Magnetic-field control of the electric polarization in BiMnO3
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Magnetic-field control of the electric polarization in BiMnO I. V. Solovyev ∗ and Z. V. Pchelkina National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan Institute of Metal Physics, Russian Academy of Sciences - Ural Division, 620041 Ekaterinburg GSP-170, Russia (Dated: November 2, 2018)We present the microscopic theory of improper multiferroicity in BiMnO , which can be summa-rized as follows: (1) the ferroelectric polarization is driven by the hidden antiferromagnetic order inthe otherwise centrosymmetric C /c structure; (2) the relativistic spin-orbit interaction is respon-sible for the canted spin ferromagnetism. Our analysis is supported by numerical calculations ofelectronic polarization using Berry’s phase formalism, which was applied to the low-energy modelof BiMnO derived from the first-principles calculations. We explicitly show how the electric po-larization can be controlled by the magnetic field and argue that BiMnO is a rare and potentiallyinteresting material where ferroelectricity can indeed coexist and interplay with the ferromagnetism. Introduction . Today, the term ‘multiferroics’ is typ-ically understood in a broad sense, as the systems ex-hibiting spontaneous electric polarization and any typeof magnetic ordering. Such materials have a great po-tential for practical applications in magnetic memories,logic, and magnetoelectric sensors, and therefore at-tracted enormous attention recently. Beside practicalmotivations, there is a strong fundamental interest inunveiling the microscopic mechanism of coupling be-tween electric polarization and magnetic degrees of free-dom. Nevertheless, the combination of ferroelectricityand ferro magnetism, what the term ‘multiferroicity’ wasoriginally introduced for, is rare. Such a combinationwould, for example, provide an easy way for manipulat-ing the electric polarization P by the external magneticfield, which is directly coupled to the net ferromagnetic(FM) moment, etc. The canonical example, where spon-taneous electric polarization was believed to coexist withthe FM ground state, is BiMnO . However, the originof such coexistence is largely unknown. Originally, theferroelectric (FE) behavior in BiMnO was attributed tothe highly distorted perovskite structure stabilized by theBi6 s “lone pairs”. However, more resent experimentalstudies (Ref. 3) and first-principles calculations (Ref. 4)suggested that the atomic displacements alone result inthe centrosymmetric C /c structure, which is incompat-ible with the ferroelectricity. In our previous papers(Refs. 5 and 6) we put forward the idea that the ferro-electricity in BiMnO could be improper and associatedwith some hidden antiferromagnetic (AFM) order. Thepurpose of this work is to provide the complete quantita-tive explanation for the appearance and behavior of theFE polarization in BiMnO . Method . The basic idea of our approach is to constructan effective Hubbard-type modelˆ H = X ij X αβ t αβij ˆ c † iα ˆ c jβ + 12 X i X αβγδ U αβγδ ˆ c † iα ˆ c † iγ ˆ c iβ ˆ c iδ (1)for the Mn3 d -bands near the Fermi level and to includethe effect of all other (“inactive”) states to the defi-nition of the model parameters of the Hamiltonian ˆ H .Thus, the model is constructed in the basis of 40 Wan- nier functions in each unit cell (including three t g - andtwo e g -orbitals for each spin and for each of the fourMn-sites), by starting from the electronic structure inthe local-density approximation (LDA). The Greek sym-bols denote the combination of spin and orbital indices.All parameters of ˆ H are defined rigorously, on the basisof the density functional theory (DFT). The details canbe found in the review article (Ref. 7) and in our pre-vious papers (Refs. 5 and 6). Briefly, the one-electronpart ( t αβij ) is derived by using the generalized downfold-ing method. One of important parameters in t αβij is thelarge (about 1.5 eV) crystal-field splitting between two e g -levels, which is caused by the Jahn-Teller distortionand manifests itself in the orbital ordering. The screenedCoulomb interactions ( U αβγδ ) are obtained by combiningthe constrained DFT technique with the random-phaseapproximation (RPA): namely, the screening by outerelectrons (such as 4 sp -electrons of transition metals) andthe change of spacial extension of the atomic wavefunc-tions upon the change of occupation numbers can be eas-ily taken into account by solving Kohn-Sham equationswithin constrained DFT approach. On the other hand,the “self-screening” by the same type of electrons, whichcontribute to other bands due to the hybridization ef-fects (for example, the 3 d -electrons in the oxygen bandwill strongly screen the Coulomb interactions in the 3 d -band near the Fermi level), is included in the perturbativeRPA treatment. The self-screening is very important insolids and substantially reduces the value of the effectiveCoulomb repulsion U (defined as the screened Slater in-tegral F ) in the 3 d -band of manganites. In BiMnO , itis only about 2.3 eV, that has important consequenceson the behavior of interatomic magnetic interactions.The model (1) is solved in the Hartree-Fock (HF)approximation: (cid:16) ˆ t k + ˆ V (cid:17) | C n k i = ε n k | C n k i , where ˆ t k is the Fourier image of ˆ t ij = k t αβij k and, if neces-sary, includes the relativistic spin-orbit coupling (SOC),ˆ V is the self-consistent HF potential, and | C n k i is theeigenvector in the basis of Wannier functions (where the x - r y - r z - r z - r plane x'y' J NN J NN J NN J NN x - r z - r y - r z - r plane x'z' J LR J LR J LR J LR J NN J NN J NN J NN ** FIG. 1. (Color online) Schematic view on the orbital order-ing and corresponding interatomic magnetic interactions inthe pseudocubic x ′ y ′ and z ′ x ′ planes. In the unit cell ofBiMnO , there are four Mn sites (indicated by numbers),which form two inequivalent groups: (1,2) and (3,4). Thenearest-neighbor FM interactions J NN operate in the hatchedbonds. The atoms involved in the long-range AFM interac-tions J LR are denoted by arrows. The inversion centers aremarked by ‘ ∗ ’. spin indices are included in the definition of n ). Once the orbital degeneracy is lifted by the strong lat-tice distortion, the HF theory provides a good approx-imation for the ground-state properties. The effect ofcorrelation interactions, which can be treated as a per-turbation to the HF solution, on the magnetic groundstate of manganites is partially compensated by the mag-netic polarization of the oxygen states: if the former tendto stabilize AFM structures, the latter favors the FMalignment. Due to this compensation, the mean-field HFtheory, formulated for the minimal 3 d -model, appears tobe rather successful for the ground state of manganites. Magnetism and the inversion symmetry breaking .First, let us explain the main idea of our work.
Whatis the possible origin of multiferroic behavior of BiMnO and how can it be controlled by the magnetic field?(1) The lattice distortion leads the orbital ordering,which is schematically shown in Fig. 1 in two pseudocu-bic planes (the orbital ordering in the y ′ z ′ -plane is similarto the one in the z ′ x ′ -plane). This orbital ordering pre-determines the behavior of interatomic magnetic interac-tions, which obey some general principles, applicable formanganites with both monoclinic ( C /c ) and orthorhom-bic ( P nma ) structure, namely: besides conventionalnearest-neighbor interactions (shown by hatched lines),one can expect some longer-range (LR) interactions be-tween remote Mn-atoms, which operate via intermediateMn-sites. These sites are shown by arrows.(2) Why should the LR-interactions exist? The answeris directly related to the fact that the on-site Coulombrepulsion U is not particularly large. Therefore, besidesconventional superexchange (SE), there are other inter-actions, which formally appear in the higher orders ofthe 1 /U -expansion and connect more remote sites. Thismechanism is rather similar to the SE interaction via in-termediate oxygen sites, but the role of the oxygen states is played by the unoccupied e g -orbitals of the intermedi-ate Mn-sites. By mapping HF total energies onto theHeisenberg model, one can obtain the following param-eters of interatomic magnetic interactions: J NN ∼ J LR ∼ − − .(3) Without SOC, the LR interactions tend to stabilizethe AFM ↑↓↓↑ structure (where the arrows denote the di-rections of spins for the four Mn-sites in the unit cell).This AFM order destroys the inversion centers (shownby ‘ ∗ ’ in Fig. 1) and thus should give rise to the FEpolarization. Since the ↑↓↓↑ structure satisfies the sym-metry operation ˆ T ⊗{ m y | R / } (where m y is the mirrorreflection y → − y associated with the one half of themonoclinic translation R , and ˆ T in the nonrelativisticcase flips the directions of spins, which are not affectedby m y ), P is expected to lie in the zx -plane. (4) Thus, the FE behavior in BiMnO should be causedby the AFM order. However, this conclusion seems tocontradict to the FM ground state of BiMnO . Thecontradiction can be reconciled by considering the rel-ativistic SOC, which is responsible for the weak ferro-magnetism. Since the FM component is additionallystabilized by the isotropic interactions J NN , the ferro-magnetism is not so “weak”, and the resulting magneticstructure, obtained in the HF calculations for the low-energy model, is strongly noncollinear (Fig. 2). It be-longs to the space group Cc , where the only nontrivialsymmetry operation is { m y | R / } and the magnetic mo-ments in the relativistic case are transformed by m y asauxiliary vectors. Thus, the net FM moment is alignedalong the y -axis, while the x - and z -components formthe AFM structure. Other magnetic configurations havehigher energies. The details can be found in Ref. 6.By summarizing this part, the C /c symmetry ofBiMnO is spontaneously broken by the hidden AFMorder. The true magnetic ground-state of BiMnO isstrongly noncollinear, where the FM order along the y -axis coexists with the AFM order, and related to it FEpolarization, along the x - and z -axes. Our scenario notonly explains the coexistence of ferroelectricity and ferro-magnetism, but also shows how the electric polarization P (and the symmetry of BiMnO ) can be controlled bythe external magnetic field B =(0 , B y ,
0) coupled to theFM moment. This basic idea was formulated in Ref. 6.In the present work we are able to provide the numericalestimates for P and to discuss its behavior in details. Electric polarization . Since the crystal structure ofBiMnO has the inversion symmetry, there will be noionic contribution to P , and the main mechanism, whichwill be considered below, is of purely electronic origin.In principle, the magneto-elastic interactions in the ↑↓↓↑ structure may cause the atomic displacements away fromthe centrosymmetric positions and give rise to the ionic FIG. 2. (Color online) Fragment of crystal and magneticstructure corresponding to the lowest HF energy. The Bi-atoms and indicated by the big light grey (yellow) spheres,the Mn-atoms are indicated by the medium grey (red) spheres,and the oxygen atoms are indicated by the small grey (green)spheres. The directions of spin magnetic moments are shownby arrows. The inversion center is marked by the symbol ‘ ∗ ’.The left lower part of the figure explains the orientation ofthe Cartesian coordinate frame. term. Nevertheless, such calculations would require thefull structure optimization, which cannot be easily in-corporated in the model analysis. The first-principlescalculations for HoMnO show that electronic and ionicterms are at least comparable. Thus, we expect that theelectronic contribution alone could provide a good semi-quantitative estimate for P . Moreover, the behavior ofelectronic contribution presents a fundamental interest asit allows one to explain how P in improper multiferroicsis induced solely by the magnetic symmetry breaking.The modern theory of electric polarization allows oneto relate the change of P to the Berry phase of Blocheigenstates. It is particularly convenient to use theformulation by Resta, where the Berry phase is com-puted on the discrete grid of k -points, generated by the N × N × N divisions of the reciprocal lattice vectors { G a } . Then, the position of each point in the Brillouinzone is specified by three integer indices (0 ≤ s a < N a ): k s ,s ,s = s N G + s N G + s N G , and components of the electric polarization in the curvi-linear coordinate frame formed by G , G and G can be obtained as ∆ P a = − V N a N N N [ γ a ( ∞ ) − γ a (0)] , (2)where V is the unit cell volume, γ = − N − X s =0 N − X s =0 Imln N − Y s =0 det S ( k s ,s ,s , k s +1 ,s ,s ) , and similar expressions hold for γ and γ . Eq. (2) im-plies that the only meaningful quantity in the bulk is thepolarization difference between two states that can beconnected by an adiabatic switching process. In the present case, S = kh C n k | C n ′ k ′ ik is the overlapmatrix, constructed from the HF eigenvectors | C n k i inthe occupied part of spectra, taken in two neighboring k -points: k = k s ,s ,s and k ′ = k s +1 ,s ,s for γ , etc. The polarization (2) was first computed in the curvilinearcoordinate frame and then transformed to the cartesianframe shown in Fig. 2. In all the calculations, we usedthe mesh of 72 × ×
36 points in the Brillouin zone.Without SOC, the AFM alignment of spins at the sites1 and 2 yields finite polarization. However, the symme-try of the system also depends on the magnetic config-uration in the sublattice 3-4. As discussed above, theelectric polarization in the ↑↓↓↑ structure lies in the zx -plane ( P x = 2 . µ C/cm and P z = 0 . µ C/cm ). The ↑↓↓↑ structure can be transformed to the ↑↓↑↓ one withthe same energy by the symmetry operation { C y | R / } (where C y is the 180 ◦ rotation around the y -axis), whichchanges the direction of P : P x ( z ) → − P x ( z ) . On theother hand, the ↑↓↓↓ structure (which has higher en-ergy) is transformed to itself by { C y | R / } , and cor-responding electric polarization will be parallel to the y -axis ( P y = 4 . µ C/cm ). Other magnetic structures,characterized by the FM alignment of spins at the sites1 and 2 (such as ↑↑↑↑ and ↑↑↑↓ ), preserve the inversionsymmetry and result in zero net polarization.Without SOC, one can easily evaluate separate contri-butions to P of the states with the spin ↑ and ↓ . Forthe ↑↓↓↑ structure, the vector of the electric polarizationtakes the following form: P ↑ , ↓ = ( P x , ± P y , P z ), where P y = 5 . µ C/cm , and the values of P x and P y are listedabove. This result is very natural, because the distribu-tion of the electron density for each spin does not haveany symmetry and, therefore, the electric polarization P ↑ , ↓ has all three components. On the other hand, theelectron density with the spin ↑ in the ↑↓↓↑ AFM statecan be transformed to the one with the spin ↓ by the sym-metry operation { m y | R / } and, therefore, P ↑ y = − P ↓ y .Thus, in the total polarization P = P ↑ + P ↓ , the x - and z -components with different spins will sum up, while thelargest y -components will cancel each other.Furthermore, one can evaluate the individual contri-butions to P coming from the t g -band, which is sepa-rated by an energy gap from the e g -band. This yields: P t g x = − . µ C/cm and P t g z = − . µ C/cm . Thus, P x P z po l a r i za ti on ( C / c m ) magnetic field along y-axis (T) a ng l e b e t w ee n M a nd M ( d e g . ) M M B y (T)M M x , M y , M z ( B ) FIG. 3. (Color online) Magnetic-field dependence of the elec-tric polarization, the angle φ between spin magnetic momentsat the Mn-sites 1 and 2, and the vector of magnetic momentat the site 1 (shown in the inset). the t g -band is polarized opposite to the e g -band, thatsubstantially reduces the value of P .The SOC results in the canting of spins away from thecollinear ↑↓↓↑ state and towards the FM configuration.It will reduce the value of P . In the HF ground-state (seeFig. 2), the angle φ between spin magnetic moments atthe sites 1 and 2 is 137 ◦ , and the electric polarization isreduced till P x = 1 . µ C/cm and P z = 0 . µ C/cm .This effect can be further controlled by the magneticfield, which is applied along the y -axis and saturates theFM magnetization. Since the absolute value of the localmagnetic moment is nearly conserved, the increase of theFM component along the y -axis will be compensated bythe decrease of two AFM components along the x - and z -axes. The corresponding FE polarization will also de-crease. Results of HF calculations in the magnetic fieldare shown in Fig. 3. Sufficiently large magnetic field( ∼
35 Tesla) will align the magnetic moments at the sites1 and 2 ferromagnetically ( φ =0) and restore the C /c symmetry. The electric polarization follows the changeof φ and complete disappears when φ =0. However, thedecline of P is much steeper: for example, P x and P z are reduced by factor two already in the moderate field B y ∼ φ ∼ ◦ . Moreover, P z is always substantially smaller than P x . Concluding remarks . We have proposed the micro-scopic theory of improper multiferroicity in BiMnO ,which is based on the inversion symmetry breaking bythe hidden AFM order. We have estimated the FE po-larization and explicitly shown how it can be controlledby the magnetic field. Our scenario still needs to bechecked experimentally, and apparently one importantquestion here is how to separate the intrinsic ferroelec-tricity in BiMnO from extrinsic effects, caused by thedefects. For example, the values of the FE polarizationobtained in the present work, although comparable withthose calculated for other improper ferroelectrics on thebasis of manganites, are substantially larger than theexperimental value 0.062 µ C/cm (at 87 K), which wasreported so far for BiMnO . Nevertheless, we believethat systematic study of manganites with the monoclinic C /c symmetry and finding conditions, which would leadto the practical realization of scenario proposed in ourwork, presents a very important direction, because itgives a possibility for combining and intermanipulatingthe ferro electricity and ferro magnetism in one sample. Acknowledgements . This work is partly supported byGrant-in-Aid for Scientific Research (C) No. 20540337from MEXT, Japan and Russian Federal Agency for Sci-ence and Innovations, grant No. 02.740.11.0217. ∗ [email protected] D. Khomskii, Physics , 20 (2009). R. Seshadri and N. A. Hill, Chem. Mater. , 2892 (2001). A. A. Belik et al. , J. Am. Chem. Soc. , 971 (2007). P. Baettig, R. Seshadri and N. A. Spaldin, J. Am. Chem.Soc. , 9854 (2007). I. V. Solovyev and Z. V. Pchelkina, New J. Phys. ,073021 (2008). I. V. Solovyev, Z. V. Pchelkina, Pis’ma Zh. Eksp. Teor.Fiz. , 701 (2009) [JETP Lett. , 597 (2009)]. I. V. Solovyev, J. Phys.: Condens. Matter , 293201(2008). I. Solovyev, J. Phys. Soc. Jpn. , 054710 (2009). The Fourier image of ˆ t ij was defined as ˆ t k = P j ˆ t ij exp( − i k · R ij ), where R ij is the radius-vectorbetween sites i and j . Such a definition guarantees thatthe eigenvectors | C n k i are periodic in both direct andreciprocal space, as it is required for calculations of P . The spin model is defined as H S = − P h ij i J ij e i · e j , where e i and e j are the directions spins. We use the following setting for the monoclinic trans-lations: R , = (sin βa, ∓ b, cos βa ) and R =(0 , , c ). Thepositions of four Mn atoms in the unit cell are specifiedby the vectors: τ = y Mn ( R − R )+ R , τ = − τ , τ = R , and τ = ( R + R ). The experimental struc-ture parameters were taken from Ref. 3. S. Picozzi et al. , Phys. Rev. Lett. , 227201 (2007). D. Vanderbilt and R. D. King-Smith, Phys. Rev. B ,4442 (1993). R. Resta, Rev. Mod. Phys. , 899 (1994). R. Resta, J. Phys.: Condens. Matter , 123201 (2010). Strictly speaking, there will be two contributions to P : oneis cause by the evalution of | C n k i in the k -space and theother one is the contribution of { W α } (the Wannier basisof the low-energy model), which is expressed in terms ofthe the matrix elements h W α | r | W β i . Since the low-energymodel is constructed by starting from the nonmagneticLDA band structure, which preserves the parity of { W α } ,these matrix elements will vanish. The interaction term with the magnetic field is given by ˆ H B = − µ B B · (2ˆ s +ˆ l ), where ˆ s and ˆ l are the operators ofspin and orbital angular momentum, respectively. A. Moreira dos Santos et al. , Solid State Commun.122