ρ3⊗σialso couple with the external magnetic\nfieldB[30]. The generators ρ3⊗σican be derived us-\ning the Peierls substitution and the Foldy and Wouthuy-\nsen transformation [31]. Details on the derivation of the\nDirac Hamiltonian [Eq. (1)] is shown in Appendix. A.\nWe do not consider another definition of ferromagnetic\norder ( Siρ0⊗σi) proposed in Ref. [22] in which the field\nSacts on electron and hole states in the same way. Ad-\nditionally, we assume a weak magnetization limit that\nsatisfies M < ∆ so as to avoid the gap closing. Note that\nthe case with M > ∆ corresponds to a Weyl semimetal\nphase [32–34]. In the following, we refer to the system\nrepresented by H0withM= 0 (M̸= 0) as the pris-\ntine (ferromagnetic) Dirac electron system. We choose\nℏ= 1 and renormalize the Fermi velocity with v= 1.\nThe Einstein notation is employed.\n1ρspace is effectively the orbital space when taking the Dirac\nHamiltonian as the effective Hamiltonian of multiorbital systems.A. Rotation matrix\nTaking the unit direction of spontaneous magnetiza-\ntion ( ˆ m≡M/M) as a reference, the wave vector kcan\nbe separated into two parts: k∥represents the parallel\ncomponent of kwith ˆ mwhereas k⊥is the perpendicular\ncomponent, that is,\nk= (k·ˆm) ˆm−(k׈m)׈m≡k∥ˆm+k⊥. (2)\nNote that k2\n⊥=k2−(k·ˆm)2. In the spher-\nical coordinate system, ˆ mis defined as: ˆ m=\n(sinθcosφ,sinθsinφ,cosθ). For simplicity, we define a\nrotation matrix R, which rotates ˆ mto ˆz:Rˆm= ˆz. The\nexplicit form of the rotation matrix Ris\nR=\n(1−cosθ) cos2φ−1 (1−cosθ) sinφcosφsinθcosφ\n(1−cosθ) sinφcosφ(1−cosθ) sin2φ−1 sin θsinφ\nsinθcosφ sinθsinφ cosθ\n.\n(3)\nNote that the inner product of kandMiskiMi=k·\nM=Mk·ˆm=Mk∥. Thus, the energy eigenstates of\nH0read\nEζ,η=ζr\nk2\n∥+ (q\n∆2+k2\n⊥+ηM)2, (4)\nwhere ζ=±1 represents the energy eigenstates [ ζ= 1 for\npositive (electron) and ζ=−1 for negative (hole) energy\nstates], and η=±1 indicates the spin state ( η= 1 for\nspin-up and η=−1 spin-down) .\n/m\n!!,#/mΔΔΔΔ\nΔ\nFIG. 1. (a) Band dispersions and (b) Fermi contours ( µ/∆ =\n3, positive energy branch) of the Dirac ferromagnet. The\nred colored bands correspond to η= +1, and blue colored\nbands correspond to η=−1. The magnetization is set to\nM/∆ = 0 .4.\nThe band dispersion of pristine Dirac Hamiltonian ( H0\nwithM= 0) is spherically symmetric, and the spin space\nis doubly degenerate. In the presence of magnetization,\nthe spin degeneracy is broken, and spin-split bands ap-\npear as shown in Fig. 1(a). Note that the spin polar-\nization of each spin state ( η=±1) is not necessarily\nparallel to the direction of M, due to the inherent SOC\nof Dirac Hamiltonian. In addition, the Fermi surface be-\ncomes axially anisotropic along the direction of magne-\ntization, which can be recognized by the Fermi contours\non the k∥−k⊥plane [see Fig. 1(b)].3\nB. Green’s function\nThe electron Green’s function G(0)(ε) = (ε− H 0)−1is\nrewritten with the generators of ρµ⊗σν[10],\nG(0)(ε) =1\nD(ε)g(0)\nµνρµ⊗σν, (5)\nwhere µ, ν = 0,1,2,3 and the denominator D(ε) =Q\nη,ζ=±1(ε−Eη,ζ). The 16 components of the numer-\nator g(0)\nµνare listed in Table I. Here, we set εk=√\nk2+ ∆2−M2.\nC. Charge and spin velocity operator\nThe velocity operator is obtained directly from the\nDirac Hamiltonian Eq. (1),\nvi=∂H0\n∂ki=ρ1⊗σi, (6)\nwhere the charge current operator is defined as ji=−evi.\ni= 1,2,3(x, y, z ) represents the Cartesian coordinates of\nthe velocity.\nThe spin velocity operator is given by the anticom-\nmutator of the velocity operator and the spin operator\nsk=ρ3⊗σk[18, 19], that is,\nvk\ni=1\n2{vi, sk}=ϵikjρ2⊗σj, (7)\nwhere ϵikjis the Levi-Civita symbol and k=\n1,2,3(x, y, z ) represents the Cartesian coordinates of the\nspin direction. We define the spin current operator as\njk\ni=ℏ\n2vk\ni.\nIII. INTRINSIC SPIN HALL CONDUCTIVITY\nThe intrinsic SHC is calculated through the Kubo for-\nmula. The correlation function between charge current\njjand spin current jk\nireads\nQk\nij(iv) =−1\nVZβ\n0du eivu⟨ˆTjk\ni(u)jj(0)⟩,\n=1\nV βX\nk,ntrh\n˜G(0)(iωn)jk\ni˜G(0)(iωn+iv)jji\n,(8)\nwhere iv→ω+i0 is analytic continuation of the response\nfrequency ωwith Matsubara frequency v= 2mπ/β and\nβis the inverse temperature. ˜G(0)(iωn) is the electron\nGreen’s function with Matsubara frequency wn= (2n+\n1)π/βand chemical potential µ,\n˜G(0)(iωn) =1\nD(iωn+µ)g(0)\nµν(iωn+µ)ρµ⊗σν.(9)The intrinsic SHC is obtained by taking the static limit\nof the correlation function Qk\nij(iv),\nσk\nij= lim\nv→0Qk\nij(iv)−Qk\nij(0)\n−v. (10)\nAssuming zero temperature ( β→ ∞ ) and evaluating\nthe Matsubara summation, the intrinsic SHC can be sep-\narated into two parts σk\nij=σk,(1)\nij+σk,(2)\nijwith\nσk,(1)\nij=e\n8πVX\nktrh\n−2GA(ε)vk\niGR(ε)vj\n+GR(ε)vk\niGR(ε)vj+GA(ε)vk\niGA(ε)vji\f\f\f\nε=µ,\n(11)\nσk,(2)\nij=e\n8πVX\nkZµ\n−∞dε trh\nGR(ε)vk\ni∂εGR(ε)vj\n−∂εGR(ε)vk\niGR(ε)vj−\u0010\nR↔A\u0011i\n, (12)\nwhere GR,A(ε) = G(0)(ε±iγ) are the retarded and\nadvanced Green’s function with a damping constant γ.\nThe detailed structure of the self-energy (damping con-\nstant) in the Dirac Hamiltonian has been considered us-\ning short-range impurities [10, 19, 20]. Here we focus on\nthe intrinsic contribution to the SHE where we first in-\ntroduce a damping constant γand later take it to the\nclean limit γ→0. The extrinsic contribution can be ob-\ntained by extending the calculation on self-energy γor\nby employing semiclassical Boltzmann equation [35] to\ninclude the effect of disorder. The intrinsic contribution\nis dissipationless and therefore it cannot be treated by the\nBoltzmann equation approach. In the zero temperature\nassumption, the integral in σk,(1)\nij reduces to a surface\nterm ε=µ, whereas, σk,(2)\nij contains integration over\nenergy up to the chemical potential µ. Consequently,\nσk,(1)\nij is often referred to as the “Fermi-surface” term,\nandσk,(2)\nijis known as the “Fermi sea” term [36].\nHere, we take one of the nonvanishing tensor compo-\nnents σ3\n21as an example, in which, the direction of mag-\nnetization is rotated arbitrarily. Through integration by\nparts, the Fermi-surface term exactly cancels part of the\nFermi sea term. See Appendix. B.3. The total intrinsic\nSHC, thus, reduces to a compact form,\nσ3\n21=σ3,(1)\n21+σ3,(2)\n21\n=2e\nVX\nkZµ\n−∞dεsgnD′(ζEη)δ(D(ε))∂ε\u0010X(0)(ε)\nD′(ε)\u0011\n,\n(13)\nwhere X(0)(ε) =sαX(0)\nα(ε) and X(0)\nα(ε) is defined as\nX(0)\nα(ε)≡g(0)\n0α(ε)∂εg(0)\n3α(ε)−∂εg(0)\n0α(ε)g(0)\n3α(ε).(14)\nEquation (13) clearly shows that the intrinsic SHE in\nthe Dirac ferromagnet is purely a Fermi sea effect. This4\nTABLE I. Expression of g(0)\nµνin the Green’s function of the Dirac ferromagnet. Indices µ, νare listed in columns and rows.\n(µ, ν) 0 1 2 3\n0 ε(ε2−ε2\nk)−2εM2−2∆εM1 −2∆εM2 −2∆εM3\n1 −2∆Mk ∥ k1(ε2−ε2\nk)−2M1Mk ∥ k2(ε2−ε2\nk)−2M2Mk ∥ k3(ε2−ε2\nk)−2M3Mk ∥\n2 0 −2ε(k2M3−k3M2) −2ε(k3M1−k1M3) −2ε(k1M2−k2M1)\n3 ∆( ε2−ε2\nk) −M1(ε2+ε2\nk) + 2Mk ∥ −M2(ε2+ε2\nk) + 2Mk ∥ −M3(ε2+ε2\nk) + 2Mk ∥\nis analogous to the intrinsic AHE: the anomalous Hall\nconductivity (AHC) is obtained by integrating the Berry\ncurvature of all bands below the Fermi surface, corre-\nsponding to an interband mixing effect independent of\nthe relaxation time [35, 37]. Through a straightforward\ncalculation, σ3\n21can be separated into two terms with\nrespect to ˆ m,\nσ3\n21≡σ3,iso\n21+m2\n1(σ3,ˆm\n21−σ3,iso\n21), (15)\nwhere we define σ3,iso\n21andσ3,ˆm\n21as\nσ3,iso\n21≡ −e∆\n4VX\nk,η,ζζZµ\n−∞dε δ(ε−ζEη)\n× \n1\nEη˜k2\n⊥+η\nM1\nEη˜k⊥−η\nMEη\n˜k3\n⊥!\n,(16)\nσ3,ˆm\n21≡ −e∆\n4VX\nk,η,ζζZµ\n−∞dε δ(ε−ζEη)\n× \n−ηM\nE3η˜k⊥−1\nE3η!\n, (17)\nand˜k⊥=p\nk2\n⊥+ ∆2.\nEquations (15)–(17) are the main results of this pa-\nper. The forms clearly show the symmetry of the in-\ntrinsic SHC in Dirac ferromagnet. σ3,iso\n21only depends\non the strength of magnetization ( M), which is the\nisotropic contribution to σ3\n21. On the contrary, the term\nσ3,ˆm\n21−σ3,iso\n21depends on the strength and the direction\nof the magnetization. The latter is the source of an\nanisotropic contribution to σ3\n21. Surprisingly, the intrin-\nsic SHC is anisotropic along the xaxis, which is parallel\nto the external electric-field ˆE(charge current). Note\nthat we take σ3\n21as an example where the external elec-\ntric field is applied along the xaxis and the polarization\nof spin current is parallel to the zaxis. σ3,iso\n21andσ3,ˆm\n21\nboth contain εintegrals with an integration limit that\napproaches minus infinity ( ε→ −∞ ). One needs to in-\ntroduce a proper energy cutoff (Λ ε) to avoid the ultravio-\nlet divergence in the effective Dirac Hamiltonian [38, 39].\nSee Appendix B.5 for the details.\nWe first present the results when the Fermi level lies\nwithin gap of the Dirac cone ( µ= 0). The SHC reads\nσ3\n21=−e∆\n4π2\u0014\nm2\n1(ln2Λε\n∆−1)−M2\n6∆2(1−4m2\n1)\u0015\n+O(˜M4).\n(18)AsMapproaches zero, we may drop the second-order\ntermO(˜M2). Interestingly, the zeroth-order term, inde-\npendent of M, is dependent on the direction of magneti-\nzation, i.e., m1. Consequently, the value of SHC differs\nwhen Mis reduced to zero from the ferromagnetic state\nalong different directions. For example, if the magneti-\nzation is oriented along the xaxis ( m1= 1), σ3\n21takes\na nonzero value when Mis reduced to zero, whereas σ3\n21\nvanishes with M→0 when the magnetization is or-\nthogonal to the xaxis ( m1= 0). Note that σ3\n21takes\na nonzero value for the pristine Dirac phase ( M= 0)\n[18, 19]. Thus, the M→0 limit of the ferromagnetic\nDirac phase does not match that of the pristine Dirac\nphase when the magnetization of the former is orthogo-\nnal to the xaxis.\n-0.3-0.2-0.100.10.2\n(a)\n-4 -2 0 2 4-9-8-7-6-5\n(b)\nFIG. 2. Chemical potential dependence of intrinsic SHC\ncontributions: (a) σ3,iso\n21and (b) σ3,ˆm\n21. The strength of mag-\nnetization ˜Mis set to 0 .1,0.5,0.8. The energy cutoff is set to\nΛε= 100. The conductivity is in the unit of σ0\ns=e∆/8π2.\nThe chemical potential is normalized by ∆, i.e., ˜ µ≡µ/∆.\nIn Fig. 2, we present the chemical potential dependence\nof the isotropic contribution σ3,iso\n21and the anisotropic\ncontribution σ3,ˆm\n21. The intrinsic SHC is even with re-\nspect to µ. This is in contrast to the intrinsic AHC in\nthe Dirac ferromagnet which is known to be odd with\nµ[10]. Both σ3,iso\n21andσ3,ˆm\n21have plateaus in the band\ngap (|µ/∆|≲1). The width of the plateau linearly re-5\nduces with increasing M. The sign of σ3,iso\n21andσ3,ˆm\n21\nat the plateau is opposite. When the chemical potential\nis placed near the edge of the band gap, σ3,iso\n21increases\ncompared to that at the plateau. σ3,iso\n21takes a maximum\natµ= ∆ for all strengths of M. A further increase in |µ|\ncauses σ3,iso\n21to decay and change its sign. In contrast,\nσ3,ˆm\n21monotonically decreases with increasing |µ|. Over-\nall, the magnitude of σ3,ˆm\n21is significantly larger than that\nofσ3,iso\n21. Thus, the intrinsic SHC of the Dirac ferromag-\nnet is dominated by the ˆ m-dependent term ( σ3,ˆm\n21). In\nthe following, we discuss the anisotropy of σ3\n21in detail.\nA. The anisotropy of intrinsic SHC\nσ3\n21is calculated and plotted as a function of magneti-\nzation direction in Fig. 3 where the strength of the mag-\nnetization and the position of the chemical potential are\nvaried. |σ3\n21|takes a maximum when M∥ˆx(m1=±1)\nand is nearly zero when M⊥ˆx(m1= 0). The SHC,\nthus, shows a strong uniaxial anisotropy where its sym-\nmetry axis is aligned along the direction of the electric\nfield (charge current). The anisotropy is not strongly in-\nfluenced by the position of the chemical potential and the\nstrength of magnetization. For the latter, σ3\n21is indepen-\ndent of Mwhen the chemical potential is inside the band\ngap ( µ= 0), see Fig. 3(a). Strikingly, the anisotropy of\nσ3\n21does not vanish when Masymptotically approaches\nzero. That is, σ3\n21takes a nonzero value when Mis re-\nduced to zero along the direction of external electrical-\nfield ˆE, whereas, σ3\n21vanishes when Mapproaches zero\nwith its direction set perpendicular to ˆE[see Fig. 4]. In\ncomparison, the AHE in the Dirac ferromagnet asymp-\ntotically vanishes with the strength of magnetization [10]\nregardless of the magnetization direction. These results,\nthus, suggest that the Dirac ferromagnet model is not\nsmoothly connected with the pristine Dirac model when\nferromagnetic ordering vanishes asymptotically. This dis-\ncrepancy is discussed in the following section.\nIV. PRISTINE VS FERROMAGNETIC DIRAC\nELECTRON SYSTEMS\nTo account for the asymptotic behavior of the intrinsic\nSHC in the Dirac ferromagnet, we compare the Hamilto-\nnians of ferromagnetic and pristine Dirac systems. The\nformer is given in Eq. (1), whereas, the latter is defined\nby\nHD=ℏvkiρ1⊗σi+ ∆ρ3⊗σ0. (19)\nThe eigenfunctions read\nΨD\nζ,η=1√\n2\nq\n1 +ζ∆\n˜kζk·σ\n|k|χηq\n1−ζ∆\n˜kχη\n, (20)\nFIG. 3. The angular dependence ( m1) of intrinsic SHC σ3\n21\nfor various chemical potentials: (a) ˜ µ≡µ/∆ = 0, (b) ˜ µ= 1,\nand (c) ˜ µ= 2. The black solid line (*) shows the results\nfor the pristine Dirac Hamiltonian [29]. The strength of the\nmagnetization ˜M≡M/∆ is set to 0 .1,0.5,0.8. The energy\ncutoff is set to Λ ε= 100. The conductivity is in the unit of\nσ0\ns=e∆/8π2.\nwhere ˜k=√\n∆2+k2and spinor χηis defined as χ+1=\n(1 0)T,χ−1= (0 1)T. Again, we choose ℏ=v= 1 for\nsimplicity.\nFor the Dirac ferromagnet, here we use a specific mag-\nnetization direction M= (0,0, M) for simplicity. The\nspecific direction of magnetization can be rotated to arbi-\ntrary direction by a global unitary transformation with-\nout loss of generality (see Appendix C for the details).\nThe ferromagnetic Dirac Hamiltonian [Eq. (1)] is, then,\nsimplified to\nHF=HD+H′(M), (21)\nwhere the magnetization-dependent part of the Hamilto-\nnian is H′(M) =Mρ3⊗σ3. The corresponding eigen-6\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-10-8-6-4-202\nFIG. 4. The magnetization strength dependence ˜Mof intrin-\nsic SHC σ3\n21for various chemical potential ˜ µ’s. The circled\nline groups correspond to the magnetization direction paral-\nlel with or perpendicular to ˆ x. The energy cutoff is set to\nΛε= 100. The conductivity is in the unit of σ0\ns=e∆/8π2.\nfunctions are\nΨF\nζ,η=1\n2\nζe−iϕq\n1 +η∆\n˜k⊥q\n1 +ζηQη\nEη\n−ηq\n1−η∆\n˜k⊥q\n1−ζηQη\nEη\ne−iϕq\n1 +η∆\n˜k⊥q\n1−ζηQη\nEη\nζηq\n1−η∆\n˜k⊥q\n1 +ζηQη\nEη\n, (22)\nwhere k1=k⊥cosϕ,k2=k⊥sinϕ,k3=k∥,Qη=˜k⊥+\nηM, and Eη=q\nk2\n∥+ (˜k⊥+ηM)2. As evident, ΨD\nζ,η\nand ΨF\nζ,ηdo not take the same form in the paramagnetic\nlimit ( M→0). This discrepancy causes the difference in\nσ3\n21for pristine and ferromagnetic Dirac electron systems\nwhen Mfor the latter asymptotically approaches zero\nalong the direction of the external electrical field.\nA. Degenerate perturbation theory\nTo trace the discrepancy between ΨD\nζ,ηand ΨF\nζ,ηin the\nparamagnetic limit, we construct the eigenfunctions of\nthe Dirac ferromagnet from ΨD\nζ,ηusing degenerate pertur-\nbation theory [40]. Here the magnetization H′is treated\nas perturbation.\nFirst, we define the following projection operator that\nmaps the total Hamiltonian to each spin-degenerate en-\nergy subspace ( ζ=±1) as\nPζ=X\nηΨD\nζ,ηΨD†\nζ,η=1\n2 \n(1 +ζ∆\n˜k)I2 ζk·σ\n˜k\nζk·σ\n˜k(1−ζ∆\n˜k)I2!\n.\n(23)\nThe projected Hamiltonian Hζin each energy subspacereads\nHζ=Pζ(HD+H′)Pζ. (24)\nThe unperturbed Hamiltonian HDis naturally diagonal-\nized in each energy subspace Hζ, whereas, the magne-\ntization H′breaks the spin degeneracy. Hence, a new\nbasis set Φ(0)\nζ,ηis required to diagonalize the perturbation\nPζH′Pζto the first order,\nPζˆH′PζΦ(0)\nζ,η=E(1)\nζ,ηΦ(0)\nζ,η, (25)\nwhere E(1)\nζ,η=ζηM˜k⊥/˜kis the first order correction of\nthe eigenenergy and Φ(0)\nζ,ηexplicitly reads\nΦ(0)\nζ,η=1\n2\nζe−iϕq\n1 +η∆\n˜k⊥q\n1 +ζη˜k⊥\n˜k\n−ηq\n1−η∆\n˜k⊥q\n1−ζη˜k⊥\n˜k\ne−iϕq\n1 +η∆\n˜k⊥q\n1−ζη˜k⊥\n˜k\nζηq\n1−η∆\n˜k⊥q\n1 +ζη˜k⊥\n˜k\n(26)\nwhich is the zeroth order ( O(M0)) correction of the eigen-\nfunction. Apparently, Φ(0)\nζ,η’s are the eigenfunctions of\nthe Dirac ferromagnet in the paramagnetic limit: Φ(0)\nζ,η=\nΨF\nζ,η|M→0. Φ(0)\nζ,ηand ΨD\nζ,ηspan the same spin-degenerate\nspace, but Φ(0)\nζ,η, in addition, contains information of the\nperturbation (i.e., the magnetization).\nThe first order correction of the eigenfunction is calcu-\nlated through the new basis,\nΦ(1)\nζ,η=Φ(0)†\nζ,−η¯H′\nζΦ(0)\nζ,η\nE(1)\nζ,η−E(1)\nζ,−ηΦ(0)\nζ,−η+X\nη′Φ(0)†\n−ζ,η′H′Φ(0)\nζ,η\nE(0)\nζ,η′−E(0)\n−ζ,ηΦ(0)\n−ζ,η′\n=−ζMk∥\n2˜k2Φ(0)\n−ζ,η, (27)\nwhere ¯H′\nζ=PζH′P−ζ(Eζ−HD)−1P−ζH′Pζis the second\norder perturbation in the subspace Hζ. Thus, up to the\nfirst order correction, the eigenenergy and eigenfunctions\nare\nEζ,η=ζ˜k+ζηM˜k⊥\n˜k+O(M2), (28)\nΨF\nζ,η= Φ(0)\nζ,η−ζMk∥\n˜kΦ(0)\n−ζ,η+O(M2), (29)\nwhich are fully consistent with the Taylor expansion of\nEζ,ηand ΨF\nζ,ηof the ferromagnetic Dirac Hamiltonian.\nClearly, when the magnetization ( M) asymptotically ap-\nproaches zero, the energy bands become spin degenerate\n[Eq. (28)], but the eigenfunctions return to Φ(0)\nζ,η, which\ndiffers from ΨD\nζ,ηby a specific gauge chosen by the direc-\ntion of magnetization.7\nB. SHC in spectral representation\nNext, we calculate the intrinsic SHC through the\nKubo formula using the three sets of eigenfunctions, i.e.,\nΨD\nζ,η,Φ(0)\nζ,η, and ΨF\nζ,η, to show the effect of different eigen-\nfunctions on the SHC. The Kubo formula for SHC in the\nspectral representation [41] is given by\nσl\nji=−e\n4X\nk,ζ,ηf(εk)Ωl,ζ,η\nji, (30)\nΩl,ζ,η\nji=−X\n(ζ′,η′)̸=(ζ,η)2Imϕ†\nζ,ηvl\njϕζ′,η′ϕ†\nζ′,η′viϕζ,η\n(Eζ,η−Eζ′,η′)2,(31)\nwhere f(εk) is the Fermi-Dirac distribution function and\nΩl,ζ,η\njiare the Berry curvaturelike terms [41, 42] for band\nindices ζ, η, where ( ζ′, η′)̸= (ζ, η) indicates summation\nover all interband transitions. In the following, we sub-\nstitute ΨD, Φ(0), and ΨFintoϕζ,η. Note that we calcu-\nlate different components of the SHC tensor with a fixed\nmagnetization direction: ˆ m∥ˆz.\nFor the eigenfunctions of pristine Dirac Hamiltonian\n(ΨD), we consider all nonvanishing components of the\ntotally antisymmetric SHC tensor. The corresponding\nBerry curvaturelike terms are\nΩD,3,ζ,η\n21 =X\n(ζ′,η′)̸=(ζ,η)∆\n(ζ−ζ′)˜k3\u0012k2\n1+k3\n2\nk2δη,η′+k2\n3\nk2δη,−η′\u0013\n=ζ∆\n2˜k3, (32)\nwhere two transition channels ζ, η→ −ζ,±ηequally con-\ntribute to ΩD,3,ζ,η\n21 due to the spin degeneracy. One may\nverify that ΩD,1,ζ,η\n32 and ΩD,2,ζ,η\n13 have similar expressions,\nwhich suggests that the ζ, η→ − ζ,±ηchannels con-\ntribute equally despite the geometry of SHC.\nFor the eigenfunctions Φ(0), the corresponding Berry\ncurvaturelike terms, denoted as ΩF0,l,ζ,η\nji , are\nΩF0,3,ζ,η\n21 = ΩF0,1,ζ,η\n32 =X\n(ζ′,η′)̸=(ζ,η)∆(η−η′)(ζη+ζ′η′)\n2(ζ−ζ′)2˜k3\n=ζ∆\n2˜k3, (33)\nΩF0,2,ζ,η\n13 =X\n(ζ′,η′)̸=(ζ,η)∆(η+η′)(ζη−ζ′η′)\n2(ζ−ζ′)2˜k3\n=ζ∆\n2˜k3. (34)\nAs evident from the first line of Eq. (33), transitions\nthat conserve spin, i.e., η′=η, vanish in the summa-\ntion. Thus, only spin-flipping transitions ζ, η→ −ζ,−η\ncontribute to ΩF0,3,ζ,η\n21 and ΩF0,1,ζ,η\n32 . In contrast, for\nΩF0,2,ζ,η\n13 , the transitions that flip the spin η′=−ηvan-\nishes as apparent from the second line of Eq. (34). Here,\nthe spin-conserving transitions ζ, η→ − ζ, ηcontributeto ΩF0,2,ζ,η\n13 . As evident, Eqs. (33) and (34) are identical\nto Eq. (32). This is because Φ(0)only differs from ΨDby\na specific gauge chosen by the direction of magnetization.\nFor the eigenfunctions of Dirac ferromagnet (ΨF\nζ,η), the\ncorresponding Berry curvaturelike terms are as follows:\nΩF,3,ζ,η\n21 = ΩF,1,ζ,η\n32\n=X\n(ζ′,η′)̸=(ζ,η)∆(η−η′)(ζηE η′Qη+ζ′η′EηQη′)\n2˜k⊥EηEη′(ζEη−ζ′Eη′)2\n=ζ∆\n2Eη˜k3\n⊥h\n˜k⊥+η\nM(k2\n∥+M2)i\n, (35)\nΩF,2,ζ,η\n13 =X\n(ζ′,η′)̸=(ζ,η)∆(η+η′)(ζηE η′Qη−ζ′η′EηQη′)\n2˜k⊥EηEη′(ζEη−ζ′Eη′)2\n=ζ∆\n2E3ηQη\n˜k⊥. (36)\nOnly spin-flipping transitions ζ, η→ ± ζ,−ηcontribute\nto ΩF,3,ζ,η\n21 and ΩF,1,ζ,η\n32 and the spin-conserving transi-\ntions ζ, η→ − ζ, ηcontribute to ΩF,2,ζ,η\n13 . Here, in the\npresence of the spin gap, ΩF,3,ζ,η\n21 and ΩF,1,ζ,η\n32 have an\nextra spin-flipping channel: ζ, η→ζ,−η. As evident,\nΩF0,2,ζ,η\n13 is also equal to ΩF,2,ζ,η\n13 in the M→0 limit,\nbut ΩF0,3,ζ,η\n21 and ΩF0,1,ζ,η\n32 do not take the same form\nwith ΩF,3,ζ,η\n21 and ΩF,1,ζ,η\n32 in the M→0 limit due to the\nabsence of the transition channel ζ, η→ζ,−ηfor the\nformer (i.e., for the eigenstates Φ(0)). Furthermore, we\nnote that the Berry curvaturelike terms all vanish with\nzero gap (∆ = 0). Thus, gap of Dirac cone ∆ is essential\nfor the nonzero SHE in the Dirac ferromagnet.\nSince we have fixed the magnetization direction along\nthezaxis, ΩF,2,ζ,η\n13 corresponds to the case with ˆ m||ˆE,\nwhereas ΩF,3,ζ,η\n21 and ΩF,1,ζ,η\n32 represent the case with\nˆm⊥ˆE. One can verify that Eq. (35) is exactly the\nsame with the integrand of Eq. (16). Thus, upon inte-\ngrating the Berry curvaturelike terms across the kspace,\nthe SHC of the Dirac ferromagnet with ˆ m⊥ˆEis near\nzero. It turns out that the transition ζ, η→ζ,−ηcom-\npensates most of the contribution from the transition\nζ, η→ − ζ,−η, causing the small SHC with ˆ m⊥ˆE.\nNote that the compensation is exact in the asymptotic\nlimit of zero magnetization.\nThe transition channels that contribute to the Berry\ncurvaturelike terms are summarized in Table II. With\nthe degenerate perturbation theory, one can trace the\norigin of the discrepancy between pristine and ferromag-\nnetic Dirac electron systems in the zero magnetization\nlimit. In the pristine Dirac model, the spin space spanned\nbyηis degenerate, thus the nonvanishing components\nof the SHC tensor share the same transition channels\n(ζ, η→ −ζ,±η). In the Dirac ferromagnet, however, the\nspontaneous magnetization H′(M) breaks the spin de-\ngeneracy. Consequently, each nonvanishing component\nof the SHC tensor possesses distinct transition channels,\neither spin-flipping or spin-conserving channels (see Ta-\nble II). Furthermore, in the presence of the spin gap,8\nTABLE II. Summary of transition channels in Ωl,ζ,η\njifor each\neigenfunction set.\nΨD\nζ,η Φ(0)\nζ,η ΨF\nζ,η\nˆm∥ˆE ζ, η → −ζ,±η ζ, η → −ζ, η ζ, η → −ζ, η\nˆm⊥ˆE ζ, η → −ζ,±η ζ, η → −ζ,−η ζ, η → ±ζ,−η\na unique spin-flipping transition channel ζ, η→ζ,−η\nemerges when ˆ m⊥ˆE, which nearly compensates contri-\nbution from the transition channel ζ, η→ −ζ,−ηon the\nSHC. Thus, the appearance of the spin-flipping transition\nchannel causes the SHC to be negligible when ˆ m⊥ˆE.\nThese results show that the change in the selection rule\nof the state transitions define the SHC and is the cause\nof the anisotropy in the SHC that persists to the M→0\nlimit.\nV. SUMMARY\nTo summarize, we have developed a minimal model\nto study the intrinsic SHE in a ferromagnetic Dirac sys-\ntem. We find the nonvanishing components of the SHC\ntensor is anisotropic. The anisotropy is defined by the\ndirection of magnetization. Specifically, the SHC is fi-\nnite when the magnetization is parallel with the external\nelectric field, whereas, it is near zero when the two are or-\nthogonal. Using the spectral representation of the Kubo\nformula, the nonvanishing components of the SHC ten-\nsor are examined in the context of transition channels.\nWe find the interband transition channels in the pristine\nDirac Hamiltonian system bifurcate into spin-conserved\nor spin-flipping ones in the presence of the exchange field,\nwhich ultimately causes the anisotropy of SHC in the\nDirac ferromagnet. Interestingly, the anisotropy per-\nsists as the magnetization approaches zero. The SHC\nis smoothly connected to the pristine Dirac phase when\nthe magnetization is reduced to zero along the applied\nelectric field. In contrast, a discontinuity in the SHC\nappears at the ferromagnetic Dirac/pristine Dirac bor-\nder when the magnetization is not parallel to the electric\nfield. The discontinuity of the SHC at the ferromagnetic\nDirac/pristine Dirac border is caused by the emergence\nof a spin-flipping transition channel spontaneously ap-\npearing with a spin gap in the Dirac ferromagnet.\nOur paper demonstrates the effect of ferromagnetic\nordering on the intrinsic SHE using a minimal model\ncombining ferromagnetism and SOC. The massive Dirac\nHamiltonian has been employed as an effective Hamil-\ntonian around the L point of semimetallic bismuth. A\nferromagnetic Dirac system can be achieved by magnetic\ndoping or the magnetic proximity effect from a neighbor-\ning layer. In the ferromagnetic Dirac phase, we placed\na limit on the strength of spontaneous magnetization to\nbe smaller than the gap of Dirac cone (i.e., M < ∆). If\nthe magnetization equals the gap, a gap closing occurs,which can be classified as a topological phase transition.\nIn the extreme case where the magnetization is larger\nthan the gap, the Dirac semimetal turns into a magnetic\nWeyl semimetal. Further theoretical study is required to\naddress the SHE in the latter system.\nACKNOWLEDGMENTS\nThis work was supported by Grants-in-Aid for Sci-\nentific Research (B) (Grants No. 21H01034 and No.\nJP18H01162) from the Japan Society for the Promotion\nof Science and by the JST-Mirai Program (Grant No.\nJPMJMI19A1).\nAppendix A: Wolff Hamiltonian\nIn this appendix, we present the derivation of the\nWolff Hamiltonian from the Kohn–Sham Hamiltonian\nand show its equivalence with the pristine Dirac Hamil-\ntonian with an isotropic velocity. We start with a general\nsingle electron Hamiltonian that includes the SOC,\nˆH=ˆp2\n2m+V(r) +ℏ\n4m2c2σ· ∇V(r)׈p. (A1)\nFollowing the k·ptheory, the Hamiltonian can be ex-\npanded around a band extremum k0,e.g., the Lpoint\nof bismuth. Accordingly, the wave function around the\nextremum k0can be constructed using the Bloch basis,\nψ(r) =X\nnZ\ndkcn(k)eik·run,k0, (A2)\nwhere cn(k) is an expansion coefficient and un,k0is a\nfunction periodic in space. The matrix elements of the\nHamiltonian around the band extremum can be rewritten\nas\n⟨n,k|ˆH|n′,k⟩cn′(k) =Encn(k) (A3)\n⟨n,k|ˆH|n′,k⟩=\u0014\nεn,k0+ℏ2k2\n2m\u0015\n+ℏk·pn,n′\nm,\n(A4)\nwhere εn,k0is the eigenenergy of the Bloch state at k0\nand the momentum matrix elements pn,n′are given as\npn,n′=(2π)3\nΩZ\ncelldru∗\nn,k0\u0012\nˆp+ℏ\n4mc2σ× ∇V(r)\u0013\nun′,k0.\n(A5)\nEquation (A5) clearly shows that the SOC generates an\nanomalous velocity term in the momentum matrix ele-\nment. Focusing on the low energy properties, we select\nthe lowest conduction and highest valance bands (both\nspin degenerate, the band gap is 2∆) to construct an ef-\nfective Hamiltonian. Under such circumstance, Eq. (A4)\nreduces to the Wolff Hamiltonian,\nH=ℏkiWj\niρ2⊗σj+ ∆ρ3⊗σ0, (A6)9\nwith the Wolff tensor W1\ni= Im( pi,1,4/m) =\n−Im(pi,3,2/m), W2\ni= Re( pi,1,4/m) =−Re(pi,3,2/m),\nandW3\ni= Im( pi,1,3/m) = Im( pi,4,2/m). The band\nindices 1 ,2 and 3 ,4 are the spin degenerate conduc-\ntion bands and the valance bands, respectively. If the\nWolff tensor is approximately isotropic, i.e., Wj\ni=δijv,\nEq. (A6) is equivalent to Eq. (1) without magnetization.\nWe note that the representation of the velocity operator\n(ρ2⊗σ) in the Wolff Hamiltonian differs from the original\nDirac Hamiltonian up to a unitary transformation.\nThe exchange coupling term in Eq. (1) is obtained us-\ning the spin magnetic moment operator [14]. To derive\nthe spin magnetic moment operator in the Dirac Hamil-\ntonian, we first rewrite the wavevector as the momentum\noperator in the pristine Dirac Hamiltonian and perform\nPeierls substitution ℏk→ˆp→ˆπ=ˆp+e/cA,\nH0=vπiρ1⊗σi+ ∆ρ3⊗σ0. (A7)\nApplying Foldy and Wouthuysen transformation [31]\ngenerated by S=v\n2∆ρ2⊗σiˆπi, the Dirac Hamiltonian can\nbe decoupled into positive and negative energy states,\nH′\n0=eiSH0e−iS\n=H0+i[S,H0] +i2\n2![S,[S,H0]] +···\n= (∆ +v2π2\n2∆)ρ3⊗σ0+ℏev2\n2∆cBiρ3⊗σi+O(v2/∆2)\n(A8)\nwhere the external magnetic field arises from the vector\npotential B=∇×A. The spin magnetic operator ρ3⊗σi\ncouples to external magnetic field oppositely in positive\n(electrons) and negative (holes) energy states.\nAppendix B: Intrinsic SHC in the Kubo formula\nThe intrinsic SHC is obtained by calculating the Fermi\nsurface and Fermi sea terms with zeroth order of damping\nconstant O(γ0).\n1. Fermi surface contribution\nSubstituting ˜GR,Ainto Eq. (11), the Fermi surface\nterm reads\nσk,(1)\nij=e\n8πVX\nkϵikltr(ρµρ2ρλρ1)tr\u0000\nσνσlστσj\u0001\n× \ngR\nµν(ε)gR\nλτ(ε)\nDR(ε)DR(ε)−2gA\nµν(ε)gR\nλτ(ε)\nDA(ε)DR(ε)+gA\nµν(ε)gA\nλτ(ε)\nDA(ε)DA(ε)!\f\f\f\nε=µ.\n(B1)\nThe numerator and denominator of Eq. (B1) can be\nexpanded on the order of γ,\ngR,A(ε) = g(ε)±iγg′(ε) +O(γ2),\nDR,A(ε) = D(ε)±iγD′(ε) +O(γ2). (B2)Note that for the denominators, we have the following\napproximation:\n1\nD2(ε) +γ2(D′(ε))2≃π\n|D′(ε)|γδ(D(ε)). (B3)\nTaking the zeroth order of γ, the Fermi surface term reads\nσ3,(1)\n21 =2e\nVX\nkδ(D(ε))\n|D′(ε)|X(0)(ε)\f\f\f\nε=µ. (B4)\n2. Fermi sea contribution\nSubstituting ˜GR,Ainto Eq. (12), the Fermi sea term\nreads\nσk,(2)\nij=e\n8πVX\nkϵikltr(ρµρ2ρλρ1)tr\u0000\nσνσlστσj\u0001\n×Zµ\n−∞dε\"\ngR\nµν(ε)∂zgR\nλτ(ε)−∂εgR\nµν(ε)gR\nλτ(ε)\n(DR(ε))2−\u0010\nR↔A\u0011#\n.\n(B5)\nFor Eq. (B5), the tensor component σ3,(2)\n21reads\nσ3,(2)\n21=−2e\nπVX\nkZµ\n−∞dεImhXR(ε)\nDR(ε)2i\n. (B6)\nTaking the zeroth order of γ, the integrand in Eq. (B6)\nis approximated as\nImhXR(ε)\nDR(ε)2i\n≈πsgnD′(ζEη)∂εδ(D(ε))X(0)(ε)\nD′(ε).(B7)\nwhere XR(ε) is expanded on the order of γ,\nXR(ε) =X(0)(ε) +iγX′(0)(ε) +O(γ2). (B8)\nThe intrinsic Fermi sea term thus reads\nσ3,(2)\n21=−2e\nVX\nkZµ\n−∞dεsgnD′(ζEη)∂εδ(D(ε))X(0)(ε)\nD′(ε)\n=−2e\nVX\nkh\nsgnD′(ζEη)δ(D(ε))\nD′(ε)X(0)(ε)\f\f\fε=µ\nε=−∞\n−Zµ\n−∞dεsgnD′(ζEη)δ(D(ε))∂ε\u0010X(0)(ε)\nD′(ε)\u0011i\n,\n(B9)\nwhere we apply integration by parts in the last equation.\nNote that the first term in Eq. (B9) with the lower bound-\nary vanishes at ε→ −∞ and its upper boundary term\nexactly cancels the Fermi surface term [Eq. (B4)]. Con-\nsequently, the total intrinsic SHC reduces to a compact\nform shown in Eq. (13).10\n3. Anisotropic SHC\nTo obtain an analytical expression of Eq. (13), we first\nrewrite δ(D(ε)),\nδ(D(ε)) =X\nη,ζ=±1δ(ε−ζEη)\n8ME η˜k⊥, (B10)\nwhere we denote ˜k⊥=p\n∆2+k2\n⊥. Hence, Eq. (13) is\nrewritten as\nσ3\n21=2e\nVX\nk,η,ζZµ\n−∞dεδ(ε−ζEη)\n64M2E2η˜k2\n⊥\n×h\n∂εX(0)(ε)−ζX(0)(ε)(1\nEη+η\nMEη\n˜k⊥)i\n.(B11)\nOwing to δ(ε−ζEη) in the integral, we replace εin\nX(0)(ε) and X′(0)(ε) with ζEη, which gives\nX(0)(ζEη) =−8∆M2h\n(1−m2\n1)Eη2+m2\n1ηM˜k⊥+m2\n1˜k2\n⊥i\n,\nX′(0)(ζEη) =−8∆ζEηh\n2M2+ηM˜k⊥−M2\n1i\n.(B12)\nEquation (B11), thus, reads\nσ3\n21=−e∆\n4VX\nk,η,ζZµ\n−∞dϵδ(ε−ζEη)h\nm2\n1 \n−ηM\nE3η˜k⊥−1\nE3η!\n+ \n1\nEη˜k2\n⊥+η\nM1\nEη˜k⊥−η\nMEη\n˜k3\n⊥!i\n, (B13)\nfrom which we obtain Eqs. (16) and (17) by separating\nthe isotropic- and magnetization-dependent parts.\n4. Calculation of σ3,iso\n21and σ3,ˆm\n21\nσ3,iso\n21andσ3,ˆm\n21both contain εintegrals with an in-\ntegration limit that approaches minus infinity ( ε→\n−∞). One needs to introduce a proper energy cutoff\nto avoid the ultraviolet divergence in the effective Dirac\nHamiltonian[38, 39]. We, therefore, separate the εinte-\ngrals in σ3,iso\n21andσ3,ˆm\n21into a divergent part [ ε∈(−∞,0)]\nand a convergent part [ ε∈(0, µ)],\nσ3,iso\n21=Diso+Ciso, (B14)\nσ3,ˆm\n21=Dˆm+Cˆm, (B15)\nwhere Ciso,CˆmandDiso,Dˆmare the convergent and\ndivergent terms of the isotropic and the ˆ m-dependent\ncontributions, respectively.\nThe divergent terms ( DisoandDˆm) represent the SHC\nwith the chemical potential exactly placed at the center\nof the gap between the positive and the negative energystates ( µ= 0). In other words, it contains contribution\nfrom the entire negative energy states ( ζ=−1, η=±1),\nDiso=e∆\n4VX\nk,η1\nEη˜k2\n⊥+η\nM1\nEη˜k⊥−η\nMEη\n˜k3\n⊥,(B16)\nDˆm=−e∆\n4VX\nk,ηηM\n˜k⊥E3η+1\nE3η. (B17)\nAs the ksummation involves a logarithmic divergence\nin three dimensional reciprocal space, we adopt a cutoff\nenergy Λ ε. Λεis defined such that the following condition\nis satisfied:\nE2\nη=k2\n∥+ (˜k⊥+ηM)2⩽Λ2\nϵ. (B18)\nWith the energy cutoff, the spin-split bands in the\nnegative-energy states are truncated by two different el-\nlipsoids [see Fig. 1(b)]. Setting ˜M=M/∆, we expand\n˜Mup to second order in ˜Mto obtain an analytical ex-\npression for DisoandDˆm(M/∆ is placed back in),\nDiso\nΛε=e∆\n4π2M2\n6∆2+O(˜M4), (B19)\nDˆm\nΛε=−e∆\n4π2\u0012\nln2Λε\n∆−1 +M2\n2∆2\u0013\n+O(˜M4).(B20)\nFor the convergent terms ( CisoandCˆm), thekintegra-\ntion is confined within the area that satisfies the following\ncondition,\nE2\nη=k2\n∥+ (˜k⊥+ηM)2⩽µ2. (B21)\nThe integration range is similar to that of the energy\ncutoff scheme. Again, the energy states are truncated by\ntwo different ellipsoids associated with the spin-up and\nspin-down bands ( η=±1). Here, either the positive- or\nthe negative-energy state ( ζ=±1) is truncated. After\nsome calculations, CisoandCˆmread\nCiso=e∆\n8π2X\nη=±1η|µ|\n2MΘ(µ)Zθβ\nθαdθcos2θ\n(sinθ−ηM\n|µ|)2\n−\u0010\n1−M2\n|µ|2(sinθ−ηM\n|µ|)2\u0011\ncosθtanh−1cosθ\n(B22)\nCˆm=−e∆\n8π2X\nη=±1Θ(µ)(cos θα+ ln tanθα\n2) (B23)\nwhere θα= sin−1∆+ηM\n|µ|,θα=π\n2, and Θ( µ) is defined as\nΘη(µ) =\u001a1,|µ|>∆−ηM\n0, (otherwise)(B24)\nNote that σ3\n21is artificially divided into the convergent\npart and the divergent part. Hence, the convergent part\nis expected to exactly cancel the divergent part when the\nchemical potential approaches negative infinity. This is\nindeed the case. The cancellation between the Cterms\nwith µ= Λ εand the Dterms justifies the energy cutoff\nscheme.11\n5. Cutoff schemes for ultraviolet divergence\nIn the pristine Dirac phase, a momentum cutoff scheme\nis typically employed to deal with the ultraviolet diver-\ngence. In such scheme, the wave-vector kis truncated\nwithin a sphere of radius Λ in the momentum space,\nk2\n∥+k2\n⊥⩽Λ2. (B25)\nNote that, in the Dirac ferromagnet, the Fermi sea of\nspin-split bands ( η=±1) form ellipsoids with axial\nanisotropy along k∥. The ellipticity of the Fermi sea\nasymptotically approaches 1 when |k| ≫ Mand the\nFermi sea of the spin-split bands tend to coincide at\nthe cutoff momentum Λ ≫M. In the momentum cut-\noff scheme, Diso\nΛandDˆm\nΛare approximated in the small\nmagnetization limit as\nDiso\nΛ=−e∆\n4π2\u0012\nln2Λ\n∆−1−M2\n6∆2\u0013\n+O(˜M4),(B26)\nDˆm\nΛ=−e∆\n4π2\u0012\nln2Λ\n∆−1 +M2\n2∆2\u0013\n+O(˜M4).(B27)\nDˆmshows exactly the same result for two cutoff schemes,\nwhereas Disois different for the two. For example, in the\nlimit of M→0,Disovanishes in the energy cutoff scheme\nbut is finite for the momentum cutoff scheme. The differ-\nence is due to the ellipticity of Fermi sea of the negative\nenergy branches, causing a gap between the Fermi sur-\nface of the two spin-split bands [see Fig. (1b)] whose area\nis on the order of M/Λ.Disocontains terms on the order\nofM−1[Eq. (B16)], which return a nonvanishing con-\ntribution to the momentum integration even in the gap\narea. Counterintuitively, the small gap between the spin-\nsplitting Fermi surfaces is significant to the whole Fermi\nsea term and does not vanish in the limit of M→0.\nAdditionally, we compare the convergent part and the\ndivergent part to check the consistency of two cutoff\nschemes. For the momentum cutoff, the convergent part\nand the divergent part are different for the isotropic con-\ntributions CisoandDiso\nΛ[see Fig. 5 (a)], which do not\nreach the same value with µ→Λ. In contrast, for the\nenergy cutoff scheme, the convergent part and the di-\nvergent part are consistent for both the isotropic contri-\nbution [see Fig. 5 (c)] and the anisotropic contribution\n[see Fig. 5 (d)]. Note that the convergent part should\nexactly cancel the divergent part at infinite negative en-\nergy (−∞or Λ), since no states contributes to SHE. This\nindicates that the energy cutoff scheme should be the ap-\npropriate approach.\nAppendix C: General magnetization direction\nTo generalize the discussion with an arbitrary mag-\nnetization direction [note that we chose M∥ˆzinHF,see Eq. (21)], we define a global unitary transformation\nwhich connects the two Hamiltonians H0andHF,\nU†H0U=HF, (C.1)\n0246810\n-0.3-0.2-0.100.1\n0 20 40 60 80 1000246810\n0 20 40 60 80 1000246810(b) (a)\n(c) (d)\nFIG. 5. Comparison between (a) Cisoand−Diso\nΛ(dashed)\n; (b) Cisoand−Diso\nΛε(dashed); (c) Cˆmand−Dˆm\nΛ(dashed);\n(d)Cˆmand−Dˆm\nΛε(dashed). Mis set to be 0 .1,0.5,0.8, cor-\nresponding to color sequence from red to purple. The cutoff\nmomentum/energy is set to be Λ = Λ ε= 100, according to\nthe limit of chemical potential ˜ µmin=−100. The divergent\nparts are taken in opposite sign for comparison. The conduc-\ntivity is in the unit of σ0\ns=e∆/8π2.\nwith unitary matrix U=uiρ3⊗σiandu=\n(sinθ/2 cosφ,sinθ/2 sinφ,cosθ/2). Note that we have\ndefined M=M(sinθsinφ,sinθsinφ,cosθ). The corre-\nsponding eigenfunctions ΨF′\nζ,ηwith such Mread\nH0ΨF′\nζ,η=Eζ,ηΨF′\nζ,η,ΨF′\nζ,η=UΨF\nζ,η. (C.2)\nSubstituting ΨF′\nζ,ηinto the Kubo formula [Eq. (31)], the\nBerry curvaturelike terms take the following form:\n(Ωl\nji)′= Ωl\nji−2umukϵjlkϵabmΩb\nai−2uiumΩl\njm\n+ 4umukuiunϵjlkϵabmΩb\nan, (C.3)\nChoosing the component Ω3\n21as an example, we obtain\n(ΩF,3,ζ,η\n21 )′= ΩF,3,ζ,η\n21 +m2\n1(ΩF,2,ζ,η\n13 −ΩF,3,ζ,η\n21 ),(C.4)\nwhich takes a similar form to that shown in Eq. 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The\nferromagnetism is induced by the electride electrons, which are loosely bound around intersti-\ntial sites and overlap with each other to form two-dimensional interlayer conduction bands. The\nsemimetallicity and two-dimensionality of the band structure are the key to understanding this\nferromagnetic instability.\n1arXiv:1511.02561v1  [cond-mat.mtrl-sci]  9 Nov 2015Introduction: Electrides are unconventional ionic materials in which there is an intrinsic\nexcess of valence electrons from the formal valence viewpoint.1{6The excess electrons \roat\nin the void space between ions and, if their energy band crosses the Fermi level, dominate\nthe transport and magnetic properties. Electrides can be conveniently classi\fed by the\ndimensionality of the electronic structure of the \roating electrons. In zero-dimensional\n(0D) electrides, the \roating electrons are localized at interstitial sites and form a narrow\nband, whereas 1D and 2D electrides have electrons \roating inside \flamentary channels and\ninterlayer gaps, respectively. For decades, all the known electrides were either 0D or 1D,\nbut recently Lee et al.7have demonstrated, through transport measurements and electronic\nstructure calculations, that the layered dicalcium nitride Ca 2N is a 2D electride having a\nsingle conduction band. The electrons in this band are well con\fned in the void space\nbetween Ca layers while being able to move freely along the layers. The concentration of\nthe 2D electrons was found to be one electron per unit cell (Ca 2N), in agreement with that\ndeduced from the formal valence [Ca 2N]+\u0001e\u0000.\nThis \fnding inspired a search for further 2D electrides using ab initio calculations. Walsh\nand Scanlon found Sr 2N and Ba 2N to be 2D electrides.8We carried out an extensive database\nscreening followed by ab initio calculations to identify, in addition to Sr 2N and Ba 2N, six\ncarbides (Y 2C and 4 flanthanide carbides Ln 2C where Ln=Gd, Tb, Dy, Ho, Er) to be\n2D electrides.9Tada et al.10investigated, in addition to Sr 2N, Ba 2N and Y 2C, some 2D\nelectrides not yet registered in current inorganic material databases. Despite their identical\ncrystal structure (anti-CdCl 2structure with unit cell M 2X, space group R\u00163m), the nitrides\nM2N and carbides M 2C signi\fcantly di\u000ber in their electronic structures. Their magnetic\nproperties are expected to di\u000ber accordingly. The nitrides have a single conduction band\nwith the character of a 2D interlayer band and are thought to be Pauli paramagnets. The\ncarbides are more complex and have two conduction bands. The lanthanide carbides Ln 2C\nare ferromagnets with a weakly ferrimagnetic spin order. Y 2C is a marginal and less clear-\ncut case requiring a careful investigation to determine its ground state. A magnetization\nmeasurement found no ferromagnetic transition down to a temperature of 2 K,11which led\nus to report only spin-nonpolarized calculations for Y 2C in our previous paper.9\nIn the present paper, we investigate the electronic structure of Y 2C in detail and show that\nits ground state, as determined by the density functional and hybrid functional methods, is\nweakly ferromagnetic. Furthermore, it is a unique type of itinerant ferromagnet in which the\n2interlayer electrons not attached to any ion, rather than the 4 delectrons of Y, are responsible\nfor the spin polarization.\nStructure: Figure 1(a) illustrates the crystal structure of Y 2C.12,13A conventional (hexag-\nonal) unit cell of Y 2C consists of nine atomic layers having the same triangular 2D lattice\nstructure, albeit being staggered. The nine layers are grouped into three units, each made\nof three closely spaced layers (Y-C-Y) stacked with a layer separation of 1.35 \u0017A. These units\nare separated by a distance (3.45 \u0017A between Y layers) larger than that between layers within\nthe same unit (1.35 \u0017A between Y and C layers). Since the standard oxidation numbers of\nY and C are +3 and -4, respectively, these layer units are positively charged ([Y 2C]2+) and\nthe excess electrons are accumulated between the units to form a quasi-2D electride.9,10The\nwhole crystal therefore has the [YCY]2+\u00012e\u0000\u0001[YCY]2+\u00012e\u0000\u0001[YCY]2+.... stacking structure.\nAs shown below, the interlayer electrons are loosely bound around the interstitial sites [in-\ndicated by a red cross in Fig. 1(a)] and extend in the layer plane to form a 2D electron\nsystem.\nMethod of Calculation: We employed two density functional-based methods using\nexchange-correlation functionals in the generalized gradient approximation (GGA) : (1)\nthe full-potential linearized augmented plane wave (FLAPW) method, as implemented in\nthe All Electron Band Structure Calculation Package (ABCAP),14and (2) the projector\naugmented wave (PAW) method with a plane wave basis, as implemented in the Vienna\nAb Initio Simulation Package (VASP).15{18The FLAPW method takes into account all the\nelectrons on an equal footing and is \\the most accurate method for electronic structure at\nthe present time.\"19The PAW method treats core electrons as frozen and is therefore more\ne\u000ecient at the cost of losing some accuracy. In our calculations, the 2 sand 2 porbitals of C\nand the 4 s, 4p, 5sand 4 dorbitals of Y were taken into account, while lower-energy orbitals\nwere frozen and ignored. The speci\fc GGA potentials used were PBE20and PW9121for (1)\nand (2), respectively. To reinforce these calculations, we also carried out plane-wave-based\ncalculations using screened Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional method,22,23\nas implemented in VASP, which goes beyond the density functional theory by mixing PBE\nexchange and exact exchange.\nFor FLAPW calculations, we employed cuto\u000b energies of 163 eV and 653 eV for the\nwave functions and charge/potential, respectively, and k-integration was carried out in the\nconventional (hexagonal) Brillouin zone (BZ) using a 12 \u000212\u00024 \u0000-centered sampling mesh.\n3FIG. 1. (Color online) (a) Crystal structure (anti-CdCl 2structure) of Y 2C. A supercell consisting\nof 3\u00023\u00021 conventional (hexagonal) unit cells is shown. The large and small spheres represent\nY and C atoms, respectively. A representative interstitial site is indicated by a red cross. A\nconventional unit cell (Y 6C3) is made of three primitive (rhombohedral) cells (Y 2C). (b) Brillouin\nzone of Y 2C. B 1is equivalent to B, and X is equivalent to Q by symmetry. (c) Band structure\nobtained by spin-nonpolarized (black solid curves) and spin-polarized (blue dashed curves and red\ndash-dotted curves) calculations using the PAW method. The zero energy is taken to be the Fermi\nlevel. (d) Fermi surfaces obtained from a spin-nonpolarized calculation (paramagnetic state). (e)\nand (f) Fermi surfaces for the minority and majority spin channels, respectively, obtained from a\nspin-polarized calculation (ferromagnetic state). The electron (hole) pockets are colored in blue\n(yellow). The darker side of the surfaces corresponds to lower energy.\n4TABLE I. Total energy decrease resulting from spin polarization \u0001 Eand magnetization m, both\nper primitive unit cell (Y 2C), obtained by di\u000berent calculation methods. Optimized lattice con-\nstants (spin-nonpolarized/spin-polarized) obtained by the PAW method are also shown.\n\u0001E m a c\n(meV) ( \u0016B) ( \u0017A) ( \u0017A)\nPAW + PBEa15 0.38 3.61/3.62 18.45/18.42\nFLAPW + PW91b12 0.36 - -\nHSE06c17 0.33 - -\naRef. 20\nbRef. 21\ncRefs. 22 and 23\nFor PAW calculations, the plane waves were cut o\u000b at 800 eV, and integration over the\nprimitive (rhombohedral) BZ was performed using a 17 \u000217\u000217 \u0000-centered mesh. The\nhybrid calculations were performed with cuto\u000b energy of 400 eV and a sampling mesh of\n16\u000216\u000216. For both spin-polarized and spin-nonpolarized cases, the structure was relaxed\nusing the PAW method until the force acting on each ion became less than 0.7 meV/ \u0017A. The\noptimized structures were used in FLAPW and hybrid functional calculations.\nResults and Discussion: Table I compares the total energy decrease resulting from spin\npolarization \u0001 Eand the magnetization mper primitive unit cell (Y 2C) obtained by the\nthree methods. Here, \u0001 Eis de\fned as Enonpol\ntot\u0000Epol\ntot, where Epol\ntotandEnonpol\ntot denote the\ntotal energies calculated with and without spin polarization, respectively. As can be seen\nfrom the Table, the results obtained by di\u000berent methods are in very good agreement and\npredict Y 2C to be a weak itinerant ferromagnet. The optimized lattice constants aandc\n(last two columns of Table I) are virtually una\u000bected by spin polarization. Hereafter, we\nwill focus on the results obtained by the PAW method unless stated otherwise.\nHaving two excess electrons per unit cell ([Y 2C]2+\u00012e\u0000]), Y 2C would be a band insulator if\nthere were no band overlap. The actual band structure calculated without spin polarization\n[black solid lines in Fig. 1(c)] is semimetallic with two conduction bands overlapping near\nthe Fermi level EF(= 0 throughout this paper). This semimetallic band structure results\nin electron and hole pockets near the BZ edge. As can be seen from the plots of Fermi\n5FIG. 2. (a) Total DOS, (b) PDOS for Y and (c) PDOS for the interstitial site calculated by\nspin-nonpolarized calculations using the PAW method. Corresponding plots obtained from spin-\npolazired calculations are shown in (a'), (b') and (c'), respectively. The positive and negative DOSs\nfor the spin-polarized case denote the majority and minority spin channels, respectively.\nsurfaces in Fig. 1(d), the electron pockets include the P 1, F and L points, and the hole\npockets include the Q \u0011X and B \u0011B1points at the boundary of the BZ. To identify the\ninterlayer states, we placed an empty sphere of 1.8 \u0017A radius at the interstitial site inside\nthe interlayer gap [red cross in Fig. 1(a)] and projected the wave functions onto this sphere\nto obtain the partial density of states (PDOS) for the interstitial site. The result shown\nin Fig. 2(c) indicates that the interlayer states span a rather narrow energy range of -1 to\n+0.4 eV. Inspection of the charge density for these states reveals that it is indeed con\fned\nin the interlayer gap with a maximum of approximate ssymmetry at the interstitial site.\nTherefore, the interlayer band may be viewed as arising from the 2D lattice of overlapping\nvirtual orbitals (quasiatoms) having ssymmetry at interstices. Figure 2 also presents the\n(a) total density of states (DOS) and (b) PDOS for Y. From the latter, one can see that the\nYdstates are hybridized with the interlayer states near E F. An expanded plot (not shown)\nreveals that the overall bandwidth of the Y dstates is about 10 eV.\nThe inclusion of spin polarization splits the interlayer bands, as shown by the blue dashed\ncurves (majority spin channel) and the red dash-dotted curves (minority spin channel) in\nFig. 1(c), causing the DOS peak to split and shift away from EF, stabilizing the ferromagnetic\nstate as can be seen in Figs. 2(a')-(c'). As the peak shifts, its shape undergoes a larger\ndistortion than normally observed in conventional itinerant magnets such as iron. In the\n6FIG. 3. (Color online) (a) Calculated magnetization density isosurface at 0.015 \u0016B=\u0017A3. The blue\n(brown) spheres represent Y (C) atoms. (b) and (c) Magnetization densities in the (001) plane\n[indicated as H in (a)] and a\u0000cplane [indicated as V in (a)], respectively. (b') and (c') represent\ncorresponding plots for valence charge density.\nferromagnetic state, there is only one majority spin band crossing the Fermi level and this\nband has electron pockets enclosing P 1, F and L [Fig. 1(e)]. For the minority spin channel,\nthe lower conduction band has hole pockets enclosing Q \u0011X and B \u0011B1, and the upper\nconduction band has small electron pockets slightly o\u000b P 1towards Z [Fig. 1(f)].\nFigure 3(a) shows the magnetization density isosurface for the ferromagnetic state. Re-\nmarkably, the bulk of the magnetization is concentrated around the interstitial sites, while\nvery small polarization exists at Y sites. (The Y ions are drawn with smaller spheres to\nmake their polarization visible.) This can be seen more clearly in the cross-sectional plots of\nmagnetization in Figs. 3(b) and (c). Here, (b) shows the cross section in the (001) horizontal\n7plane [denoted as H in Fig. 3(a)] passing through interstitial sites at its four corners, while\n(c) depicts the cross section in the a\u0000cvertical plane denoted as V in Fig. 3(a). For com-\nparison, corresponding plots of the charge density are shown in Figs. 3(b') and (c'). It can\nbe seen that the magnetization is concentrated in o\u000b-ion regions where the charge density\nhas its maxima. The magnetic moment of Y ions (evaluated using the ionic radius for Y3+\nof 0.9 \u0017A) accounts for only 7.1 % of the total magnetization. This weak polarization of Y is\nlikely to be the result of hybridization with the interlayer electrons. These features indicate\nthat the ferromagnetism of Y 2C is induced by the spin polarization of the electride electrons\n\roating between the ionic layers.\nThis picture of ferromagnetism induced by o\u000b-ion electrons is unconventional. To ascer-\ntain that the delectrons of Y are not playing the central role in the ferromagnetism, we also\ncarried out calculations with on-site Coulomb interaction Uincluded in the Y dorbitals.\nIt was found that magnetization decreases, rather than increases, as Uis increased from 0,\nwhich would be di\u000ecult to understand if the delectrons are driving the ferromagnetism.\nIn a wide range of itinerant ferromagnets, the ferromagnetism is understood to arise from\na Stoner-type instability,24i.e., energy is lowered by the exchange splitting of a sharp DOS\npeak around EF. The criterion for this instability is D(EF)I >1 where Iis the exchange\nparameter. This condition, derived originally using a simple molecular \feld argument pos-\ntulating intra-atomic exchange, has been shown to be much more general and is applicable\nto both intra-atomic and more extended exchange interactions between the electrons.25This\nStoner-type mechanism appears to operate in Y 2C as can be seen from the DOS in Fig. 2.26\nRecently, Pickard and Needs4have made an important prediction based on the density func-\ntional theory that the interstitial electrons in alkali metals at high pressures (also electrides)\nwould exhibit a Stoner-type instability towards ferromagnetism. Our results indicate that\nthe interlayer electrons in Y 2C may show a similar instability at ambient pressure.\nOne may wonder what causes the DOS to have a peak around EF. The dashed curves\nin Fig. 4 denote the DOSs of the lower conduction band (LCB) and upper conduction band\n(UCB) for the spin-nonpolarized state. Both curves drop sharply to zero in a step function\nmanner toward the edge (highlighted by a red circle), which is characteristic of a 2D electron\nsystem. As Y 2C is a semimetal, the two bands overlap near EFand cause their sum (total\nDOS, solid curve) to have a peak at EF. Such an enhancement of DOS at EFis expected\nto occur generally in semimetals with 2D conduction bands.\n8FIG. 4. (Color online) DOSs of the lower conduction band (LCB) and upper conduction band\n(UCB) obtained by a spin-nonpolarized FLAPW calculation (dashed curves). As a result of band\noverlapping, highlighted by a red circle, the total DOS (solid curve) is enhanced near EF.\nIn summary, ab initio calculations show the ground state of Y 2C to be weakly ferro-\nmagnetic, where the ferromagnetism is induced by electride electrons accumulated in the\ninterlayer void space. The magnetization measurement of polycrystalline Y 2C by Zhang et\nal.11shows no ferromagnetic transition down to 2 K. It is not yet clear whether there is a\ntransition at an even lower temperature or there is no transition. Even if the latter is the\ncase, the system should be close to a ferromagnetic instability, and spin \ructuations associ-\nated with this instability may be observable as an anomaly in spin susceptibility (deviation\nfrom the Curie-Weiss Law) and speci\fc heat,27or by techniques such as neutron di\u000braction\nand magnetic resonance. The sharp rise in susceptibility below 20 K observed by Zhang et\nal.11may signal such a \ructuation e\u000bect.\nT. I. thanks Drs. Norio Ota, Susumu Saito and Xiao Zhang for useful discussions. This\nwork was supported by the MEXT Elements Strategy Initiative to Form Core Research\nCenter and the JST ACCEL Program. The Fermi surfaces were plotted by XCrySDen.28\n1J. L. Dye, Science 301, 607 (2003).\n2J. L. Dye, Science 247, 663 (1990).\n3Alkali metals under high pressures are also predicted to be electrides. See, for example, C. J.\nPickard and R. J. Needs, Nat. Mater. 9, 624 (2010); M. Gatti, I. V. Tokatly, and A. Rubio,\n9Phys. Rev. Lett. 104, 216404 (2010) and references therein.\n4C. J. Pickard and R. J. Needs, Phys. Rev. Lett. 107, 087201 (2011).\n5S.-W. Kim and H. Hosono, Philo. Mag. 92, 2596 (2012).\n6S. Matsuishi, Y. Toda, M. Miyakawa, K. Hayashi, T. Kamiya, M. Hirano, I. Tanaka, and H.\nHosono, Science 301, 626 (2003).\n7K. Lee, S. W. Kim, Y. Toda. S. Matsuishi, and H. Hosono, Nature 494, 336 (2013).\n8A. Walsh and D. O. Scanlon, J. Mater. Chem. C 1, 3525 (2013).\n9T. Inoshita, S. Jeong, N. Hamada, and H. Hosono, Phys. Rev. X 4, 031023 (2014).\n10T. Tada, S. Takemoto, S. Matsuishi, and H. Hosono, Inorg. Chem. 53, 10347 (2014).\n11X. Zhang, Z. Xiao, H. Lei, Y. Toda, S. Matsuishi, T. Kamiya, X. Ueda, and H. Hosono, J.\nChem. Mater. 26, 6638 (2014).\n12M. Atoji and M. Kikuchi, J. Chem. Phys. 51, 3863 (1969).\n13J. P. Maehlen, V. A. Yartys', and B. C. Hauback, J. Alloys Compd. 351, 151 (2003).\n14N. Hamada, K. Terakura, and A. Yanase, J. Phys. F 14, 2371 (1984); A. Yanase and N. Hamada,\nJ. Phys. Soc. Jpn. 68, 1607 (1999).\n15G. Kresse and J. Furthm uller, Phys. Rev. B 54, 11169 (1996).\n16G. Kresse and J. Furthm uller, Comput. Mater. Sci. 6, 15 (1996).\n17P. E. Bl ochl, Phys. Rev. B 50, 17953 (1994).\n18G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).\n19R. M. Martin, Electronic Structure: Basic Theory and Practical Methods (Cambridge University\nPress, Cambridge, 2005) p. 351.\n20J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).\n21J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C.\nFiolhais, Phys. Rev. B 46, 6671 (1992).\n22J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 120, 7274 (2004).\n23A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106\n(2006).\n24E. C. Stoner, Proc. Roy. Soc. London, A 165, 372 (1938); 169, 339 (1039).\n25J. K ubler, Theory of Itinerant Electron Magnetism (Oxford University Press, Oxford, 2000)\nChap. 4.\n26This is consistent with our \fnding that the ferromagnetism of Y 2C is suppressed by shifting EF\n10away from the DOS peak in both positive and negative directions.\n27T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer-Verlag, Berlin, 1985).\n28A. Kokalj, Comp. Mater. Sci. 28, 155 (2003). Code available from http://www.xcrysden.org/.\n11"    },    {        "title": "2210.08896v1.Flat_band_induced_room_temperature_ferromagnetism_in_two_dimensional_systems.pdf",        "content": "Flat band induced room temperature ferromagnetism in two dimensional systems\nG. Bouzerar\u0003\nUniversité Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France\n(Dated: October 18, 2022)\nThe remarkable and fascinating properties of two-dimensional materials have raised them to the\nrank of most promising candidates for technological applications. In particular, the possibility\nof long-range ferromagnetic order in 2D materials is generating a growing excitement. Here, we\ndemonstrate that flat bands (FBs) may pave the way to room temperature ferromagnetism in 2D\ncompounds. Indeed, the magnetic exchanges between localized spins are largely dominated by the\nFB-FB contribution. This contribution is ferromagnetic and scales linearly with the local coupling,\nthus leading to cross-over temperatures (ferromagnetic phase/ paramagnetic phase) higher by an\norder of magnitude than those currently reported in experiments. Cross-over temperatures as high\nas600Kcould be reached in micrometer-sized FB systems.\nPACS numbers: 75.50.Pp, 75.10.-b, 75.30.-m\nSince graphene has been discovered, the interest in two\ndimensional material (2DM) has grown tremendously. In\nthe plethora of studies, two areas of research have par-\nticularly stood out: flat band (FB) physics, and mag-\nnetism in 2DM. In FB systems, the dispersionless bands\nare at the origin of unforeseen phenomena [1, 2] such as\nfractional quantum Hall states [3, 4], unconventional su-\nperconductivity [5–7], and magnetism [8–10]. FBs host\nas well an unusual type of quantum electronic trans-\nport as revealed in several studies [11–14]. The inter-\nest for ferromagnetism in 2DM such as Cr 2Ge2Te6, CrI 3\nor Fe 3GeTe 2is experiencing a boost over the past years\n[15–22]. 2DMs are undoubtedly promising candidates\nfor technological applications, in spintronics, optoelec-\ntronics and data storage. Ferromagnetism in 2DMs was\nlong ignored, because of Mermin-Wagner (MW) theo-\nrem [23] that says that in one and two dimensions, con-\ntinuous symmetries cannot be spontaneously broken at\nfinite temperature in systems with short-range interac-\ntions. The MW theorem only excludes long-range mag-\nnetic order at finite temperature in the thermodynamic\nlimit, i.e., for infinite systems. However, it does no ex-\nclude the possibility of quasi long-range ordering in finite\nsystems below a cross-over temperature T\u0003\nC[24–26]. It\nimplies that below T\u0003\nCthe spin-spin correlation length is\nmuch larger than the system size. So far, T\u0003\nCreported in\nmicrometer-sized samples has been at most about 40K\nwhich stimulates the search for strategies to gain the\nmissing order of magnitude.\nOur aim is to discuss the impact of FBs on the mag-\nnetic properties of 2D systems. One key question is\nwhether FBs may or may not promote ferromagnetism\nbeyond room temperature. To address this issue, we con-\nsider the Kondo Lieb model as it is illustrated in Fig. 1.\nThe advantages of considering the Lieb lattice (LL) are\nnumerous. First, the LL is a simple square lattice with 3\natoms/unit cell. It possesses a FB at E= 0, there is no\nneed to nano-structure or introduce defects or vacancies\nin the lattice. Finally, it could be designed experimen-\nFigure1. (Coloronline)IllustrationoftheKondoLiebModel.\nThehoppingsarerestrictedtonearestneighbourpairs,crosses\non B (C) sites correspond to the local exchange Jbetween the\nspin of the itinerant carrier ( si) and the localized one Si.Jij\nis the magnetic couplings between pairs of localized spins.\ntally in the framework of covalent-organic compounds for\ninstance [27]. The Hamiltonian reads,\nbH=X\nhiji;\u000b(tijcy\ni\u000bcj\u000b+h:c:) +JX\ni2B;Cbsi\u0001Si:(1)\ncy\ni\u000bcreates an electron with spin \u000b=\";#at site Ri. In\nthe first term, the sum runs over the lattice sites, hiji\nare restricted to nearest neighbour pairs for which the\nhoppingtij=\u0000t.Jis the local Kondo coupling between\nthe localized spin Siat site Riand that of the itinerant\ncarrierbsi. Its components are bs\u0015\ni=cy\ni\u000b\u0002\nb\u001b\u0015\u0003\n\u000b\fci\fwhere\n\u0015=x;y;andzandb\u001b\u0015are the Pauli matrices. In what\nfollows we set t= 1andJSis expressed in units of t. In\nthis study, we focus our attention on the half-filled case,\nthus the chemical potential \u0016= 0.\nThe coupling between a pair of localized spins at Ri\nandRjis given by [28],\nJij=\u0000(JS)2\n2\u0019Z+1\n\u00001=h\nG\"\nij(!)G#\nji(!)i\nf(!)d!:(2)\nThe Green’s function bG\u001b(!) = (!+i\u0011\u0000bH\u001b)\u00001, where\n\u001b=\";#,\u0011mimicsaninfinitesimalinelasticscatteringratearXiv:2210.08896v1  [cond-mat.str-el]  17 Oct 20222\nFigure 2. (Color online) Ground-state energy per unit cell\nEGS=Nas a function of JS.AFdenotes the antiferromag-\nnetic spin texture: Si= +Sez(resp. Si=\u0000Sez) on B (resp.\nC) sublattice. Fis the ferromagnetic spin texture, Si= +Sez\nonbothsublattices. Theinsetrepresentstheenergydifference\nbetween these two configurations.\nandf(!) =1\ne\f(!\u0000\u0016)+ 1is the Fermi-Dirac distribution.\nJij\u00150(resp.Jij\u00140) means antiferromagnetic (resp.\nferromagnetic) coupling.\nThe calculation of Jijrequires the knowledge of the\nground-state (GS), hence that of the underlying localized\nspin texture. The spin configuration could be the disor-\ndered state (spins randomly oriented) corresponding to\nthe high temperature phase, but we would have to deal\nwith this issue numerically only. Instead, we consider\nthe GS at T= 0Kand restrict ourself to two differ-\nent spin configurations: (i) the ferromagnetic GS (F-GS)\nwhere spins on B and C sublattices are parallel and (ii)\nthe antiferromagnetic GS (AF-GS) where spins are anti-\nparallel. Fig. 2 shows the GS energy per unit cell EGS=N\n(Nis the number of unit cells) as a function of JSfor\nboth spin configurations. First, as expected EGS=Nis\nan even function of JS. Secondly, for JS6= 0, F-GS has\nthelowestenergyandtheenergydifferencebetweenthese\ntwo spin configurations increases as jJSjincreases.\nAlthough F-GS has the lowest energy, we discuss the\nnature of the couplings for F-GS and AF-GS. The data\nare depicted in Fig. 3. In both cases, the couplings\nare found always ferromagnetic. Finding ferromagnetic\n(B;C)couplings for AF-GS is inconsistent with an an-\ntiferromagnetic ordering at T= 0K. Thus, even if we\nstart with the wrong GS, the calculated couplings re-\nveal the correct magnetic order. Interestingly, the cou-\nplings are found significantly larger for AF-GS than for\nF-GS. More precisely, for short distances they are about\n10 times larger in the first than in the second case, it\nbecomes 3 orders of magnitude when Rij\u00154a. The\nplotted data are obtained for jJSj=tbut our conclu-\nsions are general. From now on, we consider that F-GS\nis the GS at T= 0K.\n1 2 3 4 5 6 7\nRij/a10-810-710-610-510-410-310-210-1-J(Rij )\n JBB\n JBC\n JBB\n JBC |JS|=1AF-GS\nF-GSFigure 3. (Color online) Couplings (in units of t) as a function\nof the distance for jJSj=t. The circles (resp. squares) cor-\nrespond to the ferromagnetic (resp. antiferromagnetic) spin\ntexture. The open (resp. filled) symbols correspond to (B;B )\n(resp. (B;C )) couplings.\nNow, we propose to discuss in details the nature of the\ncouplings and their inter-band contributions. For that\npurpose, we first focus our attention on (B,B) pairs. For\na given distance R, we write JBB=P\na;bJBB\na;b, where\na(resp.b) is the band index in the spin \"-sector (resp.\n#-sector).aandbare\u0000,0and+, they correspond re-\nspectively to the lower dispersive band (DB), to the FB\nand to the upper DB as illustrated in Fig. 4a. JBBalong\nthex-axis and its different contributions are depicted in\nFig.4b-d forjJSj= 0:1,1and10. The fivenon vanishing\nterms correspond to the ( a,b) pairs: (\u0000;0),(\u0000;+),(0;0),\n(+;\u0000)and(0;+).JBB\n\u0000;0andJBB\n0;+are found antiferromag-\nnetic and for symmetry reasons, identical. In contrast,\nthe other three terms JBB\n\u0000;+,JBB\n+;\u0000andJBB\n0;0are ferromag-\nnetic. In the weak coupling regime ( jJSj= 0:1t), the\ndominant contribution to JBBisJBB\n0;0which is orders of\nmagnitude larger than the other contributions. A fit of\nthe data plotted in Fig. 4b, for R=a\u001d1, shows that\nJBB/1=R\u000bwhere\u000b\u00194. In the intermediate regime\n(jJSj=t), the situation differs. For short distances JBB\nreduces to JBB\n0;0whilst for larger ones it coincides with\nJBB\n+;\u0000and the sum JBB\n\u0000;0+JBB\n\u0000;++JBB\n0;0+JBB\n0;+vanishes.\nFinally, forjJSj\u001dt, these four contributions cancel out\neach other for any distance and JBB\u0019JBB\n+;\u0000as illus-\ntrutated in Fig. 4d. A fit of the data reveals that the\ncouplings decay much more rapidly than in the weak cou-\npling regime: JBB/1=R\fwhere\f\u00196. These features\nare discussed in more details in the following. Notice\nas well, that our findings are general and valid for other\ndirections and pairs of atoms.\nTo shed light on our numerical results, we go fur-\nther and analytically derive the expression of the cou-\nplings in the weak and strong coupling regime. De-\ntails can be found in the Appendices A and B. Let us\nstart with the weak coupling regime ( jJSj \u001ct) for3\n(a)(b)\n(d) (c)JS=t\n|JS|=10t |JS|=t|JS|=0.1t\nFigure 4. (a) Density of states as a function of the energy for JS=t. (b-d) (B,B) coupling ( Jtot) and its different contributions\n(Ja;b) along the x\u0000axis as a function of the distance, for three different values of jJSj=t(0:1,1and 10). The coefficient \u000f= 1\nfor antiferromagnetic couplings and \u000f=\u00001for ferromagnetic ones. a(resp.b) is the band index in the \"(resp.#) spin-sector\nandaandbcan be\u0000(lower dispersive band), +(upper dispersive band) and 0(flat band).\nwhich one needs to focus only on the FB-FB contribu-\ntion. The FB eigenvalues and eigenstates for both spin\nsectors are, E\u001b\n0=\u0006JS\n2=\u0006\u0001where +(resp.\u0000) corre-\nsponds to\u001b=\"(resp.\u001b=#) andh\t\u001b\n0j= (0;fy\ns;\u0000fx\ns),\nwheref\u0015=\u00002tcos(k\u0015a=2), with\u0015=x;yands(k) =q\nf2x(k) +f2y(k). Starting from the definition of the cou-\nplings, we get,\nJXY\n0;0(R) =\u00001\n2jJSj\f\f\f1\nNX\nkfXY(k)\ns2(k)eik:R\f\f\f2\n;(3)\nwherefXY(k) =f2\ny(k),f2\nx(k)andfx(k):fy(k)for re-\nspectively (X;Y ) = (B;B),(C;C)and(B;C). Eq.(3)\nshows that for any pairs of atoms, JXY\n0;0(R)is always fer-\nromagnetic and varies linearly with jJSj. This contrasts\nwith the standard weak coupling regime which leads to\na(JS)2dependency of the couplings. It simply means\nthattheperturbativecalculationbreaksdowninthepres-\nence of FBs. Indeed, the FB-FB contribution is simply\nabsent in the perturbative calculation since the Green’sfunctions used are those corresponding to JS= 0. In a\nrecent work, within such an approach, it has been shown\nthat the FB-DB terms introduce strong frustration ef-\nfects [29]. This is still partly correct, for jJSj\u001ct, the\nFB-DB contributions are antiferromagnetic and scale as\n(JS)2. However, JXY\n0;0(R)largely dominates and washes\nout completely the frustration effects. This explains as\nwellwhyinthenumericaldataplottedinFig.4b, theFB-\nFB term is much larger that the other contributions by\nat least one order of magnitude. The complete analytical\ncalculations as detailed in Appendix A lead to,\nJBB\n0;0(R) =JCC\n0;0(R) =\u0000jJSja4\n8\u00192R4cos2(2\u0012);(4)\nJBC\n0;0(R) =\u0000jJSja4\n8\u00192R4sin2(2\u0012):(5)\n\u0012is the angle between Rand thex-axis. Eq.(4) clearly\nexplains the 1=R4decay of the couplings found in the\nnumerical calculations. Let us define the dimensionless4\nquantityC0=JBB\n0;0\njJSj\u0001\u0002R\na\u00034. A fit of the numerical data\nplotted in Fig.4(b) gives C0=\u00000:0123which agrees very\nwell with the analytical result C0=\u00000:0126.\nWe now switch to the strong coupling regime ( jJSj\u001d\nt) for which the couplings reduce to JXY\n\u0000;+. The energy of\nthe DBs in each spin sector are E\"\n\u0006=\u0000g\u0007andE#\n\u0006=g\u0006\nwhereg\u0006=1\n2\u0002\n\u0000\u0001\u0006p\n\u00012+ 4s2\u0003\n. The corresponding\neigenvectors are respectively, h\t\"\n\u0006j=1\nD\u0006(g\u0006;fx;fy)and\nh\t#\n\u0006j=1\nD\u0007(\u0000g\u0007;fx;fy), whereD\u0006=q\ns2+g2\n\u0006. Be-\ncause, the full calculation of JXY\n\u0000;+is lengthy, the details\ncan be found in the appendix B. For jJSj\u001dt, we find,\nJBB\n\u0000;+(R) =\u00001\n\u00192(4 cos2(\u0012)\u00001)2a6\nR6t2\njJSj;(6)\nJBC\n\u0000;+(R) =\u00003\n\u00192sin2(2\u0012)a6\nR6t2\njJSj:(7)\nNotice that JCC\n\u0000;+(R)is straightforwardly obtained by re-\nplacing inJBB\n\u0000;+\u0012by\u0019\n2+\u0012. These expressions clearly\nexplain the 1=R6decay of the (B,B) couplings observed\nin the numerical calculations. As before, we define the\ndimensionless variable C1=JBB\n\u0000;+\u0001\u0002R\na\u00036\u0001jJSj\nt2. From a\nfit of the numerical data plotted in Fig.4(c) we obtain\nC1=\u00000:71, whilst Eq.(6) gives for \u0012= 0,C1=\u00000:91.\nTheagreementbetweenanalyticalandnumericalcalcula-\ntions is not as good as that found for jJSj\u001ct. However,\nfor other directions, the agreement is much better. For\n\u0012=\u0019\n2, Eq.(6) gives C1=\u00000:101and from the numeri-\ncal data one gets C1=\u00000:102. On the other hand, for\nJBC\n\u0000;+an excellent agreement is found for any values of \u0012.\nMore details are available in Appendix B.\nIn order to derive the cross-over temperature T?\nC, we\nnow turn to the calculation of the magnetic excitations.\nIt is important to point out that, because the couplings\ndecay sufficiently rapidly, in other wordsP\nRJXY(R)R2\nis finite for any (X;Y )pair, the Mermin-Wagner theorem\n[23] implies that in the thermodynamic limit (system size\n!1) the Curie temperature ( TC) vanishes. A finite TC\nin 2DM is possible only in the presence of anisotropy,\nwhich opens a gap in the Goldstone mode. However,\neven in absence of anisotropy, for a finite system one can\ndefineasizedependentcross-overtemperature T?\nC. Arel-\nevant question is, what is the order of magnitude of T?\nC\nfor a micrometer-sized sample which is typically that of\nsamples realized in laboratories. An appropriate tool to\nestimateT?\nCis the Random Phase Approximation (RPA)\n[30]. In diluted magnetic semiconductors and mangan-\nites, the comparison with Monte Carlo simulation has\nrevealed that RPA is reliable and accurate [31, 32]. The\nHeisenberg Hamiltonian reads,\nHH=1\n2X\ni6=jJijSi\u0001Sj; (8)whereJijS2=Jij. To calculate the mag-\nnetic properties, we use the equation of motion\nmethod applied to the retarded spin Green’s function,\nGS\nij;XY (!) =R+1\n\u00001GS\nij;XY (t)ei!tdtwhereGS\nij;XY (t) =\n\u0000i\u0012(t)\nh\nS+\nX;i;S\u0000\nY;ji\u000b\nand,\n:::\u000b\ndenotes the thermal av-\nerage. We only present the main results, the full pro-\ncedure is detailed in Appendix C. As expected, we find\ntwo magnon branches, !\u0006\nq=\nSz\u000b\nE\u0006\nq, where\u0000’ is the\nacoustic mode and0+0is the optical one, and,\nE\u0006\nq=fq\n+\u0006q\n(fq\n\u0000)2+ (fq\nBC)2; (9)\nwherefq\n+=1\n2(fq\nBB+fq\nCC)andfq\n\u0000=1\n2(fq\nBB\u0000fq\nCC)\nwithfq\nXX=\u0000P\nY\u0016JXY(0) +\u0016JXX(q)(X=B;C), we\nhave defined as well \u0016JXY(q) =P\nReiq:RJXY(R)and\nfq\nBC=\u0016JBC(q). As it is shown in Appendix C, the cross-\nover temperature is given by,\nkBT?\nC=1\n3(1 +1\nS)2\n41\nNX\nq6=0;\u0015=\u0006A\u0015\nq\nS2E\u0015q3\n5\u00001\n:(10)\nA\u0006\nqis the spectral weight on each magnon branch. We\ncan rewrite Eq.(10) T?\nC= (1 +1\nS)T?;cl\nC, whereT?;cl\nCis the\ncross-over temperature for classical spins ( S!1).\nLet us briefly discuss the size dependence of T?;cl\nC.\nWe consider square shaped flakes of size La\u0002La. Us-\ning the fact that for small jqj,A\u0006\nq\u00191=2,E+\nq\u0019\nE+\n0andE\u0000\nq\u0019Dq2, whereDis the spin stiffness of\nthe acoustic mode, we obtain,1\nNP\nq6=0;\u0015=\u0006A\u0015\nq\nS2E\u0015q\u0019\n1\n2S2h\n1\nE+\n0+a2\n2\u0019Dln(QC\nqmin)i\nwhereQCis a cut-off, and\nQmin=2\u0019\nLais the smallest non zero momentum. Then\nEq.(10) becomes,\nkBT?;cl\nC=2S2\nA+Bln(L\n2\u0019): (11)\nA=1\nE+\n0+a2\n2\u0019Dln(a:QC)andB=a2\n2\u0019D. As expected,\nT?;cl\nCdecays slowly as 1=ln(L), and in accordance with\nthe MW theorem, it vanishes in the limit L!1. In\nthe presence of a small anisotropy leading to a gap ( \u0001)\nopening, ln(L\n2\u0019)is replaced by1\n2ln(D\n\u0001a2)andT?;cl\nCbe-\ncomes the true Curie temperature.\nTo illustrate the reliability of RPA, we compare our\nexpression of T\u0003;cl\nCfor the nearest neighbour Heisenberg\nHamiltonian one the square lattice with that obtained\nfrom Monte Carlo simulations (MC) [22]. The system\nconsidered is a 1\u0016m\u00021\u0016mflake with a lattice spacing\na= 4\u0017A. Using Eq. (11), we get T?;RPA\nC\u00190:58Jwhilst\nit has been found T?;MC\nC\u00190:535J, whereJdenotes the\nnearest neighbour coupling between the classical spins.\nWe observe that the agreement between RPA and MC is\nsurprisingly good.5\n0.1 1 10 100\n|JS|/t0.0010.01\nL=103\nL=104\nL= 5.1040.02 |JS|/t0.24 t/|JS| *,clT    \nt\nFigure 5. (Color online) Cross-over temperature T?;cl\nCas a\nfunction ofjJSjfor three different sizes L= 103,104and\n5:104. The green dashed lines are fits for the weak and strong\ncoupling regimes for the data obtained for L= 103.\nWe confidently return to our system. In Fig.5, T?;cl\nC,\ncalculated numerically is plotted as a function of jJSj\nfor three different sizes, the biggest system contains 12:5\nbillions of atoms. As can be seen, an increase of Lhas a\nsmall impact on T?;cl\nC. This is consistent with the ln(L)\ndependency explicitly expressed in Eq.(11). We now, es-\ntimatethecross-overtemperatureinmesoscopicsamples.\nWe consider a lattice spacing aof 3\u0017Aand choose for ta\nvalueof 1eV, werecallthatingraphene t\u00192:7eV. From\nFig.5 the maximum of the cross-over temperature T?\nmax\nin a3\u0016m\u00023\u0016msample is 612K,410Kand342Kfor\nrespectively S= 1=2,S= 1andS= 3=2. These tem-\nperatures are beyond room temperature and one order of\nmagnitude higher than those reported in various 2DMs\nsuch as Cr 2Ge2Te6[15] and Fe 3GeTe 2[17].\nIn conclusion, FB states could pave the way to ambi-\nent ferromagnetism in 2D materials. It is revealed, in the\nweak and intermediate coupling regime, that the mag-\nnetic exchanges between localized spins are largely dom-\ninated by the ferromagnetic FB-FB contribution which\nscales linearly with the local coupling, contrasting with\nthe standard quadratic dependence of the other contri-\nbutions. Using reasonable physical parameters, we find\nthat cross-over temperatures well beyond 300Kcould be\nreached in micrometer-sized systems opening interesting\navenues towards technological applications.\n\u0003E-mail:georges.bouzerar@neel.cnrs.fr\n[1] D. Leykam, A. Andreanov, S. Flach, Adv Phys\nX,3,1473052 (2018).\n[2] L. Balents, C. R. Dean, D. K. Efetov and A. F. Young,\nNature Physics 16, 725(2020).[3] E. Tang, J.-W. 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Tyablicov, Methods in quantum theory of mag-\nnetism (Plenum Press, New York, 1967)\n[31] R. Bouzerar, et al., Phys. Rev. B 94, 094437 (2016).\n[32] G. Bouzerar and O. Cépas, Phys. Rev. B 76, 020401(R)\n(2007).\n[33] S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Se-\nries, and Products (Academic Press, New York, 1980),\n7th edition, edited by A. Jeffrey and D. Zwillinger.\n[34] H.B. Callen, Phys. Rev. 130, 890 (1963).\nAPPENDIX A: COUPLINGS IN THE WEAK\nCOUPLING REGIME ( jJSj\u001ct)\nIn this appendix we calculate the magnetic exchange\nbetween localized spins in the weak coupling regime\n(jJSj\u001ct). As explained in the main text, we only need\nto focus on the FB-FB contribution ( (0;0)).\nThe FB eigenvalues and eigenstates for both spin6\nsector are, E\u001b\n0=\u0006JS\n2=\u0006\u0001where +(resp.\u0000)\nis for\u001b=\"(resp.\u001b=#) andh\t\u001b\n0j= (0;fy\ns;\u0000fx\ns),\nwheref\u0015=\u00002tcos(k\u0015a=2), with\u0015=x;yands(k) =q\nf2x(k) +f2y(k). This leads to,\nJXY\n0;0(R) =\u0000jJSj\n2\f\f\f1\nNX\nkfXY(k)\ns2(k)eik:R\f\f\f2\n;(12)\nwherefXY(k) =f2\ny(k)for(X;Y ) = (B;B),f2\nx(k)for\n(X;Y ) = (C;C)andfx(k):fy(k)for(X;Y ) = (B;C).\nThis shows that the FB-FB contribution is always ferro-\nmagnetic and it varies linearly with jJSj. After linearis-\ningfXY(k)ands(k)in the vicinity of M( \u0019,\u0019), Eq.(12)\nbecomes,\n1\nNX\nkfXY(k)\ns2(k)eik:R=ei\u0019:R\nNX\nqgXY(\u0012q)eiq:R;(13)\nwheregXY(\u0012q) = sin2(\u0012q),sin(\u0012q):cos(\u0012q)andcos2(\u0012q)\nfor respectively (X;Y ) = (B;B),(B;C)and(C;C), with\ntan(\u0012q) =qy\nqx.\nFirst, we consider the case (X;Y ) = (B;B). Af-\nter replacing1\nNP\nq(::)bya2\n4\u00192Rqc\n0qdqR\u0019\n\u0000\u0019d\u0012q(::)and in-\ntroducing the Bessel functions of first kind Jn(u) =\nin\n2\u0019R\u0019\n\u0000\u0019eiucos(\u000b)cos(n\u000b)d\u000b(n= 0;1;2;::), one can write,\n1\nNX\nkfXY(k)\ns2(k)eik:R=\nei\u0019:R\n2\u0019R2\u0002\ncos2(\u0012)I1+ cos(2\u0012)I2\u0003\n; (14)\nwhere\u0012is the angle between Rand thex\u0000axis, and\nI1=R1\n0J0(u)uduandI2=R1\n0J00\n0(u)udu. The second\nderivative of J0(u)obey the equation J00\n0(u) =\u0000J0(u) +\nJ1(u)\nu. It can be shown that I1= 0andI2= 1[29, 33].\nWe finally get,\nJBB\n0;0(R) =\u0000jJSja4\n8\u00192R4cos2(2\u0012):(15)\nThis result clearly explains the 1=R4decay of the (B;B)\ncouplings found in the numerical calculations. In addi-\ntion, for a fixed distance, JBB\n0;0, is maximum along the\naxesxandy, and vanishes for \u0012=\u0019\n4. By symmetry ar-\nguments, one can immediately get JCC\n0;0(R) =JBB\n0;0(R).\nAfter repeating the same kind of calculation for (B,C)\npairs, one finds,\nJBC\n0;0(R) =\u0000jJSja4\n8\u00192R4sin2(2\u0012):(16)\nAPPENDIX B: COUPLINGS IN THE STRONG\nCOUPLING REGIME ( jJSj\u001dt)\nIn this appendix, we calculate the magnetic couplings\nbetween localized spins in the strong coupling regime,\n-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5\nθ/π-1-0.8-0.6-0.4-0.20JBB R6JS10 < R < 15\n15 < R < 20\n20 < R < 25\n25 < R < 30\nAnalytic |JS| >> tFigure 6. (Color online) jJSj:R6JBBas a function of the\nangle\u0012(anglebetween Randthex\u0000axis), inthelimitoflarge\nJS. The symbols are the numerical calculations obtained for\ndifferent distances R. The dashed black line is the analytical\nresult.\ne.g.jJSj\u001dt. As shown in the manuscript, in this case,\nthe couplings coincide with the (\u0000;+)contribution. Let\nus now derive the analytical expression of JXY\n\u0000;+(R).\nFor the spin\"-sector, the two dispersive bands are\nE\"\n\u0006=\u0000g\u0007and for#sector, they are E#\n\u0006=g\u0006where\ng\u0006=1\n2\u0002\n\u0000\u0001\u0006p\n\u00012+ 4s2\u0003\n. We recall that \u0001 =JS\n2and\ns(k) =q\nf2x(k) +f2y(k)(from Appendix A). The cor-\nresponding eigenvectors are h\t\"\n\u0006j=1\nD\u0006(g\u0006;fx;fy)and\nh\t#\n\u0006j=1\nD\u0007(\u0000g\u0007;fx;fy), whereD\u0006=q\ns2+g2\n\u0006. From\nthedefinitionofthemagneticcouplingsgiveninthemain\ntext, we get,\nJXY\n\u0000;+(R) =\u0000(JS)2\n2N2X\nk;k’hXY(k;k’)ei(k\u0000k’):R\n(g+(k) +g+(k’)):D2\n\u0000(k):D2\n\u0000(k’);\n(17)\nwherehXY(k;k’) =f2\nx(k):f2\nx(k’)for(X;Y ) = (B;B),\nhXY(k;k’) =f2\ny(k):f2\ny(k’)for(X;Y ) = (C;C)and\nhXY(k;k’) =fx(k):fy(k):fx(k’):fy(k’)for(X;Y ) =\n(B;C).\nIn the strong coupling regime, jJSj\u001dt, Eq.(17) re-\nduces to,\nJXY\n\u0000;+(R) =\u00004t2\njJSj1\nN2X\nk;k’hXY(k;k’)\ns2(k) +s2(k’)ei(k\u0000k’):R:\n(18)\nWe first consider the case (X;Y ) = (B;B). The lineari-\nsation in the vicinity of the M point of the Brillouin zone,7\n-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5\nθ/π-0.3-0.2-0.10JBCR6JS|JS| >> t\nFigure 7. (Color online) jJSjR6JBCas a function of the angle\n\u0012(angle between Rand thex\u0000axis), in the limit of large\njJSj. The symbols are the numerical calculations obtained\nfor different distances R, which are the same as those of Fig.6.\nThe dashed line corresponds to the analytical result.\nleads to,\nJBB\n\u0000;+(R) =\u00004t2\njJSj1\nN2X\nq;q’cos2(\u0012q) cos2(\u0012q’)q2q02\nq2+q02ei(q\u0000q’):R:\n(19)\nThis equation can be rewritten,\nJBB\n\u0000;+(R) =\u0000a6t2\n\u00192jJSjZqc\n0Zqc\n0q3q03dqdq0\nq2+q02F(qR;\u0012 )F(q0R;\u0012);\n(20)\nwhere,F(u;\u0012) =1\n2\u0019R\u0019\n\u0000\u0019d\u000bcos2(\u000b)eiucos(\u000b\u0000\u0012). One can\nshow that F(u;\u0012) = cos2(\u0012)J0(u)\u0000cos(2\u0012)J1(u)\nu. The\nBessel functions of first kind have been defined in Ap-\npendix A. Thus, the expression of the (B,B) couplings\nbecomes,\nJBB\n\u0000;+(R) =\u0000a6t2\n\u00192jJSjR6Z1\n0Z1\n0dudvu3v3\nu2+v2\n\u0002\u0014\nA\u0012J0(u)\u0000B\u0012J1(u)\nu\u0015\n:\u0014\nA\u0012J0(v)\u0000B\u0012J1(v)\nv\u0015\n;(21)\nwhere we have introduced, A\u0012= cos2(\u0012)andB\u0012=\ncos(2\u0012). Let us define the following set of integrals,\nFnl\n\u0016\u0017=R1\n0R1\n0dudvunvl\nu2+v2J\u0016(u)J\u0017(v). Thus, we can\nrewrite Eq.(21),\nJBB\n\u0000;+(R) =\u0000a6t2\n\u00192jJSjR6\u0002\nA2\n\u0012F33\n00+B2\n\u0012F22\n11\u00002A\u0012B\u0012F32\n01\u0003\n:\n(22)\nThe calculation of Fnl\n\u0016\u0017is detailed in the Appendix D\nwhere it has be found that F33\n00= 4,F22\n11= 1andF32\n01=\u00002. After, inserting these values in Eq.(22), we finally\nend up with,\nJBB\n\u0000;+(R) =\u00001\n\u00192(4 cos2(\u0012)\u00001)2a6\nR6t2\njJSj:(23)\nFor (B,C) pairs, one can repeat this procedure to obtain,\nJBC\n\u0000;+(R) =\u00003\n\u00192sin2(2\u0012)a6\nR6t2\njJSj(24)\nNoticeaswellthat JCC\n\u0000;+(R)isstraightforwardlyobtained\nby replacing in Eq.(23), \u0012!\u0019\n2+\u0012.\nIn Fig.6 and Fig.7 are plotted for jJSj ! 1 ,\nJBBjJSjR6andJBCjJSjR6as a function of the angle\n\u0012. First, in the case of the (B;C)couplings, we observe\nan excellent agreement between the numerical calcula-\ntions and the analytical result of Eq.(24) for any value of\n\u0012. On the other hand, for (B;B)couplings, the quality\nof agreement depends on \u0012. For\u0012=\u0019=2and\u0019=4, the\nagreement is excellent, the difference is less than 1%. On\nthe other hand, for \u0012= 0, the analytical result is about\n28%larger than the numerical value.\nAPPENDIX C: CROSS-OVER TEMPERATURE\nIN THE RPA APPROACH\nThe purpose of this appendix is to provide the analyt-\nical expression of the cross-over temperature in our two\ndimensional systems. For that purpose, a good tool is\nthe Random Phase Approximation (RPA). It has been\nshown, in three dimensional systems, that even in disor-\nderedsystems, RPAprovidesaccurateandreliable values\nof the Curie temperature compared to those obtained\nfrom Monte Carlo simulation [31, 32]. The Heisenberg\nHamiltonian reads,\nHH=1\n2X\ni6=jJijSi\u0001Sj: (25)\nTo start, we define the retarded Green’s functions,\nGS\nij;XY (!) =R+1\n\u00001GS\nij;XY (t)ei!tdt\u0011\u001cS+\nX;i;S\u0000\nY;j\u001d\nwhereGS\nij;XY (t) =\u0000i\u0012(t)\nh\nS+\nX;i;S\u0000\nY;ji\u000b\nand,\n:::\u000b\nde-\nnotes the thermal average. XandYare B or C. Then,\nwe apply the equation of motion method which leads to,\n!GS\nij;BB = 2\nSz\nB;i\u000b\n\u000eij+\u001ch\nS+\nB;i;HHi\n;S\u0000\nB;j\u001d:\n!GS\nij;CB =\u001ch\nS+\nC;i;HHi\n;S\u0000\nB;j\u001d: (26)\nThe terms of the form \u001c:::\u001dwhich contain the com-\nmutator of spin operators with HHare higher order\nGreen’s functions. The RPA consists in decoupling these\nterms in order to get a closed form of our set of equa-\ntions [30]. The commutator is given by,h\nS+\nB;i;HHi\n=8\n\u0000P\nY;lJBY\nil(S+\nB;iSz\nY;l\u0000Sz\nB;iS+\nY;l). The decoupling proce-\ndure is as follows,\n\u001ch\nS+\nB;i;HHi\n;S\u0000\nB;j\u001d!2\n4\u0000X\nY;lJBY\nil\nSz\u000b3\n5GS\nij;BB +\n\nSz\u000bX\nY;lJBY\nilGS\nlj;YB; (27)\nwhere, for symmetry reasons, we have used the fact that\nSz\nB;l\u000b\n=\nSz\nC;l\u000b\n=\nSz\u000b\n.\nAfter Fourier transformation of Eq.(26), we obtain,\n\u0014\n!I\u0000\nSz\u000b\u0012fq\nBBfq\nBC\nfq\nCBfq\nCC\u0013\u0015\u0014GS\nBB(q;!)\nGS\nCB(q;!)\u0015\n=\u0014\n2\nSz\u000b\n0\u0015\n;\n(28)\nwhere,fq\nXX=\u0000hz\nX+\u0016JXX(q)withX=B;C. In\naddition, \u0016JXY(q) =P\nReiq:RJXY(R)andhz\nX=P\nY\u0016JXY(q=0). Finally, the off-diagonal term of the\n2\u00022matrix,fq\nBC=fq\nCB=\u0016JBC(q).\nThe two magnons modes are straightforwardly ob-\ntained,!\u0006\nq=\nSz\u000b\nE\u0006\nq, where,\nE\u0006\nq=1\n2\u0014\nfq\nBB+fq\nCC\u0006q\n(fq\nBB\u0000fq\nCC)2+ 4(fq\nBC)2\u0015\n:\n(29)\nThe ’-’ sign corresponds to the acoustic mode and ’+’ to\nthe optical one. It can be easily checked that in the long\nwave length limit ( q!0),!\u0000\nq=Dq2, where D is the\nspin stiffness. After inverting Eq.(28), we find,\nGS\nBB(q;!) = 2\nSz\u000b\u0014A+\nq\n!\u0000!+q+A\u0000\nq\n!\u0000!\u0000q\u0015\n;(30)\nwhere,A\u0000\nq=E+\u0000fCC\nE+\u0000E\u0000andA+\nq= 1\u0000A\u0000\nq.\nThe magnon density of states on B sites is given by,\n\u001aB(!) =\u00001\n2\u0019\nSz\u000b\"\n1\nNX\nqGS\nBB(q;!)#\n=1\nNX\nq;\u0015=\u0006A\u0015\nq\u000e(!\u0000!\u0015\nq): (31)\nThis provides the local effective magnon occupation\nnumber at finite temperature,\n\bB=1\nNX\nq6=0;\u0015=\u0006A\u0015\nq\ne\f!\u0015q\u00001: (32)\nThe cross-over temperature T?\nC, is obtained by using the\nfact that when T!T?\nC, then\nSz\u000b\n!0and\bB!1.\nFrom the previous equation it implies,\n\bB=1\n\f?\nC2\n41\nNX\nq6=0;\u0015=\u0006A\u0015\nq\nE\u0015q3\n51\nSz\u000b:(33)Notice as well, that at finite temperature, the local\nmagnetization obey, for any quantum spin S, the follow-\ning equation [34],\n\nSz\u000b\n=(S\u0000\bB)(1 + \bB)2S+1+ (S+ 1 + \bB)\b2S+1\nB\n(1 + \bB)2S+1\u0000\b2S+1\nB:\n(34)\nThus, in the limit where T!T?\nC, Eq.(34) becomes,\n\nSz\u000b\n=1\n3S2(1 +1\nS)1\n\u001eB: (35)\nAfter combining Eq.(33) and Eq.(35), we end up with the\nanalytical expression of T?\nCwhich reads,\nkBT?\nC=1\n3(1 +1\nS)2\n41\nNX\nq6=0;\u0015=\u0006A\u0015\nq\nS2E\u0015q3\n5\u00001\n:(36)\nThis equation can be rewritten kBT?\nC= (1 +1\nS)kBT?;cl\nC,\nwhereT?;cl\nCis the cross-over temperature in the case of\nclassical spins ( S!1withJSfinite).\nAPPENDIX D: CALCULATION OF THE\nINTEGRALS Fnl\n\u0016\u0017.\nThe purpose of this appendix is to calculate the set of\nintegralsFnl\n\u0016\u0017=R1\n0R1\n0dudvunvl\nu2+v2J\u0016(u)J\u0017(v)that are\nneeded for the calculation of the magnetic couplings in\nthe limit of large jJSj. Let us first define, for any s\u00150,\nthe following function,\nGnl\n\u0016\u0017(s) =Z1\n0Z1\n0dudve\u0000s(u2+v2)unvl\nu2+v2J\u0016(u)J\u0017(v):\n(37)\nThen, after its derivation with respect to s, we obtain\nimmediately,\ndGnl\n\u0016\u0017\nds=\u0000Tn\n\u0016(s)Tl\n\u0017(s); (38)\nwhere we have introduced the functions Tn\n\u0016(s) =R1\n0xnJ\u0016(x)e\u0000sx2dx. From Ref. [33] and for <(\r)>0,\none finds,\nZ1\n0x\r+1J\r(x)e\u0000sx2dx=1\n(2s)\r+1e\u00001\n4s:(39)\nWe set\r= 0in Eq. (39), and its derivation with respect\nto s gives,\nT3\n0(s) =1\n2s2(1\u00001\n4s)e\u00001\n4s: (40)\nIn addition, if we now set \r= 1in Eq. (39), then one\ngets,\nT2\n1(s) =1\n4s2e\u00001\n4s: (41)9\nFrom Eq.(38), one can writedG33\n00\nds=\u0000(T3\n0(s))2, and with\nthe expression of T3\n0(s)given in Eq.(40), we obtain,\nG33\n00(s) = 2Z 1\n2s\n0(u2\u0000u3+1\n4u4)e\u0000udu;(42)\nwhere we have used the fact that G33\n00(s= +1) = 0.\nFinally, sinceR1\n0xne\u0000xdx=n!, we immediately get,\nF33\n00=G33\n00(s= 0) = 4: (43)To calculate F22\n11the procedure is similar. We now use\nthe fact thatdG22\n11\nds=\u0000(T2\n1(s))2. From Eq.(41), one finds\nG22\n11(s) =1\n2R1\n2s\n0u2e\u0000udu. Thus,\nF22\n11=G22\n11(s= 0) = 1: (44)\nSimilarly, we find, G32\n01(s) = 2R1\n2s\n0u2(1\u0000u\n2)e\u0000udu, lead-\ning to,\nF32\n01=G32\n01(s= 0) =\u00002: (45)"    },    {        "title": "2109.10405v1.Ferromagnetic_Double_Perovskite_Semiconductors_with_Tunable_Properties.pdf",        "content": "1 \n Ferromagnetic Double  Perovskite  Semiconductor s with Tunable Properties  \nLun Jin *, Danrui Ni, Xin Gui, Daniel B. Straus, Qiang Zhang and Robert J. Cava*  \n \nL. Jin, D. Ni, X. Gui, D. B. Straus and R. J. Cava  \nDepartment of Chemistry, Princeton University, Princeton, NJ, 08544, USA  \n* E-mails of corresponding authors : ljin@princeton.edu ; rcava@princeton.edu  \nQ. Zhang  \nNeutron Scattering Division, Oak Ridge  National Laboratory, Oak Ridge, TN, 37831, USA  \n \nKeywords : cation -ordered double perovskite s, magnetic -cation  doping, ferromagnetic \nsemiconductor s, tunable ferromagnetic ordering temperature, tunable band gap  \n \n \nWe successfully dope the magnetically silent double perovskite semiconductor Sr 2GaSbO 6 to \ninduce ferromagnetism and tune its band gap, with Ga3+ partially substituted by the magnetic \ntrivalent cation Mn3+, in a rigid cation ordering with Sb5+. Our new ferromagnetic semiconducting \nSr2Ga1-xMn xSbO 6 double perovskite, which crystallizes in tetragonal symmetry (space group I4/m) \nand has tunable ferromagnetic ordering temperature and band gap, suggests that magnetic ion \ndoping of double perovskites is a productive avenue towards obtaining materials for application in \nnext-generation oxide -based spintronic devices.  \n \n \n 2 \n 1. Introduction  \nMagnetic semiconductors combine ferromagnetic and semiconducting properties  and are therefore \npromising candidates for applications in  next generation spintronic devices.[1,2] If such materials \nare applied , then the processing, commun ication and mass storage of information can be achieved \nsimultaneously .[3,4] Given that oxide perovskites are one of the largest and most useful famil ies of \nmaterials,  and are of specific interest for use in next generation devices ,[5–7] the discovery of \nperovskite -family ferromagnetic oxide semiconductors is a high -potential field  of research . There \nare only a few existing experimental or theoretical  examples of such materials[8–11], and the \nimportance of exploring  tunable ferromagnetic semiconductors in the oxide perovskite family \ncannot be over -emphasized .  \nFerromagnetism and semiconducting properties have been known to coexist in Eu - and Mn -\ncontaining chalcogenides[12–17], in spinels[18–20] and in main -group/manganese -blended II -VI and \nIII-V compounds[21–26] for decades , leading to the observation of favorable properties associated \nwith the  interactions  between these two phenomena.  In addition to a variety of other materials that \nhave been explored for this application in recent years[27], low-dimensional chalcogenide and \nhalide compounds ( e.g. Cr2Ge2Te6, CrI 3 and VI 3) have recently been of interest ,[27–32] although \nintegrating these materials with oxide -based electronics will be challenging . Despite the fact that  \na considerable number of magnetic semiconductors ha ve been discovered  until now , perovskite -\nbased spintronic materials remain rare. Oxide perovskites are of special interest as they are \nparticularly integrable into next -generation microelectronic devices, and recent reports about \ndeveloping halide -perovskite -based spintronics[33,34] indicate that it may be  appropriate to revisit  \nthe search  for magnetic semiconductor materials in the oxide perovskite family  as well . Oxide \nperovskites have been studied intensively for decades and represent one of the major classes of 3 \n applied materials, yet ferromagnetic semiconducting oxide perovskites are rather unusual . This \ncould be explained by the Goodenough -Kanamori rules[35], which indicate  the coupling between \nequivalently filled metal ion orbitals in perovskite -like structures will be antiferromagnetic, thus \nthe vast majority of semiconducting oxide perovskites containing only one type of transition metal \nion are antiferromag nets.  \nThe preparation of cation -ordered double perovskites can be daunting, because such materials are \noften thermodynamically unfavorable with respect to simple disordered solid solutions, unless the \ncation ordering is driven by strong chemical factors .[8] We postulate that a semiconducting \nA2BB’O 6 B-cation -ordered double perovskite that lacks magnetic transition metal cations in the \nlattice, but displays a suitable semiconduct ing band gap, can serve as an excellent host material \nfor making a dilute ferromagnetic semiconductor . The ferromagnetism would then be introduced \ninto the semiconductor by partially substituting magnetic transition metal centers for one of the \nnon-magnetic B -cations . Sr 2GaSbO 6 is one such  semiconducting B-site cation ordered double \nperovskite .[36] Since Sr 2MnSbO 6[37,38] is a well-established compound, the magnetic transition \nelement ion Mn3+ was chosen to partially substitute for Ga3+ in Sr 2GaSbO 6 to attempt to induce \nferromagnetism . Thus  in this study we have prepared a series of doped A 2BB’O 6 double perovskite \nphases Sr2Ga1-xMn xSbO 6 (0.1 ≤ x ≤ 0.9) and have fully characterized them from the structural, \nmagnetic and optical points of view. We have found that the Mn3+ dopant yields both tunable band \ngaps and ferromagnetism. The findings reported here are significant since ferromagnetic \nsemicond ucting double perovskites with tunable properties are extremely rare and to the best of \nthe authors’ knowledge the current series is unique . \n2. Results  and Discussion  \n2.1 Structural Characterization  4 \n Sr2GaSbO 6 is a B -site cation ordered double perovskite with the usual double perovskite \narrangement of alternating BO 6-B’O 6 octahedra, crystallizing in tetragonal symmetry. Laboratory \nX-ray powder diffraction (XRD) data collected from Sr 2GaSbO 6 powder  is well fit by a body \ncentered tetragonal unit cell, with lattice parameters a = 5.54 Å and c = 7.90 Å, in good agreement \nwith the literature[39]. To further investigate the crystal structure , especially the oxygen positions , \nwhich  cannot be determined well from XRD, w e performed powder neutron diffraction \nmeasurements. The refinement on the time of flight (TOF) neutron powder diffraction data against \nthe reported structural model converged smoothly and provided a satisfactory agreement parameter \n(wR = 5.10 0%; GOF  = 3.83). Observed, calculated and difference plots from th e Rietveld  \nrefinement of the B -site -ordered structure of the double perovskite Sr2GaSbO 6 (space group I4/m) \nagainst the  neutron  powder diffraction data are shown in the Supporting Information (SI) Figure \nS1. The detailed structural parameters and crystallographic positions are presented in Table S1 .  \nThe initial attempt was carried out by 10% Mn -doping.  Detailed structural characterization of \nSr2Ga0.9Mn0.1SbO 6 was performed by refining ambient temperature TOF neutron powder \ndiffraction data against the structure of the parent phase Sr 2GaSbO 6[39], with the all -Ga3+ site \nreplaced  by 90% Ga3+ and 10% M n3+. The refinement converged smoothly and provided a \nsatisfactory agreement parameter  (wR = 4.179%, GOF  = 3.54). Observed, calculated and \ndifference plots from the Rietveld  refinement of Sr2Ga0.9Mn0.1SbO 6 (space group I4/m) against the  \nambient neutron  powder diffraction data are shown in Figure 1a, and the structure is depicted in \nFigure 1b . The detailed structural parameters and crystallographic positions are presented in Table \n1. Selected bond lengths for the (Ga/Mn) O6 octahedron, the SbO 6 octahedron and the SrO 12 \npolyhedron are listed in Table S 2. 5 \n The successful synthesis of  10% doped single phase Sr2Ga0.9Mn0.1SbO 6 as an ordered A 2BB’O 6 \ndouble perovskite  indicates  that the undoped parent phase Sr 2GaSbO 6 can form at least a partial \ndouble perovskite solid -solution with Sr 2MnSbO 6, so an investigation of the remainder of the \nphase diagram was carried out. We found that Sr2Ga1-xMnxSbO 6 exhibits a full -range tetragonal \nordered B site solid solution (0 ≤ x ≤ 1), most simply illustrated by continuous changes in the \nlattice parameters upon substituting Ga3+ with Mn3+. Laboratory powder XRD patterns for the \nSr2Ga1-xMnxSbO 6 (x = 0.1, 0.3, 0.5, 0.7 and 0.9) samples are stacked in Figure 2 a to show the \ncontinuous shift of the diagnostic tetragonal (200 / 020) doublet around a 2θ position of 46°. The \nabsence of the pseudo -cubic peak indicates that the BB’ cation ordering is conser ved through the \nwhole doping range . \nLattice parameters were determined by Rietveld refinements of the laboratory XRD data. The \nresulting a, c and cell volume V values are plotted against the doping level x in Sr2Ga1-xMnxSbO 6 \nin Figure 2b & c . The data for the end -member Sr 2MnSbO 6[40] are extracted from the literature. \nThe lattice parameter a exhibits an upward -opening parabolic curvature (minimum at x = 0.4) \nwhile c exhibits a downward -opening one (maximum at x = 0.8). The cell volume V possesses a \nlinear relationship with increasing x. The reported data for the Sr 2MnSbO 6 end-member[40] does \nnot fall in line with the materials reported here. One potential explanation for this observation is \nthat the all -Mn end member Sr 2MnSbO 6 may be in proximity to a structural transition, the \ndetermination of which is beyond the scope of the current study . \nFinally, s canning electron microscope (SEM) images were collected to observe the surface \nmorphologies of powder particles of the Mn -doped double perovskite s. Representative images are  \nshown in Figure S 2. The grain size tends to decrease at the elevating doping levels.  The perovskite \nphase is clearly maintained in single phase form for these materials – no impurity particles were 6 \n detected.  The eq uipped  energy -dispersive X -ray spectroscop e (EDX)  provided compositional \nresults that are consistent with the nominal ones.  \n2.2 Physical Propert y Characterization  \n2.2.1. Magnetic Properties  \nThe temperature -dependent and field -dependent magnetization data collected for the Sr2Ga1-\nxMnxSbO 6 series  and the Mn -doped single perovskite LaGa 1-xMn xO3 series, prepared for \ncomparison, are plotted as magnetic susceptibility  χ (M/H) against temperature T and magnetic \nmoment M against field H, in the Supporting Information (SI) Figures S3 – S16. The magnetic \nsusceptibility  data, over  the suitable temperature range  (selected as the straight -line part of the 1/  \nχ vs. T curves)  of the doped materials were  fitted to the Curie -Weiss law (χ = C/(T - θ) + χ0), to \nyield the Curie constants C and Weiss temperatures θ that are listed in Table S3  and depicted in \nFigure 3 a & b . The magnetization  data reported  for the end-member  Sr2MnSbO 6[38] fall in line \nwith the materials reported in this study . \nFor x = 0.1 in t he Sr2Ga1-xMn xSbO 6 series, the inverse of χ exhibits a perfect linear relationship \nwith temperature T in nearly  the whole  measured  temperature  range  1.8 ≤ T / K ≤ 300, and the \nfitted Curie constant coincides with a high -spin d4 electronic configuration for Mn3+. The Weiss \ntemperature θ has a positive value of + 3.43( 9) K, which suggests that a ferromagnetic interaction \nweakly dominates in this material . With increased Mn3+ doping concentration , a local maximum \nin the magnetic susceptibility curve starts being observed (x = 0.3) within  our measurable \ntemperature regime ( T ≥ 1.8 K).  For x = 0.4, the Curie constant  starts  to deviate  from  what  is \nexpected  for the spin-only value  and with further Mn -doping ( x ≥ 0.4), clear divergence starts to \nappear between the zero -field-cooled (ZFC) and field -cooled (FC) curves in plots of magnetic \nsusceptibility  χ (M/H) against temperature T. The Weiss temperatures θ are all positive and \nincrease in magnitude upon elevatin g the Mn3+ doping level ( Figure 3b ).  Furthermore, the value 7 \n of the ferromagnetic ordering  temperature Tc (estimated as the temperature at which the 1/χ vs. T \ncurve deviates from Curie -Weiss law linearity . Due to the character of the χ vs. T data, an exact \nmeasure of Tc is not possible by the methods employed. ) for the Sr2Ga1-xMn xSbO 6 series increases \nsystematically with increasing x. The isothermal m agnetization data collected from the Mn -doped \nmaterials at 300 K  as a function of applied field  is linear and pass es through the origin , with a \nsmall positive slope, while a nalogous data collected at 2 K have positive slopes as well, but start \nto exhibit easily observed hysteresis at 30% Mn -doping, serving as strong ev idence of \nferromagnetic behavior, consistent with the temperature -dependent magnetization data. \nAlthough the Mn -doped disordered single perovskite LaGa 1-xMn xO3 series ha s been well -\nstudied[41–45], samples of this series in the composition ran ge 0.1 ≤ x ≤ 0.5 were characterized \nmagnetically as a comparison to the Mn -doped double perovskite. The magnetization data were \nscaled to their equivalent double perovskite values (i.e. for La 2Ga2-2xMn 2xO6) for consistency. The \nmagnetic  susceptibility  data can be fitted  to the Curie -Weiss law  in suitable  temperature  regimes  \n(100 ≤ T / K ≤ 300 to 200 ≤ T / K ≤ 300 depending on the value of x). The resulting Curie constants \nfollow a similar trajectory as their double perovskite analogs but in a more linear fashion  (Figure \n3a), while the magnitude of the Weiss temperature is almost doubled in the single disordered \nperovskite series at identical Mn concentrations ( Figure 3b ).  The isothermal m agnetization data \nat 300 K  and 2 K are consistent with the temperature -dependent magnetization data and are in \ngood agreement with the literature .[41,43,45] The magnetization data collected from double - and \nsingle -perovskite s with 40% total Mn -concentration on the B -sites ( LaGa 0.6Mn 0.4O3 and \nSr2Ga0.2Mn 0.8SbO 6) are stacked in Figure 3c & d , as representative data. The temperature -\ndependent  magnetic susceptibility curves coincide with each other in the temperature range 150 ≤ \nT / K ≤ 300, while they differ  significantly at lower temperatures . 8 \n In Figure 4 , (a) the observed effective moment per formula unit μeff.obs is plotted against  the \ncalculated  effective moment per formula unit μeff.cal based  on the spin-only value  of the transition  \nmetal  magnetic  moment,  (b) and the observed magnetization M per formula unit collected at T = \n2 K under an applied field of 9 T is plotted against the calculated saturation magnetization Ms per \nformula unit . The suitably  scaled  disordered single perovskite LaGa 1-xMn xO3 series is included for \ncomparison. F or both Mn-doped  series,  the observed  μeff.obs only agrees  with the spin-only μeff.cal \nat low doping  concentrations  and they differentiate  afterwards  (Table  S3, Figure 4a ). The spin -\nonly μeff.cal are underestimations of the observed moments, suggesting that the orbital  angular  \nmomentum  exerts  some  influence  for the Mn cases. In Figure 4b , for the Mn -doped double \nperovskite Sr2Ga1-xMn xSbO 6 series , the observed magnetization M increases with the doping level \nx, but the increments gradually reduce; hence the trend line shows the sign of leveling off at higher \ndoping concentrations. In contrast to the Mn -doped double perovskite, M for the disordered single \nperovskite LaGa 1-xMn xO3 series exhibits a clear, positive linear relationship with respect to the \ndoping level x. The observed magnetizations M of the Mn -doped single perovskite and double \nperovskite are quite close at low doping concentrations, but clearly diverge afterwards, w ith the \ndifference increasing at higher doping levels . For the same Mn -doping concentration, the Tc of the \nordered double perovskite (end member Tc ≈ 170(6) K) is significantly lower than that of the \ndisordered single perovskite (end member Tc ≈ 200(6) K), and the magnetic transitions in the single \nperovskite series are much broader than those of the double perovskite series . \nA possible explanation of our overall magnetic  observations may be that in ordered double \nperovskite s, the Mn -O-M-O-Mn (M is a non -magnetic cation) super -exchange pathway dominates \nthe magnetic behavior, while in the disordered single perovskite, in addition to some Mn -O-M-O-\nMn magnetic -exchan ge pathways, Mn -O-Mn pathways are also feasible, which cannot happen in 9 \n the double perovskite except at defects. This pathway clearly should have a stronger magnetic \ncoupling strength, as non -magnetic cations (i.e., the Sb5+) are not involved. Mn -O-Mn supe r-\nexchange pathways are more likely to appear at higher doping levels in the single perovskite and \nthus Mn -O-Mn coupling in the disordered single perovskite series contributes more to the magnetic \nbehavior. This accounts for the greater differences between  the ordered and disordered perovskite \nseries in both Tc and the saturated ferromagnetic moment at the Mn -rich end of each series. This \nkind of suppression of the magnetic state has been widely observed in perovskites and their \nderivatives.[46–49]  \n2.2.2 Calculated Band Structures and Characterization of Band Gaps  \nThe calculated band  structure s of the Sr 2GaSbO 6 and Sr2MnSbO 6 double perovskites are shown in \nFigure 5 . For Sr 2GaSbO 6 a calculated direct bandgap of ~ 1 eV is clearly observed at the  point \nwhile an indirect one (~3 eV) is seen between the  and N points; a nearly flat band exists between \nthe  and X points ( Figure 5a ). The calculated band structure for ferromagnetic Sr2MnSbO 6, shown \nin Figure 5b , is different . (The spin structure for the calculations was set to that found in the \nliterature[37,38].) Firstly, it shows a bandgap for both spin directions. An indirect small bandgap \n(~0.1 eV) emerges between the  and P points for up spin while a much larger indirect one (~2 \neV) shows up between  and X point for down spin. Thus for doping by either holes or electrons \nthis material can be expected to be a spin polarized ferromagnetic semiconductor below its \nmagnetic ordering temperature. The evolution of the bandgaps of the Sr2Ga1-xMnxSbO 6 series that \nwe observe in the diffuse reflectance spectrum ( Figure 6a) is consistent with our calculations of \nthe electronic band structures . \nThe pseudo -absorbance, transferred from reflectance using the Kubelka -Munk function, is plotted \nagainst photon energy (eV) in Figure 6a  for the doped double perovskite series . The o ptical band 10 \n gaps were then analyzed based on Tauc plots  (Table S4 ). For the  undoped parent phase  Sr2GaSbO 6, \nthe band gap is found to be 3.52 eV if the indirect -transition equation was used , while  it is 3.83 eV \nif the direct -transition equation was applied instead.  The band structure calculations  (from both \nthe literature[36] and our own ) suggest this phase should have a direct band gap around 1.0 e V \n(Figure 5a ), consistent with the fact that DFT calculations of band gaps for main group element \ncompounds like this one are generally too low . \nThe size of band gap decreases with increasing  doping level x in the Sr2Ga1-xMnxSbO 6 system, \naccompanied by the samples’ darker appearances. The band gaps are significantly suppressed upon \ndoping, even 10% Mn -doping reduces the band gap to 1. 15 eV (Figure 6c) . The detailed absorption \nbehavior is complicated in the Mn-doped double perovskite series, as double absorption features \nappear in the spectrum for x = 0.1 and 0.2, but for x = 0.3 and 0.4, the absorption in the higher \nenergy regime ( near 3 to 4 eV ) has been significantly suppressed compared to earlier composit ions. \nFor 0.5 ≤ x ≤ 0.9, these  two absorptions seem to overlap and merge into  a single absorption ( lower  \nthan 1 eV) . This can be attributed to the different band structures of the two end members , \nSr2GaSbO 6 and Sr 2MnSbO 6 (Figure 5 ). For low Mn concentration s, the materials  exhibit \nabsorption features from both Sr2GaSbO 6 and Sr 2MnSbO 6, resulting in a double absorption \nbehavior , then, with elevated Mn concentrations , the Sr 2MnSbO 6 featured absorption becomes \ndominant and thus leads to a major indi rect transition  lower than 1 eV . The optical gap for this \nferromagnetic semiconductor, which should display polarized electronic states near the Fermi \nenergy, is clearly continuously variable in the semiconductor regime under  2 eV.  The band gaps \nmeasured by diffuse reflectance spectr oscopy serve as strong evidence that the Mn -doped Sr2Ga1-\nxMnxSbO 6 double perovskites are semiconductors with tunable band gaps.  11 \n For the purpose of  summariz ing the systematic changes in ferromagnetic ordering temperatures Tc \nof this series of materials, the magnetic susceptibility  (M/H), modified by a small  χ0 is plotted  \nagainst  temperature  T, hence  their first derivatives  were  taken  to yield  the embedded  plot (Figure  \n6b). The temperature  at which  the d(χT - χ0)/dT curve  deviates  from  the horizontal  line around  y = \n0 is considered  to be the Tc, as below  this temperature,  the Curie -Weiss  law is no longer  obeyed  \n(also  corresponds  to the temperature  at which  the 1/χ curve  deviates  from  linearity  in the \nSupporting  Information  (SI) Figures  S3 – S16) . Hence , Sr2Ga1-xMnxSbO 6 is found to be a \nferromagnetic semiconductor, with ferromagnetic ordering temperature Tc and size of band gap \neasily tuned by the Mn -doping concentration  (Figure 6 c). \n3. Conclusion s  \nPrevious electronic structure calculations suggested to us that the B-site-cation -ordered \nmagnetically silent oxide double perovskite Sr 2GaSbO 6 should be a semiconductor with a direct \nband gap of around 1 eV .[36] To make this material exhibit ferromagnetism, hence becoming a \npotential candidate material for spintronic devices, Mn3+ was selected to partially substitute for \nGa3+. The doped double perovskite Sr2Ga1-xMnxSbO 6 materials crystallize in the same space group \n(I4/m) as their undoped parent phase Sr 2GaSbO 6, with  the Ga3+/Mn3+ cations completely ordered \nwith Sb5+. Magnetization data collected from the doped phases show that the Mn -doped materials \nexhibit the desired ferromagnetic interactions, which become stronger with increasing Mn content . \nThe ferromagnetic ordering temperature Tc increases towards the Mn -rich end of the solid solution. \nIn addition to the magnetic properties, the Mn -doped materials all have significantly smaller band \ngaps compared to their undoped parent phase Sr 2GaSbO 6, with the size of band gaps decreasing \nwith elevated doping level x. The calculations suggest that these materials, in addition to being \nferromagnetic and semiconducting, should also display highly spin polarized ele ctronic states near 12 \n EF; an important characteristic for use in spin selective next -generation electronic devices .[50]  Our \nresults show that tunable ferromagnetic semiconducting double perovskites are viable materials \nfor implementation in next -generation electronic devices . \n \nExperimental  \nApproximately 0.5 g polycrystalline powder samples of the parent phase A 2BB’O 6 ordered double \nperovskite Sr 2GaSbO 6 and its doped phases Sr2Ga1-xMn xSbO 6 (0.1 ≤ x ≤ 0.9) were  synthesized by \na conventional solid state reaction method . Polycrystalline powder samples of the well -known B -\nsite disordered single perovskite LaGa 1-xMn xO3 (0.1 ≤ x ≤ 0.5) were  also synthesized  by this \napproach. Stoichiometric metal ratios of SrCO3 (Alfa Aesar, 99.99% ), La2O3 (Alfa Aesar, 99.99%, \ndried at 900 ° C ), Ga2O3 (Alfa Aesar, 99.999% ), Sb2O5 (Alfa Aesar, 99.998% ) and Mn 2O3 (Alfa \nAesar, 99.99%) were ground together using an agate mortar and pestl e, and then transferred into \nan alumina crucible . These reaction mixtures were first slowly (1 ° C/min) heated to 1000 ° C in air \nand held overnight to decompose the carbonate , and the n were directly annealed in air at 1300 – \n1500 ° C (3 ° C/min) for 3 periods of 72 hours with  intermittent grinding s. The reaction progress \nwas monitored using laboratory X-ray powder diffraction data collected at room temperature on a \nBruker D8 FOCUS diffra ctometer (Cu Kα) over a 2θ range between 5 ° and 70°. Once the reactions \nwere complete, laboratory XRD data with much better statistical significance, covering a 2θ range \nbetween 5 ° and 110° , were collected from each sample. Lattice parameters, atomic positions, and \natomic displacement factors using these data were determined by the Rietveld method[51] using the \nGSAS -II program . \nApproximately 3 g polycrystalline powder samples of the parent phase  Sr2GaSbO 6 and the 10% -\ndoped phase Sr2Ga0.9Mn0.1SbO 6, used for the neutron diffraction experiments, were also 13 \n synthesized via the above approach. Time -of-flight (TOF) neutron powder diffraction data were \ncollected at Oak Ridge National Laboratory’s Spallation Neutron Source , POWGEN beamline , \nusing neutron beam with a center wavelength of 0.8 Å at 300 K. Structures from this data were \ndetermined by the Rietveld method[51] using the GSAS -II program . \nThe magnetization data were collected using the vibrating sample magnetometer (VSM)  option of \na Quantum Design Physical Property Measurement System (PPMS). Temperature -dependent \nmagnetization (M) data w ere collected from finely ground powders  in an applied field (H) of 1000  \nOe. The magnetic susceptibility χ was defined as M (in emu)/ H (in Oe). F ield-dependent \nmagnetization data between H = 90000 Oe and –90000 Oe  were collected at T = 300 K and 2  K.  \nThe diffuse reflectance spectra were collected from powder samples at ambient tem perature on a \nCary 5000i UV -VIS-NIR spectrometer equipped with an internal DRA -2500 integrating sphere. \nThe data were converted from reflectance to pseudo absorbance using the Kubelka –Munk method, \nand values of band transitions were calculated from Tauc pl ots.[52] The particle morphology of \nselected compositions was investigated using an FEI XL30 field -emission gun scanning electron \nmicroscope (SEM) equipped with an Oxford X -Max 20 energy -dispersive X -ray spectroscop e \n(EDX) running on AZtec  software . \nThe band  structure s and electronic densit ies of states (DOS) were  calculated using the  WIEN2k \nprogram . The full -potential linearized augmented plane wave  (FP -LAPW) method  with local \norbitals was used .[53,54] Electron correlation w as treated via the generalized gradient \napproximation .[55] The conjugate gradient algorithm was applied, and the energy cutoff was set  at \n500 eV. Reciprocal space integrations were completed over a 6×6×4 Monkhorst -Pack k-point \nmesh .[56] Spin-orbit coupling (SOC) effects were only applied for the Sb  atom.  Spin-polarization \n(ferromagnetism with the moment oriented in the (001) direction) was only employed for the Mn 14 \n atoms.  The lattice parameters of Sr 2GaSbO 6 were obtained from M. Lufaso et al.[39] For \nSr2MnSbO 6, the crystallographic data were retrieved from the Materials Project database .[57] The \ncalculated total energy converged to less than 0.1 meV per atom .  \n \nSupporting Information  \nThe Supporting Informat ion is available from the Wiley Online Library or from the author.  \nAcknowledgements  \nThis research was primarily done at Princeton University , supported by the US Department of \nEnergy, Division of Basic Energy Sciences, grant number DE-FG02 -98ER45706. A portion of  this \nresearch  used res ources at the Spallation Neutron Source , a DOE Office of Science User Facility \noperated by the Oak Ridge National Laboratory.  The authors acknowledge the use of Princeton’s \nImaging and Analysis Center, which is partially supported through the Princeton Center for \nComplex Materials (PCCM), a National Science Foundation (NSF) -MRSEC program (DMR -\n2011750) . \nConflict of Interest  \nThe authors declare no conflict of interest . 15 \n Table 1.  Structural parameters and crystallographic positions from the refinement of neutron \npowder diffraction data collected from Sr2Ga0.9Mn 0.1SbO 6 at 300 K.  \nAtoms  x/a y/b z/c S.O.F.  Uiso equiv. ( Å2) \nSr1 0 0.5 0.25 1 0.00656  \nGa1 0 0 0 0.9032(1)  0.00349  \nSb1 0.5 0.5 0 1 0.00126  \nO1 0 0 0.2495(6)  1 0.01000  \nO2 0.2224(6)  0.2777(2)  0 1 0.00775  \nMn1  0 0 0 0.0968(1)  0.00349  \nSr2Ga0.9Mn 0.1SbO 6 space group I4/m (#87)  \nFormula weight: 461.24 g mol-1, Z = 2  \na = 5.54094 (4) Å, c = 7.93548 (4) Å, Volume = 243.635 (3) Å3 \nRadiation source: time of flight  neutrons  \nTemperature: 300 K  \nwR = 4.1 79%; GOF  = 3.5 4 \n  16 \n  \nFigure 1.  (a) Observed  (blue) , calculated  (green)  and difference (cyan ) plots  from the Rietveld  \nrefinement of Sr2Ga0.9Mn0.1SbO 6 (space group I4/m) against neutron  powder diffraction data  at \n300 K ; (b) The structural model of Sr2Ga0.9Mn0.1SbO 6, and the s elected  bonding environment of \nthe Ga 0.9Mn0.1O6 octahedron, SbO 6 octahedron and the SrO 12 polyhedron.  The noted  bond lengths \nare listed in Table S 2. \n \n17 \n  \nFigure 2.  (a) Stacked lab X -ray diffraction patterns  for selected compositions of Sr2Ga1-xMnxSbO 6 \nseries ; (b) lattice parameter a and c, (c) and cell volume V (error bars are smaller than the data \npoints) plotted for each composition of the Sr2Ga1-xMn xSbO 6 series . (Data points in red circles are \nextracted from the all -Mn end member Sr 2MnSbO 6 in literature, which does not fall in line with \nthe materials reported here .) \n \n \n \n  \n18 \n  \nFigure 3. (a) The Curie constant and (b) the Weiss temperature (error bars are smaller than the \ndata points) extracted from the fitting of paramagnetic susceptibility to the Curie -Weiss law for \neach composition of the Sr2Ga1-xMnxSbO 6 series  and the LaGa 1-xMn xO3 series (scaled based on its \nequivalent double perovskite formula) ; (c) the magnetic susceptibility χ ( M/H) against temperature \nT, with 1/χ against T plots inset; and (d) the magnetic moment M against field H for double - and \nsingle -perovskite with 40% total Mn-concentration over B -cation sites in A 2B2O6 (late members \nin each series) . (Data points in red circles are extracted from the all -Mn end member Sr 2MnSbO 6 \nin literature .)  \n19 \n  \nFigure 4. (a) The observed effective moment per formula unit plotted against the effective moment \nper formula unit  calculated from the spin -only contribution ; (b) the observed magnetization at 2  K \nunder an applied field of 9  T (error bars are smaller than  the data points) plotted against the \ncalculated saturation magnetization  for each composition of the Sr2Ga1-xMnxSbO 6 series  and the \nLaGa 1-xMn xO3 series (scaled based on its equivalent double perovskite formula) . (Data point in the \nred circle is extracted from the all -Mn end member Sr 2MnSbO 6 in literature .) \n  \n20 \n  \nFigure 5. Calculated band  structure s and electronic density of states (DOS) of (a) Sr2GaSbO 6 and \n(b) Sr2MnSbO 6 (red for up spins and blue for down spins).  \n  \n21 \n  \nFigure 6. 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Chem  2011 , 50, 656.  \n "    },    {        "title": "0912.0094v2.Controllability_of_ferromagnetism_in_graphene.pdf",        "content": "arXiv:0912.0094v2  [cond-mat.mtrl-sci]  15 Sep 2010Controllability of ferromagnetism in graphene\nTianxing Ma1,2,3,∗, Feiming Hu2, Zhongbing Huang4and Hai-Qing Lin2\n1Department of Physics, Beijing Normal University, Beijing 100875, China\n2Department of Physics and ITP, The Chinese University of Hon g Kong, Hong Kong\n3Max-Planck-Institut f¨ ur Physik Komplexer Systeme, N¨ oth nitzer Str. 38, 01187 Dresden, Germany\n4Faculty of Physics and Electronic Technology, Hubei Univer sity, Wuhan 430062, China\n(Dated: April 18, 2019)\nWe systematically study magnetic correlations in graphene within Hubbard model on a honey-\ncomb lattice by using quantum Monte Carlo simulations. In th e filling region below the Van Hove\nsingularity, the system shows a short-range ferromagnetic correlation, which is slightly strengthened\nby the on-site Coulomb interaction and markedly by the next- nearest-neighbor hopping integral.\nThe ferromagnetic properties depend on the electron filling strongly, which may be manipulated by\nthe electric gate. Due to its resultant controllability of f erromagnetism, graphene-based samples\nmay facilitate the development of many applications.\nThe search for high temperature ferromagnetic semi-\nconductors, which combine the properties of ferromag-\nnetism (FM) and semiconductors and allow for practi-\ncal applications of spintronics, has evolved into a broad\nfield of materials science[1, 2]. Scientists require a ma-\nterial in which the generation, injection, and detec-\ntion of spin-polarized electrons is accomplished with-\nout strong magnetic fields, with processes effective at or\nabove room temperature[3]. Although some of these re-\nquirements have been successfully demonstrated, most\nsemiconductor-based spintronics devices are still at the\nproposal stage since useful ferromagnetic semiconductors\nhave yet to be developed[4]. Recently, scientists antic-\nipate that graphene-based electronics may supplement\nsilicon-based technology, which is nearing its limits[5, 6].\nUnlike silicon, the single layer graphene is a zero-gap\ntwo-dimensional (2D) semiconductor, and the bilayer\ngraphene provides the first semiconductor with a gap\nthat can be tuned externally[7]. Graphene exhibits gate-\nvoltage controlled carrier conduction[7–10], high field-\neffect mobility, and a small spin-orbit coupling, making\nit a very promising candidate for spintronics application\n[11, 12]. In view of these characteristics, the study of\nthe controllability of FM in graphene-based samples is of\nfundamental and technological importance, since it in-\ncreases the possibility of using graphene in spintronics\nand other applications.\nOn the other hand, the existence of FM in graphene\nis an unresolved issue. Recent experimental and theo-\nretical results in graphene[13–15] show that the electron-\nelectron interactions must be taken into account in order\nto obtain a fully consistent picture of graphene. The\nhoneycomb structure of graphene exhibits Van Hove sin-\ngularity(VHS) in the densityofstates(DOS), whichmay\nresult in strong ferromagnetic fluctuations, as demon-\nstrated by recent quantum Monte Carlo simulations of\n∗txma@bnu.edu.cnthe Hubbard model on the square and triangular lat-\ntices [16, 17]. After taking both electron-electron inter-\naction and lattice structure into consideration, the bidi-\nmensional Hubbard model on the honeycomb lattice[18–\n20] is a good candidate to study magnetic behaviors in\ngraphene. Early studies of the bidimensional Hubbard\nmodel onthe honeycomblattice werebasedonmean field\napproximations and the perturbation theory[20]. How-\never, the results obtained are still actively debated be-\ncause they are very sensitive to the approximation used.\nTherefore, we use the determinant quantum Monte Carlo\n(DQMC) simulation technique[21, 22] to investigate the\nnature of magnetic correlation in the presence of moder-\nateCoulombinteractions. Weareparticularlyinteresting\non ferromagnetic fluctuations as functions of the electron\nfilling, because the application of local gate techniques\nenables us to modulate electron filling[7–10], which is the\nfirststepon theroadtowardsgraphene-basedelectronics.\nThe structure of graphene can be described in terms of\ntwointerpenetratingtriangularsublattices, A and B,and\nits low energy magnetic properties can be well described\nby the Hubbard model on a honeycomb lattice[18–20],\nH=−t/summationdisplay\niησa†\niσbi+ησ+t′/summationdisplay\niγσ(a†\niσai+γσ+b†\niσbi+γσ)+h.c.\n+U/summationdisplay\ni(nai↑nai↓+nbi↑nbi↓)+µ/summationdisplay\niσ(naiσ+nbiσ) (1)\nwheretandt′are the nearest and next-nearest-neighbor\n(NNN) hopping integrals respectively, µis the chemical\npotential, and Uis the Hubbard interaction. Here, aiσ\n(a†\niσ) annihilates (creates) electrons at site Riwith spin\nσ(σ=↑,↓) on sublattice A, biσ(b†\niσ) annihilates (creates)\nelectronsat the site Riwith spin σ(σ=↑,↓)on sublattice\nB,naiσ=a†\niσaiσandnbiσ=b†\niσbiσ.\nOur main numerical calculations were performed on\na double-48 sites lattice, as sketched in Fig. 1, where\nblue circles and yellow circles indicates A and B sub-\nlattices, respectively. The structure of the honeycomb\nlattice leads to the well known massless-Dirac-fermion-\nlike low energy excitations and the two VHS in the DOS2\n(a)t\nt’(b)\nΓ(0,0)M(0,π)K(-2π\n3,2π\n3)\n−3−2−10123400.511.52\n(c)\n0.751.25\nEρ(E)\n−2−101234500.511.52\n(d)\n 0.75 1.25\nE\n<n>\nFIG. 1: (Color online) (a) Sketch of graphene with double-48\nsites; (b)First Brillouin zone andthehighsymmetrydirect ion\n(redline); (c) DOS(solid darklines )and fillings /angbracketleftn/angbracketright(dashred\nlines ) as functions of energy with t′=0.10t; and (d) t′=0.20t.\n(marked in Fig. 1) at <n>=0.75 and 1.25 correspond-\ning toE=−2t′±t, respectively as t′< t/6. While when\nt′≥t/6, a third VHS appears at the lower band edge,\nwhich is a square root singularity marking the flattening\nof the energy band near Γ point. They determine much\nof system’s properties. According to the values of tand\nUreported in the literatures for graphene[20, 23, 24],\nthe ratio U/|t|maybe expected to be 2 .2∼6.0, which\nis around the range of half-bandwidth to bandwidth[25],\nwhere the mean filed theory does not work well while the\nDQMC simulation is a useful tool [22]. The exact value\noft′is not known but an ab initio calculation [26] found\nthatt′/tranges from 0.02 to 0.2 depending on the tight-\nbinding parameterizations. Therefore, it is necessary to\nstudy the ferromagnetic fluctuations within the Hubbard\nmodel on the honeycomb lattice by including t′.\nIn the followings, we show that the behaviors of mag-\nnetic correlation are qualitatively different in two filling\nregionsseparatedbytheVHSat <n>=0.75. Inthefilling\nregion below the VHS the system shows a short-ranged\nferromagnetic correlation and the on-site Coulomb in-\nteraction tends to strengthen ferromagnetic fluctuation.\nThe ferromagnetic properties depend on the electron fill-\ning, which may be manipulated by the electric gate. Fur-\nthermore, the ferromagnetic fluctuation is strengthened\nmarkedly as t′increases. Our results highlight the cru-\ncial importance of electron filling and the NNN hopping\nin graphene. The resultant controllability of FM may fa-\ncilitate the new development of spintronics and quantum\nmodulation.\nTo study ferromagneticfluctuations, wedefine the spin\nsusceptibility in the zdirection at zero frequency,\nχ(q) =/integraldisplayβ\n0dτ/summationdisplay\nd,d′=a,b/summationdisplay\ni,jeiq·(id−jd′)∝angbracketleftmid(τ)·mjd′(0)∝angbracketright(2)\nwheremia(τ)=eHτmia(0)e−Hτwithmia=a†\ni↑ai↑−a†\ni↓ai↓\nandmib=b†\ni↑bi↑−b†\ni↓bi↓. Hereχis measured in unit ofFIG. 2: (Color online) At <n>=0.25, inverse of magnetic\nsusceptibility, 1 /χ(q=Γ) versus temperature with U=3|t|,\nt′=t/10,t/6, andt/5. Fitted line 1 /χ(Γ)=α(T−Θ) are also\nshown. Inset: Magnetic susceptibility χ(q) versus qat differ-\nent temperatures with t′=0.10tandU=3|t|.\nFIG. 3: (Color online) Magnetic susceptibility χ(q) versus\nmomentum qat different t′, hereU=3|t|,<n>=0.25 and\nT=0.10|t|.\n|t|−1, andχ(Γ) measures ferromagnetic correlation while\nχ(K) measures antiferromagnetic correlation.\nWe first present temperature dependence of the mag-\nnetic correlations at <n>=0.25 with different t′andU.\nFig. 2 shows 1 /χ(q=Γ) versus temperature at U=3|t|\nwitht′=t/10,t/6, andt/5. Data for U=5|t|ast′=t/5\nare also shown. In the inset, we present χ(q) versus mo-\nmentum qat different temperatures with t′=t/10 and\nU=3|t|. It is obvious that χ(q) has strong temperature\ndependence and one observes that χ(M) andχ(K) grow\nmuch slower than χ(Γ) with decreasing temperatures.\nMoreover, 1 /χ(Γ) exhibits Curie-like behavior as tem-\nperature decreases from |t|to about 0.1 |t|. Fitting the\ndata as 1 /χ(Γ)=α(T−Θ) (solid lines in Fig.2) shows\nthat Θ is about 0.02 |t|≃580Katt′=t/5, and we also\nnote that both Θ and χ(Γ) are enhanced slightly as the\non-site Coulomb interaction is increased. Positive values\nof Θ indicate that the curves of 1 /χ(Γ) start to bend\nat some low temperatures and probably converge to zero\nasT→0,i.e.,χ(Γ) diverges. This demonstrates the\nexistence of ferromagnetic state in graphene.\nFrom Fig. 2, we may also notice that t′plays a re-\nmarkable effect on the behavior of χ(q), and results for3\nFIG. 4: (Color online) Magnetic susceptibility χ(q=Γ) (red)\nandχ(q=K) (dark) versus electron filling at U=3|t|and\nT=0.167|t|with (a) t′=0.1tand (b)t′=0.2t.\nχ(q)dependenton qwithdifferent t′atU=3|t|,T=0.10|t|\nand<n>=0.25 have been shown in Fig. 3. Clearly, χ(Γ)\ngets enhanced greatly as t′increases, while χ(M) and\nχ(K) increase only slightly. Thus, again it is significant\nto demonstrate that ferromagnetic fluctuation gets en-\nhanced markedlyas t′increases. Furthermore, the strong\ndependence of FM on t′suggests controllability of FM in\ngraphene by tuning t′.\nA great deal of current activity in graphenearises from\nits technological significance as a semiconductor material\nwhere carrier density can be controlled by an external\ngate voltage[7–9]. To understand filling dependence of\nmagnetic correlations intuitively, we present χ(Γ) (red),\nχ(K) (dark), and their ratio χ(Γ)/χ(K) (blue) versus\nfilling for (a) t′=0.1tand (b) t′=0.2tin Fig. 4, where\nU=3|t|andT=0.167|t|. There is a crossover between\nχ(Γ) and χ(K), which indicates that the behaviors of\nχ(q) are qualitatively different in two filling regions sep-\narated by the VHS at <n>=0.75, where the ratio is 1.\nThis is due to the competition between ferromagnetic\nand antiferromagnetic fluctuations. The antiferromag-\nnetic correlations are strong around the hall-filing and\nthey may dominate the shape of χ(q) in a wide filling\nrange up to the VHS. The effect of t′in enhancing ferro-\nmagnetic fluctuation can also be seen by comparing Fig.\n4 (a) with (b). At electron filling <n>=0.25, the ratio of\nχ(Γ)/χ(K) isabout maximumfor t′=0.2tandis substan-\ntial fort′=0.1t, which is the reason why did we choose\nelectron filling 0.25 in Figs. 2, and 3. From the global\npicture shown in Fig. 4, it is clear that the strength of\nferromagnetic correlation strongly depends on the elec-\ntron filling, which may be manipulated by the electric\ngates in graphene, since n∝Vg[7–9]. The filling region\nfor inducing FM required here likely exceeds the current\nexperimental ability. In fact, the challenge of increasing\nthe carrier concentration in graphene is indeed very im-\nportant and it is a topic now in progress. The second\ngate (from the top) and/or chemical doping methods are\ndevoted to achieving higher carrier density[10]. More-\nover, our results present here indicate the electron filling\nmarkedly affects the magnetic properties of graphene,and the controllability of FM may be realized in fer-\nromagnetic graphene-based samples. Furthermore, the\nchange of ferromagnetic correlation with t′may also lead\nto controllability of FM in graphene. For example, one\ncan tune t′by varying the spacing between lattice sites.\nTuningt′can also be realized in two sub triangular op-\ntical lattices[27, 28]. Due to the peculiar structure of\ngraphene, which can be described in terms of two inter-\npenetrating triangular sublattices, making controlling t′\nin principle possible in ultracold atoms system by using\nthree beams of separate laser[29].\nIn summary, wehavepresentedexact numericalresults\non the magnetic correlation in the Hubbard model on a\nhoneycomb lattice. At temperatures where the DQMC\nwereperformed,wefoundferromagneticfluctuationdom-\ninates in the low electron filling region, and it is slightly\nstrengthened as interaction Uincreases. The ferromag-\nnetic correlation showed strong dependence on the elec-\ntron filling and the NNN hoping integral. This providesa\nroutetomanipulateFMinferromagneticgraphene-based\nsamples by the electric gate or varying lattice parame-\nters. TheresultantcontrollabilityofFMinferromagnetic\ngraphene-based samples may facilitate the development\nof many applications.\nThe authors are grateful to Shi-Jian Gu, Guo-Cai Liu,\nWu-Ming Liu and Shi-Quan Su for helpful discussions.\nThis work is supported by HKSAR RGC Project No.\nCUHK 401806 and CUHK 402310. Z.B.H was supported\nby NSFC Grant No. 10674043.\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Moln´ ar, M. L. Roukes, A. Y.\nChtchelkanova, and D. M. Treger, Science 294, 1488\n(2001); K. Ando, Science 312, 1883 (2006).\n[2] A. T. Hanbicki, O. M. J. vant Erve, R. Magno, G.\nKioseoglou, C. H. Li, B. T. Jonker, G. Itskos, R. Mal-\nlory, M. Yasar, and A. Petrou, Appl. Phys. Lett. 82, 4092\n(2003); A. Murayama, M. Sakuma, Appl. Phys. Lett., 88\n122504(2006).\n[3] I.ˇZuti´ c, J. Fabian and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[4] M. Dragoman and D. Dragoman, Nanoelectronics :\nPrinciples and Devices , 2nd ed. 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A 70, 041603(R) (2004)."    },    {        "title": "1004.4425v1.High_temperature_spin_polarization_of_high_mobility_charge_carriers_in_hybrid_metal_semiconductor_structures.pdf",        "content": "arXiv:1004.4425v1  [cond-mat.mes-hall]  26 Apr 2010High-temperature spin polarization of high-mobility char ge carriers\nin hybrid metal-semiconductor structures\nMeilikhov E.Z., Farzetdinova R.M.\nKurchatov Institute, 123182 Moscow, Russia\nWe consider magnetic properties of the planar structure consistin g of a ferromagnetic metal, di-\nluted magnetic semiconductor and the quantum well (by the example of the hybrid heterostructure\nFe–Ga(Mn)As–InGaAs). In the framework of the mean-field theo ry, there is the significant ampli-\nfication of the ferromagnetism induced by the ferromagnetic meta l (Fe) in the system of magnetic\nimpurities (Mn) due to their indirect interaction via the conductivity c hannel in the quantum well.\nAs a result, the high-temperature ferromagnetism arises leading t o the spin polarization of charge\ncarriers (holes) localized in the quantum well and preserving their hig h mobility.\nIntroduction\nAmong troublesome barriers on the way to the development of the s emiconductor spintron-\nics, there are two principal ones – the lack of semiconductor mater ials and structures which\nwould be (i) ferromagnetic at high (room) temperature, and (ii) wou ld possess high enough\nmobility of charge carriers. In this connection, there may be promis ing hybrid ferromagnetic\nmetal/semiconductor structures whose magnetic properties are significantly determined by the\nhigh-temperature ferromagnetism of the constituent metal [1, 2], and heterostructures based\non diluted magnetic semiconductors with removed doping and charac terized by the high carrier\nmobility in the quasi-two-dimensional conductivity channel [3].\nHigh-temperature magnetism in the semiconductor part of the firs t type structures is con-\nditioned by inducing magnetic order in impurity atoms of the diluted mag netic semiconductor\ndue to the proximity effect [1], however the mobility of polarized charg e carriers (holes) is very\nlow at that ( ∼1-10 cm2/(V·s) [4]). To remedy this, one could spatially separate those atoms\nand holes to obtain the amplification of the above-mentioned seed ma gnetic order by means of\nthe indirect interaction of impurity atoms through the tails of carrie rs’ wave functions mainly\nlocalized in the quantum well. It has been experimentally shown that su ch a spatial separation\nresults in a high carrier mobility ( ∼103cm2/(V·s) [5]), though obtained Curie temperatures\n(with no magnetic seed induced by the proximity effect) do not excee d 250 K [6].\nIn the present paper, we consider magnetic properties of hybrid h eterostructures (of Fe–\nGa1−xMnxAs–InyGa1−yAs-type, cf. Fig. 1) where one could realize both mentioned principle s.\nHigh-mobility charge carriers (holes) are concentrated in the two- dimensional quantum well\nappearing in the narrow-band gap non-magnetic semiconductor In yGa1−yAs near its junction\nwith the wide-band gap magnetic semiconductor Ga 1−xMnxAs, whereas impurity atoms of the\nlatter are magnetized by Fe atoms. Fe film, at such, is in the ferroma gnetic single domain state\nbeing magnetized up to the saturation (along the unit vector ρ0parallel to the interface). Such\na system, as we will show could combine magnetic order with the high ca rrier mobility over a\nwide range of temperatures.\n1Mean-field model\nThe magnetization of the diluted magnetic semiconductor (which is pa rallel to the interface\ndue to the shape magnetic anisotropy) is significantly non-uniform a long the growth axis z\n(cf. Fig. 1). It could be conveniently characterized by the local ma gnetization −1/lessorequalslantj(h)≡\nM(h)/Ms⋜1, directed along the unit vector ρ0(M(h) is the local magnetization at the\ndistancehfrom the Fe/Ga 1−xMnx-interface plane, Msis the saturation magnetization). In the\nframework of the mean-field theory it is defined by the equation\nj(h) =BS/bracketleftbigg\n−W(h)\nkT/bracketrightbigg\n, (1)\nwhereBSis the Brillouin function for the spin Sof Mn atoms,\nW(h) =/summationdisplay\niwFe(Ri)+/summationdisplay\nkwMn(rk) (2)\nis the energy of the magnetic interaction of a given Mn atom with othe r parts of the structure.\nThis energy is the sum of energies wFeandwMnof its pair interactions with individual Fe and\nMn atoms, spaced at distances Riandrkfrom the specified Mn atom, respectively. Summation\nis performed over all Fe atoms (in the first sum) and all Mn atoms (in t he second sum).\nWhen calculating the first sum in (2), the interaction of Mn atoms will b e considered being\nantiferromagnetic one [1] and corresponding effective magnetic fi eld being also directed along\nthe unit vector ρ0. Because considered Fe and Mn atoms are located in media characte rized, in\ngeneral, by different lengthes of the exchange interaction, the re sult of that interaction is not\nmerely the function of the distance Ribetween those atoms. However, for not to complicate\ncalculations with non-principal details we will proceed below from the f ollowing model spatial\ndependence\nwFe(Ri) =−JFeexp[−Ri/ℓFe]ρ0·Si, (3)\nwhereJFeandℓFeare, correspondingly, the characteristic energy and length of th e considered\ninteraction for Mn atom with the spin Si. Putting the coordinate origin in the interface plane,\nwe assume that Fe layer occupies the interval −LFe< z <0, and the semiconductor film\nGa(Mn)As takes up the range 0 <z <L Mn(cf. Fig. 1). Then, in the continual approximation\n/summationdisplay\niwFe(Ri) =−JFeSnFe0/integraldisplay\nz=−LFe∞/integraldisplay\nρ=0exp/bracketleftBig\n−/radicalbig\nρ2+(h−z)2/ℓFe/bracketrightBig\n2πρdρdz, (4)\nwherenFeis the concentration of Fe atoms. From Eq. (4) it follows\n/summationdisplay\niwFe(Ri) =−4πnFeℓ3\nFeSJFeF(h), F(h) =e−h/ℓFe/bracketleftbigg\n1+h\n2ℓFe−/parenleftbigg\n1+LFe+h\n2ℓFe/parenrightbigg\ne−LFe/ℓFe/bracketrightbigg\n.\n(5)\nThe calculation of the second sum in (2) should be preceded by the fo llowing comment.\nPresently, there is no full understanding the nature of the ferro magnetism in diluted magnetic\nsemiconductors. Among mechanisms leading to the ferromagnetic o rdering of magnetic impu-\nrities’ spins they consider different types of their indirect interact ion via mobile charge carriers:\nRKKY-exchange [7], kinematic exchange [8, 9], etc. [8]. In addition, th ere is the universal\n2Bloembergen-Rowland mechanism [10], that does not require the ex istence of charge carriers\n(or their high concentration resulting in the carrier degeneracy) a nd could be drew for explain-\ning ferromagnetism in the systems like Ga(Mn)As, Ga(Mn)N [11]. In th is connection, we will\nuse the RKKY mechanism as a modelinteraction (without insisting upon its universality for\nthe considered systems).\nIn thereally two-dimensional case, when magnetic impurities are situated in the plane of\nthe two-dimensional gas of charge carriers, the energy of their R KKY interaction equals\nw0\nMn(ρk) =−JMnφ(ρk)S0·Sk, (6)\nwhereρkis the distance between a given Mn atom (with the spin S0) and any one of other Mn\natoms (with the spin Sk). The characteristic interaction energy JMnand the function φ(rik)\ndefine by the relationships [12]\nJMn=/parenleftbigg\nJ2\npdm∗a2\n0\n4π/planckover2pi12/parenrightbigg\n, φ(ρk) = (kFa0)2[J0(kFρk)N0(kFρk)+J1(kFρk)N1(kFρk)]e−ρk/ℓMn,(7)\nwhereJpdis the energy of p-dinteraction, kF= (πNs)1/2is the Fermi wave number of carriers,\nJnandNnare Bessel functions (with JMn≈0.1 eV [13]), a0≈6˚A is the side of GaAs cubic\ncell. The exponential factor in Eq. (7) takes into account the damp ing of the interaction (with\nthe characteristic length ℓMn) due to the scattering of carriers [14]. In the bulkdiluted magnetic\nsemiconductor Ga 1−xMnxAs with the actual impurity concentration x≈0.05, the hole mobility\nis rather low: µh=1-10cm2/V·s[4], that corresponds to their mean freepathalong thequantum\nwellℓ0∼a0.\nThose relations fall into the case when Mn atoms and charge carrier s, providing their inter-\naction, are placed together within the quasi-two-dimensional cond uctivity channel.\nInourcase, mobilechargecarriersarelocalizedinthequantumwellb eingspatiallyseparated\nfrom Mn atoms. The interaction of magnetic atoms occurs due to th e leakage of the carrier\nwave function into the region of their arrangement. Then, as it has been shown in [3], the\nthree-link chain of interactions works: (i) interaction of a given impu rityiwith the channel +\n(ii) transfer of the interaction along the quasi-two-dimensional ch annel + (iii) interaction of a\nremoved impurity kwith the channel. Each of them leads to the specific factor in the tot al\nexpression for the interaction energy:\nwMn(ρk,zk) =w0\nMn(ρk)·/bracketleftbiggψ2\nb(zi)\nψ2max/bracketrightbigg\n·/bracketleftbigg∝angbracketleftψ2\na∝angbracketright\nψ2max/bracketrightbigg\n·/bracketleftbiggψ2\nb(zk)\nψ2max/bracketrightbigg\n. (8)\nHere,ψmaxis the maximum value of the carrier wave function in the well (corresp onding to the\npeak of their concentration in the quasi-two-dimensional channel) ,ψb(z) is their wave function,\nleaking into themagneticsemiconductor, and ∝angbracketleftψ2\na∝angbracketrightis theaverage valueofsquared wave function\nin the well that could be found by means of the relation [3]\n∝angbracketleftψ2\na∝angbracketright=1√\n3\n1\n2a∞/integraldisplay\nLMnψ2\na(z)dz\n, (9)\nwhere\na=\n\n∞/integraldisplay\nLMnz2ψ2\na(z)dz/slashBig∞/integraldisplay\nLMnψ2\na(z)dz−\n∞/integraldisplay\nLMnzψ2\na(z)dz/slashBig∞/integraldisplay\nLMnψ2\na(z)dz\n2\n\n1/2\n3is the effective half-width of the wave function in the well.\nSpatial separationofimpurities andchargecarriersintheconsider ed structures leads, onthe\none hand, to weakening the indirect interaction, and results, on th e other hand, in increasing\nthe mean free path of carriers that promotes the strengthening of that interaction. If magnetic\nimpurities are situated out of the well, the mean carrier free path ℓincreases[16] (comparing to\nits valueℓ0corresponding to the case when they are inside the well):\nℓMn=ℓ0∝angbracketleftψ2\na∝angbracketright\n∝angbracketleftψ2\nb∝angbracketright, (10)\nwhere∝angbracketleftψ2\nb∝angbracketright= (1/2√\n3b)/integraltextLMn\n0ψ2\nb(z)dzis the averaged (over the impurity layer) value of the\nsquared wave function with the effective half-width\nb=\n\nLMn/integraldisplay\n0z2ψ2\na(z)dz/slashBigLMn/integraldisplay\n0ψ2\na(z)dz−\nLMn/integraldisplay\n0zψ2\na(z)dz/slashBigLMn/integraldisplay\n0ψ2\na(z)dz\n2\n\n1/2\n.\nTherespectiveincreaseofthecarriermobilitycouldbeverysignifica nt. Forinstance, in[5]it\nhas amounted to 2-3 orders of value that has allowed Shubnikov osc illations of the conductivity\nand quantum Hall effect in the two-dimensional channel of the single -well structure with the\nconcentration of removed (from the channel) impurities x≈0.05.\nThus, the removed doping promotes increasing the carrier mobility a nd, as a consequence,\nleads to increasing the energy of indirect interaction of magnetic imp urities.\nIntroducing the cylindrical coordinate system with the former orig in of thez-axis and the\nradius-vector ρ, parallel to the interface plane, we will characterize the magnetic o rder arising\nin the system of impurities by the reduced magnetization −1<j=j(z,r)<1, which coincides\nwith the impurity spin polarization degree. Neglecting the crystal an d surface anisotropy, one\nnotices that the shape anisotropy and the system symmetry resu lt in that the local magne-\ntization is everywhere directed along the plane of the impurity layer a nd depends on zonly:\nj=j(z).\nIn the continual approximation, the total energy of the indirect in teraction of the impurity,\nlocated in the point z=h,ρ= 0, with all surrounding impurities equals\n/summationdisplay\nkwMn(ρk,zk)≈/integraldisplay/integraldisplay\nwMn(ρ,z)2πρdρdz=\n=−2πnMnS2JMn/bracketleftbigg∝angbracketleftψ2\na∝angbracketright\nψ2max/bracketrightbigg/bracketleftbiggψ2\nb(h)\nψ2max/bracketrightbigg∞/integraldisplay\nρminφ(ρ)ρdρ·LMn/integraldisplay\nz=0/bracketleftbiggψ2\nb(z)\nψ2max/bracketrightbigg\nj(z)dz, (11)\nwhere it has been taken into account that the distance between imp urities could not be smaller\nthan the certain minimum distance, which for Mn atoms substituting G a atoms in GaAs lattice\nequalsρmin=a0/√\n2.\nLet us introduce the function\nΦ(kF,ℓ) =−π\na2\n0∞/integraldisplay\nρminφ(ρ)ρdρ, (12)\n4Then Eq. (11) becomes\n/summationdisplay\nkwMn(ρk,zk)≈2nMna3\n0S2JMnΦ(kF,ℓ)/bracketleftbigg∝angbracketleftψ2\na∝angbracketright\nψ2\nmax/bracketrightbigg/bracketleftbiggψ2\nb(h)\nψ2\nmax/bracketrightbigg\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ2\nb(z)/ψ2\nmax/bracketrightbig\nj(z)dz\n.(13)\nSubstituting (5) and (13) in Eq. (1), one comes to the equation defi ning the spatial mag-\nnetization in the magnetic semiconductor:\nj(h) =BS/bracketleftbiggµH(h)\nkT/bracketrightbigg\n=BS\nCFe(h)\nτ+CMn(h)\nτ\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ2\nb(z)/ψ2\nmax/bracketrightbig\nj(z)dz\n\n,(14)\nwhereτ=kT/JMnis the reduced temperature, CFe(h) = 4πnFea3\n0S(JFe/JMn)(ℓFe/a0)3F(h),\nCMn(h) =C0[ψ2\nb(h)/ψ2\nmax],C0= 2nMna3\n0S2Φ(kF,ℓMn)[∝angbracketleftψ2\na∝angbracketright/ψ2\nmax].\nThe solution of that equation is\nj(h) =BS/bracketleftbiggCFe(h)\nτ+γ(τ)CMn(h)/bracketrightbigg\n, (15)\nwhere the parameter γ(τ) is defined self-consistently by substituting the function (15) in th e\nrelation (14).\nLet us consider, firstly, the case nFe= 0, i.e. the system without the ferromagnetic Fe layer,\nbut with the indirect interaction of magnetic impurities via charge car riers in the quantum\nwell [3]. Then CFe(h)≡0 and the stated substitution leads to the equation\nγ=1\nτ\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ2\nb(z)/ψ2\nmax/bracketrightbig\nBS[γCMn(z)]dz\n, (16)\nwhich determines the parameter γ. It has nonzero solution in the low-temperature region\nτ <τConly, where τCis the Curie temperature.\nNear the Curie temperature, the magnetization is low ( j→0). According to (15) it is\npossible at γ→0 only. Using the series BS(x) =b1x−b3x3+...withb1= (S+1)/3S,b3>0,\none finds the solution of Eq. (16)\nγ2=b1\nb3C2\n0\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ4\nb(z)/ψ4\nmax/bracketrightbig\ndz−τ\nb1C0\n\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ8\nb(z)/ψ8\nmax/bracketrightbig\ndz\n−1\n,(17)\nwhich atγ= 0 provides the Curie temperature:\nτC=b1C0\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ4\nb(z)/ψ4\nmax/bracketrightbig\ndz\n=τ0\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ4\nb(z)/ψ4\nmax/bracketrightbig\ndz\n/bracketleftbig\n∝angbracketleftψ2\na∝angbracketright/ψ2\nmax/bracketrightbig\n.(18)\nwhereτ0= (2/3)nMna3\n0S(S+1)Φ(kF,ℓMn). It determines the temperature range for existing\nintrinsic (not induced by Fe film) ferromagnetism in Ga(Mn)As.\n5Assuming now nFe∝negationslash= 0 butℓMn= 0, we come to the situation with no indirect interaction of\nmagnetic impurities. In that case, impurities are magnetized due to t he exchange interaction\nwith the magnetically ordered system of Fe atoms only. The spatial d istribution of such an\ninduced magnetization is defined by Eq. (15), where one should assu meγ= 0. One could\nestimatethewidth Lindofthatdistribution(atthelevel 1 /2)fromthecondition BS[CFe(h)/τ] =\n1/2, which for S= 5/2 [15] gives CFe(h)/τ≈1.4. AtnFea3\n0≈2,JFe/JMn∼1,ℓFe∼a0,τ∼τC\none findsLind≈4ℓFe. Thus, the induced magnetization of Mn atoms exists in a thin layer\nof the thickness Lind∼20˚A near the Fe/Mn interface. It is just that layer which serves as a\nmagnetic seed being amplified due to the indirect inter-impurity intera ction.\nAt last, in the general case the spatial distribution of the magnetiz ation is defined by\nEq. (15), where the parameter γ(τ) is the root of the equation\n1\na0LMn/integraldisplay\nz=0/bracketleftbig\nψ2\nb(z)/ψ2\nmax/bracketrightbig\nBS/bracketleftbiggCFe(z)\nτ+γ(τ)CMn(z)/bracketrightbigg\ndz−γ(τ)τ= 0. (19)\nFormulae (15), (19) is the main result of the present work. To dete rmine the magnetization\nwith the help of those relations, it is necessary to find the wave func tionsψa(z),ψb(z) of charge\ncarriers in different parts of the considered structure.\nSuch a problem for the heterostructure being the contact of two different semiconductors\n(e.g., GaAs and GaInAs), one of which (GaAs) is diluted by Mn atoms, h as been considered\nby us early [3]. Opposite in sign charges of ionized impurities and mobile ch arge carriers\nin the well produce the electric field ε, which is directed along the normal ( z-axis) to the\nheterojunction plane and makes the potential to be non-uniform: U=U(z). The exact self-\nconsistent determination of the potential U(z) and wave functions of mobile charge carriers\nrequires the consistent solution of Schr¨ odinger and Poisson equa tions that is usually found by\nnumerical iterative methods [16] (relevant calculations could be per formed, for example, by\nmeans of the openly accessible package [17]). However, our aim is to d erive simple analytical\nexpressions describing magnetic properties of the considered sys tem. Thereby, we will use\nthe heterojunction model with the triangle well bottom and barrier top. Tests indicate that\nwave functions found with that triangle model are very close to exa ct results [3]. Respective\nexpression for the carrier potential energy has the form\nU(z) =/braceleftbigg\nU0+(z−LMn)eε,z <LMn,\n(z−LMn)eε,z >LMn(20)\n(as previously, z=LMncorresponds to the heterojunction plane, magnetic impurities are\nsituated in the region 0 <z <L Mn). The slope of the well bottom is determined by the electric\nfieldε≈(4π/κ0)eNs, produced by charges located in the well.\nAs before, we assume the carrier density being not too high, so tha t the lowest energy level\noccurs to be populated only, and the effective width of the well being so small that mixing of\nlight and heavy hole subbands could be neglected. Near the heteroj unction (z=LMn) the wave\nfunction of carriers on the lowest energy level Ehas the form [3]\nψ(z) =C\n\nψb(z)≡Ai[q(z−LMn)−ε/q2a2\n0], z <LMn\nψa(z)≡Ai(−ε/q2a2\n0)\nBi(−ε/q2a2\n0+u/q2a2\n0)Bi[q(z−LMn)−ε/q2a2\n0+u/q2a2\n0],z >LMn,\n(21)\n6where Ai(z), Bi(z) are Airy functions,\nu≡2m∗a2\n0U0//planckover2pi12, ε≡2m∗a2\n0E//planckover2pi12, q≡/parenleftbigg2m∗\n/planckover2pi12eε/parenrightbigg1/3\n, (22)\nεis the reduced energy of the populated level in the well defined by th e equation\nAi(−ε/q2a2\n0)Bi′(−ε/q2a2\n0+u/q2a2\n0)−Ai′(−ε/q2a2\n0)Bi(−ε/q2a2\n0+u/q2a2\n0) = 0,(23)\ncoefficient Cshould be found from the normalization condition.\nResults\nThough above we had to do with the concrete structure Fe–Ga(Mn )As–In(Ga)As, the qual-\nitative character of our consideration makes using accurate value s of those parameters that\ngovern its behavior to be excessive. Therefore, we assume nFea3\n0= 2,nMna3\n0= 0.15 (that\ncorresponds to x≈0.1),JFe/JMn= 1, and for other parameters we suggest typical values\nLFe=LMn= 7a0[1],ℓFe= 0.75a0,ℓ0= 3a0(that for the bulk diluted magnetic semiconductor\nGa(Mn)As corresponds to the hole mobility ∼10 cm2/V·s).\nThe temperature range of existing intrinsic (not induced by Fe film) f erromagnetism in\nGa(Mn)As, found for that set of parameters, is bounded from ab ove by a rather low Curie\ntemperature τC≈0.026 (TC≈25 K). In Fig. 2, the spatial distribution jMn(z) of the intrinsic\nlocal magnetization of Mn atoms is shown for the temperature τ, close the critical one (the\ncurve Mn ↔Mn). In the same figure, spatial distributions of Mn magnetization, induced by the\nexchange interaction with Fe atoms and decaying with moving off the in terface Fe/Ga(Mn)As\n(z= 0), are represented (curves Fe ↔Mn) for thecase when their indirect interaction is switched\noff.Atlast, curves(Fe ↔Mn+Mn ↔Mn)aretheresultofthecombinedactionofthetwomagnetic\nordering mechanisms revealing good shows of the induced ferromag netism amplification due to\nthe indirect interaction of magnetic Mn impurities. Remarkably, signifi cant amplification of\nthe induced magnetization keeps at temperatures which are higher than the Curie temperature\ncorresponding to the intrinsic ferromagnetism of Ga(Mn)As.\nMagnetization jMn(z=LMn) near the heterojunction plane is of special interest because it\nis just this value determines the spin polarization degree of charge c arriers in two-dimensional\nconductivity channel. Let N−\ns,N+\nsbe concentrations of two-dimensional holes with spins\nantiparallel and parallel to the magnetization, respectively. Then, the spin polarization degree\nξ= (N−\ns−N+\ns)/Ns(Ns=N−\ns+N+\nsis their total concentration) of holes is defined by the\neffective magnetic spin-dependent potential which for the bulk dilut ed magnetic semiconductor\nwith the uniform magnetization jMnhas the form Vmag=nMna3\n0JpdσSMnjMn[18], where σ=\n±1/2 is the hole spin. In the considered case, when charge carriers and magnetized Mn atoms\nare spatially separated, that relation should be added by the facto r allowing for the fact that\ntheir interaction occurs through the tail of the carrier wave func tion which, additionally, is\nnon-uniform within the channel region:\n∝angbracketleftVmag∝angbracketright=jMn(LMn)nMna3\n0JpdσSMn·/bracketleftbig\nψ2\na(LMn)/∝angbracketleftψ2\na∝angbracketright/bracketrightbig\n. (24)\nThe magnetic potential (24) leads to splitting the energy level Ein two spin sub-levels with\nenergiesE+=E+VmagandE−=E−Vmag. Concentrations of two-dimensional carriers at\neach of them are defined by relations N±\ns∝E±, wherefrom it follows\nξ= 2∝angbracketleftVmag∝angbracketright/E=jMn(LMn)·ξ0, (25)\n7whereξ0= 2nMna3\n0σSMn(Jpd/E)[ψ2\na(LMn)/∝angbracketleftψ2\na∝angbracketright]. Intheconsidered system ξ0∼0.5, so thatξ∼\njMn(LMn). The amplification of the magnetization in the region adjoining the he terojunction\nresults in the proportionate increasing of the spin polarization of ch arge carriers in the two-\ndimensional channel.\nFig. 3 demonstrates temperature dependencies of the magnetiza tion of Mn atoms near the\nheterojunction plane: the lower curve (Fe ↔Mn) shows the induced magnetization, the upper\ncurve (Fe ↔Mn+Mn ↔Mn) – the induced one, amplifying by the indirect interaction. The latt er\ncorresponds also (on a certain scale) to the temperature depend ence of the spin polarization\nin the hole channel. Evident magnetization jMn(LMn) (∼10%) remains up to temperatures\nτ∼20τC∼0.5, that corresponds to T∼500 K.\nIn the insert, the temperature dependence of the respective am plification factor Kj, equal to\nthe ratio of the two mentioned magnetizations, is shown. The maximu m amplification occurs\natτ≫τCand comes about Kj≈1.6. Though this effect, as such, could be important, but\nthe most interesting feature of the considered structure is the s ignificant mobility increasing of\nspin-polarized charge carriers (in two-dimensional conductivity ch annel): in comparison with\nthe mobility in the bulk Ga(Mn)As, it increases according to (10) by ℓMn/ℓ0≈25 times and\nreaches the value µh∼103cm2/V·s for the accepted parameters’ set.\nConclusions\nMagnetic properties of the planar structure Fe–Ga(Mn)As–In(G a)As, which consists of the\ndiluted magnetic semiconductor bordering upon the ferromagnetic metal (on one side) and\nupon the quantum well (on another side), are considered. In the f ramework of the mean-\nfield theory, there has been demonstrated the significant amplifica tion of the magnetization,\ninduced by the ferromagnetic metal, in the semiconductor region clo se to the interface due to\nthe indirect interaction of magnetic impurities via the conductivity ch annel. Existing evident\nhigh-temperature magnetization in the considered structure (an d, hence, the noticeable spin\npolarization of carriers, too) is provided by the interaction of Mn at oms with Fe film (keeping\nmagnetization up to ∼1000 K), and the high mobility of spin-polarized charge carriers – by\ntheir moving from the charged impurities. Such a favorable combinat ion of the two important\nparameters in the considered structures holds out a hope that th ey could be of interest as\npossible elements of different spintronic devices.\n8References\n[1] F. Maccerozzi, M. Sperl, G. Pannacione, J. Min´ ar, S. Polesya, H . Ebert, U. Wurstbauer,\nM. Hochstrasser, G. Rossi, G. Woltersdorf, W. Wegschneider, C.H . Back, Phys. Rev. Lett.,\n101, 267201 (2008).\n[2] E.Z. Meilikhov, R.M. Farzetdinova, Phys. Lett. A, 373, 2770 (2009).\n[3] E.Z. Meilikhov, R.M. Farzetdinova, JETP Letters, 87, 482 (2008); JETP, May 2010 (in\nprint).\n[4] R. Moriya, H. Munekata, J. Appl. Phys., 93, 4603. (2003).\n[5] M.A. Pankov, B.A. Aronzon, V.V. Rylkov, A.B. Davidov, E.Z. Meilikhov , R.M. Farzet-\ndinova, E.M. Pashaev, M.A. Chuev, I.A. Subbotin, I.A. Likhachev, B.N . Zvonkov,\nA.V. Lashkul, R. Laiho, JETP, 109, 274 (2009).\n[6] A.M. Nazmul, T. Amemiya, Y. Shuto, S. Sugahara, and M. Tanaka , Phys. Rev. Lett., 95,\n017201 (2005).\n[7] T. Dietl, H. Ohno, F. Matsukara, J. Cibert, and D. Ferrand, Scie nce,287, 1019 (2000).\n[8] T. Jungwirth, Jairo Sinova, J. Maˇ sek, J. Kucˇ era, A. H. MacDo nald, Rev. Mod. Phys. 78,\n809 (2006).\n[9] P.M. Krstajic, V.A. Ivanov, F.M. Peeters, V. Fleurov, and K. Kiko in, Europhys. Lett., 61,\n235 (2003).\n[10] N. Bloembergen, and T.J. Rowland, Phys. Rev., 97, 1679 (1955).\n[11] V.I. Litvinov, and V.K. Dugaev, Phys. Rev. Lett., 86, 5593 (2001).\n[12] D. N. Aristov, Phys. Rev. B. 55, 8064 (1997).\n[13] D.J. Priour, Jr., E.H. Hwang, and S. Das Sarma, Phys. Rev. Lett .,95, 037201 (2005).\n[14] P.G. de Gennes, J. Phys. Radium, 23, 630 (1962).\n[15] T.C. Schulthess, W.M. Temmerman, Z. Szotek, A. Svane, L. Pet it, J. Phys.: Condens.\nMatter,19, 165207 (2007).\n[16] T. Ando, A. Fowler, and F. Stern, Rev. Mod. Phys., 54, No. 2 (1984).\n[17] http://www.nextnano.de/; http://www.nd.edu/ gsnider/.\n[18] M.A. Boselli, I.C. da Cunha Lima, A. Ghazali, Phys.Rev. B, 68, 085319 (2003).\n9Figure captions\nFig. 1. Heterostructure Fe–Ga 1−xMnxAs–InyGa1−yAs. 2D – quasi-two-dimensional conduc-\ntivity channel in the quantum well.\nFig. 2. Spatial distributions of Mn atoms’ magnetization. (Fe ↔Mn) – induced magne-\ntization, (Fe ↔Mn+Mn ↔Mn) – induced magnetization amplified by the indirect interaction,\n(Mn↔Mn) – intrinsic magnetization. Structure parameters: u= 0.36,qa0= 0.25,kFa0= 0.1,\nLFe=LMn= 7a0,ℓ0= 3a0,nFea3\n0= 2,nMna3\n0= 0.15,ℓFe= 0.75a0,JFe/JMn= 1.\nFig. 3. Temperature dependencies of Mn atoms’ magnetization nea r the heterojunction\nplane: (Fe ↔Mn) – induced magnetization, (Fe ↔Mn+Mn ↔Mn) – induced magnetization am-\nplified by the indirect interaction. Stracture parameters are the s ame as in Fig. 2. In the inset:\ntemperature dependence of the amplification factor.\n10InGaAs \nGa(Mn)As \nFe ...z\nLMn \n0\n-LFe 2D 00.5 1.0 00.5 1.0 00.5 1.0 \n0 1 2 3 4 5 6 700.5 1.0 \n  jMn τ/τC = 2 Fe ↔Mn Fe ↔Mn+Mn ↔Mn   jMn \n τ/τC = 4 \nFe ↔Mn Fe ↔Mn+Mn ↔Mn  jMn τ/τC = 20 \nFe ↔Mn Fe ↔Mn+Mn ↔Mn \n  jMn \n \nz/a0τ/τ C = 0.98 τC=0.0256 \nMn ↔Mn Fe ↔Mn Fe ↔Mn+Mn ↔Mn 0 5 10 15 20 25 30 00.5 11.5 \n0 5 10 15 20 25 30 00.2 0.4 0.6 0.8 1.0 \nKj\nτ/τ CjMn (LMn )\nτ/τ CFe ↔Mn Fe ↔Mn+Mn ↔Mn "    },    {        "title": "0903.1726v1.Amplification_of_the_induced_ferromagnetism_in_diluted_magnetic_semiconductor.pdf",        "content": "arXiv:0903.1726v1  [cond-mat.mtrl-sci]  10 Mar 2009Amplification of the induced ferromagnetism\nin diluted magnetic semiconductor\nE.Z. Meilikhov1), R.M. Farzetdinova\nKurchatov Institute, 123182 Moscow, Russia\nMagnetic propertiesofthe planarstructure consistingofaferro magneticmetal and the diluted\nmagnetic semiconductor are considered (by the example of the str ucture Fe/Ga(Mn)As, experi-\nmentally studied in [1]). In the frameworkof the mean field theory, we demonstrate the presence of\nthe significantamplificationofthe ferromagnetism, induced bythe f erromagneticmetal in the near-\ninterface semiconductor area, due to the indirect interaction of m agnetic impurities. This results\nin the substantial expansion of the temperature range where the magnetization in the boundary\nsemiconductor region exists, that might be important for possible p ractical applications.\nIntroduction\nOne of the major roadblocks on the way to developing the semicondu ctor spintronics is now\nthe lack of semiconductor materials and structures that would be f erromagnetic at the room\ntemperature. Today, the record value of the Curie temperature is reached with the diluted\nmagnetic semiconductor Ga 1−xMnxAs (x= 0.05−0.07) and equals TC≈180 K [2]. In this\nconnection, the interest has been aroused in hybrid structures o f the metal/semiconductor type\nwhosemagneticpropertiesaretothegreatextent defined bythe high-temperatureferromagnet-\nismofthemetalbeingthepartofthestructure. Forinstance, int herecentpaper[1](selectedfor\naViewpointin Physics)theplanarstructureFe/Ga 1−xMnxAs(100)hasbeeninvestigated. Ithas\nbeenshownthattheinducedferromagneticstateinthenear-inte rfacelayerofthesemiconductor\nfilm survives even at the room temperature. The magnetization of t hat semiconductor region is\nopposite (by the direction) to the magnetization of Fe-film, that su ggests the antiferromagnetic\nexchange interaction between Fe atoms and impurity Mn atoms in GaA s matrix. The intensity\nof that interaction is determined by the relevant effective magnetic fieldHFedepending on the\nmutual disposition of a given Mn atom and different Fe atoms interact ing with the latter. In\naddition, there is the indirect exchange interaction of the ferroma gnetic nature between Mn\natoms putting into effect via mobile charge carriers (for example, by the RKKY mechanism).\nThat interaction is characterized by the effective exchange field HMndepending on the distance\nrk(k= 1,2,...) between a given Mn atom and different Mn atoms interacting with the latter.\nTherewith, it is essential that the carrier mobility in the diluted magne tic semiconductor is so\nlow (on the order of 10 cm2/V·c [3, 4]) that due to the collision broadening of the hole energy\nlevels the system becomes to be effectively three-dimensional and t here is no need to take into\naccount any effect of the size quantization [5].\nWe show that the interference of both mentioned interactions, re sponsible for the magnetic\nordering of Mn atoms, multiplies the effect of each mechanisms sever ally. In fact, the magnetic\nseeding which is the spin polarization of semiconductor magnetic atom s, induced by the fer-\nromagnet, is significantly amplified due to the indirect interaction bet ween those atoms. As a\nresult, the Curie temperature, determining the existence of the f erromagnetism in theboundary\nregion of the magnetic semiconductor increases essentially.\n1)e-mail: meilikhov@imp.kiae.ru\n1Mean field approach\nThe magnetization in the boundary layer of the diluted magnetic semic onductor is parallel\nto the interface plane and significantly non-uniform along the direct ion perpendicular to the\nlatter(z-axis). Itisconvenient tocharacterizesuchanon-uniformmagne tizationbythereduced\nlocal magnetization −1/lessorequalslantj(h)≡M(h)/Ms⋜1 (his the distance from the interface plane, Ms\nis the saturation magnetization). In the framework of the mean fie ld theory, it is defined by\nthe equation\nj(h) = tanh/bracketleftbiggµHeff(h)\nkT/bracketrightbigg\n, (1)\nwhereµis the magnetic moment of the impurity Mn atom,\nHeff(z) =/summationdisplay\niHFe(Ri)+/summationdisplay\nkHMn(rk) (2)\nis the total effective field effecting on a given Mn atom. Summation is pe rformed (in the first\nsum) over all Fe atoms, and (in the second sum) over all Mn atoms.\nWhen calculating the first sum in (2) one should take into account tha t considered Fe and\nMn atoms are, in general, situated in media with different exchange int eraction lengths. Hence,\nthe result of that interaction is not a simple function of the distance Ribetween those atoms.\nHowever, lest the calculation be complicated by unprincipled details, w e proceed below from\nthe simple exponential spatial dependence of the form\nµHFe(Ri) =jFe(Ri)JFeexp[−Ri/ℓFe], (3)\nwhereJFeandℓFeare, accordingly, the characteristic energy and the length of the interaction,\njFe(Ri) is the local magnetization of the Fe film in the point of the relevant Fe atom location.\nChoosing the coordinate origin in the interface plane, assume Fe laye r being disposed in the\nregion−LFe< z <0, and the semiconductor film is in the interval 0 < z < L Mn. Then, in the\ncontinual approximation\n/summationdisplay\niµHFe(Ri) =JFenFe0/integraldisplay\nz=−LFe∞/integraldisplay\nρ=0jFe(z)exp/bracketleftBig\n−/radicalbig\nρ2+(h−z)2/ℓFe/bracketrightBig\n2πρdρdz, (4)\nwherenFeis the concentration of Fe atoms. In principle, the magnetization jFedepends on\nthe temperature, however if the latter is significantly lower than th e ferromagnetic Curie tem-\nperature and the film itself is in the single-domain state, then jFe(z)≡1, and it follows from\n(4)\n/summationdisplay\niµHFe(Ri) = 4πnFeℓ3\nFeJFeΦ(h),Φ(h) =e−h/ℓFe/bracketleftbigg\n1+h\n2ℓFe−/parenleftbigg\n1+LFe+h\n2ℓFe/parenrightbigg\ne−LFe/ℓFe/bracketrightbigg\n.\n(5)\nBefore calculating the second sum in (2) it should be noted there is no complete understand-\ning the nature of the ferromagnetism in diluted magnetic semiconduc tors so far. Among mech-\nanisms leading to the ferromagnetic ordering of magnetic impurity sp ins various forms of their\nindirect interaction, induced by mobile charge carriers, are conside red: RKKY-exchange [6],\nkinematic exchange [7, 8], etc. [7]. In addition, there is the universal Bloembergen-Rowland\n2mechanism [9], which does not require the presence of mobile charge carriers (or their high\nconcentration leading to the carrier degeneration) and could be dr ew for describing ferromag-\nnetism in systems of the Ga(Mn)As, Ga(Mn)N type [10]. The essence of that mechanism is the\ninteraction of impurity spins via virtual holes appearing in the valence band on the transition of\nelectrons from that band to acceptor levels. The energetic thres hold of this process, concerned\nwith the finite energy ∆ of the acceptor ionization, results in the exp onential decay of the\nrelevant interaction with the inter-impurity distance. The charact eristic length of the decay is\ndefined by the de Broglie wave length λ∆=/planckover2pi1(2m∆)−1/2of holes with the energy ∆ and the\nmassm. The expression for the effective exchange field Heffof the indirect interaction of two\nparallel impurity spins separated by the distance rreads [10]\nµHeff(r) =−J2\npdm2∆\nπ3/planckover2pi14N2r2K2(2r/λ∆), (6)\nwhereJpdis the energy of the contact interaction of the impurity spin with a ho le,Nis the\nconcentration of matrix atoms, K2is the McDonald function. For long distances ( r/greaterorsimilarλ∆/2),\nK2(2r/λ∆)≈(4r/πλ∆)−1/2exp(−2r/λ∆) asymptotically. Keeping for that case in (6) the most\nessential exponential part of the interaction spatial dependenc e only, one writes it in the form\nµHeff(r) =−J0exp(−r/ℓ), (7)\nℓ=λ∆/2,J0∼J2\npd/8π3∆Nλ3\n∆.\nSimilar expression for the energy of the magnetic impurity interactio n is exploited when\ninvestigating the kinematic exchange model [7]. At last, if one turns to the RKKY interaction,\nunderthestrongscatteringofcarriers(seeabove)itsintensity dropswiththedistancesorapidly\nthat one could neglect the alternating-sign nature of that interac tion and describe it by means\nthe main exponential factor of the form exp( −r/ℓ) (ℓis the length of the spin coherency).\nIn the light of the above-stated we use the following model relation f or the indirect in-\nteraction of a given Mn atom with another Mn atom situated in the poin t with the local\nmagnetization equal to j(zk):\nµHMn(rk,zk) =j(zk)JMnexp[−rk/ℓMn]. (8)\nHereJMnandℓMnare, accordingly, the characteristic energy and the length of the indirect\ninteraction. In the continual approximation,\n/summationdisplay\nkµHMn(rk,zk) =JMnnMnLMn/integraldisplay\nz=0∞/integraldisplay\nρ=0j(z)exp/bracketleftBig\n−/radicalbig\nρ2+(h−z)2/ℓMn/bracketrightBig\n2πρdρdz=\n= 2πnMnℓ3\nMnJMnLMn/integraldisplay\n0K(z,h)j(z)dz, (9)\nwherenMnis the concentration of Mn atoms,\nK(z,h) =/parenleftbigg\n1+|h−z|\nℓMn/parenrightbigge−|h−z|/ℓMn\nℓMn. (10)\n3Unlike the infinite system, the value of the integral (9) depends on t he coordinate hof chosen\npoint.\nThe mean field equation (1) could be now written in the form\nj(h) = tanh\n1\nτ\ncFeΦ(h)+cMnLMn/integraldisplay\n0K(z,h)j(z)dz\n\n, (11)\nwherecFe= 4πnFeℓ3\nFe(JFe/JMn),cMn= 2πnMnℓ3\nMnare structure parameters, τ=kT/JMnis the\nreduced temperature. That is the non-linear integral equation de termining the profile and the\ntemperature dependence of the magnetization in the semiconduct or part of the structure. It is\nnon-local: the magnetization j(h) is defined by all points (0 < z < L) of the Ga(Mn)As film.\nResults\nNumerical solution of Eq. (11) has been found, in the same way as in [ 5], by the successive-\napproximationmethod. Thoughwehavekeptinmindtheconcretest ructureFe/Ga 1−xMnxAs[1],\nthe qualitative character of our examination makes using exact valu es of parameters, governing\nthe system behavior, to be surplus. Therefore, we accept that t he constant aof the (face cen-\ntered cubic) Felattice isabouthalf astheconstant ofthe (bodyce ntered cubic) sublattice of Ga\natoms (which are replaced by Mn atoms) and set nFea3= 2,nMna3= 0.05 (that corresponds to\nx≈0.1), and for other parameters accept the values LFe=LMn= 7a[1],ℓFe=a,ℓMn= 1.5a.\nAs for the ratio JFe/JMn, it has been varied over a wide range (see below).\nPutting cFe= 0 in (11), one could find the temperature interval of the intrinsic ( non-\ninduced by the Fe film) ferromagnetism in Ga(Mn)As. The relevant Cu rie temperature occurs\nto be equal τC≈5, orkTC= 5JMn(see below Fig. 2). For that case, in Fig. 1 (left\npanel, curve Mn ↔Mn) the spatial distribution jMn(z) of the local Mn magnetization at the\ntemperature τ= 4< τCis shown. As one would expect, it is symmetric about the middle\nplane (z=L/2) of the semiconductor layer. In the same Fig. 1, one could see the spatial\nmagnetization distribution of Mn atoms, induced by the their exchan ge interaction with Fe\natoms only, without intrinsic indirect interaction (left panel, curve M n↔Fe) forJFe/JMn= 2.5.\nSuch an induced magnetization drops rapidly with moving away from th e interface boundary\nFe/Ga(Mn)As ( z= 0). At last, the third curve (Mn ↔Mn+Mn ↔Fe) in the left panel of Fig. 1\nis the result of the combined action of both mechanisms of magnetic o rdering manifesting in\nthe high amplification of the induced ferromagnetism by the indirect in teraction of magnetic\nMn impurities. The respective magnetization gain is especially high away from the boundary\nFe/Ga(Mn)As ( z/greaterorsimilarL/2), where the induced magnetization is enlarged up to fivefold value.\nIt is remarkable that the substantial amplification of the induced ma gnetization remains\neven at temperatures being much higher the Curie temperature co rresponding to the intrinsic\nferromagnetism of Ga(Mn)As. The middle panel in Fig. 1, referring t o the temperature τ=\n6> τC, demonstrates that the gain of the induced ferromagnetism even in this case remains\nequally high, though away from the interface the absolute value of t he magnetization drops\nsomehow.\nThe magnetization gain in magnetic semiconductor stimulated by the s patially localized\nmagnetic seeding is obviously demonstrated in the right panel of Fig. 1, where as the spatial\ndependence of the effective exchange interaction of Fe and Mn ato ms we have chosen the linear\napproximation Φ(0)+Φ′(0)h= 1−h/2ℓFeof the function (5) (different from zero in the range\n40< h <2ℓFeonly). It is seen that even at the temperature τ= 6, which is essentially exceeds\nthe Curie temperature τC≈4, magnetic ordering appears even in that region of the magnetic\nsemiconductor where the induced seed magnetization is absent.\nIt is convenient to characterize the non-uniformly magnetized laye r Ga(Mn)As by the aver-\nage magnetization\n/angbracketleftjMn/angbracketright=1\nLL/integraldisplay\n0jMn(z)dz. (12)\nTemperature dependencies of that value for the intrinsic (Mn ↔Mn), induced (Mn ↔Fe) and\namplified combined (Mn ↔Mn+Mn ↔Fe) ferromagnetism are shown in Fig. 2 (left panel: ℓMn=\n1.5a,JFe/JMn= 0.5, middle panel: ℓMn= 1.5a;JFe/JMn= 5). It is seen that significant\nmagnetization of Ga(Mn)As remains even at the temperature which is by six (and more) times\nhigher than the intrinsic Curie temperature. Such a giant expansion of the temperature interval\nof existing the ferromagnetism near the interface Fe/Ga(Mn)As h as been experimentally obser-\nved in [1].\nThe effect of amplifying the induced ferromagnetism in the considere d system due to the\nindirect interaction of magnetic impurities in the semiconductor depe nds severely on the in-\ntensityJFeof the exchange interaction Mn ↔Fe and temperature. Corresponding dependencies\nare shown in Fig. 3, wherefrom it is seen that though the maximum rela tive effect (defined\nby the ratio ( jMM+MF−jMF)/jMF, wherejMF,jMM+MFare magnetisations corresponding to\nMn↔Fe and Mn ↔Fe+Mn↔Mn curves) is observed at a temperature somewhat lower than the\nintrinsic Curie temperature (in the considered case, at τ≈τC≈5), the significant (and the\nmost important) amplification effect remains even at much higher tem peratures.\nAt last, notice that varying other system parameters does not lea d to the qualitative mod-\nification of results, describing the properties of the considered sy stem. For instance, three\ntemperature dependencies /angbracketleftjMn(τ)/angbracketrightof the above-mentioned average magnetisations for the case\nℓMn= 3a,JFe/JMn= 0.5, shown in Fig. 1 (right panel), differ from the same dependencies fo r\nthe caseℓMn= 1.5a,JFe/JMn= 0.5 (see the left panel) mainly by the scale change. Conducted\ncalculations demonstrate the stability of basic results relative to va rying problem parameters.\nConclusions\nMagnetic properties of the planar structure consisting of a ferro magnetic metal and the di-\nluted magnetic semiconductor are considered (by the example of th e structure Fe/Ga(Mn)As,\nexperimentally studied in [1]). In the framework of the mean field theo ry, we demonstrate the\npresence of the significant amplification of the ferromagnetism, ind uced by the ferromagnetic\nmetal in the near-interface semiconductor area, due to the indire ct interaction of magnetic\nimpurities. This results in the substantial expansion of the tempera ture range where the mag-\nnetization in the boundary semiconductor region exists – far beyon d the limits of the interval\nlimited from above by the Curie temperature of the magnetic semicon ductor itself. Results\nmight be used for describing properties of combined nanosized syst ems ferromagnetic/diluted\nmagnetic semiconductor.\nThis work has been supported by Grant # 09-02-00579 of the Russ ian Foundation of Basic\nResearches.\n5References\n[1] F. Maccherozzi, M. Sperl, G. Panaccione, J. Mina’r, S. Polesya, H . Ebert, U. Wurstbauer,\nM. Hochstrasser, G. Rossi, G. Woltersdorf, W. Wegscheider, C. H . Back, Phys. Rev. Lett.,\n101, 267201 (2008).\n[2] M.Wang, R.P.Campion, A.W.Rushforth, K.W. Edmonds, C. T.F oxon, B.L.Gallagher,\nAppl. Phys. Lett. 93, 132103 (2008).\n[3] D. Neumaier, M. Schlapps, U. Wurstbauer, J. Sadowski, M. Reinw ald, W. Wegscheider,\nD. Weiss, arXiv:0711.3278v2 (2007).\n[4] R. Moriya, H. Munekata, J. Appl. Phys. 93, 4603 (2003).\n[5] E.Z. Meilikhov, R.M. Farzetdinova, IEEE Trans. on Magn., 44, 2871 (2008).\n[6] T. Dietl, H. Ohno, F. Matsukara, J. Cibert, and D. Ferrand, Scie nce,287, 1019 (2000).\n[7] T. Jungwirth, Jairo Sinova, J. Maˇ sek, J. Kucˇ era, A. H. MacDo nald, Rev. Mod.\nPhys.78, 809 (2006); exhaustive bibliography on this subject is placed on th e site\nhttp://unix12.fzu.cz/ms/navigate.php?cont=public in.\n[8] P.M. Krstajic, V.A. Ivanov, F.M. Peeters, V.Fleurov, and K. Kikoin , Europhys. Lett., 61,\n235 (2003).\n[9] N. Bloembergen, and T.J. Rowland, Phys. Rev., 97, 1679 (1955).\n[10] V.I. Litvinov, and V.K. Dugaev, Phys. Rev. Lett., 86, 5593 (2001).\n6Figure captions\nFig. 1. Spatial distributions of the local semiconductor magnetizat ionjMn(z) near the in-\nterface Fe/Ga(Mn)As at temperatures τ= 4< τC(left panel) and τ= 6> τC(middle and\nright panels) for JFe/JMn= 2.5. Mn↔Mn is the intrinsic magnetization of the semiconductor,\nMn↔Fe is induced one, Mn ↔Mn+Mn ↔Fe is the induced magnetization amplified by the in-\ndirect interaction. The spatial dependence of the effective excha nge interaction of Fe and Mn\natoms: left and middle panels – Eq.(5), right panel – the linear approx imation (see text).\nFig. 2. Temperature dependencies of the average magnetization /angbracketleftjMn/angbracketrightfor the intrinsic\n(Mn↔Mn), induced (Mn ↔Fe) and amplified combined (Mn ↔Mn+Mn ↔Fe) ferromagnetism of\nthe considered structure. Left panel: ℓMn= 1.5a,JFe/JMn= 0.5, middle panel: ℓMn= 1.5a,\nJFe/JMn= 5, right panel: ℓMn= 3a,JFe/JMn= 0.5.\nFig. 3. Dependencies of the average magnetization /angbracketleftjMn/angbracketrightfor the induced (Mn ↔Fe) and\namplified combined (Mn ↔Mn+Mn ↔Fe) ferromagnetism of the considered structure on the ex-\nchange (Mn ↔F) interaction energy JFeat the temperature τ= 6> τC. In the insert – tempera-\nture dependencies of the relative gain effect for the induced magne tization at various JFevalues.\n70 1 2 3 4 5 6 700.2 0.4 0.6 0.8 1.0 \n0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 700.2 0.4 0.6 0.8 1.0  jMn (z)\nz/aJFe /JMn =2.5 \n     τ =4 Mn ¨Mn M\nn ¨Fe Mn ¨Mn+Mn ¨Fe \n \nz/aM\nn ¨M\nn+M\nn ¨Fe M\nn ¨Fe \nJFe /JMn =2.5 \n     τ =6 \nMn ¨Mn \n   jMn (z)\nz/aJFe /JMn =2.5 \n     τ =6 \nMn ¨Mn Mn ¨Mn+Mn ¨Fe M\nn ¨\nFe 0 10 20 30 40 50 60 00.2 0.4 0.6 0.8 1.0 \n0 5 10 15 20 25 30 0 5 10 15 20 25 30 00.2 0.4 0.6 0.8 1.0 \n   <  j Mn >\nτJFe /JMn =0.5 \n    l Mn =3 a\nM\nn ¨M\nn+M\nn ¨Fe \nMn ¨Fe M\nn ¨M\nn \nτMn ¨Mn Mn ¨Fe JFe /JMn =5 \n l Mn =1.5 a\nMn ¨Mn+Mn ¨Fe < j Mn >\nτJFe /JMn =0.5 \n  l Mn =1.5 a\nMn ¨Mn Mn ¨Fe+Mn ¨Mn Mn \n¨Fe 0.01 0.1 1 50.01 0.1 1\n0 5 10 15 20 25 30 00.2 0.4 0.6 0.8 1.0 <jMn >\n J Fe /JMn τ=6 \nMn ¨Fe \nMn ¨Fe+Mn ¨Mn \n(j MM+MF - j MF )/j MF \nτ5JFe /J Mn =0.5 \n1.5 "    },    {        "title": "1007.2517v1.Variational_cluster_approach_to_ferromagnetism_in_infinite_dimensions_and_in_one_dimensional_chains.pdf",        "content": "arXiv:1007.2517v1  [cond-mat.str-el]  15 Jul 2010Variational cluster approach to ferromagnetism in infinite dimensions and in\none-dimensional chains\nMatthias Balzer and Michael Potthoff\nI. Institut f¨ ur Theoretische Physik, Universit¨ at Hambur g, Jungiusstraße 9, D-20355 Hamburg, Germany\nThe variational cluster approach (VCA) is applied to study s pontaneous ferromagnetism in the\nHubbard model at zero temperature. We discuss several techn ical improvements of the numerical\nimplementation of the VCA which become necessary for studie s of a ferromagnetically ordered\nphase, e.g. more accurate techniques to evaluate the variat ional ground-state energy, improved local\nas well as global algorithms to find stationary points, and di fferent methods to locate the magnetic\nphase transition. Using the single-site VCA, i.e. the dynam ical impurity approximation (DIA), the\nferromagnetic phase diagram of the model in infinite dimensi ons is worked out. The results are\ncompared with previous dynamical mean-field studies for ben chmarking purposes. The DIA results\nprovide a unified picture of ferromagnetism in the infinite-d imensional model by interlinking different\nparameter regimes that are governed by different mechanisms for ferromagnetic order. Using the\nDIA and the VCA, we then study ferromagnetism in one-dimensi onal Hubbard chains with nearest\nand next-nearest-neighbor hopping t2. In comparison with previous results from the density-matr ix\nrenormalization group, the phase diagram is mapped out as a f unction of the Hubbard- U, the\nelectron filling and t2. The stability of the ferromagnetic ground state against lo cal and short-range\nnon-local quantum fluctuations is discussed.\nPACS numbers: 71.10.Fd, 71.10.Hf, 75.10.-b, 75.10.Jm\nI. INTRODUCTION\nItinerant ferromagnetism of nanometer-sized\ntransition-metal systems deposited on non-magnetic\nsurfaces has attracted much attention recently. It is\na fascinating physical but also technological vision to\ncontrol the geometrical arrangement of the nanosystem\non the atomic scale while studying its magnetic prop-\nerties with atomic resolution.1This provokes new and\nexciting questions. It is highly interesting, for example,\nto understand how many atoms are necessary and how\nthese atoms should be arranged geometrically to ensure\na stable ferromagnetic state.\nOne important issue for magnetic nanosystems is their\nstability againstthermal fluctuations.2Thisis mainly de-\ntermined by anisotropies. Anisotropic contributions to\nthe total energy of a magnetic system can be several or-\nders of magnitude higher at a surface or for a small clus-\nter or chain as compared to a three-dimensional bulk of\nthe same material. Nevertheless, the anisotropy strength\nis usually still much smaller than the exchange coupling\nand can thus be safely disregarded for the question of\nwhether or not a ferromagnetic ground state is existing.\nThe stability of a ferromagnetic ground state is, there-\nfore, a matter of quantum fluctuations. Opposed to\nantiferromagnetic order, for example, the order param-\neter is a conserved quantity in the case of ferromag-\nnetism. It is thus mainly the quantum fluctuations of\ntheparamagnetic state which are important and which\nthe spin-polarized state is competing with.\nThe ground state of itinerant systems,3such as mono-\natomic chains of 3d transition metals,4is of particular\ninterest from a theoretical point of view. Besides the\nsubtle interplay between the kinetic energy of the itin-\nerant electrons and their Coulomb interaction, geomet-rical constraints come into play additionally. Even for a\nbulk system, however, and even for the most elementary\nmodels of itinerant ferromagnetism, such as the Hubbard\nmodel,5–7there is no simple and comprehensive physi-\ncal picture for the mechanism that drives ferromagnetic\norder.8–10\nThe physical reason which hampers a straightforward\nunderstanding of itinerant ground-state ferromagnetism\nprobably consists in the fact that the ordering and actu-\nally the formation of local magnetic moments is a strong-\ncoupling phenomenon and thus in general not capable by\nperturbative techniques. This is opposed to antiferro-\nmagnetic order, for example:11Slater or band antifer-\nromagnetism is accessible by weak-coupling approaches,\nHeisenberg or local-moment antiferromagnetism emerges\nin effective low-energy models.\nIt is therefore important to recognize that the same\nproblems already show up in the Hubbard model on\ninfinite-dimensional lattices. This limit, however, is\nrigorously accessible by dynamical mean-field theory\n(DMFT),12–15and many insights concerning itinerant\nferromagnetism could be obtained in this way.16First\nof all, quantum fluctuations are recognized as essen-\ntial. This means that a static mean-field approach, like\nHartree-Fock theory, cannot grasp the main physics and\nlargely overestimates the tendency to collective ordering.\nFurther, ferromagnetism requires a strong local Coulomb\ninteraction and must therefore be investigated by non-\nperturbative means.17–19In addition, however, subtle de-\ntails of the non-interacting electronic structure are like-\nwise important, e.g. a strongasymmetry of the local den-\nsity of states.20–22Non-local parts of the Coulomb inter-\naction, a non-local ferromagnetic Heisenberg exchange\ncoupling, for example, do affect the magnetic ground-\nstate phase diagram but are found to be of lesser im-2\nportance as compared to the Hubbard- Uin general. Fi-\nnally, ferromagnetic order strongly competes with anti-\nferromagnetism and is realized away from half-filling.\nWhile DMFT can be regarded as the optimal theoret-\nical framework to deal with strong local quantum fluctu-\nations and to understand their effect on itinerant ferro-\nmagnetism, it is still a mean-field approach. This means\nthatthefeedbackofnon-localtwo-particle,e.g.magnetic,\nexcitations on the one-particle spectrum and also on the\nthermodynamics is neglected. It is presently unclear, as\nhow severe this approximation must be regarded when\nstudying low-dimensional systems, for example. Spin-\ncharge separation,23to mention a prominent example of\na non-local quantum effect in one-dimensional chains,\ncannot be described by DMFT. As concerns ferromag-\nnetic order in one-dimensional itinerant systems, how-\never, there is reason to be more optimistic that DMFT\nmaycapturethe essentialphysics: Namely, magneticcor-\nrelations and thus the feedback of magnetic correlations\non the ground state can be expected to become less im-\nportant for fillings well below half-filling. At and around\nhalf-filling antiferromagnetic correlations dominate any-\nway. In fact, numerically exact studies by means of\nthe density-matrix renormalization group (DMRG) for\nthet1-t2one-dimensional Hubbard model24–26yield a\nground-state ferromagnetic phase diagram which shows\nstriking similarities with the DMFT results and confirm\nthe main qualitative results listed above.\nThis situation has motivated the present study which\nemploys the variational cluster approach (VCA)27,28to\ninvestigate the ferromagnetic ground-state phase dia-\ngram of the infinite- and the one-dimensional Hubbard\nmodel. The VCA is a thermodynamically consistent29\ncluster mean-field approach which determines the elec-\ntron self-energy by exploiting a general variational\nprinciple.30–32Different approximations can be con-\nstructed by the choice of different reference systems that\ndefine the space of test self-energies for the variational\nprinciple. In this way, single-site mean-field approxima-\ntions, very close to the DMFT, as well as cluster approx-\nimations can be constructed which include the local but\nalso non-local quantum fluctuations, respectively.\nBy comparison with previous DMFT and DMRG re-\nsults it should be possible to answerthe followinginterre-\nlated questions: Howsensitiveis a ferromagneticstate on\na one-dimensional chain to local quantum fluctuations?\nWhat affects its stability more, local or short-range non-\nlocal fluctuations? Is a single-site mean-field approach\nsufficient to predict stable ferromagnetic phases? How\nmuch does it improve compared to a purely static ap-\nproach? Does an inclusion of short-range non-local fluc-\ntuations improve the predictive power? Answers to these\nquestions are particularly important for future studies of\nmagnetic nanosystems in more complex geometries such\nas clusters, coupled chains, etc. and including more or-\nbitals per sites since those systems are in most cases\nnot accessible to an exact numerical treatment via the\nDMRG.A second and likewise important goal of our study is\nto advance the variational cluster approach: Its evalua-\ntionrequiresthe repeatedcalculationoftheself-energyof\nthe reference system for different one-particle parameters\nwhich serve as variational parameters. At zero temper-\nature the numerical solution has to be performed using\nexact-diagonalizationtechniques. Itisthen clearthatthe\nquality of the approximation is limited by the exponen-\ntial growth of the reference system’s Hilbert space, i.e.\nby the limited number of sites that can be taken into ac-\ncount. However, an increasing number of sites in the ref-\nerencesystematthe sametime meansthat thenumberof\nvariationalparametersincreases. To study ferromagnetic\nphases, the number of parameters is doubled because of\nthe additional spin-dependence of each of the parame-\nters. The variationally determined ground-state energy\nbecomes decreasingly sensitive with each additional vari-\national degree of freedom considered. This tightens the\nneed for extremely accurate computations. Here our goal\nistopresentanddiscussdifferent technicalimprovements\nof the VCA.\nThe paper is organized as follows: In the following\nsection II we briefly review the theoretical concept and\nthen address the different technical issues important for\na reliable numerical evaluation in Sec. III. Results for\nthe Hubbard model in infinite dimensions and in one di-\nmension are presented and discussed in Sec. IV, and a\nsummary of the main conclusions is given in Sec. V.\nII. VARIATIONAL CLUSTER TECHNIQUE\nWe consider the single-band Hubbard model5–7in one\ndimension with nearest and next-nearest neighbor hop-\npingt1andt2, respectively (except for Sec. IVB). Using\nstandard notations, the Hamiltonian reads\nH=−t1/summationdisplay\n<ij>,σc†\niσcjσ−t2/summationdisplay\n≪ij≫,σc†\niσcjσ+U/summationdisplay\nini↑ni↓\n(1)\nwhere<·>and≪ · ≫restrict the independent sums\nover lattice sites iandjto nearest and next-nearest\nneighbors, respectively. σ=↑,↓is the spin projection.\nThe strength of the on-site Hubbard interaction is given\nbyU. In the following we consider finite Hubbard chains\nconsisting of Lsites and assume periodic boundary con-\nditions. Unless stated differently, we set t1= 1 to fix the\nenergy scale.\nCalculations are performed using the variational clus-\nterapproximation27,28(VCA) which isaquantum cluster\nmean-field approach based on the self-energy functional\ntheory30–32(SFT). Central to the SFT is the self-energy\nfunctional\nΩ[Σ] = Trln/parenleftbig\nG−1\n0−Σ/parenrightbig−1+F[Σ],(2)\nwhich provides an exact functional relation between the\nself-energy Σ(with elements Σ ijσ(ωn)) and the grand\npotential Ω of Eq. (1) at temperature Tand chemical3\n1t'\n2t'εb\nεcV\nL =1c L =4c\nFIG. 1: Examples for reference systems with Lccorrelated\nsites (blue circles) and one additional bath site (red circl es)\nper correlated site generating single-site (mean-field) an d clus-\nter approximations. The (spin-dependent) on-site energie sεc\nandεbas well as the (spin-dependent) hybridization Vare\noptimized. Hopping parameters t′\n1andt′\n2are kept fixed at\ntheir physical values.\npotential µ. Furthermore, ln denotes the main branch\nof the complex logarithm, and Tr ≡T/summationtext\nnexp(iωn0+)tr\nwithtrbeingthe traceoverthespatialdegreesoffreedom\nand0+apositiveinfinitesimalwhichensuresconvergence\nof the sum over fermionic Matsubara frequencies ωn.G0\nis the free single-particle Green’s function of the model\nwhich can be assumed to be known. The functional\nF[Σ] is formallydefined as the Legendretransform ofthe\nLuttinger-Ward functional Φ[ G] which in turn is defined\ndiagrammatically33or through a functional integral.34\nThe self-energy functional is constructed such that it\nbecomes stationary at the exact (physical) self-energy of\nthe model system Eq. (1):\nδΩ[Σ] = 0. (3)\nDue to the fact that F[Σ] is not known explicitly, an ap-\nproximation must be employed to make use of this vari-\national principle. The idea of the SFT is to restrict the\nvariationoftheself-energyinEq.(3)toasubspaceoftrial\nself-energies which is spanned by the exact self-energies\nof a certain reference system. On this subspace, the self-\nenergyfunctional canbe evaluatedexactly, providedthat\nthe reference system has the same interaction part as the\noriginal model and provided that an exact (numerical)\ncomputation of the test self-energies is possible (see Refs.\n30–32 for details). Generically, a reference system is a fi-\nnite Hubbard cluster with the same Hubbard- Ubut with\nitsone-particleparameters(hoppingandon-siteenergies)\nservingto parametrizethe test self-energies. Usually, one\nselects a limited number of one-particle parameters as\nvariational parameters λ= (λ1,...,λ s). Then, the sta-\ntionarity condition Eq. (3) is approximated requiring the\nfunction\nΩ(λ)≡Ω[Σ(λ)] (4)\nto be stationary, i.e. ∂Ω(λ)/∂λ= 0.\nFor the present study of the one-dimensional Hubbard\nmodel, we consider chains with Lc≤5 correlated sites as\nreference systems. One additional uncorrelated (“bath”)\nsite is attached to each correlated site (i.e. ns= 2 local\ndegrees of freedom), see Fig. 1. Physically, this meansto take retardation effects into account, i.e. with this\nchoice of the reference system, local (temporal) fluctu-\nations are included to some degree. Local fluctuations\nare treated exactlyin the limit of ns=∞only, i.e. for a\ncontinuum of bath degrees of freedom (see Ref. 32). This\nwould correspond to the (cellular) dynamical mean-field\n(C-DMFT) approach35. It hasbeen demonstratedin var-\nious contexts,36–39however, that the main effect of local\ncorrelations is already accounted for with ns= 2, i.e.\nthe essential step is the one from a plain VCA (no bath\nsites) to an ns= 2-VCA while more bath sites give sec-\nondary corrections. This is important since only a finite\n(small) number of sites can be treated when using an\nexact-diagonalization technique at temperature T= 0\nto compute the chain self-energy. Physically, the main\npoint is that the ns= 2 reference systems already al-\nlow for a local(Kondo-type) singlet formation to screen\nthe local magnetic moments. A non-local singlet forma-\ntion is possible for reference systems with Lc≥2. This\ndescribes the feedback of non-local magnetic correlations\non the single-particle excitation spectrum. The degree to\nwhich, quite generally, spatial correlations are accounted\nfor by the VCA is controlled by the choice of Lc, ranging\nfrom a (dynamical) single-site mean-field approximation\nforLc= 1overclustermean-fieldapproachestotheexact\nsolution that is (in principle) obtained with Lc=∞.\nFor a given reference cluster (Fig. 1), we treat the on-\nsiteenergiesofthecorrelatedandofthebathsites, εcand\nεb, as variational parameters. This ensures thermody-\nnamic consistency29with respect to the particle density\nn: Even within the approximation, exactly the same re-\nsult is obtained for nwhich either can be determined via\naµ-derivative of the SFT grand potential at stationarity\nor via a frequency integral over the one-electron spectral\ndensity corresponding to G= (G−1\n0−Σ)−1with the op-\ntimal self-energy. Furthermore, to control the temporal\nfluctuations, the hybridization Vis considered as varia-\ntional parameter (see Fig. 1). For the present study it\nis important to allow for a possible spin dependence of\nall variational parameters to have thermodynamical con-\nsistency with respect to the magnetization in addition.\nAs a simplifying but excellent38assumption to limit the\nnumber of parameters, εc,εbandVare taken to be site\nindependent, and the hopping parameters within the ref-\nerence chain are fixed at their original values, i.e. t′\n1=t1\nandt′\n2=t2.\nCalculations are performed for the grand canonical en-\nsemblekeepingthe chemicalpotential µfixed. Due tothe\ndiscrete energy spectrum of the finite reference cluster,\nand due to the U(1) symmetry ofthe clusterHamiltonian\nH′, however, the cluster ground state reveals a fixed to-\ntal particle number N′within finite µranges. Therefore,\nthe cluster electron density n′=N′/(2Lc) is a discontin-\nuous function of µwhich would give rise to discontinuous\nbehavior of the self-energy and thus of all observables.\nThis fact elucidates the second major motivation for in-\ntroducing one bath site per correlated site: Bath sites\nserve as charge reservoirs. The entries i,jof the self-4\nenergy Σ ijσ(ω) are restricted to the correlated sites, and\nfor a half-filled cluster, i.e. n′= 1 orN′= 2Lc, the elec-\ntron density on the correlated sites can vary in the entire\nrange from n′\nc= 0 ton′\nc= 2 (see Ref. 38 for a detailed\ndiscussion). This is important not only for studies of\ndensity dependencies but also for ferromagnetic phases.\nWith the help ofthe bath sites, an arbitraryand continu-\nous variation of the cluster magnetization m′=n′\nc↑−n′\nc↓\ncan be achieved in the same way. We will comment on\nthis in the discussion of the results below.\nIII. NUMERICAL EVALUATION\nThe actual calculations are performed for finite Hub-\nbard chains (Hamiltonian H, Eq. (1)) with typically\nL=O(103) sites and assuming periodic boundary con-\nditions. It is convenient to consider the reference system\n(Hamiltonian H′) as being composed of Nk=L/Lciden-\nticalanddisconnected clustersconsistingof Lcsiteseach,\ni.e. the sites of the reference system form a translation-\nally invariant superlattice of Nksupersites. Via the self-\nenergy, this superlattice structure is also imposed on the\nexpectation values of observables of the original model.\nThe grand potential at zero temperature can be calcu-\nlated as:31\nΩ(λ) = Ω′−/summationdisplay\nk,nω′\nnΘ(−ω′\nn)+/summationdisplay\nk,nωn(k)Θ(−ωn(k)).(5)\nHere, Ω′is the grand potential of the reference system\nandω′\nnare the poles of the one-electron Green’s func-\ntion of the reference system. These can be calculated\nexactly by means of the Lanczos approach.40Note that\ntheω′\nndo not depend on the wave “vectors” kof the first\nBrillouin zone of the superlattice as the clusters are dis-\nconnected, and thus the k-sum in the second term simply\nyields a factor Nk.ωn(k) are the poles of the (approxi-\nmate) Green’s function of the model Eq. (1). Finally, Θ\ndenotes the Heaviside function, and the λdependence of\nΩ′and of the poles ω′\nnandωn(k) is implicit.\nThere are several technical points which are essential\nfor a reliable numerical evaluation of the VCA and there-\nwith for the interpretation of the results. One of the ma-\njor intentions ofthe present paper is to show how one can\nefficiently deal with the different finite-size effects in the\nevaluationofEq.(5), inparticularclosetoasecond-order\nphasetransition, andwith the problemoffindingstation-\nary points in a high-dimensional parameters space.\nA. Exact frequency summation\nThe first problem consists in the infinite sums over\nMatsubara frequencies in Eq. (5). For not too large\nreference systems, this is performed conveniently and\nnumerically exact by means of the so-called Q-matrix\ntechnique:38,41LetΛ′be the diagonal matrix with thepolesω′\nnof the cluster Green’s function G′as diagonal\nelements. The Lehmann representation of G′can then\nbe written as\nG′(ω) =Q1\nω−Λ′Q†, (6)\nwith an appropriate weight matrix Q(iσ),n. Note that\nQQ†=1∝negationslash=Q†Q. Using this Lehmann representation in\nthe definition Gk= (G−1\n0,k−Σ)−1of the Green’s function\noftheoriginalmodel, itiseasytoseethatthepoles ωn(k)\nofGkthen can be obtained as eigenvalues of the matrix\nM(k) =Λ′+Q†V(k)Q. (7)\nHereV(k) =ε(k)−t′is the difference between the one-\nparticle parameters of the original and the reference sys-\ntem where the matrix ε(k) is the Fourier transform of\ntwith respect to the superlattice. In practice, the effi-\nciency of the Q-matrix technique is set by the dimension\nofM(k), i.e. the number of poles of G′, which, using\nLanczos as a cluster solver, typically amounts to O(100).\nForLc<8, the repeated diagonalization of M(k) for all\nk, and for Lc>8 the Lanczos diagonalization of H′rep-\nresents the dominant contribution to the necessary total\nCPU time, respectively.\nB. Interpolative k-summation\nThek-summation is much more tedious. From Eq. (5)\nit can be read off that Ω( λ) is a non-analytic function for\nfiniteL. We first discuss the (approximate) one-electron\nexcitation energies ωn(k) of the original system. A sign\nchange of one of the energies ωn(k) as function of a vari-\national parameter λicauses a kink in Ω( λ) due to the\nΘ function. Such kinks have a negligible relative weight\nin thek-sum and can be ignored in the thermodynamic\nlimitL→ ∞(if the interacting density of states at the\nFermi edge stays finite). For finite Land in regions of\nthe parameter space where Ω( λ) is nearly flat, however,\nthe mentioned kinks may lead to artifacts or at least to\nsevere convergence problems for numerical techniques to\nfind stationary points, particularly if derivatives of Ω( λ)\nare required.\nIn principle, this finite-size effect can be controlled by\nincreasing the system size Land thereby the number of\nk-pointsNk. Although the computational effort is only\nlinear in Nk, we found it to be much more effective to\nemploy an interpolation algorithm which artificially in-\ncreases the number of k-vectors while keeping the system\nsizefixed. Fortwo adjacent k-vectorsk1andk2, we inter-\npolate between the excitation energies ωn(k1) andωn(k2)\ninstead of calculating ωn(k) for intermediate kby diago-\nnalizationof M(k). Simple linearinterpolationturnsout\nto be sufficient. The effect is a smoothing of the function\nΩ(λ) which considerably stabilizes the subsequent opti-\nmization procedure without a significant increase of the\ncomputational effort.5\n0.5 0.6 0.7 0.8\nhybridization  V-0.6252-0.625-0.6248-0.6246self-energy functional  Ω\nNk=100\nNk=1000Nk,int=200 (Nk=100)\nNk,int=1000 (Nk=100)εc = -0.14\nεb = -0.24\nFIG. 2: Self-energy functional Ω( Σ(V,εc,εb))≡Ω(V,εc,εb),\nplotted versus the hybridization strength Vwhile keeping the\non-site energies fixed at their optimal values, εc=−0.14 and\nεb=−0.24. Calculation for t1= 1 (this sets the energy scale\nthroughout the paper), t2= 0,U= 4,µ=−0.3 and using\ntheLc= 2 reference system. Solid lines represent calcula-\ntions without the interpolation method, dashed lines refer to\ncalculations with a total (original plus interpolated) num ber\nof poles Nk,int.\nWhich pairs of poles at k1andk2correspond to\neach other, respectively, is actually unknown (as long as\none does not analyze the corresponding eigenvectors of\nM(k)) but for practical purposes it is sufficient to sort\nthe respective pole sets and assume the n-th pole at k1to\ncorrespond to the n-th pole at k2, i.e. “level” crossing is\nexcluded. If the number of k-points is sufficiently large,\nthe interpolation procedure affects contributions to the\nsum over kandnin the last term of Eq. (5) only in\nthose cases where ωn(k) crosses zero. Simple continuity\nargumentsthen showthat a “level”crossingis unlikely in\nthose cases, i.e. the possible error of disregarding “level”\ncrossings is O(1/Nk). This simple idea can also easily be\ngeneralized to higher dimensions.\nAs an example to illustrate the interpolation scheme\nwe show in Fig. 2 the self-energy functional as func-\ntion of the hybridization Vfor a reference system with\nLc= 2. Convergence on the scale of the figure is ob-\ntained with Nk≈1000 (red solid line). As can be seen, a\ncomparatively smooth curve can also be obtained with\nNk= 100 original but additional 900 interpolated k-\npoints (dashed green line). The comparison shows that\nthe trend ofΩ( V) is essentiallyunaffected by the interpo-\nlation scheme. This merely produces a tiny overall shift\nof Ω(V) which is irrelevant for the determination of min-\nima and maxima. Using much less (yellow dashed) or\nno interpolated k-points (blue solid line) introduces the\nabove-mentioned kinks. For the example shown here,\nwhere the SFT grand potential is rather flat, these kinks\nwould render a reliable determination of the optimal hy-\nbridization strength impossible.C. Level crossing in the reference cluster\nLet us now turn to the second possible source of a non-\nanalytic behavior of Ω( λ), namely a sign change of one\nof the single-electron excitation energies ω′\nnof the ref-\nerence system as a function of a variational parameter.\nConsider an electron-removal process, for example. Here\nω′\nn=E0(N′)−En(N′−1)−µ≤0 where En(N′) is\nthen-th excited eigenenergy in the invariant subspace\nofH′with (total) particle number N′. If, as a function\nofλ, the excitation energy ω′\nn→0, the ground state of\nthe reference system becomes degenerate with an even-\ntually new ground state in the N′−1 subspace. Hence,\nω′\nn= 0 would indicate a level crossing and a discontinu-\nouschange of the ground state of the reference system.\nThis in turn would induce a discontinuous change of the\nself-energyand thus a discontinuity ofthe SFT grand po-\ntential Ω( λ) which was unphysical for obvious thermody-\nnamical reasons. It is therefore of utmost importance to\nkeepN′= const, i.e. to ensure that the stationary point\nof the SFT functional (and a finite environment in pa-\nrameter space) always corresponds to the same N′. The\nsame holds for z-component of the totalspin.\nIn principle, one might try to ignore a sign change\nofω′\nnas a function of the optimalλ, i.e. as a function\nof a model parameter, and formally calculate the self-\nenergy from the ground state of a subspace with given\nN′. Besides the fundamental problem that this would\nactually correspond to a non-equilibrium situation, such\na procedure also cannot work in practice: As the above\ndiscussion has shown, ω′\nn= 0 would then induce a strong\nkink in Ω( λ) which cannot be smoothed unless extremely\nlarge reference clusters with Lc→ ∞are considered.\nD. Local optimization\nTracing a stationary point of Ω( λ) as a function of a\nmodel parameter can be accomplished by a localtech-\nnique, i.e. assuming the stationary point λstto be close\nto a starting point λ0. A naive application of New-\nton’s method to find a zero of ∇Ω(λ) has turned out\nto be inefficient, however, since equipotential surfaces\nΩ(λ) = const. are usually highly anisotropic. Below\nwe briefly describe our modified algorithm which uses\nan adaptive local coordinate frame with a directionally\ndependent calculation of partial derivatives.\nLetλndenote the set of variationalparameters at iter-\nation step n. Assuming Ω( λ) to be approximately given\nby a quadratic form, the next estimate is\nλn+1=λn−H−1\nn[∇Ω(λ)]λ=λn, (8)\nwhere\nHn,ij=∂2Ω(λ)\n∂λi∂λj/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λn(9)\nis the Hessian at λn. A numerically stable evaluation of\nthe Hessian (and the gradient) is crucial here. This can6\nbe achieved iteratively by principal axis transformation:\nHn=UnDnUT\nn. (10)\nThe diagonal matrix Dncontains the eigenvalues of Hn.\nNew coordinates /tildewideλare defined via the orthogonal trans-\nformation\n/tildewideλ=UT\nnλ. (11)\nThe Hessian for the next n+1-st step is calculated in the\nnew frame:\n/tildewideHn+1,ij=∂2Ω(/tildewideλ)\n∂/tildewideλi∂/tildewideλj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideλ=/tildewideλn+1. (12)\nHere Ω(/tildewideλ)≡Ω(λ(/tildewideλ)). Inverse transformation\nHn+1=Un/tildewiderHn+1UT\nn (13)\nyields the new Hessian in the original frame which is re-\nquired for the next iteration step. The main point is that\nin principal coordinates we have\nΩ(/tildewideλ)−Ω(/tildewideλst) =1\n2/summationdisplay\ni∂2Ω(/tildewideλ)\n∂/tildewideλ2\ni/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewideλ=/tildewideλst(/tildewideλi−/tildewideλst,i)2(14)\nforλclose toλst. Hence, /tildewiderHn+1becomes almost diago-\nnal and can be calculated as a difference quotient using\ndiscrete steps which depend on the principal direction:\n∆/tildewideλi=/radicalBig\n∆Ω//tildewideHn,ii. (15)\nHere ∆Ω is a suitably chosen constant. This implies that\npartial derivatives along directions in parameter space\nwhere Ω( λ) is almost flat are computed with a large ∆ /tildewideλi,\nwhile a small ∆ /tildewideλiis used along directions where Ω( λ) is\nstrongly curved. This has turned out to be crucial for a\nnumerically stable algorithm.\nE. Global optimization\nAs a prerequisite for a local method to trace a station-\nary point, a global method must be availablewhich is ap-\nplicable even if a reasonable starting point is not known.\nExcept for global minimization algorithms which can be\napplied to minimize |∇Ω(λ)|2, for example, there is no\ngeneral global technique to find stationary points in a\nmultidimensional space unfortunately.\nIn context of the SFT, however, there is an elegant\nsolution to this problem since any local method can be\nconverted into a global one with the help of a crossover\nprocedure as has been pointed out in Ref. 42. The main\nidea is to modify the original system by switching off\nthe intercluster hopping (in the same way as it is done\nin the reference system). For this truncated system, theL =2cL =4c L =1cL =4c\nFIG. 3: Crossover from the Lc= 1 to the Lc= 4 reference\nsystem (left) and from the Lc= 2 reference system to the\none with Lc= 4 (right). Dashed lines represent intra-cluster\nhopping parameters scaled by a factor α. It is assumed that a\nstationary point of the SFT functional is known for the case\nwhere the intra-cluster hopping parameters are switched off\n(α= 0). The stationary point is traced locally while adiabat-\nically switching on the hopping parameters, 0 < α < 1. This\nyields a stationary point for the respective Lc>1 reference\nsystem ( α= 1).\nVCA trivially yields the exact solution, and the station-\nary point is trivially given by one-particle parameters of\nthe referencesystem which are equal to those of the trun-\ncated one. One then adiabatically switches on again the\ninter-cluster hopping in the truncated system, i.e. one\nreplaces tinter→αtinterand increases the parameter α\nfromα= 0 toα= 1. During this adiabatic process the\nstationary point of the reference system λst(α) can be\ntraced by means of a localoptimization method. Finally,\nλst=λst(α= 1) is the stationary point of the original\nsystem.\nHere, we present a variant of this crossovertrick which\nmakes use of the fact that it has turned out to be easy to\nglobally find a stationary solution for the Lc= 1 refer-\nence system (i.e. for the dynamical impurity approxima-\ntion). An adiabatic crossover from the Lc= 1 reference\nsystem to an Lc>1 reference system can be performed\nthen by introducing a dimensionless parameter αto scale\nthe nearest-neighbor and next-nearest-neighbor hopping\nin thereference system: t′\n1→αt′\n1andt′\n2→αt′\n2. For\nα= 0 we recover the mean-field solution, for α= 1\nwe have the VCA with Lc>1. This procedure can be\napplied to crossoverbetween two arbitraryreference sys-\ntems. Examples are given in Fig. 4.\nFor our calculations using the Lc= 4 reference sys-\ntem, we started from Lc= 1 and have changed αfrom\n0 to 1 in steps of 0.05. An example is shown in Fig.\n4. The crossover procedure can be done along two dif-\nferent routes, namely from Lc= 1 toLc= 4 directly\nand, in two crossover steps, from Lc= 1 viaLc= 2 to\nLc= 4. The resulting optimal variational parameters\nare the same for both routes as can be seen from the fig-\nure. It is physically plausible that with increasing αand\ncluster size, the values of the optimal parameters tend to\napproachthe “physicalvalues”ofthe originalsystem, i.e.\nV= 0 and εc= 0 while for Lc= 1 stronger deviations\nare necessaryto partially compensate for the effect of the\ntruncated inter-cluster hopping.7\n-0.500.5optimal variational parameters\n1 0.5 0Vεcεb\n0 0.5 1Lc=4 Lc=4 Lc=2 Lc=1\n0 0.5 1\nparameter  α\nFIG. 4: Optimal variational parameters εc,εbandVduring\nthe crossover procedure from Lc= 1 toLc= 4 (left panel,\nfrom right to left) and from Lc= 1 toLc= 4 via Lc= 2\n(middle and right panel). Both crossover procedures yield\nthe same optimal parameters at α= 1. Calculations for an\noriginal system with L= 4000 sites, t1= 1,t2=−0.2,U= 4\nandµ= 0.7.\nF. Calculations at fixed density and magnetization\nSelf-energy-functionaltheoryhasoriginallybeendevel-\noped using the grand-canonical ensemble. At zero tem-\nperature, starting from a grand-canonical Hamiltonian\nwith chemical potential µand magnetic field B,\nH=H−µ/summationdisplay\ni(ni↑+ni↓)−B/summationdisplay\ni(ni↑−ni↓),(16)\nthe SFT grand potential Ω = Ω( λ,µ,B) is a function\nof the variational parameters λ[see Eq. (4)] and of µ\nandB(and other model parameters, such as U). The\nvariational parameters are fixed by ∂Ω(λ,µ,B)/∂λ= 0\nwhile, at the respective stationary point, the derivatives\nwith respect to µandByield the expectation values of\nthe total particle number and magnetic moment:\n∝angb∇acketleftN∝angb∇acket∇ight=−∂Ω(λ,µ,B)\n∂µ,∝angb∇acketleftM∝angb∇acket∇ight=−∂Ω(λ,µ,B)\n∂B.(17)\nWe also define the electron density n=N/L=/summationtext\niσ∝angb∇acketleftniσ∝angb∇acket∇ight/Land the magnetization m=M/L=/summationtext\niσzσ∝angb∇acketleftniσ∝angb∇acket∇ight/Lwithz↑,↓=±1.\nTo construct phase diagrams it is much more conve-\nnient, however, to keep ninstead of µfixed, e.g. to study\nUdependencies at fixed density n. If there is no manifest\nparticle-hole symmetry (off half-filling or for t2∝negationslash= 0), the\ncorresponding chemical potential is not known a priori.\nFurthermore, it is highly desirable to perform calcula-\ntions at fixed minstead of Bto search for a ferromag-\nnetic phase: Starting from a paramagnetic solution with\nm= 0 and adiabatically increasing m, one simply has to\ntrace the solution and find the corresponding B=B(m).\nA spontaneous ferromagnetic solution is then indicated\nby a finite mwithB(m) = 0.Consider the twofold Legendre transformation from\nthe grand potential Ω via the free energy F= Ω+µNto\nthe Gibbs energy G= Ω+µN+BM:\nΩ(λ,µ,B)∝ma√sto→F(λ,N,B)∝ma√sto→G(λ,N,M).(18)\nAt given NandM, the SFT Gibbs energy G(λ,N,M) is\nobtainedfrom G(λ,µ,B,N,M )≡Ω(λ,µ,B)+µN+BM\nvia the original stationarity condition\n∂G\n∂λ= 0⇔∂Ω\n∂λ= 0 (19)\nand two additional conditions fixing µandB\n∂G\n∂µ= 0⇔/angbracketleftBigg/summationdisplay\niσniσ/angbracketrightBigg\n=N (20)\n∂G\n∂B= 0⇔/angbracketleftBigg/summationdisplay\niσzσniσ/angbracketrightBigg\n=M . (21)\nHence, one simply has to consider µandBin addition\ntoλas variational parameters.\nTo illustrate the method we have performed VCA cal-\nculations using the Lc= 1 reference system at fixed den-\nsityn= 0.7 and different Uclose to a second-orderphase\ntransition, see Fig. 5. The magnetization is treated as\na given quantity. The top panel shows the Gibbs free\nenergy as function of m. ForU= 19 this is a con-\nvex function as it is prescribed by thermodynamic sta-\nbility.Uc= 19.7 marks a critical point above which\nthe Gibbs energy becomes thermodynamically unstable\nwithin a certain range of magnetizations. As can be\nseen in the top panel, the phase is locally unstable for\n−0.18< m < 0.18 where the Gibbs energy is con-\ncave. A thermodynamically stable state is obtained via\na Maxwell construction. This yields the solid line. Be-\ntween−0.32< m < 0.32 the Gibbs energy is a con-\nstant which implies B=∂G/∂M = 0. Hence, an in-\nfinitesimal field B= 0+will produce a finite magne-\ntization m= 0.32. States with |m|<0.32 can real-\nized by macroscopy phase separation. We conclude that\nUc= 19.7marksa continuous transition from the param-\nagnetic state to a state with spontaneous ferromagnetic\norder. The function B(m) (lower panel in Fig. 5) can\nbe discussed analogously. Local thermodynamic insta-\nbility is indicated by a negative slope, and instability\nwith respect to a Maxwell constructed state is indicated\nby the dashed line. Finally, the same physics can be seen\nby looking at the free energy (at T= 0 equal to the\nground-state energy E=∝angb∇acketleftH∝angb∇acket∇ight −BM) given as a func-\ntion ofB(see inset for U= 21). Note that mcan be\ncomputed as a derivative of Gor via an integration of\nthe spin-dependent local density of states which, due to\nthe thermodynamical consistency of the SFT, yields the\nsame result.\nIt is interesting that the mean-field ( Lc= 1) approach\nyields a critical interaction Uc= 19.7 which is rather\nclose to the numerically exact result Uc= 18.5 obtained8\n-0.001 0 0.001F(B)\n-0.5810-0.5805\n-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4\nmagnetization  m-0.008-0.00400.0040.008magnetic field  B-5.0×10-40.05.0×10-41.0×10-31.5×10-3Gibbs free energy  G - G(m=0)\nspont. magn.\nmS=0.32U=19\nU=Uc=19.7\nU=21\nFIG. 5: Gibbs free energy G(per site) and the external\nmagnetic field Bas functions of the magnetization m. Cal-\nculations have been performed at fixed density n= 0.7 for\nt1= 1,t2=−0.2 and different U(as indicated) using the\nLc= 1 reference system ( L= 4000). For U= 21 the dashed\nline shows the result of the actual calculation. This soluti on,\nhowever, is unstable for |m|<0.32, and a solution with lower\nGis obtained by a Maxwell construction (solid line). The\ninset shows the free energy F(per site) as function of Bfor\nU= 21.\nvia density-matrix renormalization group.25Characteris-\ntic for a mean-field approach is the square-root behavior\nof the order parameter m(U) close to the critical point,\nm∝√U−Uc. This can be seen in Fig. 6, where the\nmagnetization is displayed as function of Ufor fixed den-\nsityn= 0.7. The inset shows a linear trend of m2close\ntoUc.\nG. Ferromagnetic susceptibility\nFor the calculations of m(U), as displayed in Fig. 6,\nthe simultaneous optimization of 8 variational parame-\nters is required, namely the spin-dependent on-site ener-\ngiesεcandεb, the spin-dependent hybridization strength\nVplusµandB. The number of parameters can be re-\nduced if one is interested in the phase boundaries only.\nRather than tracing a spontaneously symmetry-broken\nsolution, a second-order critical point can be found from\nthe divergence of a suitably defined susceptibility. Here19 20 21 22 23 24 25\ninteraction strength  U00.10.20.30.40.5magnetization  m\n19.5 20 20.500.020.040.06\nm2(U)\nUc=19.7mS=0.32\nFIG. 6: Magnetization mas a function of the interaction\nstrength U. ForU= 21 a spontaneous magnetization ms=\n0.32 is obtained corresponding to the blue lines in Fig. 5.\nInset:m2as a function of Uclose to Uc. The red dashed\nline is a linear extrapolation to m= 0. Parameters of the\ncalculation: see Fig. 5.\nwe consider the homogeneous static magnetic suscepti-\nbilityχ=−(1/L)∂2F/∂B2. The most obvious way\nto calculate χis to apply a small external homogeneous\nmagnetic field Band to look at the linear response m,\ni.e.χ= limB→0m/B. For this case, it is convenient to\nconsider Bas fixed which implies that only 7 variational\nparameters (for Lc= 1 orLc= 2) have to be taken into\naccount. The result for the Lc= 1 reference system is\nshown in Fig. 7 (green lines). The divergence of χat\nUc= 19.7 is consistent with the Ucextracted from the\norder parameter in Fig. 6. As it is typical for a mean-\nfield approach χ−1(see inset) is a linear function of U\nclose toUc. The same holds for Lc= 2. Again this has\nto be expected as critical phenomena should not depend\non the reference cluster size.\nFor the calculation of the susceptibility, a variational\noptimization of spin-dependent variational parameters is\nactually not necessary as was recognized by Eder.43This\ncanbeseeninthe followingway. Considerthefreeenergy\nF=F(λ,B) [Eq. (18)] where we suppress the Ndepen-\ndencein the notation. Due to the stationarityconditions,\n∂F(λ,B)/∂λ= 0, the optimal λcan be considered as a\nfunction of B, i.e.λ=λ(B). Therefore,\nd\ndB∂F\n∂λ(λ(B),B) = 0. (22)\nCarrying out the differentiation, we find\n∂2F\n∂λ∂λ(λ(B),B)dλ(B)\ndB+∂2F\n∂B∂λ(λ(B),B) = 0.(23)\nThis is a linear set of equations which can be solved by9\n16 18 20 22 24\nU-0.04-0.0200.020.04\nχ−1\nUc\n12 16 20 24 28\ninteraction strength  U-100-50050100homogeneous susceptibility  χ\nL = 16000, B = 10-4\nL = 32000, B = 10-3\nL = 4000, m = 10-2\nFIG. 7: Homogeneous static magnetic susceptibility χ=\n∂m/∂B|B=0(per site) as a function of the interaction strength\nUusing the Lc= 1 reference system ( t1= 1,t2=−0.2,\nn= 0.7). The inset shows χ−1(U). Results are obtained by\nthree different techniques. Green lines: spin-dependent op ti-\nmization of the variational parameters at fixed small field B.\nBlue lines: spin-independent parameter optimization acco rd-\ning to Eq. (26). Red lines: optimizing the Gibbs energy by\nvarying the external field at fixed small magnetization. Con-\nvergence is obtained for different respective system sizes Las\nindicated.\nmatrix inversion to get\ndλ(B)\ndB=−/bracketleftbigg∂2F\n∂λ∂λ/bracketrightbigg−1∂2F\n∂B∂λ. (24)\nNow, the susceptibility is given by χ=\n−d2F(λ(B),B)/dB2. Hence\nχ=−d\ndB/parenleftbigg∂F(λ(B),B)\n∂λdλ(B)\ndB+∂F(λ(B),B)\n∂B/parenrightbigg\n.\n(25)\nUsing Eq. (22) and the stationarity condition once more,\nwe see that the first term does not contribute, and thus\nχ=−∂2F(λ,B)\n∂λ∂Bdλ(B)\ndB−∂2F(λ,B)\n∂B2,(26)\nwheredλ(B)/dBcan be eliminated using Eq. (24). Con-\nsequently, for the calculation of χit is sufficient to\nconsider a paramagnetic state and to optimize spin-\nindependent variational parameters only. This strongly\nreduces the computational effort. Once a paramagnetic\nstationary point is found, partial derivatives according\nto Eq. (26) and Eq. (24) have to calculated with spin-\ndependent parameters λin a final step. The resulting χ\nas a function of Uis shown in Fig. 7 as the blue lines.\nA third way to determine the susceptibility is to keep\nthe magnetization mfixed at a small value and vary the\nfieldBtooptimizetheGibbsenergy G,seeredlineinFig.\n7. Here, a divergence of χis indicated by B(m∝negationslash= 0) = 0.This calculation involves spin-dependent parameter opti-\nmization and to take the field Bas a variational param-\neter in addition.\nFinite size effects play a crucial role for the calculation\nof the susceptibility in the critical regime. It turns out\nthat the numerically most expensive calculation where\nthe magnetization is kept fixed is most stable against\nfinite size effects. On the scale of Fig. 7 converged results\nare obtained for a comparatively moderate system size of\nL= 4000 sites.\nIV. RESULTS AND DISCUSSION\nA. Ferromagnetism in the Hubbard model\nApplying the Hartree-Fock approximation to the\nsingle-band Hubbard model, one is lead to the Stoner\ncriterion,44\nUρ0(0)>1, (27)\nfor the existence of a ferromagnetic instability. There-\nwith, the calculation of the free ( U= 0) local density\nof states (DOS) ρ0(ω) at the Fermi edge ω= 0 can\ngive first insights where ferromagnetism is likely to oc-\ncur. Conceptually, however, the Hartree-Fock approach\nis a static mean-field theory, and quantum fluctuations\nare neglected altogether. If at all, reliable results can\nbe derived for the extreme weak-coupling regime where\nferromagnetism is unlikely to occur.\nDespite the simplicity of the Hubbard model, only a\nfew rigorousresultson ferromagnetismareavailable.45–47\nThe Mermin-Wagner theorem48,49excludes spontaneous\nbreaking of the SU(2) symmetry for finite tempera-\ntures and dimensions lower then three. For the one-\ndimensionalcaseandnearest-neighborhopping, Lieband\nMattis50have shown that the ground state for any even\nnumber of electrons is always a non-magnetic singlet in-\ndependent of U. A ferromagnetic ground state is also\nexcluded in the low-density limit n∝ma√sto→0 irrespective of\nUas has been argued by Kanamori.7HisT-matrix ap-\nproach, however, must be based on the assumption that\nweak-coupling perturbation expansion converges. Lieb51\nhas shown that a ferromagnetic ground state is excluded\nfor a bipartite lattice with nearest-neighbor hopping at\nhalf-filling n= 1 any U >0 independent of the dimen-\nsionality. As has been demonstrated by Nagaoka52the\nground state of the half-filled model with one hole added\nis fully polarized for U=∞on bipartite lattices (for\nfcc and hcp lattices with negative hopping integrals) in\nthree or higher dimensions. While criteria for the sta-\nbility of the fully polarized state for thermodynamically\nrelevant dopings could not be obtained,53–55it is possi-\nble to reduce the parameter space left for a stable Na-\ngaoka state in the thermodynamic limit by different vari-\national approaches.56,57Mielke and Tasaki58–61proved\nthe stability of ferromagnetism for special lattices, such10\nas the Kagom´ e lattice, for which there are dispersionless\nparts of the Bloch band (“flat-band ferromagnetism”).\nIn these systems the Fermi sea is degenerate with ferro-\nmagnetic states for U= 0, and ferromagnetism becomes\nstable for U >0. M¨ uller-Hartmann62has considered\nthe one-dimensional model with a next-nearest-neighbor\nhopping such that the free band has two degenerate min-\nima. In the low-density limit a metallic ferromagnetic\nground state is obtained due to ferromagnetic exchange\nin a corresponding effective two-band model. Similarly,\nTasaki63constructed a one-dimensional Hubbard model\nwith next-nearest neighbor hopping which has a ferro-\nmagnetic (insulating) ground state at quarter filling and\nsufficiently strong U.\nB. Ferromagnetism in infinite dimensions\nA comprehensive but approximate approach to fer-\nromagnetic order is provided by dynamical mean-field\ntheory.12–14In the past several studies have addressed\nthe magnetic phase diagram of the Hubbard model on\ninfinite-dimensional lattices where the DMFT becomes\nexact. At least two routes towards ferromagnetic or-\nder could be identified: (i) On a particle-hole symmet-\nric hypercubic lattice ferromagnetism is realized for very\nstrong Coulomb interaction Uand fillings close to half-\nfilling.17–19This is reminiscent of the Nagaoka state.52\n(ii) On the other hand, a moderate Hubbard- Uis suffi-\ncientforlatticeswithafreeDOSexhibitingapronounced\nasymmetry.20–22Herea ferromagneticgroundstate is ob-\nserved in large regions of the U-nphase diagram. A\nsimple mechanism for ferromagnetic order is not appar-\nentalthoughsomeunderstandingcouldbeachieved9,10,21\nby techniques and arguments related to the Hubbard-\nI approach.5The Stoner criterion turns out to be in-\nadequate. Furthermore, also a realization of flat-band\nferromagnetism61can be found64on a Bethe lattice with\ninfinite coordination.\nWe have performed calculations for different lattices,\ni.e. for different free DOS, respectively, using the self-\nenergy-functional approach for a reference system with\none correlated and one bath site only, i.e. Lc= 1, see\nFig. 1. This is referred to as the dynamical impurity ap-\nproximation (DIA) in the following. Let us recall that\nthe DIA with an infinite number of bath sites would ex-\nactly correspond to DMFT.30While local quantum fluc-\ntuations are treated exactly within DMFT, the DIA is\nmuch simpler and includes some local fluctuations only.\nHowever, due to the presence of the bath site it allows\nfor the formation of a local (Kondo-type) singlet. Our\ngoal is here to test the DIA by comparing with available\nDMFT results for the ferromagnetic phase. This serves\nas a benchmark of the approximation. Furthermore, as\na computationally cheap method, the DIA allows for a\nmore comprehensive study of the phase diagram.\nWe start with the first route (i) towards ferromag-\nnetism and consider fillings close to half-filling and very0.0 0.2 0.4 0.6 0.8 1.0 1.2\ndoping  δ / δc0.00.20.40.60.81.0polarization  p\nδc = 14.6%  (NRG)δc = 18.5%  (DIA)\nχ-1\nFIG. 8: Polarization p=m/n (withm=n↑−n↓andn=n↑+\nn↓) as a function of the doping δ= 1−nfor the hypercubic\nlattice in infinite dimensions at U= 50.δcis the critical\ndoping. The energy scale is set by the variance σ2= 0.5 of the\nGaussian free DOS. Results of the DIA (lines) are compared\nwith data from full DMFT-NRG calculations (points) taken\nfrom Zitzler et al.18Dashed line: inverse homogeneous static\nsusceptibility χ−1as obtained from the DIA.\nstrong Coulomb interaction. Here we can compare with\nDMFT results by Zitzler et al.18which have been ob-\ntained using the numerical renormalizationgroup(NRG)\nas an impurity solver. The calculations have been car-\nried out for the hypercubic lattice in infinite dimensions.\nUsing the same conventions as in Ref. 18, the free DOS\nis given by\nρ0(ω) =1√πe−ω2. (28)\nThe variance of the Gaussian DOS σ2= 0.5 sets the\nenergy scale.\nThe results are displayed in Fig. 8. For very strong U\nthe DMRG-NRG data predict an almost fully polarized\nferromagnetic state at low dopings δ= 1−n. Note that\nas a consequence of the tails of the free DOS, the ground\nstate cannot be fully polarized in a strict sense.65How-\never, this exponentially small scale cannot be expected\nto be visible in the data. With increasing doping the\nsystem undergoes a continuous phase transition to the\nparamagnet at a critical doping δc= 14.6%. The result\nof the DIA agrees well with the full DMFT and likewise\npredicts a continuous transition from a fully polarized\nstate to the paramagnetic phase. On the rescaled plot in\nFig. 8, the agreement is even quantitative. However, the\ncritical doping for the phase transition ( δc= 0.185) is\nsignificantly overestimated as compared to DMFT-NRG\n(δc= 0.146). As this means a stronger tendency towards\nferromagnetism, one may conclude that the DIA under-\nestimates the effect of local quantum fluctuations.\nThe DIA results are consistent in themselves: The11\n0 0.2 0.4 0.6 0.8 1\nfilling  n0102030interaction strength  UFMasymmetry  a=0.98\nPMDIA\nQMC\nFIG. 9: Ground-state phase diagram Uvs. filling nfor the\nasymmetric free DOS given by Eq. (31) with half band width\nD= 2 and asymmetry parameter a= 0.98. The solid line\nrefers to the DIA. DMFT-QMC results (points) are taken\nfrom Wahle et al.21\nmagnetization mcan be calculated via the spectral theo-\nrem from the spin-dependent one-electron spectral func-\ntion, or as the derivative of the optimal grand potential\nwith respect to an external magnetic field. Both com-\nputations yield the same result as has been checked nu-\nmerically and as is clear from the formalism.29We also\nchecked numerically that the Luttinger sum rule is ful-\nfilled. Within the DIA this must be respected66- in the\nparamagnetic but also in the ferromagnetic state. In the\nspin-polarized metallic phase there are two Fermi sur-\nfaces with Fermi-surface volumes for σ=↑,↓\nVFS,σ=/summationdisplay\nkΘ(µ−ε(k)−Σσ(0)) (29)\nwhereε(k) is the Bloch band dispersion and Σ σ(ω) the\nk-independent self-energy. The Luttinger theorem then\nreads as\nVFS,σ!=∝angb∇acketleftNσ∝angb∇acket∇ight=L/integraldisplay0\n−∞dω ρσ(ω).(30)\nFor a local and realself-energy, the interactinglocal DOS\ncan be written as ρσ(ω) =ρ0(ω+µ−Σσ(ω)), and\nthe Luttinger sum rule reads µ=µ0σ+ Σσ(0). Here\nµ0σis a (spin-dependent) chemical potential of the non-\ninteracting system such that the spin-dependent particle\nnumbers are the same as for the interacting system.\nNext we consider the second route (ii) towards fer-\nromagnetism and consider a moderate Hubbard- Ubut\na free DOS with a pronounced asymmetry. Here we\ncan compare with the results of Wahle et al.21who em-\nployed the Hirsch-Fye quantum Monte-Carlo method as\nan impurity solver for DMFT. Calculations have been\nperformed for finite but low temperatures and could be\nextrapolated to extract a ground-state U-nphase dia-\ngram which is shown in Fig. 9 (points). The Bloch band0 0.2 0.4 0.6 0.8 1\nfilling  n0204060801000.20.40.60.81.0\nFM\n0.999\nPMUFM\nPM0.5 0.8\n0.9\n0.98\n0.9991 + U\n0.90.80.5\n0.98U\nFIG. 10: Phase diagram U/(1+U) andUvs.n, respectively,\nfor various asymmetry parameters a. DIA calculations for the\nmodel free DOS with D= 2 given in Eq. (31).\ndispersion ε(k) enters the DMFT (and also the DIA)\nvia the free DOS only. Hence, instead of specifying the\nlattice structure and the hopping parameters, one can\nlikewise start from a certain model free DOS as input for\nthe DMFT calculation. This has the advantage that the\neffect of the asymmetry of the free DOS can be studied\nsystematically. In Ref. 21 the following model free DOS\nhas been considered:\nρ0(ω) =c√\nD2−ω2\nD+aω. (31)\nHere,ais a parameter which controls the asymmetry\nwhile the variance stays constant. One can continuously\ntune the DOS from the symmetric case a= 0, corre-\nsponding to the semielliptic DOS of the Bethe lattice\nwith infinite coordination, overan asymmetric DOS with\nmore and more spectral weight peaked in the vicinity of\nthe lower band edge, to a DOS with an inverse square-\nroot divergence at the band edge for a= 1 eventually.\nFurthermore, in Eq. (31), c= (1 +√\n1−a2)/(πD) is\na normalization constant, and Dis the half band width\nwhichis set to D= 2to fix the energyscale. The DMFT-\nQMC results in Fig. 9 correspondto a stronglyasymmet-\nric DOS characterized by a= 0.98.\nAs is obviousfrom Fig. 9, aferromagneticgroundstate\nis realized in large areas of the phase diagram. For low12\nfillings a moderate Hubbard- Uis sufficient for ferromag-\nnetism. With increasing nthe phase boundary Uc(n)\nincreases. Note that large Uvalues cannot be accessed\neasily within the Hirsch-Fye QMC approach. It appears\nthat this phase diagram is ruled by a mechanism that is\ncompletely different from the Nagaoka mechanism that\nhas been suggested to rule the physics in case (i). Con-\ntrary to the results shown in Fig. 8, ferromagnetism be-\ncomesmorelikely with increasing doping δ= 1−nand\npersists down to very small fillings.\nFig. 9 also shows the result of our DIA calculation for\na= 0.98(solid line). Again, we find aconvincingqualita-\ntive agreement with the full DMFT. The phase boundary\nUc(n) shows the same trend but is systematically shifted\ntowards higher interaction strengths. We attribute this\ndifference partly to the very sensitive dependence of the\nresults on the asymmetry parameter. This can be seen\nin Fig. 10 where the result for the ground-state phase\ndiagram from DIA calculations for different asymmetry\nparameters aare given. It is obvious that for aclose to\nunity a tiny change of aand thus of the free DOS results\nin a strong shift of the critical U.\nUsing the DIA one can easily trace the evolution of the\nphase diagram as a function of the asymmetry parame-\nter. As can be see from Fig. 10, Uc(n) can be very small\nfora→1, i.e. for the case where the free DOS diverges\nat the lower band edge. For a <1 the critical interaction\nbecomes large and eventually Uc→ ∞forn→0. With\nincreasing n, however, the phase boundary soon devel-\nops a minimum at nminand then becomes an increasing\nfunction of n. This minimum is located at low fillings\nfor asymmetry parameters close to unity but then shifts\nto higher fillings for a less asymmetric free DOS. At the\nsame time, Uc(nmin) increases strongly. For a= 0.5 we\nfindnmin≈0.9, and the ferromagnetic phase is confined\nto asmall fillingrangecloseto half-fillingand verystrong\nCoulomb interaction.\nIt appears that the two routes towardsferromagnetism\n(Nagaoka vs. asymmetry of the free DOS) are linked\ncontinuously. For even smaller asymmetry parameters\na <0.5ferromagnetismdisappearscompletely. The sym-\nmetric case a= 0 corresponds to a Bethe lattice with\ninfinite coordination with a symmetric free DOS. Here a\nferromagnetic state cannot be stabilized. This is again\nconsistent with full DMFT (NRG) calculations.64Obvi-\nously, the stability of the ferromagnetic state not only\ndepends on the asymmetry abut is also strongly affected\nby the detailed form of the symmetric free DOS since,\nas has been discussed above, for the symmetric Gaus-\nsian free DOS corresponding to the hypercubic lattice,\nthere is again a ferromagnetic phase close to half-filling.\nIt is an open question whether the latter can really be\nattributed to the Nagaoka mechanism. On the one hand,\nthe Nagaoka mechanism needs closed loops on the lat-\ntice which are present for the hypercubic one but absent\nfor the Bethe lattice. On the other hand, the DMFT is\nsensitive to the lattice structure via the free DOS only.\nConcluding, we find that the DIA gives qualitativelyreliable results for the ferromagnetic ground-state phase\ndiagram in all cases studied, as has been corroboratedby\nthe comparison with different full DMFT calculations. A\nreference system with a single bath site appears to be\nsufficient to capture the main physics although quanti-\ntatively there is a tendency to overestimate the range\nwhere ferromagnetism is possible. While local quantum\nfluctuations are included in the DIA in a very simple\nway only, the approximation allows for the formation of\na local (Kondo-type) singlet. Together with the internal\nthermodynamical consistency of the approach and with\nthe fact that Luttinger’s sum rule is respected, this en-\nsures a reliable mean-field description of ferromagnetism.\nC. Ferromagnetism in one-dimensional chains\nGround-state ferromagnetism in the one-dimensional\nHubbard model is restricted by the Lieb-Mattis\ntheorem50whichexcludesafinite orderparameterincase\nof nearest-neighbor hopping only, irrespective of the in-\nteraction U. Including a next-nearest-neighbor hopping\nt2, however, ferromagnetismis provento exist for U=∞\nin the limit t2→0 (t2<0) for all densities.67–69This\nlimit has to be contrasted to the limit t1= 0, but finite\nt2(two-chain model) where the Lieb-Mattis theorem ap-\nplies again. In the low density limit, the ground state\n0.0 0.2 0.4 0.6 0.8 1.0\n|t2| / (|t1| + |t2|)0.00.20.40.60.81.0U / (U + Weff)\nPMFM\nHF/StonerDIA\nNagaoka\nFIG. 11: Magnetic ground-state phase diagram Uvs.t2of the\none-dimensional Hubbard model for quarter filling ( n= 0.5)\nas obtained by the dynamical impurity approximation (DIA,\nblue). The shaded area represents the parameter regime for\nwhich a continuous transition is found. t1= 1 fixes the en-\nergy unit. Only the range t2<0 is considered. Note the\nnon-linear scales with −∞< t2<0 and 0 < U < ∞.\nWeff=/radicalbig\n2t2\n1+ 2t2\n2is the effective band width of the free DOS.\nThe orange line (“Nagaoka”) marks the critical interaction\nstrength below which the fully polarized state becomes inst a-\nble against the paramagnet. The green line (“HF/Stoner”) is\nthe phase boundary according to the Stoner criterion.13\nis ferromagnetic for t2<−1/4 atU=∞.62With a fi-\nnite next-nearest-neighbor hopping, ferromagnetism oc-\ncurs in a rather large part of the U-nphase diagram as\nhas been demonstrated by DMRG calculations of Daul\nand Noack.24,25\nTostudytheeffectoflocalandofshort-rangenon-local\nquantum fluctuations on the stability of the ferromag-\nnetic ground state and to test the predictive power of\nmean-field and cluster mean-field approaches, we have\napplied the dynamical impurity approximation (DIA)\nand the variational cluster approximation (VCA) (see\nFig. 1) to the model with t2∝negationslash= 0. Note that a finite t2\ntranslates into an asymmetric free DOS and that t2∝negationslash= 0\nimplies magnetic frustration with respect to antiferro-\nmagnetic order. We exclusively consider t2<0 (t1= 1\nsets the energy scale) which implies that ferromagnetic\norder is expected to show up for fillings below half-filling.\nWe first check whether or not the Lieb-Mattis theorem\nis respected by the most simple DIA. To this end, DIA\ncalculations have been performed to map out the U-t2\nphase diagram at a fixed filling n= 0.5 (quarter filling).\nCalculations are done using chains with up to 8000 sites.\nThe critical interaction for ferromagnetic order Ucis de-\ntermined by the divergenceof the paramagneticsuscepti-\nbility.χis calculated by using a finite but small external\nfield (B= 0.01). The values for Ucobtained in this way\nare checked for selected t2by calculating the external\nfield for a given small magnetization ( m= 0.01). Devi-\nations are small, i.e. invisible on the scale of the figures\ndiscussed below, and can be neglected.\nThe resulting phase diagram is shown in Fig. 11. The\nDIA predicts a Ucwhich varies strongly with |t2|. To be\nable to display the results for 0 < U <∞and−∞<\nt2<0 in a single picture, non-linear scales for t2and\nUhave been used. The effective bandwidth defined as\nWeff=/radicalbig\n2t2\n1+2t2\n2, i.e. the standard deviation of the\nfree DOS, and t1are chosen as the relevant scales for U\nandt2.\nIn both cases where the Lieb-Mattis theorem holds,\nfort2= 0 and t2→ ∞the DIA predicts Uc→ ∞, i.e.\nthe absence of ferromagnetic order. On the other hand,\nthe static mean-field theory is clearly at variance with\nthe exact theorem as can be seen from Fig. 11 where\nthe green line (“HF/Stoner”) displays the divergence of\nthe Hartree-Fock susceptibility as determined from the\nStoner criterion. Note that the discrepancy between the\nDIA and the HF results is drastic except in the vicinity\noft2≈ −0.3 (|t2|/(|t1|+|t2|)≈0.23) where the chemical\npotential of the non-interacting system coincides with a\nvan Hove singularity in the free DOS. Here the Stoner\ncriterion correctly predicts the ferromagnetic instability\nof the ground state.\nAs one cannot expect that the Lieb-Mattis theorem\nis respected rigorously within a mean-field approach, we\nalso performed calculations for different fillings. In fact,\nforn= 0.4 a divergence of the susceptibility is found for\nt2= 0. However, the large critical value for the interac-\ntion,Uc≈30, indicates that the violation of the Lieb-0 0.05 0.1 0.15 0.2 0.25\nmagnetization  m00.0040.0080.012magnetic field  B\nPMFM\ninstable FMlocally\nFIG. 12: Maxwell construction for the discontinuous phase\ntransition for n= 0.5 andt2=−0.34 (|t2|/(|t1|+|t2|)≈0.25)\nat the critical interaction Uc= 0.43. The shaded areas have\nthe same size, i.e. ∆ G= 0 (see Eq. (32)). There is a solu-\ntion with spontaneous ferromagnetic order at m= 0.20. The\nsecond ferromagnetic solution at m≈0.14 shows a negative\nsusceptibility χ=∂m/∂B < 0 and is thus locally (and glob-\nally) unstable.\n0.4 0.45 0.5\ninteraction strength  U00.050.10.150.2magnetization  m-0.672-0.670-0.668Gibbs free energy  G0.43 0.435\n-0.6702-0.6700-0.6698\nUc\nFIG. 13: Gibbs free energies Gof the paramagnetic and of\nthe ferromagnetic solutions (top panel) and the correspond -\ning magnetizations m(bottom) as functions of the interaction\nstrength Uatt2=−0.34 and quarter filling. The red line\nshows the actual course of the stable solution. The magne-\ntization vanishes continuously at U= 0.49, and at the same\ninteraction strength the susceptibility χdiverges. The true\nphase transition is discontinuous and takes place at Uc= 0.43.14\n0 0.2 0.4 0.6 0.8 1\nfilling  n020406080100interaction strength  ULc=1 (DIA)\nHFDMRG\nPMFM\nFIG. 14: Magnetic ground-state phase diagram of the one-\ndimensional Hubbard model for t2=−0.2 as obtained by the\nDIA. DMRG data from Daul and Noack25and Hartree-Fock\n(HF) results are shown for comparison.\nMattis theorem is “weak” in the sense that it occurs for\nextremely strong interactions only.\nThe comparison with Hartree-Fock theory tells us that\nlocal quantum fluctuations are very important. On the\nother hand, the comparison with VCA results shows that\nnon-local fluctuations are not important in first place.\nFor example, for t2= 0 and for a very strong interaction,\nU= 104, we find a divergence of the susceptibility at a\ncritical filling n= 0.49 within the DIA while n= 0.42\nwithin the VCA for Lc= 2. This is the correct trend\nas the critical filling must vanish for Lc→ ∞due to\nthe Lieb-Mattis theorem. As compared to the improve-\nment of the DIA with respect to static mean-field theory,\nhowever, this appears as marginal.\nIn most caseswefind the phase transitionto be discon-\ntinuous. Fig.12givesanexample. Herethehomogeneous\nmagnetic field Bis shown as a function of the magnetiza-\ntionmfort2=−0.34andU= 0.43. This corresponds to\n|t2|/(|t1|+|t2|)≈0.25 andU/(U+Weff)≈0.22. Sponta-\nneous ferromagnetism requires a finite order parameter\nmatB= 0. There are three solutions: (i) the param-\nagnetic state at m= 0 which shows a positive suscep-\ntibilityχ=∂m/∂B, (ii) a thermodynamically unstable\nferromagnetic solution with negative χ, and (iii) a stable\nferromagnetic solution with χ >0 andm= 0.20. The\npoint|t2|/(|t1|+|t2|)≈0.25 andU/(U+Weff)≈0.22\nis just the transition point as can be seen from the area\nunder the B(m) curve, i.e. from the Maxwell construc-\ntion, but is somewhat belowthe point which is plotted\nin Fig. 11 and at which the susceptibility χdiverges:\nUc/(Uc+Weff)≈0.24.\nThis explains itself in Fig. 13 where the Gibbs free en-\nergies and the magnetizations of the three solutions are\nshown for fixed t2as a function of U. Note that for T= 0\nandB= 0 the Gibbs free energy is the ground-state en-0.5 0.6 0.7 0.8 0.9\nfilling  n020406080100interaction strength  ULc=1 (DIA)\nLc=2\nLc=3\nLc=4DMRGFM\nPM\nFIG. 15: The same as Fig. 14 but using the variational cluster\napproach with reference systems shown in Fig. 1 with 1 ≤\nLc≤4 (and the same number of bath sites, i.e. ns= 2) in\nthe range 0 .7< n < 0.8.\nergy:G=E. It can be seen that the thermodynamically\nunstable solution always has the highest Gibbs free en-\nergy. The actual phase transition therefore takes place at\nthe interaction strength U= 0.43 where the Gibbs free\nenergies of the (stable) ferromagnetic and of the para-\nmagnetic solutions are crossing. This is consistent with\nthe Maxwell construction in Fig. 12 since the difference\nin the Gibbs free energies of the paramagnet and the fer-\nromagnet is given by\n∆G=GFM−GPM=/integraldisplayFM\nPMB(m)dm .(32)\nOn the phase boundary ∆ G= 0. The divergence of\nχ, however, is related to the continuous vanishing of the\norderparameteroftheunstablesolutionat Uc= 0.49(see\nFig. 13, bottom). Note, that on the scale used in Fig.\n11 the difference between Ucand the true (first-order)\ntransition points is almost invisible.\nThet2-range in which the transition is continuous is\nmarked as the shaded area in Fig. 11. In addition, the\nfigureshowsthe t2dependenceoftheinteractionstrength\nat which the fully polarized (“Nagaoka”) state becomes\nunstable as compared to the paramagnetic state. This\nline crosses the phase transition line (diverging χ) at\n|t2| ≈0.10 (corresponding to |t2|/(|t1|+|t2|)≈0.09) and\n|t2| ≈0.39 (corresponding to |t2|/(|t1|+|t2|)≈0.28).\nThis implies that for |t2|>0.39 and for |t2|<0.09 the\nfirst-order transition is a transition from the paramag-\nnetictothefullypolarizedferromagneticstatewhileinall\nother cases the magnetization jumps to a non-saturated\nvalue at the respective transition point. Our DIA results\nare consistent with the DMRG calculations of Daul26\nwhich yield a second-order transition at t2=−0.2 and a\nfirst order transition at t2=−0.8 for quarter filling.\nFor a systematic comparison of the results of the dy-15\nnamical impurity approach with DMRG data,25we fix\nthe next-nearest-neighborhopping to t2=−0.2 and map\nout thephasediagram Uvs.filling n. The resultisshown\nin Fig. 14 in comparison with static mean-field theory.\nThe phase diagram turns out to be qualitatively similar\nto the resultfor infinite dimensions (see Fig. 9). The crit-\nicalinteraction Ucstronglyvarieswith nandbecomesex-\ntremely large for fillings close to half-filling. Despite the\nsimplicity of the reference system, the agreement with\nthe DMRG date is reasonable.\nWithin static mean-field theory, local magnetic mo-\nments are formed in the ferromagnetic state only. This\nmightbe therightpictureforthelow-densityregime. For\nfillingsn/greaterorsimilar0.5, however, static mean-field theory fails to\nreproduce the phase diagram as the tendency towards\nferromagnetic order is overestimated drastically.\nOpposed to static mean-field theory, the DIA allows\nfor local-moment formation already in the paramagnetic\nstate and captures the correlated mean-field physics of\nthe paramagnetic Mott transition at half-filling.31,37,39\nThe high values for the Hubbard interaction necessary\nto produce ferromagnetic order can be understood in\na picture where the ferromagnetic state evolves from a\nhighly correlated paramagnet with preformed but disor-\ndered local magnetic moments. With increasing Uand\nwith increasing fillings nthe local magnetic moments are\nmore and more efficiently screened by a collective (single-\nband)Kondoeffect. Thetendencytoscreenthelocalmo-\nments counteracts the formation of magnetic order and\nthus results in very strong critical interactions. Such a\nmechanism is already included in the DIA. From the rea-\nsonable agreement with the DMRG data and the strong\nimprovement with respect to static mean-field theory, we\ntherefore infer that this mechanism is essential. While\nthe correct Kondo scale cannot be captured with a sin-\ngle bath site ( ns= 2), the possibility to form a local\nsinglet within a thermodynamically consistent approxi-\nmation appears to be a key ingredient to understand the\nphase diagram.\nBesides a screening of the local moment by local fluc-\ntuations, a screening by non-local fluctuations is conceiv-\nable. This would lead to non-local singlets – or even to\nlong-range anti-ferromagnetic order. Consequently, such\na mechanism is expected to be effective for fillings close\nto half-filling where non-local antiferromagnetic correla-\ntionsareimportant. Note, however,that dueto the Lieb-\nMattistheoremthe necessitytoincludeafinite t2already\nsuppresses antiferromagnetic order by magnetic frustra-\ntion to some degree. This might explain that the effect of\nnon-local fluctuations appears to be comparatively weak\nfor intermediate fillings and significant for fillings close\nto half-filling only.\nThis can be seen in Fig. 15 where we compare the\nDIA phase diagram with the results obtained from VCA\ncalculations with finite clusters as reference systems:\nLc= 2−4 while the description of the local degrees\nis unchanged ( ns= 2, one bath site per correlated site).\nForn/lessorsimilar0.75 the critical interaction does not changemuch while for n= 0.8, the critical Uis strongly reduced\nin the cluster approach. The VCA thereby improves the\nagreement with the DMRG.\nAlthough the step from Lc= 1 toLc>1 appears to\nbe essential close to half-filling, the results of the clus-\nter approach have to be interpreted with some care since\nthe expected convergence with increasing cluster size can\nhardly be seen for Lc≤4. This reflects finite-size er-\nrors the size of which can be estimated by comparing the\nresults for different Lcamong each other. Within this\n(considerable) error there is agreement with the DMRG\nresults. On the other hand the VCA, and more impor-\ntant even the DIA, is able to predict the qualitatively\ncorrect trend for the phase diagram. It is also important\nto note within the DIA it is much easier to find, stabi-\nlize and trace magnetic solutions. For the clusters with\nLc>1 we have not been able to find solutions in the\nentire filling range for reasons discussed in Sec. IIIC.\nV. CONCLUSIONS\nThe self-energy-functional theory has been applied to\nthe Hubbard model in infinite and in one dimension to\ninvestigate spontaneous ferromagnetic order. Using dif-\nferent reference systems generating different single-site\nandclustermean-fieldapproximations,i.e.dynamicalim-\npurity and the variational cluster approximations, it is\npossible to study the effects of local and of short-range\nnon-local quantum fluctuations on the stability of the\nferromagnetic ground state.\nWe find that local fluctuations are of crucial impor-\ntance to get a qualitatively correct phase diagram and\nto respect the Lieb-Mattis theorem. Opposed to static\nmean-field theory, ferromagnetic order quite generally\nrequires substantially higher interaction strengths and\ncan be understood as evolving from preformed but disor-\ndered local magnetic moments. The extremely large crit-\nical interactions found in one-dimensional chains could\nthen be attributed to the screening of the local moments\nby local (Kondo-type) correlations which becomes more\nand more effective for increasing filling or interaction\nstrength. This local singlet formation, on a qualitative\nlevel, is already included in the DIA. Singlet formation\ndue to non-local correlations, included in the VCA, ap-\npears to be relevant for fillings close to half-filling only,\nwhile a ferromagnetic ground state can be obtained in\nlarge areas of the parameter space and down to the low-\ndensity limit in particular. The limited importance of\n(antiferromagnetic) non-local correlations is of course in-\nterrelated with the frustration of antiferromagnetic or-\nder due to a next-nearest-neighbor hopping t2, with the\nLieb-Mattis theorem, and, in the case of infinite dimen-\nsions, with an asymmetric free DOS. There is an ob-\nvious similarity of the magnetic phase diagram of the\none-dimensionalmodelwith the phasediagramininfinite\ndimensions which again suggests that local correlations\nplay the predominant role for ferromagnetic order.16\nThis is of some importance for future investigations\nof ferromagnetism in nano-sized objects, e.g. chains or\nclusters on non-magnetic substrates, as it opens a route\nto study those systems by (dynamical) mean-field meth-\nods which, as concerns the system geometry, are more\nflexible than the density-matrix renormalization group,\nfor example. The situation may be contrasted, e.g., with\nthe absence of long-rangeantiferromagnetic order in one-\ndimensional systems which is caused by non-local quan-\ntum fluctuations. In the latter case any mean-field ap-\nproach would be questionable a priori.\nThe study of infinite-dimensional lattices using the\nDIA has shown that the previously known parameter\nrangesthat are favorablefor ferromagneticorder, namely\nlow to intermediate fillings and moderate Uin case of a\nstrongly asymmetric free DOS and fillings close to half-\nfilling and extremely strong Uin case of a symmetric free\nDOS, are linked continuously. This demonstrates the dif-\nficulty to find simple “mechanisms” for ferromagnetic or-\nderintheHubbardmodel. Ferromagnetismshouldthere-\nfore be seen as a complex phenomenon the description of\nwhich necessarily requires non-perturbative and thermo-\ndynamically consistent many-body techniques.\nTo study ferromagnetism within self-energy-functional\ntheory, at least six variational parameters have to be op-\ntimized simultaneously, namely the spin-dependent one-\nparticle energies of the correlated and of the uncorre-\nlated sites andthe spin-dependent hybridizationstrength\nin addition. Two more variational parameters must be\nconsidered for calculations at fixed filling and magneti-\nzation which is convenient for the construction of phase\ndiagrams. This can be accomplished by a number of\ntechnical improvements concerning (i) an accurate treat-ment of k- and frequency summations, (ii) optimization\nalgorithmswhich adaptto the local structureofthe func-\ntional, (iii) global optimization algorithms to find a sta-\ntionary point of the functional. For the calculation of the\nstatic and homogeneous paramagnetic susceptibility, an\noptimizationofspin-independentparametersissufficient.\nThe comparisonwith dynamical mean-field theory and\nwith density-matrix renormalization-group calculations\nfor the infinite-dimensional and for the one-dimensional\nmodel, respectively, has been essential to rate the ap-\nproximations. For both, infinite dimensions and one di-\nmension, a simple DIA turns out to be sufficient for a\nqualitative and rough scan of the phase diagram. This\nmight be sufficient in view of the fact that the Hubbard\nand similar models themselves represent strong simplifi-\ncationsascomparedto arealmaterial. In onedimension,\na cluster approach including short-rangecorrelations, i.e.\ntheVCA,appearstobe necessaryforfillingsclosetohalf-\nfilling. A satisfactoryconvergencewith increasing cluster\nsize, however, could not be obtained. 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Matter 6, 481 (1994)."    },    {        "title": "1007.1774v1.Robust_half_metallic_ferromagnet_of_cubic_VGe3Si4.pdf",        "content": "arXiv:1007.1774v1  [cond-mat.mtrl-sci]  11 Jul 2010Robust half-metallic ferromagnet of cubic VGe 3Si4\nSan-Dong Guo and Bang-Gui Liu\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China and\nBeijing National Laboratory for Condensed Matter Physics, Beijing 100190, China\nTo seek half-metallic ferromagnets compatible with the GeS i semiconductor, we use state-of-the-\nart density-functional-theory methods to studycubic VGe 3Si4. After optimizing its crystal structure\nwith various magnetic orders considered, we find the ferroma gnetic structure is the ground state\nphase and investigate its stability, and then study its spin -resolved electronic structure with an\nimproved method. Our calculated results show that the cubic VGe3Si4is a half-metallic ferromagnet\nwith half-metallic gap 350 meV, and it is structurally stabl e against deformations and magnetically\nrobust against antiferromagnetic fluctuations. These resu lts make us believe that the cubic VGe 3Si4\ncould be synthesized as good thin films soon and used in practi cal spintronic devices.\nPACS numbers: 75.30.-m,71.20.-b,75.50.-y,75.90.+w\nHalf-metallic ferromagnetic materials can play impor-\ntantrolesinhigh-performancespintronicapplicationsbe-\ncause their special electronic structures can supply high\nspin-polarization at high temperatures [1, 2]. This kind\nof materials are special in the sense that one of their two\nspinchannelsismetallicandtheothernonmetallic,which\ncauses a spin-polarization of nearly 100% (there is a tiny,\nor small, modification due to the spin-orbit coupling ef-\nfect) at the Fermi energy [3]. NiMnSb, CrO 2, Fe3O4, and\nCo2MnSi are typical examples for such materials [3–7].\nBecause of the central roles of semiconductor technology\nin modern electronic devices, it has been important to\nseek high-performance half-metallic ferromagnetic mate-\nrials compatible with important semiconductors [8–21].\nGreatadvance hasbeen achievedin transition-metalsub-\nstituted III-V and II-VI semiconductors, such as CrAs,\nCrSb, MnSb, and CrTe [11–20]. For semiconductor spin-\ntronic applications, it is of much interest to explore half-\nmetallic ferromagnets based on the important group-IV\nsemiconductors such as Si, Ge, and GeSi.\nIn this Letter we use state-of-the-art full-potential\ndensity-functional-theory methods to study structural,\nelectronic, and magnetic properties of cubic VGe 3Si4\nbasedontheGeSisemiconductor. Weoptimizeitscrystal\nstructure with various magnetic orders considered, and\nthen investigate the structural and magnetic stability of\nthe ferromagnetic structure as the ground state phase\nand study its spin-resolved electronic structure with an\nimproved method. Our calculated results show that the\ncubic VGe 3Si4is a half-metallic ferromagnet with half-\nmetallic gap 0.35 eV, and it is structurally stable against\ndeformations and magnetically robust against antiferro-\nmagnetic fluctuations. More detailed results will be pre-\nsented in the following.\nWe use a full-potential linearized augmented-plane-\nwaves method within the density functional theory [22],\nas implemented in package WIEN2k [23]. We use the\ngeneralized gradient approximation[24] to optimize the\nstructures and calculate all the total energies, but we use\na modified Becke-Johnson exchange potential [25] to cal-culate the final density of states and energy bands to be\npresented because this method has been proved to give\nmuch better semiconductor gaps for various semiconduc-\ntors and insulators. Full relativistic effects are calcu-\nlated with the Dirac equations for core states, and the\nscalarrelativisticapproximationisusedforvalencestates\n[26–28]. The spin-orbit coupling is neglected because it\nhas little effect on our results. We use 2000 k points in\nthe first Brillouin zone, make harmonic expansion up to\nlmax= 10 in the atomic spheres, and set Rmt∗kmax= 7.\nThe radii of the atomic spheres of V, Ge, and Si are set\nto 2.27, 2.16, and 2.01 atomic unit, respectively. The\nvolumes are optimized in terms of total energy method,\nand the internal atomic position parameters with a force\nstandard of 3 mRy/a.u. The self-consistent calculations\nare considered to be converged when the integration of\nthe absolute charge-density difference between the input\nand out density is less than 0.0001 |e|per formula unit,\nwhereeis the electron charge.\nWe construct the cubic VGe 3Si4by substituting V for\nthe corner Ge atom in the unit cell of the crystalline\nGeSi with zincblende structure, as is shown in Fig. 1.\nThe resultant structure has space group No. 215. We\noptimize its lattice constant and internal atomic posi-\ntion parameters. The optimized lattice constant equals\n5.632˚A and the internal Si position parameter is 0.2509.\nIts total magnetic moment is 1.000 µBper formula unit,\ncan be attributed to the 3 delectron of the V4+ion. The\nVGe3Si4is a half-metallic ferromagnet, but its minority-\nspin conduction band bottom almost touches the Fermi\nlevel. In order to improve the results, we re-do the self-\nconsistentcalculation, with allthe structuralparameters,\nusing an improved exchange-correlation method includ-\ning the modified Becke-Johnson potential because this\nnew method improves semiconductor gaps and d-state\npositions for various materials [25]. We present the den-\nsity of states between -5 eV and 3 eV in Fig. 2. It is\nclear that the majority-spin channel is metallic and the\nminority-spin channel has an energy of 0.93 eV across\nthe Fermi level, and therefore the cubic VGe 3Si4is a2\nFIG. 1: (color online) The crystalline unit cell of the cubic\nVGe3Si4with space group No. 215. The largest balls (corner)\nrepresents V atoms, the medium (face-center) Ge, and the\nsmallest (in the cube) Si.\nhalf-metallicferromagnetwith half-metallicgap350meV\n[29]. Thespin-resolvedbands between-8eVand4 eVare\npresented in Fig. 3. The two majority-spin bands across\nthe Fermi level are from V 3 d-egdoublet, and the lower\nbands in the energy window can be attributed to the p\nandsstates from Ge and Si. The spin exchange splitting\nshould be driven by the V delectrons. The minority-spin\ngap of 0.93 eV across the Fermi level, and hence the half-\nmetallicity, is due to both the spin exchangesplitting and\nthe right position of V dstates in the GeSi gap.\nTo investigate the robustness of the ferromagnetic\norder against possible antiferromagnetic fluctuations,\nwe calculate the total energies for nonmagnetic struc-\nture and three kinds of antiferromagnetic configura-\ntions, namely AFG (0.5,0.5,0.5)2π\na, AFA (0,0,0.5)2π\na,\nAFC(0.5,0.5,0)2π\na. Theyareconstructedbydoublingthe\nunit cell along the [111], [110], and [001] directions, re-\nspectively. Our results show that the nonmagnetic struc-\nture is 155 meV higher in total energy per formula unit\nthanthe ferromagneticstructure, andthethreeantiferro-\nmagnetic structures are 49, 51, and 60 meV higher in to-\ntal energy per formula unit, respectively. Therefore, the\nferromagnetic order of the cubic VGe 3Si4is indeed sta-\nble against antiferromagnetic fluctuations, and these en-\nergy differences indicate that a room-temperature should\nbe reached for the Curie temperature. Since the half-\nmetallic ferromagnets from II-VI and III-V semiconduc-\ntors have small values for the elastic modulus C′, usually\nsmallerthan 10GPa, it isimportanttocalculatetheelas-\ntic moduli, B,C′, andC44, for the cubic VGe 3Si4. All\nthese calculated results are summarized in Table I. It\nis impressive that the C′value, the smallest among the\nthree moduli, equals 23.3 GPa, substantially larger than\nthose of the half-metallic ferromagnets based on II-VI\nand III-V semiconductors. This means that much better\nthin-film quality can achieved and therefore there will\nbe a much better potential for high-performance device\napplications.TABLE I: The lattice constant ( a), the Si position parameter\n(zx), the total moment ( Mtot), the half-metallic gap ( GHM),\nthe elastic constants ( B,C′,C44) of the stable ferromagnetic\nstructure, and the total energies of the three antiferromag -\nnetic (AFG, AFC, AFA) and the nonmagnetic (NF) struc-\ntures with respect to the ferromagnetic structure (FM).\na z x Mtot GHM\n5.632˚A 0.2509 1.000 µB0.35 eV\nB C′C44\n70.9 GPa 23.3 GPa 65.8 GPa\nFM AFG AFC AFA NF\n0 49. meV 51. meV 60. meV 155. meV\nIn summary, we find that the cubic VGe 3Si4is a half-\nmetallic ferromagnet with a half-metallic gap 350 meV.\nWe have optimized its crystal structure and investigated\nits crystalstructureagainstdeformationsand the robust-\nness of its ferromagnetic order against antiferromagnetic\nfluctuations. Our results show that its ferromagnetic or-\nder is stable, maybe enough to keep a Curie tempera-\nture at room temperature, and C′, the smallest among\nthe three elastic moduli, reaches 23.3 GPa, substantially\nlarger than those of the half-metallic ferromagnetic ma-\nterials based the II-VI and III-V semiconductors. These\nresults make us believe that the cubic VGe 3Si4could be\nsynthesized as thin films of enough thickness soon in the\nnear future and used in spintronic applications.\nThis work is supported by Nature Science Founda-\ntion of China (Grant Nos. 10874232 and 10774180),\nby the Chinese Academy of Sciences (Grant No.\nKJCX2.YW.W09-5), and by Chinese Department of Sci-\nence and Technology (Grant No. 2005CB623602).\n[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J.\nM. Daughton, S. von Molnar, M. L. Roukes, A. Y.\nChtchelkanova, and D. M. Treger, Science 294, 1488\n(2001).\n[2] W. E. Pickett and J. S. Moodera, Phys. Today 54, No.\n5, 39 (2001).\n[3] R. A. de Groot, F. 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Hutten, G.3\u0001 \u0000 \u0002 \u0003 \u0004 \u0005 \u0006 \u0007 \b \t \n \u000b \f \r\n\u000e\u000f\u0010\u0011\u0012\u0013\u0014\n\u0015\u0016\u0017\u0018\u0019\u001a\u001b\u001c\u001d\u001e\u001f !\"#$% & ' ( ) * + ,-\n. / 01 23 456 7 8\nFIG. 2: (color online) Spin-dependent density of states (DO S, in state/eV per formula unit) of the ferromagnetic VGe 3Si4. The\nsolid thick line represents the total DOS, and the dashed, do tted, dot-dashed, and thin solid lines describe the partial DOSs\nprojected in the Si, Ge, and V atom spheres and the interstiti al region, respectively.     \nR ΓX M ΓE F Energy (eV)  0.0  2.0  4.0\n -2.0\n -4.0\n -6.0\n -8.0     \nR ΓX M ΓE F Energy (eV)  0.0  2.0  4.0\n -2.0\n -4.0\n -6.0\n -8.0\nFIG. 3: Spin-dependent energy bands of the ferromagnetic VG e3Si4. The left-side panel shows the majority-spin bands and\nthe right-side one the minority-spin bands.\nReiss, J. S.Moodera, andM. Munzenberg, NatureMater.\n8, 56 (2009).\n[8] S. Sanvitoand N. A. 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B\n68, 054417 (2003).\n[20] M.G. Sreenivasan, K.L. Teo, M.B. A. Jalil, T. Liew,\nT.C. Chong, and A.Y. Du, IEEE Trans. Mag. 42, 2691\n(2006); G. Sreenivasan, J. F. Bi, K. L. Teo, and T. Liew,\nJ. Appl. Phys. 103, 043908 (2008).\n[21] S.-D. Guo and B.-G. Liu, Europhys. Lett. 88, 67007\n(2009).\n[22] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864\n(1964); W. Kohn and L. J. Sham, Phys. Rev. 140, A1133\n(1965).\n[23] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka,\nand J. Luitz, WIEN2k, an augmented plane wave + local\norbitals program for calculating crystal properties (Karl -heinz Schwarz, Techn. Universitat Wien, Austria), 2001.\nISBN 3-9501031-1-2.\n[24] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n[25] F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401\n(2009).\n[26] A. H. MacDonald, W. E. Pickett, and D. D. Koelling, J.\nPhys. C 13, 2675 (1980).\n[27] D. Singh, Plane waves, pseudopotentials and the LAPW\nmethod, Kluwer Academic 1994.\n[28] J. Kunes, P. Novak, R. Schmid, P. Blaha, and K.\nSchwarz, Phys. Rev. B 64, 153102 (2001).\n[29] B.-G. Liu, Lecture Notes in Physics Vol. 676(Half-\nmetallic Alloys - Fundamentals and Applications, edited\nby I. Galanakis and P. H. Dederichs, Springer Berlin\n2005), pp. 267-291."    },    {        "title": "1805.05172v1.Study_of_the_ferromagnetic_insulator_phase_in_manganites.pdf",        "content": "Study of the ferromagnetic-insulator phase in manganites\nSanjukta Paul and Sudhakar Yarlagadda\nCMP Div., Saha Institute of Nuclear Physics, HBNI, Kolkata, India\n(Dated: June 7, 2021)\nUnderstanding the coexistence of ferromagnetism and insulating behavior in manganites is an un-\nsolved problem. We propose a localized-band model involving e\u000bective intermediate-range electron-\nelectron (electron-hole) repulsion (attraction) generated by cooperative electron-phonon interac-\ntion. Double exchange mechanism, involving holes virtually hopping to nearest neighbors and back,\nproduces magnetic polarons in an antiferromagnetic environment; when these magnetic polarons\ncoalesce and percolate the system, we get a ferromagnetic insulator. Ferromagnetism gets more pro-\nnounced when the holes (doping) increases or when the ratio hopping/polaronic-energy dominates\nover superexchange-coupling/hopping.\nPACS numbers: 75.85.+t, 71.38.-k, 71.45.Lr, 71.38.Ht, 75.47.Lx, 75.10.-b\nI. INTRODUCTION\nPerovskite oxides, such as manganites, display a va-\nriety of orbital, charge, and spin orders when the par-\nent oxide is doped. While signi\fcant progress has been\nmade in characterizing most of the phenomena in bulk\ndoped materials, the understanding pertaining to fer-\nromagnetic insulator is still elusive. The doped alloy\nT1\u0000xDxMnO 3(where T refers to trivalent rare-earth el-\nements such as La, Pr, Nd, etc. and D refers to diva-\nlent alkaline elements Sr, Ca, etc.) is an antiferromag-\nnet whenx > 0:5 with the nature of the antiferromag-\nnet (i.e., A-, C-, CE-, or G-type antiferromagnet) de-\npending on the compound and the dopant value x1{3.\nContrastingly, for x < 0:5, T 1\u0000xDxMnO 3is an intrigu-\ning ferromagnetic insulator (FMI) at smaller values of x\n(i.e., 0:1.x.0:2)4{6and is a ferromagnetic metal at\nhigher dopings in the manganite systems La 1\u0000xSrxMnO 3,\nLa1\u0000xCaxMnO 3, Pr1\u0000xSrxMnO 3, and Nd 1\u0000xSrxMnO 3.\nFor modeling the diverse orderings and for exploiting\nthe functionality in these transition-metal oxides, one\nneeds e\u000bective Hamiltonians for various types of inter-\nactions. Although the importance of strong electron-\nphonon interaction (EPI) has been pointed much earlier7\nand signi\fcant progress has been made some time ago\nin numerically treating electron-phonon interaction in\nsizeable systems8, the treatment of cooperative EPI (in-\nvolving quantum phonons) was accomplished analytically\nonly more recently in two dimensions (2D)9. It has been\ndemonstrated analytically in Ref. 9 that introducing co-\noperative e\u000bects, when EPI is strong, produces nearest-\nneighbor (NN), next-nearest-neighbor (NNN), and next-\nto-next-nearest-neighbor (NNNN) interactions. Further-\nmore, incorporating spin-spin interactions along with co-\noperative strong EPI is still an unsolved analytic prob-\nlem.\nAs regards experiments pertaining to ferromagnetic-\ninsulating regions, while some suggested microscopically\nhomogeneous electronic properties10{12, others speculate\nthat coexistence of ferromagnetic metallic phases and an-\ntiferromagnetic insulating phases leads to an inhomoge-\nneous ferromagnetic insulating state13,14.We will now argue, without considering any speci\fc\nmodel, that ferromagnetic insulating phases are possible\nat low doping in manganites by presenting below gen-\neral theoretical points based on the essential features of\nmanganites.\n1) Kinetic energy (KE) is quite small at low doping be-\ncause bare hopping is small (caused by lower tolerance\nfactor15, cation disorder, compatibility of distortions16),\nand the electron-phonon coupling is strong.\n2) Potential energy [from repulsive interactions, due to\ncooperative EPI, that are intermediate-range, i.e., NN,\nNNN, NNNN, etc.] is much larger than KE; this leads\nto solid-type formation with electrons being rendered es-\nsentially immobile and site localized. The ground state\nis classical and the state of the system can be expressed\nby a single state in the occupation number basis with\nnumber density at each site either 1 or 0.\nThe fact that electrons are essentially site localized also\nfollows from the treatment in Ref. 17; then, only a local-\nized polaronic band is relevant and the upper wide band\ncannot overlap with the lower narrow polaronic band.\nFurthermore, a simple type of phase separated state\nwith ferromagnetic droplets (each containing one car-\nrier) in an antiferromagnetic matrix was shown to be\npossible in Ref. 18 . The mobility of these magnetic\npolarons is low and they are easily localized by disorder\nand Coulomb interactions.\nThus the potential energy determines the charge and\nspin order.\n3) Because of cooperative strong EPI, a NN electron-\nhole pair has a strong ferromagnetic interaction\n[t2cos2(\u0012=2)=(2EJT) withEJTbeing the cooperative\nJahn-Teller energy, tthe hopping term between the NN\nsites, and\u0012the angle between the NN core spins]. Hence,\na robust ferromagnetic cluster is produced in the vicinity\nof a hole.\n4) Our model for magnetic interaction applies to mangan-\nites with low density of localized holes. In regions away\nfrom the holes, the cooperative EPI retains essentially the\nsame orbital texture as in the undoped manganite. As a\nresult, in regions without holes, the magnetic interaction\nis A-AFM just as in the undoped manganite. As regardsarXiv:1805.05172v1  [cond-mat.str-el]  14 May 20182\nthe regions with holes, since the holes are site localized,\nthe holes only virtually hop to a NN site and back and\nthus produce ferromagnetic coupling with NN electrons.\nThis ferromagnetic coupling, between NN electron-hole\npair, is much stronger than A-AFM coupling.\n5) Presence of site localized holes produces FMI clus-\nters due to formation of magnetic polarons. A hole will\npolarize NN electrons (and realistically speaking, NNN\nand NNNN electrons as well) through virtual hopping,\nthereby producing a magnetic polaron. A collection of\ninteracting magnetic polarons will produce a FMI region.\nIt is interesting to note that FMI regions are present at\nmoderate doping in manganites that are narrow band\n(Pr1\u0000xCaxMnO 3), intermediate band (La 1\u0000xCaxMnO 3)\nand wide band (La 1\u0000xSrxMnO 3).\nRest of the paper is organized as follows. In Sec. II, in-\nvoking cooperative electron-phonon-interaction physics,\nwe obtain the e\u000bective Hamiltonian that is employed to\nunderstand the FMI phase in manganites. Next, in Sec.\nIII, we outline our calculation procedure involving Monte\nCarlo technique, used to simulate charge and spin con-\n\fgurations, and obtain the magnetization as a function\nof various parameters. Then, in Sec. IV, we discuss our\nresults obtained for systems with di\u000berent hoppings at\ndi\u000berent temperatures and dopings. Lastly, in Sec. V,\nwe conclude and o\u000ber some perspectives.\nII. EFFECTIVE HAMILTONIAN\nIn this section we focus on the analytical treatment of\nthe e\u000bective Hamiltonian which will be used for numer-\nical simulation. We are working with a 2D version of\nthe pervovskite manganite system which has Mn-O-Mn\nbonds along the xandydirections. We have e gelectrons\n(or holes) interacting with the oxygen atoms. We have\nrestricted our analysis to a system of fermions interacting\nwith the oxygens in the xy-plane via cooperative breath-\ning mode and with the out-of-plane z-direction oxygens\nthrough non-cooperative breathing mode as depicted in\nFig. 1.\nApart from the itinerant egelectrons, we also have\nat2glocalized core-spin background; the large S= 2\nspin (comprising of contribution from a egspin and three\nt2gspins) at each site is considered classical. Thus the\nHamiltonian of such a system has four interactions: the\nkinetic energy of the fermions, the fermion-lattice cou-\npling energy, the lattice energy, and the spin-spin inter-\naction energy.\nH=HKE+Hint+Hlat+HSE: (1)\nHere,\nHKE=\u0000tX\ni;j\u0014\ncos\u0012\u0012i;j;i+1;j\n2\u0013\ndy\ni+1;jdi;j+\ncos\u0012\u0012i;j;i;j+1\n2\u0013\ndy\ni;j+1di;j+ H:c:\u0015\n;(2)\n     \n         \nXy\nz(i,j)(i,j+1)\n(i+1,j)\n(i-1,j-1)(i+1,j+1)(i,j+2)\n(i+2,j)\nFIG. 1. (Color online) Schematic diagram for a 2D coopera-\ntive breathing mode (CBM) system. Hopping sites for holes\nare represented by blue solid circles, in-plane oxygen atoms\n(participating in the CBM) by black empty circles, and non-\ncooperative out-of-plane oxygen atoms by solid black circles.\nanddi;j(dyi;j) represents the annihilation (creation) op-\nerator for the fermion at site ( i;j);tis the hopping ampli-\ntude for the fermion; the hopping process is modi\fed by\n\u0012, the angle between two spins at NN sites.25,26The sec-\nond term represents the interaction between the fermions\nand the quantum phonons in the system and is expressed\nas9\nHint=\u0000g!0X\ni;jn\u0000\nay\nx;i;j+ax;i;j\u0001\n(ni;j\u0000ni+1;j)\n+\u0000\nby\ny;i;j+by;i;j\u0001\n(ni;j\u0000ni;j+1)\n+\r\u0000\ncy\nz;i;j+cz;i;j\u0001\nni;jo\n; (3)\nwhere\r=p\n2,gis the electron-fermion coupling con-\nstant,!0is the optical-phonon frequency, and ni;j=\ndy\ni;jdi;j. The displacement of the oxygen atom that is\nadjacent to the site ( i;j) in the positive x[y] direction\nis given by(ay\nx;i;j+ax;i;j)p2m!0\u0014\n(by\ny;i;j+by;i;j)p2m!0\u0015\n. In thezdirec-\ntion, the relative displacement of the two oxygen atoms\nnext to site ( i;j) is denoted by(cy\nz;i;j+cz;i;j)p\n2m!0=2withm=2 be-\ning the reduced mass of the oxygen pair. Next, the lattice\nenergy due to quantum harmonic oscillators is given by\nHlat=X\ni;j\u0010\nay\nx;i;jax;i;j+by\ny;i;jby;i;j+\u0011cy\nz;i;jcz;i;j\u0011\n;(4)3\nwith\u0011being set to be 1.\nNow, to arrive at an e\u000bective Hamiltonian which can be\nexpressed solely in terms of fermionic operators, we take\nresort to an analytic approach similar to that described\nin9. For large electron-phonon coupling and restricting\nthe system to the non-adiabatic regime, t=!0.1, the\nabove Hamiltonian His subject to a canonical trans-\nformation (i.e., modi\fed Lang-Firsov transformation) to\nproduce an unperturbed part H0and the perturbation\ntermH1. To obtain an e\u000bective Hamiltonian, we perform\nsecond-order perturbation theory (as in Refs. 9 and 28)\nand obtain\nHe\u000b=\u0000EpX\ni;jni;j+ 2VpX\ni;j\u0010\nni;jni+1;j+ni;jni;j+1\u0011\n+te\u0000(Ep+Vp)=!0X\ni;j\u0014\ncos\u0012\u0012i;j;i+1;j\n2\u0013\ndy\ni+1;jdi;j\n+ cos\u0012\u0012i;j;i;j+1\n2\u0013\ndy\ni;j+1di;j+ H:c:\u0015\n+H(2)+HSE; (5)\nwhere the polaronic energy Ep=\u0000\n4 +\r2\u0001\ng2!0= 6g2!0\nand the nearest-neighbor repulsion energy Vp=g2!0;\nthe second order perturbation theory yields the term\nH(2). The small parameter of the perturbation theory\nis\u0018h\nt2\n2(Ep+Vp)!0i1\n2as derived in Ref. 29. Now, the e\u000bec-\ntive hopping term te\u0000(Ep+Vp)=!0\u001c!0. For large g, the\ne\u000bective hopping term will be very small compared to the\nother terms in He\u000b\u0000HSE. Hence, we ignore the kinetic\nenergy of the system and treat the system as made up\nof carriers that are localized due to disorder. Then, we\nare justi\fed in treating the problem entirely classically\nwith physics being governed by the dominant potential\nenergy terms in the e\u000bective Hamiltonian. The \frst termofHe\u000bcan as well be represented in terms of electron-\nhole attraction instead of fermion-fermion repulsion. In\ngeneral\nVpX\ni;j;\u000eni;jni+\u000e;j=\u0000VpX\ni;j;\u000eni;j(1\u0000ni+\u000e;j)\n+VpX\ni;j;\u000eni;j: (6)\nThis formalism adds a constant energy term VpP\ni;j;\u000eni;j\nto the Hamiltonian and thus does not change the physics\nof the problem. Then, using Eq. (6) we can rewrite Eq.\n(5) as\nHe\u000b=\u0000EpX\ni;jni;j\u00002VpX\ni;jn\nni;j(1\u0000ni+1;j)\n+ni;j(1\u0000ni;j+1)o\n+ 4VpX\ni;jni;j\n+H(2)+HSE: (7)\nThe convention we will use throughout the paper is that\nni;jwill represent number density of a hole at the lattice\nsite (i;j) of the system. To calculate H(2), we go through\nan algebra similar to that mentioned in Appendix A of\nRef. 9 and arrive at a nearest-neighbor repulsion term\ncorresponding to the process where a particle in 2D vir-\ntually hops to its NN and comes back. When a hole\nat site (i;j) hops to its NN site, such as ( i+ 1;j), and\ncomes back, we need to keep track of the occupancy of\nthe three relevant nearest-neighbor sites of the interme-\ndiate site (i+ 1;j), i.e., the occupancy of the three sites\n(i+ 2;j) , (i+ 1;j+ 1) and (i+ 1;j\u00001). Depending on\nhow many of these three sites are \flled, the coe\u000ecient\nfor the hopping-and-returning process will be modi\fed.\nClearly, there are four such possibilities for the coe\u000e-\ncients and they will be considered below.\nA. Three NN sites of the intermediate site are \flled by electrons.\nIn Fig. 2, when the intermediate site containing an electron is surrounded by a hole and three electrons, we depict\nthe hole at site ( i;j) hopping to its NN site (the intermediate site) and returning back. The intermediate site can be\nany of the four NNs of the originating site ( i;j). A schematic view of the four possibilities is shown in Fig. 2.\nWhen a hole is at ( i;j), its energy is equal to \u0000Ep. The oxygen atoms on both the sides of the initial site are\nattracted by the hole on the initial site and hence are pulled towards the hole. When the hole virtually hops to the\nintermediate site, its energy is equal to Ep+ 2Vpbecause the oxygen distortions remain unchanged; in the energy\nof the intermediate state, Eparises due to the distortion without the hole whereas the extra energy 2 Vp(equal in\nmagnitude to the NN repulsion energy between two holes) results due to displacing the oxygen atoms towards the\ninitial site and away from the hole. Hence, change in the energy when the hole jumps from the originating site to the\nintermediate site is equal to 2 Ep+ 2Vp. Thus, the coe\u000ecient of the second order perturbation term turns out to be4\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(A)\n(i, j)(i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(B)\n(i, j)\n(i, j - 1)(i, j + 1)\n(i + 1, j -1)(i + 1, j + 1)\n(i, j - 2)(i - 1, j + 1)\n(i - 1, j -1)(i, j + 2) (C)\n(i, j)\n(i, j - 1)(i, j + 1)\n(i + 1, j -1)(i + 1, j + 1)\n(i, j - 2)(i - 1, j + 1)\n(i - 1, j -1)(i, j + 2)(D)\nFIG. 2. (Color online) Schematic diagram for the four possibilities of a hole, at an originating site ( i; j), hopping to its NN site\n(the intermediate site) and coming back (when three NN sites of the intermediate site are occupied by electrons): (A) hole at\n(i; j) hops to its right NN at ( i+ 1; j) and comes back; (B) hole at ( i; j) jumps to its left NN at ( i\u00001; j) and returns back; (C)\nhole at ( i; j) jumps to its downward NN at ( i; j\u00001) and comes back; (D) hole at ( i; j) hops to its upper NN at ( i; j+ 1) and\nreturns. A hole is represented by a blue solid circle and a particle (i.e., electron) by a blue empty circle. All lattice sites that\nare not relevant to the consideration are represented by black solid circles.\nt2\n2Ep+2Vpand the contribution to H(2)from all the possibilities corresponding to Fig. 2 is given by\nH(2)\n1=\u0000t2\n(2Ep+ 2Vp)X\ni;j\"\ncos2\u0012\u0012i;j;i+1;j\n2\u0013n\nni;j(1\u0000ni+1;j)(1\u0000ni+2;j)(1\u0000ni+1;j+1)(1\u0000ni+1;j\u00001)o\n+cos2\u0012\u0012i;j;i\u00001;j\n2\u0013n\nni;j(1\u0000ni\u00001;j)(1\u0000ni\u00002;j)(1\u0000ni\u00001;j+1)(1\u0000ni\u00001;j\u00001)o\n+cos2\u0012\u0012i;j;i;j+1\n2\u0013n\nni;j(1\u0000ni;j+1)(1\u0000ni;j+2)(1\u0000ni\u00001;j+1)(1\u0000ni+1;j+1)o\n+cos2\u0012\u0012i;j;i;j\u00001\n2\u0013n\nni;j(1\u0000ni;j\u00001)(1\u0000ni;j\u00002)(1\u0000ni\u00001;j\u00001)(1\u0000ni+1;j\u00001)o#\n: (8)\nB. Any two of NN sites of the intermediate site is \flled.\nIn Fig. 3, we depict the three possibilities corresponding to a hole at a site ( i;j) hopping to its NN site (the\nintermediate site) and returning; here, any two of the NN sites of the intermediate site are occupied by electrons.\nHenceforth, we will show all the counterpart processes of Fig. 2(A) (considering these as representative diagrams) for\nvarious possibilities. Similar processes, which will not be shown here, also occur for Fig. 2(B), Fig. 2(C), and Fig.\n2(D).\nWhen the hole virtually hops to the intermediate site, its energy is equal to Ep+ 4Vp; here, an extra repulsion\nof 2Vpis generated due to the occupancy of any one of the NN site of the intermediate site by a hole. Then, the\ncoe\u000ecient of the second order perturbation term ist2\n2Ep+4Vpand the contribution to H(2)from all the possibilities,\nsimilar to and corresponding to Fig. 3, is given by5\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(a)\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(b)\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(c)\nFIG. 3. (Color online) Schematic diagram for a hole at an originating site ( i; j) hopping to its NN site (the intermediate site)\nand returning back (when any two of the NN sites of the intermediate site are occupied by electrons). Representation of a\nhole at ( i; j) jumping to its right NN at ( i+ 1; j) and coming back when holes occupy (a) right and downward NNs of the\nintermediate site; (b) right and upward NNs of the intermediate site; (c) upward and downward NNs of the intermediate site.\nA hole is depicted by a blue solid circle and a particle by a blue empty circle. All lattice sites that are not relevant to the\nconsideration are represented by black solid circles.\nH(2)\n2=\u0000t2\n(2Ep+ 4Vp)X\ni;j\"\ncos2\u0012\u0012i;j;i+1;j\n2\u0013n\nni;j(1\u0000ni+1;j)ni+2;j(1\u0000ni+1;j+1)(1\u0000ni+1;j\u00001)\n+ni;j(1\u0000ni+1;j)(1\u0000ni+2;j)ni+1;j+1(1\u0000ni+1;j\u00001) +ni;j(1\u0000ni+1;j)(1\u0000ni+2;j)(1\u0000ni+1;j+1)ni+1;j\u00001o\n+cos2\u0012\u0012i;j;i\u00001;j\n2\u0013n\nni;j(1\u0000ni\u00001;j)ni\u00002;j(1\u0000ni\u00001;j+1)(1\u0000ni\u00001;j\u00001)\n+ni;j(1\u0000ni\u00001;j)(1\u0000ni\u00002;j)ni\u00001;j+1(1\u0000ni\u00001;j\u00001) +ni;j(1\u0000ni\u00001;j)(1\u0000ni\u00002;j)(1\u0000ni\u00001;j+1)ni\u00001;j\u00001o\n+cos2\u0012\u0012i;j;i;j+1\n2\u0013n\nni;j(1\u0000ni;j+1)ni;j+2(1\u0000ni\u00001;j+1)(1\u0000ni+1;j+1)\n+ni;j(1\u0000ni;j+1)(1\u0000ni;j+2)ni\u00001;j+1(1\u0000ni+1;j+1) +ni;j(1\u0000ni;j+1)(1\u0000ni;j+2)(1\u0000ni\u00001;j+1)ni+1;j+1o\n+cos2\u0012\u0012i;j;i;j\u00001\n2\u0013n\nni;j(1\u0000ni;j\u00001)ni;j\u00002(1\u0000ni\u00001;j\u00001)(1\u0000ni+1;j\u00001)\n+ni;j(1\u0000ni;j\u00001)(1\u0000ni;j\u00002)ni\u00001;j\u00001(1\u0000ni+1;j\u00001)\n+ni;j(1\u0000ni;j\u00001)(1\u0000ni;j\u00002)(1\u0000ni\u00001;j\u00001)ni+1;j\u00001o#\n: (9)\nC. Any one of NN sites of the intermediate site has an electron.\nIn Fig. 4, three possibilities have been shown for the process where a hole jumps to an intermediate site and comes\nback; here, any one of the NNs of the intermediate site is \flled by an electron. Extending the logic given above to\nthe present case, the coe\u000ecient of the second order perturbation term ist2\n2Ep+6Vpand the contribution to H(2)from6\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(a)\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(b)\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)(c)\nFIG. 4. (Color online) Schematic diagram for a hole at an originating site ( i; j) hopping to its NN site (the intermediate site)\nand coming back (when any one of the NN sites of the intermediate site is occupied by an electron). Depiction of a hole at ( i; j)\njumping to its right NN at ( i+ 1; j) and coming back when a particle occupies (a) downward NN of the intermediate site; (b)\nright NN of the intermediate site; (c) upward NN of the intermediate site. A hole is represented by a blue solid circle whereas\na particle by a blue empty circle. All lattice sites that are not relevant to the consideration are indicated by black solid circles.\nall the possibilities, similar to and corresponding to Fig. 4, is given by\nH(2)\n3=\u0000t2\n(2Ep+ 6Vp)X\ni;j\"\ncos2\u0012\u0012i;j;i+1;j\n2\u0013n\nni;j(1\u0000ni+1;j)ni+2;jni+1;j+1(1\u0000ni+1;j\u00001)\n+ni;j(1\u0000ni+1;j)(1\u0000ni+2;j)ni+1;j+1ni+1;j\u00001+ni;j(1\u0000ni+1;j)ni+2;j(1\u0000ni+1;j+1)ni+1;j\u00001o\n+cos2\u0012\u0012i;j;i\u00001;j\n2\u0013n\nni;j(1\u0000ni\u00001;j)ni\u00002;jni\u00001;j+1(1\u0000ni\u00001;j\u00001)\n+ni;j(1\u0000ni\u00001;j)(1\u0000ni\u00002;j)ni\u00001;j+1ni\u00001;j\u00001+ni;j(1\u0000ni\u00001;j)ni\u00002;j(1\u0000ni\u00001;j+1)ni\u00001;j\u00001o\n+cos2\u0012\u0012i;j;i;j+1\n2\u0013n\nni;j(1\u0000ni;j+1)ni;j+2(1\u0000ni\u00001;j+1)ni+1;j+1\n+ni;j(1\u0000ni;j+1)(1\u0000ni;j+2)ni\u00001;j+1ni+1;j+1+ni;j(1\u0000ni;j+1)ni;j+2ni\u00001;j+1(1\u0000ni+1;j+1)o\n+cos2\u0012\u0012i;j;i;j\u00001\n2\u0013n\nni;j(1\u0000ni;j\u00001)ni;j\u00002ni\u00001;j\u00001(1\u0000ni+1;j\u00001)\n+ni;j(1\u0000ni;j\u00001)(1\u0000ni;j\u00002)ni\u00001;j\u00001ni+1;j\u00001+ni;j(1\u0000ni;j\u00001)ni;j\u00002(1\u0000ni\u00001;j\u00001)ni+1;j\u00001o#\n(10)\nD. All the NN sites of the intermediate site have holes\n.\nHere, for the situation where all the NN sites of the intermediate site have holes, we depict in Fig. 5 a hole hopping\nto an intermediate site and coming back. Here, the coe\u000ecient of the second order perturbation term ist2\n2Ep+8Vpand7\n(i, j) (i + 1, j)(i + 2, j)\n(i + 1, j -1)(i + 1, j + 1)\n(i - 1, j)(i - 2, j)(i - 1, j + 1)\n(i - 1, j -1)\nFIG. 5. (Color online) Schematic diagram for a hole at an originating site ( i; j) hopping to its NN site (the intermediate site)\nand coming back (when all the other three NN sites of the intermediate site are occupied by electrons). Representation of a\nhole at ( i; j) jumping to its right NN at ( i+ 1; j) and coming back. A hole is represented by a blue solid circle and a particle\nby a blue empty circle. All lattice sites irrelevant to the analysis are represented by black solid circles.\nthe contribution to H(2)from all the possibilities, similar to and corresponding to Fig. 5, is given by\nH(2)\n4=\u0000t2\n(2Ep+ 8Vp)X\ni;j\"\ncos2\u0012\u0012i;j;i+1;j\n2\u0013n\nni;j(1\u0000ni+1;j)ni+2;jni+1;j+1ni+1;j\u00001o\n+cos2\u0012\u0012i;j;i\u00001;j\n2\u0013n\nni;j(1\u0000ni\u00001;j)ni\u00002;jni\u00001;j+1ni\u00001;j\u00001o\n+cos2\u0012\u0012i;j;i;j+1\n2\u0013n\nni;j(1\u0000ni;j+1)ni;j+2ni\u00001;j+1ni+1;j+1o\n+cos2\u0012\u0012i;j;i;j\u00001\n2\u0013n\nni;j(1\u0000ni;j\u00001)ni;j\u00002ni\u00001;j\u00001ni+1;j\u00001o#\n: (11)\nFrom the contributions H(2)\n1,H(2)\n2,H(2)\n3, and H(2)\n4ob-\ntained above, we express H(2)as\nH(2)=H(2)\n1+H(2)\n2+H(2)\n3+H(2)\n4: (12)\nLastly, the superexchange27term H SEgenerates A-\nAFM spin-spin exchange in manganites such as LaMnO 3\nand is given by\nHSE=\u0000JxyX\nhi;jixycos (\u0012ij) +JzX\nhi;jizcos (\u0012ij):(13)\nIt is important to note that, while the range of charge-\ncharge interaction is as far as NNNN, the range of spin-\nspin interaction is only NN.\nIII. CALCULATION PROCEDURE\nFor a numerical study, we consider a 2D lattice with pe-\nriodic boundary conditions in both directions. We treat\nthe problem fully classically using the e\u000bective Hamilto-\nnian, comprising of the e\u000bective electron-phonon inter-\naction (the charge-spin-coupled term) and the superex-\nchange interaction (the spin-spin interaction term), as\ngiven by Eq. (7). We use classical Monte Carlo technique\nand make use of the standard Metropolis algorithm toupdate the charge con\fguration as well as the spin con-\n\fguration of the system. We follow a two-step procedure\nto arrive at the \fnal charge and spin con\fgurations.\nFirstly, to deal with problem of charge con\fgurations\nthat correspond to local minima which are close in en-\nergy, we take resort to simulated annealing for the charge\ndegrees of freedom only. The spin variables are kept\nfrozen since the energy scale for the charge interactions\nis much higher than the energy scale for superexchange\ninteractions. Since we are working with low hole den-\nsities (i.e., between 0.1 and 0.3), a large number of de-\ngenerate states will appear in the charge spectrum. In\norder to obtain maximum number of such degenerate\ncon\fgurations, we employ a three-step procedure at each\ntemperature of the simulated annealing process to obtain\nthe optimized charge con\fgurations. The primary step\nis a \\single-particle-exchange\" process where we choose\nany two sites at a time|one sequentially and the other\nrandomly|and exchange their number density values\nprovided they di\u000ber by 1. Physically we exchange a par-\nticle at a site with a hole at any other site. The secondary\nstep is a \\general-two-particle-exchange\" process where\nany two random sites are selected with both being occu-\npied by particles and then their occupants are exchanged\nwith another pair of randomly chosen sites both con-\ntaining holes. Thus we actually exchange two particles8\nwith two holes at a time. The \fnal step|a \\plaquette-\nexchange\"process|is a special case of the \\general-two-\nparticle-exchange\"mechanism. Here plaquette are chosen\nsequentially; if the di\u000berence in number densities between\nthe two diagonal pairs is 2, then the number densities of\nthe diagonals are exchanged. At a particular tempera-\nture, to arrive at the \fnal lowest energy charge con\fgu-\nration at that temperature using Monte Carlo technique,\nan initial random charge con\fguration (with a \fxed num-\nber of particles) \frst goes through 4 \u0002105steps of \\single-\nparticle-exchange\"; then an equal number of steps involv-\ning \\general-two-particle-exchange\"; followed by 30 times\nthe system size number \\plaquette-exchange\"steps.\nSecondly, using the charge pro\fle generated by the\nthree-step process, we now optimize spin variables by\ntaking an initial random spin con\fguration and updating\nthrough the Metropolis algorithm. The spins being large\nin magnitude, with S= 2, are essentially classical spins\nwith~Si= (sin\u0012icos\u001ei;sin\u0012isin\u001ei;cos\u0012i). While updat-\ning the spins, we consider the full Hamiltonian He\u000band\nconsider both the charge and spin interaction energies.\nThe cos(\u0012) and\u001evalues are binned to \fx the orienta-\ntion of the classical spin vector. We have allowed equally\nspaced 40 values of cos( \u0012) in the interval ( \u00001;1) and 80\nvalues of\u001ein the usual range of (0 ;2\u0019), thus totaling to\n3200 di\u000berent possibilities. A sweep involves visiting all\nthe lattice sites sequentially and updating the spin ori-\nentation at each lattice site by the Metropolis algorithm.\nThe equilibrium number of sweeps required for medium\n(higher) temperatures is around 15 \u0002105(6\u0002105), while\nanother 15\u0002105(6\u0002105) sweeps are required for the\nthermal averaging of the total magnetization of the sys-\ntem. It is to be noted that for low hole concentrations, we\nhave many degenerate states. We calculate the magne-\ntization for typically 10 degenerate con\fgurations. The\ndegenerate states are chosen based on the charge opti-\nmization process only, fed to the full Hamiltonian He\u000b\ncontaining both charge and spin variables, and then en-\nergy is optimized to obtain the total magnetization of all\nsuch states. The magnetization/site of the system, that\nhas been plotted, is the magnetization/site averaged over\nall the degenerate states for a particular \flling of holes\nwhen spins normalized to unity.\nWe study the system for the bare hopping parame-\nter valuest= 0:2 eV andt= 0:3 eV. Our calculations\ntake the polaronic energy to be Ep= 0:43 eV and the\nnearest-neighbor repulsion energy to be Vp= 0:07 eV.\nThus, we are in the regime of strong electron-phonon\ncoupling characterized by ( Ep+Vp)=!0>>1 with the\noptical phonon frequency !0value being given as 0.05\neV< ! 0<0.1 eV. The superexchange energy coe\u000e-\ncientJz= 4:8 meV23,24; thus the superexchange energy\nis much smaller than the electron-hole pair ferromagnetic\ninteraction coupling [ t2=(2Ep+ 2Vp)]. Furthermore, the\nferromagnetic coupling Jxy= 1:4\u0002Jz. Thus, the charge\ncon\fguration can be assumed to remain constant as the\nspins are optimized. The total magnetization of the sys-\ntem is computed at various temperatures, with the high-est temperature being T= 0:1t=kB(i.e., about 330K for\nt= 0:3 eV. Henceforth, kBwill be set to unity for con-\nvenience.. The lowest temperature on the other hand is\nT= 0:001t(i.e., about 3K for t= 0:3 eV) which is much\nsmaller than Jz; thus, the system can be assumed to be in\nits ground state at T= 0:001t. Here, we should comment\nthat above T= 0:03t, the excited-state charge con\fgu-\nrations also begin to contribute to the magnetization.\nIV. RESULTS AND DISCUSSION\nWe consider a 2D lattice of dimensions 6 \u000212, (i.e.,\nwith a total of 72 sites) with periodic boundary condi-\ntions in both directions; the number of rows being l x= 6\nand the number of columns being l y= 12. Each site\nrepresents an Mn ion consisting of an electron and a pos-\nitive charge center. We study the interplay between the\nelectron-phonon interaction and the magnetic interaction\nof the spins [see Eq. (7)]. As stated earlier (in Sec. I ),\ndue to the the smallness of the kinetic energy in com-\nparison to the potential energy, the problem is treated\nfully classically. Thus the holes are site-localized and the\nsystem can be represented by a single state in the occu-\npation number basis with the number density at each site\nbeing either 1 or 0. Hence, for strong electron-phonon in-\nteraction, we have a fully insulating system resembling a\ncharge solid as shown in Fig. 7. Using e\u000bective Hamilto-\nnian in Eq. (7), we can simulate di\u000berent observables in\nthe system. We study the variation of the total magne-\ntization of the system as a function of hole doping in the\npure manganite sample.\nA. A-AFM background\nThe hopping value tis varied to study the interplay be-\ntween the electron-phonon interactions and the superex-\nchange interactions in the system. The hole doping xis\nvaried as 0:1\u0014x\u00140:3. The magnetic pro\fle still resem-\nbles that of an A-AFM system away from the holes; in\nthe NN vicinity of a hole, the spins get polarized in the\ndirection of the spin of the hole thereby forming a mag-\nnetic polaron. For di\u000berent hopping cases, the temper-\nature variation of the total magnetization of the system\nis studied. We have also considered a 2D lattice with\nsystem size 12\u000212 and carried out the magnetization\nmeasurements. It shows qualitatively similar results as\nthat of the 6\u000212 lattice as depicted in Fig. 6. How-\never, the 12\u000212 system requires a running time which\nis'5 times that of the 6 \u000212 case; also, the number of\ndegenerate states for the 12 \u000212 system is much more.\nThus, dealing with the 12 \u000212 case is computationally\nexpensive. So, we conclude that 6 \u000212 lattice can be con-\nsidered to be representative for the 12 \u000212 lattice and\nwill be used for investigating the ferromagnet-insulator\nproperties of the system.9\n0.1 0.2 0.3\n Filling fraction of holes  (x)00.51 <mi>/site6 X 12 lattice\n12 X 12 latticeDifferent system sizes;  t = 0.3 eV;  A-AFM\nFIG. 6. (Color online) Averaged per-site magnetization\n< m i>(of spins normalized to unity) as a function of hole\ndoping xfor two di\u000berent lattices (6 \u000212 and 12 \u000212) and\nfor a \fxed T= 0:001t.\nAt a very low-hole concentration '0:1 (i.e., 8 parti-\ncles on a 72-site lattice), 8 holes get distributed among\n12 columns such that NN as well as NNN and NNNN in-\nteractions are avoided. In most of the degenerate states,\nno two holes occupy the same column. Hence we have\nsite-localized holes in the system, polarizing their NN\nspins, giving rise to magnetic polarons that remain dis-\nconnected in the lattice. Due to the NN interaction Jxy,\nspins in a column try to align ferromagnetically. Thus,\nthe ferromagnetic polarons and the ferromagnetic inter-\naction in columns together give rise to an e\u000bective low\nmagnetization value (with a sizeable \ructuation) .\nAt each temperature, the value of magnetization is\nessentially unchanged between hole densities 8 =72 and\n12=72 ('0:167); this is because up to the \flling 12 =72,\nholes can still maintain to be non-interacting (on ignor-\ning the superexchage) as can be seen in Fig. 7 (a).\nAt temperatures T < J z(= 0:016tfort= 0:3 eV and\n= 0:024tfort= 0:2 eV), antiferromagnetic coupling be-\ntween columns is e\u000bective and the system has low mag-\nnetization at low concentrations (i.e., x\u001412=72). On\nincreasing the temperature up to T=Jz, the e\u000bect of\nJzdiminishes while the e\u000bect of ferromagnetic coupling\nJxy= 1:4Jzis more dominant; thus, magnetism in the\nsystem increases with increasing temperature. For higher\ntemperatures T > J xyand again at low concentrations\n(i.e.,x\u001412=72), the e\u000bect of the ferromagnetic coupling\nJxyalso diminishes, and the magnetism decreases with\nincreasing temperature.\n(a)\n(b)\n(c)\n(d)\n(e)\n(f)\n(g)\n(h)\nFIG. 7. (Color online) Charge con\fguarations in the ground\nstate of a 6 \u000212 lattice. An arbitrarily chosen degenerate\nground state, involving 72 sites, for (a) 12 holes, (b) 14 holes,\n(c) 16 holes, (d) 18 holes, (e) 22 holes, (f) 24 holes (diagonal\nstripe order), (g) 32 holes, and (h) 36 holes.\n1. t=0.3 eV case\nAs the concentration of holes increases, initially NNNN\ninteractions and later NNN interactions become relevant;\nthe NN interactions being the strongest are still avoided.\nThus longer ferromagnetic chains are formed, thereby in-\ncreasing the total magnetization of the system. So by\nx= 14=72'0:194 [see Fig. 7 (b)], magnetization for\nlower temperatures such as T= 0:001tstarts rising size-\nably; the peak magnetization value is now at a reduced\ntemperature of T= 0:01t. Byx= 18=72 = 0:25 [see Fig.\n7 (d)] NNN interactions also appear, the di\u000berent local\nmagnetic polarons start interacting with one another and\nhence form larger magnetic polarons and the peak mag-\nnetization temperature reduces to T=0.001t; here, start-\ning fromT= 0:1t, the magnetization increases with de-\ncreasing temperature. It is to be noted that there occurs\na narrow crossover region (14 =72\u0014x\u001417=72) where\nmagnetization curves for di\u000berent temperatures intersect.\nIn the crossover regime, there is a complex interplay of\nvarious competing e\u000bects: 1) aligning of di\u000berent mag-\nnetic polarons due to dominance of JxyoverJz; 2) reduc-\ntion of electron-hole spin interactions due to appearance10\n10 15 20\n Number of holes00.51 <mi>/siteT=0.001\nT=0.004\nT=0.01\nT=0.03\nT=0.05\nT=0.16 X 12 lattice;   t = 0.3 eV;  A-AFM \nFIG. 8. (Color online) Averaged per-site total magnetization\n< m i>(of spins normalized to unity) as a function of the\nnumber of holes doped for a 6 \u000212 lattice and for various tem-\nperatures (expressed in units of hopping parameter t). The\nbackground spin con\fguration is A-AFM type and hopping\nt= 0:3 eV.\nof NNNN and NNN interactions (of strengtht2\n2Ep+4Vp); 3)\ncommencing of percolation e\u000bects of magnetic polarons\ndue to large hole concentrations; and 4) disordering ef-\nfects of the temperature. At x= 18=72 (= 0:25), per-\ncolation e\u000bect of magnetic polarons is largely dominant\nover antiferromagnetic interactions. At even higher hole\nconcentrations, this e\u000bect is even more pronounced; mag-\nnetization rises faster with lowering of temperature. As\ncan be seen from Fig. 8, at lower temperatures and for\nx > 15=72, the magnetization increases faster with in-\ncreasing hole concentration. At T= 0:001t, we get an\nalmost fully ferromagnetic large cluster for x= 22=72\n'0:3 [see Fig. 7 (e)], with averaged magnetization val-\nues close to the maximum possible.\n2. t=0.2 eVcase\nFor thet= 0:2 eV situation, the crossover region\n(i.e., 13=72\u0014x\u001419=72) is wider than it is for the\nt= 0:3 eV case. The peak-magnetization temperature\noscillates in the crossover regime (see Fig. 9); further-\nmore, the curves corresponding to T.Jz= 0:024tin-\ntersect more than once in the crossover region. A plau-\nsible explanation for this can be given as follows. The\nratio of electron-hole spin interaction and antiferromag-\nnetic couplingh\u0010\nt2\n2Ep+2Vp\u0011.\nJzi\nis only 8; when NNN\nand NNNN interactions are relevant, the ratio reduces\nto\u0010\nt2\n2Ep+4Vp\u0011.\nJz= 7. Thus antiferromagnetic coupling\nbecomes more prominent than for the the t= 0:3 eV\ncase and frustration e\u000bects become relevant. It could be\n10 15 20\n Number of holes 00.51<mi>/siteT=0.001\nT=0.005\nT=0.01\nT=0.03\nT=0.05\nT=0.16 X 12 lattice;   t = 0.2 eV;  A-AFM FIG. 9. (Color online) Averaged per-site total magnetization\n< m i>(of spins normalized to unity) as a function of the\nnumber of holes doped for a 6 \u000212 lattice and for various\ntemperatures (in units of hopping t). The background spin\ncon\fguration is A-AFM and t= 0:2 eV.\ndue to frustration that, at lower temperatures (such as\nT= 0:001t), the magnetization curve drops at the higher\ncarrier concentration x= 18=72 = 0:25. This also could\nbe an indication of the superspin glass phase claimed in\nexperiments19; here, \\superspin\"refers to a spin cluster\n(i.e., a large magnetic polaron). At x\u001520=72 ('0:28),\npercolation e\u000bect of magnetic polarons dominates over\nantiferromagnetic interactions; magnetization rises with\nlowering of temperature. Finally, for the higher hole con-\ncentrationx= 22=72'0:3 and atT= 0:001t, we get a\nreasonably high magnetization value of 0.85.\nB. G-AFM background, t=0.3 eV case\nTo gain further insight, we study the interplay between\nthe strong ferromagnetic electron-hole interaction that\npolarizes the NN spins of a hole and the superexchange\nNN antiferromagnetic interaction Jz. The magnetic pro-\n\fle, away from the holes, resembles that of a G-AFM\nsystem; the holes form ferromagnetic polarons involv-\ning the hole spin and the NN spins. At temperatures\nT.Jz= 0:016tand hole \fllings up to x= 24=72, due\nto the e\u000bect of antiferromagnetic Jzcoupling on all sides,\nthe polarizations of the magnetic polarons oppose each\nother leading to a low magnetization as shown in Fig. 10.\nFor higher temperatures, due to the dominance of the dis-\nordering e\u000bect of the temperature over the superexchange\ninteraction, there is a probability for the clusters to get\nless misaligned. Hence we notice an increase in the mag-\nnetization for T > Jz. Forx= 24=72 = 1=3, we have\na diagonal stripe order as depicted in Fig. 7 (f); each\ncolumn has two holes. For this arrangement, diagonals11\n1015202530\n Number of holes 00.51 <mi>/siteT=0.001\nT=0.005\nT=0.01\nT=0.03\nT=0.05\nT=0.16 X 12 lattice;   t = 0.3 eV;  G-AFM \nFIG. 10. (Color online) Averaged per-site total magnetization\n< m i>(of spins normalized to unity) as a function of the\nnumber of holes doped for a 6 \u000212 lattice and for various\n\fxed temperatures (in units of hopping parameter t). The\nbackground spin con\fguration is G-AFM type and t= 0:3\neV.\ncontaining holes are ferromagnetic, but every such diag-\nonal (with holes) is antiferromagnetically coupled to its\nneighboring diagonal. In each column, half the spins are\nin one direction while the other half are in the opposite\ndirection leading to a very small total magnetization.\nHere too, very similar to the case of t= 0:2 eV\nwith A-AFM background, there is a crossover region;\nthe crossover occurs in the region 24 =72< x < 32=72.\nIn the crossover regime, the peak-magnetization tem-\nperature oscillates and the curves for T.Jzintersect\nthrice in the crossover region. Since all the background\nspins interact antiferromagnetically (which is in contrast\nto the A-AFM case), percolation of magnetic polarons\ndominates over antiferromagnetic interactions at an even\nlarger \flling; around x\u001532=72 = 0:44 [refer Fig. 7 (g)],\nmagnetization increases with lowering of temperature..\nIt is to be noted that, for half-\flling [see Fig. 7 (h)], we\nshould expect a fully ferromagnetic spin pro\fle with a\ncheckerboard charge structure.20\nC. Fully FM background, t=0.3 eV\nLastly, to better appreciate subtleties pertaining to\nFMI, we also study the case where the superexchange\ninteraction is fully ferromagnetic (FM) with coupling\nJz= 0:016twhent= 0:3 eV. Here, while in the NN\nvicinity of a hole the spins get strongly polarized thereby\nforming a ferromagnetic magnetic polaron, the magnetic\npro\fle is that of a weaker FM system away from the holes.\nHence, for temperatures much smaller than Jz(such as\nT= 0:001t), we have an almost fully ferromagnetic sys-\n10 15 20\n Number of holes 00.51 <mi>/siteT=0.001\nT=0.01\nT=0.03\nT=0.16 X 12 lattice;   t = 0.3 eV;  fully FMFIG. 11. (Color online) Averaged per-site total magnetization\n< m i>(of spins normalized to unity) as a function of the\nnumber of holes doped for a 6 \u000212 lattice and for various\n\fxed temperatures (in units of hopping parameter t). The\nbackground spin con\fgurataion is fully FM and t= 0:3 eV.\ntem as shown in Fig. 11, On increasing the temperature,\nthe spins get more misaligned and the magnetization re-\nduces. At lower \fllings, as the temperature increases to\nthe value 0 :03t(i.e.,T\u00192Jz), the disordering e\u000bect is\nlarge enough so that the magnetization drops consider-\nably as shown in Fig. 11; On the other hand, at higher\n\fllings and again at T= 0:03t, percolation of magnetic\npolarons counters the disordering e\u000bect and generates\nhigher magnetization values. For still higher tempera-\ntures (such as T= 0:1t), the magnetic polarons tend\nto orient in random directions because the superexchage\ncoupling is ine\u000bective, thereby reducing the magnetiza-\ntion signi\fcantly. It is interesting to note that, in all the\nthree Figs. 8, 10, and 11 plotted at t= 0:3 eV, the mag-\nnetization curves are similar because the superexchange\nis ine\u000bective and hence the nature of superexchage cou-\npling is irrelevant.\nV. CONCLUSIONS AND PERSPECTIVES\nWe studied the nature of ferromagnetic insulator in the\nexperimentally relevant doping regime of 0 :1\u0014x\u00140:3\nin bulk manganites. The magnetic interaction considered\nhere applies to manganites with low density of localized\nholes. In regions without holes, as in the undoped man-\nganites, the magnetic interaction is A-AFM; in a region\nwith a hole, the site-localized hole produces strong ferro-\nmagnetic coupling between its spin and its NN electron\nspins. We \fnd that near the doping x= 0:3, the insulator\nis almost fully ferromagnetic. Now, the critical doping at\nwhich the system becomes fully ferromagnetic depends\non the dimension; in 2D it is expected to the around12\ntwice the value of the critical doping in three dimensions\n(3D) for the following reason. In a conducting-site perco-\nlation problem, the critical concentration for conduction\nin a simple cubic lattice is 0.31 and in a square lattice it is\n0.59 [see Ref. 30]; hence, the critical doping to produce a\npercolating cluster that is a checkerboard charge-ordered\nregion is 0:5\u00020:31 in 3D and at 0 :5\u00020:59 in 2D [see Ref.\n31].\nIt was experimentally observed that a FMI phase is\nmanifested in the wide-band manganite La 1\u0000xSrxMnO 3\nin the doping region 0 :1.x.0:1821, in the\nintermediate-bandwidth La 1\u0000xCaxMnO 3in the doping\nrange 0:1\u0014x\u00140:22521,22, and in the narrow-bandwidth\nPr1\u0000xCaxMnO 3in the region 0 :1\u0014x\u00140:321. The\nfact that the FMI region persists till a higher doping\nwhen bandwidth decreases (and concomitantly electron-\nphonon coupling increases15) is consistent with the fact\nthat the tendency to localize increases as bandwidth\ndecreases17,18. The hopping values considered in this\nwork are pertinent to wide-bandwidth and intermediate-\nbandwidth manganites. While our one-band model (in-\nvolving site-localized holes) is relevant to understand\nmanganites in the FMI region, it is certainly not valid\nto study the ferromagnet metallic (FMM) phase that oc-\ncurs at higher doping in manganites; to understand the\nFMM region, we need to invoke a two-band model and/oranalyze the e\u000bect of disorder on localization.\nThe experimental managanite phase diagram reported\nin Ref. 32 reveals increasing Tcvalues at higher dop-\nings for the FMI phase in La 1\u0000xSrxMnO 3. Based on this\nphase diagram, for a \fxed T < Tc(x= 0:1), we expect\nthe magnetization to increase when the doping increases\nin the FMI region; this is consistent with the curves in\nFig. 8.\nLastly, comparing the t= 0:2 eV, A-AFM case with\nthet= 0:3 eV, G-AFM case, we conclude that the an-\ntiferromagnetic coupling Jzplays the important role of\ncausing frustrations in the system. We also point out the\npossible occurance of glassiness in the system to explain\nthe multiple intersections of the curves in the crossover\nregime atT.Jz(see Figs. 9 and 10). Such a pic-\nture is supported by the observed superspin glass phase\nin La 0:82Ca0:18MnO 3ferromagnetic insulator at T.70\nK19. Further theoretical analysis is required to clearly\nidentify and characterize a superspin glass phase at lower\ntemperatures.\nVI. ACKNOWLEDGEMENTS\nThe authors would like to thank A. Ghosh and P. 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B 51, 14103 (1995)."    },    {        "title": "2011.11901v1.Ground_state_properties_and_exact_thermodynamics_of_a_2_leg_anisotropic_spin_ladder_system.pdf",        "content": "arXiv:2011.11901v1  [cond-mat.str-el]  24 Nov 2020Ground state properties and exact thermodynamics of a 2-leg anisotropic spin\nladder system\nSk Saniur Rahaman1,∗Shaon Sahoo2,†and Manoranjan Kumar1‡\n1S. N. Bose National Centre for Basic Sciences,\nJ D Block, Sector III, Salt Lake City, Kolkata 700106 and\n2Department of Physics, Indian Institute of Technology, Tir upati, India\n(Dated: November 25, 2020)\nWe study a frustrated two-leg spin ladder with alternate iso tropic Heisenberg and Ising rung\nexchange interactions, whereas, interactions along legs a nd diagonals are Ising-type. All the inter-\nactions in the ladder are anti-ferromagnetic in nature and i nduce frustration in the system. This\nmodel shows four interesting quantum phases: (i) stripe run g ferromagnetic (SRFM), (ii) stripe\nrung ferromagnetic with edge singlet (SRFM-E), (iii) aniso tropic antiferromagnetic (AAFM), and\n(iv) stripe leg ferromagnetic (SLFM) phase. We construct a q uantum phase diagram for this model\nand show that in stripe rung ferromagnet (SRFM), the same typ e of sublattice spins (either Sor\nσ-type spins) are aligned in the same direction. Whereas, in a nisotropic antiferromagnetic phase,\nbothSandσ-type of spins are anti-ferromagnetically aligned with eac h other, two nearest Sspins\nalong the rung form an anisotropic singlet bond whereas two n earestσspins form an Ising bond.\nIn large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned,\nbut spins on different legs are anti-ferromagnetically alig ned. The thermodynamic quantities like\nCv(T),χ(T) andS(T) are also calculated using the transfer matrix method for di fferent phase. The\nmagnetic gap in the SRFM and the SLFM can be notice from χ(T) andCv(T) curves.\nI. INTRODUCTION\nThe study of quantum phase transitions in low di-\nmensionalspinsystemshasbeenafrontierareaofre-\nsearchduetoabundanceofeffectivelow-dimensional\nmagnetic materials [1–10] which exhibits a zoo of\nphases[11–23]. Theconfinementandinterplayofex-\nchange interactions in low dimensional systems like\nspin chains [6, 24, 25], spin ladders [2, 3, 26, 27]\nor two dimensional systems [28, 29] can give rise to\nvariousinterestinggroundstate(GS) properties[30–\n36]. Recently synthesized materials show that many\nof these spin-1/2 systems are frustrated even in one\ndimension (1D) [7–10], whereas the low dimensional\nsystems can be either geometrically frustrated i.e.\nantiferromagnet Heisenberg spin-1/2 on a triangu-\nlar lattice [37, 38] or exchange interaction driven\nfrustration such as 1D spin-1/2 system interacting\nwith nearest neighbor interaction J1and antiferro-\nmagneticnextnearestneighborexchangeinteraction\nJ2[12, 13, 17, 39–42]. Frustrated model Hamiltoni-\nans of one dimensional systems and zigzag geometry\n[43, 44] are extensively studied theoretically and GS\nof these systems have exotic phases like spin liquid\n[3, 45], dimer [11, 13–19], spiral/non-collinear spin\nphase [11, 14, 46], ferromagnetic phase etc.\nSpin chains and ladders can also have anisotropic\nexchange interactions [47–51] and some spin chains\n∗saniur.rahaman@bose.res.in\n†shaon@iittp.ac.in\n‡manoranjan.kumar@bose.res.incan have alternate Heisenberg and Ising exchange\ninteractions [52], whereas exchange along the leg is\nIsing type. The Heisenberg-Isingmodel has been ex-\nplored by few groups [52–54]. The simplest model\non a ladder geometry studied by Rojas et. al. [52]\nwithalternateanisotropicHeisenberg( Jx,Jx,Jz)and\nIsingtype( J0)rungexchangesandintralegexchange\ninteraction ( J1) gives interesting ground-state phase\ndiagramwith phases like frustrated phase 1 (FRU1),\nantiferromagneticphase etc. in large and small ratio\nof Ising to Heisenberg exchange interactions ( Jz/J1)\nlimits respectively. This model also shows interest-\ning sharp peak in specific heat. Verkholyak et. al.\n[53] studied an anisotropic model with Heisenberg\nrung exchange interaction ( J1) and Ising-type leg\nexchange interaction ( J2) and diagonal exchange in-\nteraction( J3). TheyshowedthatGScanexhibit dif-\nferent phases e.g. stripe leg (SL), stripe rung (SR),\nN´ eel and quantum paramagnetic (QPM) phases etc.\nin the phase diagram of J3-J1plane and the field\ndependence behavior in this model are also studied\n[54]. There are other studies of Heisenberg branched\nchainmodelwhichshowinterestingGSbehaviorand\nplateau phase in the presence of external magnetic\nfield [55].\nThethermodynamicalpropertiesoftheoneorquasi-\nonedimensionalquantumspin modelswith alternat-\ningisotropicandanisotropicunits arestudied exten-\nsively in recent times [52, 56]. In presence of alter-\nnate Heisenberg and Ising rung and Ising leg inter-\naction, the two consecutive units of the Hamiltonian\nbecomecommutingandinsuchcases,theexactther-\nmodynamical properties of these systems can be cal-culated using transfer matrix method. For example,\nthe susceptibility and other related quantities were\ncalculated exactly for an anisotropic helical single-\nchain magnet Fe2Nbusing transfer matrix method\n[56].\nIn this paper, we study a general anisotropic\nHeisenberg-Ising model on ladder geometry with al-\nternate Heisenberg and Ising exchange rung inter-\nactions, whereas the exchange interactions along\nthe leg and along the diagonal of the ladder\nare Ising type as shown in Fig. 1. The sys-\ntemexhibitsanisotropicantiferromagnetic(AAFM),\nstripe rung ferromagnetic (SRFM), stripe rung\nferromagnetic-edge (SRFM-E) and stripe leg ferro-\nmagnetic (SLFM) phases. We use exact diagonalisa-\ntion method to calculate the GS properties upto 24\nsites using Davidson algorithm [57] for diagonalisa-\ntion ofthe Hamiltonian matrix, whereasthe thermo-\ndynamical properties are studied using the transfer\nmatrixmethod [58]. Thespecificheat, magneticsus-\nceptibility, entropy and average energy are studied\nin various phases.\nThis paper is divided into four sections, and in sec-\ntion II we discuss the model Hamiltonian. In section\nIII results are discussed and is divided into four sub-\nsections. We summarise all the results and conclude\nin the section IV.\nII. MODEL HAMILTONIAN\nWe consider here a frustrated spin-1/2 ladder with\nalternating isotropic Heisenberg and Ising type in-\nteractions. For convenience, the system is divided\ninto two sublattices A and B. The A sublattice has\nHeisenberg rung interaction Jqwhile the B sublat-\n5 N/2−1\nJ\ncJ\ndJ3\nN/2−2 4 2cqJcq\nqJ\nr = 0 N/2r = N/2+1k = 1\nk = 2r = 1\nl = 1\nl = 2\nFIG. 1. (Color online). Schematic model diagram for\nspin configurations of a 2-leg anisotropic spin ladder sys-\ntem with N(= 4n) sites is shown. The interactions along\nthe odd and even rungs are Ising and Heisenberg type\nrespectively. Both along the legs and diagonals, the in-\nteractions are Ising-type. The indices k(1≤k≤2n)\nandl(= 1 or 2) in the figure represent the rung and leg\nnumber; any spin in the anisotropic or isotropic rung is\ndenoted by σk,lorSk,lrespectively. The variable ris the\ndistance from the reference spin shown within a dotted\ncircle.tice has Ising rung interaction Jc. The spins of two\nsublattices are connected by Ising-type interaction\nJcqalong the legs and also by diagonal Ising-type\ninteraction Jd. Since for the B sublattice, only the\nz-component of spins appear in the Hamiltonian, we\nrepresentthese spins by σ, whereasthe other spins S\nhave all three components. The schematic diagram\nof the spin model is shown in Fig. 1.\nThe Hamiltonian for this system (having 4 nsites)\nwith open boundary condition (OBC) is given by\nH=/summationtextn−1\ni=1Hi+Hewhere,\nHi=Jq/vectorS2i,1·/vectorS2i,2+Jc\n2(σ2i−1,1σ2i−1,2+σ2i+1,1σ2i+1,2)\n+Jcq/braceleftbig\nSz\n2i,1(σ2i−1,1+σ2i+1,1)+Sz\n2i,2(σ2i−1,2+σ2i+1,2)/bracerightbig\n+Jd/braceleftbig\nSz\n2i,1(σ2i−1,2+σ2i+1,2)+Sz\n2i,2(σ2i−1,1+σ2i+1,1)/bracerightbig\n+h\n22/summationdisplay\nl=1(2Sz\n2i,l+σ2i−1,l+σ2i+1,l),and (1)\nHe= (Jc+h\n2)2/summationdisplay\nl=1σ1,lσ1,l+Jq/vectorS2n,1·/vectorS2n,2\n+Jcq2/summationdisplay\nl=1Sz\n2n,lσ2n−1,l+Jd(Sz\n2n,1σ2n−1,2+Sz\n2n,2σ2n−1,1)\n+h\n22/summationdisplay\nl=1(2Sz\n2n,l+σ2n−1,l). (2)\nHereHeis the part of the Hamiltonian representing\nthe two edges. With the periodic boundary condi-\ntion (PBC), Hevanishes and the total Hamiltonian\nbecomes H=/summationtextn\ni=1Hiwith appropriate reduction\nof values of site index, e.g. σ2n+1,1≡σ1,1. If our\nsystem is considered to be summation over ngeo-\nmetrical units, then each unit is represented by the\nHi. It may be noted here that [ Hi,Hj] = 0 even for\nj=i+1.\nFor this work we consider Jc=Jcq= 1. The GS\nphase diagram of the system is studied here with\nrespective to the parameters JdandJq(both posi-\ntive).\nIII. RESULTS\nIn this section, fours phases are discussed in de-\ntail and to understand and characterize the phases\nand determine their boundaries, we calculate vari-\nousquantitieslikelongitudinal CL(r) =< Sz\niSz\ni+r>,\ntransverse Ct(r) =<(Sx\niSx\ni+r+Sy\niSy\ni+r)>correla-\ntions and energy crossovers. There are four major\nphases in the system: (i) stripe rung ferromagnet\n(SRFM) where the same type of sublattice spins (ei-\nther S or σ-type spins) are aligned in the same direc-\ntion, whereas other types are aligned along opposite\n2(i) SRFM\n(iii) AAFM (iv) SLFM(ii) SRFM−E\nFIG. 2. (Color online). Spin arrangements in (i) stripe\nrung ferromagnetic (SRFM), (ii) stripe rung ferromag-\nnetic edge (SRFM-E), (iii) anisotropic antiferromag-\nnetic (AAFM) and (iv) stripe leg ferromagnetic (SLFM)\nphases are shown. Arrows in the odd rungs (blue) and\neven rungs (red) represent σ-type and S type spins re-\nspectively. In subfigure (ii), the uncompensated dimer is\nshown in the box.\ndirection as shown in Fig. 2.a. (ii) In stripe rung\nferromagnetic-edge (SRFM-E) phase, bulk spins be-\nhave like SRFM phase, whereas the one of the edge\nspin pair ( S−S) behaves like isolated singlet as\nshown in Fig. 2.b and the GS is in Sz\ntot= 1 sec-\ntor where Sz\ntotis the total Szfor the entire lad-\nder. (iii) In anisotropic antiferromagnetic (AAFM)\nphase, both S and σ-type of spins are antiferro-\nmagnetically aligned with each other, two nearest\nSspins along the rung form an anisotropic singlet\nbond, whereas two nearest σspins form an Ising\nbond as shown in Fig. 2.c. The anisotropy of sin-\nglet bond decreases with increasing Jqand spins are\nhighly frustrated. (iv) In this phase, spins on each\nleg are ferromagnetically aligned but spins on other\nleg areantiferromagneticallyalignedwith eachother\n(Fig. 2.d) and therefore this frustrated arrangement\nis called stripe leg ferromagnet (SLFM).\nA. Quantum phase diagram\nIn Fig. 3 the four phases, the SRFM, the SRFM-\nE, the AAFM and the SLFM are shown separated\nby five phase boundaries for N= 24, and we no-\ntice that the phase boundaries weakly depend on\nthe system size. These phase boundaries are deter-\nmined based on energy crossovers and the correla-\ntionfunctions CL(r) andCT(r) bytuning JdandJq.\nThe large fraction of the phase space is covered by\nthe SRFM phase and the AAFM phase has second\nlargest contribution. It is interesting to note that\nthe phase boundary of the AAFM and the SLFM\nis atJd/J= 1 for large Jq. Here, the bond order0 0.5 1 1.5 2 2.5Jq00.511.522.533.54JdSRFM\nSLFM\nSRFM-E\nAAFM\nFIG. 3. (Color online). Quantum phase diagram of the\nLadder with open boundary condition is shown. Spin ar-\nrangements of the AAFM, the SLFM, the SRFM phases\nare shown inside the boxes.\n< Si.Si+1>between the two Sspin along the rung\nform a perfect singlet dimer. The correlation length\ninCL(r) shrinks to one unit cell, but this phase is\nrestricted to only this phase boundary. The strong\nsinglet dimers along the rung at Btype sublattice\n(σ−σ)areformedoneithersidesofthephasebound-\nary.\nB. Ground state energy and excitation gap\nThe GS energy EGSof the system is doubly degen-\neratein majorpart of the parameterspace, and EGS\nand the lowestexcited state in Sz\ntot= 0 and 1sectors\nare analyzed as shown in Fig. 4. The lowest state\nenergy in Sz\ntot= 0 and 1 sectors are shown in Fig.\n4.a forJq= 0.2. The lowest energy EGSinSz\ntot= 0\nsectorinitiallyincreaseswith Jdduetoenhancement\nin the frustration induced by Jdand it starts to de-\ncrease again for Jd>0.33, as the Jdbecomes dom-\ninant and frustration decreases and system goes to\nthe SRFM phase. The peak of EGSindicates the\nphase boundary. For small Jqthe phase transition\nfrom the AAFM to the SRFM seems to be sharp as\nderivative of EGSis discontinuous as shown in Fig.\n4 a. Whereas the change in EGSis continuous for\nlargeJqas shown in Fig. 4, therefore phase tran-\nsition seems to be second order. In Fig. 4.b the\nlowest excited state in Sz\ntot= 0 and the lowest state\ninSz\ntot= 1 sectorareshown with blackand red color\nline-symbols for Jq= 1 respectively. Negative value\nof red curve indicates the Sz\ntot= 1 as GS and the\nstate appears because of a singlet dimer pair for-\nmation between edge S−Sspins, if the chain starts\n30.2 0.3 0.4 0.5\nJd-6.5-6-5.5-5E/J\n0.5 1 1.5 2 2.5 3 3.5\nJd-20-18-16-14-12-10-8E/J0.40.60.81 1.2\nJd-0.5-0.2500.250.50.75\n∆(a)\nAAFM SLFM SRFM-EAAFM\nSRFMSRFM AAFM(b)\n(c)SRFM-E SRFM\nFIG. 4. (Color online). (a) Black solid line and red\ndashed line represent energies ( E/J) in two respective\nspin sectors: Sz\ntot= 0 and Sz\ntot= 1 forJq= 0.2, (b) ∆ is\nthe energy gap for Jq= 1.0. Red solid line represents the\nlowest energy gap in Sz\ntot= 0 sector and green solid line\nrepresents the energy gap of the lowest state in Sz\ntot= 1\nsector from the lowest state in Sz\ntot= 0 sector. (c) Black\nsolid line and red dashed line represent energies ( E/J)\nin two respective spin sectors: Sz\ntot= 0 and Sz\ntot= 1 for\nJq= 1.8. (E/J) and ∆ in all the subfigures are shown\nfor system size N=24.\nwithσ−σspin pair (odd rungs) and ends with S−S\nspin pair (even rungs) as considered in the system.\nIn this case, a pair of uncompensated ferromagneti-\ncallyaligned σ−σpairgivesrisetothe Sz\ntot= 1man-\nifold. The boundaries for the SRFM-E is obtained\nby onset and end of the GS with Sz\ntot= 1 as shown\nin Fig. 4.b. In Fig. 4.c all four phases and their\nboundaries are shown for Jq= 1.8. We notice that\nthe maxima of doubly degenerate GS is the phase\nboundary between the AAFM and the SLFM phase,\nwhereas, the onset and end of GS in the Sz\ntot= 1 is\nthe phase boundary of the SRFM-E phase. In the\nSRFM phase, the GS is again in Sz\ntot= 0 sector.\nIt is also evident from all three figures that EGSis\ncontinuous in large Jqlimit.\nC. Correlation functions\nTounderstandthearrangementofspinintheGS,we\nstudy the two component: longitudinal CL(r) and\ntransverse CT(r)correlationsinfourdifferentphases\nas shown in Fig. 5. The reference site is at the lower\nleg of sublattice A ( S-type spin) at mid of the lad-\nder and the arrangement of distance r is shown in\nFig. 1 . In the SRFM phase ( Jq= 0.2,Jd= 2.0),\ntheCL(r) shows long-range behavior and nearest0 2 4 6 810 12 14-0.3-0.2-0.100.10.20.3CL(r)\n0 2 4 6 810 12 14r-0.25-0.2-0.15-0.1-0.050CT(r)(a)\n(b)\nFIG. 5. (Color online). (a) Longitudinal and (b) Trans-\nverse Correlation plots are shown. Black, red, maroon\nand blue colors in both of the subfigures, represent four\nrespective phases; SRFM ( Jq= 0.2,Jd= 2.0), SRFM-E\n(Jq= 1.6,Jd= 1.25), AAFM ( Jq= 2.0,Jd= 0.4) and\nSLFM (Jq= 2.0,Jd= 1.6).\nneighboralongtherungisferromagneticallyaligned,\nwhereas nearest neighbor along the leg is antiferro-\nmagnetically aligned. The CT(r) is zero for spins,\ntherefore, GS is completely Ising like. In the SRFM-\nE(Jq= 1.6,Jd= 1.25), the correlationfunctions are\nsame as that for the SRFM except at the boundary\nwherethe CL(r) goesto zeroi.e the last pairofspins\nis decoupled from the ladder. The CT(r) is zero for\nall spins with respect to reference spin, but between\nedge rung spin pair S−Sit is -1/2. In the SLFM\nphase(Jq= 2.0,Jd= 1.6),thenearestrungspinsare\nantiferromagnetically aligned, whereas along the leg\nnearestneighborspinsareferromagneticallyaligned.\nThe nonzero value of CT(r) is restricted to nearest\nrung spin. However, in the limit Jq= 2.0,Jd= 0.4\n(AAFM phase), the CL(r) is long-range and both\nthe nearest spins along the rung and along the leg\nare antiferromagnetically aligned. The CT(r) is re-\nstricted to the only nearest rung spin and the value\nCT(r= 1) increaseswith Jqas shown in Fig. 5. It is\nalso interesting to note that the long-range behavior\nin the correlation CL(r) melts with increasing Jq.\nThe AAFM phase is interesting due to highly\nanisotropy correlations in the system and also the\nrapid variation in the correlation with Jq. To our\nsurprise, at Jd= 1, two nearest spins along the\nrung (S−Spairs) form perfect singlet dimers, and\nthe GS of the system behaves like product of Ising\nand singlet dimers. To show the GS spin arrange-\nment,CL(r) andCT(r) forJd= 0.8,1 and 1.2 for\nJq= 2.0 are plotted as a function of distance rin\nFig. 6. We notice finite value of CL(r) andCT(r)\nare restricted to nearest rung spin, whereas, CL(r)\n40 2 4 6 810 12 14-0.3-0.2-0.100.1CL(r)\n0 2 4 6 810 12 14r-0.25-0.2-0.15-0.1-0.050CT(r)(a)\n(b)\nFIG. 6. (Color online). (a) Longitudinal and (b) Trans-\nverse Correlations for three phases are shown. Maroon,\nmagenta and blue colors in both of the subfigures are for\nAAFM ( Jq= 2.0,Jd= 0.8), Perfect Dimer ( Jq= 2.0,\nJd= 1.0) and SLFM ( Jq= 2.0,Jd= 1.2) phases respec-\ntively.\nare non-collinear in nature in the neighborhood of\nJd= 1 for large Jq. For two values of Jd= 0.8 and\n1.2 forJq= 2.0,CL(r) shows non-collinear spin ar-\nrangement and the CT(r) is restricted to the same\nrung in A sublattice ( Sspin) as shown in Fig. 6.\nD. Exact thermodynamical properties\nThe spin model Hamiltonian in Eq. 1 have com-\nmuting bonds operators because Ising exchange in-\nteractions along the leg and diagonal of the ladder,\ntherefore using transfer matrix method exact solu-\ntion at finite temperature can be studied. In this pa-\nper, we study the low-temperature thermodynami-\ncal properties of our model using a suitably adapted\ntransfer matrix method. Henceforth, our transfer\nmatrix calculations assume periodic boundary con-\ndition (PBC) and we will be using the full Hamilto-\nnian without the edge part ( He= 0). The Hamil-\ntonian for a single geometrical unit (Eq.1) can be\nreduced in the following manner:\nHi=Jq\n2/parenleftbig\nS+\n2i,1S−\n2i,2+S+\n2i,2S−\n2i,1/parenrightbig\n+Jq/parenleftbig\nSz\n2i,1Sz\n2i,2/parenrightbig\n+aSz\n2i,1+bSz\n2i,2+c+d. (3)\nHere a, b, c, d can be written in terms of the pa-\nrameters Jc,Jcq,Jdandh, and the spin operator\nσ(see in appendix VI). In the equation S+,S−are\nthe creation and annihilation operators respectively\nfor spin S.\nDue to special construction of our model, we have[Hi,Hj] = 0 for any iandj. This fact helps us\nto write the partition function of the total system\nas the trace of the n-th power of a small (4 ×4)\ntransfer matrix (see the details in Appendix VI).\nThe partition function for N(= 4n) number ofspins,\nQN(β) =Tr(e−βH) withβbeing the inverse tem-\nperature can be written as,\nQN(β) =λn\n1+λn\n2+λn\n3+λn\n4,\nwhere four λ’s are the eigenvalues of the trans-\nfer matrix. If λ1is the largest eigenvalue then\nfor large N,QN(β) =λ1n(see in appendix VI).\nUsing the partition function Q4(β) (=λ1, parti-\ntion function for a geometric unit), the thermody-\nnamic quantities can be calculated using the follow-\ning standard formulas: free energy (per geometri-\ncal unit) F(T) =−kBTlogQ4(β), average energy\nE(T) =kBT2d\ndTlogQ4(β), specific heat Cv(T) =/parenleftig\n∂E(T)\n∂T/parenrightig\nv, magnetization M(T) =−∂F(T)\n∂h, mag-\nnetic susceptibility χ(T) =∂M(T)\n∂h, and entropy\nS(T) =−/parenleftig\n∂F(T)\n∂T/parenrightig\n.\nInT→0 limit, the largest eigenvalue λ1can be\nwritten as λ1=eβ(1+Jq(1+2∆ 2))\n4+eβ(4Jd−Jq+3))\n4, where\n∆2=/radicalig\n1+4(1−Jd)2\nJ2q(see in appendix VI). In the\nzero-temperature limit, the first exponential term in\nthe expression of λ1dominates over the second ex-\nponential term in the regimes corresponding to the\nAAFM and the SLFM phases, while in the regime\ncorrespondingtotheSRFMphase,theoppositehap-\npens. In this T→0 limit, the free energy takes\nthe following forms in the regimes corresponding\nto the SRFM and the AAFM phases respectively:\nFSRFM=−4Jd−Jq+3\n4andFAAFM=FSLFM=\n−1+Jq(1+2∆ 2)\n4. In this zero temperature limit, in all\nthe three regimes, the entropy and the specific heat\nare found to be zero. These results match well with\nour numerical calculations using the full expression\nofλ1(see in appendix VI).\nTo understand the thermodynamic behavior at the\nnon-zero temperatures, we calculate four thermo-\ndynamical quantities E(T),Cv(T),S(T) andχ(T)\nfor three parameter regimes and are shown in Fig.\n7. We use full expression of the largest eigenvalue\nλ1for this numerical calculation. It may be noted\nthat the different ground state phases of the sys-\ntem, which were obtained with open boundary con-\ndition for the finite system sizes, may not have di-\nrect consequences in our low-temperature thermo-\ndynamic results as the thermodynamic calculations\nare done with periodic boundary condition for ther-\nmodynamically large system. Here, our main pur-\npose of studying the thermodynamical quantities is\nto see how these quantities change across the pa-\n500.5 11.5\nT/J-4-3-2-10E(T)00.5 11.5\nT/J00.511.522.53Cv(T)\n00.5 11.5\nT/J00.511.522.53S(T)\n00.5 11.5\nT/J00.10.20.30.40.5χ(Τ)(d)(a)\n(c)(b)\nAAFMSLFMSRFM\nFIG. 7. (Color online). (a) Specific heat ( Cv(T)), (b)\nEntropy ( S(T)), (c) Magnetic susceptibility ( χ(T)) and\n(d) Average Energy ( E(T)) plots are shown. Black,\nred and green curves in each of the subfigures, repre-\nsent three phases: AAFM ( Jq= 2.0,Jd= 0.5), SLFM\n(Jq= 2.0,Jd= 2.5) and SRFM ( Jq= 0.2,Jd= 2.5)\nrespectively.\nrameter regimes of interest. The Cv(T) of the three\ndifferent phases show different features as shown in\nFig. 7.a. In the AAFM region where Jdis weak\nandJqis dominant, Cv(T) shows a small peak near\ntheT→0, which may be because of small gap due\nto small excitation gap in Sz\ntot= 0 sector, and then\nthere is broad maxima at higher temperature, which\nis similar to the Heisenberg spin dimer system. The\nweak singlet dimer is formed along the rung of S−S\nspins and that may give a broad peak at moderate\ntemperature. The Cv(T) in the SLFM phase shows\nvery sharp peak and long tail, but have vanishing\nsmall value for T/J <0.09 due to finite energy gap\nin the system. In the SRFM phase, this quantity is\nvanishingly small for T/J <0.35 due to large mag-\nnetic gap which makes the system to thermalise at\nhigher temperature and a relatively higher peak at\nT/J= 0.75. The entropy S(T) is in some sense is\nthe measure of thermalisation, which in three dif-\nferent phases of the system are shown in Fig. 7.b.\nIn the AAFM phase, there is a small non-magnetic\ngap. Whereas, in other two phases S(T) is vanish-\ningly small for T/J <0.1 due to large energy gap\nand thereafter it increases monotonically.\nThe magnetic susceptibility χ(T) in these three\nphases are shown in Fig. 7.c and all the χ(T) have\nsmall values in all three phases for T/J <0.1. It\nhas a broad maxima and small gap in the AAFM\nphase due to the formation of singlet dimer, and for\nbreaking the weak singlet dimer it costs finite en-\nergy, therefore,singlet-tripletgapisfinite. The χ(T)\nin the SRFM phase has dominant Ising interaction,\ntherefore, there is a finite energygap and sharp peaksimilar to the 1D Ising system. In the SLFM phase\nthere is large magnetic gap as it requires breaking of\nstrong rung interaction, and this leads to small χ(T)\nat low temperature and exponential increase in the\nχ(T). The average internal energy E(T) shows a\nlinear variation with Tin the AAFM phase, but al-\nmost constant value of E(T) forT <0.09 indicates\nthe gap in the SLFM phase as shown in Fig. 7.d. In\nthe SRFM phase, variation of the energy is almost\nconstant for T/J <0.35 due to large energy gap and\nit varies linearly with Tthereafter.\nIV. SUMMARY AND CONCLUSION\nIn this paper, we consider a very general anisotropic\nHeisenberg-Ising model on ladder geometry with al-\nternate Heisenberg and Ising exchange rung interac-\ntions, whereas the exchange interactions along the\nleg and along the diagonal of the ladder are Ising\ntype. We construct a quantum phase diagram of the\nmodel Hamiltonian in Sec.II, and have shown that\nthere are four quantum phases: (i) the AAFM, (ii)\nthe SRFM, (iii) the SRFM-E and (iv) the SLFM\nwhich appear due to competing interactions and\nanisotropy in the system. The GS is doubly degen-\nerateand have finite magnetic gap in most ofthe pa-\nrameter space and to our surprise, exact dimer state\nalong the rung in A sublattice (rungs with isotropic\nexchange interactions) appears for Jd= 1 and large\nJqlimit. However, weak dimer appears along rung\nof spinSnear toJd= 1.\nThe thermal properties of this system are also stud-\nied analytically using the transfer matrix method.\nFour temperature dependent properties like specific\nheatCv(T), average internal energy E(T), entropy\nS(T) and magnetic susceptibility χ(T) are studied\nin three different phases: the SRFM, the SLFM and\nthe AAFM. In largeJq\nJdregime (AAFM phase), Cv\nshows a small peak at small Tdue to a small exci-\ntation gap, whereas it has vanishingly small χupto\nT <0.09 due to finite magnetic gap in the SLFM\nphase. Due to large excitation gap in the SRFM\nphase, all four quantities vanish for T <0.35.\nIn conclusion, we have studied a highly anisotropic\nmodel on a ladder geometry and the model Hamil-\ntonian exhibits four interesting GS phases. The\nthermodynamic quantities like Cv(T),χ(T),E(T)\nandS(T) are also studied using the transfer ma-\ntrix method. This model may be realized in Cu\nor Vi based materials having magnetic interaction\nconfined in ladder like geometry and the material\nshould also have large anisotropy to ensure the Ising\nexchange.\n6V. ACKNOWLEDGEMENTS\nMK thanks DST India for a Ramanujan Fellowship\nSR/S2/RJN-69/2012. MK thanks SMST Depart-\nment of IIT (BHU) for the hospitality during his\nvisit. SS thanks SNBNCBS for supporting him un-\nder EVLP during his stay at the Centre when this\nwork was started.VI. APPENDIX\nThe partition function for Nsites,QN(β) with\nHamiltonian Hcan be written as-\nQN(β) =Tr/parenleftbig\ne−βH/parenrightbig\n(4)\nwhere, Tr means trace of the matrix, β= 1/(kBT)\nandkBis the Boltzmann constant. Using explicit\nconfigurationbasis forthe system, Eq. 4is rewritten\nin the following form,\nQN(β) =/summationdisplay\n{σ,S}<···,σ2i−1,1,σ2i−1,2,S2i,1,S2i,2,··· |e−βH| ···,σ2i−1,1,σ2i−1,2,S2i,1,S2i,2,···>,\nhere the summation is over all possible configurations {σ,S}of the system. For a given configuration,\n|···,σ2i−1,1,σ2i−1,2,S2i,1,S2i,2,···>represents a basis state. Since for our system, the Hamiltonians c or-\nresponding to different units commute with each other, we further get,\nQN(β) =/summationdisplay\nσ<···,σ2i−1,1,σ2i−1,2,··· |n/productdisplay\ni=1Ti| ···,σ2i−1,1,σ2i−1,2,···>\nwhereTi=/summationtext\n{S}i< S2i,1,S2i,2|e−βHi(σ,S)|S2i,1,S2i,2>. Here the summation is over {S}iwhich\nrepresents all possible configurations of spins S2i,1andS2i,2(from the ithunit). It may be noted that Tidoes\nnot contain the components of spin Soperators and it has only σvariables, namely, σ2i−1,1,σ2i−1,2,σ2i+1,1\nandσ2i+1,2.\nThis form is well-known with Tibeing the trans-\nfer operator. Introducing identity operators I=/summationtext\n{σ}i|σ2i−1,1,σ2i−1,2>< σ2i−1,1,σ2i−1,2|between\nsuccessive Toperators, we can finally write the par-\ntition function as the trace of the n-th power of a\nsmall (4×4) transfer matrix P. We have,\nQN(β) =Tr(Pn),wherenis the number of geometrical units. The\nelements of the transfer matrix are given by\nP(σ2i−1,1,σ2i−1,2),(σ2i+1,1,σ2i+1,2)=< σ2i−1,1,σ2i−1,2|Ti|σ2i+1,1,σ2i+1,2> (5)\nBefore we construct and diagonalise the Pmatrix,\nwe first need to carry out the trace over the configu-\nrations{S}ito find out the form of Ti. SinceTi=/summationtext\n{S}i< S2i,1,S2i,2|e−βHi(σ,S)|S2i,1,S2i,2>, ifwe\ntake the eigenstate basis of Hi, we will get Tias the\nsummation overexponential ofeigenvalues of −βHi.\nNext we calculate eigenvalues of Hioperator.\nBy considering,\na=Jd/parenleftbig\nσz\n2i−1,2+σz\n2i+1,2/parenrightbig\n+Jcq/parenleftbig\nσz\n2i−1,1+σz\n2i+1,1/parenrightbig\n+h\nb=Jd/parenleftbig\nσz\n2i−1,1+σz\n2i+1,1/parenrightbig\n+Jcq/parenleftbig\nσz\n2i−1,1+σz\n2i+1,1/parenrightbig\n+\nh\nc=Jc\n2/parenleftbig\nσz\n2i−1,1σz\n2i−1,1+σz\n2i+1,1σz\n2i+1,2/parenrightbig\nd=h\n2/parenleftbig\nσz\n2i−1,1+σz\n2i−1,2+σz\n2i+1,1+σz\n2i+1,2/parenrightbig\n,\nHamiltonian (Eq. 3 ) for the ithgeometrical unit\ncan be written as-\n7Hi=Jq\n2/parenleftbig\nS+\n2i,1S−\n2i,2+S−\n2i,1S+\n2i,2/parenrightbig\n+Jq/parenleftbig\nSz\n2i,1Sz\n2i,2/parenrightbig\n+aSz\n2i,1+bSz\n2i,2+c+d\nBy taking f=c+d, we can write down the following Hamiltonian matrix in the eigenstate ba sis ofSz\n2i,1Sz\n2i,2\noperator,\nHi=\nJq\n4+(a+b)\n2+f 0 0 0\n0−Jq\n4+(a−b)\n2+fJq\n20\n0Jq\n2−Jq\n4−(a−b)\n2+f 0\n0 0 0Jq\n4−(a+b)\n2+f\n.\nThe Hamiltonian matrix comes up with its four\neigenvalues from three Sz\nSSsectors based on S-S\npairs-\n(i) From Sz\nSS= 1sector (formed by S-S pair)\nθ1= (f+Jq\n4)+(a+b)\n2\n(ii) From Sz\nSS=−1sector (formed by S-S\npair)\nθ2= (f+Jq\n4)−(a+b)\n2\n(iii) From Sz\nSS= 0sector (formed by S-S pair)\nθ3= (f−Jq\n4)+√\nJ2q+(a−b)2\n2\nθ4= (f−Jq\n4)−√\nJ2q+(a−b)2\n2.\nWe note that the eigenvalues ( θk) are functions\nofσvariables, namely σ2i−1,1,σ2i−1,2,σ2i+1,1and\nσ2i+1,2. Using these eigenvalues, we rewrite Tias,\nTi=/summationdisplay\n{S}i< S2i,1,S2i,2|e−βHi(σ,S)|S2i,1,S2i,2>\n=4/summationdisplay\nk=1e−βθk.\nWithout magnetic field (h=0), the Transfer Matrix\n(P) takes the following form (using Eq. 5)-\nP=\np q q r\nq s u q\nq u s q\nr q q p\n,here,\np= 2e−β\n4[Q−1Coshβ(1+Jd)+QCosh(βJq\n2)]\nq= 2[Q−1Coshβ((1+Jd)/2)+QCosh(βJq∆1\n2)]\nr= 2e−β\n4[Q−1+QCosh(βJq\n2)]\ns= 2eβ\n4[Q−1+QCosh(βJq∆2\n2)]\nu= 2eβ\n4[Q−1+QCosh(βJq\n2)]\n∆1=/radicaligg\n1+(1−Jd)2\nJ2q\n∆2=/radicaligg\n1+4(1−Jd)2\nJ2q\nQ=eβJq\n4.\nThe above Transfer Matrix has 4 simple eigenvalues\nλ1,λ2,λ3,λ4as follows\nλ1=(p+r+s+u)\n2\n+/radicalbig\n(p+r+s+u)2+16q2−4(p+r)(s+u)\n2\nλ2=(p+r+s+u)\n2\n−/radicalbig\n(p+r+s+u)2+16q2−4(p+r)(s+u)\n2\nλ3= (p−r)\nλ4= (s−u).\nIt is to be noted that λ1is the largest eigenvalue\nhere.\nIn the special case with T→0 limit, the\nlargest eigenvalue can be approximated as-\nλmax= (p+r+s+u).\nExplicitly, we have,\n8λmax=\n(2eβ(3Jq−1)\n4+eβ(4Jd+3−Jq)\n4+5eβ(1−Jq)\n4+eβ(1+Jq(1+2∆ 2)\n4).\n[1] S. 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El Sherbini1\n1Department of Physics, Cairo University, Cairo, 12613, Egy pt\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federati on\n3Dubna State University, Dubna, 141982, Russian Federation\n4Department of Electrical and Electronic Systems Engineeri ng, Utsunomiya University, Utsunomiya, Japan.\nWe investigate the coupling between ferromagnet and superc onducting phase dynamics in\nsuperconductor-ferromagnet-superconductor Josephson j unction. The current-voltage character-\nistics of the junction demonstrate a pattern of subharmonic current steps which forms a devil’s\nstaircase structure. We show that a width of the steps become s maximal at ferromagnetic reso-\nnance. Moreover, we demonstrate that the structure of the st eps and their widths can be tuned\nby changing the frequency of the external magnetic field, rat io of Josephson to magnetic energy,\nGilbert damping and the junction size.\nThis paper is submitted to LTP Journal.\nI. INTRODUCTION\nJosephson junction with ferromagnet layer (F) is\nwidely considered to be the place where spintronics and\nsuperconductivity fields interact1. In these junctions\nthe supercurrent induces magnetization dynamics due\nto the coupling between the Josephson and magnetic\nsubsystems. The possibility of achieving electric con-\ntrol over the magnetic properties of the magnet via\nJosephson current and its counterpart, i.e., achieving\nmagnetic control over Josephson current, recently at-\ntracted a lot of attention1–7. The current-phase rela-\ntion in the superconductor-ferromagnet-superconductor\njunction (SFS) junctions is very sensitive to the mutual\norientation of the magnetizations in the F-layer8,9. In\nRef.[10] the authors demonstrate a unique magnetization\ndynamics with a series of specific phase trajectories. The\norigin of these trajectories is related to a direct coupling\nbetween the magnetic moment and the Josephson oscil-\nlations in these junctions.\nExternal electromagnetic field can also provide a cou-\npling between spin wave and Josephson phase in SFS\njunctions11–17. Spin waves are elementary spin excita-\ntions which considered to be as both spatial and time\ndependent variations in the magnetization18,19. The fer-\nromagnetic resonance(FMR) correspondsto the uniform\nprecession of the magnetization around an external ap-\nplied magnetic field18. This mode can be resonantly ex-\ncited by ac magnetic field that couples directly to the\nmagnetization dynamics as described by the Landau-\nLifshitz-Gilbert (LLG) equation18,19.\nIn Ref.[18] the authors show that spin wave resonance\nat frequency ωrin SFS implies a dissipation that is mani-\nfested as adepressionin the IV-characteristicofthe junc-\ntion when /planckover2pi1ωr= 2eV, where/planckover2pi1is the Planck’s constant,\ne is the electron charge and Vis the voltage across the\njunction. The ac Josephson current produces an oscil-\nlating magnetic field and when the Josephson frequencymatches the spin wave frequency, this resonantly excites\nthe magnetization dynamics M(t)18. Due to the non-\nlinearity of the Josephson effect, there is a rectification\nof current across the junction, resulting in a dip in the\naverage dc component of the suppercurrent18.\nIn Ref.[13] the authors neglect the effective field due\nto Josephson energy in LLG equation and the results re-\nveal that even steps appear in the IV-characteristic of\nSFS junction under external magnetic field. The ori-\ngin of these steps is due to the interaction of Cooper\npairs with even number of magnons. Inside the ferro-\nmagnet, if the Cooper pairs scattered by odd number of\nmagnons, no Josephson current flows due to the forma-\ntion of spin triplet state13. However, if the Cooper pairs\ninteract with even number of magnons, the Josephson\ncoupling between the s-wave superconductor is achieved\nand the spin singlet state is formed, resulting in flows of\nJosephsoncurrent13. In Ref.[20]weshowthat takinginto\naccount the effective field due to Josepshon energy and\nat FMR, additional subharmonic current steps appear in\nthe IV-characteristic for overdamped SFS junction with\nspin wave excitations (magnons). It is found that the po-\nsition of the current steps in the IV-characteristics form\ndevil’s staircase structure which follows continued frac-\ntion formula20. The positions of those fractional steps\nare given by\nV=\nN±1\nn±1\nm±1\np±..\nΩ, (1)\nwhere Ω = ω/ωc,ωis the frequency of the external ra-\ndiation, ωcis the is the characteristic frequency of the\nJosephson junction and N,n,m,pare positive integers.\nIn this paper, we present a detailed analysis for the\nIV-characteristics of SFS junction under external mag-\nnetic field, and show how we can control the position\nof the subharmonic steps and alter their widths. The\ncoupling between spin wave and Josephson phase in SFS\njunction is achieved through the Josephson energy and\ngauge invariant phase difference between the S-layers. In\nthe framework of our approach, the dynamics of the SFS2\njunction isfully describedbytheresistivelyshuntedjunc-\ntion (RSJ) model and LLG equation. These equations\nare solved numerically by the 4thorder Runge-Kutta\nmethod. The appearance and position of the observed\ncurrent steps depend directly on the magnetic field and\njunction parameters.\nII. MODEL AND METHODS\nF\nss\nHacxyz\nH0I\nI\nFIG. 1. SFS Josephson junction. The bias current is applied\nin x-direction, an external magnetic field with amplitude Hac\nand frequency ωis applied in xy-plane and an uniaxial con-\nstant magnetic field H0is applied in z-direction.\nIn Fig 1 we consider a current biased SFS junction\nwhere the two superconductors are separated by ferro-\nmagnet layer with thickness d. The area of the junction\nisLyLz. An uniaxial constant magnetic field H0is ap-\nplied in z-direction, while the magnetic field is applied in\nxy-plane Hac= (Haccosωt,Hacsinωt,0)withamplitude\nHacand frequency ω. The magnetic field is induced in\nthe F-layer through B(t) = 4πM(t), and the magnetic\nfluxes in z- and y-direction are Φ z(t) = 4πdLyMz(t),\nΦy(t) = 4πdLzMy(t), respectively. The gauge-invariant\nphase difference in the junction is given by21:\n∇y,zθ(y,z,t) =−2πd\nΦ0B(t)×n, (2)\nwhereθis the phase difference between superconducting\nelectrodes, and Φ 0=h/2eis the magnetic flux quantum\nandnis a unit vector normal to yz-plane. The gauge-\ninvariantphasedifference in terms ofmagnetizationcom-\nponents reads as\nθ(y,z,t) =θ(t)−8π2dMz(t)\nΦ0y+8π2dMy(t)\nΦ0z,(3)\nwhere Φ 0=h/(2e) is the magnetic flux quantum.\nAccordingtoRSJ model, the currentthroughthe junc-\ntion is given by13:\nI\nI0c= sinθ(y,z,t)+Φ0\n2πI0cRdθ(y,z,t)\ndt,(4)\nwhereI0\ncis the critical current, and R is the resistance\nin the Josephson junction. After taking into account thegaugeinvarianceincludingthemagnetizationoftheferro-\nmagnetandintegratingoverthejunction areatheelectric\ncurrent reads13:\nI\nI0c=Φ2\nosin(θ(t))sin/parenleftBig\n4π2dMz(t)Ly\nΦo/parenrightBig\nsin/parenleftBig\n4π2dMy(t)Lz\nΦo/parenrightBig\n16π4d2LzLyMz(t)My(t)\n+Φ0\n2πRI0cdθ(y,z,t)\ndt. (5)\nThe applied magnetic field in the xy-plane causes pre-\ncessionalmotionofthemagnetizationinthe F-layer. The\ndynamics of magnetization Min the F-layer is described\nby LLG equation\n(1+α2)dM\ndt=−γM×Heff−γ α\n|M|[M×(M×Heff)](6)\nThe total energy of junction in the proposed model is\ngivenby E=Es+EM+EacwhereEsistheenergystored\nin Josephson junction, EMis the energy of uniaxial dc\nmagnetic field (Zeeman energy) and Eacis the energy of\nac magnetic field:\nEs=−Φ0\n2πθ(y,z,t)I+EJ[1−cos(y,z,t)],\nEM=−VFH0Mz(t),\nEac=−VFMx(t)Haccos(ωt)−VFMy(t)Hacsin(ωt)(7)\nHere,EJ= Φ0I0\nc/2πis the the Josephson energy, H0=\nω0/γ,ω0is the FMR frequency, and VFis the volume of\nthe ferromagnet. We neglect the anisotropy energy due\nto demagnetizing effect for simplicity. The effective field\nin LLG equation is calculated by\nHeff=−1\nVF∇ME (8)\nThus, the effective field Hmdue to microwave radiation\nHacand uniaxial magnetic field H0is given by\nHm=Haccos(ωt)ˆex+Hacsin(ωt)ˆey+H0ˆez.(9)\nwhile the effective field ( Hs) due to superconducting part\nis found from\nHs=−EJ\nVFsin(θ(y,z,t))∇Mθ(y,z,t).(10)\nOne should take the integration of LLG on coordinates,\nhowever, the superconducting part is the only part which\ndepends on the coordinate so, we can integrate the ef-\nfective field due to the Josephson energy and insert the\nresult into LLG equation. Then, the y- and z-component\nare given by\nHsy=EJcos(θ(t))sin(πΦz(t)/Φ0)\nVFπMy(t)Φz(t)/bracketleftbigg\nΦ0cos(πΦy(t)/Φ0)\n−Φ2\n0sin(πΦy(t)/Φ0)\nπΦy(t)/bracketrightbigg\nˆey, (11)\nHsz=EJcos(θ(t))sin(πΦy(t)/Φ0)\nVFπMz(t)Φy(t)/bracketleftbigg\nΦ0cos(πΦz(t)/Φ0)\n−Φ2\n0sin(πΦz(t)/Φ0)\nπΦz(t)/bracketrightbigg\nˆez. (12)3\nAs a result, the total effective field is Heff=Hm+\nHs. In the dimensionless form we use t→tωc,ωc=\n2πI0\ncR/Φ0is the characteristic frequency, m=M/M0,\nM0=∝ba∇dblM∝ba∇dbl,heff=Heff/H0,ǫJ=EJ/VFM0H0,hac=\nHac/H0, Ω =ω/ωc, Ω0=ω0/ωc,φsy=4π2LydM0/Φo,\nφsz=4π2lzdM0/Φo. Finally, the voltage V(t) =dθ/dtis\nnormalized to /planckover2pi1ωc/(2e). The LLG and the effective field\nequations take the form\ndm\ndt=−Ω0\n(1+α2)/parenleftbigg\nm×heff+α[m×(m×heff)]/parenrightbigg\n(13)\nwith\nheff=haccos(Ωt)ˆex+(hacsin(Ωt)+ΓijǫJcosθ)ˆey\n+ (1+Γ jiǫJcosθ)ˆez, (14)\nΓij=sin(φsimj)\nmi(φsimj)/bracketleftbigg\ncos(φsjmi)−sin(φsjmi)\n(φsjmi)/bracketrightbigg\n,(15)\nwherei=y,j=z. The RSJ in the dimensionless form is\ngiven by\nI/I0\nc=sin(φsymz)sin(φszmy)\n(φsymz)(φszmy)sinθ+dθ\ndt.(16)\nThe magnetization and phase dynamics of the SFS\njunction can be described by solving Eq.(16) together\nwith Eq.(13). To solve this system of equations, we em-\nploy the fourth-order Runge-Kutta scheme. At each cur-\nrent step, we find the temporal dependence of the volt-\nageV(t), phase θ(t), andmi(i=x,y,z) in the (0 ,Tmax)\ninterval. Then the time-average voltage Vis given by\nV=1\nTf−Ti/integraltext\nV(t)dt, whereTiandTfdetermine the in-\nterval for the temporal averaging. The current value is\nincreased or decreased by a small amount of δI (the bias\ncurrent step) to calculate the voltage at the next point\nof the IV-characteristics. The phase, voltage and mag-\nnetization components achieved at the previous current\nstep are used as the initial conditions for the next cur-\nrent step. The one-loop IV-characteristic is obtained by\nsweeping the bias current from I= 0 toI= 3 and back\ndown to I= 0. The initial conditions for the magnetiza-\ntion components are assumed to be mx= 0,my= 0.01\nandmz=/radicalBig\n1−m2x−m2y, while for the voltage and\nphase we have Vini= 0,θini=0. The numerical param-\neters (if not mentioned) are taken as α= 0.1,hac= 1,\nφsy=φsz= 4,ǫJ= 0.2 and Ω 0= 0.5.\nIII. RESULTS AND DISCUSSIONS\nItiswell-knownthatJosephsonoscillationscanbesyn-\nchronized by external microwave radiation which leads\nto Shapiro steps in the IV-characteristic22. The position\nof the Shapiro step is determined by relation V=n\nmΩ,\nwheren,mare integers. The steps at m= 1 are calledharmonics, otherwise we deal with synchronized subhar-\nmonic (fractional) steps. We show below the appearance\nof subharmonics in our case.\nFirst we present the simulated IV-characteristics at\ndifferent frequencies of the magnetic field. The IV-\ncharacteristics at three different values of Ω are shown\nin Fig 2(a).\nFIG. 2. (a) IV-characteristic at three different values of Ω.\nFor clarity, the IV-characteristics for Ω = 0 .5 and Ω = 0 .7\nhave been shifted to the right, by ∆ I= 0.5 and ∆ I= 1,\nrespectively with respect to Ω = 0 .2; (b) An enlarged part\nof the IV-characteristic with Ω = 0 .7. To get step voltage\nmultiply the corresponding fraction with Ω = 0 .7.\nAs we see, the second harmonic has the largest step\nwidth at the ferromagnetic resonance frequency Ω = Ω 0,\ni.e., the FMR is manifested itself by the step’s width.\nThere are also many subharmonic current steps in the\nIV-characteristic. We have analyzed the steps position\nbetween V= 0 and V= 0.7 for Ω = 0 .7 and found dif-\nferent level continued fractions, which follow the formula\ngiven by Eq.(1) and demonstrated in Fig.2(b). We see4\nthe reflection of the second level continued fractions 1 /n\nand 1−1/nwithN= 1. In addition to this, steps with\nthird level continued fractions 1 /(n−1/m) withN= 1\nis manifested. In the inset we demonstrate part of the\nfourth level continued fraction 1 −1/(n+ 1/(m+1/p))\nwithn= 2 and m= 2.\nIn case of external electromagnetic field which leads to\nthe additional electric current Iac=AsinΩt, the width\nof the Shapiro step is proportional to ∝Jn(A/Ω), where\nJnis the Bessel function of first kind. The preliminary\nresults (not presented here) show that the width of the\nShapiro-like steps under external magnetic field has a\nmore complex frequency dependence20. This question\nwill be discussed in detail somewhere else.\nThe coupling between Josephson phase and magneti-\nzation manifests itself in the appearance of the Shapiro\nsteps in the IV-characteristics at fractional and odd mul-\ntiplies of Ω20. In Fig.3 we demonstrate the effect of the\nratio of the Josephson to magnetic energy ǫJon appear-\nance of the steps and their width for Ω = 0 .5 where the\nenlarged parts of the IV-characteristics at three differ-\nent values of ǫJare shown. As it is demonstrated in\nthe figures, at ǫJ= 0.05 only two subharmonic steps\nappear between V= 1 and V= 1.5 (see hollow ar-\nrows). An enhanced staircase structure appears by in-\ncreasing the value of ǫJ, which can be see at ǫJ= 0.3\nand 0.5. Moreover, an intense subharmonic steps appear\nbetween V= 1.75 andV= 2 forǫJ= 0.5. The posi-\ntions for these steps reflect third level continued fraction\n(N−1)+1/(n+1/m)withN=4 andn=1 [see Fig.3(b)].\nLet us now demonstrate the effect of Gilbert damping\non the devil’s staircase structure. The Gilbert damping\nαis introduced into LLG equation23?to describe the\nrelaxation of magnetization dynamics. To reflect effect\nof Gilbert damping, we show an enlarged part of the IV-\ncharacteristic at three different values of αin Fig.4.\nThewidthofcurrentstepat V= 2Ωisalmostthesame\nat different values of α(e.g., see upward inset V= 2Ω).\nThe subharmonic current step width for V= (n/m)Ω (n\nis odd,mis integer) is decreasing with increasing α. In\naddition a horizontal shift for the current steps occurs.\nWe see the intense current steps in the IV-characteristic\nfor small value of α= 0.03 (see black solid arrows). With\nincrease in Gilbert damping (see α= 0.1, 0.16 and 0 .3)\nthe higher level subharmonic steps disappear. It is well-\nknownthatatlargevalueof αtheFMRlinewidthbecome\nmore broadening and the resonance frequency is shifted\nfrom Ω 0. Accordingly, the subharmonic steps disappear\nat large value of α. Furthermore, using the formula pre-\nsented in Ref.[20] the width at Ω = Ω 0for the fractional\nand odd current steps is proportional to (4 α2+α4)−q/2\n×(12+3α2)−k/2, whereqandkare integers.\nFinally, we demonstrate the effect of the junction size\non the devil’s staircase in the IV-characteristic under ex-\nternalmagneticfield. Thejunction sizechangesthe value\nofφsyandφsz. In Fig.5(a) we demonstrate the effect of\nthe junction thickness by changing φsz(φsyis qualita-\nFIG. 3. (a) An enlarged part of the IV-characteristic at\ndifferent values of ǫJin the interval between V= 1 and V=\n1.5; (b)Thesameintheintervalbetween V= 1.75andV= 2.\nFor clarity, the IV-characteristics for ǫJ= 0.3, and 0 .5 have\nbeen shifted to right, by ∆ I= 0.07, and 0 .14, respectively\nwith respect to the case with ǫJ= 0.05.\ntively the same).\nWe observe an enhanced subharmonic structure with\nincrease of junction size or the thickness of the ferro-\nmagnet. In Ref.[13] the authors demonstrated that the\ncritical current and the width of the step at V= 2Ω as a\nfunction of Lz/Lyfollow Bessel function of first kind. In\nFig.5(b), we can see the parts of continued fraction se-\nquences for subharmonic steps between V= 1 andV= 2\natφsz=φsy= 6. Current steps between V= 1 and\nV= 1.5 reflect the two second level continued fractions\n(N−1)+ 1/nandN−1/nwithN= 3 in both cases,\nwhile for the steps between V= 1.5 andV= 2 follow\nthe second level continued fraction ( N−1) + 1/nwith\nN= 4.\nFinally, wediscussthepossibilityofexperimentallyob-5\nFIG. 4. An enlarged part of IV-characteristic for four differ -\nent values of Gilbert damping for Ω = 0 .5. The inset shows an\nenlargedpartofcurrentstepwithconstantvoltage at V= 2Ω.\nserving the effects presented in this paper. For junction\nsized= 5nm, Ly=Lz= 80nm, critical current I0\nc≈\n200µA, saturation magnetization M0≈5×105A/m,\nH0≈40mT and gyromagnetic ratio γ= 3πMHz/T,\nwe find the value of φsy(z)=4π2Ly(z)dM0/Φ0= 4.8 and\nǫJ= 0.1. With the same junction parameters one can\ncontrol the appearance of the subharmonic steps by tun-\ning the strength of the constant magnetic field H0. Esti-\nmations showthat for H0= 10mT, the value of ǫJ= 0.4,\nand the fractional subharmonic steps are enhanced. In\ngeneral, the subharmonic steps are sensitive to junction\nparameters, Gilbert damping and the frequency of the\nexternal magnetic field.\nIV. CONCLUSIONS\nIn this work, we have studied the IV-characteristics\nof superconductor-ferromagnet-superconductor Joseph-\nson junction under external magnetic field. We used a\nmodified RSJ model which hosts magnetization dynam-\nics in F-layer. Due to the external magnetic field, the\ncouplingbetweenmagneticmomentandJosephsonphase\nis achieved through the effective field taking into account\nthe Josephson energy and gauge invariant phase differ-\nence between the superconducting electrodes. We have\nsolvedasystemofequationswhichdescribethe dynamics\nof the Josephson phase by the RSJ equation and magne-\ntization dynamics by Landau-Lifshitz-Gilbert equation.\nThe IV-characteristic demonstrates subharmonic current\nsteps. The pattern of the subharmonic steps can be con-\ntrolled by tuning the frequency of the ac magnetic field.\nWe show that by increasing the ratio of the Josephson to\nmagneticenergyanenhancedstaircasestructureappears.\nFinally, we demonstrate that Gilbert damping and junc-\nFIG. 5. (a) IV-characteristic at three different values of\nφsz= 0.7,3,6 andφsy=φsz. (b) An enlarged part of the IV-\ncharacteristic at φsz=φsy=6. The hollow arrows represent\nthe starting point of the sequences. To get step voltage we\nmultiply the corresponding fraction by Ω = 0 .5.\ntion parameters can change the subharmonic step struc-\nture. The observed features might find an application in\nsuperconducting spintronics.\nV. ACKNOWLEDGMENT\nWe thank Dr. D. V. Kamanin and Egypt JINR col-\nlaboration for support this work. The reported study\nwas partially funded by the RFBR research Projects No.\n18-02-00318 and No. 18-52-45011-IND. Numerical cal-\nculations have been made in the framework of the RSF\nProject No. 18-71-10095.6\nREFERENCES\n∗majed@sci.cu.edu.eg\n†shukrinv@theor.jinr.ru\n1J. Linder and K. Halterman, Phys. Rev. B 90, 104502\n(2014).\n2Yu. M. Shukrinov, A. Mazanik, I. Rahmonov, A. Botha,\nand A. Buzdin, EPL122, 37001 (2018).\n3Yu. M. Shukrinov, I. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n4A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n5A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n6F. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005).\n7A. A. Golubov, M. Y. Kupriyanov, and E. IlIchev, Rev.\nMod. Phys. 76, 411 (2004).\n8M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Phys. Rev.\nB95, 184508 (2017).\n9I. Bobkova, A. Bobkov, and M. Silaev, Phys. Rev. B 96,\n094506 (2017).\n10Y. M. Shukrinov, I. Rahmonov, and K. Sengupta, Phys.\nRev. B99, 224513 (2019).\n11M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D.\nKoelle, R. Kleiner, and E. 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Rev.\nB 97, 224514 (2018).\n21K. K. Likharev, Dynamics of Josephson junctions and cir-\ncuits, Gordon and Breach science publishers -Switzerland\n(1986).\n22S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n23T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443-\n3449 (2004).\n24M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601(2009)."    },    {        "title": "2212.12383v2.Surface_ferromagnetism_in_a_chiral_topological_semimetal_CoSi.pdf",        "content": "arXiv:2212.12383v2  [cond-mat.mes-hall]  27 Dec 2022Surface ferromagnetism in a chiral topological semimetal C oSi.\nN.N. Orlova,1A.A. Avakyants,1A.V. Timonina,1N.N. Kolesnikov,1and E.V. Deviatov1\n1Institute of Solid State Physics of the Russian Academy of Sc iences,\nChernogolovka, Moscow District, 2 Academician Ossipyan st r., 142432 Russia\n(Dated: December 29, 2022)\nDespite the chiral topological semimetal CoSi is known as bu lk diamagnetic, it shows unusual\nsurface ferromagnetism of debatable origin. The ferromagn etic ordering has been attributed to the\ndistorted bonds, the superlattice of ordered vacancies, or even to topological surface textures due\nto the spin polarization in the neighboring Fermi arcs. We ex perimentally compare magnetization\nreversal curves for initially oxidized CoSi single crystal s and cleaved samples with a fresh, oxide-free\nsurface. While the oxidized CoSi samples do not show sizable ferromagnetism, the fresh CoSi surface\ngives a strong ferromagnetic response, which is accompanie d by the pronounced modulation of the\nangle dependence of magnetization, as it can be expected for easy and hard axes in a ferromagnet.\nIn addition to the first order reversal curves analysis, this observation allows us to distinguish\nbetween different mechanisms of the ferromagnetic ordering in CoSi single crystals. We conclude\nthat the surface states-induced RKKY interaction between d istorted bonds near the sample surface\nis responsible for the strong ferromagnetic multi-domain b ehavior for freshly cleaved samples.\nPACS numbers: 71.30.+h, 72.15.Rn, 73.43.Nq\nINTRODUCTION\nThe concept of a chiral topological semimetal [1–3] is\nusually regarded as modification of Weyl one by addition-\nally broken mirror symmetry. While Weyl semimetals\nare characterized by multiple pairs of Weyl nodes with\nsmall separation in momentum space [4], there is only\none pair of chiral nodes of opposite Chern numbers with\nlarge separation in chiral semimetals, which results in ex-\ntremely long surface Fermi arcs [5, 6]. Chiral topological\nsemimetals also host new types of massless fermions with\na large topological charge [7, 8], which lead to numerous\nexotic physical phenomena like unusual magnetotrans-\nport [9], lattice dynamics [10], and a quantized response\nto circularly polarized light [11].\nChiral crystals are known, e.g., as monosilicides of Cr,\nMn, Fe, and Co with the simple cubic B20 crystal struc-\nture [12–15]. Chiral symmetry is provided by the neu-\ntral 21screw axis in the P2 13 space group [16]. Among\nthese materials, CoSi is the mostly investigated one, so\nthe bulk band structure and the presence of long surface\nFermi arcs have been experimentally confirmed for CoSi\n[5, 17–19].\nOne can expect a complicated response of topologi-\ncal semimetals on the external magnetic field due to the\nspin-momentum locking and, therefore, spin polarization\nof topological surface states. For example, spin polar-\nization of the Fermi arcs in TaAs lies completely in the\nplane of the (001) surface and reaches 80%[20]. For\nthe chiral topological semimetal CoSi, recent theoreti-\ncal studies have shown that the spin-orbit interaction\nlifts the spin degeneracy of the surface states leading to\ntheir in-plane spin polarization on the (001) surface, with\nstrongly correlated and predominantly antiparallel spin\ntextures in the neighboring Fermi arcs [21]. In general,spin textures are known in magnetic materials as surface\nskyrmions [22–30]] or spin helix structures [31, 32].\nAmong the family of chiral crystals, MnSi is a fer-\nromagnet with known low temperature helimagnetic\norder and skyrmion magnetism near the Curie tem-\nperature [13], FeSi is a small-gap semiconductor with\nan anomalous temperature-dependent magnetic mo-\nment [33], while CoSi is a diamagnetic semimetal [34].\nIn the latter case, unusual surface ferromagnetism is re-\nported for nanowires, policrystalline films [35–37], or\neven CoSi single crystals [38]. Skyrmion lattice has\nbeen experimentally shown for the polycrystalline CoSi\nfilms [37, 39, 40]. Also, unconventional magnon modes\nhave been reported as a joint effect of surface ferromag-\nnetism and spin-orbit coupling in CoSi [41].\nFor the CoSi surface ferromagnetism, there is no clear\nunderstanding of the ordering mechanism. The Curie\npoint is near the room temperature (T c= 328 K), which\nis the highest one among all B20-type ferromagnets [37].\nOne of the mechanisms is due to the distorted bonds: the\ntransition metal (Co) d-orbital electron spin up and spin\ndown populations become asymmetric from the exchange\ninteractions near the CoSi surface [42, 43]. As an alterna-\ntive, the superlattice of ordered vacancies is considered a s\na source of ferromagnetic ordering [36]. Topological sur-\nface textures should also be considered due to the spin\npolarization in the neighboring Fermi arcs [21, 44]. More-\nover, the surface states-induced RKKY interaction is ex-\npected to be available [45–47] when the topological sur-\nface states couple with the lattice-related ordering (e,g,\nthe above mentioned distorted bonds or ordered vacan-\ncies). Thus, one should to have a keystone experiment to\ndistinguish between the different proposed mechanisms.\nHere, we experimentally compare magnetization rever-\nsal curves for initially oxidized CoSi single crystals and2\ncleaved samples with a fresh, oxide-free surface. While\nthe oxidized CoSi samples do not show sizable ferromag-\nnetism, the fresh CoSi surface gives a strong ferromag-\nnetic response, which is accompanied by the pronounced\nmodulation of the angle dependence of magnetization, as\nit can be expected for easy and hard axes in a ferromag-\nnet. This observation is compared with the expected one\nfor different mechanisms of the ferromagnetic ordering in\nCoSi.\nSAMPLES AND TECHNIQUES\nThe initial CoSi material was synthesized from cobalt\nand silicon powders by 10◦C/h heating in evacuated sil-\nica ampules up to 950◦РЎ. The ampules were held at\nthis temperature for two weeks and then cooled down to\nroom temperature at 6◦C/h rate. The obtained mate-\nrial was identified as CoSi with some traces of SiO 2by\nX-ray analysis. Afterward, CoSi single crystals are grown\nfrom this initial load by iodine transport in evacuated sil-\nica ampules at 1000◦. X-ray diffractometry demonstrates\ncubic structure of the crystals, also, X-ray spectral anal-\nysis confirms equiatomic ratio of Co and Si in the com-\nposition, without any SiO 2traces. The quality of our\nCoSi crystals have been verified in a number of transport\nexperiments [41, 48, 49].\nTo investigate magnetic properties of small CoSi sin-\ngle crystal samples, we use Lake Shore Cryotronics 8604\nVSM magnetometer, equipped with nitrogen flow cryo-\nstat. A CoSi crystal is mounted to the sample holder by\na low temperature grease, which has been tested to have\na negligible magnetic response.\nFor the magnetization measurements, we use small\n(about 2.52-3.91 mg) CoSi single crystals, see the inset\nto Fig. 1 (a) as an example. Initially, the crystals have\nbeen exposed to air (at ambient conditions) for several\nmonths, so the surface is covered by the native oxide.\nAfter the first step of magnetization measurements, ev-\nery crystal has been cleaved to obtain a fresh, oxide-free\nsurface, see the inset to Fig. 2 (a). The cleaved sample\nhas been immediately mounted to the flow cryostate for\nthe second step of magnetization measurements.\nWe obtain hysteresis loops at different temperatures\nby the standard method of the magnetic field gradual\nsweeping between two opposite saturation values. We\nalso perform first order reversal curves (FORC) analysis,\nwhich provides additional information on the magnetic\nphases and their interaction. For the FORC analysis the\nmagnetization curves are recorded as a two-dimensional\nmap with the reversal field Hrand demagnetization field\nH[50, 51]. Then the obtained ρ(H,Hr)map is usually\nredrawn in (Hu,Hc)coordinates, where Hu=1\n2(H+\nHr)is the interaction field and Hc=1\n2(H−Hr)is the\ncoercitivity field, see Refs. [50, 52] for details.\nThe FORC density distribution ρ(Hu,Hc)is known to-3.0 -1.5 0.0 1.5 3.0-30-1501530\n0 180 360384042\n0 180 360405060\n-1.0 -0.5 0.0 0.5 1.0-6-4-20246\nM (µemu)\nH (kOe)(b)\nM (µemu )\nα (deg)H = 5 kOeM (µemu )\nα (deg)H = 15 kOeM (µemu)\nH (kOe)(a)\nFigure 1. (Color online) Magnetization curves M(H)at 100 K\ntemperature for two initially oxidized CoSi crystals, 3.91 mg\nand 2.52 mg for (a) and (b), respectively. The qualitative\nbehavior is very similar, it well corresponds to the known\none [34–38] . The main M(H)dependence corresponds to\nthe paramagnetic response, M(H)does not demonstrate any\nclear orientation dependence, as depicted in the insets. Th e\nsignal amplitude at fixed magnetic field does not scale with\nthe sample mass, so M(H)mostly reflects the contribution\nof the surface Co oxides [55, 56], despite a weak hysteresis\nwithin±1kOe interval.\nbe convenient for analysis [53, 54]. The closed contours of\nthe density distribution peak are usually associated with\nsingle-domain regime, while multi-domain material gives\nopen contours that diverge towards the Huaxis. In gen-\neral, presence of more than one peak in ρ(Hu,Hc)map\ncorresponds to multiple magnetic phases. Vertical shift\nof the peaks characterizes the type of interaction between\nthe phases: it is dipolar for the positive shift values while\nthe exchange interaction appears as the negative ones.\nEXPERIMENTAL RESULTS\nWe start magnetization measurements from the ini-\ntially oxidized samples. Fig. 1 shows magnetization loops\nat 100 K temperature for two CoSi crystals, 3.91 mg and\n2.52 mg for (a) and (b), respectively, the qualitative be-\nhavior is very similar. There is a weak hysteresis within\n±0.75kOe interval of magnetic field. The main M(H)\ndependence corresponds to the paramagnetic response,\nwhich is the known contribution of the surface Co ox-\nides [55, 56]. The sample magnetization M(H)does not\ndemonstrate any specific orientation dependence, as it is\nshown in the insets to Fig. 1. The signal amplitude at\nfixed magnetic field does not scale with the sample mass,\nwhich also correlates with the surface oxide response.\nThus, the qualitative M(H)behavior well corresponds\nto the known one for CoSi semimetal [34–38].\nAs a second step, we repeat the magnetization mea-\nsurements after cleaving the initial samples from Fig. 1\non the smaller fragments with fresh, oxide-free, CoSi sur-\nface.\nTo our surprise, both samples demonstrate one order of3\n-2 -1 0 1 2-40-2002040\n-3.0 -1.5 0.0 1.5 3.0-20-1001020\n180 36010203040M (µemu)\nH (kOe)(a)\nM (µemu)\nH (kOe)(b)\n  0  degree\n 90 degreeM (µemu )\nα (deg)H = 1 kOe\nFigure 2. (Color online) Ferromagnetic response from the\nfresh CoSi surface. The initial samples from Fig. 1 are cleav ed\non smaller fragments, 1.86 mg and 0.48 mg in (a) and (b),\nrespectively. Both samples demonstrate one order of magni-\ntude increase in Mvalue, there is now prominent hysteresis\nwithin±2kOe field range, which is accompanied by well-\ndefinedM(H)saturation in higher fields. Inset to (a) shows\nangle dependence of magnetization M(α)with 180◦periodic\nmodulation, reflecting well-defined easy and hard axes in a\nferromagnet. For the second sample in (b), the angle depen-\ndence is shown as two hysteresis loops for these two (easy and\nhard) field orientations.\nmagnitude increase in Mvalue, while the sample dimen-\nsions are diminished to 1.86 mg and 0.48 mg in Fig. 2 (a)\nand (b), respectively. Sample cleaving has also dramatic\neffect on the qualitative M(H)behavior: there is now\nprominent hysteresis within ±2kOe field range, which is\naccompanied by well-defined M(H)saturation in higher\nfields. Thus, the cleaved samples demonstrate standard\nferromagnetic response.\nFerromagnetic response from the fresh CoSi surface is\nalso supported by the angle dependence of magnetization\nin the inset to Fig. 2 (a). We observe strong modulation\nof theM(α)with 180◦periodicity, as it can be expected\nfor easy and hard axes in a ferromagnet. For the second\nsample in Fig. 2 (b), the angle dependence is shown as\ntwo hysteresis loops for two (easy and hard) field orien-\ntations. As usual, the coercivity field is not sensitive to\nthe field direction, while the saturation level varies withi n\n30% of magnitude, similarly to the angle dependence in\nthe inset to Fig. 2 (a).\nThe Curie point is above the room temperature (T c=\n328 K) for surface ferromagnetism in CoSi [37], so the\nferromagnetic response is expected to be practically in-\nsensitive to the temperature much below the Curie point,\nas we confirm in Fig. 3. The hysteresis loops well coin-\ncide from 80 K to 180 K, both the loop width and the\nsaturation level are temperature-independent within this\ninterval.\nAdditional information on the ferromagnetic state can\nbe obtained from FORC data in Fig. 4. The raw\nFORC curves and the calculated FORC density diagram\nρ(Hu,Hc)are shown in (a,c) and (b,d), respectively, at\n100 K temperature for two cleaved samples from Fig. 2.-2 -1 0 1 2-40-2002040M (µemu)\nH (kOe) 80K\n 100K\n 120K\n 140K\n 160K\n 180K\nFigure 3. (Color online) The hysteresis loops for the cleave d\nCoSi sample from Fig. 2 (a), obtained at 80 K, 100 K, 120 K,\n140 K, 150 K, 160 K and 180 K temperatures. The curves\nare practically insensitive to temperature much below the\nknown Curie point T c= 328 K for surface ferromagnetism in\nCoSi [37]. This temperature dependence confirms the stan-\ndard ferromagnetic behavior of the fresh CoSi surface.\nEvery sample shows a single peak in ρ(Hu,Hc), which\nis centered at low Hcvalues with so-called open con-\ntours at the Huaxis. This behavior is usually regarded\nas a fingerprint of the multi-domain regime for a ferro-\nmagnet [53, 54]. The peak center is slightly shifted to\nthe positive values of the interaction field Hu, which\nis corresponds to the dipolar interaction between do-\nmains [53, 54].\nDISCUSSION\nAs a result, while the samples with oxidized surface\nshow mostly paramagnetic response, the cleaved samples\ndemonstrate strong ferromagnetic multi-domain behav-\nior with definite orientations for easy and hard axes. This\nbehavior is even more surprising, since also the absolute\nM(H)values are strongly increased for cleaved (i.e. di-\nminished) samples.\nBulk single crystals of CoSi are known to be diamag-\nnetic with temperature-independent susceptibility [34].\nThis behavior can be confirmed for the initial (oxi-\ndized) samples in Fig. 1: after subtracting the param-\nagneticM(H)linear dependence in high fields, we ob-\ntain−0.3∗106emu/g diamagnetic susceptibility in good\ncorrespondence with the previously reported data [34].\nOn the other hand, the unusual ferromagnetism was\nobserved in CoSi single-crystal nanowires [35, 36], thin\npolycrystalline films [37] and some bulk single crys-\ntals [38]. The origin of this ferromagnetism is still de-\nbatable.4\n-2 -1 0 1 2-16-80816M (µemu)\nH (kOe)(c) (d)-1 0 1-40-2002040M (µemu)\nH (kOe)(a) (b)\n0.00 0.25 0.50 0.75 1.00-1.0-0.50.00.51.0\nHu (kOe)\nHc (kOe)00.00 0.25 0.50 0.75 1.00-1.0-0.50.00.51.0Hu (kOe)\nHc (kOe)0\nFigure 4. (Color online) (a,c) The raw FORCs curves for two\ncleaved samples from Fig. 2 (a) and (b), respectively, at 100 K.\n(b,d) FORC density diagrams ρ(Hu,Hc)for these samples.\nSingle peak in ρ(Hu,Hc)confirms a single magnetic phase in\nthe sample, while open contours at the Huaxis indicate the\nmulti-domain regime [53, 54]. Thus, a fresh CoSi surface\ndemonstrates strong ferromagnetic multi-domain behavior .\n(i) Ferromagnetic ordering can appear due to the dis-\ntorted and dangling bonds near the sample surface. The\ntransition metal (Co) d-orbital electron spin up and spin\ndown populations become asymmetric from the exchange\ninteractions near the CoSi surface [42, 43]. The unit cell\nof CoSi contains four Co and four Si atoms, with each Co\n(Si) atom covalently bonded to six nearest neighboring Si\n(Co) atoms. In the Co (Si) – terminated (001) surface,\nevery Co (Si) atom and the underlying Si (Co) one forms\na zigzag atomic chain [14].\n(ii) Another source of ferromagnetic ordering is the\nsuperlattice of defects (ordered vacancies) in CoSi sin-\ngle crystals [35, 36]. While it is difficult to distinguish\nexperimentally, theoretical simulations suppose that the\ninternal ordered vacancies is the dominant contribution\nin CoSi single crystal nanowire ferromagnetism. [36].\n(iii) Topological surface structures should also be con-\nsidered for CoSi topological semimetal. The spin textures\nwere predicted even for weakly spin-split Fermi surfaces,\nso the chiral cubic symmetry enforces perfectly paral-\nlel spin-momentum locking [57, 58]. Skyrmion lattice\nhas been experimentally shown for polycrystalline CoSi\nfilms [37, 39, 40].\nOur experiment indicates, that a fresh surface gives the\ndominant contribution to the ferromagnetism of a CoSi\nsingle crystal. Thus, the surface effects (i) and (iii) shoul d\nbe considered as the main reason for strong ferromagnetic\nbehavior in Fig. 2, while we can not exclude bulk ordered\nvacancies (ii) as a source of narrow loops in Fig. 1 for the\noxidized samples. To distinguish between the surface or-\ndering mechanisms (i) and (iii), we wish to note that thestructure of the distorted and dangling bonds near the\nsample surface can be seriously corrupted by the surface\noxidation , while the topological effects (iii) are usually\nconsidered to have protection from disorder. Thus, the\nmechanism (i) well correspond to our experiment. On\nthe other hand, one can not expect a strong ferromag-\nnetic response from the distorted bonds near the sample\nsurface [42, 43].\nThis apparent inconsistence indicates to a joint effect\nof the surface-induced ordering mechanisms (i) and (iii).\nNaturally, the RKKY interaction is expected to be avail-\nable for the surface band of Weyl semimetals, and, there-\nfore, also for chiral ones [45–47]. In particular, two dis-\ntorted bonds can be regarded as magnetic impurities,\nwhich are placed on the surface of the semimetal. Consid-\nering the spin-exchange interaction (s-d model) between\nimpurities and host electrons, the system Hamiltonian\ncan be written [45–47] as\nH=HWSM−J0/summationdisplay\ni=1,2Sisi\nwhereHWSM is the low-energy Hamiltonian of Weyl\nsemimetal, J0is the strength of the exchange interac-\ntion,Siis the spin of impurity at site i, andsirefers to\nthe spin of host electrons [47]. Mediated by the itiner-\nant host electrons, an indirect exchange interaction (i.e. ,\nRKKY interaction) between two impurities is generated,\nwhich can be rewritten in the form of\nHRKKY=JS1S2\nRKKY interaction mediated by surface states in Weyl\nsemimetals can be induced by different mechanisms [45–\n47]. For impurities deposited in the direction perpendicu-\nlar to the Weyl points splitting, the surface contributions\ndecay much more slowly with impurity distance than that\nof bulk contribution [47].\nSince the topological surface states are confirmed for\nCoSi [5, 17–19, 21], the surface states-induced RKKY\ninteraction can be responsible for the enhancement of\nthe initially weak ferromagnetism of the distorted bonds.\nThe latter is sensitive to the surface oxidation, while\nthe surface states-induced RKKY interaction provides\nhigh absolute M(H)values, which are strongly increased\nfor the cleaved (i.e. significantly diminished) samples in\nFig. 2.\nAs an additional argument, we do not observe the so\ncalled bow-tie type hysteresis loops, which are usually\nascribed to skyrmions [59–62]. From the FORC data in\nFig. 4, we should exclude any sizable input from the inde-\npendent surface phase, since there is only single magnetic\nphase for the samples with clean crystal surface. This\nwell correlates with the theoretical statement, that the\nsurface states-induced RKKY interaction survives only\nwhen the surface states couple with bulk states (or other\nsurface states of different spins) [45, 46]. This does not5\ncontradict to the previous experiments, while skyrmions\nhave been demonstrated for the polycrystalline samples\nonly [37, 39, 40].\nAs a result, our experiment allows to distinguish be-\ntween different mechanisms of the ferromagnetic order-\ning in CoSi single crystals, so the surface states-induced\nRKKY interaction between distorted and dangling bonds\nnear the sample surface is responsible for the strong fer-\nromagnetic multi-domain behavior with definite orienta-\ntions for easy and hard axes for freshly cleaved samples.\nACKNOWLEDGEMENT\nWe wish to thank S.S Khasanov for X-ray sample char-\nacterization.\n[1] B. Bradlyn, J. Cano, Z. Wang, M. G. Vergniory, C. Felser,\nR. J. Cava, and B. A. 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Uzunov\nCP Laboratory, Institute of Solid State Physics, Bulgarian Academ y of Sciences,\nBG-1784 Sofia, Bulgaria.\nPacs: 74.20.De, 74.25.Dw, 64.70.Tg\nKeyword : unconventional superconductivity, quantum phase transition,\nstrongly correlated electrons, multi-critical point, phase diagram .\nAbstract\nA general quasi-phenomenological theory that describes ph ases and phase\ntransitions of ferromagnetic superconductors with spin-t riplet electron Cooper\npairing is presented. The theory is based on extended Ginzbu rg-Landau expan-\nsion in powers of superconducting and ferromagnetic order p arameters. A simple\nform for the dependence of theory parameters on the pressure ensures a correct\ntheoretical outline of the temperature-pressure phase dia gram where a stable\nphase of coexistence of p-wave superconductivity and itinerant ferromagnetism\nappears. Thisnewtheory isinanexcellent agreement withth eexperimental data\nfor intermetallic compounds, for example, UGe 2, URhGe, UCoGe, and UIr that\nare experimentally proven to be itinerant ferromagnets exh ibiting spin-triplet\nsuperconductivity. The mechanism of appearance of superco nductivity due to\nitinerant ferromagnetism ( M-trigger effect) is established and demonstrated. On\nthe basis of the same theory, basic features of quantum phase transitions in\nthis type of ferromagnetic superconductors are explained i n agreement with the\nexperimental data. The theory allows for a classification of the spin-triplet ferro-\nmagnetic superconductors in two different types: type I and ty pe II. The classi-\nfication is based on quantitative criteria, i.e., on simple r elations between theory\nparameters. Both theory and experiment indicate that the tw o types of p-wave\nferromagnetic superconductors are well distinguished by e ssential differences in\ntheir physical properties.\n1. Introduction\nThere exists an intensive research on Cooper pairing of fermions in c ondensed mat-\nter. Both experimental and theoretical studies reveal fascinat ing phenomena of super-\nconductivity, unconventional superfluidity and coexistence of su perconductivity and\nferromagnetism. The fermion Cooper pairs in s-wave state are referred to as conven-\ntional pairing and, hence, the respective superfluidity or superco nductivity are labeled\nconventional . The unconventional (non- s-wave) pairing, fore example, the p-wave\npairing creates another type of similar phenomena – unconventiona l superfluidity and\nunconventional superconductivity of p-wave type. These phenomena can be described\nwithin the general scheme of spin-triplet states of fermion pairs. T he phenomenon of\nunconventional (spin-triplet) Cooper pairing of fermions was first ly described for the\n1case of superfluid Helium 3 [1]. Here we focus the attention to unconv entional super-\nconductivity of spin-triplet type which allow for electron pairing in p-wave states with\nspin equal to unity.\nThe spin-triplet superconducting phases are described in the fram ework of the general\nGinzburg-Landau (GL) effective free energy functional with the h elp of the symme-\ntry groups theory and detailed thermodynamic analysis [2, 3]; see als o [4, 5]. This\nunconventional superconductivity has been discovered in itineran t ferromagnets – the\nintermetallic compounds UGe 2[6], URhGe [7], UCoGe [8] and UIr[9].\nAt low temperature ( T∼1 K) all these compounds exhibit coexistence of p-wave su-\nperconductivity and itinerant ( f-band) electron ferromagnetism (in short, FS phase).\nIn the mentioned compounds, the FS phase appears only in the ferr omagnetic phase\ndomain of the T−Pdiagram. At given pressure P, the temperature TF(P) of the\nnormal-to-ferromagnetic phase (or N-FM) transition is never lowe r than the tempera-\ntureTFS(P)oftheferromagnetic-to-FSphase(orFM-FS)transition. Thisis consistent\nwith the point of view that the superconductivity in these compound s is triggered by\nthe spontaneous magnetization M, as explained in details in [10]). The main system\nproperties are affected by a term in the Ginzburg-Landau (GL) exp ansion of the form\nM.(ψ×ψ), which represents the interaction of M={Mj;j= 1,2,3}with the com-\nplex superconducting vector field ψ={ψj}. This term triggers ψ/ne}ationslash= 0 for certain T\nandPvalues (M-trigger effect [10]).\nThe mechanism of formation of pairs in these systems is under conje ctures. The elec-\ntron band theory of such compounds is quite complex and the densit y of statesDn(E)\nattheFermisurface EFcanhardlybeevaluatedindetails. Inthissituation, thereliable\ntheoretical interpretation of experiments is performed within the general phenomeno-\nlogical theory of GL type. In such theory, the Landau parameter s,µ= (a,b,...)\nshould depend on the shape of Dn(E). While this dependence is unknown, one may\nreliably assume the dependence of Dn(E) on temperature Tand pressure P, and hence,\nthe double functional dependence µ=µ[Dn(T,P)]. Although unknown, the latter jus-\ntifies the function µ(T,P), i.e., the dependence of the Landau parameters on Tand\nP. This dependence can be reliably proposed on the basis of heuristic a rguments and\nexperimental data. So, let us focus on the phenomenological appr oach.\nThespin-tripletferromagneticsuperconductorsaredescribedb ythequasi-phenomenological\ntheory based on an extended GL in powers of the fields ψandM(for details, see\n[5, 10, 11, 12]). In this Report, the theory and related topics are b riefly presented; for\ndetails, see [5, 10, 11, 12]. New aspects and outstanding problems a re briefly indicated.\n2. Theory\nHere we hold the consideration in the lowest (mean-field) approximat ion. The latter is\nenough to elucidate the main system properties in a quite correct wa y. The free energy\nper unit volume, F/V=f(ψ,M), can be written in the form\nf(ψ,M) =as|ψ|2+bs\n2|ψ|4+us\n2|ψ2|2+vs\n23/summationdisplay\nj=1|ψj|4(1)\n+afM2+bf\n2M4+iγ0M·(ψ×ψ∗)+δM2|ψ|2.\n2N\nBFM\nFSAC\n01234T (K)\n0.5 1 1.5\nP (GPa)\nFigure 1: Low-temperature part of the T−Pphase diagram of UGe 2, shown in Fig. 1. The\npoints A, B, C are located in the high-pressure part ( P∼Pc∼1.6 GPa). The FS phase\ndomain is shaded. The solid lines AC and BC show the first-orde r transitions N-FS, and\nFM-FS, respectively. The dotted line shows a second-order F S-FM transition, the dashed\nline stands for the N-FM transition.\nThe material parameters satisfy bs>0,bf>0,as=αs(T−Ts), andaf=αf[T−\nTf(P)]. The terms proportional to usandvsdescribe, respectively, the anisotropy of\nthe spin-triplet electron Cooper pairs and the crystal anisotropy . Next,γ0∼J(with\nJ >0 the ferromagnetic exchange constant) and δ >0 are parameters of the ψ-M\ninteraction terms. Previous mean-field studies [10] have shown tha t the anisotropy\nrepresented by the usandvsterms in Eq. (1) slightly perturb the size and shape of\nthe stability domains of the phases, while similar effects can be achieve d by varying\nthebsfactor in the bs|ψ|4term. For these reasons, in the present analysis we ignore\nthe anisotropy terms, setting us=vs= 0, and consider bs≡b >0 as an effective\nparameter. Then, without loss of generality, we use the gauge M= (0,0,M).\nA convenient dimensionless free energy can now be defined by ˜f=f/(bfM4\n0), where\nM0= [αfTf0/bf]1/2>0isthevalueof Mcorrespondingtothepuremagneticsubsystem\n(ψ≡0)atT=P= 0andTf0=Tf(0). Onscalingtheorderparametersas m=M/M0\nandϕ=ψ/[(bf/b)1/4M0] we obtain\n˜f=rφ2+φ4\n2+tm2+m4\n2+2γmφ1φ2sinθ+γ1m2φ2, (2)\nwhereφj=|ϕj|,φ=|ϕ|, andθis the phase angle between the complex ϕ2andϕ1.\nThe dimensionless constants are t= [˜T−˜Tf(P)],r=κ(˜T−˜Ts) withκ=αsb1/2\nf/αfb1/2,\nγ=γ0/[αfTf0b]1/2, andγ1=δ/(bbf)1/2. The reduced temperatures are ˜T=T/Tf0,\n˜Tf(P) =Tf(P)/Tf0,˜Ts(P) =Ts(P)/Tf0.\n3. Results\nThe analysis involves making simple assumptions for the Pdependence of the t,r,γ,\nandγ1parameters in Eq. (2). Specifically, we assume that only Tfhas a significant P\ndependence, described by ˜Tf(P) = (1−˜P), where ˜P=P/P0andP0is a characteristic\npressure deduced later; for UGe 2,P0∼Pc- the critical pressure above which both\nmagnetic and supercionducting order vanish at T∼0 (see Fig. 1); for some substances\nthis is not the case, but then another dependence of TfonPtakes place. Within this\n3simple variant, all other theory parameters are considered P-independent.\nThe simplified model in Eq. (2) is capable of describing the main thermod ynamic\nproperties of spin-triplet ferromagnetic superconductors[10]. F or example, in Fig. 1\nthe low temperature part of the T−Pdiagrams of UGe 2is shown. The result is in a\nremarkable agreement with the experimental shape of the same dia gram [6]. The type\nof the phase transitions and the multicritical points (A, B, C) also co rrespond to the\nexperimental observations. There are three stable phases: (i) t he normal (N) phase,\ngiven byφ=m= 0; (ii) thepureferromagnetic(FM) phase, given by m= (−t)1/2>0,\nφ= 0; and (iii) the FS phase, given by φ2\n1=φ2\n2= (γm−r−γ1m2)/2,φ3= 0, where\nsinθ=−1 andmsatisfies\n(1−γ2\n1)m3+3\n2γγ1m2+/parenleftbigg\nt−γ2\n2−γ1r/parenrightbigg\nm+γr\n2= 0. (3)\nAlthough Eq. (3) is complicated, some analytical results follow, e.g., w e find that\nthe second order phase transition line ˜TFS(P) separating the FM and FS phases (the\ndotted lines in Figs. 1) is the solution of\n˜TFS(P) =˜Ts+γ1\nκt(TFS)+γ\nκ[−t(TFS)]1/2. (4)\nUnder certain conditions, which are satisfied for UGe 2, theTFS(P) curve has a maxi-\nmumat˜Tm=˜Ts+(γ2/4κγ1)withpressure Pmfoundbysolving t(Tm,Pm) =−(γ2/4γ2\n1).\nUnder certain conditions, Eq. (2) describes pure superconductin g phases, too. To date\nthere are no experimental observation of such phase, which mean s that the respective\nconditions are not satisfied by the parameters of the currently kn own superconductors.\nNegative values of Tsarepossible, andthey describe aphase diagramtopologyinwhich\nthe FM-FS transition line terminates at T= 0 forP <P c. This might be of relevance\nfor other compounds, e.g., URhGe.\nAs in experiment[6], the dashed line in Fig. 1 slopes to T= 0 at another critical\npressureP0c. This is possible in UGe 2because for this compound the condition γ <γ1\nis satisfied. Such systems are spin-triplet ferromagnetic superco nductors of type I,\nwhereas systems which fulfill the condition γ >γ1are of type II [10, 12]. The quantum\nphasetransitionsat P0candPchaveremarkableproperties, asshownin[11]. Depending\non the system properties, TC– the temperature locating point C, can be either positive\n(when a direct N-FS first order transition is possible), zero, or neg ative (when the\nFM-FS and N-FM phase transition lines terminate at different zero-t emperature phase\ntransition points). The last two cases correspond to Ts<0. All these cases are possible\nin spin-triplet ferromagnetic superconductors. The zero temper ature transition at Pc0\nis found to be a quantum critical point, whereas the zero-tempera ture phase transition\natPcis of first order. As noted, the latter splits into two first order pha se transitions.\nThis classical picture may be changed through quantum fluctuation s [10].\nThe quantum phase transitions at P0candPchave remarkable properties. An investi-\ngation [11] performed by renormalization group methods revealed a fluctuation change\nin the order of the zero temperature first order phase transition atPcto a continuous\nphase transition belonging to anentirely new class of universality. Ho wever, thisoption\n4exists only for magnetically isotropic order (Heisenberg symmetry) and is unlikely to\napply in the known spin-triplet ferromagnetic superconductors, w hich are magnetically\nanisotropic.\nThe application of the theory was demonstrated on the example of U Ge2, but the\nsame theory has ample volume of options to describe various real su perconductors of\nthe same type and is not restricted to a particular compound, or, t o a particular group\nof such materials.\nEven in its simplified form, this theory has been shown to be capable of accounting\nfor a wide variety of experimental behavior. A natural extension t o the theory is\nto add aM6term which provides a formalism to investigate possible metamagnetic\nphase transitions and extend some first order phase transition line s, as required by\nexperimental data. Another modification of this theory, with rega rd to applications to\nother compounds, is to include a Pdependence for some of the other GL parameters.\nAmong the outstanding problems are: local gauge effects on the vo rtex phase and the\nphasetransitions, theoutlineoftheuppercriticalmagneticfield Hc2(T,P),thermaland\nmagnetic properties, and the description of T−Pdiagrams with topologies, which are\nobserved in experiments. Such studies require an extension of the theory by including\nM6term, the magnetic induction B=H+4πM, and a more precise P-dependence\nof some theory parameters.\nAcknowledgements: The author is grateful to A. Harada and S. M. Hayden for\nvaluable discussions of experiments.\nReferences\n[1] D. Vollhardt and P. W¨ olfle (1990) The Superfluid Phases of Helium 3 , Taylor &\nFrancis, London.\n[2] G. E. Volovik and L. P. Gor’kov (1985) Sov. Phys. JETP 61843 [Zh. Eksp. Teor.\nFiz.88(1985) 1412].\n[3] E. J. Blagoeva, G. Busiello, L. De Cesare, Y. T. Millev, I. Rabuffo, a nd D. I.\nUzunov (1990) Phys. Rev. B426124.\n[4] D. I. Uzunov (1990)in Advances in Theoretical Physics , ed. by E. Caianiello; World\nScientific, Singapore, p. 96.\n[5] D. I. Uzunov (2010) Theory of Critical Phenomena , World Scientific, Singapore\n(Second edition); First edition (1993).\n[6] S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Hase lwimmer, M.J.\nSteiner, E. Pugh, I. R. Walker, S.R. Julian, P. Monthoux, G. G. Lonz arich, A.\nHuxley. I. Sheikin, D. Braithwaite, and J. Flouquet (2000) Nature406587.\n[7] D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flouquet, J -P.. Brison, E.\nLhotel, and C. Paulsen (2001) Nature413613.\n5[8] N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P. Klaasse, T. Gortenmul-\nder, A. de Visser, A. Hamann, T. G¨ orlach, and H. v. L¨ ohneysen( 2007) Phys. Rev.\nLett.99067006 .\n[9] T. Akazawa, H.Hidaka, H.Kotegawa, T. C.Kobayashi, T. Fujiwa ra, E. Yamamoto,\nY. Haga, R. Settai, and Y. Onuki (2005) Physica B 359-361 1138.\n[10] D. V. Shopova and D. I. Uzunov (2003) Phys. Lett. A 313139; D. V. Shopova\nand D. I. Uzunov (2005) Phys. Rev. B 72024531; D. V. Shopova and D. I. Uzunov\n(2009)Phys. Rev. B 79064501.\n[11] D. I. Uzunov (2006) Phys. Rev. B 74134514; D. I. Uzunov (2007) Europhys. Lett.\n7720008.\n[12] D. I. Uzunov (2012) in Superconductors , ed. by A. Gabovich; Intech, Rijeka, Ch.\n17, pp 415-440.\n6"    },    {        "title": "1502.00118v1.Room_temperature_local_ferromagnetism_and_nanoscale_domain_growth_in_the_ferromagnetic_semiconductor_GeFe.pdf",        "content": " 1 \nRoom -temperature local ferromagnetism and nanoscale domain growth in the \nferromagnetic semiconductor Ge 1 – xFex \n \nYuki K. Wakabayashi,1 Shoya Sakamoto,2 Yukiharu Takeda,3 Keisuke Ishigami,2 Yukio \nTakahashi ,2 Yuji Saitoh,3 Hiroshi  Yamagami,3 Atsus hi Fujimori,2 Masaaki Tanaka ,1 and \nShinobu Ohya1 \n1Department of Electrical Engineering and Information Systems, The University of \nTokyo, 7 -3-1 Hongo, Bunkyo -ku, Tokyo 113 -8656, Japan  \n2Department of Physics, The University of Tokyo, Bunkyo -ku, Tokyo 113 -0033, Japan  \n3Synchrotron Radiation Research Unit, JAEA, Sayo, Hyogo 679 -5148, Japan  \n \nAbstract  \nWe investigate the local electronic structure and magnetic properties of the group -IV-\nbased ferromagnetic semiconductor, Ge 1 – xFex (GeFe), using soft X -ray magnetic circular \ndichroism. Our results show that the doped Fe 3 d electrons are strongly hybridize d with \nthe Ge 4 p states, and have an unusually large orbital magnetic moment relative to the spin \nmagnetic moment; i.e., morb/mspin ≈ 0.3. We f ind that local ferromagnetic domains , which \nare formed through ferromagnetic exchange interactions in the high -Fe-content regions \nof the GeFe films , exist at room temperature, well above the Curie temperature of 20  – \n100 K. We demonstrate  the first observation of the intriguing nanoscale domain growth \nprocess in which ferromagnetic domains expand as the temperature d ecreases, followed \nby a transition of the entire film into a ferromagnetic state at the Curie temperature .\n  2 \nA major issue that must be addressed for the realiz ation of semiconductor spintronic \ndevices  is to achieve room -temperature  ferromagnetism in ferromagnetic semiconductors \n(FMS s) based on the widely used III -V and group -IV materials. In Ga1 – xMn xAs \n(GaMnAs) , which  is a particularly well -studied FMS, the highest Curie temperature ( TC) \never reported is 200 K [1]. In GaMnAs, TC is limited by the presence of interstitial Mn \natoms, which are antiferromagnetically coupled to the substitutional Mn atoms [2]. \nRecently, however, the group -IV-based FMS, Ge 1 – xFex (GeFe), has been reported to \nexhibit several attractive features [3 – 5]. It can be grown epitaxially on Si and Ge \nsubstrates without the formation of intermetallic precipitates, and is therefore compatible \nwith mature Si process technology. Unlike GaMnAs, with GeFe, interstitial Fe atoms do \nnot lead to a decrease in TC [6], and TC can be easily increased to above 200 K by thermal \nannealing [7]. Furthermore, TC does not depend on the carrier concentration, and thus TC \nand resistivity can be controlled separately [8], which is a unique feature that is only \nobserved with G eFe and is a considerable advantage in overcoming the conductivity \nmismatch problem between ferromagnetic metals and semiconductors in spin -injection \ndevices. Despite these attractive features, a detailed microscopic understanding of the \nferromagnetism in GeFe , which is  vitally important for room -temperature applications , is \nlacking. Here, we investigate the local magnetic behavior of GeFe  using X -ray magnetic \ncircular dichroism (XMCD), which is a powerful technique for element -specific detection \nof magnetic moments [9 – 12]. We find that local ferromagnetic domains remain in the \nGeFe films even at room temperature, i.e., well ab ove TC; it follows that GeFe potentially \nhas strong ferromagnetism, which persist s even at room temperature. Furthermore, we \npresent the first observation s of the intriguing feature that ferromagnetic domains, which \nare formed above TC via the ferromagneti c exchange interaction in high -Fe concentration \nregions of the films, develop and grow as the temperature decreases, and that all of them  \ncoalesce at temperatures below TC. Such a nanoscale domain growth process is a key \nfeature in understanding materials that exhibit homogeneous  ferromagnetism  (i.e., where \nthe film is free from any ferromagnetic precipitates)  despite the inhomogeneous  \ndistribution of magnetic atoms in the film [6,7].  \nWe carried out XMCD measurements on two samples (labeled A and B) consist ing \nof a 2-nm-thick Ge capping layer, and a 120-nm-thick Ge0.935Fe0.065 layer with a 30-nm-\nthick Ge buffer layer grown on a Ge(001) substrate by low-temperature molecular beam \nepitaxy (LT -MBE) [Figs. 1(a) and 1(b) ]. The Ge buffer and Ge cap layer s were grown at  3 \n200°C, and the  Ge0.935Fe0.065 layer of sample A was grown at 160°C, whereas that of \nsample B was grown at 240°C [6]. From the Arrott plots of the magnetic field (H) \ndependence of the magnetic circular dichroism (MCD) measured with visible li ght with \na photon energy of 2.3 eV (corresponding to the L-point energy gap of bulk Ge), we found \nTC = 20 K and 100 K for samples A and B, respectively. Detailed crystallographic \nanalyses showed that the GeFe films are single crystalline, with a diamond -type structure \nand nanoscale spatial Fe concentration fluctuations of 4%  – 7% (sample A) and 3%  – \n10% (sample B) [6]. We found that TC is higher when the fluctuations in the Fe \nconcentration are larger.  In addition, channeling Rutherford backscattering and \nchanneling particle -induced X -ray emission measurements showed that ~85% (~15% ) of \nthe doped Fe atoms exist at the substitutional (tetrahedral interstitial) sites in both samples \nA and B, and that the interstitial Fe concentration is not related to TC [6]. This indicates  \nthat there are no ferromagnetic precipitates with different crystal structures in our films. \nWe performed X -ray absorption spectroscopy (XAS) and XMCD measurements at the \ntwin-helical  undulator beamline BL23SU of SPring -8 [13]. The XAS spect ra were \nobtained in total electron yield mode. To remove the oxidized surface layer, the samples \nwere briefly etched in dilute hydrofluoric acid ( HF) prior to loading into the XAS \n(XMCD) vacuum chamber.  \nWe measured XAS spectra [ μ+, μ–, and (μ+ + μ–)/2] at the L2 (~721 eV) and L3 (~708 \neV) absorption edges of Fe in sample A [Fig. 1(c)] and B [Fig. 1(d)] at 5.6 K with μ0H = \n5 T applied perpendicular to the film surface. Here, μ+ and μ– refer to the absorption \ncoefficients for photon helicity parallel and antiparallel to the Fe 3 d majority spin \ndirection, respectively. In both films, the three peaks a, b, and  c are observed at the Fe L3 \nedge in the XAS spectra [ see also the insets in Figs. 1(c) and 1(d)]. We found that the \nsmall peak c was suppressed by  etching the surface with dilute HF, indicating that this \npeak, which  can be assigned to the Fe3+ state, originates from a small quantity of surface \nFe oxide [14], which  remains even after surface cleaning. Meanwhile, p eaks a and b are \nassigned to the Fe atoms in GeFe [15,16]. \nWe measured the XMCD ( = μ+ – μ–) spectra at the Fe L2 and L3 absorption edges of \nsamples A [Fig. 1(e)] and B [Fig. 1(f)] at 5.6 K with various H applied perpendicular to \nthe film  surface . Here, we discuss the XMCD  intensit ies at 707.66 eV (X) and 708.2 eV \n(Y), which correspond to the photon energies of peaks a and b in the XAS spectra, \nrespectively. When normalized to 707.3 eV , the XMCD spectra with various H differ, and  4 \nthe intensity at X gr ows faster than that at  Y as H increase s, as shown in the insets of Figs. \n1(e) and 1(f). As shown in F igs. 1(c) and (d), the shapes of the XAS spectra at the Fe L3 \nedge are similar between samples A and B, which have  almost the same interstitial Fe \nconcentrations (i.e., 15% of the total Fe content [6]) ; therefore,  we can assign  the XMCD \nintensity  at X to the substitutional Fe atoms and the paramagnetic component of the \nXMCD intensity  at Y to the interstitial Fe atoms.  We do not observe fine structure s due \nto multiplet splitting at the Fe L3 edge in both samples, which would be observed  if the \n3d electrons were localized and were not strongly hybridized with other orbitals [17]. \nThese observations indicate that the Fe 3 d electrons are strongly  hybridize d with the Ge \n4p states [18]. \nWe determine the orbital magnetic moment, morb, and the spin magnetic moment, \nmspin, of the substitutional  Fe atoms from the XAS and XMCD spectra at the L2,3 edge \nregion of Fe using the XMCD s um rules [19 – 23] [see Section I of Supplemental Material  \n(SM)]. As shown in Figs. 2(a) and 2(b), both mspin and morb (and therefore the total \nmagnetic moment M = mspin + morb) are larger in sample B ( TC = 100 K) than in sample A \n(TC = 20 K) over the entire temperature region when 𝜇0H = 5 T. For sample A, morb/mspin \n= 0.31 ± 0.02, and for sample B , morb/mspin = 0.30 ± 0.03, both of which are positive and \nsignificantly larger than that of bulk Fe (where morb/mspin ~ 0.043 [19]); the o rbital angular \nmomentum in GeFe is not quenched. The observation that the spin and orbital angular \nmomentum are parallel suggests that the Fe 3 d shell is more than half  filled. This implies  \nthat the Fe atoms are in the 2+ state rather than in the 3+ state, in which the Fe 3 d shell is \nhalf-filled and the orbital angular momentum vanishes. This large morb is a characteristic \nproperty of GeFe, and excludes the possibility of the existence of ferromagnetic Fe metal \nprecipitates in our films.  \nFigure 2(c) shows the H dependence of the XMCD intensity at energy X and a \ntemperature of 5.6 K (blue curve), the MCD intensity measured with visible light of 2.3 \neV at 5 K (red dotted curve), and the magnetization measured using a superconducting \nquantum interference  device (SQUID) at 5 K (green dotted curve) for sample B. The \nshapes of these curves show excellent agreement with each other; it follows that the spin \nsplitting of the valence band composed of the Ge 4 p orbitals is induced by the Fe 3 d \nmagnetic moment, wh ich originates from the substitutional Fe atoms , through the  p-d \nhybridization. The lower panels of Fig. 2 show the effective magnetic -field (Heff) \ndependence of the XMCD intensity measured at X for samples A (d) and B (e) at various  5 \ntemperatures. Here, M is also plotted (filled red symbols), and μ0Heff is obtained by \nsubtracting the product of M and the sheet density of the substitutional Fe atoms from \nμ0H to eliminate the influence of the demagnetization field. The scale of the vertical axis \nof the XMCD intensity is adjusted so that it represents M at each temperature.  The insets \nshow clear hysteresis below TC in both samples. The XMCD  – Heff curves show large \ncurvature above TC in both samples [see the main panel s of Figs. 2(d) and 2(e)], indicating \nthat part of the film is superparamagnetic (SPM) above TC. It indicates  that ferromagnetic \ndomains form in nanoscale high -Fe concentration regions at temperatures above TC, and \nthus M can be described by  \n𝑀=5.2𝑓SPML(𝑚SPM𝜇0𝐻eff\n𝑘B𝑇) + (1 -𝑓SPM)𝐶\n𝑇𝜇0𝐻eff, (1) \nwhere fSPM and mSPM are fitting parameters expressing the fraction of substitutional Fe \natoms which participat e in the SPM component , and the magnetic moment per \nferromagnetic domain,  respectively. Also, C is the Curie constant per substitutional Fe \natom (see Section III of the SM), and L is the Langevin function . Here,  5.2 is the ideal  \nsaturated value of M; i.e., M = mspin + (morb/mspin) × mspin, where mspin = 4 μB (for Fe2+) \nand morb/mspin ≈ 0.3 [Figs. 2(a) and 2(b)] when  all the substitutional Fe atoms are \nmagnetically active. The first and second terms in Eq. (1) correspond to the SPM and \nparamagnetic components, respectively. In Figs. 2(d) and 2(e), the black dashed curves \ncorrespond to the best fit obtained with  Eq. (1). For sample B, the M – Heff curves at \ntemperatures in the range 100  – 300 K are well reproduced by  Eq. (1), which indicates \nthat the ferromagnetic  – SPM transition occurs at TC = 100 K. With sample A, the M – \nHeff curves at temperatures above TC (i.e., T > 20 K) are well reproduced by  Eq. (1), \nexcept fo r T = 20 K, which is probably due to the onset of ferromagnetism. These good \nfits up to room temperature indicate that ferromagnetic interactions within the nanoscale  \nhigh-Fe concentration region s remain at room temperature in both samples.  \nThe residual M, which  is obtained from a linear extrapolation of M from the high \nmagnetic field region to Heff = 0 at 5.6 K, is 1.2 μB per Fe atom in sample A, and 1.5 μB \nper Fe atom  in sample B. This result suggests that only ~23% (=  1.2/5.2) and ~29% (=  \n1.5/5.2) of the substitutional Fe atoms are magnetically active in samples A and B, \nrespectively. In Figs. 2(d) and 2(e), t he high -field magnetic susceptibilities ∂𝑀/\n𝜕(𝜇0𝐻eff) (μB/T per Fe atom) at 4 T and 5.6 K are 0.15 in sa mple A and 0.10 in sample \nB. Because  ∂𝑀/𝜕(𝜇0𝐻eff) at 4 T per substitutional parama gnetic Fe atom should be 0.37  6 \n(see Section IV of the SM), this result indicates  that the ratios of paramagnetic Fe atoms \nto the total number of Fe atoms are only ~41%  (= 0.15/0.37)  and ~27%  (= 0.10/0.37) , \nrespectively. This means that some fraction of the moment of the Fe atoms is missing, \nand thus suggests that there are Fe atoms th at couple antiferromagnetically with the \nferromagnetic Fe atoms in the films. This is also supported by the weak spin -glass \nbehavior observed in GeFe at very low temperatures [7].  \nWe see a similar trend in the temperature dependence of the fitting paramete rs fSPM \nand mSPM in both films; i.e., fSPM and mSPM both increase with decreasing temperature  (Fig. \n3). This result means  that the ferromagnetic domains, which form only in the nanoscale \nhigh-Fe concentration regions at room temperature [Fig. 4(a)] , expand toward lower Fe \nconcentration regions  with decreasing temperature [Fig. 4(b)], and finally the entire film \nbecomes ferromagnetic at TC [Fig. 4(c)].  This appears to be a characteristic feature of \nmaterials that exhibit homogeneous  ferromagnetism, d espite the inhomogeneous  \ndistribution of magnetic atoms in the film [6,7]. As shown in Fig. 3, fSPM and mSPM are \nlarger in sample B than in sample A, which is attributed to the difference in spatial  \nfluctuations of the Fe concentration, which are 4% – 7% i n sample A and 3%  – 10% in \nsample B  [6]. The larger  the nonuniformity of the Fe distribution is , the larger \nferromagnetic domains, fSPM , and mSPM become,  and the domains  can more easily connect \nmagnetically , resulting in a higher  TC. \nIn summary, we have in vestigated the local electronic structure and magnetic \nproperties of the doped Fe atoms in the Ge0.935Fe0.065 films using XAS and XMCD. The \nFe atoms appear in the 2+ state, with the 3 d electrons strongly hybridized with the 4 p \nelectrons in Ge; this results  in a delocalized 3 d nature and long -range ferromagnetic \nordering, leading to the excellent agreement between the H dependence of magnetization, \nMCD, and XMCD. Using the XMCD sum rules, we obtained the M – Heff curves, which \ncan be explained by the coexistence of SPM and paramagnetic ordering at temperatures \nabove TC. The fitting results clearly show that the local ferromagnetic domains , which \nexist at room temperature,  expand with decreasing temperature, leadin g to a  \nferromagnetic transition of the entire system at  TC. The nonuniformity of the Fe \nconcentration plays a crucial role for  the formation of the magnetic domains, and our \nresults indicate that strong ferromagnetism is inherent to GeFe, and persists at r oom \ntemperature.  \n  7 \nACKNOWLEDGEMENTS  \nWe would like to thank T. Okane for support with the experiments. This work was partly \nsupported by Grants -in-Aid for Scientific Research (22224005, 23000010, and \n26249039) including the Specially Promote d Research , and the Project for Developing \nInnovation Systems from MEXT.  This work was performed under the Shared Use \nProgram of JAEA Facilities (Proposal No. 201 4A-E31) with the approval of the \nNanotechnology Platform Project supported by  MEXT . The synchr otron radiation \nexpe riments were performed at the JAEA beamline BL23SU in SPring -8 (Proposal No. \n2014A3881 ). Y. K. Wakabayashi and Y. Takahashi acknowledge financial support from \nJSPS through the Program for Leading Graduate Schools (MERIT). S. Sakamoto \nacknowledge s financial support from JSPS through the Program for Leading Graduate \nSchools (ALPS). 8 \nReferences  \n[1] L. Chen , X. Yang , F. Yang , J. Zhao , J. Misuraca , P. Xiong , and S. von Molnár , Nano  \nLett. 11, 2584  (2011).  \n[2] K. M. Yu, W. Walukiewicz, T. Wojtowicz, I. Kuryliszyn, X. Liu, Y . Sasaki, and J. K. \nFurdyna, Phys. Rev. B 65, 201303 (2002).  \n[3] Y . Shuto, M. Tanaka, and S. Sugahara, phys. stat. sol. 3, 4110 (2006).  \n[4] Y . Shuto, M. Tanaka, and S, Sugahara, Appl. Phys. Lett. 90, 132512 (2007).  \n[5] Y . Shuto, M. Tanaka, and S. Sugahara, Jpn. J. Appl. Phys. 47, 7108 (2008).  \n[6] Y . K . Wakabayashi, S. Ohya, Y . Ban, and M. Tanaka, J. Appl. Phys. 116, 173906 \n(2014).  \n[7] Y . K. Wakabayashi, Y . Ban, S. Ohya and M. Tanaka, Phys. Rev. B 90, 205209 (2014).  \n[8] Y . Ban, Y . Wakabayashi, R. Akiyama, R. Nakane, and M. Tanaka, AIP Advances 4, \n097108  (2014).  \n[9] D. J. Keavney, D. Wu, J. W. Freeland, E. J. Halperin, D. D. Awschalom, and J. Shi, \nPhys. Rev. Lett. 91, 187203 (2003).  \n[10] K. W. Edmonds, N. R. S. Farley, T. K. Johal, G. V . D. Laan, R. P. Campiom, B. L. \nGallegher, and C. T. Foxon, Phys. Rev.  B 71, 064418 (2005).  \n[11] D. J. Keavney, S. H. Cheung, S. T. King, M. Weinert, and L. Li, Phys. Rev. Lett. 95, \n257201 (2005).  \n[12] V . R. Singh, K. Ishigami, V . K. Verma, G. Shibata, Y . Yamazaki, T. Kataoka, A. \nFujimori, F. -H. Chang, D. -J. Lin, C. T. Che n, Y . Yamada, T. Fukumura, and M. \nKawasaki, Appl. Phys. Lett. 100, 242404 (2012).  \n[13] Y . Saitoh, Y . Fukuda, Y . Takeda, H. Yamagami, S. Takahashi, Y . Asano, T. Hara, K. \nShirasawa, M. Takeuchi, T. Tanaka, and H. Kitamura, J. Synchrotron Rad. 19, 388 \n(2012)  \n[14] T. J. Regan, H. Ohldag, C. Stamm, F. Nolting, J. Luning, J. Stohr, and R. L. White, \nPhys. Rev. B 64, 214422 (2001).  \n[15] R. Kumar, A. P. Singh, P. Thakur, K. H. Chae, W. K. Choi, B. Angadi, S. D. Kaushik, \nand S. Panaik, J. Phys. D: Appl. Phys. 41, 155 002 (2008).  \n[16] E. Sakai, K. Amemiya, A. Chikamatsu, Y . Hirose, T. Shimada, and T. Hasegawa, J. \nMagn. Magn. Mater, 333, 130 (2013).  \n[17] G. V . D. Laan and I. W. Kirkman, J. Phys.: Condens. Matter 4, 4189 (1992).  \n[18] I. A. Kowalik, A. Persson, M. A. Nino,  A. Na, A. Navarro -Quezada, B. Faina, A. \nBonanni, T. Dietl, and D. Arvanits, Phys. Rev. B 85, 184411 (2012).  \n[19] C. T. Chen, Y . U. Idzerda, H. -J. Lin, N. V . Smith, G. Meigs, E. Chaban, G. H. Ho, \nE. Pellegrin, and F. Sette, Phys. Rev. Lett. 75, 152 (1995).  \n[20] Y . Takeda, M. Kobayashi, T. Okane, T. Ohkochi, J. Okamoto, Y . Saitoh, K. \nKobayashi, H. Yamagami, A. Fujimori, A. Tanaka, J. Okabayashi, M. Oshima, S. \nOhya, P. N. Hai, and M. Tanaka, Phys. Rev. Lett. 100, 247202 (2008).  \n[21] K. Mamiya , T. Koide, A. Fujimori, H. Tokano, H. Manaka, A. Tanaka, H. Toyosaki, \nT. Fukumura, and M. Kawasaki, Appl. Phys. Lett. 89, 062506 (2006).  \n[22] J. Stohr and H. Konig, Phys. Rev. Lett. 75, 3748 (1995).  \n[23] C. Piamonteze, P. Miedema, and F. M. F. de Groot, P hys. Rev. B 80, 184410 (2009).    9 \nFig. 1. (Color) Schematic diagrams showing the structures of sample  A (a) and sample B \n(b). (c), (d) XAS spectra of μ– (blue curve), μ+ (red curve),  and (μ+ + μ–)/2 (black curve) \nat the L2 (~721 eV) and L3 (~708 eV) absorption edges of Fe for sample  A (c) and sample \nB (d). The measurements were made  with a magnetic field of μ0H = 5 T applied \nperpendicular to the film surface at a temperature of 5.6 K. The insets show a magnified \nplot of the spectra at the Fe L3 edge.  (e), (f) XMCD (=  μ+ – μ–) spectra at the L2 and L3 \nabsorption edges of Fe for sample  A (e) and sample B (f) measured at 5.6 K with μ0H = \n0.1 T (red curve), 1 T ( brown  curve), 3 T (green curve), and 5 T (blue curve) applied \nperpendicular to the film  surface . The insets show a magnified plot  of the spectra at the \nFe L3 edge, in which the XMCD data are normalized to 707.3 eV .  \n \n 10 \nFig. 2. (Color) (a ), (b) The temperature  dependence of mspin + morb (green squares),  mspin \n(blue triangles),  morb (violet circles),  and morb/mspin (red rhombuses) for sample A (a) and \nsample B (b) with an applied magnetic field of μ0H = 5 T. (c) The H dependence of the \nXMCD intensity (blue solid curve) at X shown in  Fig. 1 (707.66 eV) at 5.6 K, the MCD \nintensity at 5 K with a photon energy of 2.3 eV corresponds  to the L-point energy gap of \nbulk Ge (red dotted curve), and the magnetization measured using a  SQUID at 5 K (green \ndotted curve) for sample B. (d),(e) The dependence of the XMCD intensity measured at \nX on the effective magnetic field  Heff for sample A (d) and sample B (e) at various \ntemperatures. The total magnetization (M = mspin + morb) obtained using the XMCD sum \nrules is also plotted (filled red symbols ). We scaled the vertical axis of the XMCD \nintensity so that it represents M at each temperature. In all measurements, H was applied \nperpendicular to  the film  surface . \n \n \n \n \n \n 11 \nFig. 3. (Color  online ) (a), (b) The temperature  dependence of the best -fit parameters fSPM \nand mSPM obtained for sample  A (a) and sample B (b). The red  (or dark gray) , gray, and \nwhite areas indicate ferromagnetic (FM), FM + SPM + paramagnetic (PM), and SPM + \nPM regions,  respectively.  \n \n \n \n \n \n \n \n \n \n \n \n 12 \n \nFig. 4 . (Color) (a ) – (c) Schematic diagrams showing  the magnetic states in the \nGe0.935Fe0.065 films with zero magnetic field at room temperature  (i.e., T = 300 K ) (a), TC \n< T < 300 K  (b), and T < TC (c). The small black , red, and blue arrows correspond to the \nmagnetic moments  of the paramagnetic , ferromagnetic , and antiferromagnetically  \ncoupled substitutional Fe atoms , respectively.  The red areas indicate ferromagnetic  \nregions. Antiferromagnetically  coupled Fe atoms are thought to exist all  over the  film at \ntemperatures below TC.  \n 13 \nSupplemental Material for  Room -temperature local ferromagnetism and nanoscale \ndomain growth in the ferromagnetic semiconductor Ge 1 - xFex \n \nYuki K. Wakabayashi,1 Shoya Sakamoto,2 Yukiharu Takeda,3 Keisuke  Ishigami,2 Yukio \nTakahashi,2 Yuji Saitoh,3 Hiroshi Yamagami,3 Atsushi Fujimori,2 Masaaki Tanaka,1 and \nShinobu Ohya1 \n \n1Department of Electrical Engineering and Information Systems, The University of Tokyo, \n7-3-1 Hongo, Bunkyo -ku, Tokyo 113 -8656, Japan  \n2Department of Physics, The University of Tokyo, Bunkyo -ku, Tokyo 113 -0033, Japan  \n3Synchrotron Radiation Research Unit, JAEA, Sayo, Hyogo 679 -5148, Japan  \n \nI.  Estimation of the spi n magnetic moment and the orbital magnetic moment of \nsubstitutional  Fe atoms using the X -ray m agnetic -circular -dichroism sum rules  \n \nWe obtain the spin magnetic moment mspin and the orbital magnetic moment morb \nfrom the spectra of the X -ray absorption spectroscopy (XAS) and X -ray magnetic circular \ndichroism (XMCD) in the energy region near the L2 and L3 absorption edges of Fe using \nthe XMCD sum rules.S3  Figure S1(a) shows the XAS spectra (solid curves) and the \nXAS signals integrated from 690 eV (dashed curves) of sample A.  Figure S1(b) shows \nthe XMCD spectra (solid curves) and the XMCD signals integrated from 690 eV (dashed \ncurves) of sample A.  Here, the measurements were carried out with a magnetic field \n𝜇0H = 5 T applied perpendicular to the film surface at 5.6 K (black curves), 20 K (blue \ncurves), 50 K (lig ht blue curves), 100 K (green curves), 150 K (orange curves), 250 K \n(pink curves), and 300 K (red curves).  Figure S2 shows the same data measured for \nsample B.   For the XMCD sum -rules analyses, we define th e values of r, p, and q as the \nfollowing equations at each temperature.  \n𝑟= ∫(𝜇++𝜇−)\n2𝑑𝐸𝐸3+𝐸2,     (S1) \n𝑝= ∫(𝜇+−𝜇−)𝑑𝐸𝐸3,    (S2) \n𝑞= ∫ (𝜇+−𝜇−)𝑑𝐸𝐸3+𝐸2,    (S3) \nwhere E3 (690-718 eV) and E2 (718-760 eV) represent the integration energy range s for \nthe L3 and L2 absorption edges, respectively.  Here, μ+ (μ-) and E represent the absorption  14 \ncoefficient for the photon helicity parallel (antiparallel) to the Fe 3 d majority spin \ndirection and the i ncident photon energy, respectively.   We can obtain mspin and morb of \nsubstitutional Fe atoms using  the XMCD sum rules, which are expressed as follows:  \n𝑚orb= −2𝑞\n3𝑟(10 − 𝑛3𝑑),   (S4) \n 𝑚spin+7𝑚T= −3𝑝−2𝑞\n𝑟(10 − 𝑛3𝑑),     (S5) \nwhere 𝑛3𝑑 and 𝑚T are the number of 3 d electrons on the  Fe atom and the expectation \nvalue of the intra -atomic magnetic dipole operator , respectively . Because the \nparamagnetic component observed at Y in Figs. 1(e) and 1(f) in the main text, which  \noriginates from the intersti tial Fe atoms, is negligibly small (See Sec tion II of the \nSupplemental Material), the integrated values of the XMCD spectra p [Eq. ( S2)] and q \n[Eq. ( S3)] can be attributed only to the substitutional Fe atoms.  Meanwhile, because the \nXAS signals have both c ontributions of the substitutional and interstitial Fe atoms, we \nreduced the integrated XAS intensity r [Eq. ( S1)] to 85% of its raw value (85% is the \napproximate ratio of the substitutional Fe atoms to that of the total Fe atoms in both \nsamples A and BS4) when using  the XMCD sum rules.   We neglect the expectation value \nof the intra -atomic magnetic dipole operator, because it is negligibly small for Fe atoms  \nat the Td symmetry  site.S5  Also, we assume 𝑛3𝑑 to be 6 and the correction factor for \nthe mspin to be 0.88 for Fe2+.S6 \nBy the above calculations  using  the temperature dependence of XAS and XMCD \nspectra shown in Figs. S1 and S2,  we obtain ed the temperature dependence of mspin and \nmorb of substitutional Fe atoms as shown in Figs. 2(a) and 2(b)  in the main text.  \n \n \n \n \n  15 \n \n \nFig. S1. (a) XAS [= ( μ+ + μ-)/2] spectra (solid curves) and the XAS signals integrated \nfrom 690 eV (dashed curves) of sample A.  (b) XMCD (= μ+ - μ-) spectra (solid curves) \nand the XMCD signals integrated from 690 eV (dashed curves) of sample A.  These \nmeasurements were carried out with a magnetic field 𝜇0H = 5 T applied perpendicular \nto the film surface  at 5.6 K (black curves), 20 K (blue curves), 5 0 K (light blue curves), \n100 K (green curves), 150 K (orange curves), 250 K (pink curves), and 300 K (red curves).  \n \n \nFig. S2. (a) XAS [= ( μ+ + μ-)/2] spectra (solid curves) and the XAS signals integrated \nfrom 690 eV (dashed curves) of sample B.  (b) XMCD (= μ+ - μ-) spectra (solid curves) \nand the XMCD signals integrated from 690 eV (dash ed curves) of sample B.  These \nmeasurements were carried out with a magnetic field 𝜇0H = 5 T applied perpendicular \n 16 \nto the film surface at 5.6 K (black curves), 50 K (light  blue curves), 100 K (green curves), \n150 K (orange curves), and 250 K (pink curves).  \n \n \nII.  Influence of the paramagnetic XMCD component on the XMCD sum -rules  \nanalyses  \n \nFigures S3 show s the XMCD spectra of samples A (a) and B (b) normalized to 707.3 \neV measured at 5.6 and 300 K with  magnetic fields of  0.1 and 5 T applied perpendicular \nto the film surface .  In both films, all the line shape s of the XMCD spectr a are almost \nthe same, which means that the paramagnetic component observed at Y in Figs. 1( e) and \n1(f) of the main text is negligibl y small  for the XMCD sum -rules analyses  and that the \nintegrated XMCD signal can be attributed only to the substitutional Fe atoms.  \n \n \nFig. S3. XMCD spectra of samples A (a) and B (b) normalized to 707.3 eV  measured at \n5.6 a nd 300 K with magnetic fields of 0.1 and 5 T applied perpendicular to the film surface . \n \nIII.  Estimation of the Curie constant per substitutional Fe atom  \n \nThe Curie constant per substitutional Fe atom is obtained using the following \nequations:  \n𝐶= 𝜇B2\n3𝑘B𝑛B2,     (S6) \n 17 \n𝑛B= [3\n2+ 𝑆(𝑆+1) − 𝐿(𝐿+1)\n2𝐽(𝐽 + 1)]√𝐽(𝐽+1),    (S7) \nwhere 𝜇B, 𝑘B, 𝑛B, S, L, and 𝐽 represent the Bohr magneton, the Boltzmann constant , \nthe effective Bohr magneton number,  the spin angular momentum, the orbital angular \nmomentum, and the total angular momentum,  respectively.   We obtained 𝑛B assuming \nS = 2 (for Fe2+), L = 1.2 ( L = 2S × morb/mspin, where morb/mspin ≈ 0.3 as shown in Figs. \n2(a) and 2(b) in the main text), and J = 3.2 (= L + S because the spin and orbital angular \nmomenta of a substitutional Fe atom are parallel) in Eq. ( S7).  Thus, 𝑛B is estimated to \nbe 5.96.  \n \nIV .  Estimation of the high -field magnetic susceptibility of a substitutional \nparamagnetic  Fe atom  \n \nAt very low temperature below ~20 K, the effective magnetic -field Heff dependence \nof the total magnetization M (= mspin+morb) of one substitutional paramagnetic  Fe atom is \nexpressed by the Langevin function.  Thus, the Heff dependence of M of one \nsubstitutio nal paramagnetic  Fe atom at 5.6 K is obtained by substituting 5.2 μB, 1, and 5.6 \nK in mSPM, fSPM, and T of Eq. (1) in the main text, respectively (Fig. S4).  We \napproximated the high -field magnetic susceptibility ∂𝑀/𝜕(𝜇0𝐻eff) (μB/T per Fe) at 4 T \nby th e slope of the M-Heff line from 4 T to 5 T (black dashed line in Fig. S4).  In this \nway, ∂𝑀/𝜕(𝜇0𝐻eff) at 4 T is estimated to be 0.37 μB/T per one substitutional \nparamagnetic  Fe.  In Figs. 2(d) and 2(e) in the main text, t he ∂𝑀/𝜕(𝜇0𝐻eff) value s at \n4 T and 5.6 K in samples A and B are 0.15 and 0.10, respectively ; it follows that the ratios \nof paramagnetic Fe atoms to the total number of Fe atoms are  ~41%  (= 0.15/0.37)  in \nsample A and ~27%  (= 0.10/0.37)  in sample B.  \n  18 \n \nFig. S4. Effective magnetic -field Heff dependence of the total magnetization M (= \nmspin+morb) per one substitutional paramagnetic  Fe at 5.6 K  obtained using Eq. (1) in the \nmain text.  \n \nReferences  \nS3 C. T. Chen, Y . U. Idzerda, H. -J. Lin, N. V . Smith, G. Meigs, E. Chaban, G. H. Ho, E. \nPellegrin, and F. Sette, Phys. Rev. Lett. 75, 152 (1995).  \nS4 Y . K. Wakabayashi, S. Ohya, Y . Ban, and M. Tanaka, J. Appl. Phys . 116, 173906 \n(2014).  \nS5 J. Stohr and H. Konig, Phys. Rev. Lett. 75, 3748 (1995).  \nS6 C. Piamonteze, P. Miedema, and F. M. F. de Groot, Phys. Rev.  B 80, 184410 (2009).   \n"    },    {        "title": "2110.06478v1.Theoretical_Study_of_Temperature_Dependence_of_Spin_Susceptibility_in_Anisotropic_Itinerant_Ferromagnets.pdf",        "content": "Theoretical Study of Temperature Dependence of Spin \nSusceptibility  in Anisotropic Itinerant Ferromagnets  \n \n \nDaisuke Miura * and Akimasa Sakuma \n \nDepartment of Applied Physics, Tohoku University , Sendai 980 -8579 , Japan  \n \nWe develop ed a framework for directly calculating the spin \nsusceptibility  of anisotropic  itinerant ferromagnets in the full \ntemperature range  within  the coherent potential approximation in the \ndisordered local moment picture .  As a test of our formulation, we  \ndemonstrate  the computation for the temperature ( 𝑇) dependence of  \nthe spin susceptibility for Rashba -type ferromagnets  with the Curie \ntemperature 𝑇C . In a certain parameter, we found  that the inverse \ntransverse su sceptibility  1/𝜒⊥≃const.   for 𝑇<𝑇C  and 1𝜒⊥⁄ ∝\n𝑇−𝑇C  for 𝑇>𝑇C , which  reflects characteristics of itinerant \nferromagnets  including a spin -orbit interaction.  \n \nThe magnetic susceptibility 𝜒  is one of the most fundamental properties of magnetic \nmaterials, whose microscopic description has attracted the interest of many researchers  [1–5]. \nHistorically, demonstrating the Curie –Weiss (CW) law has been a touchstone issue to confirm \nthe reasonableness of models and approximations used in theoretical stud ies on finite -\ntemperature magnetism. It has  long been known that the transverse spin fluctuation plays a \ncentral role in the CW behavior in the localized  electron systems such as ferromagnetic \ninsulators, which can easily be recognized by the Heisenberg model  [6]. On the other hand, \nderiv ing the CW law on itinerant  electron systems such as ferromagnetic metals  was a lengthy  \nprocess . Primarily , this is  for the itinerant electrons ; the magnitude of local  moment on each \nsite is not a conserved quantity against the temperature change and the e xternal field, which is \ngoverned by the electron -electron correlation effects. For weak ferromagnets,  in 1972,  Murata \nand Doniach  [7] have first presented the phenomenological description of 𝜒 in which the self-\nconsistent mean field theory was improved to include the mode -mode coupling, and the first \nmicroscopic one was given by Moriya and Kawabata in 1973  [8,9] . Furthermore, a unified \npicture including a treatment of strong fe rromagnets was proposed by Moriya and Takahashi in \n1978  [10]. In addition to the transverse spin fluctuation  considered in the Heisenberg model, \nthese theories considered  the longitudinal spin fluctuation through the mode -mode coupling of \nspin-waves and successfully demonstrated the qualitative characteristic of itinerant electron \nsystem s, especially of weak ferromagnets.  \nRecently, in both practical a nd industrial viewpoint s, strong ferromagnets having a large 2 \n magnetization and a high 𝑇C are required. The disordered local moment (DLM) analysis  [11–\n16], by using the first -principles calculation , has shown at quantitative level s that many strong \nferromagnetic metals retain robust local magnetic moments even above the  𝑇C, implying  that \nthe transverse spin fluctuation greatly  governs the CW behavior in these systems. In fact, first -\nprinciples calculations for the magnetic susceptibility of Fe  [17–20] and Ni  [18,20]  have been \nperformed on the basis of combined theory  [17,21]  between the density functional theory and \nthe coh erent potential approximation (CPA) in the DLM picture in finite temperatures . There , \nthe CW law was successfully observed for both Fe and Ni , although  the calculations for the \nCurie temperature of Ni resulted in an underestimation indicating the failure o f the DLM picture.  \nIn addition to the CW law in 1/𝜒 , the importance of finite temperature magnetism of \npractical materials lies in a temperature dependence of magnetic anisotropy constants  (MACs)  \nbelow 𝑇C . In this temperature range, several authors have performed first-principles \ncalculations for magnetocrystalline anisotropy  (MA)  in the DLM picture for metallic \nferromagnets for FePt   [22,23] , FePd  [23,24] , CoPt, MnAl, and FeCo  [23]. In addition,  the \nauthors demonstrated that the temperature dependence of the MACs  are roughly proportional \nto the second power of the magnetization , as indicated by experiments for FePt  [25]. The second \npower law cannot be understood by single ion models predicting  the third power law (so -called \nAkulov –Zener –Callen –Callen law)  [26–31], which suggests a characteristic of ferromagnetic \nmetals  that the DLM picture appears  to capture well. Again,  the transverse susceptibility 𝜒⊥ \ncan reflect the temperature dependence of MA below the  𝑇C. For example, if a material has \nuniaxial MA, then the 𝜒⊥ may ha ve a finite value depending on the temperature, whereas 𝜒⊥ \ndiverges below the  𝑇C in the absence of MA. However , theoretical stud y of 𝜒⊥ in the full \ntemperature range  has not yet been reported for anisotropic itinerant ferromagnetism.  \nThe purpose of this work is to provide  a practical framework for directly  computing  the \ntemperature dependence of the magnetic susceptibility of  anisotropic  ferromagnetic metals  in \nthe full temperature range  from the microscopic viewpoint  based on the DLM and CPA theory . \nFor confirming the validity of our formulation, we also show  the temperature dependence of  \nmagnetization curves , from whose gradient  we can estimate the  magnetic susceptibilit y. \nHowever,  the computational load fo r obtaining  the magnetization curves  is overly  heavy ; thus, \nwe employ the two -dimensional tight -binding Rashba model  [32] for describing itinerant \nelectrons with MA, which has been valued as a theoretically tractable model for investigating \nanisotropic magnetic metals  [33–35]. The usage of  the CPA -Green function method  enables  us \nto easily extent the present theory to first -principles calculation s in the future, which has been 3 \n widely used  for evaluating various physical quantities  such as electrical conductivities  [36,37] , \nGilbert damping constants  [38,39] , magnetic anisotropies  [24,40] , and so on.  \nLet us consider  the tight -binding Rashba Hamiltonian  in the presence of an external  uniform \nstatic  magnetic field  𝑯=(𝐻x,𝐻y,𝐻z) on the two -dimensional square  lattice  given  by \n ℋ≔∑𝑐𝒌†(𝜖̂𝒌+𝑣̂𝑯)𝑐𝒌\n𝒌−𝛥ex∑𝒆𝑖⋅𝑐𝑖†𝛔̂𝑐𝑖𝑁\n𝑖=1, (1) \nwhere 𝑐𝒌 is the annihilation operator of the electron  with a wave vector 𝒌=(𝑘x,𝑘y) in the \nspinor representation , the 𝒌  summation is perfor med in the first Brillouin zone,  𝑐𝑖≔\n(1\n√𝑁)∑𝑐𝒌exp(i𝒌⋅𝑹𝑖)𝒌 , 𝛔̂≔(σ̂x,σ̂y,σ̂z) for the Pauli matrices  σ̂𝛼 (a quantity with the hat  \nis a 2×2 matrix in the spin space ), and  𝛥ex denotes the strength of the exchange field whose \ndirection is a classical unit vector 𝒆𝑖 allocated at a lattice vector 𝑹𝑖 in the 𝑁 sites. In the \nlanguage of  the functional integral method under the Hubbard -Stratonovich transformation, \n𝛥ex corresponds to the saddle point concerning  the amplitude of auxiliary exchange field s. The \nkinetic and Zeeman parts in the first term in Eq.  (1), respectively,  are defined by  \n 𝜖̂𝒌≔−2𝑡(cos𝑘x𝑎+cos𝑘y𝑎)1̂+𝜆(σ̂xsin𝑘y𝑎−σ̂ysin𝑘x𝑎), (2) \n 𝑣̂𝑯≔𝑔μB\n2 𝝈̂⋅μ0𝑯≡𝑡𝝈̂⋅𝒉, (3) \nwhere 𝑡  is the hopping integral between the two nearest -neighbor  sites, 𝑎  is the lattice  \nconstant,  1̂ is the unit matrix,  𝜆 is the strength of the Rashba spin-orbit interaction ( SOI), 𝑔 \nis the g factor of an electron , μB is the Bohr magneton , μ0 is the  permeability in vacuum , and  \n𝒉 stands for  the d imensionless  external magnetic field . In the DLM -CPA theory , it is assume d \nthat a local moment  at each site  is located in a configuration {𝒆𝑖}, and {𝒆𝑖} is determined by \na functional integral method based on the single -site approximation at each temperature ; \nsubsequently , a Green ’s function is obtained  from  the CPA condition  [11,15] . In the present  \nmodel , the single -electron Green ’s function  [35] between sites 𝑖 and 𝑗 is given by \n 𝐺̂𝑖𝑗𝑯(𝑧):=1\n𝑁∑ei𝒌⋅(𝑹𝑖−𝑹𝑗)𝑔̂𝒌𝑯(𝑧)\n𝒌, (4) \nwhere 𝑧  is a complex variable , and 𝑔̂𝒌𝑯(𝑧)≔[𝑧1̂−𝜖̂𝒌−𝑣̂𝑯−𝛴̂𝑯(𝑧)]−1 . The self -energy \n𝛴̂𝑯(𝑧) is determined by the CPA condition:  \n ⟨𝑡̂𝒆𝑯(𝑧)⟩𝒆𝑯=(00\n00), (5) \nwhere 4 \n  ⟨⋯⟩𝒆𝑯≔∫d𝒆 𝑤𝒆𝑯⋯, (6) \n ∫d𝒆≔∫d𝜃sin𝜃𝜋\n0∫ d𝜙2𝜋\n0 in terms of 𝒆≔(sin𝜃cos𝜙,sin𝜃sin𝜙,cos𝜃), (7) \n 𝑡̂𝒆𝑯(𝑧)≔1\n1̂−[−𝛥ex𝒆⋅𝛔̂−𝛴̂𝑯(𝑧)]𝐺̂00𝑯(𝑧)[−𝛥ex𝒆⋅𝛔̂−𝛴̂𝑯(𝑧)], (8) \n 𝑤𝒆𝑯≔e−𝛽(Ω𝒆𝑯−𝐹𝑯), (9) \n Ω𝒆𝑯≔−1\n𝛽∑Trln{1̂−[−𝛥ex𝒆⋅𝛔̂−𝛴̂𝑯(𝑧𝑛𝑯)]𝐺̂00𝑯(𝑧𝑛𝑯)}\n𝑛, (10) \n 𝐹𝑯≔−1\n𝛽ln∫d𝒆 e−𝛽Ω𝒆𝑯, (11) \nwhere 𝛽≔(kB𝑇)−1 for the temperature 𝑇 , 𝑧𝑛𝑯≔iℏ𝜔𝑛+𝜇𝑯 , 𝜔𝑛≔𝜋(2𝑛+1)/(𝛽ℏ) , \nthe 𝑛 summation is performed over all integers, and 𝜇𝑯 is the chemical potential . \n  The spin magnetization  𝑴𝑯, in units of magnetic field,  is represented regarding  the single -\nelectron Green’s function  as \n 𝑴𝑯=−𝑔μB\n2𝑎3𝛽∑Tr𝛔̂𝐺̂00𝑯(𝑧𝑛𝑯)\n𝑛≡𝑔μB\n2𝑎3𝝈𝒉, (12) \nwhere  𝝈𝒉 denotes  the spin concentration vector  (dimensionless) . Here w e assume  the form of \n𝝈𝟎=(0,0,𝜎s) with 𝜎s>0 for 𝑇<𝑇C, that is , we set the model parameters s uch that the \nsystem exhibits uniaxial ly anisotropic  ferromagnets . In addition, we must  refer to the problem \nof which be fixed, the chemical potential or the electron concentration . Although  the magnetism \nand the MA of the ground state , at least,  must be ferromagnetic and perpendicular to the lattice \nplane, both strongly depend on the electron concentration  as shown by previous theoretical \nstudies on Rashba magnets  [33–35]. Therefore, we adjust 𝜇𝑯  such that the electron \nconcentration  is fixed to an appropriate value 𝑛e: \n 𝑛e𝑯=𝑛e, (13) \nwhere 𝑛e𝑯 is represented by  \n 𝑛e𝑯=1\n𝛽∑Tr𝐺̂00𝑯(𝑧𝑛𝑯)\n𝑛. (14) \nFrom Eq.  (12), we obtain the formally microscopic expressions for the longitudinal and \ntransverse spin susceptibilities  (dimensionless)  as \n 𝜒∥:=lim\n𝑯→𝟎𝜕𝑀 z𝑯\n𝜕𝐻z=𝑔2μB2μ0\n4𝑡𝑎3𝜒̃z, (15) \nand \n 𝜒⊥:=lim\n𝑯→𝟎𝜕𝑀 x𝑯\n𝜕𝐻x=𝑔2μB2μ0\n4𝑡𝑎3𝜒̃x, (16) 5 \n respectively, and the normalized susceptibility has been defined by  \n 𝜒̃𝛼:=lim\n𝒉→𝟎𝜕𝜎𝛼𝒉\n𝜕ℎ𝛼=−lim\n𝒉→𝟎1\n𝛽∑Trσ̂𝛼𝜕𝐺̂00𝑯(𝑧𝑛𝑯)\n𝜕ℎ𝛼𝑛. (17) \nAs indicated , we simply must  compute the on -site Green function 𝐺̂00𝑯(𝑧𝑛𝑯) in principle , but \nthe high-precision computation  is required  for reducing  an error owing  to the numerical \ndifferentia tion, especially  whose compu tational load for 𝜒⊥ is heavy . Therefore,  developing a \nmethod  to directly comput e the derivative of 𝐺̂00𝑯(𝑧𝑛𝑯)  is desired for future realistic \ncalculations.  It can be  achieved by the following pro cedure.  \n  Firstly, from the definition  (4), the derivative of 𝐺̂00𝑯(𝑧𝑛𝑯) is represented  by \n 𝜕𝐺̂00𝑯(𝑧𝑛𝑯)\n𝜕𝒉=−1\n𝑁∑𝑔̂𝒌𝑯(𝑧𝑛𝑯)(𝜕𝜇𝑯\n𝜕𝒉1̂−𝑡𝝈̂−𝜕𝛴̂𝑯(𝑧𝑛𝑯)\n𝜕𝒉)𝑔̂𝒌𝑯(𝑧𝑛𝑯)\n𝒌, (18) \nand 𝜕𝜇𝑯/𝜕𝒉 is determined by \n 𝜕𝜇𝑯\n𝜕𝒉=(1\n𝛽𝑁∑Tr𝑔̂𝑘𝑯(𝑧𝑛𝑯)2\n𝑛𝒌)−11\n𝛽𝑁∑Tr𝑔̂𝑘𝑯(𝑧𝑛𝑯)2(𝑡𝝈̂+𝜕𝛴̂𝑯(𝑧𝑛𝑯)\n𝜕𝒉)\n𝑛𝒌, (19) \nwhich is derived by  𝜕𝑛e𝑯/𝜕𝒉=𝟎  from the condition (13) and the definition (14). The final \nform of the  equation  to be solved is obtained by differentiating both sides of  the CPA condition \n(5) with respect to 𝒉: \n ∂𝛴̂𝑯(𝑧𝑛𝑯)\n∂𝒉−⟨𝑡̂𝒆𝑯(𝑧𝑛𝑯)(𝜕𝐺̂00𝑯(𝑧𝑛𝑯)\n𝜕𝒉−𝐺̂00𝑯(𝑧𝑛𝑯)𝜕𝛴̂𝑯(𝑧𝑛𝑯)\n𝜕𝒉𝐺̂00𝑯(𝑧𝑛𝑯))𝑡̂𝒆𝑯(𝑧𝑛𝑯)⟩\n𝒆𝑯\n=−⟨𝑡̂𝒆𝑯(𝑧𝑛𝑯)Tr∑(𝜕𝐺̂00𝑯(𝑧𝑚𝑯)\n𝜕𝒉−𝐺̂00𝑯(𝑧𝑚𝑯)𝜕𝛴̂𝑯(𝑧𝑚𝑯)\n𝜕𝒉𝐺̂00𝑯(𝑧𝑚𝑯))𝑡̂𝒆𝑯(𝑧𝑚𝑯)\n𝑚⟩\n𝒆𝑯\n. (20) \nConsequently, we can obtain  a closed  equation for ∂𝛴̂𝑯(𝑧𝑛𝑯)∂𝒉⁄ from Eqs. (18), (19), and \n(20), which is free from  the numerical differentiation . After the limit of  𝒉→𝟎, the equation  \nfor ∂𝛴̂𝑯(𝑧𝑛𝑯)∂𝒉⁄ is consisted of  only the quantities  based on  the zero -magnetic field CPA . \nHere , in Eq. (19) for 𝑇<𝑇C and 𝒉→0, we notice that  𝜕𝜇𝑯𝜕ℎx→0 ⁄  but 𝜕𝜇𝑯𝜕ℎz ⁄  does \nnot vanish  as noted  in Ref.  [41].  \n  Let us  consider a st rong Rashba  SOI case  [42] in which the parameters are set as 𝛥ex𝑡⁄=\n0.2  and 𝜆𝑡⁄=1 , and 𝑛e=0.25 ; the Curie temperature was estimated at kB𝑇C𝑡⁄=\n3.7×10−4 from the computation of the  temperature dependence of 𝜎s. Fortunately,  because \nof the simplicity  of the present model , we can directly perform the CPA computation  for finite \n𝒉 . Figure 1 shows  the calculated spin-concentration  curves  along  the hard axis , i.e. , 𝒉=\n(ℎx,0,0), by Eq. (12) for several temperatures . For each temperature,  we can define a  magnetic \nfield ℎA such that 𝜎z(ℎA,0,0)=0. In the range of ℎx<ℎA, the magnetization  rotates in the 𝑧-6 \n 𝑥  plane  as indicated  by |𝝈𝒉|≃const  . On the other hand , in the range of  ℎx≥ℎA , the \nmagnetization is perfectly parallel to  the external magnetic field and  the magnitude slowly  \ngrows  with increasing  ℎx  primarily because of  suppressing the spin fluctuation . Here, we \nshould refer the ℎx dependence  of 𝜎x(ℎx,0,0) at a near zero temperature , 𝑇=0.005 =𝑇1 in \nFig. 1 . Although 𝜎x(ℎx,0,0) appears  to be in a constant  in the high -field range of ℎx>ℎA in the \npresent scale of  the vertical axis in  Fig. 1,  the gradient  — the high -field spin susceptibility  \n𝜒high ≔𝜕𝜎x(ℎx,0,0)/𝜕ℎx  — is no n-zero, whose value is evaluated at 𝜒high ≃0.179  by the \nlinear fitting to 𝜎x(ℎx,0,0)  with respect to ℎx . This finite value  can be  understood as a \ncharacteristic of the itinerant electron system.  More specifically,  ℎx  enhances  the spin \npolarization of the density of states of the electron via the Zeeman energy , and its effect  on \n𝜒high  is represented by  𝜒high ≃4𝜌↑𝜌↓(𝜌↑+𝜌↓) ⁄   for ℎx≪𝛥ex/𝑡  where 𝜌𝜎≔\n−(𝑡𝜋⁄)ℑ [𝐺̂00𝟎(𝜇𝟎+i0)]𝜎𝜎 at 𝑇=0. By the expression, we can obtain 𝜒high ≃0.175 from  \nthe calculated values of  𝜌↑=0.0909  and 𝜌↓=0.0841 , and it is  in good agreement with the \nabove fitting value . Next,  we can find a high linearity  on the 𝜎x𝒉-ℎx curves  in the cases  of 𝑇=\n𝑇1,𝑇2,𝑇3,𝑇4(<𝑇C) and ℎx<ℎA, and can observe that its gradient  at ℎx=0, i.e., 𝜒̃x, does \nnot depend on temperature , in which 𝜒̃x describes the response regarding  the magnetization \nrotatio n. In the para magnetic case (𝑇=𝑇5,𝑇6,𝑇7(≥𝑇C)), the external  magnetic  field, even at \nnear zero strength , acts on the magnetization as a suppresser for the spin fluctuatio n (not as a \ndriving force of the rotation ), and thus 𝜒̃x  directly  reflects  the spin fluctuation , and \nconsequently i ts temperature dependence  is led to  the CW law.  Figure 2 shows the spin \nsusceptibilities  calculated on the basis of  Eqs. (15)–(20), for 𝜆𝑡⁄=1 and 1/√2. As expected  \nfrom the above discussion, 𝜒̃x is a constant  for 𝑇<𝑇C and exhibits the CW law for 𝑇≥𝑇C. \nThe comparison between  the results for  𝜆𝑡⁄=1 and 1/√2 suggests 𝜒̃x−1∝𝜆2, which may \nbe interpret ed as below discussion.  Now , we approximate  ℎA  by the equation , 𝜎s=\n𝜎x(ℎA,0,0)≃𝜒̃xℎA, that is,  \n ℎA=𝜎s\n𝜒̃x, (21) \nand if estimating the MAC  from 𝐾u=𝑡𝜎sℎA(2𝑎3) ⁄   by regarding ℎA  as the magnetic \nanisotropy  field, which correspond s to the area of the triangle formed by 𝑀z𝑯-μ0𝐻x and 𝑀x𝑯-\nμ0𝐻x curves , then  we have  \n 𝐾u=𝑡\n2𝑎3𝜎s2\n𝜒̃x. (22) \nIgnoring the effect of the Rashba SOI on the temperature dependence of t he saturation \nmagnetization , we may estimate  the 𝜆  dependency  of 𝜒̃x−1  from  one of 𝐾u . Because t he \nperturbative treatment for 𝐾u with respect to 𝜆 gives that 𝐾u∝𝜆2 [33,34] , we can accept  the \nrelation 𝜒̃x−1∝𝜆2. Furthermore,  let us  consider  what  Eq. (22) suggests  from the viewpoint of  7 \n the temperature dependence of the MAC.  Although the Akulov –Zener –Callen –Callen law  \npredicts 𝐾u∝𝜎s3 for single -site uniaxial MA systems  [26–31], several previous studies have \nindicated  𝐾u∝𝜎s2  in ferromagnetic metals  [22–25]. Here, r ecalling that t he present model  \ndescribes anisotropic ferromagnetic metals , it is rea sonable that the temperature dependence of \n𝜒̃x−1 becomes  a constant , i.e., 𝐾u∝𝜎s2. In the anisotropic Heisenberg model, the temperature \ndependenc ies of 𝜒̃x−1  and 𝜒̃z−1  have been  demonstrated by several authors  [43–45]. \nEspecially,  Fröbrich and Kuntz  have obtained  a constant  𝜒̃x and a slightly decreasing 𝜒̃x in \ninter-site and single -site anisotrop ic models , respectively, for 𝑇<𝑇C. This suggests  that the \nfinite -temperature magnetism of the present model  involves the situation presented by  the \nHeisenberg model with  the inter-site anisotropy.  Lastly, we  remark the effect of the SOI on 𝜒̃z−1 \nin this model.  The longitudinal susceptibility is a response  for the external magnetic field along \nthe easy axis  yielded by the SOI , thus the Zeeman effect and the SOI  do not compete.  \nConsequently , the SOI  acts as an additional effective magnetic field, only affecting  the Curie \ntemperature  as indicated by the shift of the zero points of 𝜒̃z−1. \n In conclusion,  we have  develope d a computational method for directly calculating the spin \nsusceptibility of itinerant ferromag nets with the uniaxial MA. It is based on the DLM -CPA \ntheory  and includes the finite temperature effect  over the full range.  To demonstrate , we have \nperformed  the computation  for the temperature dependence of the spin susceptibility  in the \nRashba -type ferromagnet , and especially, we have confirmed that the transverse susceptibility \n𝜒̃x has a finite value reflecting the MAC  for 𝑇≤𝑇C and obeys the CW law  for 𝑇>𝑇C. 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In the present work, t his large  SOI has been introduced  \nonly for easily showing  both the CW law and a finite value of χ̃x in the  vertical -axis scale in \nFig. 2 . Furthermore, we notice that , with respect to even such large λ, the purturbative \ntreatment  for the spin susceptibility  remains  valid  as discussed later . \n[43] P. J. Jensen, S. Knappmann, W. Wulfhekel, and H. P. Oepen, Phys. Rev. B 67, 184417 (2003).  \n[44] P. Fröbrich and P. J. Kuntz, Phys. Stat. Sol. 241, 925 (2004).  \n[45] P. Fröbrich and P. J. Kuntz, J. Phys. Condens. Matter 16, 3453 (2004).  \n \n \n 0510152025303540\n0123456Fig. 1 The calculated spin concentrations as functions of external magnetic fields ݄୶\tfor several temperatures. The\nsolid, dashed, and dotted lines are ߪ୶,ߪ,a n d||, respectively. Each number reflects the index of the\ntemperatures given by ܶଵൌ 0.005 ,ܶଶൌ0 . 5 ,ܶଷൌ0 . 8 ,ܶସൌ 0.095 ,ܶହൌ1,ܶൌ1 . 5 ,a n dܶൌ3in units of ܶେ.\nNote that ߪൌ0and||ൌߪ୶\tforܶହ,ܶ,a n dܶ.Spin concentrations×10-3\n×10-31\n1\n22\n3\n3\n44\n5\n6\n7\n୶\n0.00.10.20.30.40.50.60.70.80.91.0\n01234567Inverse spin susceptibilities×10-4୶\n\nFig. 2 The calculated inverse spin susceptibilities as functions of\ntemperature ܶ. The solid and dotted lines correspond to the results\nfor⁄ݐߣൌ1and1/2, respectively."    },    {        "title": "1908.07652v1.Novel_universality_class_for_the_ferromagnetic_transition_in_the_low_carrier_concentration_systems_UTeS_and_USeS_exhibiting_large_negative_magnetoresistance.pdf",        "content": "arXiv:1908.07652v1  [cond-mat.str-el]  21 Aug 2019APS/123-QED\nNovel universality class for the ferromagnetic transition in the low carrier\nconcentration systems UTeS and USeS exhibiting large negat ive magnetoresistance∗\nNaoyuki Tateiwa1,†Yoshinori Haga1, Hironori Sakai1, and Etsuji Yamamoto1\n1Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai,\nNaka, Ibaraki 319-1195, Japan\n(Dated: August 22, 2019)\nWe report the novel critical behavior of magnetization in lo w carrier concentration systems UTeS\nand USeS that exhibit the large negative magnetoresistance around the ferromagnetic transition\ntemperatures TC∼85 and 23 K, respectively. UTeS and USeS crystallize in the sa me orthorhom-\nbic TiNiSi-type crystal structure as those of uranium ferro magnetic superconductors URhGe and\nUCoGe. We determine the critical exponents, βfor the spontaneous magnetization Ms,γfor the\nmagnetic susceptibility χ, andδfor the magnetization isotherm at TCwith several methods. The\nferromagnetic states in UTeS and USeS have strong uniaxial m agnetic anisotropy. However, the\ncritical exponents in the two compounds are different from th ose in the three-dimensional Ising\nmodel with short-range magnetic exchange interactions. Si milar sets of the critical exponents have\nbeen reported for the uranium ferromagnetic superconducto rs UGe 2and URhGe, and uranium\nintermetallic ferromagnets URhSi, UIr and U(Co 0.98Os0.02)Al. The universality class of the ferro-\nmagnetic transitions in UTeS and USeS may belong to the same o ne for the uranium compounds.\nThe novel critical phenomenon associated with the ferromag netic transition is observed not only in\nthe uranium intermetallic ferromagnets with the itinerant 5felectrons but also in the low carrier\nconcentration systems UTeS and USeS with the localized 5 felectrons. The large negative magne-\ntoresistance in UTeS and USeS, and the superconductivity in UGe2and URhGe share the similarity\nof their closeness to the ferromagnetism characterized by t he novel critical exponents.\nPACS numbers:\nI. INTRODUCTION\nMuch interest has been focused on novel physical phe-\nnomena in uranium metallic compounds with 5 felec-\ntrons such as “hidden order”in URu 2Si2, unconventional\nsuperconductivity in UPt 3or UBe 13, and ferromagnetic\nsuperconductivity in UGe 2, URhGe, and UCoGe[1–4].\nMeanwhile, relatively few studies have been conducted\nfor the magnetism and the electrical conductivity in ura-\nnium semimetals or semiconductors. This is in contrast\nwith rare earth magnetic semiconductors, for example,\neuropium chalcogenides EuX (X = O, S, Se, and Te)\nwhere the interplay between 4 fand conduction electrons\nplaysanimportantroleforanomalousphysicalproperties\nsuch as a negative magnetoresistance[5].\nVery recently, the superconductivity has been discov-\nered in uranium compound UTe 2[6]. We have studied\nuranium dichalcogenides UTeS, USeS, and β-US2that\nshow the large magnetoresistanceat low temperatures[7–\n11]. Figure 1 (a) represents the orthorhombic TiNiSi-\ntype crystal structure ( Pnma) of the uranium dichalco-\ngenides. The structure is the same as those of uranium\nferromagnetic superconductors URhGe and UCoGe[4].\nNote that UTe 2crystalizes in a different orthorhombic\nstructure ( Immm)[12]. Figure 1(b) shows the tempera-\n∗Phys. Rev. B 100, 064413 (2019)\n†Electronic address: tateiwa.naoyuki@jaea.go.jpturedependencies ofthe electricalresistivity ρin the ura-\nnium dichalcogenides under magnetic fields of 0 T and 7\nT applied parallel to the crystallographic c-axis[10, 11].\nThe electrical current Jwas applied along the b-axis.\nUTeS is a semimetal with a low carrier density in the\norder of 1025m−3[10]. USeS and β-US2are narrow-gap\nsemiconductors[7,9,11]. UTeSandUSeS showferromag-\nnetic transitionsat TC= 85 and 23 K, respectively[9, 10].\nβ-US2does not order magnetically down to 0.5 K[13].\nFigs. 1(c) and 1(d) show the magnetization at 5.0 K in\nUTeS and at 2.0 K in USeS, respectively. The ferromag-\nnetic states have strong uniaxial magnetic anisotropy.\nThe effect of the magnetic field on the electrical resis-\ntivity is strong in the uranium dichalcogenides as shown\nin Fig. 1(b). In particular, the resistivity values in\nUSeS and β-US2decrease largely with increasing field\nat low temperatures[11]. The magnitudes of the trans-\nverse magnetoresistance are comparable with those in\nperovskite-typemanganeseoxides[14]. Theapplicationof\nthe pressure above 1 GPa induces a ferromagnetic state\ninβ-US2[15]. The large magnetoresistance in the ura-\nnium dichalcogenides can be regarded as a novel phys-\nical phenomenon that appears around a ferromagnetic\nphase boundary. The mechanism of the magnetoresis-\ntance has not been fully understood yet[11]. In this pa-\nper, we report the novel critical behavior of the mag-\nnetization in UTeS and USeS, and its similarity to those\nin the uranium ferromagneticsuperconductors UGe 2and\nURhGe[16].2\nabcUST(a)\nFIG. 1: (a)Representation of the orthorhombic TiNiSi-type\ncrystal structure in UTS (T: Te, Se, S). (b)Temperature de-\npendencies of the electrical resistivity ρunder magnetic fields\nof 0 and 7 T in UTeS[10], USeS, and β-US2[11]. Magnetic\nfield dependencies of the magnetization in magnetic fields ap -\nplied along the a,b, andc-axes (c) at 5.0 K in UTeS[10] and\n(d) at 2.0 K in USeS.\nII. EXPERIMENT\nSingle crystals of UTeS and USeS were grown by\nthe chemical transport using bromine as a trans-\nport agent[10, 11]. Neither impurity phase nor off-\nstoichiometric chemical composition larger than about\n1%wasdetected intheX-raydiffractionandthe Electron\nProbe Micro Analysis (EPMA). Magnetization was mea-\nsuredinacommercialsuperconductingquantuminterfer-\nence device (SQUID) magnetometer (MPMS, Quantum\nDesign). Themagneticfield µ0Hextwasappliedalongthe\nmagnetic easy caxis in the orthorhombic structure. We\ndetermine the internal magnetic field µ0Hby subtract-\ning the demagnetization field DMfromµ0Hext:µ0H=\nµ0Hext-DM. Thedemagnetizationfactors D=0.50and\n0.46 were estimated from the macroscopic dimensions of\nthe single crystals of UTeS and USeS, respectively.\nIII. RESULTS\nIn the asymptotic critical region near TCwhere the\nmean field theory fails, the magnetic correlation length\nξ=ξ0|(T−TC)/TC|−νdiverges. Here, νis the critical\nexponent. The spontaneous magnetization Ms, the ini-\ntial susceptibility χ, and the magnetization at TCfollow\nuniversal scaling laws[17].\nMs(T)∝ |t|β(T < T C) (1)\nχ(T)−1∝ |t|γ′\n(T < T C),|t|γ(TC< T) (2)\nM(µ0H)∝(µ0H)1/δ(T=TC) (3)/s49/s46/s53/s120/s49/s48/s49/s54\n/s48/s46/s48 /s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s56 /s52 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s51/s68/s32/s73/s115/s105/s110/s103/s40/s100/s41/s32/s32/s85/s83/s101/s83\n/s84/s32/s61/s32/s50/s49/s46/s52/s32/s75/s32\n/s50/s52/s46/s54/s32/s75/s32/s68/s84/s32/s32/s61\n/s32/s48/s46/s56/s32/s75/s50/s120/s49/s48/s49/s48\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s52/s48 /s50/s48 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s40/s97/s41/s32/s32/s85/s84/s101/s83\n/s77/s70/s84/s32/s61/s32/s56/s49/s46/s48/s32/s75/s32\n/s56/s57/s46/s48/s32/s75/s32/s68/s84/s32/s32/s61/s32\n/s32/s32/s49/s46/s48/s32/s75\n/s49/s120/s49/s48/s49/s54\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s50/s48 /s49/s48 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s40/s99/s41/s32/s32/s85/s84/s101/s83\n/s32/s32/s51/s68/s32/s73/s115/s105/s110/s103/s84/s32/s61/s32/s56/s49/s46/s48/s32/s75/s32\n/s56/s57/s46/s48/s32/s75/s32/s68/s84/s32/s32/s61/s32\n/s32/s32/s49/s46/s48/s32/s75/s51/s120/s49/s48/s49/s48\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s56 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s77/s70/s40/s98/s41/s32/s32/s32/s85/s83/s101/s83\n/s84/s32/s61/s32/s50/s49/s46/s52/s32/s75/s32\n/s50/s52/s46/s54/s32/s75/s32/s68/s84/s32/s32/s61\n/s32/s48/s46/s56/s32/s75\nFIG. 2: Magnetic isotherms in the form of M1/βvs.\n(H/M)1/γwith the mean field theory ( β= 0.5 and γ= 1.0)\n(a) for 81.0 K ≤T≤89.0 K in UTeS and (b) for 21.4 K ≤\nT≤24.6 K in USeS. Isotherms with the short-range (SR) 3D\nIsing model ( β= 0.325 and γ= 1.241) in (c) UTeS and in (d)\nUSeS.\nHere,tis the reduced temperature t= (T−TC)/TC.β,\nγ,γ′andδare the critical exponents.\nFigures 2(a) and 2(b) show the magnetic isotherms in\nthe form of M1/βversus (H/M)1/γwith the mean field\n(MF) theory ( β= 0.5 and γ= 1.0) for 81.0K ≤T≤89.0\nK in UTeS and for 21.4 K ≤T≤24.6 K in USeS, respec-\ntively. The data do not form straightlines. This suggests\nthat the mean field theory is not suitable to describe the\nmagnetization around TC. Figs. 2(c) and 2(d) represent\nthe isotherms in the form of M1/βvs. (H/M)1/γwith\nthe 3D Ising model with short-range (SR) exchange in-\nteractions( β= 0.325and γ= 1.241)for UTeS and USeS,\nrespectively. The isotherms are curved, suggesting that\nthe 3D Ising model is also not appropriate.\nWeanalyzedthedatawiththefollowingArrott-Noakes\nequation of state[18]:\n(H/M)1/γ= (T−TC)/T1+(M/M1)1/β(4)\n, whereT1andM1are material constants.\nFigures 3(a) and 3(b) show the modified Arrott plot\n(MAP) in the form of M1/βvs. (H/M)1/γfor UTeS and\nUSeS, respectively. The isotherms become straight if the\nappropriate values of TC,β, andγare chosen. We deter-\nmine these parameters from fits of Eq. (4) to the data\nfor 81.0 K ≤T≤89.0 K and 1.2 T ≤µ0H≤7.0 T in\nUTeS, and those for 21.4 K ≤T≤24.6 K and 0.4 T ≤\nµ0H≤3.0 T in USeS. The values of TC,β, andγare3\n/s51/s120/s49/s48/s49/s55\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s49/s53 /s49/s48 /s53 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s40/s98/s41/s32/s32/s85/s83/s101/s83\n/s84/s32/s61/s32/s50/s49/s46/s52/s32/s75/s32\n/s50/s52/s46/s54/s32/s75/s32/s68/s84/s32/s32\n/s61/s32/s48/s46/s56/s32/s75/s54/s120/s49/s48/s49/s54\n/s48/s77/s49/s47/s98/s32/s91/s65/s47/s109/s93/s49/s47/s98\n/s52/s48 /s50/s48 /s48\n/s40/s72/s47/s77/s41/s49/s47/s103/s40/s97/s41/s32/s32/s85/s84/s101/s83\n/s84/s32/s61/s32/s56/s49/s46/s48/s32/s75/s32\n/s56/s57/s46/s48/s32/s75/s32/s68/s84/s32/s32/s61/s32\n/s32/s32/s49/s46/s48/s32/s75\n/s49/s48/s53/s77/s32/s91/s65/s47/s109/s93\n/s48/s46/s49 /s49\n/s109/s48/s72/s32/s91/s84/s93/s84/s32/s61/s32/s56/s49/s46/s48/s32/s75/s32\n/s68/s84/s32/s32\n/s61/s32/s48/s46/s53/s32/s75\n/s56/s57/s46/s48/s32/s75/s40/s99/s41/s32/s85/s84/s101/s83\n/s56/s57/s49/s48/s53/s50/s77/s32/s91/s65/s47/s109/s93\n/s48/s46/s49 /s49\n/s109/s48/s72/s32/s91/s84/s93/s84/s32/s61/s32/s50/s49/s46/s52/s32/s75/s32\n/s68/s84/s32/s32\n/s61/s32/s48/s46/s50/s32/s75\n/s50/s52/s46/s54/s32/s75/s40/s100/s41/s32/s85/s83/s101/s83\nFIG. 3: Modified Arrott plot (MAP) of magnetic isotherms\n(a) for 81.0 K ≤T≤89.0 K in UTeS and (b) for 21.4 K\n≤T≤24.6 K in USeS. Data represented as close circles in\n(a) and (b) are analyzed with the Arrott-Noakes equation\nof state [Eq. (4)]. Solid lines show fits to the data in the\nhigher magnetic field region with a linear function. Magneti c\nfield dependencies of the magnetization in (c) UTeS and in\n(d) USeS. Bold circles indicate the critical isotherms at 85 .0\nK and 23.2 K for UTeS and USeS, respectively. Solid lines\nrepresent fits to the data represented as closed bold circles\nwith Eq. (3).\ndetermined as TC= 84.88±0.05 K,β= 0.309±0.003,\nandγ= 0.998±0.003 for UTeS, and TC= 23.09±0.03\nK,β= 0.300±0.003, and γ= 1.00±0.02for USeS. The\nparameters are shown in Table I. The data used for the\nanalyses are represented as closed circles in Figs. 3(a)\nand 3(b). The data points in the MAPs generally form\nstraight lines but those in the low magnetic field region\ndeviate from the lines. There are several reasons for the\ndeviation such as the movement of domain walls or sam-\nple inhomogeneities. In addition, there might be an error\nin the calculated value of the demagnetization factor D.\nThe origins of the deviation have been discussed in Ref.\n19, although it has not been completely understood yet.\nSolidlinesinFigs. 3(a)and3(b)representfitstothedata\nin the high magnetic field region with a linear function\nin order to obtain the spontaneous magnetic moment Ms\nand the magnetic susceptibility χ. The temperature de-\npendencies of the quantities will be used in the analysis\nwith the Kouvel-Fisher method.\nWe determine the critical exponent δfrom the critical\nisotherm at TC. The value of δis determined as δ= 4.21/s49/s48\n/s48 /s99/s32/s45/s49/s49/s120/s49/s48/s53\n/s48/s77/s115/s32/s91/s65/s47/s109/s93/s84/s67/s40/s97/s41/s32/s85/s84/s101/s83\n/s45/s49/s52\n/s48/s77/s115/s32/s40/s100/s77/s115/s47/s100/s84/s41/s45/s49/s32/s91/s75/s93\n/s56/s56 /s56/s52\n/s84/s32/s91/s75/s93/s52\n/s48/s99/s45/s49/s32/s40/s100/s99/s45/s49/s47/s100/s84/s41/s45/s49/s32/s91/s75/s93/s84/s67/s32\n/s85/s84/s101/s83/s45/s54\n/s48/s77/s115/s32/s40/s100/s77/s115/s47/s100/s84/s41/s45/s49/s32/s91/s75/s93\n/s50/s52 /s50/s50\n/s84/s32/s91/s75/s93/s50\n/s48/s99/s45/s49/s32/s40/s100/s99/s45/s49/s47/s100/s84/s41/s45/s49/s32/s91/s75/s93/s84/s67\n/s85/s83/s101/s83/s50\n/s48/s99/s32/s45/s49/s49/s120/s49/s48/s53\n/s48/s77/s115/s32/s91/s65/s47/s109/s93/s84/s67/s40/s98/s41/s32/s85/s83/s101/s83\nFIG. 4: Temperature dependencies of Ms(T) andχ−1de-\ntermined from the MAPs (upper panels) and Kouvel-Fisher\nplots (lower panels) in (a) UTeS and in (b) USeS.\n±0.04for UTeS and 4.34 ±0.04for USeS from fits to the\nisothermsat85.0KforUTeSandat23.2KforUSeS with\nEq. (3)asshowninFigs. 4(c)and4(d), respectively. The\ndata shown as closed circles are analyzed. The exponents\nβ,γ, andδshould be related by the Widom scaling law\nδ= 1+γ/β[20]. The value of δis estimated as 4.23 ±\n0.06 for UTeS and 4.33 ±0.10 for USeS using the βand\nγvalues in the MAPs. These values are consistent with\nthose determined from the critical isotherms.\nNext, we analyze the data with the Kouvel-Fisher\nmethod where the critical exponents can be determined\nmore accurately[21]. The solid lines in Figs 3(a) and\n3(b) intersect with the vertical axis at M1/β=M1/β\nsfor\nT < T Cand with the transverse axis at ( H/M)1/γ=\n(1/χ)1/γforTC< T. The values of Ms(T) andχ(T) can\nbe obtained by inserting the βandγvalues. Figures 4(a)\nand 4(b) show the temperature dependencies of Ms(T)\nandχ−1(T) for UTeS and USeS, respectively. Solid lines\nrepresent fits to the data with Eqs. (1) and (2). We de-\ntermine the critical exponents βandγwith the Kouvel-\nFisher (KF) method where temperature-dependent ex-\nponentsβ(T) andγ(T) are introduced as follows[21]:\nMs(T)[dMs(T)/dT]−1= (T−TC−)/β(T) (5)\nχ−1(T)[dχ−1(T)/dT]−1= (T−TC+)/γ(T) (6)\nThe quantities β(T) andγ(T) are equal to the critical\nexponents βandγ, respectively, in the limits T→TC\nandH→0. The slopes of Ms(T)[dMs(T)/dT]−1and4\n/s55/s120/s49/s48/s53\n/s48/s77/s47/s124/s116/s124/s32/s98/s32/s91/s65/s47/s109/s93\n/s52/s48/s48/s48 /s50/s48/s48/s48 /s48\n/s40/s109/s48/s72/s41/s47/s124/s116/s124/s40/s32/s98/s43/s103/s41/s32/s91/s84/s93/s84/s32/s60/s32/s84/s67/s84/s67/s32/s60/s32/s84/s85/s84/s101/s83 /s40/s97/s41\n/s54/s120/s49/s48/s53\n/s48/s77/s47/s124/s116/s124/s32/s98/s32/s91/s65/s47/s109/s93\n/s49/s48/s48/s48 /s48\n/s40/s109/s48/s72/s41/s47/s124/s116/s124/s40/s32/s98/s43/s103/s41/s32/s91/s84/s93/s84/s32/s60/s32/s84/s67/s84/s67/s32/s60/s32/s84/s85/s83/s101/s83/s40/s98/s41\n/s49/s48/s53/s50/s52/s54/s56/s77/s47/s124/s116/s124/s32/s98/s32/s91/s65/s47/s109/s93\n/s49/s48 /s32/s49/s48/s48/s48\n/s40/s109/s48/s72/s41/s47/s124/s116/s124/s40/s32/s98/s43/s103/s41/s32/s91/s84/s93/s84/s32/s60/s32/s84/s67\n/s84/s67/s32/s60/s32/s84\n/s50/s120/s49/s48/s53/s51/s52/s53/s54/s55/s77/s47/s124/s116/s124/s32/s98/s32/s91/s65/s47/s109/s93\n/s49/s48/s48 /s49/s48/s48/s48\n/s40/s109/s48/s72/s41/s47/s124/s116/s124/s40/s32/s98/s43/s103/s41/s32/s91/s84/s93/s84/s32/s60/s32/s84/s67\n/s84/s67/s32/s60/s32/s84\nFIG. 5: Renormalized magnetization mas afunction ofrenor-\nmalized field hfollowing Eq. (7) below and above TCfor (a)\nUTeS and for (b) USeS. Solid lines represent best-fit polyno-\nmials.\nχ−1(T)[dχ−1(T)/dT]−1-Tplots at TCyield the βand\nγvalues, respectively. The fits to the data with Eqs.\n(5) and (6) are shown as solid lines in the low panels of\nFigs. 4(e) and 4(f), respectively. The parameters are\ndetermined as β= 0.315±0.003 and γ= 0.996±0.003,\nandTC= 85.09 ±0.04 K for UTeS, and β= 0.293 ±\n0.003,γ= 0.989 ±0.003, and TC= 23.18 ±0.03 K for\nUSeS. Here, TCis defined as TC= (TC++TC−)/2.\nIt may be possible to speculate that the 3D Ising uni-\nversality class below TCis changed to the mean field one\naboveTCin UTeS and USeS. Scaling theory enables us\nto determine separately the values of γ’ forT < T Cand\nγforTC< T. A reduced equation of state close to TC\nwas predicted in the scaling theory as follows[17]:\nm=f±(h) (7)\nHere,f+andf−are regular analytical functions for\nTC< TandT < T C, respectively. The renormalized\nmagnetization mis defined as m≡ |t|−βM(µ0H,t) and\nthe renormalized field hash≡µ0H|t|−(β+γ). Two uni-\nversal curves are formed in the plot of mvs.hwhen the\ncorrect values of β,γ’,γ, andTCare chosen. The data\nin the temperature ranges t=|(T−TC)/TC|<0.05\nfor UTeS and 0.08 for USeS are shown in Figs. 5(a) and\n5(b), respectively. The analyses yield the values of TC\nand the critical exponents as TC= 85.09±0.03 K,β=\n0.318±0.002,γ′= 1.03±0.02, and γ= 1.04±0.02 in\nUTeS, and TC= 23.18 ±0.02 K,β= 0.300 ±0.002,γ′\n= 1.00±0.02, and γ= 1.02±0.02 in USeS. This result\nsuggests that the sets of the critical exponents in the two\ncompounds are common below and above TC. Note that\nthe magnetic isotherms in the forms of the mean field\ntheory and the 3D Ising model with SR exchange inter-\nactions do not form straight lines below and above TCas\nshown in Figs. 2(a)-2(d). The strongly asymmetric crit-\nical region or the change of the universality class across\nTCcan be ruled out.IV. DISCUSSION\nTable I shows the critical exponents β,γ′,γ, andδin\nUTeS and USeS, and those in mean field theory and var-\nious theoretical models with SR exchange interactions\nof a form J(r)∼e−r/b[17, 22, 23]. The exponents in\nthe uranium ferromagnetic superconductors UGe 2and\nURhGe[16], and some uranium ferromagnets URhSi[24],\nUIr[25,26], andU(Co 0.98Os0.02)Al[27,28]arealsoshown.\nThe sets of the exponents in UTeS and USeS are similar\nto those of the uranium ferromagnets. The ferromag-\nnetic states of these ferromagnets have strong uniaxial\nanisotropy. However, the critical exponents of the com-\npounds differ from those in the 3D Ising model with SR\nexchange interactions. The βvalues are relatively close\nto those of the 3D models. Meanwhile, the values of γ\nare close to unity, expected one in the mean field theory.\nWe discuss the mean-field behavior of the magne-\ntization in the uranium ferromagnetic superconductor\nUCoGe[29]. The extent of the asymptotic critical region\n∆TGwhere the mean field theory fails can be estimated\nby the Ginzburg criterion[30]. ∆ TGin three dimensions\nis given as ∆ TG/TC=k2\nB/[32π2(∆C)2ξ06][31, 32]. Here,\n∆Cis the jump of the specific heat at TCin units of\nerg·cm−3K−1andξ0is the bare correlation length. The\nvalue of ∆ TGfor UCoGe was estimated as less than 1\nmK using reported ∆ Candξ0values[16, 29, 33]. It is\nnatural that the mean field behavior of the magnetiza-\ntion is observed because most of magnetic data might be\ntaken outside the very narrow region around TC. The\nlonger magnetic correlation length may be originated\nfrom the strong itinerant character of the 5 felectrons\nin UCoGe[24]. We have previously reported the critical\nexponents in UGe 2[16]. The value of ∆ TGwas estimated\nas∼100K.Itcanbeconcludedthatthedatausedforthe\ndetermination of the critical exponents were taken inside\nthe asymptotic critical region in UGe 2. Meanwhile, it is\nimpossible to estimate ∆ TGfor UTeS and USeS since the\nmagnetic correlation length ξhas not been reported so\nfar. In this study, the data for 81.0 K ≤T≤89.0 K and\n1.2 T≤µ0H≤7.0 T in UTeS, and those for 21.4 K ≤T\n≤24.6 K and 0.4 T ≤µ0H≤3.0 T in USeS are analyzed\nto determine the critical exponents. If the data up to 7.0\nT in USeS are analyzed, they do not form straight lines\nin the MAP and nor do universal curves in the scaling\nanalysis for any values of the critical exponents. We re-\npeated the analysis and found that the upper limit of the\ncritical region is about 3.0 T for USeS. The consistency\nin the obtained exponents determined by different meth-\nods suggests the reliability of them. We conclude that\nthe data used for the analyses were collected inside the\nasymptotic critical regions of each compound.\nFigures 6(a) and 6(b) show the normalized sponta-\nneous magnetic moment Ms/M0and the magnetic sus-\nceptibility χ/Cas a function of the reduced tempera-\nture|t|(=|(T−TC)/TC|) determined from the MAPs\nin UTeS, USeS, UGe 2[16], URhGe[16], and URhSi[24].\nHere, constants M0andCare obtained from fits to5\nTABLE I: Critical exponents β,γ,γ′, andδof UTeS and USeS, and those in the mean field theory and various theoretical\nmodels with short-range (SR) exchange interactions[17, 22 , 23]. The exponents in UGe 2[16], URhGe[16], URhSi[24], UIr[25],\nand U(Co 0.98Os0.02)Al[27] are also shown. Abbreviations: RG- φ4, renormalization group φ4field theory; MAP, Modified Arrott\nplot; CI, Critical isotherm; KF, Kouvel-Fisher method.\nMethod TC(K) β γ′(T < T C)γ(TC< T) δ Reference\nMean field 0.5 1.0 3.0\nd= 2,n=1 Onsager solution 0.125 1.75 15.0 [17, 22]\nd= 3,n=1 RG- φ40.325 1.241 4.82 [17, 23]\nd= 3,n=2 RG- φ40.346 1.316 4.81 [17, 23]\nd= 3,n=3 RG- φ40.365 1.386 4.80 [17, 23]\nUTeS MAP, CI 84.88 ±0.05 0.309 ±0.003 0.998 ±0.003 4.21 ±0.04 this work\nKF 85.09 ±0.04 0.315 ±0.003 0.996 ±0.003\nScaling 85.09 ±0.03 0.318 ±0.002 1.03 ±0.02 1.04 ±0.02\nUSeS MAP, CI 23.09 ±0.03 0.300 ±0.003 1.00 ±0.02 4.34 ±0.04 this work\nKF 23.18 ±0.03 0.293 ±0.003 0.989 ±0.003\nScaling 23.18 ±0.02 0.300 ±0.002 1.00 ±0.02 1.02 ±0.02\nUGe2 Scaling, CI 52.79 ±0.02 0.329 ±0.002 1.00 ±0.02 1.02 ±0.02 4.16 ±0.02 [16]\nURhGe Scaling, CI 9.47 ±0.01 0.302 ±0.001 1.00 ±0.01 1.02 ±0.01 4.41 ±0.02 [16]\nURhSi Scaling, CI 10.12 ±0.02 0.300 ±0.002 1.00 ±0.02 1.03 ±0.02 4.38 ±0.04 [24]\nUIr MAP, CI 45.15 ±0.2 0.355 ±0.05 1.07 ±0.1 4.04 ±0.05 [25]\nU(Co0.98Os0.02)Al MAP, CI 25 0.33 1.0 4.18 [27]\nthe|t|-dependencies of Msandχwith formulas Ms(t)\n=M0|t|βandχ(t) =C/|t|γ, respectively. The data\nfor the latter three compounds are from our previous\nstudies[16, 24]. Although the spontaneous magnetiza-\ntionMsshows the critical behavior expected for the 3D\nmagnets, the magnetic susceptibility χdoes the mean-\nfield-like behavior. Many theoretical studies have been\ndone for critical phenomena of ferromagnetic transitions\nin ferromagnetic materials. However, this unusual be-\nhavior cannot be explained with previous theoretical ap-\nproaches. We discuss this issue from following six view-\npoints.\n(1) It is well known that the long-range nature of mag-\nnetic exchange interactions affects the critical behavior\nof the magnetization around TC. The theoretical values\nof the critical exponents in Table I are those of theoret-\nical models with short-range (SR) exchange interactions\nofa form J(r)∼e−r/b[17, 22, 23]. The strengthofthe ex-\nchange interaction J(r) decreases rapidly with increasing\ndistance r. When the range of the exchange interaction\nbecomes longer, the critical exponents ofthe models shift\ntoward those of the mean field theory. This problem was\nstudied by Fischer et al.with a renormalization group\napproach for systems with the exchange interaction of a\nformJ(r)∼1/rd+σ[34]. Here, σis the range of exchange\ninteraction and dis the dimension of the system. They\nshowed that the model is valid for σ <2 and derived\na theoretical formula for the exponent γ= Γ{σ,d,n}.\nHere,nis the dimension of the order parameter and the\nfunction Γ is given in Ref. 34. We calculate the criti-\ncal exponents using the formula and scaling relations for\ndifferent sets of {d:n}(d,n= 1, 2, 3) in order to\nreproduce the exponents in the uranium ferromagnets.\nHowever, there is no reasonable solution of σ.\n(2) The critical phenomenon of the magnetization isalso affected by the classical dipole-dipole interaction as\nhas been studied for rare earth metal gadolinium ( TC=\n292.7K and the spontaneousmagnetic moment ps= 7.12\nµB/Gd)[35]. Theinteractionmaynothaveastrongeffect\non critical phenomena in uranium ferromagnets since the\nstrength of the effect is proportional to p2\ns[36]. The value\nofpsis determined as 1.62 µB/U at 5.0 K for UTeS and\n1.09µB/U at 2.0 K for USeS. These are much smaller\nthanthatofGd. Moreover,the exponentsintheuranium\nferromagnets are not consistent with those of theoretical\nstudies for the critical phenomenon associated with the\nisotropic or anisotropic dipole-dipole interaction[37, 38].\n(3) We discuss the critical exponents from the view-\npointofthelocalmomentmagnetism. Theferromagnetic\nstates in the uranium ferromagnets can be regarded as\nthe anisotropic 3D Ising system or the anisotropic next-\nnearest-neighhor 3D Ising (ANNNI) system. However,\nthe critical exponents in the uranium ferromagnets are\nnot consistent with those obtained in numerical calcula-\ntions for the two systems[39, 40].\n(4) The temperature dependencies of the spontaneous\nmagnetic moment psand the magnetic susceptibility χ\nobtained analytically or numerically in the spin fluctua-\ntion theoriesarenot consistentwith thosein the uranium\nferromagnets[41, 42]. The spin fluctuation theories can-\nnot be applied to physical phenomena in the asymptotic\ncritical region.\n(5) The unconventional critical phenomenon in UGe 2\nand URhGe has been discussed by Singh, Dutta, and\nNandy with a nonlocal Ginzburg-Landau model focus-\ning on magnetoelastic interactions[43]. It was claimed\nthat their calculated results are comparable with those\nof UGe 2and URhGe. It is hoped that the almost mean-\nfield behavior of χis completely reproduced.\nThe itinerant picture of the 5 felectrons is basically6\n/s49/s48/s49/s48/s48/s99/s47/s67\n/s48/s46/s48/s49 /s48/s46/s49\n/s124/s116/s124/s32/s40/s61/s32/s124/s40/s84/s45/s84/s67/s41/s47/s84/s67/s124/s41/s51/s68/s32/s72/s101/s105/s115/s101/s110/s98/s101/s114/s103\n/s32/s32/s32/s51/s68/s32/s88/s89\n/s32/s32/s32/s32/s32/s51/s68/s32/s73/s115/s105/s110/s103\n/s77/s70/s32/s40/s98/s41/s48/s46/s48/s49/s48/s46/s49/s49/s77/s115/s47/s77/s48\n/s48/s46/s49 /s48/s46/s48\n/s124/s116/s124/s32/s40/s61/s32/s124/s40/s84/s45/s84/s67/s41/s47/s84/s67/s124/s41/s32/s50/s68/s32/s73/s115/s105/s110/s103\n/s51/s68/s32/s73/s115/s105/s110/s103\n/s51/s68/s32/s88/s89\n/s51/s68/s32/s72/s101/s105/s115/s101/s110/s98/s101/s114/s103\n/s77/s70\n/s32/s85/s71/s101/s50\n/s32/s85/s82/s104/s71/s101\n/s32/s85/s82/s104/s83/s105/s32\n/s32/s85/s84/s101/s83\n/s32/s85/s83/s101/s83/s32/s40/s97/s41\nFIG. 6: (a) Normalized spontaneous magnetic moment\nMs/M0and (b) magnetic susceptibility χ/Cas a function of\nreduced temperature |t|(=|(T−TC)/TC|) for UTeS, USeS,\nUGe2[16], URhGe[16], and URhSi[24]. Dotted lines indicate\ncalculated curves for various theoretical models.\nappropriate to describe the ferromagnetism in uranium\nintermetallic compounds[44]. Meanwhile, the dual na-\nture of the 5 felectrons in UGe 2has been experimentally\nsuggestedin the Muonspin rotationspectroscopy[45,46].\nThe concept of the duality of the 5 felectrons has been\na basis in theoretical studies for the superconductivity in\nUGe2[47], URhGe[48], and UPd 2Al3[49]. Previously, we\npointed out relevance between the dual nature of the 5 f\nelectrons and the novel critical behavior of the magne-\ntization in UGe 2, URhGe, and URhSi[16, 24]. A novel\ncritical phenomenon can be expected due to two corre-\nlation lengths of the localized and itinerant components\nof the 5felectrons and a Hund-type coupling between\nthem. However, this scenario cannot be applied to UTeS\nand USeS with the localized 5 felectrons. The soft X-\nray photoelectron spectroscopy showed that the 5 flevelis situated about 750 meV below the Fermi energy in\nUTeS[50]. Thepresentstudyshowsthatthenovelcritical\nphenomenon of the ferromagnetic transition is observed\nnot only in the uranium intermetallic compounds where\nthe ferromagnetism is carried by the itinerant 5 felec-\ntrons but also in UTeS and USeS with the localized 5 f\nelectrons. The large negative magnetoresistance in UTeS\nand USeS, and the uranium ferromagnetic superconduc-\ntivity in UGe 2and URhGe are observed in the vicinity\nof the ferromagnetism characterized by the novel critical\nexponents.\nIt has been long thought that the p-wave supercon-\nductivity in the uranium ferromagnetic superconductors\nis driven by longitudinal spin fluctuations developed in\nthe vicinity of the ferromagnetic state described with the\n3D Ising model. Meanwhile, our studies suggest that\nthe ferromagnetic correlation between the 5 felectrons\ndiffers from that of the 3D Ising system in the ura-\nnium ferromagnets including the superconductors. Re-\ncent uniaxial experiment on URhGe and its theoretical\ninterpretation suggest that a pairing mechanism other\nthan that driven by Ising-type longitudinal fluctuations\ntake a certain role for the superconductivity[51, 52].\nIt would be interesting to study the dynamical mag-\nnetic property of the uranium dichalcogenides. It was\nclaimed that the superconductor UTe 2is on the verge of\nthe ferromagnetism since the critical exponents are close\nto values expected for a ferromagnetic quantum critical\npoint[6,53]. Magneticfluctuationsobservedinmuonspin\nrelaxation/rotation ( µSR) measurements on UTe 2may\ntake an important role for anomalous behaviors of the\nupper critical field Hc2or the unconventional supercon-\nductingorderparameterwithpointnodessuggestedfrom\nthe thermal transport, heat capacity and magnetic pen-\netration depth measurements[54–56]. The group of the\nuranium dichalcogenides would be an interesting plat-\nform for the study of both the large magnetoresistance\nand the superconductivity in terms of the ferromagnetic\ncorrelation between the 5 felectrons.\nV. SUMMARY\nIn summary, we study the novel critical behavior of\nmagnetization in uranium semimetal UTeS and semicon-\nductor USeS exhibiting a large transverse magnetoresis-\ntance around the ferromagnetic transition temperatures.\nThe critical exponents in the two compounds differ from\nthoseinthe3DIsingmodelwithshort-rangeexchangein-\nteractions in spite of uniaxial magnetic anisotropy in the\nferromagnetic states. The critical exponents are similar\ntothoseinuraniumferromagneticsuperconductorsUGe 2\nandURhGe, andsomeuraniumferromagnetsURhSi, UIr\nand U(Co 0.98Os0.02)Al. The universalityclassforthe fer-\nromagnetic transition in UTeS and USeS may belong to\nthe same one for the uranium ferromagnets. The novel\ncritical phenomenon of the ferromagnetic transition ap-\npears not only in the uranium intermetallic ferromag-7\nnets with the itinerant 5 felectrons but also in UTeS\nand USeS with the localized 5 felectrons. There is simi-\nlarity between the large magnetoresistance in UTeS and\nUSeS, and the superconductivity in UGe 2and URhGe\nof their closeness to the ferromagnetism characterized by\nthe novel critical exponents.VI. ACKNOWLEDGMENTS\nThis work was supported by Japan Society for the\nPromotion of Science (JSPS) KAKENHI Grant No.\nJP16K05463, JP16KK0106, and JP17K05522.\n[1] J. A. Mydosh and P. M. Oppeneer, Rev. Mod. Phys. 83,\n1301 (2011).\n[2] C. Pfleiderer, Rev. Mod. 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Rev. B 96, 104501 (2017).8\n[53] T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 91, 214407\n(2015).\n[54] S. Sundar, S. Gheidi, K. Akintola, A. M. Cˆ ot´ e, S. R.\nDunsiger, S. Ran, N. P. Butch, S. R. Saha, J. Paglione,\nand J. E. Sonier, arXiv:1905.06901\n[55] S. Ran, I-L. Liu, Y. S. Eo, D. J. Campbell, P. Neves, W.T. Fuhrman,S.R.Saha, C.Eckberg, H.Kim, J.Paglione,\nD. Graf, J. Singleton, and N.P. Butch, arXiv:1905.04343.\n[56] T. Metz, S. Bae, S. Ran, I-L. Liu, Y. S. Eo, W. T.\nFuhrman, D. F. Agterberg, S. Anlage, N. P. Butch, and\nJ. Paglione, arXiv:1908.01069."    },    {        "title": "1807.10387v1.Phase_separation_and_proximity_effects_in_itinerant_ferromagnet_superconductor_heterostructures.pdf",        "content": "Phase separation and proximity e\u000bects in itinerant ferromagnet - superconductor\nheterostructures\nC. Martens,1A. Bill,2,\u0003and G. Seibold1,y\n1Institut F ur Physik, BTU Cottbus{Senftenberg, PBox 101344, 03013 Cottbus, Germany\n2Department of Physics & Astronomy, California State University Long Beach, Long Beach, CA 90840, USA\n(Dated: November 28, 2021)\nHeterostructures made of itinerant ferromagnets and superconductors are studied. In contrast\nto most previous models, ferromagnetism is not enforced as an external Zeeman \feld but induced\nin a correlated single-band model (CSBM) that displays itinerant ferromagnetism as a mean-\feld\nground state. This allows us to investigate the in\ruence of an adjacent superconducting layer on\nthe properties of the ferromagnet in a self-consistent Bogoliubov-de Gennes approach. The CSBM\ndisplays a variety features not present in the Zeeman exchange model that in\ruence the behavior\nof order parameters close to the interface, as e.g. phase separation and the competition between\nmagnetism and superconducting orders.\nPACS numbers: 74.45+c, 75.70.Cn, 74.78.Fk, 75.30.Cr\nI. INTRODUCTION\nSuperconductivity and ferromagnetism were long be-\nlieved to be mutually exclusive phases because in a\nconventional superconductor the spin part of the wave-\nfunction is a singlet which is easily broken by the strong\nmagnetization of the ferromagnet. However, there are sit-\nuations where the coexistence is possible as in an itiner-\nant ferromagnet where the spin-up and spin-down bands\nare split by an e\u000bective exchange \feld. Following the idea\nof Fulde, Ferrell, Larkin and Ovchinnikov1one can con-\nstruct a Cooper pair which is composed of two electrons\nwith opposite spins and momenta,but acquires a \fnite\ntotal momentum due to the exchange \feld splitting and\nresulting di\u000berent Fermi momenta. Another possibility\nis the formation of 'spin triplet' Cooper pairs where the\nassociated orbital part is antisymmetric either in the ex-\nchange of the electrons position or the time coordinates,\nthe latter being called 'odd frequency pairing'.2\nAn ideal playground to study these possibilities and\nmany related questions are heterostructures composed\nof superconducting and magnetic layers. Besides their\nrelevance for studying fundamental properties of the in-\nterplay between superconductivity and magnetism such\nsystems have also a strong relevance for applications in\nspintronics due for example to the possibility of acting as\nspin-polarized current sources.3\nThe attractive feature of nanohybrid structures is that\nthe two phases are spatially separated and interact solely\nvia the proximity e\u000bect. In a S/F/S junction (S being\na superconductor, F a ferromagnet) Cooper pairs tun-\nnel through the S/F interfaces and, for a thin enough\nferromagnetic layer, may realize a Josephson junction.\nFirst studies on such systems where conducted by\nBuzdin and collaborators4who predicted for example the\nexistence of \u0019-junctions. Here the oscillatory behavior\nof the superconducting state wave function leaking into\nthe ferromagnet may lead to a reversal of the Joseph-\nson current (the \u0019\u0000state) with respect to the ordinaryJosephson e\u000bect (correspondingly called the 0 \u0000state) if\nthe thickness of the ferromagnet is chosen such that there\nis a sign change of the wave function on either side of the\njunction.\nIn a series of papers Bergeret, Efetov and Volkov,5\npredicted that under certain conditions, such as for ex-\nample the presence of inhomogeneities in the magnetiza-\ntion, a triplet component with non-zero spin projection\nalong the quantization axis, m=\u00061 of the pair amplitude\nmight arise, even though the Cooper pairs generated in\nthe superconducting parts of the junction are singlets.\nA similar conclusion was reached in Ref. 6. Singlet and\nspin-zero triplet projection pairs ( m= 0) entering a fer-\nromagnet are subject to the pair breaking e\u000bect and are\nthus short-range components decaying exponentially over\na few nanometers, except at every change of direction of\nthe magnetization where they are regenerated through\nthe cascade e\u000bect.7,8In contrast, non-zero spin projection\ntriplet components are comparatively long range because\nuna\u000bected by magnetism and have in fact been observed\nin a number of experiments (see e.g. Refs. 9{16).\nThe creation of such long-ranged triplet components\nout of a singlet superconductor requires a rotation of the\nmagnetization which can be either realized via magnetic\nmultilayers,11,13conical magnets as Ho10,11or Heusler\nalloys.17On the theoretical side the conditions for which\na long range triplet component of the order parameter\ncan be observed in superconductor { magnetic nanojunc-\ntions has been worked out in some detail in the di\u000busive\nregime (i.e. on the basis of Eilenberger and/or Usadel\nequations; see e.g. Refs.5, 7, 8, 18{22 and references\ntherein) and in the clean limit within the Bogoliubov-de\nGennes (BdG) approach (see e.g.Refs. 23{34).\nMore recently other sources for the generation of a\nlong-ranged triplet component have been analyzed, in\nparticular it has been shown that the physical mecha-\nnism of the singlet- triplet conversion can be linked to\nthe local SU(2) invariance of magnetized systems with\nspin-orbit interaction.35\nIn most of the studies of S/F junctions a strong fer-arXiv:1807.10387v1  [cond-mat.supr-con]  26 Jul 20182\nromagnet is considered with a N\u0013 eel temperature much\nlarger than Tcso that the in\ruence of superconductiv-\nity on the magnetism is negligible. However, it was\nsuggested5,36that under certain conditions for a ferro-\nmagnetic \flm attached to a S it is favorable to be in\na 'cryptoferromagnetic state',37i.e. a segregation of the\nferromagnet into small-size domains, smaller than the su-\nperconducting coherence length.\nIn this paper we aim to investigate the in\ruence of S on\nthe magnetism in a S/F system in the clean limit on the\nbasis of the Bogoliubov-de Gennes approach. We will\ndescribe both the superconductor and the ferromagnet\non equal footing i.e. the latter is not implemented by an\nexternal exchange \feld but also described by an itinerant\nmodel with ferromagnetic exchange Jbetween the charge\ncarriers (cf. also Refs. 38 and 39). Our model is therefore\nimplemented on a \fnite lattice in contrast to previous\ncontinuum studies of proximity e\u000bects within the BdG\napproach.28{34\nThe approach has the advantage that it allows for the\nstudy of both proximity e\u000bects, S order parameter in the\nF and ferromagnetic order parameter in the S as a func-\ntion of the exchange J, thus covering the range from soft\nto hard F's. Moreover, we will investigate the inverse\nproximity e\u000bect, in particular the weakening of the fer-\nromagnetic order in the F close to the interface.\nPrevious studies of heterostructures on discrete lattices\nwithin BdG have addressed the subgap conductance at\nFS interfaces,23the decay of d-wave correlations inside\nthe F within an extended Hubbard model,24or the gen-\neration of parallel spin triplets in conical magnetizations\nsuch as Holmium.26\nOur paper is organized as follows. The next section\noutlines the system, model and approximations of our\ninvestigations. The resulting charge density, magnetiza-\ntion and order parameter pro\fles together with the cor-\nresponding spectral properties are discussed in Secs. III\nand IV. Finally, the \fndings are summarized in Sec. V.\nII. MODEL AND FORMALISM\nWe consider a two-dimensional superconducting sys-\ntem sandwiched between two ferromagnetic layers as\nshown in Fig. 1. We use periodic boundary conditions\nalong x- and y-directions so that the S system is actually\nconnected to the same ferromagnet on the left and right.\nTranslational invariance is assumed along the y-direction.\nThe calculations are done at zero temperature.\nOur model hamiltonian reads\nH=H0+HU+Hint; (1)\nFSFxyLSLxLyFIG. 1. Sketch of the SF bilayer proximity system with pe-\nriodic boundary conditions along x- and y-directions. Lx;y=\nNx;yawhere adenotes the lattice constant which in the fol-\nlowing is set to a\u00111.\nwhere\nH0=X\nij;\u001btijcy\ni;\u001bcj;\u001b+X\ni;\u001b(Vloc\ni\u0000\u0016)cy\ni;\u001bci;\u001b\n+X\ni;\u001bhi;z\u001bcy\ni;\u001bci;\u001b (2)\nis the single-particle part composed of the kinetic energy\nand a local term. The latter consists of the chemical po-\ntential\u0016and a local energy Vloc\niwhich is implemented\nin order to account for the local orbital energy but also\nto tune the charge density in the S layer. We consider\nelectrons on a Nx\u0002Nysquare lattice and c(y)\ni;\u001bannihi-\nlates (creates) a particle with spin \u001b=\u00061 at site Ri;\nthe site index is a two-dimensional vector, i= (ix;iy).\nThe hopping matrix element of the \frst term is set to\ntij=\u0000t <0 for nearest neighbors and tij= 0 other-\nwise. We also consider a constant Zeeman term \u0018hi;z\nto describe the local magnetization at site i. This com-\nponent of the Hamiltonian is used below to compare the\nresults for the itinerant F system studied in this paper,\nwith previous investigations where F was modeled with\nan external exchange \feld h.\nThe second part of Eq. (1),\nHU=X\niUini;\"ni;#; (3)\nis a local interaction with ni;\u001b=cy\ni;\u001bci;\u001b. Within the\nsuperconducting layers we take Ui\u0011US<0, i.e. an\nattractive Hubbard interaction, in order to support sin-\nglet superconductivity. In the ferromagnetic layers Ui\narises as the \frst term in an expansion of the long-range\nCoulomb interaction in Wannier functions and therefore\nUi\u0011UF\u00150 in these regions.\nFinally, we model the magnetic interaction in the fer-\nromagnetic layer as\nHint=\u0000X\nhijiJij[sisj+ninj]; (4)\nwhereni=ni;\"+ni;#,s\u000b\ni=P\n\u001b\u001b0cy\ni;\u001b\u001c\u000b\n\u001b\u001b0ci;\u001b0with the\nPauli matrices \u001c\u000b, andhijilimits the summations to\nnearest neighbor sites. The interaction Eq. (4) orig-\ninates from the nearest neighbor Coulomb interaction3\ntermsJ=hijj1=rjjii=hiij1=rjjji>0 and has been\nderived in Refs. 40{42 from the expansion of the long-\nrange Coulomb interaction in Wannier functions. In the\nexpansion we have neglected a density dependent 'corre-\nlated hopping' term arising from Coulomb matrix con-\ntributionshiij1=rjijibetween sites RiandRj, which\nhas been shown to further stabilize ferromagnetism.43,44\nBased on a variational Ansatz it has been argued that the\nmodel explains the occurrence of weak metallic ferromag-\nnetism in materials with a partially \flled nondegenerate\nband, as in Sc 3In45and metallic hydrogen.46Note, how-\never, that the model does not capture ferromagnetism\nin transition metals, where magnetism is usually gener-\nated from localized electrons.43,46Nevertheless, as shown\nbelow, the theory contains many features that are also\npresent in double exchange models. No speci\fc material\nis considered here and Eq. (4) is chosen as a convenient\nway to describe itinerant ferromagnetism.\nIn the heterostructure of Fig. 1, the ferromagnetic cou-\nplingJijis only \fnite in the F layer. Furthermore, mag-\nnetism and charge density solely vary along the xdirec-\ntion and are constant along the ydirection. Therefore,\nEq. (4) is rewritten in the form\nHint=\u0000X\ni=(ix;iy)Jx\ni[sisi+^x+nini+^x] (5)\n\u0000X\ni=(ix;iy)Jy\ni[sisi+^y+nini+^y]:\nIn the calculations of later sections it is assumed that\nJx\ni=Jy\ni=Jis a constant within the F layer since a\nmoderate anisotropy did not seem to a\u000bect the results\nsigni\fcantly.\nSince the system is translationally invariant along the\nydirection we perform the corresponding Fourier trans-\nform\nci;\u001b=1p\nNyX\nixcix;\u001b(ky) exp(\u0000ikyiy); (6)\nso that the kinetic term Eq. (2) reads ( t>0)\nH0=\u0000tX\nix;ky;\u001bh\ncy\nix;\u001b(ky)cix+1;\u001b(ky) +h:c:i\n(7)\n+X\nix;ky;\u001b[\u00002tcos(ky)\u0000\u0016]cy\nix;\u001b(ky)cix;\u001b(ky):\nWe apply the transformation, Eq. (6), to the inter-\naction terms, Eqs. (3-5). We then approximate these\nterms in mean-\feld. This includes the anomalous singlet\n(Gor'kov) correlations f0(i) =hci;#ci;\"ithat are induced\nin the S regions where Ui<0 but leak into the F layer\ndue to the proximity e\u000bect. The problem then can be\ndiagonalized by means of the Bogoliubov-Valatin trans-\nformation\ncix;\u001b(ky) =X\nph\nuix;\u001b(p;ky)\rp;ky\u0000\u001bv\u0003\nix;\u001b(p;ky)\ry\np;kyi\n;\n(8)and the integer plabels the eigenvalue. In-\ntroducing the basis vector ~\tn(p;ky) =\n[un;\"(p;ky);un;#(p;ky);vn;\"(p;ky);vn;#(p;ky)] one has to\nsolve the following eigenvalue problem for each value of\nky\nHij(ky)~\tj(p;ky) =\"p(ky)~\ti(p;ky); (9)\nwhere the hamiltonian is composed of a local and an\nintersite part\nHixjx(ky) =Tix;jx(ky) (\u000ejx;ix+1+\u000ejx;ix\u00001)+Vix(ky)\u000ejx;ix:\n(10)\nThe explicit structure of these operators is given in ap-\npendix A.\nIII. FERROMAGNETIC SYSTEM\nBefore presenting the results for the SF heterostructure\nwe brie\ry discuss the homogeneous magnetic system, i.e.\nUi\u0011UF>0 in Eq. (3) and Jx\ni=Jy\ni\u0011Jin Eq.\n(5). This case is instructive for the later analysis of the\ncompetition between F and S in the interface regions. A\nmore extensive discussion of the magnetic system can be\nfound in Refs. 40{42, and 46. Details of the calculations\nare provided in appendix B.\nThe chosen values of parameters, UF=tandJ=t, are\ngeneric but describe realistic systems. For example, UF\nis up to the order of the bandwidth, while Jis typically\nless thanUF.40Calculations are performed on a lattice\nwithNx\u0002Ny= 420\u0002420 sites.\nThe model already displays rich physics for a sin-\ngle magnetic layer. Depending on the parameter values\nthe system is paramagnetic, ferromagnetic, antiferromag-\nnetic or shows electronic phase separation.40{42,46\nPanels (a,b) of Fig. 2 report the ground state energy\nE(q) for a spiral modulation Si=S0exp (iq\u0001Ri) as a\nfunction of the spiral wave-vector qwhich is taken along\nthe diagonal direction ( qx=qy).S0is a variational pa-\nrameter.\nAt half-\flling and UF=t> 0,J=t= 0 the system shows\nantiferromagnetic spin-density wave order [ q=QAF=\n(\u0019;\u0019)] which due to perfect nesting occurs for in\fnitesi-\nmally small values of the repulsive interaction UF=t.47\nUpon increasing the ferromagnetic exchange J=t> 0 a\nsecond energy minimum develops at q= (0;0) which\nabove some critical J=tthat depends on UF=tcorre-\nsponds to the ferromagnetic ground state [cf. Fig. 2(a)].\nAs can be seen from panel (b) of Fig. 2 the same holds for\ndoping away from half-\flling where for su\u000eciently large\nrepulsionUF=tandJ=t= 0 the commensurate AF is re-\nplaced by a spiral, but with some incommensurate mod-\nulation Qspiral = (q;q). In the regime of small doping\n(n\u001c1) and small UF=t(<1=N(EF)) the system would\nbe a paramagnet for J=t= 0 and ferromagnetism can\nbe induced above some critical J=t. The corresponding4\n01 2 3 qx=qy-1.7-1.65-1.6-1.55-1.5E(q)/N [t]0\n1 2 3 qx=qy-1.8-1.75-1.7-1.65-1.6-1.55E(q)/N [t]0.2\n0.4 0.6 0.8 1 n-2-1.5-1-0.50chem. pot. [t]0\n2 4 6 8 U\nF/t00.20.40.60.8J [t]a) n=1, UF/t=5J/t=0.1\nJ/t=0.15J/t=0.2J/t=0.2 J/t=0.1\nb) n=0.9, UF/t=5J/t=0.15\nc)\nferro\npara\nd) UF/t=2J/t=0.7\nJ/t=0.5n=0.2n=0.6\nFIG. 2. Ground state energy vs. spin-spiral modulation vec-\ntorq(Si=S0exp (iq\u0001Ri)) along the diagonal direction of\nthe Brillouin zone at half-\flling (a) and density n= 0:9 (b).\nNote that for other UF=tresults are qualitatively similar but\nenergy variations decrease with decreasing UF=t. (c) Phase\ndiagram in the ( UF=t; J=t )-plane for density n= 0:2. (d)\nChemical potential vs. density. The horizontal dotted lines\nare determined by the Maxwell construction. The compress-\nibility diverges at the local extrema.\nphase diagram is displayed in Fig. (2c) for concentrations\nn= 0:2 andn= 0:6. Upon increasing UF=tthe transi-\ntion line approaches the value for the standard Stoner\ncriterionUF= 1=N(EF) atJ= 0 (N(EF) is the density\nof states at the Fermi energy EF).\nIn Figure 2(d) we also demonstrate that the model\nhas an instability region with respect to phase separa-\ntion which can be deduced from the dependence of the\nchemical potential \u0016on the density n. The compressibil-\nity\u0014=@n=@\u0016 diverges at the local extrema of \u0016(n) and\nbecomes negative in between. The phase separation re-\ngion innis determined by the Maxwell construction (dot-\nted horizontal line). The two curves in Fig. 2d indicate\nthat the phase separation region decreases with decreas-\ningJ=t. Note that the occurrence of phase separation is\nnota peculiar feature of the present model. This phe-\nnomenon also appears in double exchange models that\nare for example used for the description of magnetism in\nmanganites (cf. Ref. 48 and references therein).\nIV. SUPERCONDUCTING-MAGNETIC\nHETEROSTRUCTURE\nThe results of this section have been obtained on lat-\ntices with 120\u000280 sites and periodic boundary conditions\nin both directions. In the S region (40 \u0014ix\u001480) singlet\nsuperconductivity is generated with a negative US=\u00002t.\nFor this value the coherence length can be estimated as\u0018S\u00194 in units of the lattice spacing, i.e. much smaller\nthan the linear size of the system.\nThe remaining sites pertain to the F region with local\nonsite repulsion UF>0 and ferromagnetic exchange in-\nteractionsJx\ni\u0011Jx,Jy\ni\u0011JyandJx=Jy=J. This\ndescription of the F layer will be referred to as the cor-\nrelated ferromagnetic model (CFM). For comparison, we\nalso use an exchange \feld hzto model the F. The latter\nis referred to as the exchange \feld model (EFM).\nSince we consider an itinerant F we treat supercon-\nductivity and magnetism on equal footing. Hence, we\npresent in this section results for the charge density, the\nmagnetization and pair correlations in both the F and\nthe S.\nIn addition to the normal proximity e\u000bect, two distinct\nphenomena appear in these hybrid structures: the inverse\nproximity e\u000bect and phase separation. In the \frst, the\nS correlations suppress the magnetization inside the F\nnear the SF interface. In general, ni6=miin such sit-\nuation; the F is nowhere fully polarized. Moreover, the\ncoupling between magnetic and charge degrees of free-\ndom leads to a concomitant reduction of the nnear the\nSF interface. By contrast, when the system undergoes\nphase separation, the system is fully polarized ( ni=mi)\ndeep in the F and the superconducting state is a\u000bected\nby the itinerant electrons of the F.\n1. Charge density, magnetization and pair correlations\nFigure 3 reports the charge density n(panel a), mag-\nnetizationm(panel b), and singlet pair correlations f0\n(panel c) in the heterostructure for varying exchange con-\nstantJ=t. Since this parameter also in\ruences the local\nHartree-potential in the F layers a change of Jalters the\ncharge distribution between the S and F regions. To be\nable to compare results for di\u000berent values of Jwe there-\nfore adjust the local potential Vlocin the S regions in\nsuch a way that the charge density is the same for all\nJ=t-values deep inside the S layer; hence, in panel (a) of\nFig. 3 the charge densities noverlap in S for all J=t. Since\nmost of the physics occurs close to the interface between\nthe S and the F layer, Fig. 4 zooms into this region to\nshow the behavior of ni,miandf0(i). In addition, the\n\fgure also reports the result for the EFM (dashed lines)\nforhi;z= 3t. The value of hzis \fxed in such a way that\nit reproduces the same magnitude of the magnetization\nas the CFM deep inside the F region for J=t= 0:5.\nFrom Figs. 3 and 4 one can distinguish three di\u000ber-\nent regimes. For the parameters of the system, these are\nJ=t.0:3, 0:3.J=t.0:55 andJ=t&0:55. At low\nJ=t(.0:3) ferromagnetism disappears; this paramag-\nnetic regime was discussed in the previous section. Above\nthis transition the inverse proximity e\u000bect regime [black\nopen circles and red squares in panels (a-d)] is e\u000bective.\nThe S correlations completely suppress the magnetiza-\ntion in F over a signi\fcant distance from the interface.\nThis regime is also characterized by a partial depletion of5\n0,20,40,60,81charge niJ/t=0.35\nJ/t=0.4\nJ/t=0.5\nJ/t=0.6\n0\n0,20,40,60,8magnet. miJ/t=0.7\n20\n40 60 80 100 120 x-position x\ni00,020,040,06singlet OP f0(i)90\n100110 a)b)\nc)\nFF S \nFIG. 3. Charge density (a), magnetization (b) and singlet\npair correlations (c) as a function of xand various values of\nthe exchange coupling J=tin the ferromagnetic region. For\nall cases the local potential Vlocis adjusted in such a way as\nto obtain a similar charge density deep in the S layer. The\nthin dashed green line reports the result within the EFM for\nhi;z=t= 3 (see text). Parameters: UF=t= 2, US=t=\u00002,\nn= 0:625.\ncharge density over similar depth in the F, resulting from\nthe coupling between charge and spin. The third regime\nis the phase separation regime found at high values of\nJ=t&0:55 (blue triangles and gold stars). Phase sepa-\nration was found in the homogeneous system of Sec. III\n[see Fig. 2(d)]. It is characterized by full polarization\ndeep in the F ( i.e.ni=mi) and a concomitant deple-\ntion of the charge density and the magnetization in the\nF over moderate distance away from the interface.\nWe note that for intermediate values, J=t\u00190:5 (green\ndiamonds), the charge and magnetic pro\fles adjust to\nreach equal value already within the charge/spin corre-\nlation length\u00181=kF(\u00192\u00003 lattice constants) from\nthe SF interface. This steep rise is close to the result of\nthe conventional EFM (dashed green line in Figs. 3,4),\nwhere the transition is driven by the abrupt onset of the\nmagnetization inside the ferromagnetic layer.\nFigure 4(c) reveals the behavior of the magnetization\nin the S. The magnetization decays exponentially with a\ncorrelation length (naturally) independent of J=t. The\noverall magnitude is determined by the value of miat\n8085 90 95 100 00,20,40,60,8charge ni75\n80 85 90 95 100 00,20,40,60,8magnetization m\ni60\n70 80 site i\nx1e-061e-050,00010,0010,010,1magnetization |mi|75\n80 85 90 95 100 site i\nx1e-101e-091e-081e-071e-061e-050,00010,0010,01singlet OP |f\n0(i)|J/t=0.35J/t=0.4\nJ/t=0.5\nJ/t=0.6\nJ/t=0.7\na)b) c)\nd)\nFIG. 4. Close look near the interface of the charge den-\nsity (a), magnetization (b,c) and singlet pair correlations (d)\nshown in Fig. 3 for various values of the exchange coupling\nJ=tin the F. For all cases the local potential Vlocis ad-\njusted as described in Fig. 3. Shown for comparison as a\nthin dashed green line is also the pair correlation in the EFM\nforhi;z=t= 3. The S correlations inside the F in panel d have\nbeen \ftted with Eq. (11) (light red and black lines; see text\nfor \ftting parameters). Parameters: UF=t= 2, US=t=\u00002,\nn= 0:625.\nthe interface, which is largest for intermediate values of\nJ=t, wheremiis not suppressed by S correlations and\nphase separation is not relevant (diamond green lines in\nFig. 4).\nFigs. 3c and 4d display the decay of the superconduct-\ning order parameter f0(i) in the F. Panel 3(c) shows the\noverall behavior of the singlet order parameter for the\nsame range of J=tvalues while the decay inside the F\nis detailed in the inset, and in panel (d) of Fig. 4 on a\nlogarithmic scale. The latter \fgure shows the stark con-\ntrast between the inverse proximity ( J=t.0:55) and the\nphase separated ( J=t> 0:55) regimes. Both cases can be\nmodeled by the following expression\nf0\u0018e(\u0000x=\u0018N)\nxcos\u0012x\n\u0018F\u0013\n; (11)\nwhere\u0018N=vF=2\u0019Tand\u0018F=vF=2\u0019m(x) are the para-\nmagnetic and ferromagnetic coherence lengths. A close\nlook at the curves in Fig. 4 shows that the magnetization\ntakes non-zero values from the interface on; for exam-\nple, forJ=t= 0:35 the value of m(x) is small but \fnite\nalready for 80 < x\u001491. The above expression for f0,\nEq. (11), can be used to obtain an excellent piecewise \ft\nof the numerical data; one divides the space into x<x 0\nandx > x 0withx0\u001882 (x0\u001890) forJ=t= 0:35\n(J=t= 0:4). ForJ=t= 0:35 for example, the curve in\nregions 80 < x\u0014x0can be \ftted with \u0018N> LFand\n\u0018F= 30, whereas for x > x 0we have\u0018N= 33 but\n\u0018F= 0:6. Forx < x 0the pair correlation undergoes a\nsmooth exponential decay that is expected of a paramag-6\nnet (the magnitude of m(x) is very small in this region),\nwhereas farther away the behavior is characteristic of a\nhomogeneous ferromagnet. These results are consistent\nwith previous \fndings for singlet pair correlations in the\nEFM, in a paramagnet (Ref. 27) and a homogeneous fer-\nromagnet (see for example Ref. 4). Refs. 28 and 29 pro-\nvided a complementary analysis of the decay of f0within\nthe EFM. Two almost identical length scales were intro-\nduced that inversely scale with the polarization of the F\nand are\u0018(kF;\"\u0000kF;#)\u00001. The oscillatory behavior is\ndue to the interference of up- and down excitations in\nthe pair amplitude.\nIn the opposite case of large exchange coupling J=t\n(phase separation regime) the pair correlations \frst fol-\nlow the small- J=tbehavior up to some distance x0away\nfrom the interface, followed by a much stronger decay\ndeeper inside the F ( x > x 0). The length x0is deter-\nmined by the point where the magnetization reaches full\npolarization and therefore increases with J=tdue to the\nincreasing low density domain inside the F.\n8085 90 95 100 00,20,40,60,8charge ni75\n80 85 90 95 100 00,20,40,60,8magnetization m\ni60\n70 80 site i\nx1e-061e-050,00010,0010,010,1magnetization |mi|75\n80 85 90 95 100 site i\nx1e-101e-091e-081e-071e-061e-050,00010,0010,01singlet OP |f\n0(i)|J/t=0.2J/t=0.5\nJ/t=0.7\na)b) c)\nd)\nFIG. 5. Charge density (a), magnetization (b,c) and sin-\nglet pair correlations (d) close to the interface as in Figs. 3,4\nbut for a larger value of the Coulomb potential in the F,\nUF=t= 5. The phase separation regime is substantially ex-\ntended when compared to Figs. 3, 4. Both pair correlations\nin the F and the charge and magnetic con\fgurations in S\nare rapidly suppressed away from the interface. Parameters:\nUF=t= 5,US=t=\u00002,n= 0:625.\nThe results of Figs. 3, 4 were obtained for UF=t= 2. A\nlarger local correlation stabilizes the F and increases the\nrange of values of J=tfor which the F is fully polarized\n(see Fig. 5). As a result, even for values of J=t\u00190:2\nclose to the onset of ferromagnetism, pair correlations\nare not able to signi\fcantly suppress magnetism; the in-\nverse proximity e\u000bect is almost absent. Nevertheless, this\nreduction of the intermediate regime does not imply a\ncorresponding extension of the phase separation regime\nto lower values of J=t. AtJ=t\u00180:5 one still observes a\nbehavior similar to the EFM. The phase separation insta-\nbility for large J=t&0:5 persists and the pair amplitudedecay is again shifted away from the interface (panel (d)\nof Fig. 5). The distance from the interface over which\nthe magnetization is suppressed is about the same than\nobserved in Fig. 4. As expected, the behavior of the\nmagnetization in the S is una\u000bected by the change in\nUF=tin the F.\n2. Spectral properties and pair correlations\nThe proximity e\u000bect is also re\rected in spectral prop-\nerties such as the local density of states (LDOS),\n\u001aloc(xi;!) =1\nNX\np;ky;\u001b[jui;\u001b(p;ky)j\u000e(!\u0000\"p(ky))\n\u0000jvi;\u001b(p;ky)j\u000e(!+\"p(ky))];\nwhich within the BdG formalism and the EFM has\nbeen analyzed in Refs. 28 and 29.\nFig. 6a shows the LDOS deep in the S and the F. No-\nticeable are the standard BCS coherence peaks at the\ngap edges in the S (near != 0, black curve at x= 60);\na small numerical 'pair-breaking parameter' \u000f= 0:02t\nhas been introduced for numerical reasons which is re-\nsponsible for the small \fnite LDOS inside the gap. The\noverall structure of the LDOS is otherwise characteristic\nof a two-dimensional square lattice with its logarithmic\nvan-Hove singularity at the band center.\nDeep inside the ferromagnet [red curve at x= 110 in\nFig. 6(a)] the van Hove singularity is split due to the\nformation of subbands (peaks near !=t=\u00063). Note\nthat the apparent \"noise\" in the data is not due to the\nlack of precision of the calculation, but are oscillations\noriginating from the discreteness of the lattice.\nIt is instructive to investigate the dynamical singlet\npair correlations (Gor'kov function)\nf0(ix;t) =1\n2[hcix;\"(t)cix;#(0)i\u0000hcix;#(t)cix;\"(0)i] (12)\ninside the ferromagnet for di\u000berent exchange parameters\nJ=t. The imaginary part of the Fourier transform reads\nF(ix;!) = ImZ1\n\u00001dtei!tf0(ix;t) (13)\n=\u0019X\np;ky\u0002\nui;\"(p;ky)v\u0003\ni;#(p;ky) +ui;#(p;ky)v\u0003\ni;\"(p;ky)\u0003\n\u0002h\rp;ky\ry\np;kyi\u000e(!+\"p(ky))\n\u0000\u0019X\np;ky\u0002\nui;\"(p;ky)v\u0003\ni;#(p;ky) +ui;#(p;ky)v\u0003\ni;\"(p;ky)\u0003\n\u0002h\ry\np;ky\rp;kyi\u000e(!\u0000\"p(ky)):\nFigure 7 shows the position and frequency dependence\nof the singlet correlations, Eq. (13), as an intensity plot\nwhich visualizes the decay of the pair correlations inside\nthe ferromagnet. Note that the Gor'kov functions are\nasymmetric in !,f0(\u0000!) =\u0000f0(!), and in a clean S7\n-4-20246800.20.40.6LDOS [1/t]x=60x=110\n0\n-0.2 -0.4 -0.6 -0.8 -10.150.1\n0.05\n0\n-0.05\n-0.1\n0\n-0.2 -0.4 -0.6 -0.8 -10.20.1\n0\nF(x,ω) [1/t]J/t=0.35J/t=0.5\nJ/t=0.7\n0\n-0.2 -0.4 -0.6 -0.8 -10.030\n-0.03\n0\n-0.2 -0.4 -0.6 -0.8 -1ω \n[t]0.050\nF(x,ω) [1/t]J/t=0.35J/t=0.5\nJ/t=0.7\n0\n-0.2 -0.4 -0.6 -0.8 -1ω\n [t]0.020.01\n0\n-0.01\n-0.02\nb) x=80c) x=81\nd) x=85e) x=90\nf) x=100\na)\nFIG. 6. Local density of states (a) inside the S (black;\nx= 60) and the F (red; x= 110) for J=t= 0:5. Panels (b-f):\nimaginary part of the singlet Gor'kov pair correlation F(ix; !)\ndependence on frequency at speci\fc points in the heterostruc-\nture and for various exchange couplings as indicated in the\npanels. The correlations at x= 80 are in the S while those\nforx\u001581 are in the F. Note that the singlet order parameter\nforJ=t= 0:5 and x= 81 has opposite sign as compared to the\nother couplings (cf. panel (c) of Fig. 3). For better compari-\nson we have therefore multiplied the J=t= 0:5 in panel (c) by\n\u00001, symbolized by the dashed line. Parameters: UF=t= 2,\nUS=t=\u00002,n= 0:625.\nsystem show a\u00061=p\n!\u0000\u0001 singularity at the gap edges\n[\u0001(x) =f0(ix;t= 0) being the superconducting gap].\nOnly the! <0 part ofF(ix;!) is shown in Figs. 6 and\n7.\nA main di\u000berence between the inverse proximity\nregime and the phase separated regime is immediately\napparent at small !=twhen comparing Figs. 7(a) and\n7(b): pair correlations extend deep into the F for small\nJ=t[Fig. 7(a)]. The region in the F where these pair\ncorrelations are present also shrinks with increasing !=t.\nIn the region where the magnetization is suppressed, ei-\nther because of the inverse proximity e\u000bect (small J=t)\nor phase separation (large J=t) the low energy F(ix;!)\ncontinuously extends from the S region into the F. This\nis clearly visible in Fig. 7(a) at low !=twhere the inten-\nsity plot shows pronounced pair correlations (indicated\nby solid red color) close to the interface. By contrast,\nthe Gor'kov function starts oscillating within a partially\npolarized region of the ferromagnet as indicated by the\nalternating red-blue pattern of Fig. 7(a). Similarly, os-\ncillations of F(ix;!) are seen at \fxed position ixas a\nfunction of !=tand we now analyze this !-dependence\nofF(ix;!) in more detail at various distances from the\ninterface in the F, shown in panels (b-f) of Fig. 6.\nThe singlet pair correlations F(ix;!) shown in panel\n(b) of Fig. 6 are calculated at the interface, on the S side,\nx= 80. The peak position, indicating the size of the S\ngap is largest (and sharpest) for J=t= 0:7 in agreement\nwith the larger singlet order parameter (cf. panel (c) of\nFig. 3).\nFIG. 7. Position and frequency dependent singlet correla-\ntions, Eq. (13), for (a) J=t= 0:35 and (b) J=t= 0:7. Pair\ncorrelations oscillate both in space and frequency and extend\ndeeper into the F for small J=t. Parameters US=t=\u00002,\nUF=t= 2n= 0:625.\nOn the other side of the interface, in the F ( x= 81,\npanel (c) of Fig. 6) one still observes a sizable gap for all\ncouplings, which however, is now largest for J=t= 0:35\nwhere the magnetization is suppressed close to the in-\nterface. Further away from the interface, at x= 85,\npanel (d) shows that for J=t= 0:5 the magnetic sys-\ntem is already completely polarized, and the pair cor-\nrelations are suppressed on the scale of the plot. In-\nterestingly, at this same location, the phase separated\nsolution (J=t= 0:7) and the inverse proximity solution\n(J=t= 0:35) still reveal a low energy peak and thus the\noccurrence of a proximity induced gap. Moreover, at\nlarger energies ( !=t\u0018\u00000:3) a second broad peak ap-\npears with opposite sign. For J=t= 0:35 this peak turns\nout to be related to the onset of frequency oscillation in\nF(ix;!) observed further away from the interface, as seen\nin panels (e) and (f). In the di\u000busive limit and within\nthe EFM4,5it was shown that the occurrence of such\noscillations in frequency is a direct consequence of the\nexchange \feld h. Similarly, in the BdG approach they\narise from the superposition of the di\u000berent excitations\nin the spin-up and down bands and thus disappear when\nthe system is completely polarized. This explains why in\npanel (e) and (f), at x= 90;100, only pair correlations\nin the inverse proximity regime ( J=t= 0:35) are \fnite.\nNote that the frequency integration of F(ix;!) yields the\nlocal (equal time) Gor'kov function \u0001( x) =f0(ix;t= 0)\natxwhich vanishes in the F region and thus requires a\ncancellation of the \fnite contributions to F(ix;!).\nFinally, we note in panels (d-f) that there are smaller\noscillations, both in magnitude and frequency, super-\nposed to the large oscillation of F(ix;!) just mentioned.\nThese are the same small oscillations found in all curves\nof panels (b,c) and are related to the discrete spatial lat-\ntice.\nWe used everywhere minstead ofSzso i changed it in\nthe following paragraph and removed the sentence that\nrefers to the z-axis choice. The results of this section IV\nwere obtained for singlet pair correlations generated in8\nthe S and leaking into the F. A similar behavior is also\nfound for the m= 0 triplet correlations brought about\nin the F region. In the SF hybrid structure with periodic\nboundary conditions studied here, there are no m=\u00061\ntriplet components since the magnetic inhomogeneities\nare found in the magnitude of the magnetization while its\norientation is \fxed. This contrasts the magnetic and su-\nperconducting inhomogeneities discussed here from those\nof previous work.7,8,20{22\nV. CONCLUSION\nWe have analyzed proximity e\u000bects in a FS het-\nerostructure in which the F is described within an ex-\ntended Hubbard-type model where the ferromagnetic ex-\nchange arises from intersite contributions of the Coulomb\ninteraction. Such a description allows the self-consistent\ntreatment of both superconducting and magnetic order\nparameters which gives rise to features not present in ap-\nproaches where the exchange \feld is \fxed inside the F. In\nparticular, we have found that for small exchange inter-\nactions and onsite correlations the magnetization close\nto the interface may be suppressed by the S correlations\nwhich signi\fcantly alters the decay of the pair correla-\ntions inside the F. Similar for large exchange couplings\nthe system shows an instability towards phase separa-\ntion which is also realized close to the interface with a\nconcomitant suppression of the magnetization. As a con-\nsequence the S correlations extend far inside the phase\nseparated region and only get suppressed when the mag-netization recovers at some distance from the interface.\nThe correlations leak much deeper than for the e\u000bec-\ntive \feld model; this may signi\fcantly a\u000bect the Joseph-\nson current and is being investigated. Such an instabil-\nity towards phase separation is also inherent in double-\nexchange models for ferromagnetism which are usually\nconsidered to be appropriate for transition metals48.\nTherefore, we expect that these aspects of our results\nare also valid in such systems and are planned in future\ninvestigations. In the present paper we have restricted\nourselves to collinear magnetic structures. However, mi-\ncroscopic magnetic models usually show also more com-\nplex magnetic structures in some part of the phase di-\nagram, as for example the antiferromagnetic spirals in\npanel (b) of Fig. 2 for J=t= 0:1. A heterostructure where\nthe ferromagnet displays a spiral rotation would also in-\nducem= 1 triplet components inside the F. As a result,\nwe expect that the pair correlations inside the F have a\npronounced in\ruence on the periodicity of such magnetic\nstructures. Work in this direction is in progress.\nVI. ACKNOWLEDGEMENTS\nCM and GS thank the DAAD for \fnancial support.\nAB gratefully acknowledges funding provided by the Na-\ntional Science Foundation (DMR-1309341).\nAppendix A\nThe matrices de\fned in Eq. (10) are given by\nTi;i+x(ky) =0\nBB@\u0000t+ 2Jx\ni(kx\ni;\"+kx\ni;#)\u00030 0 2 Jx\nipx(i)\n0\u0000t+ 2Jx\ni(kx\ni;\"+kx\ni;#)\u00032Jx\nipx(i+x) 0\n0 2 Jx\nipx(i)t\u00002Jx\ni(kx\ni;\"+kx\ni;#) 0\n2Jx\nipx(i+x) 0 0 t\u00002Jx\ni(kx\ni;\"+kx\ni;#)1\nCCA(A1)\nand\nVi(ky) =0\nBB@\u00002tcos(ky) +v\";\"(i)v\";#(i) 0 Uif0(i) + 2Jy\nipy(i)\nv#;\"(i)\u00002tcos(ky) +v#;#(i)Uif0(i) + 2Jy\nipy(i) 0\n0 Ui\u0001\u0003\ni+ 2Jy\ni[py(i)]\u00032tcos(ky)\u0000v\";\"(i)v\u0003\n\";#(i)\nUi\u0001\u0003\ni+ 2Jy\ni[py(i)]\u00030 v\u0003\n#;\"(i) 2tcos(ky)\u0000v#;#(i)1\nCCA(A2)\nwith the following abbreviations\nv\u001b;\u001b(i) =Ui\n2(ni\u0000\u001bmi) +Vloc\ni\u0000\u0016+hi;z\u001b\n\u0000Jx\ni\u0000x(ni\u0000x+\u001bmi\u0000x)\u0000Jx\ni(ni+x+\u001bmi+x)\n\u00002Jy\ni(ni+\u001bmi) + 4Jy\niRe[(ky\ni;\"+ky\ni;#)eiky]\nv\";#(i) =\u0000UihS\u0000\nii\u00002Jx\ni\u0000xhS\u0000\ni\u0000xi\u00002Jx\nihS\u0000\ni+xi\n\u00004Jy\nihS\u0000\nii\nni=X\n\u001bhni;\u001bi\nmi=X\n\u001b\u001bhni;\u001bi(\u001b=\u00061)\nhS\u0000\nii=hcy\ni;#ci;\"i=1\nNyX\nkyhcy\ni;#(ky)ci;\"(ky)ikx\ni;\u001b=hcy\ni;\u001bci+x;\u001bi=1\nNyX\nkyhcy\ni;\u001b(ky)ci+x;\u001b(ky)i\nky\ni;\u001b=hcy\ni;\u001bci+y;\u001bi=1\nNyX\nkye\u0000ikyhcy\ni;\u001b(ky)ci;\u001b(ky)i\npx(i) =hci;#ci+x;\"i=1\nNyX\nkyhci;#(ky)ci+x;\"(ky)i\npy(i) =hci;#ci+y;\"ie\u0000iky+hci+y;#ci;\"ieiky:9\nAppendix B\nIn Sec. III we discuss the magnetic state of the F alone,\nin the correlated single band model, Eq. (9). Spiral mag-\nnetic solutions with the Ansatz hS\u0006\nii=S0exp(\u0006iqRi)\nare obtained by factorizing Eqs. (3, 4) with respect to\nthe operators\nS+(\u0000)\nq =X\nkcy\nk+q\"(#)ck#(\"):Since the charge density for these solutions is constant\nHartree terms are neglected as they only shift the en-\nergy by a constant value. 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Dagotto, Nanoscale Phase Separation and Colossal\nMagnetoresistance , Springer (2003)."    },    {        "title": "1010.1075v1.Ferromagnetic_Resonance_in_Spinor_Dipolar_Bose__Einstein_Condensates.pdf",        "content": "arXiv:1010.1075v1  [cond-mat.quant-gas]  6 Oct 2010Ferromagnetic Resonance in Spinor Dipolar Bose–Einstein C ondensates\nMasashi Yasunaga and Makoto Tsubota∗\nDepartment of Physics, Osaka City University, Sumiyoshi-k u, Osaka 558-8585, Japan\n(Dated: April 27, 2022)\nWe used the Gross–Pitaevskii equations to investigate ferr omagnetic resonance in spin-1 Bose–\nEinstein condensates with a magnetic dipole-dipole intera ction. By introducing the dipole interac-\ntion, we obtained equations similar tothe Kittel equations used torepresent ferromagnetic resonance\nin condensed matter physics. These equations indicated tha t the ferromagnetic resonance originated\nfrom dipolar interaction, and that the resonance frequency depended upon the shape of the conden-\nsate. Furthermore, spin currents driven by spin diffusions a re characteristic of this system.\nPACS numbers: 03.75.Mn, 03.75.Nt\nI. INTRODUCTION\nMagnetic resonance (MR) as a physical concept has\nbeen applied in various fields, enabling physical, chem-\nical, and medical experiments to obtain information on\nnuclear spin and electron spin systems. The concept has\nalso provided valuable information to help understand\nthe unknown structures of many condensed matter sys-\ntems [1].\nThe use of MR in the study of ferromagnets, e.g.\nNickel, Cobalt, and Iron, began in the 1940s. Grif-\nfiths observed that the Land´ e’s g-factor of electrons in\nferromagnets was far from the well known value, 2 [2].\nIn order to understand these anomalous results, Kittel\ntheoretically introduced a demagnetizing field into the\nequation representing the motion of the magnetization\nM= (Mx,My,Mz), obtaining an equation valid in an\nexternal magnetic field H0ˆz, withMz0=H0/Nzand de-\nmagnetizing fields [3], thereby obtaining the Kittel equa-\ntion,\ndM\ndt=γn[M×H]. (1)\nHere,γnis the nuclear gyromagnetic ratio, and H=\n(−NxMx,−NyMy,H0−NzMz) is given by the demagne-\ntizing factors Ni. By linearizing the magnetization M=\nM0+δMfromthestationarymagnetization M0=Mz0ˆz,\nKittel obtained a precession of the magnetization and a\nprecessing frequency, i.e.resonance frequency,\nω2=γ2\nn{H0+(Ny−Nz)Mz0}{H0+(Nx−Nz)Mz0},(2)\nwhich explained the anomalous g-factor. Furthermore,\nhe found that the resonance frequency depends on the\nshape of a ferromagnet because Nidepends on the shape\n[3]. Thus, ferromagnetic resonance (FMR) was estab-\nlished, and the workenabled numerousadditionalstudies\n[4].\nMR also plays an important role in quantum conden-\nsate systems. In superfluid3He, the dynamics of the spin\n∗Electronic address: tsubota@sci.osaka-cu.ac.jpvector and the d-vector are represented by the Leggett\nequation, which couples these vectors through magnetic\ndipole-dipole interactions [5]. The equation also shows\nnot only an MR typical of condensed matter, but also a\nnew MR that cannot be described using the equations of\nmotion for general paramagnets and ferromagnets. This\nMR was used to find AandBphases [6]. Parallel ring-\ning, which is an oscillation of longitudinal magnetization,\nwas also observed [7].\nSince the discovery of atomic Bose–Einstein conden-\nsates(BECs)[8,9], BECshavebeenstudiedinopticsand\natomic and condensed matter physics. We have intro-\nducedMRintoBECstorealizemagneticresonanceimag-\ning, a popular method of nondestructive testing. Spinor\nBECs are expected to be suitable for MR, since they\nhave not only internal degrees of freedom but also mag-\nnetic properties. In particular, we are interested in mag-\nnetic dipole-dipole interactions (MDDI) in spinor BECs,\nwhich have been actively studied. The interaction be-\ntween spins has a characteristic symmetry of rotation\nand spin, which is expected to result in a new quan-\ntum phase [10–12] and Einstein–de Haas effects [13]. Ex-\nperimentally, Griesmaier et al.realized spinor dipolar\ncondensates using52Cr atoms, which have a larger mag-\nnetic moment than alkali atoms [14]. The shape of the\ncondensates clearly represented the anisotropy of the in-\nteraction [15, 16]. Thus, MDDI has opened new areas of\nspinor condensate research.\nAs an introduction to MR in BECs, we numerically\nstudied spin echo in dipolar BECs with spin-1 [19]. The\nspin echo is a typical phenomenon of MR, discovered by\nHahn [17] and developed by Carr and Purcell [18]. Previ-\nously, we calculated the transition from Rabi oscillations\nto internal Josephson oscillations in spinor condensates\n[20]. In this paper, we consider MDDI in spin-1 BECs,\nexamining FMR by analyzing the Gross–Pitaevskii (GP)\nequations.\nIn section II, we derive Kittel-like equations from the\nGP equations, and analyze them. In section III, using\na single-mode approximation, we derive Kittel equations\nfrom the Kittel-like equations. The MDDI of the Kittel\nequationsisconsideredasthe originofthe demagnetizing\nfield, which is phenomenologically introduced in Eq. (1).\nIn section IV, we numerically solve the GP equations,2\nobtaining resonance frequencies that depend upon the\nshape of the condensates, and spin currents driven by\nspin diffusion which is given by the MDDI. Finally, Sec.\nV is devoted to our conclusions.\nII. FORMULATION\nIn this section, we derive the equations of motion for\nspinsfromthespin-1GPequationswithanexternalmag-\nnetic field and an MDDI [19].\ni/planckover2pi1∂ψα\n∂t=/parenleftbigg\n−/planckover2pi12\n2M∇2+V−µ+c0n/parenrightbigg\nψα\n−gµBHiFi\nαβψβ+c2FiFi\nαβψβ\n+cdd/integraldisplay\ndr′δij−3eiej\n|r−r′|3Fi(r′)Fj\nαβψβ.(3)\nHere,Vis the trapping potential, µis the chemical\npotential, and the total density n=/summationtext\niniis given\nbyni=|ψi|2. The external magnetic field is H=\n(Hx,Hy,Hz), and the components Fi\nαβof the spin ma-\ntricesˆFiare for spin-1. The interaction parameters are\nc0= (g0+2g2)/3 andc2= (g2−g0)/3 forgi= 4π/planckover2pi12ai/M\nrepresented by s-wave scattering lengths ai. The dipolar\ncoefficient is cdd=µ0g2\neµ2\nB/4π, and the unit vector is\ne= (ex,ey,ez) = (x−x′,y−y′,z−z′)/|r−r′|.\nUnder the homogeneous magnetic field H=Hˆz, the\nequations can be rewritten as,\ni/planckover2pi1∂ψ1\n∂t=/parenleftbigg\n−/planckover2pi12∇2\n2M+V−µ+c0n/parenrightbigg\nψ1−gµBHψ1\n+c2{(n1+n0−n−1)ψ1+ψ∗\n−1ψ2\n0}+D1,\n(4a)\ni/planckover2pi1∂ψ0\n∂t=/parenleftbigg\n−/planckover2pi12∇2\n2M+V−µ+c0n/parenrightbigg\nψ0\n+c2{(n1+n−1)ψ0+2ψ∗\n0ψ1ψ−1}+D0,\n(4b)\ni/planckover2pi1∂ψ−1\n∂t=/parenleftbigg\n−/planckover2pi12∇2\n2M+V−µ+c0n/parenrightbigg\nψ−1+gµBHψ−1\n+c2{(n−1+n0−n1)ψ−1+ψ∗\n1ψ2\n0}+D−1.\n(4c)\nThese dipolar terms are represented as,\nD1=/parenleftbiggψ0√\n2d−+ψ1dz/parenrightbigg\n,\nD0=/parenleftbiggψ1√\n2d++ψ−1√\n2d−/parenrightbigg\n,\nD−1=/parenleftbiggψ0√\n2d+−ψ−1dz/parenrightbigg\n,\nwith the integrations d±=dx±idyanddzgiven by,\ndi=cdd/integraldisplay\ndr′Fi(r′)\n|r−r′|3{1−3ei/summationdisplay\njej}.(5)The spin density vectors Fiare defined as,\nFx=Ψ†ˆFxΨ\n=/planckover2pi1√\n2{ψ∗\n0(ψ1+ψ−1)+ψ0(ψ∗\n1+ψ∗\n−1)},(6a)\nFy=Ψ†ˆFyΨ\n=i/planckover2pi1√\n2{ψ∗\n0(ψ1−ψ−1)−ψ0(ψ∗\n1−ψ∗\n−1)},(6b)\nFz=Ψ†ˆFzΨ=/planckover2pi1(|ψ1|2−|ψ−1|2). (6c)\nHere,Ψ= (ψ1,ψ0,ψ−1)Tis the spinor wave function.\nDifferentiating Eq. (6) with respect to time and utiliz-\ning Eq. (4), we can obtain the Kittel-like equation,\n∂F\n∂t=K+γe[F×Heff] (7)\nwith the gyromagnetic ratio γe=gµB//planckover2pi1of an electron.\nThe first term K= (Kx,Ky,Kz) becomes,\nKx=/planckover2pi1\n2Mi1√\n2{(ψ1+ψ−1)∇2ψ∗\n0−ψ∗\n0∇2(ψ1+ψ−1)\n+ψ0∇2(ψ∗\n1+ψ∗\n−1)−(ψ∗\n1+ψ∗\n−1)∇2ψ0},\nKy=/planckover2pi1\n2Mii√\n2{(ψ1−ψ−1)∇2ψ∗\n0−ψ∗\n0∇2(ψ1−ψ−1)\n−ψ0∇2(ψ∗\n1−ψ∗\n−1)+(ψ∗\n1−ψ∗\n−1)∇2ψ0},\nKz=/planckover2pi1\n2Mi(ψ1∇2ψ∗\n1−ψ∗\n1∇2ψ1\n+ψ∗\n−1∇2ψ−1−ψ−1∇2ψ∗\n−1).\nThe effective magnetic fields Heff=H+Hdd=\n(Hx\neff,Hy\neff,Hz\neff)consistoftheexternalmagneticfieldand\nthe dipolar field Hdd, given by,\nHx\neff=−cdd\ngµBdx,\nHy\neff=−cdd\ngµBdy,\nHz\neff=H−cdd\ngµBdz.\nNote that Eq. (7) does not depend on spin exchange\ninteraction, which refers to the second term with c2in\nEq. (3). Generally, the interaction affects a spin through\nthe effective magnetic fields of the other spins. However,\nexchange interaction does not appear in Heff. Therefore,\nthe isotropic exchange interaction does not affect MR in\nthese condensates.\nWe can redefine Eq. (7) as,\n∂Fk\n∂t=/planckover2pi1\n2Mi∇2Fk−∇·jk+γe[F×Heff]k,(8)\nwhere,\njx=/planckover2pi1√\n2Mi(ψ∗\n0∇(ψ1+ψ−1)+(ψ∗\n1+ψ∗\n−1)∇ψ0),\njy=/planckover2pi1√\n2M(ψ∗\n0∇(ψ1−ψ−1)−(ψ∗\n1−ψ∗\n−1)∇ψ0),\njz=/planckover2pi1\nMi(ψ∗\n1∇ψ1−ψ∗\n−1∇ψ−1).3\nThe equation of motion (8) for spins describes the prop-\nerties of spin dynamics in a ferromagnetic fluid. The\nfirst, second, and third terms of Eq. (8) represent spin\ndiffusion, spin current, and spin precession around Heff,\nrespectively.\nComparing Eq. (8) with Eq. (1), we noticed several\ndifferences. First, Eq. (8) was directly derived from\nthe GP equations, whereas Eq. (1) is a phenomenologi-\ncal equation of magnetization. The spin density vectors\nin Eq. (8) are microscopically affected by other spins\nthrough the dipolar fields in the effective magnetic fields.\nOn the other hand, the magnetization in Eq. (1) is af-\nfected by demagnetizing fields originating from macro-\nscopicallypolarizedmagnetizationin the condensed mat-\nter. Namely, Eq. (8) can describe the macroscopic de-\nmagnetizing field resulting from the microscopic dipolar\nfield. This is a very important difference between these\nequations.\nWe initially investigated the physics of the first and\nsecond terms of Eq. (8). To simplify the discussion, we\nconsidered the equation under the condition Heff=0.\nThus, we derived the continuity equations,\n∂Fi\n∂t+∇·Ji= 0, (9)\nwhereJk=jk−/planckover2pi1/(2Mi)∇Fkisaneffectivecurrentterm,\nJx=−i/planckover2pi12\n2√\n2M{ψ∗\n0∇(ψ1+ψ−1)+(ψ∗\n1+ψ∗\n−1)∇ψ0\n−ψ0∇(ψ∗\n1+ψ∗\n−1)−(ψ1+ψ−1)∇ψ∗\n0},(10a)\nJy=/planckover2pi12\n2√\n2M{ψ∗\n0∇(ψ1−ψ−1)−(ψ∗\n1−ψ∗\n−1)∇ψ0\n+ψ0∇(ψ∗\n1−ψ∗\n−1)−(ψ1−ψ−1)∇ψ∗\n0},(10b)\nJz=−i/planckover2pi12\n2M(ψ∗\n1∇ψ1−ψ1∇ψ∗\n1−ψ∗\n−1∇ψ−1+ψ−1∇ψ∗\n−1).\n(10c)\nEquation (9) can also be rewritten as,\nd\ndt/integraldisplay\nVFidV=/integraldisplay\nV∇·JidV=/integraldisplay\nSJi·ndS,\nby using the volume integral and the surface integral,\nwhose unit vector nis vertical to the surface for Stokes’\ntheorem. The equation indicates that the expectation\nvalue of the spin matrix ∝an}b∇acketle{tˆFi∝an}b∇acket∇i}ht=/integraltextdVFiin the volume V\nis conserved for the spin probability flux Jileaving and\nentering the surface.\nUnderHeff∝ne}ationslash= 0, the Kittel-like equation can be re-\nduced to the following equation,\n∂Fi\n∂t+∇·Ji= [F×Heff]i, (11)\nwhere the right side of the equation breaks the conserva-\ntion law of spin density. Therefore, the Kittel-like equa-\ntionshavetwodynamics: spin precessionswith frequency\ngiven by the effective magnetic field and spin currents\nwithout spin conservation. The spin currents of the sys-\ntem will be discussed in Sec. IVBIII. FMR UNDER SINGLE-MODE\nAPPROXIMATION\nIn order to study the basic properties of the second\nterm in Eq. (7), we introduced the single-mode approxi-\nmation,\nψi(r,t) =√\nNξi(t)φ(r)exp/parenleftbigg\n−iµt\n/planckover2pi1/parenrightbigg\n,(12)\nwhereφsatisfies the eigenvalue equation ( −/planckover2pi12∇2/2M+\nV+c0n)φ=µφwith the relation/integraltextdr|φ|2= 1. The\napproximation is effective when the shapes of the con-\ndensates are determined by the spin-independent terms,\nnamely|c0| ≫ |c2|[21]. For87Rb and23Na, the re-\nlation is satisfied. Under this approximation, the first\nterm of Eq. (7) vanishes, and we obtain the Kittel equa-\ntion for the spatially independent spin density vector\nS= (Sx,Sy,Sz),\ndS\ndt=γe[S×HSMA\neff], (13)\nwhere,\nSx=/planckover2pi1√\n2{ξ∗\n0(ξ1+ξ−1)+ξ0(ξ∗\n1+ξ∗\n−1)},\nSy=i/planckover2pi1√\n2{ξ∗\n0(ξ1−ξ−1)−ξ0(ξ∗\n1−ξ∗\n−1)},\nSz=/planckover2pi1(|ξ1|2−|ξ−1|2),\nand the effective magnetic field HSMA\neff =\n(−Nx\nddSx,−Ny\nddSy,H−Nz\nddSz) is given by\nNi\ndd=cdd\ngµBN/integraldisplay /integraldisplay\ndrdr′|φ(r)|2|φ(r′)|2\n|r−r′|3{1−3ei/summationdisplay\njej}.\n(14)\nEquation (13) also indicates that the spin vector Spre-\ncesses around HSMA\neff. The precession frequency reveals\nthe characteristic dynamics. Next, we consider a small\ndeviationδS= (δSx,δSy,δSz) around the stationary so-\nlution,S0=S0ˆzwithS0=H0/Nz\ndd, of Eq. (13), namely\nS=S0+δS. Introducing this representation into Eq.\n(13) and linearizing the equation, we derived the follow-\ning equations,\nd\ndtδSx=γe{H+(Ny\ndd−Nz\ndd)S0}δSy,\nd\ndtδSy=−γe{H+(Nx\ndd−Nz\ndd)S0}δSx,\nd\ndtδSz= 0,\nwhich give the resonance frequency,\nω2=γ2\ne{H+(Nx\ndd−Nz\ndd)S0}{H+(Ny\ndd−Nz\ndd)S0}(15)\nThespinprecesseswiththe resonancefrequency ω, which\ndepends on the dipolar terms Ni\ndd.4\nHere, we consider the single particle density distri-\nbution|φ(r)|2∝e−(x2+y2+λzz2)/a2, whereλzis the as-\npect ratio, and discuss simple situations. For the spher-\nical case of λz= 1, the integration (14) results in\nNx\ndd=Ny\ndd=Nz\ndd, givingω=γeH. The dipolar\nfields are canceled because of the isotropy, so that the\nspin precesses with Larmor frequency. For the circular\nplane (infinite cylinder) case of λz=∞(0), we obtain\nω=γe{H−(Nx\ndd−Nz\ndd)S0}forNx\ndd=Ny\ndd.\nIn this representation, it seems that the microscopic\ndipolar fields, Eq. (14), act as a macroscopic demagne-\ntizingfield tocompareEq. (2) with (15). We believethat\nthe origin of the demagnetizing field is an MDDI. If the\nabove discussion is correct, the dipolar coefficients Ni\ndd\nshoulddepend ontheshapeofthecondensates. However,\nthe single-mode approximation in spinor dipolar BECs\nis not effective in large-aspect-ratio condensates, as dis-\ncussed by Yi and Pu [22]. Therefore, we must consider\nthe spin dynamics beyond the approximation.\nIV. FMR FOR NUMERICAL CALCULATION\nA. Precession dependence on the aspect ratio λ\nIn this section, we discuss FMR by numerically calcu-\nlating the two-dimensional Eq. (3) under the condition\nof87Rb, namely c0≫ −c2>0. We begancalculatingthe\nspin precessions by applying a π/20 pulse to the ground\nstate, whose spins were polarized to the uniform mag-\nnetic field H=Hˆztrapped by V=Mω2\nx(x2+λ2y2)/2\nwithgµBH//planckover2pi1ωx= 20 and an aspect ratio λ=ωy/ωx.\nWe investigated the dynamics of ∝an}b∇acketle{tFx∝an}b∇acket∇i}htforλ= 0.5,1,\nand 1.5 with and without the MDDI. From t= 0 to\nπ/(20γeH), aπ/20 pulse was applied. Then, the spins\nwere tilted by π/20 radians from the zaxis with pre-\ncession. After turning off the pulse, the spins precessed\naround the zaxis, conserving ∝an}b∇acketle{tFz∝an}b∇acket∇i}ht. We define the nota-\ntion∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=λaand∝an}b∇acketle{tFi∝an}b∇acket∇i}htλ=λaas indicating the expectation\nvalues ofFiwith and without an MDDI in the trap with\nλ=λa.\nFirst, the typical motions of spins are shown in Fig. 1.\nInvestigatingthe time development of ∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=0.5,∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=1,\nand∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=1.5, we obtained the differences between their\nprecession frequencies, as shown in Fig. 1 (a) and (b).\nThe differences appeared at frequencies below the Lar-\nmor frequency, given by H. For 0 ≤t≥2, no devi-\nation between the precessions was observed, but devia-\ntions clearly appeared as more time elapsed. In order\nto demonstrate that the λdependence was given not by\nHbut byHdd, we show precessions for the same aspect\nratios without the MDDI in Fig. 1 (c) and (d). The\nprecession frequency did not change without the MDDI\nfor different values of λ. Therefore, the dipolar frequency\nωdd=γeHdddepends upon the shape of the condensate.\nNext, we examined the effects of the MDDI on the\nprecessions in Fig. 2. Comparing ∝an}b∇acketle{tFx∝an}b∇acket∇i}htdd\nλwith∝an}b∇acketle{tFx∝an}b∇acket∇i}htλ,!\"#$!\"#%\"\"#%\"#$\n\" % $!\"#$!\"#%\"\"#%\"#$\n&\" &% &$π/20pulse/angbracketleftFx/angbracketrightdd\nλ=0.5/¯h/angbracketleftFx/angbracketrightdd\nλ=1/¯h\n/angbracketleftFx/angbracketrightdd\nλ=1.5/¯h!\"# !$#\nωxt ωxt!\"#$!\"#%\"\"#%\"#$\n\" % $!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!%# !&#\n/angbracketleftFx/angbracketrightλ=1/¯h/angbracketleftFx/angbracketrightλ=1.5/¯h\n/angbracketleftFx/angbracketrightλ=0.5/¯h\nFIG. 1: (Color online) The time development of /angbracketleftFx/angbracketrightdd\nλ, (a)\nand (b), and /angbracketleftFx/angbracketrightλ, (c) and (d). The red solid, blue dashed,\nand green dotted lines show the results of λ= 0.5, 1, and 1 .5\nrespectively. The gray zone represents the duration of a π/20\npulse.\nwe observed that the MDDI caused an effective mag-\nnetic field, because the frequency of the precession with\nthe MDDI deviated from that without the MDDI in\nFig. 2 (a) to (f). Assuming that ∝an}b∇acketle{tFi∝an}b∇acket∇i}htdd\nλ=1− ∝an}b∇acketle{tFi∝an}b∇acket∇i}htλ=1\nis represented approximately to Acosγe(H+Hdd)t−\nAcosγeHtwith an amplitude A, we extracted the dipole\nfrequency from the waveform. Since the waveform be-\ncame−2Asinωddt/2sin(ωL+ωdd/2)t, the beat consisted\nof the large frequency ωL+ωdd/2 and the small fre-\nquencyωdd/2. From Fig. 2 (h), we estimated these\nfrequencies to obtain ωdd/ωL≃6.5,9,and11×10−3for\nλ= 0.5,1,and1.5 respectively.\nFigure 3 shows the λdependence of ωdd/ωL. From the\nresults, however, we cannot safely conclude that the λ\ndependence of the frequencies is given by changing the\nshape of the condensates, since the dipolar frequencies\nmay be given by change of the density with the shape.\nFMR in condensed matters has been discussed in con-\ndensed matter of uniform density, even with changing\nshape. On the other hand, atomic BECs have tunable\ndensity and shape. Therefore, our calculations indicate\ncharacteristic of FMR in atomic cold gases.\nB. Spin current\nWe observed spin currents driven by spin diffusion,\nwhich was caused by a rdependence of the dipolar field.\nFigure 4 shows the projections of Fonto thex-yplane5\n!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!\"#$!\"#%\"\"#%\"#$\n\" % $π/20pulse\nωxt ωxt!\"#\n!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!$#\n!\"#$!\"#%\"\"#%\"#$\n\" % $!%#\n!\"#$!\"#%\"\"#%\"#$\n&\" &% &$!&#\n!\"#$!\"#%\"\"#%\"#$\n\" % $!'# !(#\n!\"#$!\"#%\"\"#%\"#$\n\" &\" %\" '\" $\" (\"!)# !*#\n!\"#$!\"#%\"\"#%\"#$\n\" & % ' $ ( )λ=0.5\nλ=1\nλ=1.5λ=0.5\nλ=1\nλ=1.5\nFIG. 2: (Color online) Comparing the precession with and\nwithout the MDDI. (a) and (b), (c) and (d), and (e) and (f)\nshow the precession for λ= 0.5, 1, and 1 .5, respectively. The\nsolid anddashedlines are /angbracketleftFx/angbracketrightλand/angbracketleftFx/angbracketrightdd\nλ. (g)and(h)repre-\nsent (/angbracketleftFx/angbracketrightdd\nλ=0.5−/angbracketleftFx/angbracketrightλ=0.5)//planckover2pi1(solid), ( /angbracketleftFx/angbracketrightdd\nλ=1−/angbracketleftFx/angbracketrightλ=1)//planckover2pi1\n(dot), and ( /angbracketleftFx/angbracketrightdd\nλ=1.5−/angbracketleftFx/angbracketrightλ=1.5)//planckover2pi1(dashed), respectively.\nforλ= 1.5 andωxt= 12.7. The precession with the\nMDDI lost homogeneity of the spin directions, whereas\nthe precession without the MDDI maintained this ho-\nmogeneity. This is because the precession frequency\nhas anrdependence, specifically, ω(r) =γeHeff(r) =\nγe(H+Hdd(r)).\nThe dipole interaction drives the spin diffusion, which\nis shown in Fig. 5. The figure shows Fx/|Fxy|= cosφas\na function of xaty= 0, where φis the angle between\nthe spin vector and the xaxis. In the dynamics with!\"!#!\"!#$!\"!%!\"!%$!\"!&\n!!\"$##\"$%%\"$&\nλωdd\nωL\nFIG. 3:λdependence of ωdd/ωL.\nxy\n03.637×10−3\n0!\"# !$#1.964×10−3\nFIG. 4: (Color online) Projection of Fonto the x-yplane\nforλ= 1.5 andωxt= 12.7. The figures show the results (a)\nwith MDDI and (b) without that. The vectors are nondimen-\nsionalized.\nthe dipole interaction for λ= 1.5 (a) and 1 (b), the\nspin densities lost their angular coherence, whereas the\ndynamicswithout the dipole interactionsmaintained this\ncoherence ( (c) and (d)).\nThe spin diffusion drives the spin current Jkin Eq.\n(10), which is shown in Fig. 6. In order to explain how\nthe spin current is driven by the spin diffusion, we con-\nsidered the amplitudes of the wave functions ψj=fjeiϕj\nas,\nψ1(r,t) =/radicalbig\nn(r,t)\n2(1+cosθ(r,t))eiϕ1(r,t),\n(16a)\nψ0(r,t) =/radicalbigg\nn(r,t)\n2sinθ(r,t)eiϕ0(r,t),(16b)\nψ−1(r,t) =/radicalbig\nn(r,t)\n2(1−cosθ(r,t))eiϕ−1(r,t),\n(16c)6\n!\"!#$%##$%\"\n!\"#!%# % \"#!\"#!\"!#$%##$%\"\n!\"#!%# % \"#!$#\n!%#\nx xλ=1λ=1.5\nωxt=0.128\n163240\n4848\n!\"!#$%##$%\"\n!\"#!%# % \"#λ=1\n0.128\n1632\n40\n0.12 4048 32 16\n8!&#\nλ=1.5\n!\"!#$%##$%\"\n!\"#!%# % \"#0.12 40832 16 48cosφ cosφ\nFIG. 5: Dynamics of a cross-section of Fx/|Fxy|aty= 0,\nwhere|Fxy|=/radicalbig\nF2x+F2y. From the relation Fx=|Fxy|cosφ,\nthe parameter represents cos φ. The results with the MDDI\n(a) and without it (b) are shown for λ= 1.5, and (c) and (d)\nshow results for λ= 1. The xaxis are nondimensionalized by/radicalbig\n/planckover2pi1/Mωx\nwhere the forms show the ground state of the ferromag-\nnetic state [23]. The amplitude is represented by nand\nthe angleθbetween the spin and the zaxis. We intro-\nduced this representation to demonstrate that the spin\ncurrent is derived from the spin diffusion. Of course, we\nconfirmed the validity of the ferromagnetic representa-\ntion under the pulse and magnetic field by calculating θ\ndirectly. Therefore, it can be utilized for the polarized\nspin state studied in our work. The amplitudes f±1were\nformed to represent Fz=n/planckover2pi1cosθ, andf0was deter-\nmined to satisfy the relation n=/summationtext\nj|ψj|2. For example,\n(n1,n0,n−1) = (n,0,0) led toFz=n/planckover2pi1withθ= 0, and\n(n1,n0,n−1) = (n/4,n/2,n/4) resulted in Fz= 0 with\nθ=π/2. The wave function can only express the fer-\nromagnetic states, i.e.the form cannot represent the\nantiferromagnetic state ( n1,n0,n−1) = (n/2,0,n/2) or\nthe polar state ( n1,n0,n−1) = (0,n,0). This restriction\nof the wave function is caused by the first representation\nFz=n/planckover2pi1cosθ.\nBy introducing this representation into Eqs. (6) and\n(10), we can redefine as follows,\nFx=n/planckover2pi1sinθ(cosϕrcosϕ−cosθsinϕrsinϕ),\nFy=−n/planckover2pi1sinθ(cosϕrsinϕ+cosθsinϕrcosϕ),and,\nJx=n/planckover2pi12\n4M/braceleftbigg\nsinθ(1+cosθ)cos(ϕ1−ϕ0)∇ϕ1\n+ sinθ(1−cosθ)cos(ϕ−1−ϕ0)∇ϕ−1\n−2sinθ(cosϕrcosϕ−cosθsinϕrsinϕ)∇ϕ0\n+ 2(cosϕrsinϕ+cosθsinϕrcosϕ)∇θ/bracerightbigg\n,(17a)\nJy=−n/planckover2pi12\n4M/braceleftbigg\nsinθ(1+cosθ)sin(ϕ1−ϕ0)∇ϕ1\n−sinθ(1−cosθ)sin(ϕ−1−ϕ0)∇ϕ−1\n+ 2sinθ(cosϕrsinϕ+cosθsinϕrcosϕ)∇ϕ0\n+ 2(cosϕrcosϕ−cosθsinϕrsinϕ)∇θ/bracerightbigg\n,(17b)\nJz=n/planckover2pi12\n4M{(1+cosθ)2∇ϕ1−(1−cosθ)2∇ϕ−1},\n(17c)\nwhereϕr= (ϕ1+ϕ−1−2ϕ0)/2 andϕ= (ϕ1−ϕ−1)/2 are\nrelative phases. Since the relation ϕr= 0 was satisfied in\nour calculations, we used the relation in Eqs. (17), and\nthe spin density vectorformed an azimuthal angle ϕwith\nthexaxis. Then, we derived the spin components Fx=\nn/planckover2pi1cosϕsinθ,Fy=n/planckover2pi1cosϕsinθ, andFz=n/planckover2pi1cosθ. We\ncan therefore rewrite the spin density currents,\nJx=n/planckover2pi12\n4M(4cosϕsinθ∇ϕ0\n+2cosϕsinθcosθ∇ϕ−2sinϕ∇θ),(18a)\nJy=−n/planckover2pi12\n4M(4sinϕsinθ∇ϕ0\n+2sinϕsinθcosθ∇ϕ+2cosϕ∇θ),(18b)\nJz=n/planckover2pi12\n4M{4cosθ∇ϕ0+4(1+cos2θ)∇ϕ},(18c)\nwhich are driven by the gradients of the angles, ϕandθ,\nandthephase ϕ0. IntheprecessionswithMDDI,thegra-\ndients occurred because of the dipolar fields Hdd(r). As\na result, the spin currents were clearly driven, as shown\nin Fig. 6. For ωxt= 0.12, the spin vectors were coher-\nent just after the applied π/20 pulse (Fig. 6 (a)). The\nspin densities, FxandFy, then flowed to the center of\nthe condensates from Fig. 6 (b) to (c). Then, the den-\nsities reversed, and diffused outward from Fig. 6 (d) to\n(e). This oscillation was repeated. Of course, we cannot\nobtain the spin current without the dipolar interactions,\nsince the gradients of θandϕwere not caused; the dy-\nnamics are shown in Fig. 7.\nIn order to investigate the spin fluid dynamics, we cal-\nculated the spin current Jxfor Eq. (17), as shown in\nFigs. 8 and 9. These figures represent Jxfrom the pre-\nvious calculations with λ= 1 and 1.5 respectively. De-\nspite the difference in the ratio, we observed two com-\nmon properties in these figures. The direction of the\ncurrents changed rapidly, corresponding to the large pre-\ncession frequency, and the magnitudes changed slowly7\n1.961×10−3\n0\n!\"# !$# 3.152×10−3\n0\n 0\n3.447×10−3!%#\n3.168×10−3\n0\n!&#2.113×10−3\n0\n!'# !(#3.043×10−3\n0\nωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=48\nFIG. 6: (Color online) Dynamics of Fprojected onto the x-\nyplane for λ= 1.5 with dipolar interaction.\nwith the small dipolar frequency, as shown in Fig. 10,\nwhich shows the time development of the xcomponent\nofJx(x= 4,y= 0). This figure indicates that the oscil-\nlation of the current direction occurred with the preces-\nsion frequency, which varied in magnitude with changing\ndipolar frequency. Eq. (11) also indicates that the spin\ndensity was not conserved because of the effective mag-\nnetic field. Therefore, the spin currents can be driven\nfrom a source and sink in the center of the condensates,\nas in Figs. 8 and 9. The two common properties were\ninsensitive to the value of λ. However, the change in spin\ndensity for λ= 1.5 exhibited quadratic pole motion in\na scissors-like mode for mass density [24], which can be\nunderstood as an oscillationbetween the spin density mi-\ngrating to the yaxis from the xaxis and back again, as\nshown in Figs. 6 (a) to (c). Therefore, the spin collec-\ntive mode was caused by spin diffusions induced by the\nMDDI. Therefore, the spin current causes the dynam-\nics of spin scissors-like mode, which was observed as a\nshrinking and expansion of the spin density in Fig. 6.\nThe shrinking and expansion were common features for\nλ= 1 and 1.5. However, the spin currents were affected\nby the symmetry of the traps, as shown in Figs. 8 and 9.\nFrom the calculations, we expected that the spin cur-\nrent would be observable when using the spinor BECs.\nRecently, spin current is focused from fields of spintron-\nics. However, it is difficult to observe the spin current in\nmetals and condensed matter. Atomic BECs, a macro-\nscopic quantum phenomenon, can show the spin current\nclearly and directly in the dynamics of the spinor den-\nsities. Therefore, we should attempt to observe various\nspin currents utilizing tunable experimental parameters,\ni.e.interaction parameters, trap frequencies, and the\nnumber of particles.\nV. CONCLUSION\nWe investigated the properties of magnetic resonance\nin spinor dipolar BECs by calculating the GP equations,\nobtainingKittel-likeequationsastheequationsofmotion!\"# !$# !%#\n!&# !'# !(#0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3\n0\n1.965×10−3ωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=48\nFIG. 7: (Color online) Dynamics of Fprojected onto the x-\nyplane for λ= 1.5 without dipolar interaction.\nωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=481.95×10−4\n0\n!\"# !$# !%#\n!&# !'# !(#0\n2.69×10−4\n0\n9.8×10−5\n0\n4.6×10−5\n02.73×10−4\nFIG. 8: (Color online) Dynamics of the spin currents Jxpro-\njected onto the x−yplane for λ= 1 with dipolar interaction.\nThe vectors are nondimensionalized.\nfor the spin density vector. The equations revealed two\nproperties. One is the dynamics of the spin fluid, and\nthe other is precession under the effective magnetic field\nconsisting of the external magnetic fields and the dipolar\nfields. The magnetic resonancewith the properties of the\nspin fluid was characteristic of this system.\nIn order to extract properties from the GP equations,\nwe studied the law of conservation of spin density cur-\nrent without effective magnetic fields by first deriving\nthe continuity equations from the GP equation, obtain-\ning representations of the spin current. Second, we ana-\nlytically evaluated the precession dynamics described by\nthe Kittel equations derived from the GP equations us-\ning a single-mode approximation, where the Kittel equa-\ntions show conventional FMR. The analysis clearly in-\ndicated that the origin of the FMR in the BECs is like\nthe dipolar field, whereas the origin of the resonance in\nthe Kittel equations for condensed matter is the demag-\nnetizing field. Comparing the FMR of the BEC with\nthat of the condensed matter, we concluded that the ori-\ngin of the resonance was not the spin exchange interac-\ntion that causes magnetism in condensed matter, but the8\nωxt=0.12 ωxt=8 ωxt=16\nωxt=32 ωxt=40 ωxt=48!\"# !$# !%#\n!&# !'# !(#0\n3.59×10−4\n0\n2.76×10−4\n0\n2.13×10−4\n0\n8.5×10−5\n05.5×10−5\nFIG. 9: (Color online) Dynamics of the spin currents Jxpro-\njectedontothe x−yplanefor λ= 1.5withdipolarinteraction.\n!\"#\"\"\"$\"\"#\"\"\"$\n\" %\" $\" &\" '\" (\"!\"#\"\"\"$\"\"#\"\"\"$\n\" %\" $\" &\" '\" (\"{Jx}x(x=4,y=0)\nωxtλ=1 λ=1.5\nωxt!\"# !$#\n!\n! \" #!\n! \" #\nFIG. 10: Dynamics of the xcomponent of Jxatx= 4 and\ny= 0. The inter figures are the results for ωxt= 0 to 4.anisotropy of the MDDI. Finally, we numerically calcu-\nlated the GP equations, representing the dynamics with\nthe twocommon properties. The characteristicdynamics\nshowed that the effective magnetic field introduced spin\ndiffusion into the Larmor precession, driving the spin-\ncurrent-like scissors modes.\nThe relation between the spin current and FMR has\nnot yet been discussed for typical FMR. Therefore, it is\nimportant to study spin current in condensates. We also\nbelieve that the study of spin current will be useful for\nthe development of spintronics, because it is difficult to\ndirectly observe spin currents in condensed matter spin-\ntronics.\nVI. ACKNOWLEDGMENT\nM. Y. acknowledges the support of a Research Fellow-\nship of the Japan Society for the Promotion of Science\nfor Young Scientists (Grant No. 209928). M. T. ac-\nknowledges the support of a Grant-in Aid for Scientific\nResearch from JSPS (Grant No. 21340104).\n[1] C. P. Slichter, Principles of Magnetic Resonance , (Berlin:\nSpringer–Verlag, 1990).\n[2] J. H. E. Griffiths, Nature 158, 670 (1946).\n[3] C.Kittel, Phys.Rev. 71, 270(1947) ; ibid.73, 155(1948).\n[4] C. Kittel, Introduction to Solid State Physics, 8th ed.\n(John Wiley and Sons Inc., USA, 2005)\n[5] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975).\n[6] D. Vollhard and P. W¨ olfle, The Superflluid Phasess of\nHelium 3 (Taylor and Francis, London, 1990).\n[7] R. A. Webb, R. L. Kleinberg, and J. C. Wheatley, Phys.\nRev. Lett. 33, 145 (1974).\n[8] K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van\nDruten, D. S. Durfee, D. M. Kurnand W. Ketterle, Phys.\nRev. Lett. 75, 3969 (1995).\n[9] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E.\nWieman and E. A. Cornell, Science 269, 198 (1995).\n[10] S. Yi, L. You and H. Pu, Phys. Rev. Lett. 93, 040403\n(2004)\n[11] Y. Kawaguchi, H. Saito and M. Ueda, Phys. Rev. Lett.\n97, 130404(2006).\n[12] H.M¨ akel¨ aandK.-A.Suominen, Phys.Rev.A 75, 033610\n(2007).\n[13] Y. Kawaguchi, H. Saito and M. Ueda, Phys. Rev. Lett.96, 080405 (2006).\n[14] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T.\nPfau, Phys. Rev. Lett. 94, 160401 (2005).\n[15] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau,\nS. Giovanazzi, P. Pedri, and L. Santos, Phys. Rev. Lett.\n95, 150406 (2005).\n[16] T. Lahaye, T. Koch, B. Fr¨ ohlich, M. Fattori, J. Metz, A.\nGriesmaier, S. Giovanazzi, and T. Pfau, Nature 448, 672\n(2007).\n[17] E. L. Hahn, Phys. Rev. 80, 580 (1950).\n[18] H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954).\n[19] M. Yasunaga and M. Tsubota, Phys. Rev. Lett. 101\n220401 (2008).\n[20] M. Yasunaga and M. Tsubota, J. Low Tem. Phys. 158,\n51 (2010).\n[21] W. Zhang, D. L. Zhou, M. -S. Chang, M. S. Chapman,\nand L. You, Phys. Rev. A 72, 013602 (2005).\n[22] S. Yi and H. Pu, Phys. Rev. Lett. 97, 020401 (2006).\n[23] T. -L. Ho, Phys. Rev. Lett. 81, 742 (1998).\n[24] C. J. Pethick and H. Smith, Bose-EInstein Condensation\nin Dilute Gases, 2nd ed. (Cambridge, New York, 2008)."    },    {        "title": "2011.05566v1.Reduction_of_back_switching_by_large_damping_ferromagnetic_material.pdf",        "content": "arXiv:2011.05566v1  [cond-mat.mes-hall]  11 Nov 2020Reduction of back switching by large damping ferromagnetic material\nTomohiro Taniguchi1, Yohei Shiokawa2, and Tomoyuki Sasaki2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nResearch Center for Emerging Computing Technologies, Tsuk uba, Ibaraki 305-8568, Japan,\n2TDK Corporation, Advanced Products Development Center, Ic hikawa, Chiba 272-8558, Japan\nRecent studies on magnetization dynamics induced by spin-o rbit torque have revealed a weak\ndependence of the critical current for magnetization switc hing on the damping constant of a ferro-\nmagnetic free layer. This study, however, reveals that the d amping constant nevertheless plays a key\nrole in magnetization switching induced by spin-orbit torq ue. An undesirable switching, returning\nto an initial state, named as back switching, occurs in a ferr omagnet with an easy axis parallel\nto the current direction. Numerical and theoretical analys es reveal that back switching is strongly\nsuppressed when the damping constant of the ferromagnet is l arge.\nLow-damping ferromagnetic materials has been inves-\ntigated for spintronics applications [1–6]. These materi-\nals are interesting because the critical current for excit-\ning magnetization dynamics by spin-transfer torque[7, 8]\nin two-terminal devices is typically proportional to the\ndampingconstant[9,10], andtherefore, low-dampingfer-\nromagnetic materials help to reduce power consumption.\nThe value of the damping constant in typical ferromag-\nnets, such as Fe, Co, Ni, and their alloys, is on the order\nof 10−3−10−2, even after the effect of spin pumping\n[11, 12] is taken into account [5, 6]. Studies on three-\nterminal devices manipulated by spin-orbit torque [13–\n24], however, have revealed that the dependence of the\ncritical current on the damping constant is weak when\nthe easy axis of the ferromagnet is perpendicular to the\nfilm plane [25, 26] or parallelto the currentdirection [27];\nthese systems are named type Z and type X, respectively,\nin Ref. [19]. Then, a question arises as to what the role\nof the damping constant is in three-terminal devices.\nThe purpose of this study is to investigate the rela-\ntion between the magnetizationstate manipulated by the\nspin-orbit torque and the magnetic damping constant by\nsolving the Landau-Lifshitz-Gilbert (LLG) equation. We\nfocus onthe type-Xsystembecause the switchingmecha-\nnism in the system is not fully understood yet, unlike the\nothersystems[9,10,25,26]. Anundesirablereturnofthe\nmagnetization,calledbackswitchinginthispaper,occurs\nin a ferromagnet, where, although the spin-orbit torque\nbrings the magnetization close to the switched state, it\nreturns to the initial state after turning off the current.\nAs a result, the phase diagramofthe magnetization state\nas a function of the current and external magnetic field\nalternately shows switched and non-switched states. It\nis revealed that the back switching appears as a result of\nthe magnetization precession around the perpendicular\naxis after the current is turned off. The back-switching\nregion is strongly suppressed by using large-damping fer-\nromagnetic materials.\nThe system studied in the work is an in-plane mag-\nnetized ferromagnet placed on a nonmagnetic metal and\nschematically shown in Fig. 1(a). Electric current flow-\ning in the nonmagnet in the xdirection generates pure\nspincurrentthatisinjectedintotheferromagnetwiththez\nxy\njm\nHappl\nj=300 MA cm-2Happl=500 Oemx1\n0\n-1\nTime (ns)0 1(a) (b)\nFIG. 1: (a) Schematic illustration of the system. The spin\nHall effect in the bottom nonmagnet generates spin current\nwith a polarization along the ydirection and excites the mag-\nnetization ( m) dynamics in the top ferromagnet. The easy\naxis of the ferromagnet is parallel to the xaxis. An external\nmagnetic field Happlis applied in the zdirection. (b) Exam-\nple of the time evolution of mxforj= 300 MA cm−2and\nHappl= 500 Oe.\nspin polarization in the ydirection. The spin current ex-\ncitesthespin-transfertorqueactingonthemagnetization\nin the ferromagnet and induces magnetization dynamics.\nThezaxis is perpendicular to the film plane. We denote\nthe unit vectors pointing in the magnetization direction\nin the ferromagnet and along the k-axis (k=x,y,z) as\nmandek, respectively. The magnetization dynamics are\ndescribed by the LLG equation\ndm\ndt=−γm×H−γHsm×(ey×m)+αm×dm\ndt,(1)\nwhereγandαarethe gyromagneticratioandthe Gilbert\ndamping constant, respectively. The magnetic field H\nconsists of the in-plane magnetic anisotropy field HKin\nthexdirection, the demagnetization field −4πMin thez\ndirection, and an external field Happlin thezdirection:\nH=HKmxex+(Happl−4πMmz)ez.(2)\nThe strength of the spin-transfer torque is given by\nHs=/planckover2pi1ϑj\n2eMd, (3)\nwhereϑis the spin Hall angle in the nonmagnet, whereas\nManddare the saturation magnetization and thickness\nof the ferromagnet. The electric current density is de-\nnoted as j. The values of the parameters used in this2\n\"mx.dat\" u 1:2:3 \"mx_relax.dat\" u 1:2:3 (a)\nApplied field, Happl (kOe)Current density, j (MA cm -2 )\n0 1 2 0 1 202004006008001000 1\n0\n-1 mx\nApplied field, Happl (kOe)1\n0\n-1 mx\nS0 S1 S2\njc(b)\nCurrent density, j (MA cm -2 )\n02004006008001000\nFIG. 2: Phase diagrams of mx(a) at a fixed point in the\npresence of current and (b) in a relaxed state after turning o ff\nthe current. The initial state is m0−. The white dotted line in\n(a) represents the theoretical formula of the critical curr ent\ndensity derived in Ref. [27]. Switching regions are distin-\nguished by the labels S n(n= 0,1,2) in (b). The damping\nconstant αis 0.005.\nstudy were taken from typical experiments [13–23] as\nM= 1500 emu c.c.−1,HK= 200 Oe, γ= 1.764×107\nrad Oe−1s−1,ϑ= 0.4, andd= 1.0 nm.\nThe magnetic field His related to the magnetic energy\ndensityEviaE=−M/integraltext\ndm·H,\nE=−MHapplmz−MHK\n2m2\nx+2πM2m2\nz.(4)\nThe energy density Ehas two minima at m0±=\n±/radicalbig\n1−m2\n0zex+m0zez, wherem0z=Happl/(HK+4πM).\nThroughout this paper, the initial state is set to be m0−,\nwhich points in the negative xdirection. Accordingly, we\ncall the other stable state, m0+pointing in the positive x\ndirection, the switched state. Thus, we are interested in\nexperiments where the initial state is reset to m0−dur-\ning each trial ofmagnetization switching. By convention,\nwe will focus on switching by a positive current and field\nregion.\nFigure 1(b) shows an example of the time evolution of\nmxin the presence of current. The damping constant is\nα= 0.005. The results indicate that the magnetization\nsaturates to a fixed point. Figure 2(a) is a phase diagram\nsummarizing the fixed point of mxin the presence of cur-\nrent, where the vertical and horizontal axes represent the\ncurrent density jand the external magnetic field Happl.\nThe magnetization stays near the initial state [ mx≃ −1\nshown in black in Fig. 2(a)] in a relatively small cur-\nrent region, where its boundary is well explained by the\ncritical-currentformula jc[27] shownby the white dotted\nline in Fig. 2(a). On the other hand, the magnetization\nstate above the critical current satisfies mx>0. There-\nfore, one might suppose that the magnetization relaxes\nto the switched state, mx≃+1, after turning off the\ncurrent.\nHowever, the phase diagram after turning off the cur-\nrentshowninFig. 2(b) revealsacomplicateddependence\nof the relaxed state on the current and external magnetic\nfield. Therearestripesdistinguishingtheswitched( m0+,\nyellow) and non-switched ( m0−, black) states. This re-\nsult indicates that the magnetization returns to the ini-tial state under certain conditions. We name this phe-\nnomenon back switching and investigate its relation to\nthe damping constant in the following.\nLet us briefly comment onanotherphenomenon, called\nbackhopping [28–32], to avoid any confusion. Back-\nhopping is a phenomenon in which, after magnetization\nswitchinginafreelayerisachieved,anundesirablereturn\nto the original resistance-state occurs at a high-bias volt-\nage. There are two differences between back switching\nandbackhopping. First, backhoppingintwo-terminalde-\nvices originates from magnetization switching in the ref-\nerence layer. On the other hand, back switching relates\nto the magnetization dynamics in the free layer. Second,\nbackhopping has often been investigated by sweeping the\ncurrent. On the other hand, we reset the initial state of\nthe magnetization at each trial because we are interested\nin designing the operation conditions of memory devices.\nBack switching originates from the fact that the fixed\npoint of the magnetization in the presence of current is\nnot parallel to the easy axis of the magnetization, as dis-\ncussed below, and thus, it will occur in not only type-X\ndevices, but also type-Z devices [25, 26].\nNote that the horizontal stripe in the relatively low\ncurrent region of Fig. 2(b), slightly above jc, was ana-\nlyzed in Ref. [27], and therefore, it will be excluded from\nthe following discussion. We label the other switched\nregions shown by the vertical stripes as the S n-region\n(n= 0,1,2,···); see Fig. 2(b). The role of the integer n\nwill be clarified below.\nFigure 3(a) shows the dynamic trajectory of the mag-\nnetization obtained by numerically solving Eq. (1),\nwherej= 300 MA cm−2andHappl= 500 Oe correspond\nto the S 0region. The red line represents the trajectory\nfrom the initial state to the fixed point in the presence of\nthe current, whereas the blue line shows the relaxation\ndynamics after turning off the current. The fixed point\nsatisfying dm/dt=0in the presence of the current is\nindicated by a yellow circle. Note that the fixed point is\nfar away from the energetically stable states, m0±. This\nis because the spin-transfer torque drives the magnetiza-\ntion in the ydirection. The black solid line in Fig. 3(b)\nshows the time evolution of mxafter turning off the cur-\nrent. Starting from the fixed point located in the region\nofmx>0, the magnetization relaxes to the switched\nstatem0+after showing several precessions around it.\nFigures 3(c) and 3(d), on the other hand, show an exam-\nple of back switching, where the parameter Happl= 750\nOe corresponds to the region sandwiched by the S 0and\nS1-regions. In this case, the magnetization after turning\noff the current shows a precession around the zaxis. As\na result, even though the fixed point in the presence of\nthe current is in the region of mx>0, the magnetization\nreturns to the initial state ( m0−). Let us then move to\nthe next switching region, the S 1-region. The dynamic\ntrajectory, as well as mx, shown in Figs. 3(e) and 3(f),\nindicates that, starting from the fixed point in the region\nofmx>0, the magnetization returns once to the region\nofmx<0 before relaxing to the switched state. Now3\n(a) (b)\nj=300 MA cm-2Happl=500 Oe\nj=300 MA cm-2Happl=750 Oe\nj=300 MA cm-2Happl=1000 Oemz1\n0\n-1\n0-1\n-111\n0mx mymx1\n0\n-1\nTime (ns)0 1m0+m0-\n(e) (f)\nmz1\n0\n-1\n0-1\n-111\n0mx mymx1\n0\n-1\nTime (ns)0 1m0+m0-(c) (d)\nmz1\n0\n-1\n0-1\n-111\n0mx mymx1\n0\n-1\nTime (ns)0 1m0+m0-\nNormalized energy (×10-3) Normalized energy (×10-3) Normalized energy (×10-3)1\n0\n-1\n-2\n-3\n-4\n12\n0\n-1\n-2\n-3\n-4\n1234\n0\n-1\n-2\n-3\n-4\nFIG. 3: (a) Dynamictrajectory in theS 0-region;Happl= 500\nOe, and j= 300 MA cm−2. The red line represents the\ntrajectory from the initial state to the fixed point (yellow\ncircle) in the presence of the current, whereas the blue line\ncorresponds to the relaxation dynamics after turning off the\ncurrent. The symbols m0±indicate the locations of the initial\n(m0−) and switched ( m0+) states. (b) Time evolutions of mx\n(black solid line) and normalized energy density ε(dashed red\nline) after turning off the current. The red triangle indicat es\nthe time at which εbecomes zero. (c) Dynamic trajectory\nand (d) time evolution of mxandεin the region sandwiched\nby the S 0- and S 1-regions; Happl= 750 Oe, and j= 300 MA\ncm−2. (e) Dynamic trajectory and (f) time evolution of mx\nandεin the S 1-region; Happl= 1000 Oe, and j= 300 MA\ncm−2.\nthe meaning of the integer n(n= 0,1,2,···) we used to\ndistinguish the switched regions becomes clear: it repre-\nsents how many times the magnetization returns to the\nregion of mx<0 before relaxing to the switched state.\nThese results imply that the precession around the z\naxis after turning off the current is the origin of back\nswitching. Such a precession is induced by the precession\ntorque,−γm×H. In particular, the precession around\nthezaxis occurs when the energy density at the fixed\npoint of the magnetization in the presence of the current\nis larger than the saddle-point energy given by\nEd=−MH2\nappl\n8πM. (5)In fact, a fixed point of the LLG equation is given by\nmx=HapplHs\nH2s−HK4πM, mz=−HapplHK\nH2s−HK4πM.(6)\nSubstituting Eq. (6) into Eq. (4), the energy density at\nthis fixed point is\nEj=MH2\napplHK\n2(H2s−HK4πM). (7)\nNote that H2\ns−HK4πM >0 because the back switch-\ning appears in the current region above jcforHappl=\n0, which is given by jc= [2eMd/(/planckover2pi1ϑ)]√HK4πM[27].\nTherefore, Eq. (7) is alwayspositive, whereasthe saddle-\npoint energy density given by Eq. (5) is negative. Ac-\ncordingly, the fixed point in the presence of the current is\nin the unstable region corresponding to an energy larger\nthan the saddle-point energy. As a result, after turn-\ning off the current, the magnetization starts to precess\naround the zaxis, as mentioned above. Simultaneously,\nthe magnetization loses energy due to damping torque,\nwhere the dissipated energy density is given by\n∆E(t) =αγM\n1+α2/integraldisplayt\n0dt′/bracketleftBig\nH2−(m·H)2/bracketrightBig\n.(8)\nHere,tis the time after the current is turned off. When\nthe condition,\nEj−∆E(t) =Ed, (9)\nis satisfied, the magnetization relaxes to the nearest sta-\nble state. The point here is that the magnetization al-\nternately comes close to two stable states, m0±, due to\nthe precession around the zaxis before Eq. (9) is sat-\nisfied. As a result, both m0+andm0−can be the final\nstate, which leads to back switching. The conclusions of\nthe discussion are confirmed by the dashed red lines in\nFigs. 3(b), 3(d), and 3(f), where are the time evolutions\nof the energy density. Here, we introduce the normalized\nenergy density,\nε(t) =E(t)−Ed\n4πM2. (10)\nTheenergydensity Eisafunction oftime with the initial\ncondition of E(t= 0) =EjbecauseE, given by Eq. (4),\ndepends on the magnetization direction, and the magne-\ntization changes direction in accordance with the LLG\nequation. The condition given by Eq. (9) is ε(t) = 0\nin terms of the normalized energy density. The times at\nwhichεbecomes zero are indicated by the red triangles\nin the figures. It is clear from Figs. 3(b), 3(d), and 3(f)\nthat the final state of the magnetization is determined by\nwhether mxis positive or negative when εbecomes zero.\nIt is expected that using a large-damping ferromag-\nnetic material will result in a reduction of the back-\nswitching region. Remember that back switching occurs\ndue tothe precessionaroundthe zaxis. When the damp-\ning constant is large, the energy dissipation given by Eq.4\n\"mx_relax.dat\" u 1:2:3 \"mx.dat\" u 1:2:3 mx1\n0\n-1 \nTime (ns)0 1\nNormalized energy (×10 -3 )\nNormalized energy (×10 -3 )\n2\n0\n-2 \n-6 -4 mx1\n0\n-1 \nTime (ns)0 124\n0\n-2 \n-6 -4 \n(c)(a)\nApplied field, Happl (kOe)Current density, j (MA cm -2 )\n0 1 2 0 1 202004006008001000 1\n0\n-1 mx\nApplied field, Happl (kOe)1\n0\n-1 mx(d)(b)\nCurrent density, j (MA cm -2 )\n02004006008001000\nFIG. 4: Time evolutions of mx(black solid lines) and nor-\nmalized energy density ε(red dashed lines) after turning off\nthe current, where j= 300 MA cm−2and (a)Happl= 750 Oe\nand (b) 1000 Oe. The damping constant αis 0.050. The red\ntriangles indicate the time at which εbecomes zero. Phase\ndiagrams of mx(c) at a fixed point in the presence of current\nand (d) in a relaxed state after turning off the current.\n(8) becomes large within a short time t, and the condi-\ntion given by Eq. (9) is immediately satisfied before the\nmagnetizationreturnstothe regionof mx<0bythepre-\ncession around the zaxis. Accordingly, back switching\ndoes not occur. On the other hand, when the damping\nconstant is small, it takes a long time to dissipate the\nenergy to satisfy Eq. (9), during which time the mag-\nnetization shows the precession around the zaxis. As a\nresult, back switching appears.\nTo verify this picture, we evaluated the phase diagram\nof the magnetization state for a relatively large damping\nconstant, α= 0.050, which is ten times large than that\nused above. Figures 4(a) and 4(b) show the time evo-\nlutions of mxandεafter turning off the current, where\nHapplis (a) 750 Oe and (b) 1000 Oe. Contrary to the\ndynamics shown in Figs. 3(d) and 3(f), the magnetiza-tion in Figs. 4(a) and 4(b) does not return to the region\nofmx<0 because the energy dissipates quickly due to\nthe large damping torque. As a result, the magnetiza-\ntion immediately relaxes to the switched state. Figures\n4(c) and 4(d) summarize the magnetization state in the\npresence of current and after turning off the current, re-\nspectively. Comparing Fig. 4(c) with Fig. 2(a), it is\nclear that the phase diagram of the magnetization state\nin the presence of the current is approximately indepen-\ndent of the damping constant. On the other hand, the\nback-switching region is strongly suppressed, as can be\nseen by the comparing Fig. 2(b) and 4(d). This is be-\ncause the large damping torque immediately dissipates\nthe energy from the ferromagnet and leads to a fast re-\nlaxationtothe switched state, due to whichthe S 0-region\ndominates in the phase diagram.\nThe existence of the back switching gives an upper\nlimit of a write-current margin for memory applications,\nand therefore, it restricts the device design and/or ma-\nnipulationconditions. Ontheotherhand, backswitching\nmight be applicable to other devices such as a random\nnumber generator. The analysis in this study provides\nfruitful insights for the development of spintronics appli-\ncations utilizing spin-orbit torque.\nIn conclusion, the phase diagram of the magnetization\nstate in a type-X spin-orbit torque device was calculated\nasafunction ofthe electriccurrentdensity and the exter-\nnal magneticfield. The magnetizationstate afterturning\noff the current has stripe structures alternately showing\nswitched and non-switched states. Such a non-switched\nstate, named back switching here, occurs as a result of\nthe magnetization precession around the perpendicular\naxis after the current is turned off. 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Blanter,4and B.J. van Wees5\n1WPI Advanced Institute for Materials Research,\nTohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan\n2Institute for Materials Research and CSIS, Tohoku University, 2-1-1 Katahira, Sendai 980-8577, Japan\n3Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 10090, China\n4Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, the Netherlands\n5Physics of Nanodevices, Zernike Institute for Advanced Materials,\nUniversity of Groningen, 9747 AG Groningen, the Netherlands\n(Dated: April 24, 2023)\nWe discuss spin-wave transport in anisotropic ferromagnets with an emphasis on the zeroes of\nthe band edges as a function of a magnetic \feld. An associated divergence of the magnon spin\nshould be observable by enhanced magnon conductivities in non-local experiments, especially in\ntwo-dimensional ferromagnets.\nI. INTRODUCTION\n\\Magnonics\" is the study of the elementary excitations\nof the magnetic order, i.e. spin waves and their quanta\ncalled \\magnons\"[1{3]. It is believed to be competitive\nin future information, communication, and thermal man-\nagement technologies [4, 5].\nIn the exchange interaction-only continuum model the\nspin wave dispersion is a parabola that shifts linearly\nwith an applied magnetic \feld [1{3]. The gain of angu-\nlar momentum of the ground state by \ripping a single\nelectron is ~. The associated change of the magnetic mo-\nmentum is\u00002\u0016B, where\u0016Bis the Bohr magneton. The\nexchange energy cost of a single spin \rip is minimized by\nspreading this excitation over the whole system, forming\na spin wave or its quantum, the magnon.\nMagnetic dipolar interactions and spin-orbit interac-\ntions strongly a\u000bect the spin wave dispersion of ferro-\nmagnets. Crystal anisotropies are the main consequence\nof the latter, and cause, for example, magnon gaps in\nthe absence of an applied magnetic \feld. Only quite re-\ncently have researchers realized that the magnon spin is\nnot a universal constant. Ando et al. [6] reported en-\nhanced spin pumping by an elliptic magnetization pre-\ncession, which, as we show below, can be interpreted as\nan enhanced magnon spin. Flebus et al. [7] reported\nthat the exchange magnon polaron, i.e. the hybrid state\nof a magnon and a phonon, carries a spin between 0 and\n~. Kamra and Belzig [8] predict super-Poissonian shot\nnoise in the spin pumping from ferromagnets into metal-\nlic contacts based on a magnon spin that is enhanced\nfrom its standard value of ~by anisotropy \\squeezing\".\nThese authors start from a lattice quantum spin Hamil-\ntonian with local and magneto-dipolar anisotropies and\npredicted magnon spins of around 4 ~for the fundamental\n(Kittel) mode of an iron \flm. Kamra et al. [9] address\nthe emergence of the magnon spins in antiferromagnets\nat weak applied magnetic \felds. Neumann et al. [10]\nintroduced an \\orbital contribution\"to the magnon mag-\nnetic moment. Yuan et al. [11] review the history of the\nmagnon spin concept and its enhancement by quantumsqueezing\nMagnon currents can be injected into ferromagnetic in-\nsulators by heavy metal contacts, electrically by means\nof the spin Hall e\u000bect or by thermal gradients (spin See-\nbeck e\u000bect). Vice versa , magnons can pump a spin cur-\nrent into a heavy metal contact and be detected by an\ninverse-spin-Hall voltage. Both e\u000bects may be combined\nto study magnon transport in magnetic insulators [12].\nFilms of ferrimagnetic yttrium iron garnet are well suited\nfor magnon transport studies and can be grown with high\nquality down to a few monolayers [13]. The same tech-\nnique also works well for antiferromagnetic hematite [14{\n16].\nde Wal et al. [17] studied non-local magnon transport\nin an antiferromagnetic van der Waals \flm with a per-\npendicular N\u0013 eel vector. An in-plane magnetic \feld cants\nthe two sublattices until the material becomes ferrimag-\nnetically ordered at the \\spin-\rip\" transition, not unlike\nthe in-plane spin texture of hematite at high \felds. In\nthe absence of additional anisotropies, the band gap of\nthe spin wave dispersion vanishes at this point, i.e. the\nmagnons become \\soft\".\nHere we make a step back by realizing that magnons\ncan be \\soft\" in simple ferromagnets as well. We \fnd\nthat the magnon spin can be strongly a\u000bected by the as-\nsociated non-analyticities of the dispersion relation and\nmay even diverge. We illustrate the general concept of\n\\soft magnons\" and \\magnon spin\" for the three generic\nmagnetic con\fgurations in Figure 1 in which applied\nmagnetic \felds and uniaxial anisotropies compete. We\npredict observable enhancements of magnon transport in\nthe proximity of the soft magnon con\fgurations. These\nresults may partly explain the strong enhancement of\nthe magnon transport at the spin-\rip transition in anti-\nferromagnets [18].\nSections II and III set the stage by re-deriving results\nfor the ground and excited states of long energy excita-\ntions in magnetic systems. In Section II we analyze the\nground state by minimizing the classical magnetic free\nenergy. We solve the linearized Landau-Lifshitz equa-\ntions in Section 3 for the spin wave dispersion relations,\n\fnding results that agree with those obtained from quan-arXiv:2304.10709v1  [cond-mat.mes-hall]  21 Apr 20232\nFIG. 1. Three con\fgurations of ferromagnets with uniaxial\nmagnetic anisotropies parameterized by the constant Kand\ntheir magnetization MBin the direction of the applied mag-\nnetic \feldB. Case (A) is an easy-plane ferromagnet with a\nmagnetic \feld along z(red). In case (B) the \feld lies in the\neasy plane (green). In case (C) the \feld is normal to the easy\nz-axis and\u0012is the tilt angle of the magnetization.\ntum spin models. In Section IV we address the magnon\nspin in a way we have not found in the literature. We\nshow that the Hellman-Feynman theorem can be a use-\nful tool to get hands on the magnon magnetic moment,\nbut only when \feld and magnetization are collinear. In\nSection V we discuss possible experimental signatures of\nthe large magnon spins close to the kinks in the magnon\ndispersion. Section VI contains a critical discussion of\nthe model and recommendations for future work.\nII. SPIN HAMILTONIANS AND\nFERROMAGNETIC GROUND STATES\nWe consider Hamiltonians for local spins ^Siand mag-\nnetic moments ^Mi=\u0000\r^Sion lattice sites i:\nH=\r~X\ni^Si\u0001B\u0000X\nijJij(^Si\u0001^Sj)\u0000\u0016KX\ni(^Sz\ni)2;(1)\nwhereJijis the exchange integral between spins on sites\ni6=j,\u0000\ris the gyromagnetic ratio of an electron, and B\nis a constant external magnetic \feld. We focus here on\nJij>0 that leads to ferromagnetic order. \u0016Kis a uniax-\nial anisotropy parameter along the Cartesian z-direction,\n\u0016K > 0\u0000\u0016K < 0\u0001\ncorresponds to an easy-axis (easy-plane)\nferromagnet.\nThe ground state that minimizes the total energy of\nthe macroscopic system is ferromagnetic. The total\nspin of the system ^S=P\ni^Sithen becomes a classi-\ncal vector Scorresponding to a magnetization density\nM=\u0000(\r~=\n)S, where \n is the crystal volume. The as-\nsociated energy density without irrelevant constant termsreads\nE(M) =\u0000M\u0001B\u0000\u0016K\nM(Mz)2+D\n2M(rM)2+Edmag(M);\n(2)\nwhereM=jMj= (\r~S=\n) andS=jSj. The spin-\nwave sti\u000bness Drepresents the exchange energy cost of\nspatial deformations in the continuum limit that depends\non crystal structure and the exchange parameters Jij.\nThe magnetizations of the ground states are constant in\nspace with zero exchange energy. The demagnetization\nenergy in magnetic \flms with surface normal nalong\nzcan then be absorbed into the anisotropy \feld K=\n(S\u0016K)=(\r~) +M. We discuss here three con\fgurations:\n(A) Easyxy-plane anisotropy ( K < 0) and the magnetic\n\feld normal to the plane, (B) easy xy-plane anisotropy\nwith an in-plane magnetic \feld (Kittel problem), and (C)\neasyz-axis anisotropy ( K > 0) with a magnetic \feld in\nthey-direction. Other con\fgurations such as in-plane\neasy axis etc. give similar results.\nWe are interested in the discontinuities that emerge at\ncritical \felds Bc= 0 for case (B) and Bc= 2jKjfor (A)\nand (B).\nIn case (A) B=Bz;where zis the unit vector along\nthe anisotropy axis, the energy and magnetizations are\ndiscontinuous at Bc= 2jKj\nE(A)\n0\nM=(\n\u0000B2\n4jKj\njKj\u0000BforB < 2jKj\nB > 2jKj(3)\nMz\nM=\u001aB\n2jKj\n1forB < 2jKj\nB > 2jKj(4)\nIn case (B) the energy\nE(B)(M) =\u0000MyB+jKj\nMM2\nz (5)\nis minimal for M0= (0;M;0) for allB6= 0:Finally,\nin (C) a \feld B=Byalong they-axis tilts the magne-\ntization into the yz-plane with M0=M(0;sin\u0012;cos\u0012),\nwhere\u0012is the angle with the out-of-plane direction. Eq.\n(2)\nE(C)(\u0012)\nM=\u0000Kcos2\u0012\u0000Bsin\u0012: (6)\nis minimized for\nsin\u0012=\u001aB\n2K\n1forB <B c\nB >B c: (7)\nAboveBc= 2K;the \feld and magnetization are aligned.\nIII. SPIN WAVE DISPERSION\nHere we consider the frequency dispersion relation\nfor the elementary excitations in homogeneous extended3\nFIG. 2. Spin wave frequency of the magnon band edges for\nthe con\fgurations (A)-(C) in Fig. 1. The dashed black line\n!0=\r=B\u0000Kis the asymptotic form of the case (B) at large\n\felds.\nmagnets that can be bulk crystals, thin \flms, or two-\ndimensional systems. The excitation frequencies are\nsharply de\fned in the limit of small amplitude oscilla-\ntions, in which spin waves can be mapped on a set of non-\ninteracting harmonic oscillators. The magnon Hamilto-\nnian is the lowest-order term in the Holstein-Primako\u000b\nexpansion of the spin Hamiltonian, which can subse-\nquently be diagonalized by a Bogoliubov transformation\n[1{3].\nHere we chose to start from the Landau-Lifshitz\nequation _M =\u0000\r(M\u0002Be\u000b), in which Be\u000b=\n\u0000@E(M)=@MandE(M) is the classical magnetic free\nenergy introduced above. Writing Mq=M0+mqand\nto leading order in the small transverse excitation ampli-\ntudes mq\u0001M0= 0, the spin wave frequencies !qfor wave\nvector qare the solutions of\n(i!q\u0000Be\u000b\u0002)Mq=0: (8)\nFor simplicity, we compute !qwithout dipolar correc-\ntions that cause well-known anisotropic corrections for\nq6= 0, but do not a\u000bect the band edges in Figure 3. The\nresults agree with those obtained from the lowest-order\nHolstein-Primakov expansion of the corresponding spin\nHamiltonians.\nWhen the magnetic \feld is normal to the easy plane\n(A), the spin wave dispersion reads\n!(A)\nq=\r(q\nDq2\u0000\n2jKj\u0000B2\n4K2+Dq2\u0001\nB\u00002jKj+Dq2forB < 2jKj\nB > 2jKj\n(9)\nThe spin wave frequency of the uniform precession\n(q= 0) vanishes for magnetic \felds below Bcbecause thetorques generated by the anisotropy and applied \felds\ncancel. The energies for all magnetization directions that\nlie on a cone with opening angle \u0012are the same, so the\nground state is highly degenerate.\nThe Kittel problem (B) leads to\n!(B)\nq=\rp\n(B+Dq2+ 2K) (B+Dq2): (10)\nThe anisotropy qualitatively changes the dispersion at\nsmall magnetic \felds from !(B)\n0=\rBforK= 0 to\n!(B)\n0\u0018\rp\nBwhenK > B . The anisotropy breaks the\naxial symmetry and mixes the anti-clockwise circular pre-\ncession mode with positive frequencies and the forbidden\nclockwise one with negative frequencies, which leads to\nthe square root dependence rather than 2 \rK, the linearly\nextrapolated value from the high-\feld region.\nIn con\fguration (C) (as in (A)), magnetization and\n\feld are not collinear for \felds B <B c. By rotating the\ncoordinate system around the x-axis by\u0012such that M0\n0\nis alongz0, we can impose the magnon approximation by\nsolving for small amplitude oscillations m0\nq\u0001M0\n0= 0. The\nresult is\n!(C)\nq=\r(q\n1\n2K(2K+Dq2) (4K2\u0000B2+ 2DKq2)p\n(B+Dq2\u00002K) (B+Dq2)\nforB < 2jKj\nB > 2jKj: (11)\nWe observe that the B-dependence of the collinear con-\n\fguration equals that of the Kittel mode shifted by 2 jKj\nto higher magnetic \felds.\nIV. MAGNON SPIN AND\nHELLMANN-FEYNMAN THEOREM\nAn excited state jqiwith energy \"q=~!qof an arbi-\ntrary spin Hamiltonian carries a spin magnetic moment\n\u0016q=hqj^Mjqi\u0000M0;where ^M=\u0000\r~P\ni^Si:Consider\na spin system with zero-\feld Hamiltonian HSthat in-\nteracts with constant applied magnetic \feld Bby the\nZeeman interaction\nH=HS\u0000^M\u0001B: (12)\nThe Hellmann-Feynman theorem then states that for\nsimplicity in the absence of textures,\n\u0016q=\u0000@\"q\n@BkM0\nM(13)\nwhereBk=B\u0001M0=M. For our classical spin system,\nwe replace\"qwith \u0001Eq=E(M0+\u0016q)\u0000E(M0). When\nthe magnetization and the applied \feld are parallel, we\ncan simply read o\u000b the spin of the excited state from\nthe magnon spectrum as a function of the applied \feld.\nWhen M,B, the situation is more complicated and the4\nFIG. 3. Magnon spin at the band edges for the con\fgurations\n(A)-(C) in Fig. 1. In (C) the magnon spin is tilted for B <\n2jKjwith a component in the y- (full curve) and z-directions\n(dashed curve).\nHellmann-Feynman theorem requires additional calcula-\ntions. Con\fgurations (A) and (C) at \felds B < 2K;for\nexample, acquire an \\orbital correction\" [10]\n\u0016q=\u0000d\"q\ndB+@\"q\n@\u0012@\u0012\n@B; (14)\nwhere\u0012is the equilibrium tilt angle. The implicit de-\npendence on the \feld enters here with an opposite sign\ncompared to Ref. [10].\nTo leading order, the solutions of the LL equation are\ntransverse, i.e. mq\u0001M0=O(m2\nq). If we transform back\ninto the time domain, the transverse components oscil-\nlate with the spin wave frequency and average out to\nzero. A magnon moment along the magnetization direc-\ntion persists in the time-average as\n\u0016q\u0019\u00001\n2\u0012jmqj\nM\u00132M0\nM: (15)\nThe amplitudes of the solutions of the LL equation are\ncontinuous but can be quantized by requiring that the\nminimum excitation energy of a mode with energy \"qis\ndiscrete:\n\u0001Eq=E(Mq)\u0000E0=~!q: (16)\nBy normalizing the mode amplitudes in this way we can\ncompute the spin of a single magnon for our three con-\n\fgurations (A-C).\nIn case (A), the moment of a single magnon \u0016(A)\nq=\n\u00002\u0016BforB > B cfrom the Hellmann-Feynman. How-\never, spin and magnetization are not collinear for B <\nBc. The LL equation then leads to a magnon magneticmoment along the equilibrium magnetization direction of\n\u0016(A)\nq\n2\u0016B=\u0000jKj\u0000B2\n4jKj+Dq2\nq\nDq2(2jKj\u0000B2\n2jKj+Dq2): (17)\nthat diverges when q!0 because the degeneracy of all\nmagnetization directions on the cone with angle \u0012allows\ncoherent precession of the magnetization without an en-\nergy cost.\nIn case (B), we can compute the magnon spin simply as\na derivative of the frequency with respect to the applied\nmagnetic \feld from the Hellmann-Feynman theorem Eq.\n(13) since BkM:\n\u0016(B)\nq\n2\u0016B=\u0000B+K+Dq2\np\n(B+Dq2+ 2K) (B+Dq2): (18)\nIn the zero \feld limit \u0016(B)\n0=\u00002\u0016B=p\nK=(2B) diverges.\nThe anisotropy mixes right and left precession modes to\ncreate an elliptic motion that in the limit of zero mag-\nnetic \feld leads to a linear x-polarized motion in the\neasy plane. As in case (A) the restoring torque vanishes\nin this limit, which allows the magnon amplitude to be-\ncome large.\nAlso in the high \feld regime B >B cof the case (C), we\ncan use the Hellmann-Feynman theorem without having\nto worry about \\orbital\" terms\n\u0016(C)\nq\n2\u0016B=\u0000\u0000\nB+Dq2\u0001\n+\u0000\nB+Dq2\u00002K\u0001\np\n(B+Dq2\u00002K) (B+Dq2)forB >B c;\n(19)\nwhich is a shifted version of \u0016(B)\nq.\u0016(C)\n0diverges for\nB#Bcbecause the torque in the out-of-plane z-direction\nvanishes. The magnon spin for the canted con\fguration\n(B <B c) can be computed directly from the solutions of\nthe LL equation. Along the canted equilibrium magneti-\nzation direction\n\u0016(C)\nq\n2\u0016B=\u0000p\nDq2+ 2Kp\n2K\u0002\np\n4K2\u0000B2+ 2DKq2\u0000\n8K2\u0000B2+ 4DKq2\u0001\n(4K2\u0000H2)4K+ (16K2+ 4KDq2\u00003H2)Dq2(20)\nwhich at the band edge q= 0 diverges like 1 =pBc\u0000B.\nIn Figure 3 we plot the in-plane projection of the magnon\nspin at the band edges along the y-direction, i.e. the spin\ncomponent \u0016(C)\n0sin\u0012that can be injected and detected\nby Pt contacts, as well as the z-component along the easy\naxis.\nV. TRANSPORT\nWe now discuss some physical consequences of the re-\nsults derived above.5\nIn case (A) the magnons are gapless up to Bc= 2jKj:\nThe absence of an energy cost of the in-plane magnetiza-\ntion amplitude implies that the equilibrium direction is\narbitrary and depends on the history, remaining in-plane\nanisotropies, or disorder. Even small energy gains of the\nmagneto-dipolar interactions break a uniform magnetiza-\ntion down into domains. At Bcthe magnon spin jumps to\nits standard value of \u00002\u0016B. In case (B) interesting e\u000bects\nmay occur when the magnon spin diverges at vanishing\nmagnetic \felds but it is again di\u000ecult to control the equi-\nlibrium magnetic order. Therefore, the soft magnon in\nthe \feld dependence of case (C) at the critical \feld looks\nmost interesting. While the \ructuations and the associ-\nated magnon spin become large, the \fnite applied \feld\nhinders the breaking of the magnetic order even when\nthe magnon gap vanishes. Hence, we focus on case (C)\nwith an applied magnetic \feld that approaches the crit-\nical value from above. We address magnon transport in\nbulk materials and in two-dimensional ferromagnets.\nA. Spin pumping\nThe magnon spin a\u000bects the spin pumping [6] and spin\npumping shot noise [8]. The spin pumped into the metal\nby the dynamics of an insulating or metallic ferromagnet\nin units of ampere reads\nJs=e\n2\u0019\u0014gr\nM2\n0M\u0002_M+gi\nM0_M\u0015\n(21)\nwheregr+igiis the dimensionless interface spin-mixing\nconductance [19]. Inserting the LL equation for case (C)\nandB#Bcleads to a dc spin current:\nJ(dc)\ns=\u0000e\n2\u0019\rgr\nM2\n0(M0\u0001Be\u000b)\u00160\n=egr\n2\u0019N\nM0\u00160!0=egr\n2\u00192\u0016BN\nM0\rBc: (22)\nwhereNis the number of magnons in the Kittel mode.\nSince the divergence of the magnon spin is canceled by\nthe vanishing resonance frequency, we do not expect any\nanomalies of the spin pumping when the magnons be-\ncome soft.\nB. Magnon conductivity and spin Seebeck e\u000bect\nNext, we consider di\u000buse dc magnon transport in\na bulk ferromagnet under gradients of temperature or\nmagnon chemical potential with a special focus on the\nmagnon conductivity of the magnon soft mode in con\fg-\nuration (C). We employ the relaxation time approxima-\ntion to the linearized Boltzmann equation in which the\nmagnon current is polarized along the equilibrium mag-\nnetization M0:We disregard the magneto-dipolar inter-\nactions on the dispersion relation since these cancel to\na large extent in magnon transport except at very lowtemperatures. The magnon conductivity \u001b(nd)\nmin electric\nunits of Sm2\u0000n[13] and spin Seebeck coe\u000ecient S(nd)\nmin\nunits of V/m then read\n\u001b(nd)\nm=e2\u001c(nd)\nr\nkBTZdqn\n(2\u0019)n\u0000\u0016q\n2\u0016Bv2\nq(fP)2e~!q\nkBT; (23)\n\u001b(nd)\nmS(nd)\nm=e\u001c(nd)\nr\nkBT2Zdqn\n(2\u0019)n\u0000\u0016q\n2\u0016B~!qv2\nq(fP)2e~!q\nkBT;\n(24)\nwherefP=h\nexp\u0010\n~!q\nkT\u0011\n\u00001i\u00001\nis the Planck distribu-\ntion,vq=@!q=@kyis the group velocity in the transport\ny-direction and nis the spatial dimension. The average\nrelaxation time \u001c(nd)\nr can be very di\u000berent in 2 and 3\ndimensions even when everything else remains the same\n[13]. Typical frequencies are in the GHz regime, so as-\nsuming that temperatures at not too low, kBT\u001d~!q:\nThe combination \u0016q!qin the integrand of the spin See-\nbeck coe\u000ecient is analytic when the magnon turns soft.\nSo we do not expect anomalies in the thermally driven\nspin transport.\nAn artifact of the above equations, derived for a con-\ntinuum model of the lattice, a constant relaxation time\napproximation, and the high-temperature limit, is the di-\nvergence of Eq. (23) at high wave numbers. When K= 0\nand in three dimensions\n\u001b(3d)\nm=e2\n~22\n3\u00192\u001c(3d)\nrkBT\n \nQ1+Bq\n2 (DQ21+B)\u00003\n2r\nB\nDarctan r\nD\nBQ1!!\n;\n(25)\nwhereQ1is an ultraviolet cut-o\u000b, at room tempera-\nture provided by the onset of strong magnon-phonon\nscattering at THz frequencies. Including the easy axis\nanisotropy K > 0 causes a logarithmic divergence at the\ncritical \feld B#Bcthat can be regulated by a low-\nmomentum cut-o\u000b Q0that can be rationalized by a resid-\nual in-plane anisotropy or disorder. To leading orders for\nlargeQ1and smallQ2\n0:\n\u001b(3d)\nm!e2\n~22\n3\u00192\u001c(3d)\nrkBT \nQ1\u0000p\n2\n16r\nK\nDlnDQ2\n0\n8K!\n(26)\nThe last term is the conductivity enhancement by the\nsoft mode magnon.\nIn two dimensions and K= 0\n\u001b(2d)\nm=e2\n~21\n\u0019\u001c(2d)\nrkBTe2\u001c(2d)\nrkBT\n\u00191\n2\u0014\nln\u0012\n1 +DQ2\n1\nB\u0013\n+B\nB+DQ21\u0015\n(27)\nthe divergence is logarithmic, but with vanishing magnon\ngapB!0 there is now also a logarithmic infrared di-\nvergence for a constant magnon spin, which is caused6\nby the enhanced magnonic density of states at the band\nedge. Including the divergent magnon spin when K > 0,\nto leading order in a small Q0and largeQ1, and for\nB#Bc\n\u001b(2d)\nm!e2\u001c(2d)\nrkBT\n\u0019\"\nln r\n2D\nKQ1!\n+r\nK\n2D1\n4Q0#\n(28)\nThe high-momentum divergence is still logarithmic, but\nthe low-momentum one becomes algebraic. This implies\na dimensionally enhanced magnon transport around the\ncritical magnetic \feld in van der Waals ferromagnets with\nperpendicular magnetization.\nVI. DISCUSSION\nSeveral processes regulate the enhanced \ructuations\nthat cause the divergence of the magnon spin reported\nhere, but should not destroy the predicted enhancement\nof spin pumping and spin transport. The situation is not\nunlike that of ferrimagnets with singular points in the\nphase diagram caused by angular or magnetic momen-\ntum compensation that causes interesting enhancements\nof e.g. domain wall velocities, in their vicinity [20].\nThe magnon spin of the Kittel mode can be observed\nby spin pumping under ferromagnetic resonance. Ando\net al. [6] indeed reported that the ellipticity of the mag-\nnetization precession under ferromagnetic resonance con-\nditions increases the spin pumping, a \fnding that we in-\nterpret as evidence for an enhanced magnon spin. How-\never, the resonance frequency also enters the pumped\nspin current, and the product spin \u0002frequency remains\nwell-behaved for soft magnons. Similarly, the spin See-\nbeck e\u000bect is hardly a\u000bected.\nMagnon transport using heavy metal contacts for spin\ninjection and detection can be carried out only on mag-\nnets that are also good electrical insulators. The magnon\nconductivity can be measured reliably when the injector\nand detector separation does not exceed the magnon dif-\nfusion length, which requires high magnetic quality. YIG\n\flms can be tuned to a perpendicular magnetization by\ndoping and epitaxial strain [21]. The hexagonal barium\nferrite (BaFe 12O19) has a perpendicular anisotropy \feld\nof 1.7 teslas and damping constant \u000b= 10\u00003[22]. Other\noptions are electrically insulating van der Waals ferro-\nmagnets with perpendicular magnetization such as CrI 3\n[23]. With minor adaptions, the present formalism can be\nused as well for magnetic \flms (strips) with an in-plane\neasy crystal (demagnetization) anisotropy axis and an\nin-plane magnetic \feld applied at right angles to it.\nWe estimate the magnitude of the expected e\u000bects\nby adopting the spin wave sti\u000bness of YIG D= 5\u0001\n10\u000017Tm2;a perpendicular anisotropy K= 1 T;a high\nmomentum cut-o\u000b frequency of 1 THz and an in-plane\nanisotropy of 1 mT :This leads to an enhancement of themagnon conductivity at the soft magnon point of \u00182 for\nthree-dimensional and \u00185 for two-dimensional magnets.\nThe divergences reported here occur only in materi-\nals with weak residual anisotropies, i.e. a su\u000eciently\nsmall cut-o\u000b Q0. Moreover, non-parabolicities render\nthe magnon approximation invalid when the spin wave\namplitudes become large by large magnon spins and/or\nnumbers. The implied magnon interactions may act as\na brake on the dynamics. The predicted numbers at the\ncritical points should therefore be taken with a grain of\nsalt, but the enhancement of the magnon thermal con-\nductivity close to the softening of the magnon modes\nshould persist even when these factors are taken into ac-\ncount.\nVII. CONCLUSIONS\nWe considered the spin waves of ferromagnetic\nmagnons as a function of an applied magnetic \feld, fo-\ncusing on the singular points at which the band edges\n(nearly) vanish. The more general conclusions such as\nthe relation between the Hellmann-Feynman theorem for\ncollinear systems also hold for antiferromagnets. We pre-\ndict enhanced non-local magnon transport caused by the\ndivergence of the magnon spin for magnets with a mag-\nnetic \feld applied perpendicular to a uniaxial magnetic\nanisotropy, which is stronger in two than in three dimen-\nsions. The e\u000bect should contribute to the observation\nof magnon transport above and down to the \\spin-\rip\"\ntransition at which the spin sublattices are forced to align\nferromagnetically [17] and we expect enhanced non-local\nsignals carried by the soft acoustic magnon as in case (C).\nHowever, more work is necessary to fully understand the\nexperiments [18].\nThe present calculations of the transport properties ad-\ndress the linear response at elevated temperatures, disre-\ngarding the e\u000bect of non-linear terms that are likely to be-\ncome important. Higher harmonic generation is easier for\n\roppy magnetic order but is at vanishing applied \felds\ncomplicated by spontaneous magnetic textures [24, 25],\nwhich may not interfere when the soft magnon is shifted\nto su\u000eciently large magnetic \felds. It should also be in-\nteresting to study the non-linearities of propagating soft\nmagnons by microwave spectroscopy [26].\nACKNOWLEDGMENTS\nP.T. and G.B. acknowledge the \fnancial support\nby JSPS KAKENHI Grants No. 19H00645 and No.\n22H04965. BW was supported by the European Union's\nHorizon 2020 program under Grant Agreement No.\n785219 and 881603, the NWO Spinoza prize, and ERC\nAdvanced Grant 2DMAGSPIN (Grant agreement No.\n101053054). Akash Kamra kindly commented on the\nmanuscript.7\n[1] A.G. Gurevich and G.A. Melkov, Magnetization Oscilla-\ntions and Waves, (CRC Press, London, 1996).\n[2] D.D. Stancil and A. Prabhakar, Spin Waves, (Springer,\nNew York, 2009).\n[3] S.M. Rezende, Fundamentals of Magnonics (Springer,\nCham, 2020).\n[4] A. Barman et al ., The 2021 Magnonics Roadmap, J.\nPhys.: Condens. Matter 33, 413001 (2021).\n[5] A. V. Chumak et al., Advances in Magnetics: Roadmap\non Spin-Wave Computing, IEEE Trans. Mag. 58, 1\n(2022).\n[6] K. Ando, S. Takahashi, J. Ieda, Y. 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Nanotech. 15, 563 (2020)..\n[17] D.K. de Wal, A. Iwens, T. Liu, P. Tang, G.E.W.\nBauer, and B.J. van Wees, Long distance magnon\ntransport in the van der Waals antiferromagnet CrPS 4,\narXiv:2301.03268\n[18] P. Tang, unpublished.\n[19] Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, B.I.\nHalperin, Nonlocal magnetization dynamics in ferromag-\nnetic heterostructures, Rev. Mod. Phys. 77, 1375 (2005).\n[20] L. Caretta, M. Mann, F. B uttner, K. Ueda, B. Pfau,\nC. M. G unther, P. Hessing, A. Churikova, C. Klose, M.\nSchneider, D. Engel, C. Marcus, D. Bono, K. Bagschik, S.\nEisebitt, and G. S. D. Beach, Fast current-driven domain\nwalls and small skyrmions in a compensated ferrimagnet,\nNat. Nanotechnol. 13, 1154 (2018).\n[21] L. Soumah, N. Beaulieu, L. Qassym, C. Carr\u0013 et\u0013 ero, E.\nJacquet, R. Lebourgeois, J.B. Youssef, P. Bortolotti,\nV. Cros, and A. Anane, Ultra-low damping insulating\nmagnetic thin \flms get perpendicular, Nat. Comm. 9,\n3355(2018).\n[22] L. Alahmed and P. Li, Chapter 3 in: Magnetic Materials\nand Magnetic Levitation, edited by D.R. Sahu and V.\nN. Stavrou (IntechOpen Limited, London, 2021), DOI:\n10.5772/intechopen.92277 .\n[23] B. Huang, G. Clark, E. Navarro-Moratalla, et al. , Layer-\ndependent ferromagnetism in a van der Waals crystal\ndown to the monolayer limit, Nature 546, 270 (2017).\n[24] C. Koerner, R. Dreyer, M. Wagener, N. Liebing, H. G.\nBauer, and G. Woltersdorf, Frequency multiplication by\ncollective nanoscale spin-wave dynamics, Science 375,\n1165 (2022).\n[25] G. Lan, K. Liu, Z. Wang, F. Xia, H. Xu, T. Guo, Y.\nZhang, B. He, J. Li, C. Wan, G.E.W. Bauer, P. Yan,\nG. Liu, X. Pan, X. Han, and G. Yu, Coherent frequency\nmultiplication by interacting magnons in spin textures\n(unpublished).\n[26] L. Sheng, M. Elyasi, J. Chen, W. He, Y. Wang, H. Wang,\nH. Feng, Y. Zhang, I. Medlej, S. Liu, W. Jiang, X. Han,\nD. Yu, J.-P. Ansermet, G.E.W. Bauer, and H. Yu, Non-\nlocal Detection of Interlayer Three-Magnon Coupling,\nPhys. Rev. Lett. 130, 046701 (2023)."    },    {        "title": "1009.0643v1.Josephson_current_and__π__state_in_ferromagnet_with_embedded_superconducting_nanoparticles.pdf",        "content": "arXiv:1009.0643v1  [cond-mat.supr-con]  3 Sep 2010Josephson current and π−state in ferromagnet\nwith embedded superconducting nanoparticles\nA. V. Samokhvalov(1), A. I. Buzdin(2)\n(1)Institute for Physics of Microstructures,\nRussian Academy of Sciences, 603950 Nizhny Novgorod, GSP-1 05, Russia\n(2)Institut Universitaire de France and Universit´ e Bordeaux I,\nCPMOH, UMR 5798, 33405 Talence, France\nAbstract\nOn the basis of Usadel equations we investigate superconduc tor/ferromagnet/superconductor\n(S/F/S) hybrid systems which consist of superconductingna nostructures (spheres, rods)embedded\nin ferromagnetic metal. The oscillations of the critical cu rrent of the S/F/S Josephson junctions\nwith the thickness of ferromagnetic spacer between superco nducting electrodes are studied. We\ndemonstrate that the πstate can be realized in such structures despite of a dispers ion of the\ndistances between different parts of the electrodes. The tran sitions between 0 and πstates at some\nthickness of ferromagnetic spacer can be triggered by tempe rature variation.\nPACS numbers:\n1I. INTRODUCTION\nThe particularity of the proximity effect in superconductor/ferro magnet (S/F) hybrid\nstructures is the damped oscillatory behavior of the Cooper pair wa ve function inside the\nferromagnet1,2(for the reviews see3,4). In some sort it is a manifestation of the Larkin-\nOvchinnikov-Fulde-Ferrell (LOFF) state induced in ferromagnet ( F) near the interface with\nsuperconductor (S). In contrast with original LOFF, which is poss ible only in the clean\nsuperconductors, the damped oscillatory S/F proximity effect is ve ry robust and exists also\nin the diffusive limit. This special type of the proximity effect is at the or igin of the π\nJosephson S/F/S junction1,2which has at the ground state the opposite sign of the super-\nconducting orderparameter inthebanks. Firstlythe π-junctionwasobserved atexperiment\nin Ref.5and since then a lot of progress has been obtained in the physics of π- junctions and\nnow they are proven to be promising elements of superconducting c lassical and quantum\ncircuits6. Different manifestations of unusual proximity effect and the πstates have been\nobserved experimentally in various layered S/F hybrids7–9. The proximity induced switching\nbetween the superconducting states with different vorticities in mu ltiply connected hybrid\nS/F structures was suggested recently in10,11. Theoretical studies and experiments12–15both\ndemonstrated that in the diffusive limit the spin-flip and spin-orbit sca ttering lead to the\ndecrease of the decay length and the increase of the oscillating per iod. In addition the spin-\nflip scattering may generate the temperature induced transition f rom 0 to πstate of the\njunction14,15.\nNaturally at the first stage of the work on the S/F/S junctions the systems with planar\ngeometry and well controlled F-layer thickness were considered. H owever, now when the π\nstate proven to be very robust vs different types of the impurities scattering13,15(magnetic\nand non-magnetic), interface transparency16,17it may be of interest to address a question\nhow the πjunction could realize in the S/F/S systems with a bad defined thickne ss of F-\nspacer, in particular for two superconducting particles imbedded in ferromagnet or between\na flat superconducting electrode and a small superconducting nan oparticle (such situation\ncouldbeof interest for theSTM-like experiments with asupercondu cting tip). This question\nis non-trivial because the transition from 0 to πstate occurs at the very small characteristic\nlength3ξf=/radicalbig\nDf/h, whereDfis the diffusion constant in ferromagnetic metal and his the\nferromagnetic exchange field, and the typical values of ξfare in the nanoscopic range. We\n2could expect that when the variation of the distance between differ ent parts of S-electrodes\nis of the order of ξftheπstate would disappear. Our calculations show that it is not the\ncase and once again the πstate occurs to be very robust and the transition between 0 and π\nstates is always present at some distance and also can be triggered by temperature variation.\nIn this paper we present the results of a theoretical study of the peculiarities of the\nproximity effect and Josephson current in S/F hybrids which consist of superconducting\nnanostructures placed in electrical contact with a ferromagnetic metal. The paper is orga-\nnized as follows. In Sec. II we briefly discuss the basic equations. In Sec. III we calculate\nthe Josephson current in two model hybrid S/F/S systems. The fir st system consists of two\nsuperconducting rod-shaped electrodes imbedded in ferromagne t. The second one is a S/F\nbilayer with a superconducting spherical particle at the surface of the ferromagnetic layer.\nWe examine the temperature dependence of the critical current o f S/F/S junction between\ntheflatsuperconductingelectrodeandtheS-particletakingintoa ccountthespin-flipscatter-\ning. For both cases the S/F interface transparency between sup erconducting nanoparticles\nand ferromagnet is assumed to be low to prevent from supercondu ctivity destruction due to\nproximity. We summarize our results in Sec. IV\nII. MODEL AND BASIC EQUATIONS\nSince the models of S/F/S junctions we are going to study consist of superconducting\nparticles embedded in a ferromagnetic matrix or placed on a ferroma gnetic substrate, we\nstart from a description of the damped oscillatory behavior of the C ooper wave function\ninduced by such particle in a ferromagnet.\nWe assume the elastic electron-scattering time τto be rather small, so that the critical\ntemperature Tcand exchange field hsatisfy the dirty-limit conditions Tcτ≪1 andhτ≪1.\nIn this case a most natural approach to calculate Tcis based on the Usadel equations18for\nthe averaged anomalous Green’s functions FfandFsfor the F- and S-regions, respectively\n(see3for details). These equations are nonlinear but can be simplified when the tempera-\nture is close to the critical temperature Tcor at any temperature of the F-layer when the\ntransparency of S/F interface is low. In the F-region the linearized Usadel equations take\nthe form\n−Df\n2∇2Ff+(|ω|+ıhsgnω+1/τs)Ff= 0, (1)\n3whereDfis the diffusion coefficient in ferromagnetic metal and ωare the Matsubara fre-\nquencies, ω= 2πT(n+ 1/2), andτsis the magnetic scattering time. We consider the\nferromagnet with strong uniaxial anisotropy, in which case the mag netic scattering does not\ncouple the spin up and spin down electron populations. Restricting ou rselves to the case of\nsuperconducting inclusions with cylindrical or spherical symmetry ( cylindrical rod-shaped\nor spherical particles of radius Rs), one can easily find the following solutions of Eq. (1),\nwhich describe the distribution of anomalous Green’s function Ffin ferromagnet ( r≥Rs)\nsurrounding a superconducting cylinder\nFc\nf(r) =AK0(qr), (2)\nor superconducting sphere\nFs\nf(r) =Aexp(−qr)/qr, (3)\nwhereK0(z) is the Macdonald function,\nq=/radicalBig\n2/Df/radicalbig\n|ω|+ıhsgnω+1/τs (4)\nis the characteristic wave number of the order parameter variatio n in the F-metal, and the\namplitude Ais determined by the boundary conditions at the S/F interface ( r=Rs)19:\nσs∂rFs=σn∂rFf, Fs=Ff−γbξn∂rFf. (5)\nThe S/F interface between a particle and ferromagnet is assumed t o be characterized by\nthe dimensionless parameter γb=Rbσn/ξnrelated to the boundary resistance per unit area\nRb. Hereξs(n)=/radicalbig\nDs(f)/2πTcis the superconducting (normal-metal) coherence length,\nσsandσnare the normal-state conductivities of the S- and F-metals, and ∂rdenotes a\nderivative taken in the radial direction. We will assume that the rigid b oundary condition\nγb≫min{ξsσn/ξnσs,1}is satisfied, when the inverse proximity effect and the suppression\nof superconductivity is S-metal can be neglected20,21. As a result, the pair amplitude Fs(r)\nat the S/F interface is equal to the one far from the boundary:\nFs(Rs) =∆√\nω2+∆2=∆\nωGn, (6)\nwhere ∆ is the superconducting order parameter, and\nGn=ω√\nω2+∆2(7)\n4is normal Green’s function. Using the solution (2) or (3) we obtain fr om the Eqs. (5) and (6)\nthe value of Ffat the S/F boundary r=Rs, and the amplitudes Afor both cases. Finally,\nthe expressions\nFc\nf(r) =∆\nωGnK0(qr)\nK0(qRs)+γbξnqK1(qRs), (8)\nFs\nf(r) =∆\nωGnRse−q(r−Rs)\nr(1+γbξn(q+1/Rs)). (9)\ndescribe the the damped oscillatory behavior of anomalous Green’s f unctionFfin ferromag-\nnet surrounding the superconducting cylinder or sphere, respec tively.\nThe general expression for the supercurrent density is given by\n/vectorJs=iπTσn\n4e/summationdisplay\nω,σ=±/parenleftBig\n˜Ff∇Ff−Ff∇˜Ff/parenrightBig\n, (10)\nwhere˜Ff(r,ω) =F∗\nf(r,−ω).\nIII. CRITICALCURRENT OFJUNCTIONS WITHSUPERCONDUCTING PAR-\nTICLES\nNow we proceed with calculations of the Josephson critical current for two examples of\nmesoscopic hybrid S/F systems. The first one is two identical super conducting cylindrical\nrod-shaped electrodes surrounded by a ferromagnetic metal (s ee Fig. 1). The second one is\na S/F bilayer with a superconducting particle at the surface of the f erromagnetic layer (see\nFig. 3).\nA. S/F/S junction between two superconducting rod\nConsider two superconducting cylinders of a radius Rsembedded in ferromagnet as it\nis shown in Fig. 1. The distance between the cylinder axes is d >2Rs. These rod-shaped\nelectrodes form a Josephson junction in which the weak link between two superconductors\nis ensured by ferromagnetic neighborhood. The supercurrent\nIs(ϕ) =Icsin(ϕ) (11)\nflowing acrossthis structure depends onthephase difference ϕbetween theorder parameters\nof the rods:\n∆1,2= ∆e±iϕ/2. (12)\n5Rs-d/2 d/2 ay\nx\n/c68ei /2/c106/c68e-i /2/c106\nF\nFIG. 1: (Color online) Schematic representation of the F/S h ybrid system under consideration:\ntwo identical superconducting cylindrical rod-shaped ele ctrodes of radius Rssurrounded by a fer-\nromagnetic metal. The axes of superconducting cylinders ar e assumed to be parallel. Figure shows\nthe cross section of the structure by the plane ( x,y) perpendicular to the cylinder axis.\nFor large enough distance between the superconducting cylinders (a=d−2Rs>2ξf), the\ndecay of the Cooper pair wave function in ferromagnet in the first a pproximation occurs\nindependently near either of the electrodes and can be described b y the solution (8). There-\nfore the anomalous Green function Ff(r) in ferromagnet nearby the plane x= 0 may be\ntaken as the superposition of the two decaying functions (8), tak ing into account the phase\ndifference ϕ22:\nFf(x,y) =∆\nωGnK0(qr+)eiϕ/2+K0(qr−)e−iϕ/2\nK0(qRs)+γbξnqK1(qRs), (13)\nwherer±=/radicalbig\n(x±d/2)2+y2. Using the expression (10), we obtain the sinusoidal current-\nphase relation (11) in the S/F/S Josephson junction between two s uperconducting rod-\nshaped electrodes for the case of low transparent S/F interface s. For the critical current of\nsuch Josephson structure, we have\nIc=2πTσn\ne/summationdisplay\nω>0∆2\nω2G2\nn\n×Re\n\naq\n[K0(qRs)+γbξnqK1(qRs)]2∞/integraldisplay\n−∞dyK0(qr0)K1(qr0)\nr0\n\n, (14)\nwherer0=/radicalbig\ny2+d2/4. In the limit of large Rs≫ξfa curvature of the electrodes is not\nessential and the formula (14) coincide with the corresponding exp ressions for the critical\ncurrent of S/F/S layered structures with a large interface trans parency parameter γbprevi-\n6/s48 /s50 /s52 /s54 /s56/s49/s48/s45/s51/s49/s48/s45/s49/s49/s48/s49\n/s48/s46/s53 /s49/s46/s48/s49/s48/s45/s50/s49/s48/s48\n/s49/s48\n/s49/s73\n/s99/s32/s47/s32/s73\n/s48\n/s97/s32/s47/s32\n/s102/s49/s48/s48\n/s97\n/s48\n/s32/s73\n/s99/s32/s47/s32/s73\n/s48\n/s100/s32\n/s102/s32/s49/s48/s49/s48/s48\n/s49\nFIG. 2: (Color online) Influence of the electrode radius Rson the dependenceof the critical current\nIc(15) on the distance abetween two superconducting rod-shaped electrodes embedd ed in F-metal\n(1/τs= 0). The numbers near the curves denote the corresponding va lues of the radius Rsin the\nunits ofξf.. The inset gives the zoomed part of the Ic(a) line , marked by the dashed box.\nously obtained in Refs.15. The critical current equation (14) can be simplified for h≫πTc\nandRs,r0≫ξfand may be written as\nIc=I0dRs\nξf∞/integraldisplay\n−∞dye−2(√\ny2+d2/4−Rs)/ξf\ny2+d2/4cos/parenleftBigg\n2/radicalbig\ny2+d2/4−Rs\nξf+π\n4/parenrightBigg\n, (15)\nI0=πσn∆ξ2\nf\n2√\n2eγ2\nbξ2ntanh/parenleftbigg∆\n2T/parenrightbigg\n. (16)\nNote that our approach is valid for large enough distance between t he superconducting\ncylinders a >2ξfand the first thansition into the πstate ata0is described only qualita-\ntively. It has been demonstrated23that for the planar S/F/S junction with low interface\ntransparency the first transition into the πstate occurs at F layer thickness smaller than\nξf( its actual value depends on the the exchange field hand transparency parameter γb\n).The similar situation is expected for the embedded superconductin g particles and to find\nthe corresponding interparticle distance we need to solve our prob lem exactly.\nThe dependence of the critical current Icas a function of the distance abetween the\n7superconducting cylindrical electrodes calculated from Eq. (15) is presented in Fig. 2 for\nseveral values of the radius Rs. From the figure, we see that with increasing the distance\na, the S/F/S junction undergoes the sequence of 0- πandπ-0 transitions when the value of\nIcchanges its sign from positive to negative and vice versa. We may rou ghly estimate that\nthe first transition from 0 to πstate in S/F/S junction formed by rod-shaped electrodes\noccurs at the thickness of F-layer a0∼0.5ξf, similar to a S/F/S junction with a low S/F\ninterface transparency in the ordinary layered geometry16,23. We observe that the distances\nacorresponding to 0- πandπ-0 transitions grow slightly with decrease of the cylinders radius\nRsdue to a dispersion of the distances between different parts of the electrodes (see the inset\nin Fig. 2).\nB. S/F/S junction in S/F bilayer with a superconducting particle\nAsa second example we consider ferromagnetic film of a thickness don asuperconducting\nplate with a transparent S/F interface. The S/F/S Josephson jun ction is assumed to be\nformed between the flat superconducting electrode and a small su perconducting half-sphere\nof radius Rsembedded into ferromagnet, as it is shown in Fig. 3. The center of th e sphere\nis placed at the surface of the F-film. As before, there is the tunne l barrier ( γb≫1) at the\nS/F interface between the superconducting particle and ferroma gnetic metal.\nSince we consider a S/F bilayer with a transparent interface then th e complete nonlinear\nUsadel equation in the F-layer has to be employed. Using the usual p arametrization of the\nnormal and anomalous Greens functions Gf= cosΘ fandFf= sinΘ f, the Usadel equation\nis written as\n−Df\n2∇2Θf+/parenleftbigg\n|ω|+ihsgnω+cosΘf\nτs/parenrightbigg\nsinΘf= 0. (17)\nNote that Eq. (17) transforms into the linear equation (1) in the limit of small Θ f≪1.\nFor simplicity we restrict ourselves to the case of thick F-layer ( d≫ξf) then the decay\nof superconducting order parameter occurs independently near each S/F interface. In that\ncase, the behavior of the anomalous Green’s function near each int erface can be treated\nseparately, assuming that the F-layer thickness is infinite. Following Ref.1415, the analytical\nsolution of the equation (17) for flat transparent interface at z=dcan be written as\n/radicalbig\n1−ε2sin2(Θf/2)−cos(Θf/2)/radicalbig\n1−ε2sin2(Θf/2)+cos(Θ f/2)=f0e2q(z−d), (18)\n8/c71/c114\nz\nzc/c68e-i /2/c106\n/c68ei /2/c106rn\nRs\nFd\nFIG. 3: (Color online) Schematic representation of the S/F/ S Josephson junction between the flat\nsuperconducting electrode and a small superconducting sph erical particle of radius Rsembedded\ninto ferromagnet. Dashed line shows the cross-section of th e paraboloid Γ by the plane ( ρ, z) (24):\nρ2=x2+y2. Here/vector nis a unit vector along the normal to the surface Γ. Figure show s the cross\nsection by the plane.\nwhere\nε2= (1/τs)(|ω|+ıhsgnω+1/τs)−1.\nTheintegrationconstant f0shouldbedeterminedfromtheboundaryconditionatthesurface\nz=d. As before the rigid boundary conditions is assumed to be valid at z=d:\nΘf(d) = arctan∆\nω. (19)\nFrom Eqs. (18),(19) we get\nf0=(1−ε2)F2\nn/bracketleftBig/radicalbig\n(1−ε2)F2\nn+1+1/bracketrightBig2, (20)\nFn=|∆|\nω+/radicalbig\nω2+|∆|2. (21)\nLinearizing the solution (18) for Θ f≪1 we obtain the anomalous Green’s function in\nferromagnet (0 ≤z≤d) induced by flat superconductig electrode:\nΘf≃4Fn/radicalbig\n(1−ε2)F2\nn+1+1eq(z−d). (22)\n9The total anomalous Green’s function in F-layer far from both the S /F interfaces may be\ntaken as superposition of the two decaying functions (9), (22), t aking into account the phase\ndifference in each superconducting electrode\nFf=∆\nωGnRse−q(r−Rs)+iϕ/2\nr(1+γbξn(q+1/Rs))+4Fneq(z−d)−iϕ/2\n/radicalbig\n(1−ε2)F2n+1+1, (23)\nwherer=/radicalbig\nx2+y2+z2.\nTo derive the general expression for the critical current Icwe have to calculate the total\nJosephson current flowing through a virtual surface Γ: the point s of the surface Γ are\nequidistant from both electrodes of the junction. The surface Γ is a paraboloid which form\nis described by the equations:\nz=zc−(x2+y2)/4zc, zc= (Rs+d)/2. (24)\nUsing the solution (23) and Eq.(10), one can arrive at a sinusoidal cu rrent-phase relation\n(11) with the critical current (see A for details):\nIc=I0zcRsT\nξfTcRe∞/summationdisplay\nn=0/braceleftBigg\nFn(∆Gn/ω)e−q(d−Rs)\n[/radicalbig\n(1−ε2)F2n+1+1][1+ γbξn(q+1/Rs)]\n×\nqzc/integraldisplay\n0due−2qu\nu+zc+1\n2zc/integraldisplay\n0due−2qu\n(u+zc)2\n\n\n, (25)\nwhereI0= 64π2Tcσnξf/e.\nThe dependence of the critical current Ic(25) as a function of the thickness a=d−Rsof\nthe ferromagnetic spacer separating the superconducting plate and the particle is presented\nin Fig. 4 for several values of the particle radius Rsand the magnetic scattering time τs. It\nis clearly seen from Fig. 4 that with a decrease of the particle radius Rs, the position a0of\nthe first zero of the critical current is shifted towards larger valu es of the distance abetween\nsuperconducting electrodes. Figure 4b demonstrates the influen ce of magnetic scattering on\nthe proximity effect and the critical current in S/F bilayer with the pa rticle: decrease of\nthe magnetic scattering time τsleads to the decrease of decay length and increase of the\noscillation period of the anomalous Green’s function Ff14. This results in much stronger\ndecrease of the critical current in the S/F/S junction with increas e of the thickness a, if the\nmagnetic scattering time τsbecomes relatively small τ−1\ns≥h.\nFigure 5 shows the temperature dependence of the S/F/S junctio n critical current Ic(25)\nat several values of the thickness of the ferromagnetic spacer b etween the superconducting\n10/s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s48/s45/s55/s49/s48/s45/s53/s49/s48/s45/s51/s49/s48/s45/s49\n/s48/s49/s49/s48/s73\n/s99/s32/s47/s32/s73\n/s48\n/s40/s32/s100/s32/s45/s32/s82\n/s115/s32/s41/s32/s47/s32\n/s102/s49/s48/s48/s40/s97/s41\n/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s48/s45/s49/s49/s49/s48/s45/s56/s49/s48/s45/s53/s49/s48/s45/s50\n/s49/s49/s48/s73\n/s99/s32/s47/s32/s73\n/s48\n/s40/s32/s100/s32/s45/s32/s82\n/s115/s32/s41/s32/s47/s32\n/s102/s49/s48/s48/s40/s98/s41\n/s48\n/s97\n/s48\nFIG. 4: (Color online) Thedependenceof thecritical curren tIc(25) on thedistance d−Rsbetween\nthe superconducting plate and the particle of radius Rsembedded in F-metal for different values of\nthe radius Rsand the magnetic scattering time τs(T/Tc= 0.5,h= 3πTc,γb= 10): (a) hτs= 100;\n(b)hτs= 0.5. The numbers near the curves denote the corresponding valu es of the radius Rsin\nthe units of ξf.\nelectrodes. The F-spacer thickness ais chosen close to the first transition from 0 to πstate:\na∼a0≃3ξf. The nonmonotonic dependences Ic(T) demonstrate 0 - πtransition due to a\nchange of the temperature. The transition temperature T∗(Ic(T∗) = 0) seems to be very\nsensitive to the size of the superconducting particle. It should be n oted however that the\ntemperature T∗is determined rather by the scale a=d−Rsthen by the scales dorRs,\n11/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s57/s53/s49/s49/s46/s48/s53/s73\n/s99/s32/s47/s32/s73\n/s48/s32/s91/s49/s48/s45/s55\n/s93\n/s84/s32/s47/s32/s84\n/s67/s40/s97/s41\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s57\n/s51/s46/s48\n/s51/s46/s49/s73\n/s99/s32/s47/s32/s73\n/s48/s32/s91/s49/s48/s45/s55\n/s93\n/s84/s32/s47/s32/s84\n/s67/s40/s98/s41\nFIG. 5: (Color online) (a) The dependence of the critical cur rentIc(25) on the temperature T\nfor different values of the radius Rs/ξf= 0.95,1.0,1.05 and the fixed F-spacer thickness a= 3ξf.\n(b) The dependence of the critical current Ic(25) on the temperature Tfor different values of the\nF-spacer thickness a/ξf= 2.9,3.0,3.1 and fixed radius Rs=ξf. The calculation parameters are\nh= 3πTc,hτs= 0.5,γb= 10.\nseparately.\n12IV. CONCLUSION\nTo sum up, we have analyzed the Josephson effect in S/F/S hybrid st ructures with a bad\ndefined thickness of F-spacer. As an example, we have calculated t he Josephson current\nbetween two rod-shaped superconducting electrodes embedded in ferromagnet or between\nflat superconducting electrode and the small superconducting na noparticle at the surface of\nthe F-layer. For the both cases we have demonstrated the possib ility of the realization of π\njunctions in such hybrid systems. We have studied dependence of t he transitions between\n0 andπstates both on the size of superconducting particles and the temp erature. The π\nstate has been proven to be very robust with respect to a geomet ry of the S/F/S junction.\nIn the dirty limit the transition into πstate is determined rather by the thickness of the F-\nspacer between superconducting electrodes then by a shape of t he electrodes. Naturally our\ncalculations can be easily generalized to the different shape of the S p articles (for example\nspherical) with similar conlusions.\nA set of the superconducting particles embedded in a ferromagnet ic matrix realize a\nJosephson network. Depending on the geometry of this network a nd its state (0 or π) it\nmay reveal a spontaneous current similar to that observed in supe rconducting arrays of π\njunctions24. For example the equilibrium phase difference for triangular 2D πjunctions\nnetwork is equal to 2 π/3 which corresponds to the current state. For typical paramete rs\nNb/CuNi hybrid system ( Tc= 9K,ξf≈2nm,ρn= 1/σn≈60µΩcm14one can get from\n(25) the following estimate of the Josephson energy EJ=φ0Ic/2πcof the S/F/S junction:\nEJ/Tc∼104(Ic/I0)<1, i.e. an observation of spontaneous currents near Tcis expected\nto be masked by strong temperature fluctuations. Despite of this restriction we believe\nthat intrinsically–frustrated superconducting networks induced by the proximity effect can\nbe experimentally observable in such S/F/S composites. In particula r, a two-dimensional\nJosephson network of πjunctions may serve as a laboratory to study the phase transition s\nwith continuous degeneracy25.\nThe possibility to fabricate the regular 2D and 3D arrays of Josephs onπjunctions and\nmonitor the transitions between 0 and πstates simply varying the temperature open inter-\nesting perspectives to study a very reach physics of different pha se transition in such systems\ndue to interplay between fluctuations, frustration, disorder and dimensionality.\n13V. ACKNOWLEDGMENTS\nWe are indebted to A. S. Mel’nikov for useful discussions. This work w as supported, in\npart, by the Russian Foundation for Basic Research, by Russian Ag ency of Education under\nthe Federal Program ”Scientific and educational personnel of inn ovative Russia in 2009-\n2013”, byInternational Exchange ProgramofUniversite Bordea uxI, byFrenchANR project\n”ELEC-EPR”, and by the program of LEA Physique Theorique et Mat iere Condensee.\nAppendix A: Josephson current in S/F bilayer with a superconducting particle\nThe general expression for the supercurrent is given by the Eq. ( 10), where the anomalous\nGreens function Ffnearby the surface Γ (see Fig. 3) may be written as\nFf=A1eqz−iϕ/2+A2e−q√\nρ2+z2+iϕ/2, (A1)\nA1=4Fne−qd\n/radicalbig\n(1−ε2)F2n+1+1, A 2=(∆Gn/ω)eqRs\n1+γbξn(q+1/Rs). (A2)\nwherer2=ρ2+z2, and the functions Gn,Fnare determined by expressions (7) and (21),\nrespectively. At the surface Γ the function Ffand the projection of the vector ∇Ffalong\nthe normal\n/vector n=ρ/radicalbig\nρ2+4z2c/vector ρ0+2zc/radicalbig\nρ2+4z2c/vector z0\nto the surface are\nFf|Γ=/parenleftbigg\nA1eqzc−iϕ/2+A24zcRs\nρ2+4z2ce−qzc+iϕ/2/parenrightbigg\ne−qρ2/4zc, (A3)\n(∇Ff,/vector n)|Γ=2zce−qρ2/4zc\n/radicalbig\nρ2+4z2c\n×/bracketleftbigg\nqA1eqzc−iϕ/2−A24zcRs\nρ2+4z2c/parenleftbigg\nq+4zc\nρ2+4z2c/parenrightbigg\ne−qzc+iϕ/2/bracketrightbigg\n. (A4)\nSubstitution of Eqs. (A3), (A4) into the expression for the super current (10) and taking into\naccount the symmetry relations q(−ω) =q∗(ω),A1,2(−ω) =A∗\n1,2(ω) leads to the following\nformula\n(/vectorJs,/vector n)/vextendsingle/vextendsingle/vextendsingle\nΓ=Jc(ρ) sinϕ, (A5)\nJc(ρ) =32πTσn\neRe/summationdisplay\nω>0/braceleftbiggz2\ncRsA1A2\n(ρ2+4z2\nc)3/2/parenleftbigg\nq+4zc\nρ2+4z2\nc/parenrightbigg\ne−qρ2/4zc/bracerightbigg\n. (A6)\n14Further integration of supercurrent density (A6) over the surf ace Γ\nIc=/integraldisplay\nΓdSJc=π\nzc2zc/integraldisplay\n0dρρ/radicalbig\nρ2+4z2\ncJc(ρ). (A7)\nresults in the expressions (25) for the critical current Icof the S/F/S Josephson junction\nbetween superconducting plate and particle.\nReferences\n1A. I. Buzdin, L. N. Bulaevskii, and S. V. Panyukov, Pis’ma Zh. Eksp. Teor. Fiz. 35, 147 (1982)\n[JETP Lett. 35, 178 (1982)].\n2A. I. Buzdin and M. Y. Kuprianov, Pis’ma Zh. Eksp. Teor. Fiz. 5 3, 308 (1991) [JETP Lett. 53,\n321 (1991)].\n3A. I. Buzdin, Rev. Mod. Phys., 77, 935 (2005).\n4A. A. Golubov, M. Yu. Kupriyanov, and E. Il’ichev, Rev. Mod. P hys. 76, 411 (2004).\n5V. V. Ryazanov, V. A. Oboznov, A. Yu. Rusanov, A. V. Veretenni kov, A. A. Golubov, and J.\nAarts, Phys. Rev. Lett. 86, 2427 (2001)\n6A. K. Feofanov, V. A. Oboznov, V. V. Bol’ginov, J. Lisenfeld1 , S. Poletto, V. V. Ryazanov, A.\nN. Rossolenko, M. Khabipov, D. Balashov, A. B. Zorin, P. N. Dm itriev, V. P. Koshelets and A.\nV. Ustinov, Nature Physics, 6, 593 (2010).\n7J. S. Jiang, D. Davidovic, D. H. Reich, and C. L. Chien, Phys. R ev. Lett. 74, 314 (1995).\n8A. S. Sidorenko, V. I. Zdravkov, A. A. Prepelitsa, C. Helbig, Y. Luo, S. Gsell, M. Schreck, S.\nKlimm, S. Horn, L. R. Tagirov, and R. Tidecks, Ann. Phys. 12, 37 (2003).\n9I. A. Garifullin, D. A. Tikhonov, N. N. Garifyanov, L. Lazar, Yu. V. Goryunov, S. Ya. Khleb-\nnikov, L. R. Tagirov, K. Westerholt, and H. Zabel, Phys. Rev. B66, R020505 (2002).\n10A. V. Samokhvalov, A. S. Melnikov, and A. I. Buzdin, Phys. Rev . B76, 184519 (2007).\n11A. V. Samokhvalov, A. S. Melnikov, J-P. Ader, and A. I. Buzdin , Phys. Rev. B 79, 174502\n(2009).\n12L. Cretinon, A. K. Gupta, H. Sellier, F. Lefloch, M. Faur´ e, A. Buzdin, and H. Courtois, Phys.\nRev. B 72, 024511 (2005).\n1513E. A. Demler, G. B. Arnold, and M. R. Beasley, Phys. Rev. B 55, 15174 (1997)\n14V. A. Oboznov, V. V. Bolginov, A. K. Feofanov, V. V. Ryazanov, and A. I. Buzdin, Phys. Rev.\nLett. 96, 197003 (2006).\n15M. Faure, A. I. Buzdin, A. A. Golubov, and M. Yu. Kupriyanov, P hys. Rev. B 73, 064505\n(2006).\n16A. Buzdin and I. Baladie, Phys. Rev. B 67, 184519 (2003).\n17M. Weides, M. Kemmler, E. Goldobin, D. Koelle, R. Kleiner, H. Kohlstedt, and A. Buzdin,\nAppl. Phys. Lett. 89, 122511 (2006).\n18L. Usadel, Phys. Rev. Lett. 25507 (1970).\n19M. Yu. Kuprianov and V. F. Lukichev, Zh. Eksp. Teor. Fiz. 94, 139 (1988) [Sov. Phys. JETP\n67, 1163 (1988)].\n20V. N. Krivoruchko and E. A. Koshina, Phys. Rev. B 66, 014521 (2002).\n21F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. B 69, 174504 (2004).\n22K. K. Likharev, Rev. Mod. Phys. 51, 101 (1979).\n23A. Buzdin, JETP Lett 42, 583 (2003).\n24S. M. Frolov, M. J. A. Stoutimore, T. A. Crane, D. J. Van Harlin gen, V. A. Oboznov, V. V.\nRyazanov, A. Ruosi, C. Granata, and M. Russo, Nature Physics 4, 32 (2008).\n25S. E. Korshunov, Usp. Fiz. Nauk 49, 225 (2006).\n16"    },    {        "title": "0805.3925v2.Magnetic_anisotropy_in_ferromagnetic_Josephson_junctions.pdf",        "content": "arXiv:0805.3925v2  [cond-mat.supr-con]  15 Jul 2008Magnetic anisotropy in ferromagnetic Josephson junctions\nM. Weides\nCenter of Nanoelectronic Systems for Information Technolo gy and Institute of Solid State Research,\nResearch Centre J¨ ulich, D-52425 J¨ ulich, Germany\n(Dated: December 3, 2018)\nMagnetotransport measurements were done on Nb /Al2O3/Cu/Ni/Nb superconductor-insulator-\nferromagnet-superconductor Josephson tunnel junctions. Depending on ferromagnetic Ni interlayer\nthickness and geometry the standard (1 d) magnetic field dependence of critical current deviates\nfrom the text-book model for Josephson junctions. The resul ts are qualitatively explained by a\nshort Josephson junction model based on anisotropy and 2 dremanent magnetization.\nPACS numbers: 74.25.Fy 74.45.+c 74.50.+r, 74.70.cn\nSuperconductivity (S) and ferromagnetism (F) in thin\nlayered films have now been studied during some decades\n[1]. In SF bilayers the superconductivity may be non-\nuniform[ 2], i.e. the Cooperpairwavefunction extends to\nthe ferromagnet with an oscillatory behavior. In Joseph-\nson junctions (JJs) based on s-wave superconductors the\nphase coupling between the superconducting electrodes\ncan be shifted by πwhen using a ferromagnetic barrier\nwith an appropriate chosen thickness dF, i.e. SFS or\nSIFS-type junctions (I: insulating tunnel barrier). Only\nin recent years the experimental realization of πJJs was\nsuccessful. In particular, the πcoupling was demon-\nstratedbyvaryingthe temperature[ 3,4,5], the thickness\nof the F-layer [ 5,6,7,8] or measuring the current-phase\nrelation of JJs incorporated into a superconducting loop\n[9,10,11]. The coupling can also change within a single\nJJ by a step-like F-layer, i.e. one half is a 0 JJ and the\nother half is a πJJ [12,13].\nFor useful classical or quantum circuits based on\nSFS/SIFS JJs a large critical current density jc(small\nJosephson penetration depth λJ) and a high IcRprod-\nuct are needed [ 14,15]. Up to now the limiting fac-\ntor is the low jcdue to strong Cooper pair breaking in-\nside F-layer. Alloys of magnetic and non-magnetic atoms\nsuch as NiCu face problems of clustering [ 16] and strong\nmagnetic scattering [ 5,8]. Promising experiments using\nstrong ferromagnet transition metals [ 6,17,18,19,20]\nwere published.\nShape anisotropy of magnetic interlayer may provoke a\nnot flux-closeddomainstructure andconsequentlya shift\nof critical current diffraction pattern Ic(H) [19]. In ex-\nperiments [ 3,5,7,19,20] the SFS/SIFS JJs had nearly\nmirror-symmetrical Ic(H), i.e. the effective shift along\nH-axis is small, usually less than one flux quantum Φ 0.\nThis is explained by a multi-domain state of F-layer with\na very small net magnetization. However, up to now the\n2dnature of thin-film magnetism was disregarded.\nIn this Letter the Ic(H) dependence for remanent 2 d\nmagnetization of F-layer is systematically studied. First,\nthe maximal flux from F-layer is estimated. Second, the\nIc(H) pattern considering 2 din-plane magnetization is\ncalculated for different aspect ratios. Third, the Ic(H)\npattern is measured alongboth magnetic axes for various\njunction geometries and dF./s48 /s49 /s50/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48 /s49 /s50/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48 /s49 /s50/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s48 /s49 /s50/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s48 /s48/s46/s53/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s48/s46/s48 /s48/s46/s53/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s50\n/s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s48/s46/s48 /s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s77 /s32/s32\n/s50/s53/s32/s120/s32/s50/s48/s48/s32 /s109/s50/s53/s48/s32/s120/s32/s49/s48/s48/s32 /s109/s50/s51/s48/s32/s120/s32/s51/s48/s32 /s109/s50/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s120/s32/s40 /s109/s84 /s41\n/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s121 /s32/s40/s109/s84 /s41/s61/s48/s46/s49/s52/s50/s32 /s84 /s47/s109/s61/s48/s46/s48/s55/s49/s32 /s84 /s47/s109/s61/s48/s46/s50/s56/s52/s32 /s84 /s47/s109/s61/s48/s32 /s84 /s47/s109/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s99/s114/s105/s116/s105/s99/s97/s108/s32/s99/s117/s114/s114/s101/s110/s116/s32 /s73\n/s99/s47/s73/s48 /s99\n/s48/s48/s46/s50/s53/s48/s48/s48/s46/s53/s48/s48/s48/s48/s46/s55/s53/s48/s48/s49/s46/s48/s48/s48\nFIG. 1: (Color online) Calculated surface plot of Ic(Hx,Hy)\nfor different geometries and remanent magnetizations α. The\nmagnetization vector /vectorMpoints from top right to bottom left\ncorner (arrow), I0\ncis shifted in opposite direction.\nThe maximal shift of Ic(Hy) is estimated for a strong\nmagnet, i.e. Ni, being magnetized fully in-plane and\nalongy-axis. The atomic magnetic momentum is 0 .6µB\n[21], the specific density ρis 8.9g/cm3(bulk) and mag-\nnetization µ0M= 0.64 T. A cross-section of length\nLx= 100µm and F-layer thickness dF= 3nm encloses\na magnetic flux Φ M=dFLxµ0M. The total magnetic\nflux Φ through the JJ is the applied field flux Φ H=\n(2λL+dF)LxHy(Londonpenetrationdepth λL= 90nm)\nplus Φ M, i.e. Φ = Φ H±ΦM= 8.85Φ0Hy/mT±92Φ0.\nTheIc(Hy) pattern is shifted by 92 periods from the\ncenter. This simple calculation neglects dead magnetic\nlayer [22], as found in SFS/SIFS JJs [ 5,8,20], and de-\nmagnetizingbydomains. RealNi filmstendtoformcom-\nplex magnetization profiles (in/out-of-plane) and domain\nstructures as function of dF[23]. Integral magnetization\nmeasurements in SF layers show a complex behavior as a\nfunctionoftemperature,appliedfield, andsamplehistory\n[24,25],e.g. SFstructuresspontaneouslyaltertheirstray\nfield by changingmagnetic domain distribution [ 26]. The\nlocal magnetization depends on stray fields from neigh-\nbor domains, flux focusing from S-electrodes [ 19,27] or\non bias induced spin accumulation at F/S interface [ 28].\nIt is generally agreed that the average magnetization in\nSFS/SIFS JJs is much smaller than the maximum mag-2\n/s45/s50 /s45/s49 /s48 /s49 /s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s45/s48/s46/s49/s48 /s45/s48/s46/s48/s53 /s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s45/s48/s46/s54 /s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s72\n/s120/s61/s48/s72\n/s121/s61/s48\n/s72\n/s121/s61/s48\n/s72\n/s121/s61/s48\n/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s121/s32/s40/s109 /s84/s41 /s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s121/s32/s40/s109 /s84/s41/s32 /s61/s48/s32 /s84/s47/s109 /s32/s32 /s32 /s61/s48/s46/s48/s55/s49/s32 /s84/s47/s109 /s32 /s32 /s61/s48/s46/s49/s52/s50/s32 /s84/s47/s109 /s32/s32 /s32 /s61/s48/s46/s50/s56/s52/s32 /s84/s47/s109/s32/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s99/s114/s105/s116/s105/s99/s97/s108/s32/s99/s117/s114/s114/s101/s110/s116/s32 /s73\n/s99/s47 /s73/s48 /s99\n/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s121/s32/s40/s109 /s84/s41/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s120/s32/s40/s109 /s84/s41\n/s72\n/s120/s61/s48/s72\n/s120/s61/s48/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s120/s32/s40/s109 /s84/s41/s32/s32/s110/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s99/s114/s105/s116/s105/s99/s97/s108/s32/s99/s117/s114/s114/s101/s110/s116/s32 /s73\n/s99/s47 /s73/s48 /s99\n/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s120/s32/s40/s109 /s84/s41/s51/s48/s32/s120/s32/s51/s48/s32 /s109/s50\n/s53/s48/s32/s120/s32/s49/s48/s48/s32 /s109/s50\n/s50/s53/s32/s120/s32/s50/s48/s48/s32 /s109/s50\nFIG. 2: (Color online) Calculated Icas function of 1 dmag-\nnetic field, Ic(Hx,0) (top) and Ic(0,Hy) (bottom) for the\nthree geometries and various α. A substantial deviation from\nideal (α= 0T/m) pattern appears for already small magne-\ntization vector /vectorM=α(Lx,Ly).\n/s45/s48/s46/s54 /s45/s48/s46/s53 /s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s49/s48/s48/s48/s49/s50/s48/s48\n/s106\n/s99/s61/s50/s51/s46/s51/s177/s48/s46/s52/s32 /s65/s47/s99/s109/s50\n/s32/s40 /s61/s49/s46/s55/s37/s41/s100\n/s70/s61/s50/s46/s50/s32 /s110/s109\n/s32/s32/s32/s32 /s76\n/s120/s32/s120/s32 /s76\n/s121/s32/s40 /s109/s50\n/s41/s32/s32/s32 /s76\n/s120/s47\n/s74/s32/s120/s32 /s76\n/s121/s47\n/s74\n/s32/s51/s48/s32/s120/s32/s51/s48/s32/s32/s32/s32/s32/s40/s48/s46/s52/s32/s120/s32/s48/s46/s52/s41\n/s32/s53/s48/s32/s120/s32/s49/s48/s48/s32/s32/s32/s40/s48/s46/s54/s53/s32/s120/s32/s49/s46/s51/s41\n/s32/s50/s53/s32/s120/s32/s50/s48/s48/s32/s32/s32/s40/s48/s46/s51/s50/s32/s120/s32/s50/s46/s54/s41\n/s86\n/s99/s61 /s49 /s32 /s86\n/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s121/s32/s40 /s109/s84 /s41\n/s32/s32/s99/s114/s105/s116/s105/s99/s97/s108/s32/s99/s117/s114/s114/s101/s110/s116/s32 /s73\n/s99/s32/s40 /s65 /s41\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s53/s49/s48/s49/s53/s50/s48\n/s72\n/s121\n/s72\n/s120/s100\n/s70/s61/s51/s46/s57/s32 /s110/s109\n/s51/s48/s32/s120/s32/s51/s48/s32 /s109/s50\n/s32\n/s86\n/s99/s61 /s49 /s32 /s86/s53/s48/s32/s120/s32/s49/s48/s48/s32 /s109/s50\n/s86\n/s99/s61 /s48/s46/s50 /s32 /s86\n/s50/s53/s32/s120/s32/s50/s48/s48/s32 /s109/s50\n/s86\n/s99/s61 /s48/s46/s53 /s32 /s86/s50/s53/s32/s120/s32/s50/s48/s48/s32 /s109/s50\n/s32\n/s86\n/s99/s61 /s48/s46/s53 /s32 /s86/s50/s53/s32/s120/s32/s50/s48/s48/s32 /s109/s50\n/s86\n/s99/s61 /s48/s46/s50 /s32 /s86/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s53/s48/s49/s48/s48\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s53/s48/s32/s120/s32/s49/s48/s48/s32 /s109/s50\n/s86\n/s99/s61 /s49 /s32 /s86\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32 /s72\n/s120/s44/s121/s32/s40 /s109/s84 /s41/s99/s114/s105/s116/s105/s99/s97/s108/s32/s99/s117/s114/s114/s101/s110/s116/s32 /s73\n/s99/s32/s40 /s65 /s41\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\nFIG. 3: (Color online) Measured Ic(Hy) pattern of SIFS JJs\nfor thin (top) and thick (bottom) Ni layer and different ge-\nometries. Ic(Hx) pattern of thick Ni layer is plotted in gray.\nOnset of magnetic anisotropy effects in Ni layer is between\n2.2 and 3.9nm.\nnetization estimated above. But as shown below even a\nremanent 2 dmagnetization of 1% of the maximal value\nmay notable change the Ic(H) pattern.\nA qualitativemodel for Ic(H) in presenceofa uniform,fixed 2dmagnetization /vectorMis derived. The short JJ model\nIc(H) =I0\nc/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglesin(πΦ\nΦ0)\nπΦ\nΦ0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleand Φ = Φ H±ΦM\nis modified by 2 ddistributions of applied field flux Φ H\nand magnetization Φ Mwith\nΦ =/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg\n(2λL+dF)/parenleftbigg\n0Hx\nHy0/parenrightbigg\n±/parenleftbigg\nα0\n0α/parenrightbigg/bracketrightbigg\n×/parenleftbigg\nLx\nLy/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\n/vectorM=α(Lx,Ly) is assumed to be orientated in-plane\n(Meissner-screening of S-electrodes) along the diagonal\n(i.e. longest) axis of sample (rough approximation for\nmagnetic shape anisotropy). The magnetization |/vectorM|is\nseveral orders of magnitude smaller than the upper limit\ngiven by a fully saturated magnetic layer. Note that the\neasy axis of ferromagnetic film can be determined by the\nmagnetic field during deposition, too. The model for Φ M\nis just exemplary for the effect of 2 din-plane magnetiza-\ntion inIc(H).\nIn Fig.1the surface plot of Ic(Hx,Hy) is depicted for\nvariousαand geometries. The position of I0\ncis shifted\nfrom the center ( Hx=Hy= 0mT) in opposite direction\nof/vectorM. Fig.2depictsIc(Hx,0) andIc(0,Hy) pattern.\nThese graphs resemble standard 1 d Ic(H) (H=Hx,Hy)\nmeasurements. For some |/vectorM| /negationslash= 0 T a single-peaked\nIc(Hx),Ic(Hy) pattern is calculated, which -on first\nglance- resembles a shifted |sin(H)/H|Fraunhofer pat-\ntern. However, the maximum Icis smaller than the\nrealI0\ncand the height of side-maxima do not obey the\nexpected value. For example the Ic(Hx,0) pattern of\n30×30µm2sample and α= 0.071 T/m is shifted by\nless than Φ 0and its maximum Icis already reduced to\n∼0.8I0\nc. This simple Ic(Hx,Hy) modelmayqualitatively\nexplainsomeexperimentalobservations(Fig. 3)onSIFS-\ntype JJs with Ni interlayer.\nForexperiment JJs with similar areas,i.e. 30 ×30,50×\n100 and 25 ×200µm2, were fabricated. The deposition\nand structuring of JJs is described in Ref. 29. The\nSIFS multilayer was magnetron sputter deposited with\nNi thickness dFranging from 1 −6nm. The tunnel bar-\nrier was formed for 30min at a partial oxygen pressure\nof 0.1 mbar. After oxidation a 2 nm Cu film was in-\nserted. All JJs were deposited in a single run by shift-\ning the substrate and target to obtain a wedge-shaped\nNi-layer. Normal state and subgap resistance indicate a\nsmall junction to junction variation. The IV and Ic(Hx),\nIc(Hy) characteristics were measured at 4 .2 K for two\nsets of samples ( dF= 2.2,3.9nm). Cooldown was done\nin zero field and thermal cycling up to ≈15 and 300K\nto check reproducibility. Transport measurements were\nmadein aliquid Hedip probeusinglow-noisehomemade\nelectronics and room-temperature voltage amplifier. The\nmagnetic fields ( Hx,0), (0,Hy) were applied in-plane of\nthe sample and parallel the junctions axis (Fig. 1). The\nvoltagecriteria VcforIc(H)determinationwas0 .2−1µV.\nA lower subgap resistance for dF= 3.9nm sample leads3\nto larger offset currents. Positive and negative current\nbranch of IVC had similar magnetic field dependence\n+Ic(Hy)≃ |−Ic(Hy)|. Magnetic field wassweptbetween\n±1.5mT for all samples. All junctions had their lateral\nsizes comparable or smaller than the Josephson penetra-\ntion length λJ, except the longest sample ( dF= 2.2nm,\n25×200µm2), whose Lyis not strictly inside the short\nJJ limit.\nThedF= 2.2 nm samples showed very regular Ic(Hy)\npattern. All maximum Ic’s were nearly centered and\nthe spread of jcwas∼1.7%, as determined from the\nmaximum Ic’s. The Ic(Hx) pattern is symmetric, too\n(not shown). The oscillation period of Ic(Hy) were de-\ntermined by magnetic cross-section ∼1/Lx, and nearly\nindependent ofaspect ratio. No indication fora distorted\nsupercurrent transport due to alloying at the Nb /Ni\ninterface [ 30] can be found. Effects due to magnetic\nanisotropy were not detectable, either because the sam-\nples were still inside the dead magnetic layer, or the\nanisotropy was absent or totally out-of-plane.\nThedF= 3.9nmsampleshadcompletelydifferent Ic(Hy)\npattern showing in-plane magnetic anisotropy with some\ncharacteristic features. All maximum Ic’s were shifted\nfrom the center, and the amplitude of shift increased\nwithLy, i.e.≈0.24mT for 30 ×30µm2,≈0.5mT for\n50×100µm2, and≈0.8mTfor25 ×200µm2samples. The\ndirectionofshift variesbetween sampleseveniftheywere\ncooled and measured at the same time (random polarity\nof magnetic configuration). The position of main peak of\nIc(Hy) was reproducible after thermal cycling to 300 K.\nThe width of main maxima (measured at offset line) was\nnot strictly ∼1/Lx, and varies from sample to sample.\nThe pattern were asymmetric, i.e. the height of same-\norder side maxima differed, probably due to non-uniform\nflux guidance in F-layer and re-orientation of domains.\nBy rotating the magnetic field by 90◦, i.e. measuring in\nIc(Hx) mode, low Ic’s, being nearly independent of Hx,\nwere detected. Even the squared shaped 30 ×30µm2\nJJ had an almost flat Ic(Hx) pattern. This indicates\nsome magnetic crystallographic anisotropy along y-axis,\nprobably caused by magnetron sputter deposition. Small\ndeviations of Ic(0,0) forIc(Hx,0) andIc(0,Hy) measure-ments can be related to variations of magnetic configu-\nration by the unshielded sample handling at 300 K. A\nconsiderable spread of maximum Ic(Fig.3) can be al-\nready seen for JJs with same geometry, which is even\nincreased by considering the maximum jcfor different\ngeometries. Simulations (Fig. 2) show that already a\nmoderate magnetization /vectorM(α <0.1 T/m) yields very\ndifferent maximum Ic’s ofIc(Hx) andIc(Hy). A sample\nto sample variation of direction and amplitude of /vectorMex-\nplains the data spread of Fig. 3. Obviously, this leads to\nvery large variations in the Ic(dF) dependence.\nIn literature, the Ic(H) pattern of SFS/SIFS JJs with\ncomparable strong magnets were either shown for sam-\nples with thin dF[19], or had deviations from the ideal\n|sin(H)/H|form. For example, the maximum Icin Ref.\n20, Fig. 5 (inset) is too small compared to the first side-\nmaxima. These samples were small in area ( /lessorequalslant1µm2),\nand the F-layer could have been in single domain state.\nAsIcvaried smoothly with H, either /vectorMrotated softly,\nor a multi-domain structure with averaged 2 dmagneti-\nzation existed. For both cases the 2 dnature of remanent\nmagnetization may have suppressed the maximum Ic-\ndetermined from Ic(H) pattern- below the real I0\nc.\nIn summary, the Ic(H) pattern along both field axis\nof SIFS JJs with Ni interlayer were measured. Assum-\ning magnetic anisotropy the characteristic features, i.e.\nshift or absence of central peak, can be qualitatively re-\nproduced by simulations. As conclusion, the 1 d Ic(H)\npattern in presence of magnetic anisotropy can not yield\ntherealI0\nc. Future experiments on SFS/SIFS JJs should\nbe done by 2-axis Ic(Hx,Hy) scan of JJs with well-\ncontrolled magnetic interlayer domain configuration. 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Mal’shukov1and Arne Brataas2\n1Institute of Spectroscopy, Russian Academy of Sciences, 14 2190, Troitsk, Moscow oblast, Russia\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nWe show that injecting nonequilibrium spins into the superc onducting leads strongly enhances\nthe stationary Josephson current through a superconductor -ferromagnet-superconductor junction.\nThe resulting long-range super-current through a ferromag net is carried by triplet Cooper pairs\nthat are formed in s-wave superconductors by the combined eff ects of spin injection and exchange\ninteraction. We quantify the exchange interaction in terms of Landau Fermi-liquid factors. The\nmagnitude and direction of the long-range Josephson curren t can be manipulated by varying the\nangles of the injected polarizations with respect to the mag netization in the ferromagnet.\nPACS numbers: 72.25.Dc, 71.70.Ej, 73.40.Lq\nI. INTRODUCTION\nStudies of hybrid structures combining supercon-\nducting and ferromagnetic components attract much\nattention due to their unique, rich, and complex\nphysical properties that are promising in a number\nof potential applications1. The interface of an s-\nwave superconductor with a ferromagnet is charac-\nterized by an unusual proximity effect that is spa-\ntially oscillating and can lead to a sign reversal of\nthe criticalcurrentthroughsuperconductor-ferromagnet-\nsuperconductor(SFS) Josephsonjunctions. Sucharever-\nsal is equivalent to a π-shift in the current-phase relation\nfor the Josephson current. This interesting property is\na motivation for using the so-called π-junctions as el-\nements of superconducting quantum circuits for poten-\ntial application in quantum computing2. However, the\nproximity effect in ferromagnets does not reach far. Two\ncritical tasks are to extend its range and to find a way to\nmanipulate the π-junction in order to switch the device\nbetween its various phase states. In contrast, Cooper-\npairs can be transferred over relatively long distances\neven in ferromagnets, if they are in a triplet state with\n±1 projections of their total spin onto the spin quantiza-\ntion axis. Various mechanisms have been proposed that\nconvert a singlet pair into a triplet pair, such as a spa-\ntially dependent magnetization3, spin-flip scattering at\nFS interfaces4, and precessing magnetization5. A num-\nber of works in this direction has been reviewed in Ref.1.\nIn this work, we will show that these tasks can be ful-\nfilledviatheproductionandmanipulationofalong-range\nproximity effect by injecting spins into superconducting\nleads. The novelty of our idea is based on the impor-\ntant, and so far unaddressed, role played by the electron-\nelectron interaction in SFS. Our insight is that the com-\nbined effects of spin-injection and electron-electron inter-\naction generate a long-range proximity effect despite the\nstrong exchange field in the ferromagnet. The conven-\ntional wisdom is that spin polarized electrons can only\nexist as excitations in s-wave superconductors, since the\nCooper pairs do not carry a spin. However, we will\ndemonstrate that this simple picture, which is based onthe neglect of electron-electron interactions beyond su-\nperconducting pairing correlations, misses qualitatively\nimportant effects. Quantitatively, in simple metals, the\nexchange interaction of itinerant carriers is noticeable\nandcanbedescribedintermsofLandauFermi-liquidfac-\ntors. Although the exchange interaction does not cause\nferromagnetism in s-wave superconductors, it causes a\ntransfer of spin polarization from the quasi-particle exci-\ntations to the condensate, in the form of polarized triplet\nCooper pairs. When such a triplet pairing is generated\nby the combined effects of spin-injection and exchange\ninteraction, these pairs subsequently tunnel through the\nferromagneticlayer via the long-rangeproximity effect, if\nthe spin polarizations in the leads and the layer are not\ncollinear. Only at this stage, which includes the so far\nunaddressed important electron-electron interaction, the\nsituation becomes similar to proposals of Ref.1,3 where\nan inhomogeneous magnetization gives rise to the long-\nrange effect provided by ±1 triplets. The relative angles\nbetween the spin polarizations in the superconducting\nleads and in the ferromagnet can be varied by control-\nling the injected spin polarizations, making it possible to\nvary the magnitude and sign of the Josephson current.\nThis enables manipulations of π-junctions. In addition\nto the Josephson supercurrent, which is driven by the\ndifference in the condensate phases, there is also a dissi-\npative DC current. The latter is induced by the spin po-\nlarization flow through the ferromagnetic layer with spin\ndependent conductivity. This dissipativecurrentalsocan\nbe manipulated by varying the injected polarization an-\ngles. As it will be shown, at some angles it vanishes, so\nthat the dissipative and supercurrents can be measured\nindependently.\nVariouseffectsofaninjected spinpolarizationandspin\ncurrent on the electric transport in SFS junctions6,7and\nother superconducting systems8–11have been recently\nconsidered. Despite this interest, the fact that the ex-\nchange interaction transfers the spin polarization from\nthe quasiparticles to the condensate has not been ad-\ndressed so far.\nThe article is organized by the following way. In Sec.II\nan expression is derived connecting the triplet compo-\nnents of the anomalous Green function to the nonequi-2\nFIG. 1: (Color online) A sketch of the system. The electric\ncurrent flows in normal leads N through contacts with ferro-\nmagnetic leads F Land F R. Spin density is injected from F L\nand F Rinto N and further penetrates across tunneling barri-\ners into superconductors S Land S R. The Josephson current\nflows between these leads through a ferromagnetic layer F.\nArrows show possible magnetizations of the ferromagnets.\nlibrium spin polarization in superconducting leads. In\nSec.III the Josephson and dissipative currents are calcu-\nlated. Finally, our results are discussed in Sec.IV.\nII. TRIPLET ELECTRON PAIRING FUNCTION\nINDUCED BY SPIN INJECTION\nHow to efficiently inject a spin polarization into para-\nmagnetic metals is well known12. A nonequilibrium spin\naccumulation is induced by the electric current through\na paramagnetic-ferromagneticinterface. We consider the\nscenario that the spin polarization further diffuses from\na paramagnetthrough a resistive barrierinto a supercon-\nducting lead, so that the electric circuit where the spin\ninjection takes place is effectively separated from the su-\nperconducting circuit. We assume that the steady state\nspin polarizations are generated in both superconduct-\ning leads, in the vicinity of the F-layer. The sketch of\nthe system is shown in Fig.1. For clarity, we simplify\nthe problem by assuming that the FS contacts contain\na barrier, so that the proximity effect is weak. We also\nassume that the spin relaxation time τspinin the leads is\nlong, so that the spin diffusion length lspinis large com-\npared to the SN contact sizes and the coherence length.\nConsequently, the spin densities sL(R)and the order pa-\nrameters ∆ L(R)only vary slowly in space near the left\n(L) and right ( R) contacts.\nThe electronic transport through an SFS system,\nwhose characteristic dimensions are larger than the elas-\ntic mean free path, can be described in terms of Us-\nadel equations for angular averaged Green’s functions\ng(for a review see13). These functions are matrices in\nthe Keldysh, spin, and Nambu spaces. We choose the\nspin and Nambu spaces so that the one-particle destruc-\ntion operators are c1k↑=ck↑,c1k↓=ck↓,c2k↑=c†\n−k↓,\nc2k↓=−c†\n−k↑, where the labels 1 and 2 denote theNambu spinor components, while ↑and↓are the spin\nindices. The Keldysh component gKof the Green func-\ntion can be represented as13\ngK=grh−hga, (1)\nwheregrandgaaretheretardedandadvancedfunctions,\nrespectively, and the distribution function his a diagonal\nmatrix in the Nambu space.\nIn order to determine the distribution hin the super-\nconducting leads, the interfaces between these leads and\nthespin-polarizednormalmetalsmustbeconsidered. We\nusestandardboundaryconditionsrelatingfluxesthrough\nS-N (S-F) interfaces to Green functions in superconduc-\ntors and normal metals (ferromagnets). It is assumed\nthat the spin relaxation rates in the superconducting\nleads are slow enough ( lspin≫rsnσs) and the leak-\nage of the spin polarization through the SF boundary is\nsufficiently slow rsn/Asn≪rsf/Asf, where 1 /rsnand\n1/rsfare the interface conductances (per unit square) of\nSN and SF interfaces, AsnandAsfare the SN and SF\ncontact areas, and σsis the normal-state conductivity of\nthe superconductor’s lead. With these assumptions, the\ndistribution functions in the superconductor, h(s), and\nnormal metal, h(n), are equal to each other, h(s)=h(n).\nWe further assume that nonequilibrium spins in N-leads\nare thermalized with chemical potentials µ↑andµ↓for\nthe two spin directions. Therefore, denoting by the sub-\nscripts 11 and 22 the corresponding matrix elements in\nthe Nambu space, we get for h↑(↓)≡h(s)\n11↑(↓)=h(n)\n11↑(↓)\nand¯h↑(↓)≡h(s)\n22↑(↓)=h(n)\n22↑(↓)\nh↑(↓)=¯h↑(↓)= tanhω−µ↑(↓)\n2kBT. (2)\nAt the same time, the retarded ( gr) and advanced ( ga)\nGreenfunctions havethe sameforms asin an equilibrium\nsuperconductor.\nOur calculation so far re-iterates the conventional wis-\ndom of spin-injection in superconductors: the effects are\nlimited to a spin-dependent statistical distribution func-\ntion, whiletheretardedandadvancedGreenfunctionsdo\nnotchange. Inthispicture, spininjectiondoesnotleadto\nthe appearance of triplet correlations in the condensate\nwave-function, which would cause long-range Josephson\ntunneling through a ferromagnetic layer. Fortunately,\nthere is a mechanism to generate triplet correlations in\nspin-polarized superconducting leads, which others have\nso far overlooked. The electron-electron exchange in-\nteraction provides a coupling between a spin accumu-\nlation and the spectral properties of superconductors,\nin that spin polarized quasiparticles produce an effec-\ntive Zeeman field. The latter, in its turn, gives rise to\ntripletcorrelationsthat aredescribed viathe correspond-\ning spin components of the anomalous functions gr\n12and\nga\n12. In Fermi-liquid theory, the effective Zeeman energy\nisǫxc(σN), where Nis a unit vector parallel to the in-\njected spin polarization S=NSand\nǫxc= GS/2NF. (3)3\nThe spin-accumulation magnitude is\nS=−NF\n4(1+G)/integraldisplay\ndωTr[(1+τ3)\n2σzgK],(4)\nwhereτ3andσzare the Pauli matrices acting in the\nNambu and spin spaces, respectively, and NFis the den-\nsity of states at the Fermi level. The renormalization\nfactor 1/(1+G), where G is the exchange Landau-Fermi\nliquid parameter, appears when the spin-density of Eq.\n4 is expressed in terms of a semiclassical Green func-\ntion integrated over energy14. This factor is not quali-\ntatively important in our case, since G is not too close\nto the paramagnet instability G = −1.15The exchange\nCoulomb interaction in metals gives rise to a negative G.\nFor example, the calculated value is -0.17 in Al16. The\nspindensity(4) stronglydependsontemperature, mostly\nvia the temperature dependence of the superconducting\ngap in the energy spectrum. In order to determine Sand\n∆ in both leads, Eq. (4) have to be solved together with\ntheS-depended selfconsistency equation for ∆.9\nVia the effective Zeeman energy of Eq.3, the retarded\nand advanced Green functions become spin-dependent.1\nIndeed, choosing the quantization axis along N, the\nanomalous functions fr\n↑↓=gr\n12↑↑andfr\n↓↑=−gr\n12↓↓be-\ncome\nfr\n↑↓(↓↑)=±|∆|exp(iφ)/radicalbig\n(ω∓ǫxc+iδ)2−|∆|2,(5)\nwhere the phase φof the order parameter ∆ equals φL\nandφRat the left and right contacts, respectively. The\ntriplet component of this function with 0-spin-projection\nonto the z-axis isfr\n0= (fr\n↑↓+fr\n↓↑)/√\n2, while the triplet\ncomponents with ±1-projections vanish, fr\n±1=fr\n↑↑(↓↓)=\n0. Theadvancedfunction, aswellastheconjugatedfunc-\ntionsf†, are determined from symmetry relations.\nIt ismoretransparentto discussthe Greenfunctions in\na basis where the spin quantization axis is parallel to the\nmagnetization in the ferromagnetic layer, which is along\nz, as shown in Fig. 1. We assume that the spin polariza-\ntionsintheleftandrightleadsarerotatedwithrespectto\nthis axis by the angles θLandθR, respectively. We follow\nthe convention that the three components of the triplet\nf0,f1,f−1arerelated to a 3Dvector a= (ax,ay,az) with\naz=f0,ax= (f−1−f1)/√\n2 anday=i(f−1+f1)/√\n217.\nHence, in the geometry shown in Fig. 1, after a rota-\ntion ofaaround the y-axis, we get in the new basis\nf′\n0=f0cosθandf′\n1=−f′\n−1=−f0sinθ/√\n2. So, by\nusing Eq. (5) the triplet components in the left and right\nsuperconducting leads are\nf±1R(L)=−sinθR(L)\n2(f↑↓+f↓↑), (6)\nwherethelabels randahavebeenomitted fromhereand\nthe same magnitudes of ǫxcare assumed in both leads.In the new basis, the distribution function (2) is\nhL(R)=h↑(1+σzcosθL(R))\n2+h↓(1−σzcosθL(R))\n2+\nσysinθL(R))h↑−h↓\n2. (7)\nIII. THE JOSEPHSON AND DISSIPATIVE\nCURRENTS\nWhat we have established is that the superconduct-\ning leads acquire triplet pairing correlations determined\nby non-equilibrium spin polarizations whose directions\nare tilted with respect to the ferromagnet’s magnetiza-\ntion in the SFS junction. We will show that the current\nthroughsuchatriplet pairing-ferromagnet-tripletpairing\nsystem consists of two parts, a dissipative contribution\ncontrolled by the non-equilibrium distribution of spins\nin the device, and a super-current driven by the phase\ndifference between superconductors and provided by the\ntriplet components of the superconducting condensates\nin the left and right leads.\nLet us first consider the dissipative current. It can be\nexpressed in terms of the distribution function hfinside\nthe ferromagnet. Due to precession in the exchange field\nBex, the spins that are not parallel to it decay quickly\non the length-scale/radicalbig\nDf/Bex, whereDfis the diffusion\nconstant. Therefore, only the components of hfthat are\nparallel and anti-parallel to z, denoted as hf↑andhf↓, re-\nmain finite inside the ferromagnet, if the junction length\nL≫/radicalbig\nDf/Bex. When the spin relaxation length is\nlarger than L, in the linear approximation these collinear\ncomponents satisfy the spin-conserving diffusion equa-\ntionDfσ∇2\nxhfσ= 0, where σ=↑,↓, that takes into ac-\ncount spin-dependent diffusion coefficient in a strong fer-\nromagnet. The solution of this equation is a linear func-\ntion ofxwhose slope is obtained from the boundary con-\nditions∓rsfσσfσ∇xhfσ|x=xL(R)=hL(R)σ−hfσ|x=xL(R),\nwherehL(R)σare given by the first two terms of Eq. (7).\nTaking into account that rsfσandσfσcan depend on the\nelectron spin and assuming equal barrier transmittances\nat L and R contacts we obtain\nhfσ=hRσ+hLσ\n2+hRσ−hLσ\n1+2γσx\nL, (8)\nwherexR(L)=±L/2 andγσ= (rsfσσfσ/L)≫1. Using\nEqs. (8) and (7) we compute the dissipative part of the\ncurrent through the junction:\njd=/summationdisplay\nσ/integraldisplay\ndωσfσ∇xhfσ=\nδµ\neL/parenleftbiggσf↑\n1+2γ↑−σf↓\n1+2γ↓/parenrightbigg\n(cosθR−cosθL).(9)\nThis current is proportional to the difference in the spin-\nup and spin-down conductances (2 rsfσ+L/σfσ)−1of the\ntotal ferromagnetic layer, including the interfaces; this4\nis the well known19connection between spin and elec-\ntric transport in ferromagnets. The electric current at-\ntains its maximum when cos θR=−cosθL=±1, and\nvanishes at θR=θL, as well as at θR,θL=±π/2.\nSuch an angular dependence has a simple physical ex-\nplanation. The electric current (9) is proportional to\nthe spin-current through the junction. The latter attains\nits maximum when the nonequilibrium spin polarizations\nin the ferromagnetic layers are oppositely directed and\nit vanishes if these polarizations are collinear and have\nequal magnitudes. The spin current obviously also van-\nishes if these polarizationsare perpendicular to the ferro-\nmagneticmagnetizationaxis, sinceperpendicularcompo-\nnents do not penetrate deep into ferromagnet. The spin\nflow through the junction is accompanied by energy dis-\nsipation. It is determined by the Ohmic losses in the\nferromagnet during transport of spin polarized electrons\nbetween the leads having spin dependent electrochemical\npotentials. The dissipative current of Eq. (9) is inde-\npendent of the superconducting phases φLandφR. We\nassume that the electric potentials of both contacts are\nequal. Iftheloadispresentinthecircuit, thespincurrent\nwill induce a voltage difference. The latter, in its turn,\ncan cause periodic oscillations of the Josephson current6.\nWhen/radicalbig\nDf/Bexis much shorter than the junction\nlengthLand the coherence length, the up and down-\nspin Fermi surfaces become decoupled. In this regime,\nthe supercurrent jsthrough the junction is determined\nby the decoupled tunneling of ±1 triplet Cooper pairs at\ntheir respective ferromagnet’s Fermi surfaces. Unlike the\ndissipative current, the spin-dependence of the electron\ndiffusion coefficients and conductivities is not so impor-\ntant, at least in the case when the Bex≪EF. Therefore,\nin the leading approximation we set Df↑=Df↓=Df,\nand a similar relation for the conductivities. Further-\nmore, in the linear approximation, only the first term of\nEq. (8) has to be taken into account. Moreover, since\nthe Josephson current is determined by part of the dis-\ntribution function that is odd in frequency, from Eqs.\n(8), (7) and (2) only the spin-independent part hf↑+hf↓\ncontributes to the current. It is given by\njs=σf\n16e/summationdisplay\nm=±1/integraldisplay\ndω/bracketleftbig\n(fr\nm∇xfr†\nm−∇xfr\nmfr†\nm)−\n(fr→fa)](hf↑+hf↓)\n2, (10)\nwherefr\nm(m=±1) are the retarded triplet compo-\nnents of the anomalous function in ferromagnet and\nfa\nm(ω) =fr\nm(−ω), while fr†\nm(ω) =fr∗\nm(−ω). When the\nspin-relaxation length is much larger than L, and within\nthe linearized approximation, fr\n±1obey1\nDf∇2\nxfm+2iωfm= 0, (11)\nwith the boundary conditions18rsfσf∇xfm|x=xR(L)=\n±fmR(L), where fmR(L)are given by Eqs. (5-6). Af-\nter transforming the integral in Eq. (10) into a sum overFIG. 2: (Color online) The Josephson current as a function\nof the spin potential δµmeasured in units of the unperturbed\nsuperconducting gap ∆ 0, atL=ζ, where ζ=/radicalbig\nDf/2|∆0|\nandI0= (2ζπkBT/er2\nsfσf)×10−2\nthe frequencies ωn=πkBT(2n+ 1),jscan be finally\nrepresented in the form\njs= sin(φL−φR)LsinθRsinθL\neσfR2\nsfK, (12)\nwhere\nK=|∆|2kBT/summationdisplay\nωn>0,ν=±11\nkνLsinh(kνL)×\n/bracketleftBigg\n1/radicalbig\n(ωn+iHν)2+|∆|2−1/radicalbig\n(ωn−iH−ν)2+|∆|2/bracketrightBigg2\n(13)\nandkν=/radicalbig\n2(ωn+iνδµ)/Df, withHν=ǫxc+νδµand\n2δµ=µ↑−µ↓.\nIV. DISCUSSION\nAs follows from Eq. (12), the Josephson current de-\npends on the directions of the nonequilibrium spin po-\nlarizations in the superconducting leads. The current\nreaches its maximum when the spin accumulations in the\nleads are perpendicular to the magnetization in the fer-\nromagnet, θR=θL=π/2. It reverses its sign when\nthe spin polarization in one of the leads flips its direc-\ntion. Therefore, in the setup shown in Fig. 1, the junc-\ntion can be switched into the π-state by simply revers-\ning the electric current through one of the FN contacts.\nIt should be noted that the dissipative current given by\nEq.(9) vanishes when the relative angles are such that5\nthe supercurrent reaches its maximum. Hence, the dissi-\npative transport can be turned off, a feature that can be\nimportant for practical purposes. Eqs. (12) and (13) also\nimply that the long-range proximity effect, described via\njs, vanishes when the exchange interaction ǫxc= 0. The\ndependence of jsonthe spin-potential δµisshownin Fig.\n2. A finite spin-potential causes variations of the order\nparameter ∆ and spin density entering in Eq. (12) which\nhave been found from a pair of self-consistent equations.\nIn our calculation of ∆ and S, we neglected the exchange\nfieldǫxc, assuming that ǫxc≪δµ. This is a realistic\nassumption, taking into account that S < δµN Fin Eq.\n(3) and|G|is considerably less than 1 in some supercon-\nducting metals (e.g. Al). In this limit, the dependence of\n∆ onδµis formally the same as in a thermally equilib-\nrium superconductor subject to a Zeeman splitting equal\ntoδµ. Such a scenario is well studied in the literature\n(see e.g.21). In Fig. 2, we see that the critical current\nchanges sign at some values of δµ. This is caused by\ninjection of nonequilibrium spins into the ferromagnetic\nlayer. As a result, the distribution function in Eq. 10 is\ndifferentfromtheequilibriumdistribution. Thereissome\nsimilarity of this effect with a current reversal observed\nin Josephson transistors20. At the lower temperature,\nthe supercurrent versus spin-potential is more peaked in\nthe range of higher δµ, since the spin density increases\nsharply together with ǫxcin this range. At even higher\nδµ, the superconductivity is destroyed by spin injection.\nThat causes a sudden drop of the current. We believethat this narrow range can be easily observed experi-\nmentally in the set up shown in Fig. 1, because δµcan\nbe fine tuned by varying the current through the normal\nleads.\nFig. 2 is calculated at L=/radicalbig\nDf/2|∆0|, that is the\ncharacteristic length of the ±1-triplet proximity effect\nin the range of temperatures considered. This length is\nobviously much larger than the s-wave Cooper pair pen-\netration depth/radicalbig\nDf/Bexand therefore clearly demon-\nstrates how the range of the proximity effect becomes\nmuch longer by spin injection into the superconducting\nleads.\nIn conclusion, spin injection into s-wave superconduc-\ntors can dramatically increase the stationary Josephson\ncurrent in SFS system. This enhancement is provided\nby±1 triplet components of the electron pairing func-\ntion. They are generated in superconducting leads by\nexchange fields that are noncollinear with the ferromag-\nnet magnetization. These fields, in turn, are induced by\nan injected spin polarization. Besides a strong effect on\nthe Josephson current, spin injection also gives rise to a\ndissipative current that at zero bias potential is induced\ndue to spin dependence of the ferromagnet conductivity.\nBoth Josephson and dissipative currents can be manipu-\nlated by varying the injected spin directions in the leads\nenabling control of π-junctions.\nAnatoly Mal’shukov gratefully acknowledges the hos-\npitality of NTNU.\n1F. S. Bergeret, A. F. Volkov, K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005); A. I. Buzdin, Rev. Mod. Phys, 77,\n935 (2005)\n2A. K. Feofanov, et. al., Nature Phys. 6, 593 (2010); L.\nB. Ioffe, V. B. Geshkenbein, M. V. Feigel’man, A. L.\nFauch‘ere, and G. Blatter, Nature 398, 679 (1999); A. V.\nUstinov and V. K. Kaplunenko, J. Appl. Phys. 94, 5405\n(2003); T. 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Luo\net. al., Phys. Rev. B 79, 140508 (2009)\n12F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. S. Basel-\nmans, and B. J. van Wees, Nature (London) 416, 713\n(2002); S. O. Valenzuela and M. Tinkham, Nature (Lon-\ndon)442, 176 (2006).\n13J. Rammer, H. Smith, Rev. Mod. Phys. 58, 323 (1985)\n14J. M. Serene, D. Rainer, Phys. Rep. 101, 221 (1983)\n15L.P. Pitaevskii, E.M. Lifshitz . Statistical Physics, Part 2.\nVol. 9. Butterworth-Heinemann (1980).\n16J. A. Leiro, Solid. State Comm., 93, 953 (1995)\n17L.D. Landau, E.M. Lifshitz . Quantum Mechanics: Non-\nRelativistic Theory. Vol. 3. Pergamon Press (1977)\n18M. Yu. Kupriyanov and V. F. Lukichev, Zh. Eksp. Teor.\nFiz.94, 139 (1988) [Sov. Phys. JETP 67, 1163 (1988)].\n19A. G. Aronov, Pis’ma Zh. Eksp. Teor. Fiz. 24, 37 (1976),\nJETP Lett. 24, 32 (1976); M. Johnson and R. H. Silsbee,\nPhys. Rev. Lett. 55, 1790 (1985).\n20L. N. Bulaevskii, V. V. Kuzii, and A. A. Sobyanin, Solid\nState Commun. 25, 1053 (1977); J. J. A. Baselmans, A.\nMorpurgo, B. J. van Wees, and T. M. Klapwijk, Nature\n397, 43 (1999).\n21Yu. A. Izyumov, Yu. N. Skryabin, Phys. Stat. Sol. (b) 61,\n9 (1974)"    },    {        "title": "0802.4009v1.Ferromagnetic_to_spin_glass_cross_over_in__La_Tb___2_3_Ca__1_3_MnO__3_.pdf",        "content": "arXiv:0802.4009v1  [cond-mat.str-el]  27 Feb 2008Ferromagnetic to spin glass cross over in (La,Tb) 2/3Ca1/3MnO 3\nC. Raj Sankar, S. Vijayanand, and P. A. Joy∗\nPhysical and Materials Chemistry Division, National Chemi cal Laboratory, Pune 411008, India\nIn the series La 2/3−xTbxCa1/3MnO3, it is known that the compositions are ferromagnetic for\nsmaller values of xand show spin glass characteristics at larger values of x. Our studies on the\nmagnetic properties of variouscompositions intheLa 2/3−xTbxCa1/3MnO3series showthatthecross\nover from ferromagnetic to spin glass region takes place abo vex≈1/8. Also, a low temperature\nanomaly at 30 K, observed in the ac susceptibility curves, di sappears for compositions above this\ncritical value of x. A mixed phase region coexists in the narrow compositional r ange 0.1 ≤x≤\n0.125, indicating that the ferromagnetic to spin glass cros s over is not abrupt.\nI. INTRODUCTION\nThere has been a wide interest in the study of the sub-\nstituted perovskite-type manganites, La 1−xAxMnO3, in\norder to understand the different aspects of the complex\nmagnetic behaviour exhibited by these compounds.1,2,3\nThe double exchange interactions involving Mn3+and\nMn4+ions give rise to ferromagnetism in the substituted\nmanganites. The changes in the Mn-O-Mn bond angle,\nfrom structural distortions, are very crucial in determin-\ning the strength of the ferromagnetic interactions. High-\nest Curie temperature in the La 1−xCaxMnO3series is\nobtainedfor x≈1/3andhencetherearemanystudiesre-\nported forthe compositionLa 2/3A1/3MnO3. Manyinter-\nestingnewmagneticbehaviorsareobservedwhen La3+is\npartially substituted by other trivalent rare-earth ions in\nLa2/3Ca1/3MnO3,4,5,6,7,8,9though there are no changes\nin the Mn3/Mn4+ratio after substitution. Hwang et al.4\nfound a direct correlationbetween the Curie temperature\nand the average ionic radius of the La-site ions, where\nthe Curie temperature decreases with decreasing aver-\nage ionic radius in La 0.7−xRxCa0.3MnO3(R = Pr, Y),\nindicating the role of lattice effects in determining the\nferromagnetic properties. Partial substitution of La by\nsmaller ions leads to a decrease in the Mn-O-Mn bond\nangle and this affects the long range ferromagnetic ex-\nchange interactions.6,7,8\nFor La 0.67−xTbxCa0.33MnO3(in the re-\nports, the chemical composition is used as\n(La1−xTbx)2/3Ca1/3MnO3), a gradual decrease in\nthe Curie temperature is observed with increasing\nconcentration of Tb.7,9This is also associated with a\nbroadening of the magnetic transition and ultimately a\nspin glass behaviour is observed at larger concentrations\nof Tb.10,11In La 0.67−xTbxCa0.33MnO3, the evolution\nof a spin glass (SG) or a cluster glass (CG) state is\nthought to be due to the competing interactions of\nferromagnetic (FM) and antiferromagnetic (AFM) types\nor the random distribution of Mn-O-Mn bond angle\nwhich suppresses the exchange strength between the\nMn ions significantly.7,8,11At sufficiently large values\nofx(x= 0.22), short range ordered magnetic clusters\nare formed with typical magnetic coherence length of\naround 18 ˚A, at low temperatures.10\nThe reported studies on La 0.67−xTbxCa0.33MnO3areperformed on ferromagnetic compositions ( x≤0.1) or\nspin glass compositions ( x >0.15) and therefore no in-\nformation is available on the changeover region from fer-\nromagnetic to spin glass characteristics. In order to\nexplore the critical concentration of the Tb ions which\nresults in the formation of magnetic clusters leading\nto spin glass characteristics, a series of compositions\nwith very close values of xbetween 0.1 and 0.15 in\nLa0.67−xTbxCa0.33MnO3have been studied. The criti-\ncal concentration is found to be x≈1/8, and interesting\nmagnetic properties are observed for compositions close\nto the cross over region.\nII. EXPERIMENTAL\nThe polycrystalline La 0.67−xTbxCa0.33MnO3compo-\nsitions were synthesized by the conventional solid state\nroute from La 2O3, Tb4O7, CaCO 3and MnO 2by mix-\ning these oxides in the required stoichiometry for x= 0,\n0.03, 0.07, 0.10, 0.11, 0.12, 0.125, 0.13, 0.15, 0.20, and\n0.25. The well-mixed powders were initially heated at\n1273 K for 48 h, and subsequently at 1473 K for 48h,\n1573 K for 48 h and finally at 1623 K for 24h, with in-\ntermediate grindings at every 24 h steps to ensure the\nsample homogeneity. Finally the powder samples were\npelletized and sintered at 1673 K for 24 h. The samples\nwere characterized by powder X-ray diffraction using Cu\nKαradiation and Ni filter. The Mn4+contents in the\ncompounds were estimated by the iodometric titration\nmethod.12The magnetization measurements were per-\nformed on a vibrating sample magnetometer. Temper-\nature variation of the ac magnetic susceptibility of the\nsamples was measured by the mutual inductance method\nin a field of 2 Oe and at a frequency of 210 Hz.\nIII. RESULTS AND DISCUSSION\nThe samples were characterized for their phase purity\nby X-raydiffraction. All samples showedsinglephase na-\nture and the diffraction patterns were indexed on the dis-\ntorted orthorhombic structure with space group Pbnm.6\nThe lattice parameters were obtained by least squares\nrefinement of the diffraction patterns and found to be2\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s48/s46/s48\n/s32/s48/s46/s48/s51\n/s32/s48/s46/s48/s55\n/s32/s48/s46/s49/s48\n/s32/s48/s46/s49/s50/s53\n/s32/s48/s46/s49/s51/s51/s48/s32/s75/s47\n/s109/s97/s120\n/s84/s32/s40/s75/s41\nFIG. 1: Temperature variation of the ac susceptibility of so me\ncompositions in La 0.67−xTbxCa0.33MnO3. The numbers in-\ndicate the values of x.\ncomparable for those of the corresponding compositions\nreported previously.6,11The Mn4+content was found to\nbe matching with the expected value of 33% for all the\ncompositions.\nThe ac susceptibility curves of the unsubstituted and\nsome Tb-substituted compositions are shown in Fig. 1.\nWith increasing concentration of Tb, the magnetic tran-\nsition temperature is decreased, and the susceptibility\ncurves show a cusp-like feature when the concentration\nis increased beyond x= 0.1. The shapes and features of\nthe ac susceptibility curves of the Tb-substituted com-\npositions (for x≤0.1 and x >0.15) are similar to\nthat reported earlier.6,9,10Although well-defined mag-\nnetic transitions are observed for the samples with low\nconcentrations of Tb, there is a distinct anomalous fea-\nture in the susceptibility curves of these compositions at\nlow-temperatures. A decrease in the susceptibility is ob-\nserved below ∼30 K, as soon as a small amount of Tb is\nincorporated ( x= 0.03).\nAs shown in Fig. 2, for x= 0.1, a small step in the\nsusceptibility curve is observed below the main magnetic\ntransition. For x= 0.11, there is a cusp like feature ob-\nserved at a higher temperature which is succeeded by the\nnormal flat curve, and finally a drop in the susceptibility\nbelow 30 K. This is a complex feature and is not com-\nmonly reported for the manganite compositions. As the\nconcentration of Tb ions is further increased, it can be\nseen that the cusp-like feature becomes more prominent,\nthe flat region gradually vanishes, but a dip in suscep-\ntibility is still observed at 30 K for the compositions up\ntox= 0.125. For x >0.125 there is only a cusp-like\nfeature in the ac susceptibility curve as observed for spin\nglasssystems. The composition with x=0.33is reported\nto show spin glass behavior.7Neutron diffraction studies\nhave indicated ferromagnetically ordered state for x=\n0.067 at low temperatures, whereas short range ordered\nclusters are found for x= 0.2 (the authors have used the/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s47\n/s109/s97/s120\n/s84/s32/s40/s75/s41/s32/s48/s46/s49\n/s32/s48/s46/s49/s49\n/s32/s48/s46/s49/s50\n/s32/s48/s46/s49/s50/s53\n/s32/s48/s46/s49/s51\n/s32/s48/s46/s49/s53\n/s32/s48/s46/s50/s48\nFIG. 2: Temperature dependence of the ac susceptibility of\nthe compositions in La 0.67−xTbxCa0.33MnO3, for 0.1 ≤x≤\n0.2. The numbers indicate the values of x.\nchemicalformulaas(La 1−xTbx)2/3Ca1/3MnO3, sothatx\n= 0.1 and 0.3, respectively, correspond to x= 0.067 and\n0.2 in La 0.67−xTbxCa0.33MnO3).11These clusters show\nspin glass-type of behavior due to competition between\nFM and AFM types of exchanges.6,9\nForthe compositions with x >0.125, thereis a gradual\nshift of the cusp towards lower temperatures as the value\nofxis increased. Also, it was found that the relative\nsusceptibility value alsodecreasedwith the increasingTb\nconcentration. These observations point out the evolu-\ntion of short range ordering due to the decreasing Mn-\nO-Mn bond angle and consequent formation of magnetic\nclusters as the concentration of Tb is increased. This\nclustering is likely to be due to the breaking or weaken-\ning of the double exchange closer to the Tb-sites caused\nby the local structural distortion and the spin glass like\nfeature originates from the formation of such clusters.\nThe decreasing temperature corresponding to the cusp,\nidentified as T ghenceforth, suggests the confinement of\nmagnetic clusters to shorter length scales as the number\nof Tb centers is increased (increasing values of x).\nAs shown in Fig. 3, the ferromagnetic transition tem-\nperature decreases almost linearly with increasing x, up\ntox= 0.125. For the compositions showing both fer-\nromagnetic and spin glass characteristics (0.1 ≤x≤\n0.125),TcandTgare determined as illustrated in the\ninset of Fig. 3 for x= 0.11. Here, the χ-T data for x=\n0.15 (spin glass composition) is shifted towards the right\nside and that of x= 0.07 (ferromagnetic composition) is\nshifted towards the left side along the x-axis, to match\nwith the observed data of the mixed phase composition\nand the curves are normalized with respect to the maxi-\nmum values. Tcis taken as the mid point ofthe magnetic\ntransition. T gchanges abruptly around x= 0.125 and\nvaries from 66 K for x= 0.13 to 43 K for x= 0.25.Tg\nhas been found to be almost independent (40-50 K) of x\nfor higher Tb concentrations.6Thus, the lowest value of3\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s53/s48 /s49/s48/s48 /s49/s53/s48\n/s32/s84\n/s67\n/s32/s84\n/s103/s84/s32/s40/s75/s41\n/s120 /s32/s105/s110/s32/s76/s97\n/s48/s46/s54/s55/s45/s120/s84/s98\n/s120/s67/s97\n/s48/s46/s51/s51/s77/s110/s79\n/s51/s32/s120 /s32/s61/s32/s48/s46/s49/s49\n/s32/s120 /s32/s61/s32/s48/s46/s48/s55\n/s32/s120 /s32/s61/s32/s48/s46/s49/s53/s84\n/s103\n/s84\n/s99/s32\n/s32/s84/s32/s40/s75/s41\nFIG. 3: Variation of TcandTgas a function of xin\nLa0.67−xTbxCa0.33MnO3. Inset: illustration of the method\nof extracting TcandTg, forx= 0.11, using the data for x=\n0.15 (for Tg) andx= 0.07 (for Tc).\npossible Tgis larger than the temperature (30 K) where\na decrease in the ac susceptibility is observed for 0 < x≤\n0.125.\nThe decrease in the susceptibility below 30 K as soon\nas a small amount of Tb is incorporated in the lat-\ntice of La 2/3Ca1/3MnO3has been ascribed to spin glass\nbehavior.9However, neutron diffraction studies showed\nferromagnetic ordering for x= 0.067 down to 7 K.11\nThus, the feature at 30 K for 0.03 ≤x≤0.125 is not\nlikely to be spin glass transition. It is possible that at\nsmall concentrations the Tb3+ions are randomly dis-\ntributed in the lattice, the double exchange is disturbed\naround the Tb centers and therefore tiny magnetic clus-\ntersareformedwithreducedMn-O-Mnangle. Thesetiny\nmagnetic clusters remain isolated until x= 0.125 (1/8)\nabove which larger magnetic clusters are formed due to\nthe breaking or considerable weakening of the three di-\nmensional long range ordering. Thus, the temperature\nat which a decrease in the susceptibility is observed, due\nto these small clusters, remains the same until x= 1/8.\nThe three dimensional ordering is affected when x >1/8.\nHowever, there is another possibility that, at small con-\ncentrations, the Tb ions form a TbMnO 3like local envi-\nronment in the lattice of La 0.67−xTbxCa0.33MnO3. For\nTbMnO 3, anincommensurate-commensuratephasetran-\nsition, which is accompanied by a ferroelectric transition,\nassociated with a lattice modulation, is observed close to\n∼30 K and large magnetic field controlled polarization\neffects are reported at this temperature.13,14\nA local phase separation exists in the x= 0.1 sample,\nas evidenced by a small step- like magnetic transition, in-\ndicating that ferromagnetic clusters are started forming\nat this value of x. It is possible that, above this value of\nx, some magnetic clusters with short range ordering are\nseparated whose size decreases with increasing x. Thus,\nat intermediate values of x, the lattice is consisting of/s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53/s45/s57/s48/s45/s54/s48/s45/s51/s48/s48/s51/s48/s54/s48/s57/s48\n/s48/s46/s48/s48 /s48/s46/s48/s54 /s48/s46/s49/s50 /s48/s46/s49/s56 /s48/s46/s50/s52/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s65/s116/s32/s49/s50/s32/s75/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s120 /s32/s61/s32/s48/s46/s48\n/s32/s120 /s32/s61/s48/s46/s49/s48\n/s32/s120 /s32/s61/s48/s46/s49/s51\n/s32/s120 /s32/s61/s32/s48/s46/s49/s53\n/s32/s120 /s32/s61/s32/s48/s46/s50/s48\n/s65/s116/s32/s49/s53/s32/s107/s79/s101/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s120 /s32/s105/s110/s32/s76/s97\n/s50/s47/s51/s45/s120/s84/s98\n/s120/s67/s97\n/s49/s47/s51/s77/s110/s79\n/s51\nFIG. 4: The M-H curves of La 0.67−xTbxCa0.33MnO3for dif-\nferent values of x. Inset: variation of magnetization at 15 kOe\nas a function of x.\nlarger ferromagnetic clusters with sufficiently long range\nordering and smaller short range ordered clusters. Also,\nfor the largerlong range orderedpart, the magnetic tran-\nsition temperature decreases due to the decrease in the\nMn-O-Mn bond angle. Thus, the clustering may be seen\nto start when x= 0.11, where a cusp is also observed\nalong with the normal magnetic features.\nFig. 4 shows the dc magnetization curves of different\ncompositions, measured at 12 K, up to a maximum field\nof 15 kOe. The magnetization is saturated above 10\nkOe for x≤0.1. The variation of the magnetization\nat the maximum measured field of 15 kOe, as a function\nofxin La2/3−xTbxCa1/33MnO3is shown in the inset\nof Fig. 4. The magnetization remains almost the same\nup tox= 0.1 and then decreases above this value of\nx. Also, the magnetization is not saturated for x >\n0.1 in the measured field range. Recent studies sug-\ngest the existence of quantum critical point (QCP) ef-\nfect in La 0.67Ca0.33Mn1−xGaxO3.15QCP is defined as\na second order transition accompanied by the change of\na non-thermal parameter. The observation of QCP in\nLa0.67Ca0.33Mn1−xGaxO3system is as predicted by the\ntheoretical calculations and the QCP in this system was\nexpected for a value of 10-20% of Ga substitution.16The\nsubstitution of a nonmagnetic ion like Ga at the Mn-\nsublattice of the perovskite-type oxide causes the local-\nization of the electronic states suppressing the double ex-\nchange mechanism. For La 0.67Ca0.33Mn1−xGaxO3, the\nspontaneous magnetic moment calculated from the ex-\nperimental neutron diffraction patterns recorded at 1.5\nK decreased for x >0.1 and vanished for x >0.16, in-\ndicating electron localization which suppresses the dou-\nble exchange mechanism. The present dc magnetization\ndata on La 2/3−xTbxCa1/33MnO3shows almost a similar\ntrend, except for the finite value of the magnetization at\nhigher values of x, suggesting the possible existence of\nQCP in this system also. However, this needs to be ver-4\n/s48 /s49 /s50 /s51/s48/s50/s48/s52/s48/s54/s48\n/s120 /s32/s61/s32/s48/s46/s49/s50/s53/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s48 /s53 /s49/s48 /s49/s53/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s120 /s32/s61/s32/s48/s46/s49/s51\n/s32/s32\n/s48 /s53 /s49/s48 /s49/s53/s48/s50/s48/s52/s48/s54/s48\n/s120 /s32/s61/s32/s48/s46/s49/s53/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s72/s32/s40/s107/s79/s101/s41/s48 /s53 /s49/s48 /s49/s53/s48/s50/s48/s52/s48\n/s120 /s32/s61/s32/s48/s46/s50/s48/s32\n/s72/s32/s40/s107/s79/s101/s41\nFIG. 5: The virgin magnetization (red curves) and part of the\nhysteresis loop (blue curves) of x= 0.125, 0.13, 0.15, and 0.2\ninLa0.67−xTbxCa0.33MnO3. Notethatthe Hscale is different\nforx= 0.125.\nified with the help of neutron diffraction measurements,\nas made in the case of the Ga substituted system.\nAnother interesting behavior observed in the low tem-\nperature M−Hmeasurements for the compositions im-\nmediately above x= 1/8 is an irreversible jump to a fer-\nromagnetic state at higher magnetic fields, as shown in\nFig. 5. This is observedonly for the virgin magnetization\nmeasurements. Up to x= 0.125, a normal feature is ob-\nserved, where the virgin magnetization curve lies inside\nthe hysteresis loop. For x= 0.2, the behaviour is similar\nto that observed for some typical spin glass systems,12\nwhere the virgin magnetization curve initially lies out-\nside the loop and then merges with the loop at higher\nfields. On the other hand, for x= 0.13 and 0.15, the en-\ntire virgin magnetization curve lies outside the hysteresis\nloop above a certain small field (this small crossing field\nis observed for x= 0.2 and 0.25 also, and increases with\nx), and an anomalous step-like feature is observed in the\nvirginmagnetizationcurve,similartothat ofametamag-\nnetic transition. However, this transition is completelyirreversible. The field above which a broad step is ob-\nserved is larger for x= 0.15 compared to that for x=\n0.13. After the magnetic field is increased in the negative\ndirection to -15 kOe and when brought back to +15 kOe\nthrough H = 0, the transition is not observed. This is\nan irreversible ferromagnetic transition in the sense that\nthe step-like feature is never obtained when the measure-\nmentswererepeatedimmediatelyorevenafteratimegap\nof 30 minutes. In the subsequent measurements, the first\npart of the curve always lies inside the hysteresis loop,\nlike that for the compositions for x≤0.125. It appears\nthat an irreversible magnetic field induced phase transi-\ntion is occurred. Similar characteristicswerereported for\nsomePr-basedsubstituted manganitecompositions. Dho\nand Hur explained this behavior in terms of the reorien-\ntation ofthe Mn spins pinned by localizedPrmoments,17\nwhereas Woodward et al.18explained the observations in\nterms of an avalanche behavior. It may be noted that\nthe first part of the virgin curve is similar to that of the\nspin glass composition x= 0.2, in the present case, and\ntherefore, it is possible that the second jump is due to a\nfield induced growth of the larger ferromagnetic clusters.\nOnce the clusters are grown, it is not possible to revert\nback to the original state due to the unavailability of suf-\nficient thermal energy. The original state is found only\nwhen the temperature is raised above the peak tempera-\nture and then cooled back in zero field.\nIV. CONCLUSIONS\nThe present studies made on a series of close compo-\nsitions in La 0.67−xTbxCa0.33MnO3indicate that single\nphase ferromagnetic compositions are possible for x <\n0.1, mixed long range ordered ferromagnetic and short\nrange ordered magnetic clusters coexist for 0.1 ≤x≤\n0.125 and spin glass like phases are formed for larger val-\nues ofx. The ferromagnetic clusters present in the com-\npositions immediately above the cross over region show\nmagneticfield induced growthand givelargermagnetiza-\ntionathigherfields. Furtherdetailedstudiesarerequired\nto understand the complex magnetic behavior shown by\nthese Tb substituted manganite compositions at the in-\ntermediate and the cross over regions.\n∗Electronic address: pa.joy@ncl.res.in\n1Y. Tokura, Colossal Magnetoresistive Oxides (Gordon and\nBreach, Singapore, 2000).\n2E. 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B 70, 174433 (2004)."    },    {        "title": "2202.01611v1.On_the_Divergence_of_the_Ferromagnetic_Susceptibility_in_the_SU_N__Nagaoka_Thouless_Ferromagnet.pdf",        "content": "arXiv:2202.01611v1  [cond-mat.str-el]  3 Feb 2022On the Divergence of the Ferromagnetic Susceptibility in th e SU(N)\nNagaoka-Thouless Ferromagnet\nRajiv R. P. Singh1and Jaan Oitmaa2\n1Department of Physics, University of California Davis, CA 9 5616, USA\n2School of Physics, The University of New South Wales, Sydney 2052, Australia\n(Dated: February 4, 2022)\nUsing finite temperature strong coupling expansions for the SU(N) Hubbard Model, we calculate\nthe thermodynamic properties of the model in the infinite- Ulimit for arbitrary density 0 ≤ρ≤1\nand allN. We express the ferromagnetic susceptibility of the model a s a Curie term plus a ∆ χ,\nan excess susceptibility above the Curie-behavior. We show that, on a bipartite lattice, graph by\ngraph the contributions to ∆ χare non-negative in the limit that the hole density δ= 1−ρgoes\nto zero. By summing the contributions from all graphs consis ting of closed loops we find that the\nlow hole-density ferromagnetic susceptibility diverges e xponentially as exp∆ /TasT→0 in two\nand higher dimensions. This demonstrates that Nagaoka-Tho uless ferromagnetic state exists as a\nthermodynamicstateofmatteratlowenoughdensityofholes andsufficientlylowtemperatures. The\nconstant ∆ scales with the SU(N) parameter Nas 1/Nimplying that ferromagnetism is gradually\nweakened with increasing Nas the characteristic temperature scale for ferromagnetic order goes\ndown.\nINTRODUCTION\nThe Hubbard model [1–3] is a central model for de-\nscribing the behavior of electrons in solid state systems\nand has had a huge impact in our understanding of con-\ndensed matter physics [4–6]. Strong correlations, arising\nfrom the on-site repulsion in the Hubbard model, can\nbe used to understand many basic solid state phenom-\nena including metal-insulator transitions, antiferromag-\nnetism, superconductivity, spin-liquids and itinerant fer-\nromagnetism.\nNagaoka-ThoulessFerromagnetismisaclassicproblem\nin itinerant magnetism [7]. Nagaoka [8] and Thouless\n[9] independently showed that when the Hubbard repul-\nsionUis large enough, a single hole introduced into a\nsystem with one-particle per site, polarizes the system\naround the hole. There have been several variational\nand numerical studies [10–14] of Nagaoka-Thoulessferro-\nmagnetism, especially in the ground state of the system.\nAtfinite temperatures, rigorousmathematicalarguments\nhave been made to show that magnetization in a field ex-\nceeds the pure paramagnetic value [15, 16] and Dynami-\ncal Mean-Field Theory (DMFT) [17] was used to obtain\na phase diagram in the density-temperature plane.\nThe cold atomic gases in optical lattices provide a new\nmotivation for study of the Hubbard model [18–23]. In\nthese systems, it is possible to build an ensemble that is\nwell described by the Hubbard model and where the mi-\ncroscopic parameters such as Uandtcan be controlled\nand a priori well understood. Furthermore, cold atomic\ngases allow one to change the number of fermion species\nfrom two to a larger Nand thus study the Hubbard\nmodel with SU(N) symmetry [24–29] for different values\nofN.\nFinite temperature strong-couplingexpansion is a nat-\nural way to address the magnetic behavior of the Hub-bard model at finite temperatures, at various hole densi-\nties, in the thermodynamic limit [30–32]. These expan-\nsionscanbedevelopedinthegrandcanonicalensembleat\nfixed fugacity ζ= expβµin powers of βt,w= exp−βU,\nand1/βU. After changingvariablesfrom fugacitytopar-\nticle density ρ, one can obtain temperature dependent\nthermodynamic properties at various densities. For Uof\norder or larger than the bandwidth they allow one to re-\nlate the thermodynamics of the Hubbard model at low\ntemperatures to a generalized Heisenberg or t-J model\n[33–36]. The expansions simplify in the limit U→ ∞,\nin which case many terms can be set to zero and can\nbe used to study the problem of Nagaoka-Thouless fer-\nromagnetism.\nThe first few terms of the expansion suffice to give\nan accurate numerical description of the thermodynamic\nproperties of the model at temperatures larger than the\nhopping parameter t. And, as shown previously for the\nSU(2) t-J model [37–39], series extrapolationmethods al-\nlowoneto goto muchlowertemperatures. But, anumer-\nical extrapolation is difficult to control reliably down to\nT= 0. Here, we are interested in the entire temperature\nrange 0< T <∞. We show that, close to half filling, i.e.\nin the limit δ= 1−ρgoing to zero, the thermodynamic\nuniform magnetic susceptibility can be computed all the\nway toT→0 by summing over the loop graphs in each\norder of perturbation theory. This calculation provides\na lower bound for the susceptibility and leads to a func-\ntion which diverges exponentially to infinity as exp∆ /T\nas the temperature goes to zero. This shows that, for\nlarge enough U, the Nagaoka-Thoulessferromagnetic be-\nhavior is a thermodynamic phenomena at low density of\nholes and low enough temperatures. These results are\ntrue for any N >1 of the SU(N) models [40] and in any\ndimension greater than one. However, the constant ∆\nscales as 1 /N, that is the temperature scale for the tran-2\nsition goes down as Nincreases. For the SU(2) case, our\nresults are in agreement with DMFT which also found\nthat the transition temperature goes to zero as δ→0\n[17].\nMODEL AND METHODS\nTheSU(N)HubbardmodelisdefinedbyaHamiltonian\nH=H0+V, where the unperturbed Hamiltonian H0is\nan on-site term:\nH0=U/summationdisplay\nini(ni−1)\n2−µ/summationdisplay\nini−h/summationdisplay\ni(n1i−ni\nN),(1)\nwithnithe total number operator for particles on site i\nandµis the chemicalpotential. The last term his a spin-\npolarizing field that lowers the energy when the particle\nis in the first spin state n1i= 1 and raises it for all other\nstates and has an overall zero trace. The perturbation V\nis the hopping term:\nV=−t/summationdisplay\n<i,j>N/summationdisplay\nα=1(C†\ni,αCj,α+h.c.), (2)\nwhere the sum < i,j > runs over nearest-neighbor pairs\nof sites of a lattice and the sum over αruns over the\nNspecies of Fermions. The total number of fermions\nof each species is a constant of motion. Thus both the\nchemical potential and field terms commute with the rest\nof the Hamiltonian.\nUsing the formalism of thermodynamic perturbation\ntheory [30, 31],the logarithm of the grand partition func-\ntion, per site, can be expended as\nlnZ=lnz+∞/summationdisplay\nr=1/integraldisplayβ\n0dτ1/integraldisplayτ1\n0dτ2.../integraldisplayτr−1\n0dτr\n<˜V(τ1)...˜V(τr)>N(3)\nwherezis the single-site partition function,\n˜V=eτH0Ve−τH0, (4)\nand,\n< X >= Tre−βH0X/Tre−βH0. (5)\nIn each order, the terms in the expansion can be ex-\npressed in terms of various graphs on the lattice as:\nlnZ= lnz+/summationdisplay\nGLGz−Ns(βt)NbXG,(6)\nIntheexpression,thegraph GhasNssitesand Nbbonds.\nLGis the lattice constant of the graph defined as the\nextensive part of the graph count, per lattice site. The\nweight-factor XGisthereducedcontributionofthegraphobtained from an evaluation of the traces which depends\nonβt,βU, fugacity ζ, fieldhandN.\nIn theU→ ∞limit, no double occupancy is allowed\nand the weight-factor for a graph with Ns-sites reduces\nto\nXG=Ns−1/summationdisplay\nn=1xG\nnζn. (7)\nHerexG\nnis a polynomial in the SU(N) parameter Nof\nordern.\nFrom the partition function, the particle density (per\nsite) can be obtained via the relation\nρ=ζ∂\n∂ζlnZ. (8)\nThermodynamic quantities such as Internal energy per\nsite,e, and entropy per site, s, are obtained using the\nrelations\ne=−(∂\n∂βlnZ)ζ, (9)\nand\ns=−β(∂\n∂βlnZ)ζ−ρlnζ+lnZ. (10)\nThe ferromagnetic susceptibility per site is defined by\nthe second derivative of ln Zwith respect to the spin-\npolarizing field h. It is given by\nχ=1\nβ∂2\n∂h2lnZ (11)\nThe field term in the Hamiltonian is defined solely for\ncalculating the susceptibility. Otherwise, we will restrict\nall calculations to h= 0.\nSINGLE-SITE TERM AND SERIES EXPANSIONS\nIn thelimit of U→ ∞the single-sitepartitionfunction\nto order h2becomes z=z0+h2z1, where,\nz0= 1+Nζ (12)\nand\nz1=β2ζ\n2N−1\nN(13)\nFor all calculations other than the susceptibility, we can\nseth= 0. In zeroth order the particle density is given\nby\nρ0=Nζ\n1+Nζ. (14)3\nThe zeroth order susceptibility per site is given by\nχ=2\nβz1\nz0= (N−1\nN2)βρ. (15)\nThis is a Curie law as no double occupancy means we\nhave local moments at all temperatures.\nWe will define excess susceptibility, over and above the\nCurie-law as\nχ=Cρ\nT+∆χ, (16)\nwith Curie constant Cequal toN−1\nN2. Note that in this\nequation the density is the full density not the bare den-\nsity obtained in zeroth order. Our goal is to calculate\n∆χ.\nIn our studies, we will restrict ourselves to bipartite\nlattices. All the graphs that contribute to the zero-field\npartition function on a bipartite lattice to eighth order\ntogether with their weights for arbitrary Nare given in\nSupplementary materials.\nFIG. 1: Two classes of graphs that contribute to the suscepti -\nbility expansion. (a) Tree graphs, with noclosed loops. Eve ry\nbond must be doubled in order to have a non-zero trace con-\ntribution. In the single hole sector, each fermion moves bac k\nandforthastheholes movesaroundthegraph. Everyspincan\nbe independently of any spin species. Such graphs only con-\ntribute to the Curie-law and do not contribute to the excess\nsusceptibility at all. (b) Graphs consisting of closed loop s. In\nthe single hole sector, as the hole traverses the loop, each s pin\nmoves to its neighboring position. In order to have a non-zer o\ntrace contribution, all fermions must be of the same species .\nThus these graphs have a maximum relative contribution to\nthe excess susceptibility.\nNear one-particle per site all properties can be ex-\npanded in powers of the hole density δ0, given by\nδ0= 1−ρ0=1\n1+Nζ=1\nz0, (17)\nFor the Nagaoka-Thouless problem, we are interested in\nthelimit δ= 1−ρgoingtozero. Thus, wewillkeepterms\nlinear in δ0and drop all terms proportional to higher\npowers of δ0. These linear in δ0terms come from exactlyone hole in each cluster. We should note that this does\nnot mean we are looking at a single hole in the ther-\nmodynamic system. Our formalism implies that we are\nstudying the limit of low hole density as similar behavior\nwill be happening independently all over the system.\nIn the large Ulimit, the weight of a graph XGis a\npolynomial in ζof orderNs−1 , where Nsis number of\nsites in the cluster. The restriction to lowest power of δ0\nreduces the weight factor for a graph to:\nXG=xGζNs−1. (18)\nThe coefficients xG, which depend on Nand the field h,\nturn out to be always positive as can be seen from the\nexplicit calculations to eighth orderin the supplementary\nmaterials. These terms correspond either to a single hole\nmovingback and forth on a tree like graphwith no closed\nloops or a single hole moving in closed loops. In both\ncases they are positive. We will see that this will lead us\ntothe resultthat the contribution toexcesssusceptibility\nfrom every graph is non-negative. This means that even\na partial summation of graphs is a lower bound on the\nexcess susceptibility.\nIn this limit, the relation between the full density func-\ntion and fugacity becomes\nρ=ρ0+/summationdisplay\nGLG(βt)Nb(Ns−1−Nsρ0)xGζNs−1\nzNs\n0(19)\nThe excess susceptibility is given by\n∆χ=/summationdisplay\nGLG(βt)Nb(C1G−C2G),(20)\nwhere\nC1G=1\nβ∂2\n∂h2(XG\nzNs), (21)\nand\nC2G= (Ns−1−Nsρ0)C\nTXG\nzNs. (22)\nFor tree graphs, with no closed loops (see Fig. 1), the\ncoefficient xGin zerofieldisproportionalto NNs−1. This\nreflects the fact that in the absence of closed loops ev-\nery spin can independently be of any species. For these\ngraphs the contribution to excess susceptibility vanishes\nidentically. Physically this is a reflection of the fact that\nsusceptibility of independent spins is already contained\nin the Curie law. Thus, we only need to consider those\nweights where the power of Nin a graph with Nssites\nis less than Ns−1. In all these terms at least some of\nthe spins are constrained to be of the same species. Even\nthe smallest such constraint can be shown to lead to a\npositive contribution to the susceptibility.\nAt the other extreme are those terms where zero-field\nxGscales linearly with N. This implies that every spin4\nin the graph must be of the same species to contribute to\na non-zero trace. An example is a graph consisting of a\nsingleclosedloop(SeeFig.1). Itmusthavethisbehavior.\nAs a hole traversesaroundthe loop, everyFermion in the\nloop movesto its neighboringposition and hence must be\nof the same species as its neighbor to contribute to the\ntrace. These graphs contribute maximally to the excess\nsusceptibility. It can be shown that for a single loop of\nlengthlthe zero-field weight-factor is\nXG=2l\nl!N ζl−1(23)\nThe excess susceptibility contribution from this graph\ncan be shown to be\nC\nT(βt)l\nl!2l(l−1)(l−2)Nζl−1\nzl\n0. (24)\nExpressing this in terms of ρ0gives\nC\nT(βt)l\nl!2l(l−1)(l−2)ρl−1\n0δ0\nNNs−2. (25)\nIt is well known that for large l, the number of polygons\nof even length lembedded in a bipartite lattice scale as\n[41]\npl=Aµl\npla−3, (26)\nwhere the constant µpcalled the connectivity constant\nis known approximately for most lattices [42]. Ignoring\nthe weak dependence on the exponent awhich will only\naffect the prefactor, the contributions of polygons can be\nsummed to obtain an excess susceptibility of\n∆χ∝N2δ0\nρ0C\nTexp∆\nT, (27)\nwith ∆ =tρ0µp\nN.Ignoring the slowly varying prefactor,\nthis shows that the excess susceptibility diverges expo-\nnentially as T→0. We believe the primary role of the\nadditional terms not included in this summation is to\ndecorate these graphs and renormalize the bare density\nρ0to the full density ρ.\nThisresultprovidesalowerboundtothemagneticsus-\nceptibility and implies that the Nagaoka-Thouless ferro-\nmagnet is a thermodynamic phase of matter for low hole\ndensity and low enough temperatures. The characteristic\ntemperaturescaleatwhichthesusceptibilitybecomesex-\nponentially large is inversely proportional to the SU(N)\nparameter N. Thus, the tendency for ferromagnetism\ngradually weakens with increase in N. These results are\nin agreement with the earlier dynamical mean-field the-\nory results for the SU(2) case in that the ferromagnetic\nphase boundary was found to go to zero temperature\nas the hole density goes to zero [17]. They are also in\nagreement with mathematical arguments that show the\nexistence of finite magnetization in a field that exceeds\nthe paramagnetic value at any temperature [16].The extension of these results to finite U/tand finite\nhole doping can be done numerically as was done for\nthe SU(2) t-J models some time ago [37–39, 43]. Those\nstudies show that at small enough J/tand close to half\nfilling the peak in the magnetic susceptibility shifts to\nq= 0. However, from a small number of terms in the\nexpansion it is more difficult to rigorously establish the\ndivergence of the susceptibility.\nDISCUSSIONS AND CONCLUSIONS\nIn this paper we have used finite temperature strong\ncoupling expansions for the Hubbard model to revisit the\nproblem of Nagaoka-Thouless ferromagnetism at large U\nandsmall holedopingnearone-particleper site. We have\nshown that the ferromagnetic susceptibility of the sys-\ntem diverges exponentially as exp∆ /TasT→0. 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B 41 9397 (1990)."    },    {        "title": "1502.04934v1.Direct_Observation_of_Ferromagnetic_State_in_Gold_Nanorods_Probed_using_Electron_Spin_Resonance_Spectroscopy.pdf",        "content": "arXiv:1502.04934v1  [cond-mat.mes-hall]  17 Feb 2015Direct Observation of Ferromagnetic State in Gold Nanorods\nProbed using Electron Spin Resonance Spectroscopy\nYuji Inagaki∗and Tatsuya Kawae\nDepartment of Applied Quantum Physics, Faculty of Engineer ing,\nKyushu University, Fukuoka 819-0395, Japan\nNatsuko Sakai and Yuta Makihara\nDepartment of Material Physics and Chemistry,\nFaculty of Engineering, Kyushu University, Fukuoka 819-03 95, Japan\nHiroaki Yonemura and Sunao Yamada\nDepartment of Applied Chemistry, Faculty of Engineering,\nKyushu University, Fukuoka 819-0395, Japan\nAbstract\nX-band electron spin resonance (ESR) spectroscopy has been performed for gold nanorods\n(AuNRs) of four different sizes covered with a diamagnetic sta bilizing component, cetyltrimethy-\nlammmonium bromide. The ESR spectra show ferromagnetic fea tures such as hysteresis and reso-\nnance field shift, depending on the size of the AuNRs. In addit ion, the ferromagnetic transition is\nindicated by an abrupt change in the spectra of the two smalle st AuNRs studied. A large g-value\nin the paramagnetic region suggests that the ferromagnetis m in the AuNRs originates from strong\nspin-orbit interaction.\n1I. INTRODUCTION\nIn the past decade, rod-shaped gold nanoparticles, called Au nano rods (AuNRs), have\nattracted much attention because of their potential applications in sensing, imaging, and in\nvivo photothermal cancer therapy1–4. These applications are based on the inherent tunable\noptical properties of the AuNRs by changing the aspect ratio ardefined by the ratio of the\nlength to the diameter5. In contrast, there have been few reports thus far concerning the\nmagnetic properties of AuNRs6, although there have been many studies on the magnetism\nof spherical Au nanoparticles (AuNPs)7–12.\nRecently, Yonemura et al.examined the effects of magnetic processing and observed\nthe magnetic orientation and the side-by-side aggregation of AuNR s/poly(styrenesulfonate)\ncomposites, AuNRs with three different aspect ratios ( ar= 8.3, 5.1, and 2.5), and Au\nnanowires under a magnetic field13–15, which implies that the magnetic response of AuNRs\nis positive, although bulk Au has been believed to be diamagnetic for a lo ng time. However,\nthe detailed properties and origin of magnetism in AuNRs is not yet well understood.\nGenerally, magnetic properties have been investigated through ma gnetization measure-\nments using highly sensitive magnetometers, such as a commercial S QUID magnetometer.\nIn nano-sized particles coated with a stabilizing polymer, however, it is difficult to extract\nthe net magnetization precisely owing to the superposition of a large amount of diamagnetic\ncomponents16. We focus on the electron spin resonance (ESR) technique to elimina te the\nsignal from the stabilizing polymer because ESR is insensitive to diamag netism. Moreover,\nthe skin depth of Au at the X-band microwave region ( ∼9 GHz) is about 0.8 µm, which is\nmuch larger than the size of AuNRs in the present study. These fea tures indicate that ESR\nis a powerful tool for investigating the magnetic properties of AuN Rs.\nIn the present study, X-band ESR measurements were performe d in detail for AuNRs\nof four different sizes, where arwas varied from 2.5 to 8.3. Ferromagnetic features were\nobserved in all AuNRs. In addition, ferromagnetic transitions were clearly detected in the\ntwo smallest AuNRs studied. From the ESR parameters deduced fro m the conduction\nelectron spin resonance (CESR) at high temperatures, strong sp in-orbit interaction and\nresultant large effective mass of conduction electrons are pointed out for the origin of the\nmagnetism in the AuNRs on the basis of Elliott-Yafet theory.\n2TABLE I. Dimensions, aspect ratio, estimated content of Au, CTAB and other metals for each\nAuNR\nsize (nm) arAu(%) CTAB(%) Ag(%)\ns-AuNR 4.6 φ×11.6 2.5 3.6 Br:22.6, C:59.3, H:10.8, N:3.7 -\ns′-AuNR 5.0 φ×20.0 4.0 41.3 Br:16.7, C:31.4, H:5.8, N:1.9 2.9\nm-AuNR 7.2 φ×36.6 5.1 3.1 Br:22.9, C:59.4, H;11.0, N:3.6 -\nl-AuNR 7.7 φ×63.8 8.3 1.9 Br:22.7, C:60.6, H:11.2, N:3.7 -\nII. EXPERIMENTAL\nAuNRs studied here are named as s-, s′-, m-, and l-AuNR from small to large values\nofar. All AuNRs are prepared by the soft template method using cetyltr imethylamm-\nmonium bromide (CTAB)17. The s- and s′-AuNR are prepared using a reducing agent,\ntriethylamine, with or without acetone. The m-AuNR is prepared usin g a combination of\nchemical reduction and photoreduction. The l-AuNR is prepared us ing two kinds of re-\nducing agents, sodium borohydride and triethylamine. Typical TEM im ages of AuNRs are\ngiven in ref.13. X-ray fluorescence and CHN elemental analyses confirm that AuN Rs contain\nno contamination of other magnetic metals, as listed in Table I, which e nsures that the\nmagnetic response in the present study originates from the Au ato ms in AuNRs. ESR mea-\nsurements are performed using an X-band microwave system (JEO L ES-SCEX) equipped\nwith a continuous-He-flow-type cryostat (Oxford ESR910) oper ating down to T∼5 K.\nIII. RESULTS AND DISCUSSION\nFigures 1(a) and 1(b) show the temperature dependence of ESR s pectra observed for\ns-AuNR and s′-AuNR, respectively. Two spectra plotted with gray lines in Fig. 1(a) corre-\nspond to the blank signals recorded at 7 K and 295 K, which originate f rom the background\nfrom instruments including the sample holder (quartz tube) and the CTAB. This indicates\nthat the absorptions at around 320 mT are not intrinsic signals from the AuNR. At a glance,\nit seems that the spectra from the two AuNRs show quite similar temp erature dependences.\nHowever, on closely observing both spectra, several differences can be discerned. At room\ntemperature, the ESR spectrum of s-AuNR consists of two compo nents: a sharp absorption\n30 100200300400500600700 \nH (mT) 286 K \n200 K \n100 K \n6 K \n(d) 0 100 200 300 400 500 600 \nH (mT) (b) 297 K \n251 K \n202 K \n150 K \n120 K \n100 K \n80 K \n62 K \n49 K \n41 K \n30 K \n21 K \n12.5 K \n5.5 K \n0 100 200 300 400 500 Derivative Intensity (arb. units) \nH (mT) 295 K \n200 K \n140 K \n103 K \n75 K \n60 K \n51 K \n40 K \n32 K \n16.5 K \n9.7 K \n6.0 K \nblank 7 K blank 295 K \n(a) \n0 100 200 300 Derivative intensity (arb.units) \nH (mT) 300 K \n250 K \n200 K \n100 K \n88 K \n75 K \n62 K \n45 K \n32 K \n15 K \n10 K \n6.8 K 150 K \n(c) \nFIG. 1. (Color online) Temperature dependence of ESR spectr a in (a) s-AuNR, (b) s′-AuNR (c)\nm-AuNR and (d) l-AuNR. The absorption around 320 mT is due to i nstrumental background, as\nobserved from the blank spectra recorded at 295 K and 7 K shown by the gray lines in (a).\ncentered at s 1= 275 mT and a broad one around s 2= 230 mT, which is strongly suppressed\nwhen the temperature is decreased. In contrast, the spectra o f s′-AuNR have no broad\ncomponent and show only a sharp absorption centered at s′\n1= 289 mT. Thus, the sharp ab-\nsorptions at s 1and s′\n1are regarded as intrinsic properties of AuNRs, and their temperat ure\nvariations are examined below.\nThe temperature dependences of g-values estimated from resonance fields and line widths\n∆Hfor s1and s′\n1are summarized in Fig. 2. The resonance field and line width are nearly\nindependent of temperature for both samples above ∼80 K, while the spectra show a drastic\nshift of the resonance field and broadening of the width below 55 K fo r s-AuNR and 75\n41.5 2.0 2.5 3.0 \n020 40 60 80 100 \n0 50 100 150 200 250 300 g-value ΔH (mT) \nT (K) TcsTcs' \nFIG. 2. Temperature dependence of g-value (circle) and line width ∆ H(square). Open and solid\nsymbols represent the results for s-AuNR and s′-AuNR, respectively.\nK for s′-AuNR. The g-value of s-AuNR (s′-AuNR) increases with decreasing temperature\nand reaches 3.19 (3.07) at the lowest temperature T∼6 K. For s-AuNR (s′-AuNR), the\nline width shows a maximum at approximately 40 K (60 K), which is followed by a gradual\nnarrowing with decreasing temperature. This series ofbehaviors is atypical featureobserved\nin the magnetic ordering process. Hence, we conclude that the res onance field shift and\nbroadening of the width in s- and s′-AuNRs are caused by magnetic ordering with transition\ntemperatures of Ts\nc∼55 K and Ts′\nc∼75 K, respectively.\nThe results for m- and l-AuNRs are depicted in Figs. 1(c) and 1(d), r espectively. In\nm-AuNR, the ESR spectrum cannot be observed in the high-temper ature region. When the\ntemperature is decreased below T∼100 K, a narrow absorption with ∆ H= 8 mT appears\nat around 150 mT. The intensity of the spectrum grows with decrea sing temperature, while\nboth the resonance center and line width are almost independent of temperature down to T\n= 6.8 K. This temperature dependence will be discussed later.\nIn contrast, l-AuNR shows broad ESR spectra in the entire temper ature range, as shown\nin Fig. 1(d). The resonance field at ∼70mT in l-AuNR is the lowest among all AuNRs, while\nthe line width of ∼130 mT is the broadest. It is significant that a hysteresis emerges in all\nthe spectra of l-AuNR below ∼60 mT in the magnetic field sweep, as indicated by arrows. A\nhysteresis between magnetizing and demagnetizing processes is a c haracteristic feature of a\n50 50 100 150 200 250 300 350 400 Derivative intensity (arb. units) \nH (mT) g = 2 \nFIG. 3. (Color online) ESR spectra of all AuNRs at T ∼6K. Red, blue, green and orange colors\ncorrespond to s-, s′-, m- and l-AuNR, respectively. The vertical dotted line ind icates the resonance\nfield forg=2.\nferromagnet with domain structures and/or magnetic anisotropy18,19. Note that the domain\nstructure and anisotropy give rise to the line broadening in ESR spec tra. In other words,\nthese features observed in l-AuNR are well explained by assuming a la rge scale ferromagnet.\nNext, we discuss the systematic resonance shift in all AuNRs. The s pectra at the lowest\ntemperature T∼6 K are plotted together in Fig. 3 to make a comparison between all\nAuNRs. In a ferromagnet with a cylindrical shape like the present Au NRs, ferromagnetic\nresonance (FMR) occurs at a resonance field that depends on the demagnetizing form factor\nowing to the aspect ratio ar, as well as on the magnitude of moment Msand anisotropy K.\nIn fact, the magnitude of shift increases with ar in the present systems; the shift is estimated\nto be approximately 248, 176, 114, and 118 mT for l-, m-, s′-, and s-AuNR, respectively,\nwhich corresponds to the order of arexcept for s- and s′-AuNR. The reversal between them\nmay be caused by the resonance shift dominated by an effective anis otropy field represented\nbyK/Ms. Thus, a small anisotropy Kcan give rise to large effective field if Msis small. As\nfor the anisotropy, the following qualitative discussion can be made. In randomly oriented\nferromagnetic species, a powder pattern is expected in the FMR sp ectrum. Although a\nclear powder pattern was not recorded for all AuNRs, finite ESR int ensity can be seen like\na tail in a higher field range than the main absorption peak, as repres ented in l-AuNR. It\nis difficult to refer about other AuNRs because of the superposition of background signal\nat around 320 mT. However, a small intensity like shoulder is visible at a bout 270 mT in\n6s′-AuNR, as indicated by the vertical arrow in Fig. 3. Such FMR spectr a correspond to the\ncase with a negative anisotropy in the cubic symmetry20. Detailed frequency dependence of\nESR is required to obtain further information about the FMR parame tersMsandK.\nThe observed systematic shift, which is generally not expected in ot her antiferromagnets,\nparamagnets, and ferromagnets with a spherical shape, provide s further evidence to the\nexistence of ferromagnetic states at low temperatures in all the p resent AuNRs. On the\nbasis of the hysteresis and systematic shift of the resonance field , it is reasonable to consider\nthat all AuNRs are in the ferromagnetic state at low temperatures . To our knowledge, this\nis the first time that a ferromagnetic state has been found in AuNRs .\nIn the final part of this Letter, we examine the ESR spectra in the p aramagnetic region.\nThe spectra of s- and s′-AuNRs above Tccan be understood as CESR ones. In the CESR\nregion,g-values are estimated to be 2.34 ±0.09 and 2.26 ±0.06 for s- and s′-AuNRs, respec-\ntively. These values are considerably larger than the value of 2.11 re ported for bulk Au\nbut comparable to that of 2.26 ±0.02 for small particles of Au with a mean diameter of 3\nnm21,22. For CESR, the difference ∆ gbetween the g-value of real metals and that of ideal\nfree electrons (2.0023) gives an approximate value of the spin-orb it interaction through the\nequation ∆ g∼λ/∆E. Here,λis the spin-orbit coupling constant and ∆ Eis the difference in\nenergy between the 6s band and the nearest 5d band23. Thus, the large ∆ gobserved in the\npresent measurements indicates the large contribution of the orb ital moment in the 5d band\nto conduction electrons in the 6s band, which governs the magnetic properties of AuNRs\nand leads to the ferromagnetic state at low temperatures. A stro ng spin-orbit coupling was\nalso confirmed in recent studies of X-ray magnetic circular dichroism in bulk Au as well as\nAuNPs24,25.\nThe strong spin-orbit interaction also causes a significant broaden ing of line width. The\nline width of CESR in metals is closely related to the spin-lattice relaxatio n time, i.e., the\nspin flip rate by phonons. Accordingly, ∆ Hvaries linearly with temperature. Therefore,\nat high temperatures, CESR is hardly detected in not only bulk metals but also m-AuNR.\nThis is a reason why we could not observe the ESR in m-AuNR above 100 K. In contrast,\nnarrow ESR absorptions are obtained for both s- and s′-AuNRs, which show rather gentle\ntemperature dependences without a marked increase in the width. These features may be\nexplained by considering the system size of AuNRs. In small systems , there exists a lower\nlimit of phonon mode frequency given by ν=vs/2L, whereLis the largest dimension of\n710 510 610 710 810 910 10 10 11 10 12 10 13 \n10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0ΔH/ρ (mT/ Ω/m) \nΔg2Au \nPd \nNa KRb Cs \nCu Ag \nAl \ns'-AuNR \ns-AuNR \nFIG. 4. Beuneu-Monod plot, which connects ∆ g2and ∆H/ρ, for pure metals given in ref.27and\nfor two systems in the present study. The value ∆ H/ρof most metals differs from the fit, which\nis indicated by the solid and dashed lines, by less than one or der of magnitude.\nthe system and vsis the speed of sound in the crystal; this results in an increase in the\nspin-lattice relaxation time of conduction electrons26. As a result, narrow absorptions are\nobserved in s- and s′-AuNRs. This scenario is qualitative and does not account for the\nnearly equal line widths of s-AuNR and s′-AuNR; other factors should be considered for the\nquantitative explanation.\nNevertheless, owing to the narrow line widths, we could successfully observe CESR of\nAuNR, which allows identification of the characteristic feature of th e present CESR results\nthrough sorting with other metals in the so-called BeuneuMonod plot shown in Fig. 427,28.\nThe BeuneuMonod plot is an empirical plot that connects CESR param eters ∆gand ∆H\nfor pure metals via ∆ H/ρ=α∆g2, whereρis the resistivity and αis a metal-dependent\nconstant. Most metals follow a straight-line fit in the log-log plot of ∆ H/ρvs. ∆g2by less\nthan one order of magnitude, as indicated by the solid and dashed line s in Fig. 4. The values\nused in this plot are taken at a temperature of approximately TD/7, where TDis the Debye\ntemperature. In the case of bulk Au, TD/7 corresponds to 20 K. The CESR parameters\nfor the two systems in the present study (s- and s′-AuNR) are not available for such a low\ntemperature, which is less than Tc. Thus, we plot the estimate taken in the temperature\n8range between Tcand room temperature, while the resistivity is fixed at the bulk value a t 20\nK. Error bars for both axes originate from the temperature varia tion of ∆ gand ∆H. In the\nplot, the two data points corresponding to s- and s′-AuNR deviate significantly fromthose of\nother metals, but they are located near the point corresponding t o Pd. It is well known that\nPd is close to satisfying the Stoner criterion even in a bulk form. Inde ed, ferromagnetism is\nrealized by the downsizing of Pd29–31. Accordingly, it is confirmed that the two AuNRs are\nalso close to the ferromagnetic state.\nTo reproduce the results for s-AuNR, s′-AuNR, and Pd, a smaller αis needed. This\nsuggests that the low value of αis closely related to the realization of ferromagnetism.\nAccording to the Elliott-Yafet theory, the coefficient αis given by ne2/γm∗, wheren,e, and\nm∗are the density, charge, and effective mass of the conduction elec tron, respectively, and γ\nis the magnetomechanical ratio. Therefore, the origin of the devia tion caused by the narrow\n∆Hand the large ∆ gis suggested to be an enhancement of the effective mass. This is a\nreasonable conclusion because a large effective mass is realized in a ty pical ferromagnet of\nNi, as revealed recently by high-resolution angle-resolved photoem ission studies32.\nIn summary, we performed ESR measurements for AuNRs of four s izes with different\naspect ratios. The ESR spectra at low temperatures are explained in the context of ferro-\nmagnetic resonance for all AuNRs. Detailed frequency dependenc e of FMR measurements\nwill enable further quantitative discussion on the size- and shape-d ependent magnetism of\nAuNRs. In the two smallest AuNRs, we detected ferromagnetic tra nsitions in the ESR spec-\ntra atT∼60 K, which offers an opportunity to explore the critical behavior of the phase\ntransition in nano-rod systems. The CESR above Tcsuggests that the strong spin-orbit\ninteraction is responsible for ferromagnetism in the AuNR systems.\nACKNOWLEDGMENTS\nThe authors are grateful to T. Asano and T. Sakurai for the help of ESR experiments\nand useful discussion. This work was partially supported by a Grant -in-Aid for Scientific\nResearch, No.23540392, No.25220605 and No.25287076. The auth ors thank to Dr. Daigou\nMizoguchi (Dai Nippon Toryo Co. Ltd.) for providing them with four d ifferent types of\nAuNRs. 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