We extend the band theory of linear orbital magnetoelectric coupling to treat
crystals under finite electric fields. Previous work established that the
orbital magnetoelectric response of a generic insulator at zero field comprises
three contributions that were denoted as local circulation, itinerant
circulation, and Chern-Simons. We find that the expression for each of them is
modified by the presence of a dc electric field. Remarkably, the sum of the
three correction terms vanishes, so that the total coupling is still given by
the same formula as at zero field. This conclusion is confirmed by numerical
tests on a tight-binding model, for which we calculate the field-induced change
in the linear magnetoelectric coefficient.Comment: 4 pages, 2 figure

Andrei Malashevich

David Vanderbilt

Ivo Souza

T. O’Dell

English

arXiv

Crossref

Physical Review B

Orbital magnetoelectric coupling at finite electric field

We extend the band theory of linear orbital magnetoelectric coupling to treat crystals under finite electric fields. Previous work established that the orbital magnetoelectric response of a generic insulator at zero field comprises three contributions that were denoted as local circulation, itinerant circulation, and Chern-Simons. We find that the expression for each of them is modified by the presence of a dc electric field. Remarkably, the sum of the three correction terms vanishes, so that the total coupling is still given by the same formula as at zero field. This conclusion is confirmed by numerical tests on a tight-binding model, for which we calculate the field-induced change in the linear magnetoelectric coefficient.Peer Reviewe

Malashevich, Andrei

Vanderbilt, David

Souza, Ivo

Digital.CSIC

PHYSICAL REVIEW B 83, 092407 (2011)Orbital magnetoelectric coupling at finite electric fieldAndrei Malashevich*Department of Physics, University of California, Berkeley, California 94720, USADavid VanderbiltDepartment of Physics & Astronomy, Rutgers, the State University of New Jersey, Piscataway, New Jersey 08854, USAIvo SouzaCentro de Fı´sica de Materiales and DIPC, Universidad del Paı´s Vasco, 20018 San Sebastia´n, Spain andIkerbasque, Basque Foundation for Science, E-48011 Bilbao, Spain(Received 24 January 2011; published 28 March 2011)We extend the band theory of linear orbital magnetoelectric coupling to treat crystals under finite electric fields.Previous work established that the orbital magnetoelectric response of a generic insulator at zero field comprisesthree contributions that were denoted as local circulation, itinerant circulation, and Chern-Simons. We find thatthe expression for each of them is modified by the presence of a dc electric field. Remarkably, the sum of thethree correction terms vanishes, so that the total coupling is still given by the same formula as at zero field. Thisconclusion is confirmed by numerical tests on a tight-binding model, for which we calculate the field-inducedchange in the linear magnetoelectric coefficient.DOI: 10.1103/PhysRevB.83.092407 PACS number(s): 75.85.+t, 03.65.Vf, 71.20.PsMagnetoelectrics are magnetic insulators whose dielectricpolarization P changes linearly under a small applied magneticfield B and, conversely, whose magnetization M changeslinearly with a small applied electric field E .1,2 This linearmagnetoelectric (ME) coupling is described by the responsetensor3αij = ∂Mj∂Ei =∂Pi∂Bj, (1)which is odd under both spatial inversion (P) and time-reversal(T ) symmetries. Thus ME materials must be acentric anddisplay magnetic order.In crystals where only one of the two symmetries, P orT , is present, it may still be possible to induce a linearME effect by applying an external field which breaks thatsymmetry. So, for example, a centrosymmetric insulatingantiferromagnet placed in a (strong) electric field loses itsinversion center. Likewise, a nonmagnetic ferroelectric crystalloses time-reversal symmetry when subject to a magnetic field.In both cases the symmetry is sufficiently lowered that thetensor α becomes nonzero.It is useful to view these field-induced effects as higher-order ME responses of the unbiased crystal.4 Two quadraticME effects can be defined in this way. Going to next order inmagnetic field yields the tensorβijk = ∂αij∂Bk= ∂2Pi∂Bj∂Bk, (2)which is odd under P and even under T . Going instead to nextorder in the electric field givesγijk = ∂αji∂Ek =∂2Mi∂Ej ∂Ek , (3)which is even under P and odd under T . Reference 4 lists theform of these tensors for all the crystal classes. While mostinvestigations of ME couplings in solids have focused on thelinear response α for a reference state of the crystal at zeroelectric and magnetic fields, the quadratic responses β andγ have also been measured in materials where α vanishes bysymmetry. In particular the electric-field-induced effect, whichconstitutes the primary focus of this work, was first measuredby O’Dell in yttrium iron garnet.5The ME response can be divided into four contributions,depending on whether the response is frozen ion (purelyelectronic) or lattice mediated, and whether it is spin ororbital in character. We will refer to the frozen-ion part of theorbital response as the orbital magnetoelectric polarizability(OMP).6,7 While the OMP is typically a small contributionto the ME response in conventional magnetoelectrics, it wasrecently realized that, under certain conditions of surfacepreparation, Z2-odd topological insulators8 should displaya large, quantized OMP response.6,9 This is a remarkableprediction, especially considering that in this class of materialsT symmetry is preserved in the bulk (it must, however, bebroken on the surface). This topological magnetoelectric effecthas triggered a great deal of interest in orbital magnetoelectriccouplings in solids.The microscopic theory needed to calculate the OMP at zeroelectric and magnetic fields from first principles was workedout in Refs. 7 and 10. In addition to the so-called Chern-Simonsterm responsible for the topological ME effect,6,9,11 it wasfound that two more (Kubo) terms contribute to the OMP inconventional magnetoelectrics in which T and P symmetriesare broken spontaneously in the bulk.In this work we generalize the band theory of OMP ofperiodic insulators7,10 to finite electric fields. That is, weevaluate the coefficient α at nonzero E ,αij (E) = ∂Mj∂Ei∣∣∣∣B=0. (4)(Henceforth, the condition B = 0 will be implied throughout.It is also understood that from now on α denotes the OMP092407-11098-0121/2011/83(9)/092407(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW B 83, 092407 (2011)part of the entire ME response.) A principal result of our workis the conclusion that the zero-field expression for the totalOMP remains valid at finite electric field, while the above-mentioned Chern-Simons and Kubo terms separately acquirefield-induced contributions. We confirm our formal results bynumerical tests on a tight-binding model.Our derivation of a formula for α(E) proceeds along thelines of Ref. 7. We start from the expression given therein forthe orbital magnetization of a generic band insulator under afinite electrical bias. It comprises three terms,Mj (E) = MLCj (E) + M ICj (E) + MCSj (E), (5)whereMLCj = −η2jpq∫d3k Im 〈˜∂punk|H 0k |˜∂qunk〉, (6)M ICj = −η2jpq∫d3k Im{〈unk|H 0k |umk〉〈˜∂pumk |˜∂qunk〉} ,(7)andMCSj =eη2Ej∫d3k pqr tr[Ap∂qAr − 2i3 ApAqAr]. (8)The common prefactor in these formulas is η = −e/h¯(2π )3(e > 0 is the magnitude of the electron charge), and a sum isimplied over repeated Cartesian (pqr) and valence-band (mn)indices. The cell-periodic part of the field-polarized Blochstate12 is denoted by |unk〉, ∂j is the partial derivative withrespect to the j th component of the wave vector k, and thetilde indicates a covariant derivative ∂˜j = Qk∂j , where Qk =1 − |unk〉〈unk| (sum implied over n). The Hamiltonian H 0k isdefined asH 0k = e−ik·rH0eik·r, (9)where H0 is the zero-field part of the crystal Hamiltonian. InEq. (8) the symbol Ap denotes the Berry connection matrixAmnkp = i〈umk|∂punk〉, (10)and the trace is over the valence bands.Equations (6) and (7) describe respectively the local anditinerant circulation contributions to the magnetiztion,7 whileEq. (8) is the Chern-Simons term. At variance with the othertwo terms, whose dependence on the electric field is onlyimplicit, MCS displays an explicit linear dependence on E . Itis therefore expedient to introduce a new quantity MCS1 via therelationMCSj (E) ≡ EjMCS1 (E), (11)where the subscript “1” serves as a reminder that MCS1 entersthe expression for M multiplied by E to the first power.All three magnetization terms, MLC, MIC, and MCS, areinvariant under gauge transformations within the valence-bandmanifold, although in the case of MCS this invariance is onlymodulo a quantum of indeterminacy.9 In the limit that E goesto zero, MCS vanishes and Eq. (5) reduces to the expressionfor the spontaneous orbital magnetization.13As already mentioned, all terms in Eq. (5) can contribute tothe linear ME coupling, Eq. (4), so thatαij (E) = αLCij (E) + αICij (E) + αCSij (E). (12)The derivation of the expressions for these objects is straight-forward though somewhat lengthy. It essentially repeats thesteps in Appendix B of Ref. 7, where the derivation was carriedout for the LC and IC (“Kubo”) terms under the assumptionthat E = 0 (the CS term is trivial at E = 0). At E = 0 one mayshow that each of the terms in Eq. (12) consists of a “zero-field”part plus a “field-correction” part having an explicit lineardependence on E ,αij (E) = α0,ij (E) + Ejα1,i(E). (13)The field-correction terms for the LC and IC contributionscan be traced back to Eqs. (B7) and (B8) in Ref. 7, whichat E = 0 acquire extra terms. As for the Chern-Simonscontribution, differentiating Eq. (11) with respect to Ej yieldsαCS0,ij = δijMCS1 and αCS1,i = ∂MCS1 /∂Ei .Thus, we arrive at the resultsαLC0,ij (E) = ηjpqIm∫d3k(〈˜∂punk|(∂qH0k)|D˜iunk〉− 12〈˜∂punk|(DiH0k)|˜∂qunk〉), (14)αIC0,ij (E) = ηjpqIm∫d3k(〈˜∂punk|D˜iumk〉〈umk|(∂qH0k)|unk〉− 12〈˜∂punk |˜∂qumk〉〈umk|(DiH0k)|unk〉), (15)αCS0,ij (E) = δij ηe2∫d3k pqr tr[Ap∂qAr − 2i3 ApAqAr],(16)αLC1,i (E) = ηe∫d3k pqrRe[〈D˜iunk |˜∂pumk〉〈˜∂qumk |˜∂runk〉],(17)andαLC1,i (E) = αIC1,i(E) = − 12αCS1,i . (18)In the above expressions, Di is the partial derivative withrespect to the ith component of the electric field. Theterms containing DiH 0k in Eqs. (14) and (15) are screeningcorrections which are present in self-consistent calculations.Equations (14)–(16) for the zero-field terms are essentiallyrewritten from Ref. 7. It should be emphasized, however,that in the present context these expressions depend on theelectric field implicitly via the wave functions. The explicitfield dependence is given by the field-correction terms,Eqs. (17) and (18). Remarkably, these terms are not inde-pendent and add up to zero when inserted into Eq. (12). Weconclude, therefore, that the expression for the total OMPderived in Refs. 7 and 10 assuming E = 0 remains validfor E = 0. This constitutes one of our principal results. Theexplicit expression given in Eq. (17) for the field-correctionterms is the other main result of this work. It is usefulif one is interested in the field dependence of the separate092407-2BRIEF REPORTS PHYSICAL REVIEW B 83, 092407 (2011)gauge-invariant contributions to the OMP. Because it containsthree k derivatives and one field derivative, this quantity iseven under P and odd under T , just like the coefficient γdefined by Eq. (3). This is reasonable since, as one can seefrom Eq. (13), αLC/IC/CS1 gives a contribution to γ LC/IC/CS andshould therefore have the same symmetry properties.As a check of our analytic derivation, we have implementedthe formula for α(E) in a tight-binding model, and used it tocalculate the nonlinear ME coefficient γzzz at E = 0. Since thetensor γ vanishes in T -invariant systems, we need a modelwhere T is spontaneously broken, and we chose that of Ref. 7.This is a spinless model with eight sites per primitive cellarranged on a 2 × 2 × 2 cube, where T symmetry is brokenby complex nearest-neighbor hoppings, and we have used thesame on-site energies and nearest-neighbor hoppings tabulatedin that work. (This choice of parameters also breaks P , so thatthe linear ME tensor α is nonzero already at E = 0, but thisis not essential for our present purposes.) As in Ref. 7 the twolowest bands were treated as occupied, and the phase ϕ of oneof the complex hoppings was chosen as a control parameterfor plotting purposes.The technical details of the tight-binding implementationof Eqs. (6)–(8) and (14)–(17) can be found in Ref. 7. Theonly significant difference with respect to that work is thatthe field derivative |D˜iunk〉 of the cell-periodic Bloch statesmust be evaluated at finite E . Under these circumstances theusual “sum-over-states” formula13 cannot be employed, andone must instead minimize a suitably defined functional.14We shall calculate the zzz component of γ from the firstequality in Eq. (3). Combining with Eq. (12) we findγ = γ LC + γ IC + γ CS. (19)The CS term is the simplest to evaluate, as the derivative ofEq. (13) with respect to Ez can be taken analytically. Thezero-field and field-correction terms therein both contribute anamount αCS1,z(0) to γ CSzzz (0). Thus,γ CSzzz (0) = 2αCS1,z(0) = −4αLC1,z(0), (20)where the second equality follows from Eq. (18). The quantityon the right-hand side can be evaluated directly from Eq. (17).For the LC and IC terms we calculate the derivative of thezero-field terms in Eq. (13) using finite differences and obtainγ LC/ICzzz (0) αLC/IC0,zz (Ez) − αLC/IC0,zz (−Ez)2Ez + αLC1,z(0). (21)In practice we evaluate the first term from Eqs. (14) and (15),using small positive and negative fields along z of magnitudeEz = 1.0 × 10−5 V/m.The results of the above calculations were compared witha finite-difference determination of the second field derivativeof M,γzzz(0) = ∂2Mz∂E2z∣∣∣∣E=0 Mz(Ez) − 2Mz(0) + Mz(−Ez)E2z,(22)using the k-space expressions from Ref. 7 for the LC, IC,and CS terms in Eq. (5). The results obtained in this mannercan be taken as a reference, since the k-space expression forFIG. 1. Decomposition of γzzz of Eq. (19) into γ LC (solid lines),γ IC (dashed lines), and γ CS (dotted lines) calculated using Eqs. (20)and (21). Symbols denote the same contributions evaluated usingEq. (22).M(E) has been carefully tested by comparing with real-spacecalculations on bounded samples cut from the bulk crystal.7The agreement between the two sets of calculations canbe seen in Fig. 1, where the LC, IC, and CS contributions toγzzz are plotted separately as functions of ϕ. In this calculationγ CSzzz is about an order of magnitude smaller than γ LCzzz . FromEqs. (20) and (21) it then follows that the field-correctionterms contribute little, especially in the case of γ LCzzz . Furthernumerical tests focusing on those small terms are thereforedesirable.FIG. 2. (a) Right-hand side (solid line) and left-hand side(symbols) of Eq. (23). Squares and circles denote the LC and ICcontributions, respectively. (b) Equation (20) (solid line) and Eq. (22)for the CS contribution (crosses), both multiplied by a factor of −1/4for visual check of Eq. (18) by comparison to (a).092407-3BRIEF REPORTS PHYSICAL REVIEW B 83, 092407 (2011)In order to isolate the field-correction terms in γ LCzzz and γ ICzzz,we subtract the zero-field terms from the total:[∂2MLC/ICz∂E2z− ∂αLC/IC0,zz∂Ez]E=0= αLC/IC1,z (0). (23)In Fig. 2(a) we plot, as a function of ϕ, the two sides ofthis equation. The field derivatives on the left-hand side areevaluated by finite differences, while the right-hand side iscalculated from Eq. (17). It is clear that the field-correctionterms in Eq. (13) are nonzero, and the good agreement betweenthe three curves demonstrates that for both LC and IC they aregiven by Eq. (17).The CS contribution does not need additional tests since, asnoted above, the contributions to γ CSzzz from the zero-field andfield-correction terms are identical. However, we reproducein Fig. 2(b) the CS curve from Fig. 1 multiplied by a factor−1/4, so that the correctness of Eq. (18) can be verified bydirect visual inspection. This completes the numerical checksof the k-space formula for α(E).To summarize, we have extended the recently developedband theory of orbital magnetoelectric response to treatcrystals under a finite electrical bias. The theory presentedin this work may be especially useful in calculations of thesecond-order magnetoelectric effect defined by Eq. (3). Whileit is possible in principle to calculate the second derivative ofM by finite differences, the numerical stability is likely to beimproved by taking one of the field derivatives analytically,leaving only one derivative to be performed numerically.We have demonstrated that in order to calculate the totalOMP at finite electric field, one may use the same equations(14)–(16) that were previously derived for zero field. This istrue even though the individual local-circulation, itinerant-circulation, and Chern-Simons contributions do separatelyacquire field-correction terms. At present, we are not awareof any simple argument that could have anticipated the exactcancellation of these terms in the expression for the totalOMP.We thank Sinisa Coh for stimulating discussions. The workwas supported by NSF under Grants No. DMR-0706493 andNo. DMR-1005838. Computational resources were providedby NERSC.*andreim@berkeley.edu1T. O’Dell, The Electrodynamics of Magneto-Electric Media (North-Holland Amsterdam, 1970).2M. Fiebig, J. Phys. D 38, R123 (2005).3In this work we use SI units. For a discussion of various conventionsand choices of units used to define a magnetoelectric tensor, see Ref.11 and references therein.4E. Ascher, Philos. Mag. 17, 149 (1968).5T. H. O’Dell, Philos. Mag. 16, 487 (1967).6A. M. Essin, J. E. Moore, and D. Vanderbilt, Phys. Rev. Lett. 102,146805 (2009).7A. Malashevich, I. Souza, S. Coh, and D. Vanderbilt, New J. Phys.12, 053032 (2010).8M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045(2010).9X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424(2008).10A. M. Essin, A. M. Turner, J. E. Moore, and D. Vanderbilt, Phys.Rev. B 81, 205104 (2010).11S. Coh, D. Vanderbilt, A. Malashevich, and I. Souza, Phys. Rev. B83, 085108 (2011).12I. Souza, J. ´In˜iguez, and D. Vanderbilt, Phys. Rev. Lett. 89, 117602(2002).13D. Ceresoli, T. Thonhauser, D. Vanderbilt, and R. Resta, Phys. Rev.B 74, 024408 (2006).14X. Wang and D. Vanderbilt, Phys. Rev. B 75, 115116 (2007).092407-4

Orbital magnetoelectric coupling at finite electric field

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