Superexchange theory of electronic polarization driven by relativistic spin-orbit interaction at the half-filling

TL;DR

This paper derives an analytical expression for electronic polarization driven by relativistic spin-orbit coupling in magnetic materials, using Berry-phase theory and superexchange models, to explain ferroelectric and magnetoelectric effects.

Contribution

It introduces a novel superexchange-based analytical formula for electronic polarization influenced by spin-orbit interaction, linking symmetry and magnetic structure.

Findings

The theory accurately predicts symmetry properties of polarization in various multiferroics.

It successfully explains the linear magnetoelectric effect in Cr₂O₃ and BiFeO₃.

The polarization magnitude is sometimes underestimated, indicating other contributing effects.

Abstract

By applying Berry-phase theory for the effective half-filled Hubbard model, we derive an analytical expression for the electronic polarization driven by the relativistic spin-orbit (SO) coupling. The model itself is constructed in the Wannier basis, using the input from the first-principles electronic structure calculations in the local-density approximation, and then treated in the spirit of the superexchange theory. The obtained polarization has the following form: ${\bf P}_{ij} = \boldsymbol{\epsilon}_{ji} \boldsymbol{\cal P}_{ij} \cdot [\boldsymbol{e}_i \times \boldsymbol{e}_j]$, where $\boldsymbol{\epsilon}_{ji}$ is the direction of the bond $\langle ij \rangle$, $\boldsymbol{e}_i$ and $\boldsymbol{e}_j$ are the directions of spins in this bond, and $\boldsymbol{\cal P}_{ij}$ is the pseudovector containing all the information about the crystallographic symmetry of the considered system. The expression describes the ferroelectric activity in various magnets with noncollinear but otherwise nonpolar magnetic structures, which would yield no polarization without SO interaction, including the magnetoelectric (ME) effect, caused by the ferromagnetic canting of spins in the external magnetic field, and spin-spiral multiferroics. The abilities of this theory are demonstrated for the the analysis of linear ME effect in Cr$_2$O$_3$ and BiFeO$_3$ and properties multiferroic MnWO$_4$ and $\beta$-MnO$_2$. In all considered examples, the theory perfectly describes the symmetry properties of the induced polarization. However, in some cases, the values of this polarization are underestimated, suggesting that other effects, besides the spin and electronic ones, can also play an important role.

Equations

$$P_{ij}^a=\sum _b{\mathfrak{d}}_{ij}^{ab}[{\mathbf{e}}_i\times {\mathbf{e}}_j{]}^b,$$

$${\mathbf{P}}_{ij}\propto {\mathbf{ϵ}}_{ji}\times [{\mathbf{e}}_i\times {\mathbf{e}}_j],$$

$$\mathbf{P}=-\frac{e}{V}\int \mathbf{r}{w}^2(\mathbf{r})d\mathbf{r},$$

$${\mathbf{P}}_{ij}={\mathbf{ϵ}}_{ji}{\mathcal{P}}_{ij}\cdot [{\mathbf{e}}_i\times {\mathbf{e}}_j],$$

$$|\Psi^{-}\rangle=\left(\begin{array}[]{l}-\sin\frac{\theta}{2}e^{-i\phi}\\ \phantom{-}\cos\frac{\theta}{2}\\ \end{array}\right)|\Psi\rangle$$

$$|\Psi^{+}\rangle=\left(\begin{array}[]{l}\cos\frac{\theta}{2}\\ \sin\frac{\theta}{2}e^{i\phi}\\ \end{array}\right)|\Psi\rangle,$$

$$|{\tilde{\Psi }}^{-}⟩=|{\Psi }^{-}⟩-\bar{\xi }|{\Psi }^{+}⟩⟨{\Psi }^{+}|\left(\hat{\mathbf{L}}-[{\hat{\bar{h}}}_{\mathrm{cf}},\hat{\mathbf{L}}]\right)\cdot \hat{\mathbf{S}}|{\Psi }^{-}⟩$$

$$|{\tilde{\Psi }}^{+}⟩=|{\Psi }^{+}⟩+\bar{\xi }|{\Psi }^{-}⟩⟨{\Psi }^{-}|\left(\hat{\mathbf{L}}+[{\hat{\bar{h}}}_{\mathrm{cf}},\hat{\mathbf{L}}]\right)\cdot \hat{\mathbf{S}}|{\Psi }^{+}⟩,$$

$$\delta E_{ij}\simeq -U⟨{\tilde{\Psi }}_i^{-}|{\hat{\bar{t}}}_{ij}+\frac{1}{2}[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ij}]|{\tilde{\Psi }}_j^{+}⟩⟨{\tilde{\Psi }}_j^{+}|{\hat{\bar{t}}}_{ji}-\frac{1}{2}[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ji}]|{\tilde{\Psi }}_i^{-}⟩+(i\leftrightarrow j).$$

$$\delta E_{ij}\simeq J_{ij}\left(1-{\mathbf{e}}_i\cdot {\mathbf{e}}_j\right)+{\mathbf{d}}_{ij}\cdot [{\mathbf{e}}_i\times {\mathbf{e}}_j],$$

$$J_{ij}=-U{\mathrm{Tr}}_L\left\{{\hat{\bar{t}}}_{ij}{\hat{\bar{t}}}_{ji}\right\}$$

$${\mathbf{d}}_{ij}=\xi {\mathrm{Tr}}_L\left\{{\hat{\bar{t}}}_{ij}[[{\hat{\bar{h}}}_{\mathrm{cf}},i\hat{\mathbf{L}}],{\hat{\bar{t}}}_{ji}]\right\}$$

$$w_i^2(\mathbf{r})=\sum _j{w}_{ij}^2\delta (\mathbf{r}-\Delta {\mathbf{\tau }}_{ji}),$$

$$\mathbf{P}=\frac{1}{2}\sum _{ij}{\mathbf{P}}_{ij},$$

$${\mathbf{P}}_{ij}=-\frac{e\Delta {\mathbf{\tau }}_{ji}}{V}\left(w_{ij}-w_{ji}\right).$$

$$w_{ij}\simeq |⟨{\tilde{\Psi }}_j^{+}|{\hat{\bar{t}}}_{ji}-[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ji}]|{\tilde{\Psi }}_i^{-}⟩{|}^2.$$

$${\mathbf{P}}_{ij}=\frac{e\Delta {\mathbf{\tau }}_{ji}}{V}\bar{\xi }{\mathrm{Tr}}_L\left\{[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ij}][[{\hat{\bar{h}}}_{\mathrm{cf}},i\hat{\mathbf{L}}],{\hat{\bar{t}}}_{ji}]+[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ji}][[{\hat{\bar{h}}}_{\mathrm{cf}},i\hat{\mathbf{L}}],{\hat{\bar{t}}}_{ij}]\right\}\cdot [{\mathbf{e}}_i\times {\mathbf{e}}_j],$$

$${\mathcal{P}}_{ij}=\frac{e|\Delta {\mathbf{\tau }}_{ji}|}{V}\bar{\xi }{\mathrm{Tr}}_L\left\{[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ij}][[{\hat{\bar{h}}}_{\mathrm{cf}},i\hat{\mathbf{L}}],{\hat{\bar{t}}}_{ji}]+[{\hat{\bar{h}}}_{\mathrm{cf}},{\hat{\bar{t}}}_{ji}][[{\hat{\bar{h}}}_{\mathrm{cf}},i\hat{\mathbf{L}}],{\hat{\bar{t}}}_{ij}]\right\}.$$

Full text

I Introduction

The relativistic spin-orbit (SO) interactions is responsible for many spectacular phenomena in condensed matter physics, which are widely employed in many technological applications. Particularly, being a natural mechanism connecting spin and orbital degrees of freedom, it provides a unique possibility for the mutual control of various spin and lattice-related properties. Every year, growing interest in this problem leads to the discovery new and more sophisticated schemes of such control SO_review .

One of the interesting topics is the effects of the SO coupling in noncentrosymmetric substances. In magnetic systems, it leads to the famous antisymmetric Dzyaloshinskii-Moriya (DM) interaction $\boldsymbol{d}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ between spins in the noncentrosymmetric bond $\langle ij\rangle$, where $\boldsymbol{e}_{i}$ and $\boldsymbol{e}_{j}$ are the directions of these spins Dzyaloshinskii_weakF ; Moriya_weakF . The DM interaction is generally responsible for the noncollinear spin order. Alternatively, in some magnetic architectures, the noncollinear alignment of spins can break the inversion symmetry, which will be immediately manifested in the ferroelectric (FE) activity. The classical example of such activity is the magnetoelectric (ME) effect, where the noncollinearity is induced by the external magnetic field Dzyaloshinskii_ME . The interest in this problem has reemerged a decade ago, after the discovery of new generation of multiferroic materials, where the inversion symmetry is broken by some complex and, in many cases, noncollinear magnetic order MF_review . Nevertheless, the microscopic understanding of mechanisms resulting in finite electric polarization in this case is still far from being complete, even despite of significant progress in this direction.

Historically, the first phenomenological expression for the electric polarization, which can be induced by a noncollinear spin order, was introduced by Moriya in 1968 on the basis of general symmetry considerations Moriya_pol . In each magnetic bond, such polarization has the form:

[TABLE]

which is similar to the expression for DM exchange interaction, where the vector $\boldsymbol{d}_{ij}$ is replaced by the tensor $\mathfrak{d}_{ij}^{ab}$ with $a$ and $b$ denoting $x$, $y$, or $z$.

The microscopic derivation of expression for the electric polarization, which is driven by the relativistic SO coupling in noncollinear magnetic substances, was given in Ref. KNB . However, it should be understood that the microscopic model considered Ref. KNB deals with very special example of electronic structure of the transition-metal (TM) oxides, consisting of the $t_{2g}$ levels with some particular scheme of filling, which are split by the SO coupling and interact via intermediate oxygen (O) states in the single undistorted TM-O-TM bond. Thus, the analysis is hardly to be complete. Nevertheless, on the basis of these considerations, the authors of Ref. KNB have concluded that the electric polarization should behaves as

[TABLE]

where $\boldsymbol{\epsilon}_{ji}$ is the unit vector in the direction of TM site $j$ relative to the TM site $i$. It is referred to as the spin-current mechanism of the electric polarization, which is widely used today for the analysis of experimental data MF_review . Similar conclusion was drawn in Ref. Mostovoy , being based on the phenomenological Ginzburg-Landau theory. The expression (2) does not depend on the specific crystallographic symmetry of the considered system, so that one can have a wrong impression that the electric polarization in all noncollinear magnets should behave in a similar way. Nevertheless, this expression is formally consistent with the general definition (1), given by Moriya, and can be reduced to it by introducing the tensor $\mathfrak{d}_{ij}^{ab}=-\varepsilon_{abc}\epsilon_{ji}^{c}$, where $\varepsilon_{abc}$ is the antisymmetric symbol of Levi-Civita. It is often claimed that the microscopic mechanism responsible for such behavior is similar to the inverse DM mechanism, proposed in Ref. SergienkoPRB : similar to what how the off-centrosymmetric oxygen displacement in the bond TM-O-TM gives rise to the noncollinear alignment of spins Dzyaloshinskii_weakF ; Moriya_weakF , one can expect the opposite (magnetostrictive-like) effect, where the noncollinear magnetic alignment should lead to the off-centrosymmetric atomic displacement responsible for the additional magnetic energy gain: $\sum_{ij}\boldsymbol{d}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$. However, it should be understood that these mechanisms are quite different (though complementary to each other): Ref. KNB deals with the purely electronic effect, while Ref. SergienkoPRB deals with the lattice effect.

The most rigorous theoretical basis for the analysis of electronic polarization is provided by the Berry-phase theory, which relates the polarization with the expectation value of the position operator in the state specified by localized Wannier functions KSV ; Resta ; W_review :

[TABLE]

where -$$e<0 is the electron charge, $V$ is the unit-cell volume, and $w^{2}({\bf r})=\sum_{n=1}^{M}|W_{n}({\bf r})|^{2}$ is the total weight of Wannier functions for the $M$ occupied states. Each Wannier function is centered near certain site of the lattice and can have tails spreading to the neighboring sites. The relative weight of these tails depends on the magnetic state. This is how the Wannier function bears the information about the magnetic configuration at the neighboring sites. Thus, the understanding of magnetic-state dependence of the electronic polarization is essentially the understanding of how the magnetic order and relativistic SO interaction leads to the asymmetric deformation of the Wannier functions around each magnetic site Barone ; DEpol ; Yamauchi_rew . It should not be confused with the asymmetric distribution of the electron density at each magnetic site, because the electron density is a superposition of the weights of the Wannier functions centered at this and neighboring sites, which can lead to the incorrect answer KSV ; Resta .

In our previous work DEpol we have applied this strategy to the analysis of electronic polarization caused by the nonrelativistic double exchange mechanism. In that case, competing magnetic interactions of both relativistic and nonrelativistic origin result in highly asymmetric magnetic structure, which breaks the inversion symmetry. The SO interaction plays an important role in this asymmetry: for instance, it is responsible for the single-ion anisotropy, which deforms the homogeneous spin-spiral texture in multiferroic manganites DEpol ; PRB11 (the so-called bunching effect REbunching ). This deformation gives rise to the polarization ${\bf P}_{ij}\propto(\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j})$, which depends on the SO coupling only indirectly, via the noncentrosymmetric distribution of the directions of spins, while the proportionality coefficient between ${\bf P}_{ij}$ and $(\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j})$ does not depend on the SO coupling. This double exchange mechanism has allowed us to rationalize many aspects of the behavior of electric polarization in multiferroic manganites DEpol .

In this article we consider the proper spin-current mechanism. In some sense, the situations is the opposite to the double exchange mechanism, considered in Ref. DEpol . Namely, we will deal with some noncollinear magnetic structures, which are stabilized by nonrelativistic means: it can be either the spin-spiral structure arising from the competition of several isotropic exchange interactions or a canted spin structure, inherent to the ME effect, where the the collinear antiferromagnetic (AFM) order is deformed by the external magnetic field. Without SO coupling all these magnetic structures can be transformed to themselves by combining the spacial inversion with some appropriate rotation of the spin system as the whole PRB13 . Therefore, the electric polarization will vanish. Nevertheless, the situation may change after switching on the SO coupling, which does not deform the spin texture itself (or, at least, such deformation can be neglected), but can deform the Wannier functions, resulting in their asymmetry and finite electronic polarization.

Our analysis will be applied to the effective Hubbard model derived from the first-principles electronic structure calculations and using the local-density approximation (LDA) as the starting point for such derivation review2008 . We consider the simplest case of the half-filling, which also allows us to get rid of additional complications related to the orbital degrees of freedom. Furthermore, the on-site Coulomb repulsion is the largest parameter in our model, so that other parameters can be treated as a perturbation in the spirit of the superexchange (SE) theory PWA . We will use this strategy in order to derive an analytical expression for the DM exchange interactions and electronic polarization. We will show that the correct expression for electronic polarization, which driven by the spin-current mechanism in the framework of the Berry-phase theory KSV ; Resta , has the following form:

[TABLE]

where the pseudovector $\boldsymbol{\cal P}_{ij}$ contains all the information about the individual symmetry of the lattice. Thus, there is at least one important addition to the phenomenological expression (2): the polarization does depend on the symmetry of the lattice, as it should be. Moreover, the functional dependence is different and there is no direct coupling between $\boldsymbol{\epsilon}_{ji}$ and $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$. Furthermore, by defining $\mathfrak{d}_{ij}^{ab}=\epsilon_{ji}^{a}{\cal P}_{ij}^{b}$, it is also straightforward to see the form of this expression is consistent with Eq. (1), proposed by Moriya Moriya_pol . We will show that this expression is very general and describes not only the behavior of polarization in the spin-spiral magnets, but also the ME effect, caused by the ferromagnetic (FM) canting of spins in otherwise collinear AFM states of a special symmetry, which is not captured by phenomenological Eq. (2).

Another important issue is whether the spin-current mechanism alone is able to reproduce experimental values of the ME effect and electric polarization in real materials. Additional mechanisms, which are widely discussed in the literature, are the lattice deformation Malashevich2008 ; Bousquet ; Malashevich (in line with the proposal SergienkoPRB ), orbital contribution to the ME coupling Malashevich ; Scaramucci , and hidden deformation of the magnetic texture with broken inversion symmetry PRB11 ; PRB13 . By using realistic model, derived from the first-principles calculations, we will show that the situation can be very different: In some cases, the spin-current mechanism alone reproduces the experimental polarization reasonably well. In other cases (e.g., in Cr2O3), it captures only the symmetry properties of the polarization, while the numerical values can be off by several order of magnitude, suggesting the importance of other mechanisms Bousquet ; Malashevich ; Scaramucci .

The rest of the article is organized as follows. In Sec. II we will present our formalism based on the SE theory, which is applied to antisymmetric DM exchange interactions and electric polarization in Secs. II.1 and II.2, respectively. The details of these derivations are given in the Supplemental Material SM . In Sec. III we will consider practical applications of this formalism to the linear ME effect in Cr2O3 and BiFeO3 (Secs. III.1 and III.2, respectively) and FE activity caused by the spin-spiral order in multiferroic MnWO4 and $\beta$-MnO2 (Sec. III.3 and III.4, respectively). Finally, in Sec. IV, we will summarize our work.

II Formalism

In this section we will sketch the main details of derivation of analytical expressions for the DM exchange interactions and electric polarization, following the SE theory in the lowest order of perturbation with respect to the transfer integrals $\hat{t}_{ij}$ PWA . The technical details can be found in the Supplemental Material SM . The simplest microscopic model, capturing the physics of the spin-current mechanism, reads $\hat{H}=\hat{h}+\hat{t}$, where $\hat{h}\equiv\hat{h}_{\rm ex}+\hat{h}_{\rm cf}+\hat{h}_{\rm so}$ is the on-site part, including the interaction $\hat{h}_{\rm ex}=\frac{U}{2}\boldsymbol{e}\cdot\hat{\boldsymbol{\sigma}}$ with the internal exchange field in the direction $\boldsymbol{e}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ ($\hat{\boldsymbol{\sigma}}$ being the vector of Pauli matrices), the crystal-field splitting $\hat{h}_{\rm cf}$, and the SO interaction $\hat{h}_{\rm so}=\frac{\xi}{2}\hat{\bf L}\cdot\hat{\boldsymbol{\sigma}}$, while $\hat{t}\equiv[\hat{t}_{ij}]$ is the inter-site part. More specifically, $\hat{H}$ can be viewed as a mean-field Hamiltonian (for instance, the one obtained from the solution of the Hubbard model in the Hartree-Fock approximation), where $\hat{h}_{\rm ex}$ describes the averaged exchange splitting for the half-filled ionic shell, driven by the effective interaction $U$, and the crystal field $\hat{h}_{\rm cf}$ also includes the effects of nonsphericity of the Coulomb and exchange potential. The form of $\hat{h}_{\rm ex}$ implies that spins are decoupled from the orbital degrees of freedom, which do not adjust the reorientation of spins. Thus, what we consider here is the canonical “spin-current” model, which include direct contributions of the orbital magnetization in neither DM interactions nor electric polarization. Parameters of such microscopic model, formulated in the Wannier basis W_review , can be derived from the first-principles electronic structure calculations review2008 . For practical purposes we use the linear muffin-tin orbital (LMTO) method LMTO .

The basic idea of the SE theory is to start from the atomic limit and treat $\hat{h}_{\rm so}$ and $\hat{t}$ as a perturbation. Then, the wavefunctions of $\hat{H}_{0}=\hat{h}_{\rm ex}+\hat{h}_{\rm cf}$ for the occupied the occupied ($-$) and unoccupied ($+$) spin states are given by

[TABLE]

and

[TABLE]

respectively, where $|\Psi\rangle$ is the column of eigenvectors of $\hat{h}_{\rm cf}$ with the eigenvalues $\{\varepsilon_{n}\}$. More specifically, $|\Psi\rangle$ is the $M$-dimensional vector in the subspace of orbital states, while $|\Psi^{\pm}\rangle$ are $2M$-dimensional vectors in the space of spin and orbital states. Then, corresponding eigenvectors in the first order of the SO interaction will be given by

[TABLE]

and

[TABLE]

where $\bar{\xi}=\xi/U$, $\hat{\bar{h}}_{\rm cf}=\hat{h}_{\rm cf}/U$, and $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$. Moreover, in the conventional perturbation theory expression, we further expand $(\varepsilon_{n}-\varepsilon_{m}\pm U)^{-1}$ with respect to $\hat{\bar{h}}_{\rm cf}$. Then, the first terms in $(\dots)$ correspond to $\hat{h}_{\rm cf}=const$ (the constant energy shift), while the second terms appear in the first order of $\hat{\bar{h}}_{\rm cf}$. In practical calculations, we use the effective $\xi$, which also incorporates the change of the Coulomb and exchange potential in the first order of the SO interaction, as obtained in the self-consistent linear response (SCLR) theory SCLR .

II.1 Exchange Interactions

The exchange interactions in the bond $\langle ij\rangle$ describe the energy change $\delta E_{ij}$ in the second order of $\hat{\bar{t}}_{ij}=\hat{t}_{ij}/U$, where the transfer integrals connect the occupied and unoccupied states of the sites $i$ and $j$:

[TABLE]

This expression is also valid in the first order of $\hat{\bar{h}}_{\rm cf}$. Then, after tedious but rather straightforward algebra, it can be rearranged as (see Ref. SM for details)

[TABLE]

where

[TABLE]

is the isotropic exchange coupling, which does not depend on the SO intercation, and

[TABLE]

is the DM coupling, which appears in the first order of the SO interaction. Other exchange interactions, including the symmetric anisotropic one, appear only in higher orders of the SO coupling. ${\rm Tr}_{L}$ in Eqs. (6) and (7) denotes the trace over $M$ orbital indices.

Finally, we note the following properties:

(i) $\boldsymbol{d}_{ij}$ is the antisymmetric pseudovector: $\hat{I}\boldsymbol{d}_{ij}=\boldsymbol{d}_{ij}$, and $\boldsymbol{d}_{ji}=-\boldsymbol{d}_{ij}$;

(ii) The values of the DM interactions depend on the crystal-field splitting and vanish when $\hat{\bar{h}}_{\rm cf}=const$. Then, Eq. (7) can be interpreted in the following way: since $[\hat{\bar{h}}_{\rm cf},i\hat{\bf L}]$ is the measure of unquenched orbital magnetization (or the observable orbital magnetization in the presence of the crystal field), the DM interactions $\boldsymbol{d}_{ij}$ is a probe of the orbital magnetization at the site $j$ by the electron hoppings from the site $i$ (and vice versa).

II.2 Electronic Polarization

We start with the general expression for the electric polarization (3) in terms of the Wannier functions for the occupied states. Moreover, we adopt it for the lattice model and assume that all weights of $w$ are localized in the lattice points: i.e., if $w_{i}$ are occupied Wannier functions centered at the site $i$, their weights are distributed as

[TABLE]

where $\Delta\boldsymbol{\tau}_{ji}=\boldsymbol{R}_{j}-\boldsymbol{R}_{i}$ is the position of the site $j$ relative to the site $i$. Then, the electronic polarization (3) can be related to the asymmetric transfer of the weights of the Wannier functions in each bond DEpol :

[TABLE]

where

[TABLE]

In the SE theory, the quantities $w_{ij}$ are evaluated in the first order of perturbation theory for the Wannier functions with respect to $\hat{\bar{t}}_{ij}$, starting from the atomic limit:

[TABLE]

Then, using tedious but rather straightforward algebra, one can obtain the following expression for ${\bf P}_{ij}$ (see Ref. SM for details):

[TABLE]

which can be further rearranged as Eq. (4) whith $\boldsymbol{\epsilon}_{ji}=\frac{\Delta\boldsymbol{\tau}_{ji}}{|\Delta\boldsymbol{\tau}_{ji}|}$ and

[TABLE]

Thus, we note the following:

(i) Unlike $\boldsymbol{d}_{ij}$, $\boldsymbol{\cal P}_{ij}$ is the symmetric pseudovector: $\hat{I}\boldsymbol{\cal P}_{ij}=\boldsymbol{\cal P}_{ij}$, while $\boldsymbol{\cal P}_{ji}=\boldsymbol{\cal P}_{ij}$, where the latter property comes from the definition of ${\bf P}_{ij}$ [Eq. (9)];

(ii) Similar to the DM interactions, the electronic polarization crucially depends on $\hat{\bar{h}}_{\rm cf}$ and vanishes when $\hat{\bar{h}}_{\rm cf}=const$;

(iii) There is a fundamental difference from phenomenological expression (2) KNB ; Mostovoy . Namely, the spin-dependent cross product $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ does not couple directly to $\boldsymbol{\epsilon}_{ji}$. It couples to the pseudovector $\boldsymbol{\cal P}_{ji}$, which contains all the information about particular crystallographic symmetry of the considered system. The directional dependence of ${\bf P}$ is specified by the vectors $\boldsymbol{\epsilon}_{ji}$, which are modulated by the scalar products $\boldsymbol{\cal P}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$. This important addition will allow us to resolve several controversies related to the symmetry properties of the electric polarization induced by the noncollinear magnetic alignment.

III Results and Discussions

III.1 Perpendicular Magnetoelectric Effect in Cr2O3

We start our discussion with the canonical example of ME effect in antiferromagnetic Cr2O3 Dzyaloshinskii_ME , which crystallizes in the corundum structure (the space group is $R\bar{3}c$) Cr2O3_structure . The formal configuration of the Cr-ions in Cr2O3 is $3d^{3}$. According to the electronic structure calculations in LDA, Cr2O3 has well isolated Cr $t_{2g}$ band near the Fermi level, which accommodates 3 electrons per one Cr site (Fig. 1).

Thus, as a first approximation, we consider the simplest $t_{2g}$ model at the half filling and try to apply this model for the analysis of the ME effect in Cr2O3. The model itself has been constructed in the basis of Wannier functions, and the parameters of this model have been derived as described in Ref. review2008 . The obtained transfer integrals and the crystal-field splitting perfectly reproduce Cr $t_{2g}$ band structure in LDA. The matrix of screened Coulomb interactions, evaluated in the framework of constrained random-phase approximation (RPA) Ferdi04 , can be approximated in terms of two Kanamori parameters Kanamori : the intraorbital Coulomb repulsion ${\cal U}=3.15$ eV and the exchange interaction ${\cal J}=0.67$ eV. Then, the effective interaction responsible for the intraatomic exchange splitting between the minority- and majority-spin states can be evaluated $U={\cal U}+2{\cal J}$. The crystal-field splitting of atomic $t_{2g}$ levels is about $100$ meV PRB06 . Other parameters can be found elsewhere footnote2 . As we will see below, the model has serious limitations for the quantitative description of the ME effect in Cr2O3. Nevertheless, we consider it for the explanatory purposes.

The corundum structure of Cr2O3 has four interconnected Cr sublattices, which can be antiferromagnetically arranged as A1, A2, and A3 (see Fig. 2).

Among them, the magnetic space group of A1 contains the spacial inversion $\hat{I}$ as it is, while in A2 and A3 $\hat{I}$ is combined with the time reversal $\hat{T}$. Thus, the A1 structure allows for the weak ferromagnetism Dzyaloshinskii_weakF , while A2 and A3 are expected to exhibit the perpendicular ME effect, when the AFM structure is deformed by the external magnetic field Dzyaloshinskii_ME as explained in Fig. 2. The magnetic ground state of Cr2O3 is A3, which was also confirmed by our calculations. The directions of magnetic moments are parallel to $\boldsymbol{c}=\boldsymbol{z}$.

Eqs. (8) and (10) allow us to rationalize the behavior of electronic polarization by separating the contributions of atomic pairs around each Cr site. Around site 1, the largest contributions to ${\bf P}$ comes from the atomic pairs in three coordinations spheres, formed by the atoms 2, 3, and 4, which are displayed in Fig. 3, and where the notations of atomic types is the same as in Fig. 2.

First, we note that the FM bond will not contribute to the perpendicular ME effect: even in the external field $\boldsymbol{H}$ such spins remain ferromagnetically aligned and, therefore, the cross product $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ will vanish. Moreover, in the case of perpendicular ME effect, the cross products $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ will be the same in all equivalent bonds.

Another important aspect is the symmetry. In order to estimate the ME coupling constant, we first evaluate the parameters $\boldsymbol{\cal P}_{ij}$, which obey the symmetry properties of the $R\bar{3}c$ group and contains all the information about the individual symmetry of the Cr2O3 lattice. For instance, the pseudovectors $\boldsymbol{\cal P}_{ij}$ in the nearest-neighbor (NN) bond $\langle 13\rangle$ (and equivalent to it bonds), parallel to the $\boldsymbol{c}$ axis, will vanish due to the joint effect of the threefold rotation and the glade reflection, which transform this bond to itself. Then, for the bonds $\langle 12_{\rm I}\rangle$, $\langle 13_{\rm I}\rangle$, and $\langle 14_{\rm I}\rangle$ (see Fig. 3) the calculated parameters $\boldsymbol{\cal P}_{ij}$ are $(-0.066,0.007,0)$, $(0,0.009,0)$, and $(-0.002,0.010,0)$, respectively (in $\mu{\rm C}/{\rm m}^{2}$). The parameters for other bonds can be obtained from $\boldsymbol{\cal P}_{12_{\rm I}}$, $\boldsymbol{\cal P}_{13_{\rm I}}$, and $\boldsymbol{\cal P}_{14_{\rm I}}$ using the symmetry operations of the $R\bar{3}c$ group. Moreover, due to the threefold rotation about $\boldsymbol{c}$, we will have the following property: $\sum_{j}\boldsymbol{\cal P}_{ij}=0$. Nevertheless, the combination $\sum_{j}\boldsymbol{\epsilon}_{ji}\boldsymbol{\cal P}_{ij}$, which specifies the value of the electronic polarization (4), can be finite, which will lead to the finite ME effect. All these properties do not depend on the type of the AFM order and will hold for A1, A2, and A3.

Then, we consider the behavior of polarization vectors $\boldsymbol{p}_{i}=\sum_{j}\partial\boldsymbol{P}_{ij}/\partial e^{x}$, induced by the FM canting of spins along $\boldsymbol{x}$ and associated with each magnetic site for different types of the AFM order (see Fig. 2). In the A1 phase, due to the FM alignment of spins in the bonds connecting the atomic types 1 and 2 (3 and 4), these bonds will not contribute to $\boldsymbol{p}_{i}$. Thus, one have to consider all possible connections of the sites 1 and 2 with the sites 3 and 4. Moreover, since the sites 1 and 2 (3 and 4) are transformed to each other by $\hat{I}$ without flipping the directions of spins, we will have: $\hat{I}\boldsymbol{\epsilon}_{ji}\boldsymbol{\cal P}_{ij}=-\boldsymbol{\epsilon}_{j^{\prime}i^{\prime}}\boldsymbol{\cal P}_{i^{\prime}j^{\prime}}$ but $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]=[\boldsymbol{e}_{i^{\prime}}\times\boldsymbol{e}_{j^{\prime}}]$, and, therefore, $\boldsymbol{p}_{2}=-\boldsymbol{p}_{1}$ ($\boldsymbol{p}_{4}=-\boldsymbol{p}_{3}$), where $i^{\prime}$ ($j^{\prime}$) is the inversion image of $i$ ($j$). Thus, as expected Dzyaloshinskii_ME ; Dzyaloshinskii_weakF , the FM canting of spins in the phase A1 will lead to the antiferroelectric behavior with no net polarization. Our analysis provides a transparent microscopic explanation of this effect.

In the phases A2 and A3, however, the spins 1 and 2 (3 and 4) are coupled antiferromagnetically. Therefore, these bonds will contribute to $\boldsymbol{p}_{i}$. Moreover, in addition to $\hat{I}\boldsymbol{\epsilon}_{ji}\boldsymbol{\cal P}_{ij}=-\boldsymbol{\epsilon}_{j^{\prime}i^{\prime}}\boldsymbol{\cal P}_{i^{\prime}j^{\prime}}$, in this case we will have: $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]=-[\boldsymbol{e}_{i^{\prime}}\times\boldsymbol{e}_{j^{\prime}}]$ (due to the $\hat{I}\hat{T}$ symmetry of the phases A2 and A3, the inversion will also flip the directions of spins) and, therefore, $\boldsymbol{p}_{2}=\boldsymbol{p}_{1}$ ($\boldsymbol{p}_{4}=\boldsymbol{p}_{3}$). This is a microscopic explanation of the ME effect, which is expected in the phases A2 and A3.

The direction of polarization, however, requires additional symmetry considerations, and this is the point where the external field $\boldsymbol{H}$ comes into play. For instance, if (without field) all $\boldsymbol{e}_{i}$ are parallel to $\boldsymbol{z}$ and the field $\boldsymbol{H}$ is parallel to $\boldsymbol{x}$ (see Fig. 2), $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ will be parallel to $\boldsymbol{y}$ and, according to Eq. (4), we have to consider the behavior of ${\cal P}_{ij}^{y}$ and $\boldsymbol{\epsilon}_{ji}$ under the glade reflection $\{\hat{m}_{y}|\boldsymbol{c}/2\}$ ($\hat{m}_{y}$ being the mirror reflection y\rightarrow-$$y), connecting the sites 1 and 4 (2 and 3). In the A2 phase, this transformation is combined with $\hat{T}$ and, therefore, will additionally flip the direction of spins. Then, it is straightforward to show (similar to the above considerations for $\hat{I}$) that $\{\hat{m}_{y}|\boldsymbol{c}/2\}$ leads to the the additional symmetry properties: $p_{1}^{x}=-p_{4}^{x}$ and $p_{1}^{y}=p_{4}^{y}$ ($p_{1}^{x}=p_{4}^{x}$ and $p_{1}^{y}=-p_{4}^{y}$) for A2 (A3). This explains why ${\bf P}$ in A2 and A3 will be parallel to, respectively, $\boldsymbol{y}$ and $\boldsymbol{x}$.

These behavior cannot be properly described by the phenomenological expression (2) KNB ; Mostovoy : in the case of ME effect, the cross product $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ is the same for all equivalent bonds. Then, the bonds $\langle 14_{\rm I}\rangle$ - $\langle 14_{\rm VI}\rangle$ will not contribute to ${\bf P}$ because $\sum_{j}\boldsymbol{\epsilon}_{ji}=0$ (see Fig. 3). For other types of bonds $\sum_{j}\boldsymbol{\epsilon}_{ji}$ is finite and parallel to $\boldsymbol{z}$. Therefore, according to Eq. (2), for $\boldsymbol{H}||\boldsymbol{x}$ the induced polarization should be always parallel to $\boldsymbol{x}$. This could explain the direction of the polarization in the A3 phase, but not in the A2 one.

Finally, we evaluate the matrix element of the ME tensor $\alpha_{\perp}$ using the numerical value of $p_{x}=0.08$ $\mu{\rm C}/{\rm m}^{2}$ for the A3 phase and the chain rule: $\alpha_{\perp}\equiv\frac{\partial P^{x}}{\partial H^{x}}=\frac{\partial P^{x}}{\partial e^{x}}\frac{\partial e^{x}}{\partial H^{x}}$, where $\frac{\partial e^{x}}{\partial H^{x}}$ is estimated using parameters of the Heisenberg model $E_{\rm H}=-\sum_{i>j}J_{ij}\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}$, obtained in the theory of infinitesimal spin rotations review2008 ; JHeisenberg , as $\frac{\partial e^{x}}{\partial H^{x}}=-\frac{M}{J_{0}}$ ($M\approx 3$ $\mu_{\rm B}$ being the spin magnetic moment and $J_{0}=\sum_{j}J_{ij}\approx-116$ meV) footnote1 . It should be noted that for half-filled Mott insulators, the orbitals degrees of freedom and inactive and parameters of exchange interactions obtained in the SE theory, Eq. (6), are typically well consistent with the ones obtained in the more general theory of infinitesimal spin rotations JHeisenberg . This justifies the perturbative treatment of the transfer integrals and the crystal-field splitting in the framework of the SE theory. However, the obtained value of $\alpha_{\perp}\sim 2\times 10^{-4}$ ${\rm ps}/{\rm m}$ is very small, which simply means that the considered spin-current effect is not the main mechanism of the ME coupling in Cr2O3. This is in line with modern understanding of the ME effect in Cr2O3, which is known include other important contributions beyond the considered model. Particularly, the lattice effect is very important Bousquet ; Malashevich . Moreover, the orbital magnetization, which is neglected in the considered half-filled $t_{2g}$ model, can also contribute to $\alpha_{\perp}$ Malashevich ; Scaramucci . We expect that much better agreement with experimental data can be obtained by considering a more general model, describing the behavior of all Cr $3d$ bands in the basis of Cr $t_{2g}$ and $e_{g}$ Wannier orbitals (see Fig. 1). For instance, we have found that the DM interactions are also strongly underestimated in the $t_{2g}$ only model in comparison with the five-orbital model, where $\boldsymbol{d}_{ij}$ can be computed using Green’s function perturbation theory SCLR . Apparently, the half-filled $t_{2g}$ model is a crude approximation both for DM exchange interactions and electric polarization in Cr2O3, and a more relevant microscopic model should include explicitly the effect of the $e_{g}$ band. For Cr2O3, it implies the consideration of several new contributions to the electronic polarization, which are no longer described by Eq. (10) at the half-filling. Below, we will consider several example of $3d^{5}$ compounds which are described by a more general model, which explicitly includes both $t_{2g}$ and $e_{g}$ states at the half filling, and argue that such model generally provides much better description for the electronic polarization.

III.2 Linear Magnetoelectric Effect in BiFeO3

BiFeO3 is the well known type-I multiferroic, where the FE activity is manly related to the off-centrosymmetric atomic displacements of Bi and O, while the magnetism originates from the half-filled $3d$ shell of Fe. The good aspect of BiFeO3 is that the FE and AFM transition temperatures are high (1100 K and 650 K, respectively), which makes it promising for practical applications MF_review . In the bulk, due to DM exchange interactions operating in the noncentrosymmetric $R3c$ structure, BiFeO3 forms an incommensurate long-periodic spin spiral texture. The DM interactions overcome the effect magnetocrystalline anisotropy favoring the conventional G-type antiferromagnetism Sosnowska ; BiFeO3Jeong ; BiFeO3Matsuda . Nevertheless, the latter can be stabilized in the thin films of BiFeO3, where the magnetocrystalline anisotropy can be substantially increased. An interesting aspect of the G-phase is that it allows for the linear ME effect, where the electric polarization rises linearly with the applied magnetic field, whereas in the spin-spiral phase, this effect is averaged to zero by the spin-spiral modulation. The linear ME coupling $\alpha$ in the BiFeO3 films was first studied experimentally in Ref. BiFeO3_Rivera . However, the obtained value of $\alpha$ was rather moderate (of the order of $0.51$ ${\rm ps}/{\rm m}$). The interest in this problem has reemerged again after report of giant ME coupling, being of the order of 3 ${\rm V}/({\rm cm~{}Oe})$ BiFeO3_Wang . Even higher value of 24 ${\rm V}/({\rm cm~{}Oe})$ (corresponding to $3\times 10^{4}$ ${\rm ps}/{\rm m}$ BiFeO3_Stevenson ) was reported later in the composite films including BiFeO3 BiFeO3_Lorenz .

In this section we will investigate the linear ME effect in BiFeO3. The computational details and parameters of the model, constructed in the basis of five Fe$3d$ states near the Fermi level, can be found in the previous publication SCLR .

The behavior of $\boldsymbol{\cal P}_{ij}$ can be understood on the cluster, where the central Fe site is surrounded by its six nearest neighbors (Fig. 4).

In fact, other bonds also produce a sizable contribution to the ME effect in BiFeO3. However, as expected, the main contribution comes from the nearest neighbors. Moreover, the bonds between Fe sites of the sate type (either 1 or 2) are ferromagnetically coupled and, therefore, do not contribute to the ME effect (see discussions in Sec. III.1). For the bond $\langle 12_{\rm I}\rangle$, corresponding to $\Delta\boldsymbol{\tau}_{2_{\rm I}1}=(-a,0,\frac{c}{2})$ (where $a=3.222$ Å and $c=4.625$ Å are the rhombohedral lattice parameters), we obtain $\boldsymbol{\cal P}_{12_{\rm I}}=(7.88,-3.08,-1.68)$ $\mu{\rm C}/{\rm m}^{2}$. The parameters for other bonds can be obtained using the symmetry operations of the group $R3c$, similar to the DM exchange interactions, which were considered in details in Ref. SCLR . These parameters $\boldsymbol{\cal P}_{ij}$ are more than two orders of magnitude larger than the ones obtained in the $t_{2g}$ model for Cr2O3. Again, due to the threefold rotational symmetry, it holds $\sum_{j}\boldsymbol{\cal P}_{ij}=0$. However, when $\boldsymbol{\cal P}_{ij}$ is combined with $\boldsymbol{\epsilon}_{ji}$ in Eq. (4), one can expect finite $\boldsymbol{p}_{i}$.

In our analysis of the ME effect, we assume that the magnetocrystalline anisotropy confines the spins in the $\boldsymbol{xy}$ plane. To be specific, we consider here only the case of $\boldsymbol{L}||\boldsymbol{y}$, where $\boldsymbol{L}=\frac{M}{2}(\boldsymbol{e}_{1}-\boldsymbol{e}_{2})$ is the AFM order parameter (Fig. 5), but, due to the $R3c$ symmetry, similar analysis holds also for $\boldsymbol{L}||\boldsymbol{x}$.

Then, we consider the effect of the magnetic field, which cants the spins in the direction of either $\boldsymbol{z}$ or $\boldsymbol{x}$.

In the first case ($\boldsymbol{H}||\boldsymbol{z}$), the active component of $\boldsymbol{\cal P}_{ij}$, which is selected by $[\boldsymbol{e}_{1}\times\boldsymbol{e}_{2}]$, is ${\cal P}_{ij}^{x}$. Then, by combing it with $\boldsymbol{\epsilon}_{ji}$, using the symmetry operation of the $R3c$ group, and noting that $\frac{\partial}{\partial e^{z}}[\boldsymbol{e}_{1}\times\boldsymbol{e}_{2}]=2$, it is straightforward to show that $p_{1}^{x}\approx-16a{\cal P}_{12_{\rm I}}^{y}/\sqrt{4a^{2}+c^{2}}$, while $p_{1}^{y}=p_{1}^{z}=0$. This NN contribution accounts for 65% of total $p_{1}^{x}=-78.2$ $\mu{\rm C}/{\rm m}^{2}$, obtained after summation over all bonds. In the BiFeO3 structure, the Fe sites 1 and 2 are transformed to each other by the symmetry operation $\{\hat{m}_{y}|(0,0,\frac{3c}{2})\}$, which keeps the sign of $\epsilon^{x}$, but changes the one of ${\cal P}^{x}$. Moreover, in the AFM phase, this transformation flips the directions of spins. Altogether it leads to the property $p_{2}^{x}=p_{1}^{x}$ and net electric polarization in the magnetic field.

In the second case ($\boldsymbol{H}||\boldsymbol{x}$), the active component is ${\cal P}_{ij}^{z}$, which leads to the properties: $p_{1}^{x}=p_{1}^{y}=0$ and $p_{1}^{z}\approx-12c{\cal P}_{12_{\rm I}}^{z}/\sqrt{4a^{2}+c^{2}}$ (note that in this case $\frac{\partial}{\partial e^{x}}[\boldsymbol{e}_{1}\times\boldsymbol{e}_{2}]=-2$). This NN contribution accounts for 60% of total $p_{1}^{z}=19.7$ $\mu{\rm C}/{\rm m}^{2}$. Similar to $\boldsymbol{H}||z$, it is straightforward to show that $p_{2}^{z}=p_{1}^{z}$, also resulting in finite ME effect.

Thus, the induced electronic polarization satisfies the condition ${\bf P}\sim[\boldsymbol{H}\times\boldsymbol{L}]$, being in total agreement with results of the Ginzburg-Landau theory BiFeO3_Popkov . Finally, we evaluate matrix elements of the ME tensor, $\alpha_{\parallel}=\frac{\partial P^{z}}{\partial H^{x}}$ and $\alpha_{\perp}=\frac{\partial P^{x}}{\partial H^{z}}$ (for $\boldsymbol{L}||\boldsymbol{y}$), using the same procedure as for Cr2O3 and parameters of exchange interactions $J_{ij}$ reported in Ref. SCLR , which are consistent with experimental data and reproduce the experimental value of Néel temperature ($T_{\rm N}$). Then, using the obtained value of $J_{0}\approx-241$ meV and $M\approx 5$ $\mu_{\rm B}$, we will find $|\alpha_{\parallel}|=0.03$ ${\rm ps}/{\rm m}$ and $|\alpha_{\perp}|=0.12$ ${\rm ps}/{\rm m}$. These results are consistent (at least, by an order of magnitude) with direct calculations of electronic polarization for the model Hartree-Fock Hamiltonian without invoking the perturbation theory for the SO coupling, and also the experimental value of $0.51$ ${\rm ps}/{\rm m}$, reported in Ref. BiFeO3_Rivera . The giant enhancement of the ME coupling, which was reported in Refs. BiFeO3_Wang ; BiFeO3_Lorenz , probably requires additional mechanisms, such as the structural and magnetic reconstruction in the critical electric field, as was proposed in Refs. BiFeO3_Popkov ; BiFeO3_Lisenkov ; BiFeO3_Kulagin .

When the spins lie in the $\boldsymbol{xy}$ plane, there is also an “intrinsic ME effect” due to the FM canting of spins ($\sim 0.5^{\circ}$ SCLR ; EdererSpaldin ) in the direction perpendicular to $\boldsymbol{L}$, which is caused by DM exchange interactions without any magnetic field. Below $T_{\rm N}$, it leads to the polarization change $\Delta P^{z}$, which can be estimated, using the obtained values of $\boldsymbol{\cal P}_{ij}$, as $~{}0.2$ $\mu{\rm C}/{\rm m}^{2}$.

Below we will critically examine the main approximations of our theory by considering the DM exchange interactions, which can be easily computed by using other techniques. Parameters of DM interactions, obtained in the SE theory for bare $\xi_{0}=53.1$ meV, $\boldsymbol{d}_{12_{\rm I}}=(0.106,-0.287,0.140)$ meV agree reasonably well with $\boldsymbol{d}_{12_{\rm I}}=(0.145,-0.418,0.177)$ meV, derived using more general Green’s function perturbation theory method for the same model SCLR . Both superexchange and Green’s function methods are the first-order theories with respect to the SO coupling. Nevertheless, the Green’s function method does not employ additional approximations, such as the perturbation theory expansion with respect to the transfer integrals and the crystal-field splitting. The reasonably good agreement obtained for the DM parameters demonstrates that such approximations are indeed justifiable. The conclusion is not so trivial because, for the five-orbital model, the $t_{2g}$-$e_{g}$ level splitting in the octahedral environment is not small, being about $1.7$ eV. Nevertheless, it is smaller than the effective interaction $U\approx 5.8$ eV. Another important effect is the polarizability of the electron system by the SO interaction SCLR , which in our case is taken into account only approximately, by using the effective coupling $\xi=123$ meV instead of $\xi_{0}$, where $\xi$ was derived by fitting results of the SCLR calculations for matrix elements of the “screened” SO interactions with different projections spins. The “screened” SO interaction includes the bare contribution as well as all the self-consistent change of the Coulomb and exchange potential, treated on the mean-field level in the first order of the SO coupling. Thus, the use of $\xi$ instead of $\xi_{0}$ simply scales the DM parameter $\boldsymbol{d}_{12_{\rm I}}$ by about factor of 2. Although it captures the main tendency, it does not describe all details of $\boldsymbol{d}_{12_{\rm I}}=(0.494,-1.450,0.330)$ meV, obtained by combining SCLR with Green’s function perturbation theory, which is the most rigorous method for the evaluation of DM interactions SCLR . Thus, our SE theory for the DM interactions and ME coupling is probably only semi-quantitative one. However, we believe that it should not change the main conclusions, particularly regarding the comparison with the experimental data for BiFeO3.

Finally, we would also like to stress that the phenomenological expression (2) fails to describe the ME effect in BiFeO3: for the canted spin structure, inherent to the ME effect, $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ is the same for all neighboring bonds surrounding each Fe site. On the other hand, it holds $\sum_{j}\boldsymbol{\epsilon}_{ji}=0$. Thus, no ME effect would be expected if ${\bf P}_{ij}\propto\boldsymbol{\epsilon}_{ji}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$, which is obviously not true.

III.3 Noncollinear spin order and ferroelectric polarization in MnWO4

MnWO4 has attracted a considerable attention as an example of the spin-spiral magnet, which was theoretically suggested to be multiferroic Heyer , where this multiferroic behavior was indeed observed experimentally Heyer ; Taniguchi ; Arkenbout , and studied in many details after that Mitamura ; Olabarria1 ; Olabarria2 ; Xiao . Finite polarization was observed in the so-called noncollinear AF2 phase which is realized in the temperature interval 7.6 K $<T<$ 12.5 K and described by the propagation vector ${\bf q}_{\rm AF2}=(-0.214,\frac{1}{2},0.457)$ Taniguchi . The spins rotate in the plane formed by the monoclinic $\boldsymbol{b}$ axis and one of the axes $\boldsymbol{a}^{*}$ in the $\boldsymbol{ac}$ plane (see Fig. 6), the direction of which is specified by the single-ion anisotropy. The electric polarization is parallel to $\boldsymbol{b}$ axis, but can be realigned along $\boldsymbol{a}$ by applying the external magnetic field parallel to $\boldsymbol{b}$. In our previous work (Ref. PRB13 ) we have suggested that the FE activity in MnWO4 may be related to the deformation of the spin-spiral texture, which explicitly breaks the inversion symmetry. The computational details and parameters of the effective low-energy model, constructed for the half-filled Mn$3d$ bands of MnWO4, can be also found in Ref. PRB13 .

It is interesting to note that, unlike in the magnetoelectric Cr2O3 and BiFeO3, the direction of polarization in MnWO4 is described by the phenomenological model (2). Indeed, for the spin rotation plane formed by $\boldsymbol{a}^{*}$ and $\boldsymbol{b}$, the cross product $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ is parallel to $\boldsymbol{c}^{*}$, which is another vector in the $\boldsymbol{ac}$ plane being perpendicular to $\boldsymbol{a}^{*}$. Then, for ${\bf q}_{\rm AF2}=(-0.214,\frac{1}{2},0.457)$, there are two types of neighboring bonds formed by noncollinear spins for which $\boldsymbol{\epsilon}_{ji}||\boldsymbol{a}$ and $\boldsymbol{\epsilon}_{ji}||\boldsymbol{c}$. In both cases the expression ${\bf P}_{ij}\propto\boldsymbol{\epsilon}_{ji}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ yields ${\bf P}_{ij}||\boldsymbol{b}$, which agrees with the experimental situation Taniguchi . Nevertheless, below we will show that such agreement is rather fortuitous and the actual reason behind it is the specific $P2/c$ symmetry of MnWO4.

The behavior of pseudovectors $\boldsymbol{\cal P}_{ij}$, reflecting the symmetry properties of MnWO4, is explained in Fig. 7.

The vectors are long-ranged and not restricted by the nearest neighbors. For instance, we have found sizable parameters spreading up to twelfth coordination sphere. Similar behavior was found for isotropic exchange interactions (being in total agreement with the experimental data Ye ) and is related to the long-range character of the transfer integrals PRB13 . Due to the twofold rotation about the monoclinic $\boldsymbol{b}$ axis ($\hat{C}_{b}^{2}$), which is one of the symmetry operations of the $P2/c$ group (apart from a translation), ${\cal P}^{y}$ in equivalent bonds will have identical signs, while ${\cal P}^{x}$ and ${\cal P}^{z}$ will have opposite signs. Moreover, if the bond connect two Mn sites of the same type (either $I$ or $II$), $\hat{C}_{b}^{2}$ will transform it to equivalent bond, separated by a translation. Therefore, for this type of bonds we will have additional condition: ${\cal P}^{x}={\cal P}^{z}=0$.

Then, we consider the effect of noncollinear spin-spiral texture with ${\bf q}=(-\frac{1}{4},\frac{1}{2},\frac{1}{2})$ (Fig. 6), which is close ${\bf q}_{\rm AF2}$ realized in the experimental FE AF2 phase Taniguchi . We will use this model mainly for numerical estimations, while our symmetry considerations are more general and valid also for the experimental ${\bf q}_{\rm AF2}$. First we consider perfect spiral structure texture. The effect of deformation of the spin spiral, which was proposed in Ref. PRB13 , will be investigated below.

The spin-spiral structure itself is stabilized by competing isotropic exchange interactions PRB13 . However, its spacial orientation depends on the single-ion anisotropy and DM interactions, which also compete with each other: the former tends to align the spins in the $\boldsymbol{ac}$ plane (and cant them off the $\boldsymbol{a}$ axis by about $40^{\circ}$) PRB13 ; Taniguchi , while the main DM vectors $\boldsymbol{d}_{ij}$ also lie in the $\boldsymbol{ac}$ plane ($\in\boldsymbol{ac}$) PRB13 . Thus, in order to minimize the energy of DM interactions, some of the spins should be parallel to the $\boldsymbol{b}$ axis ($||\boldsymbol{b}$), which conflicts with the small single-ion anisotropy. Moreover, the DM exchange interactions compete with the isotropic ones. Thus, the situation is indeed very subtle and the magnetic structure is rather fragile. Nevertheless, this is a very important point because, as we will see in a moment, the spacial orientation of the spin-spiral plane can control both magnitude and direction of the electric polarization.

First, we consider the experimental situation where the spin-spiral plane is formed by the $\boldsymbol{b}$ axis and one of directions ($\boldsymbol{a}^{*}$) in the $\boldsymbol{ac}$ plane Taniguchi . Then, considering the magnetic structure in Fig. 6, half of the spins is parallel to $\boldsymbol{b}$ and another half belongs to $\boldsymbol{ac}$. This means that for the bonds $\langle ij\rangle$ with unparallel spins, $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ will also belong to $\boldsymbol{ac}$ and, therefore, the active components of $\boldsymbol{\cal P}_{ij}$ are ${\cal P}_{ij}^{x}$ and ${\cal P}_{ij}^{z}$. Then, for the equivalent bond $\langle i^{\prime}j^{\prime}\rangle$, which is obtained from $\langle ij\rangle$ by $\hat{C}_{b}^{2}$, we will have the following properties: ${\cal P}_{i^{\prime}j^{\prime}}^{x}=-{\cal P}_{ij}^{x}$, ${\cal P}_{i^{\prime}j^{\prime}}^{z}=-{\cal P}_{ij}^{z}$, and $[\boldsymbol{e}_{i^{\prime}}\times\boldsymbol{e}_{j^{\prime}}]=-[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$. The latter property holds because $\hat{C}_{b}^{2}$ reverses the direction of propagation of the spin-spiral along $\boldsymbol{a}$ and $\boldsymbol{c}$. Therefore, if the bond $\langle ij\rangle$ is along the propagation direction, the bond $\langle i^{\prime}j^{\prime}\rangle$ lies in the opposite direction. Thus, according to Eq. (4), the finite polarization is possible in the direction, which does not change under $\hat{C}_{b}^{2}$, keeping the sign of corresponding projection of the vector $\boldsymbol{\epsilon}_{ji}$. For the considered geometry of the spin spiral, this direction is $\boldsymbol{b}$ ($=\boldsymbol{y}$), in agreement with the experimental data Taniguchi . However, the absolute value of polarization depends on the orientation of spins in the $\boldsymbol{ac}$ plane. Indeed, according to Eq. (4), if $\boldsymbol{e}_{i}=(0,1,0)$ and $\boldsymbol{e}_{j}=(\sin\beta,0,\cos\beta)$, the polarization behaves as $P_{ij}^{y}\sim({\cal P}_{ij}^{x}\cos\beta-{\cal P}_{ij}^{z}\sin\beta)$. The dependence of total polarization $P^{y}=\sum_{j}P_{ij}^{y}$ on $\beta$, obtained using the numerical values of ${\cal P}_{ij}^{x}$ and ${\cal P}_{ij}^{z}$, is displayed in Fig. 8.

Thus, one can conclude the follows:

(i) The finite polarization in MnWO4 can be indeed induced by the spiral magnetic order. In this sense, the conclusion of our previous work PRB13 about crucial importance of inhomogeneity (or deformation) of the spin-spiral order was exaggerated;

(ii) The absolute value of $P^{y}$ strongly depends on the orientation of spins in the $\boldsymbol{ac}$ plane. The maximal value of about 25 $\mu{\rm C}/{\rm m}^{2}$ is comparable with experimental 50 $\mu{\rm C}/{\rm m}^{2}$ Taniguchi . However, it does not mean that this maximal value is realized for the same angle $\beta$, which minimizes the total energy of the system. In fact, the directions of spins are controlled by anisotropic interactions, which are small in MnWO4 PRB13 . Therefore, the situation is very fragile. This probably explains the large spread of the values of electric polarizations reported in electronic structure calculations, which are typically underestimated in comparison with the experimental data PRB13 ; Tian ; Shanavas .

Then, we consider the situation when all spins lie in the $\boldsymbol{ac}$ plane and also form the spin spiral with the propagation vector ${\bf q}=(-\frac{1}{4},\frac{1}{2},\frac{1}{2})$. This behavior was observed experimentally in the magnetic filed $\boldsymbol{H}||\boldsymbol{b}$, which causes the spin-flop-like transition and orients the spins in the $\boldsymbol{ac}$ plane, also changing the direction of experimental polarization from ${\bf P}||\boldsymbol{b}$ to mainly ${\bf P}||\boldsymbol{a}$ Taniguchi . In this case, $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ is parallel to $\boldsymbol{b}$ and the active component of $\boldsymbol{\cal P}_{ij}$ is ${\cal P}_{ij}^{y}$. Then, for two bonds $\langle ij\rangle$ and $\langle i^{\prime}j^{\prime}\rangle$, which are transformed to each other by $\hat{C}_{b}^{2}$, we will have: ${\cal P}_{i^{\prime}j^{\prime}}^{y}={\cal P}_{ij}^{y}$ and $[\boldsymbol{e}_{i^{\prime}}\times\boldsymbol{e}_{j^{\prime}}]=-[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$. Therefore, the finite polarization is possible in the directions, which are reversed by $\hat{C}_{b}^{2}$. This means that the polarization should lie in the $\boldsymbol{ac}$ plane. The direction of polarization in the plane is not specified by the symmetry and is the matter of numerical calculations. Using the numerical values of ${\cal P}_{ij}^{y}$ (Fig. 7), we obtain $P^{x}=36.6$ $\mu{\rm C}/{\rm m}^{2}$ and $P^{z}=9.4$ $\mu{\rm C}/{\rm m}^{2}$. In agreement to the symmetry arguments Olabarria2 , our theory also predicts small polarization along $\boldsymbol{c}$, which could be verified experimentally. This conclusion is formally consistent with the phenomenological model (2). Nevertheless, we would like to emphasize that the actual reason for such behavior, both for ${\bf P}||\boldsymbol{b}$ and ${\bf P}\in\boldsymbol{ac}$, is the specific symmetry of MnWO4 and the existence of the twofold rotation $\hat{C}_{b}^{2}$ among symmetry operations of the space group $P2/c$.

Finally, we discuss the effect of spin-spiral inhomogeneity on the electronic polarization $P^{y}$ in the ground state, which was proposed in Ref. PRB13 . This inhomogeneity is caused by the competition of isotropic and DM exchange interactions, which breaks the inversion symmetry and makes two Mn sublattices in MnWO4 inequivalent (shown by different colors in Ref. 6). Particularly, for the ${\bf q}=(-\frac{1}{4},\frac{1}{2},\frac{1}{2})$ structure, half of the spins will remain parallel to the $\boldsymbol{b}$ axis, while another half will split in two groups, forming different angles $\beta$ with respect to the $\boldsymbol{c}$ axis ($69^{\circ}$ and $56^{\circ}$, respectively) PRB13 . Then, there will be four types of Mn sites with distinct neighborhood: 1, 2, 5 and 6 in Fig. 6, which yield four distinct values of ${\bf P}_{i}=\sum_{j}{\bf P}_{ij}$, respectively: $25.3$, $20.0$, $11.8$, and $21.2$ $\mu{\rm C}/{\rm m^{2}}$. The total polarization in this case is the average value of these four, yielding $19.6$ $\mu{\rm C}/{\rm m^{2}}$, which is consistent with the value of electric polarization $|{\bf P}|$ of homogeneous spin-spiral with the average $\beta=61.5^{\circ}$ (see Fig. 8). Thus, the spin-spiral inhomogeneity does not seem to make a significant effect on the value of ${\bf P}$ in MnWO4, contrary to manganites, where the polarization is driven by nonrelativistic double exchange mechanism DEpol .

III.4 Symmetry constraints on the direction of polarization in spin-spiral MnO2

The rutile ($\beta$-) phase of MnO2 is another interesting example. Due to competing first- and second-neighbor AFM exchange interactions, it develop the incommensurate spin-spiral order below $T_{\rm N}\approx 92$ K MnO2_Yoshimori ; MnO2_Sato . The spin spiral propagates along the tetragonal $\boldsymbol{c}$ axis (=$$\boldsymbol{z}). Therefore, from the viewpoint of spin-current theory, it could be another potential multiferroic compound KNB ; Mostovoy , though has never been considered in this context. In this section, we will show that the multiferroic effect can be indeed expected in the rutile phase of MnO2. Moreover, the behavior of electronic polarization obeys the phenomenological rule ${\bf P}\propto\boldsymbol{c}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ KNB ; Mostovoy . Nevertheless, we will argue that the actual reason behind it is related to the specific $P4_{2}/mnm$ symmetry of MnO2, which imposes the symmetry constraints on the properties of $\boldsymbol{\cal P}_{ij}$.

We use the experimental parameters of the crystal structure, reported in Ref. MnO2_structure . There are two Mn sites in the primitive cell, which are connected by the symmetry operations of the space group $P4_{2}/mnm$. Like in Cr2O3, we consider the minimal model comprising of half-filled $t_{2g}$ states near the Fermi level (Fig. 9).

In this case, the crystal-field splitting of $t_{2g}$ levels is pretty large (about $370$ meV). The Kanamori parameters of screened intraorbital Coulomb and exchange interaction are $3.0$ and $0.72$ eV, respectively. Other parameters can be found elsewhere footnote2 . The isotropic exchange interactions between first and second neighbors, located at $(0,0,\pm c)$ and $(\pm\frac{a}{2},\pm\frac{a}{2},\pm\frac{c}{2})$ ($a$ and $c$ being the tetragonal lattice parameters), are $-16.4$ meV and $-12.3$ meV, respectively. Like in other considered systems, the theories of SE interactions, Eq. (6), and infinitesimal spin rotations, Ref. JHeisenberg , give very close values of $J_{ij}$. The obtained exchange interactions support the appearance of spin-spiral superstructure with ${\bf q}\approx(0,0,\frac{1}{7})$ (comprising of 7 primitive cells), in agreement with the analysis of experimental data MnO2_Yoshimori and results of first-principle calculations MnO2_Lim . Moreover, the magnetocrystalline anisotropy energy confines the spins in the $\boldsymbol{xy}$ plane.

The $P4_{2}/mnm$ space group imposes the symmetry constrains on the properties of $\boldsymbol{\cal P}_{ij}$, which are explained in Fig. 10.

The parameters $\boldsymbol{\cal P}_{ij}$ between nearest neighbors vanish identically due to the $mmm$ symmetry in the bonds $(0,0,\pm c)$. Then, due to the symmetry operations $\{\hat{C}^{4}_{c}|(\frac{a}{2},\frac{a}{2},\frac{c}{2})\}$ and $\{\hat{m}_{x}|(\frac{a}{2},\frac{a}{2},\frac{c}{2})\}$ ($\hat{C}^{4}_{c}$ being the fourfold rotation about the tetragonal axis $\boldsymbol{c}$), transforming the second-neighbor bonds to themselves, the corresponding parameters $\boldsymbol{\cal P}_{ij}$ will have the following form: $\boldsymbol{\cal P}_{ij}=(\pm{\cal P},\pm{\cal P},0)$ (see Fig. 10). Therefore, it is straightforward to see that the spin spiral, propagating along $\boldsymbol{c}$ ($=\boldsymbol{z}$) and rotating in the $\boldsymbol{xy}$ plane, does not induce any polarization because the active component ${\cal P}_{ij}^{z}$ is identically equals to zero. For other bonds with lower symmetry, some of ${\cal P}_{ij}^{z}$ can be finite. However, the phases of $\boldsymbol{\epsilon}_{ji}{\cal P}_{ij}^{z}$ alternate for the equivalent types of bonds, again resulting in no net polarization.

However, when the spins rotate in the $\boldsymbol{yz}$, the active component is ${\cal P}_{ij}^{x}$, which is finite. Moreover, by combining the phases of ${\cal P}_{ij}^{x}$ with the ones of $\boldsymbol{\epsilon}_{ji}$, it is straightforward to see that $P^{x}=P^{z}=0$, while $P^{y}$ can be finite. Using obtained parameters ${\cal P}_{ij}^{x}$ we estimate $P^{y}$ for ${\bf q}\approx(0,0,\frac{1}{7})$ as $2$ $\mu{\rm C}/{\rm cm}^{2}$. Similar conclusion holds when the spins rotate in the $\boldsymbol{zx}$ plane.

Thus, we expect no FE activity in the magnetic ground state of MnO2. However, small polarization can be induced by rotating the spins to either $\boldsymbol{yz}$ or $\boldsymbol{zx}$ plane. It can be done by applying the external magnetic field along either $\boldsymbol{x}$ or $\boldsymbol{y}$ axis. Thus, our finding can be verified experimentally. The result is formally consistent with the phenomenological expression ${\bf P}\propto\boldsymbol{c}\times[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$ KNB ; Mostovoy . However, it should be understood that, similar to MnWO4, the actual reason behind it is the specific symmetry of the rutile phase of MnO2.

IV Summary and Conclusions

We have derives an analytical expression for the electronic polarization driven by the SO interaction in noncollinear magnets. For these purposes we have considered the Berry-phase theory of electric polarization and applied it to the Hubbard model at the half filling. Thus, our analysis is limited by the spin-current mechanism and do not involve additional complications caused by the orbital degrees of freedom. Moreover, all derivations are performed in the spirit of the SE theory, which is valid in the first order of the SO coupling and in the lowest order of $\hat{t}_{ij}/U$, similar to the analysis of DM exchange interactions Moriya_weakF .

We have found that the electric polarization in each bond is given by Eq. (4), which is substantial revision of the phenomenological expression (2). Namely, the electronic polarization in Eq. (4) explicitly depends on the symmetry of the lattice (similar to the DM exchange interactions $\boldsymbol{d}_{ij}$ Dzyaloshinskii_weakF ): this dependence is described by the pseudovector $\boldsymbol{\cal P}_{ij}$, which is coupled to the cross product $[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$, depending on the directions of spins. Thus, this coupling describes how the symmetry of the lattice interferes with the symmetry of the noncollinear arrangement of spins. The direction of the polarization itself is specified by the unit vectors $\boldsymbol{\epsilon}_{ji}$ in the direction connecting two magnetic sites, which are modulated by the scalar $\boldsymbol{\cal P}_{ij}\cdot[\boldsymbol{e}_{i}\times\boldsymbol{e}_{j}]$. We argue that, even though the direction of electric polarization in some spin-spiral magnets can be described by the phenomenological expression (2), the actual reason behind it is the specific symmetry properties of each considered system, which are described by the pseudovectors $\boldsymbol{\cal P}_{ij}$. Moreover, we have shown that the spin-current mechanism is much more generic and operates not only in spin-spiral compounds, but also in other types of noncollinear magnets, where the phenomenological expression (2) breaks down. Particularly, absolutely the same mechanism may lead to the ME effect induced by the ferromagnetic canting of spins in the external magnetic field.

Another important factor, which plays a crucial role even at the half-filling, is the crystal-field splitting. We have shown that without crystal field, both DM exchange interactions and electronic polarization will vanish. However, the crystal field may have other interesting consequences. For instance, it leads to the asphericity in the distribution of the charge density around each transition-metal site and, if the latter is located not in the centrosymmetric position (like for all considered here compounds), one can also expect ionic contribution to the electronic polarization, which can be also derived from the Berry-phase theory, as was demonstrated recently in Ref. Ba2CoGe2O7 for Ba2CoGe2O7. This is also consistent with the phenomenological analysis by Moriya Moriya_pol , who expressed the total polarization as the sum of ionic contributions and the ones originating from the bonds. The ionic contributions were also evaluated in the present work and found to be at least one order of magnitude smaller than the “anomalous” pair contributions, which are given by Eq. (4) and related to fine details of the electronic structure Picozzi .

Using parameters of the effective Hubbard model, derived from the first-principles electronic structure calculations, we have evaluated the spin-current contribution to the electronic polarization for the series of ME materials (Cr2O3 and BiFeO3) and multiferroics compounds with the spin-spiral structure (MnWO4 and MnO2). We have shown that, although Eq. (4) excellently reproduces the symmetry properties of polarization, its numerical values can be severely underestimated, depending on the material. Particularly, the largest disagreement was found for the ME effect in Cr2O3, which suggest the importance of other (lattice and orbital) contributions, in agreement with the previous finding Bousquet ; Malashevich ; Scaramucci .

We have also clarified the microscopic origin of FE activity in the spin-spiral phase of MnWO4: although the spin spiral in this case is deformed by competing isotropic and antisymmetric DM exchange interactions, which explicitly breaks the inversion symmetry PRB13 , this deformation seems to have a minor effect on the value of electronic polarization. The main contribution to the polarization comes from the spin-current term, given by Eq. (4), which also describes the change of the direction of polarization, depending on the spacial orientation of the spin spiral.

Finally, we have predicted the FE activity in the rutile phase of MnO2 when the spin spiral is rotated our of the tetragonal $\boldsymbol{xy}$ plane.