Ferroelectricity induced by interatomic magnetic exchange interaction

Xiangang Wan,1† Hang-Chen Ding,2 Sergey Y. Savrasov3 and Chun-Gang Duan2,4‡ 1Department of and National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China 2Key Laboratory of Polar Materials and Devices, Ministry of Education, East China Normal University, Shanghai 200062, China 3Department of Physics, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA 4National Laboratory for Infrared Physics, Chinese Academy of Sciences, Shanghai 200083, China

† Electronic address: [email protected] ‡ Electronic address: [email protected] or [email protected]

Multiferroics, where two or more ferroic order parameters coexist, is one of the hottest fields in condensed matter physics and materials science1-9. However, the coexistence of and conventional ferroelectricity is physically unfavoured10. Recently several remedies have been proposed, e.g., improper ferroelectricity induced by specific magnetic6 or charge orders2. Guiding by these theories, currently most research is focused on frustrated magnets, which usually have complicated magnetic structure and low magnetic ordering temperature, consequently far from the practical application. Simple collinear magnets, which can have high magnetic transition temperature, have never been considered seriously as the candidates for multiferroics. Here, we argue that actually simple interatomic magnetic exchange interaction already contains a driving for ferroelectricity, thus providing a new microscopic mechanism for the coexistence and strong coupling between ferroelectricity and magnetism. We demonstrate this mechanism by showing that even the simplest antiferromagnetic (AFM) insulator MnO, can display a magnetically induced ferroelectricity under a biaxial strain.

The combination of different ferroic properties, proposed to explain the ferroelectricity in magnetic spiral especially ferroelectricity and (anti), structures. Electric polarization can also be induced by provides additional degree of freedom to control collinear order in the frustrated magnet with several magnetic and dielectric properties of the material. Such species of magnetic ions6,15-17. There are also proposals to functionality is of great potential in applications to realize multiferroic state in composite systems or next-generation fast, portable and low-energy materials with nanoscale inhomogeneity18,19. consumption data storage and processing devices. Nevertheless, currently all of the known magnetically Unfortunately, magnetism and ferroelectricity tend to be driven single-phase multiferroics require either mutually exclusive, as conventional ferroelectric Dzyaloshinskii-Moriya interaction (DMI)20,21, which is perovskite oxides usually require transition metal (TM) small in strength, or competing exchange interactions in ions with a formal configuration d0, whereas magnetism, real space. Therefore, they generally have complex in contrast, needs TM ions with partially filled d shells10. magnetic order, low transition temperature (below As a consequence, simultaneous occurrence of several ten K) and small electric polarization (generally ferromagnetism and ferroelectricity is hard to be two to three orders of magnitude smaller than typical achieved, especially at room temperature. There are ferroelectrics), making them still far away from practical indeed some exceptions in Bismuth related magnetic applications. Therefore searching new mechanism for oxides, e.g., BiFeO3 and BiMnO3. In these compounds, multiferroicity is of both fundamental and technological however, the magnetism and ferroelectricity are widely interest. believed to have different origins, rendering that the In this Letter, we demonstrate that regardless the magnetoelectric coupling rather weak9. spin-orbit coupling, simple interatomic magnetic To gain a strong magnetoelectric coupling, vast exchange interaction already provides a strong driving efforts have been devoted to search the improper force to break the inversion symmetry of the system, ferroelectricity where electric dipoles are induced by which is necessary for the occurrence of ferroelectricity. magnetism6. Phenomenological theory suggests that It is therefore a new microscopic mechanism for the spatial variation of is essential for the coexistence and strong coupling between ferroelectricity magnetically induced electric polarization11. Several and magnetism, and all of the magnetic insulators with microscopic mechanisms12-14, all emphasizing the simple magnetic structure need to be revisited. Using importance of spin-orbital coupling, have also been band structure calculations, we numerically confirm this

1 new mechanism by illustrating that even simplest for M1/M2 and O site. t1/t2 are the hopping integrals antiferromagnets, i.e. MnO, can display a magnetically between M1/M2 and O. Based on the standard induced ferroelectricity under biaxial strain. Schrieffer-Wolff transformation22, we can eliminate the O orbital and obtain the effective hopping integral between M1 and M2 site: tt 12 teff ∝ (2) εεdo− It is well known that the hopping integral is inversely − y − y proportional to the bond-length: tr11∼ , tr22∼ , where r1(2) is the bond length of M1(2)-O, y is a positive value and strongly depends on both the bond type and the Figure 1 | Illustration of ferroelectricity induced by participated orbital23. A longitudinal displacement of O indirect magnetic exchange interaction between ion u will then change the effective hopping to anion-mediated magnetic cations. Here M and M are 11 1 1 2 t ∝ =C (3) magnetic cations, O is anion. When O , eff yy yy εεd −+0 ()()ru ru − ()() ru + ru − which originally sits in the middle of M1 and M2 ions, is where C is a parameter and not sensitive to u, r is the shifted along the M1(2)-O bond direction (x) by a small distance between magnetic ion and center O-site. It is displacement u, the magnetic exchange interactions of interesting to notice that regardless the parameter C and the system will increase and may support the O off-center the superscript y in Eq. (3), a displacement u which movement. An electric dipole is then formed, as shown breaks inversion symmetry always increases the effective by an arrow in the picture. hopping between the magnetic ions. To study the interatomic magnetic exchange To see the effect of u on the magnetic ordering interaction and the associated magnetic ordering energy, energy, we take the superexchange, which is one of the we consider a three case, where a diamagnetic ion most common mechanisms in magnetic insulators, as an such as oxygen ion sits between two transition metal example. It is well known that for superexchange the 24 ions, as shown in Fig.1. As revealed by Sergienlo and interatomic exchange interaction can be written as : Dagotto13, DMI provides a driving force for the oxygen 4t 2 J =− eff (4) ion to shift perpendicularly to the spin chain. Due to the U small strength, however, the energy gain from DMI is where U is the Coulomb interaction for the magnetic usually less than the ordinary elastic energy, orbital. Thus, an off-center displacement of O ion can consequently only a few compounds show magnetically enhance the interatomic exchange interaction. Usually 6 induced ferroelectricity . Here, we consider the effect of the magnetic ordering energy has the form of longitudinal displacement of diamagnetic ions (shown by − J SS⋅ , thus for the AFM case, the energy gain up ∑ij ij i j the dot line in Fig.1) on the magnetic ordering energy. As is well known, in the above case, the distance between to the second order of the longitudinal displacement of O magnetic ions usually is much larger than the radii of d/f ions u is 112y orbital which carry magnetic moments, thus the direct Δ∝−Eu + ≈− 2 (5) exchange is negligible, and the hybridization between ()()ru+−22442yyyy ru r r+ magnetic and diamagnetic ions is essential for the Thus we prove that regardless the exact formula for the indirect magnetic exchange coupling. Therefore, we first hopping dependence on distance, an off-center distortion discuss the hopping integrals in this system. For can definitely lower the total energy by an amount ∼ u 2 . simplicity, here we only show the case of one orbital per Above we have discussed the superexchange in ion, and it is easy to generalize our results to multi-orbital one-band case. For multi-band superexchange and even cases. The kinetic energy then can be written as: double exchange mechanism25, the exchange coupling J Hcccccc=++εεε+++ is also proportional to the effective hopping, just the kin∑∑∑ d d11σσ d o o σσ o d d 2 σ d 2 σ σσσ relationship between J and teff in these mechanisms may ++++tcchctcchc (++ . .) ( . .) (1) not be as simple as shown in Eq. (4). Keeping in mind 12∑∑do12 d o σσ that the effective hopping is a function of 1(ru22− ), an where the operator cc+ (), cc+ (), cc+ () dd11σ σ dd22σ σ ooσ σ off-center distortion will therefore always lower the magnetic energy. Moreover, regardless the specific form creates (annihilates) a spin σ at M1, M2 and O site, respectively. ε d and ε o are the on-site energies

2 Figure 2 | Energy difference as a function of O offcenter displacement. a, Calculated energy differences between zero-temperature AFM and FM state of MnO for a series of O off-center displacement at different biaxial strains: no strain (red square), -3% (blue triangle), -5% (green circle). b, Calculated lattice instabilities of MnO under different biaxial strains at zero-temperature AFM and high-temperature PM states (magenta open star). Zero-temperature results come from DFT+U calculations with U=3.0 eV, and high-temperature result is from LDA+DMFT simulation with U=3.0 eV and T=300 K. 2 28 of J (teff), the corresponding energy gain is still ～ u , state . Thus we utilize the DFT+U method to check as in the case of the single band superexchange. This is whether a reasonable biaxial strain can induce a nontrivial conclusion, since now we see that both the ferroelectricity in the ground state of MnO. Here, we magnetic exchange energy change and the ordinary concern the off-center (001)-direction motion of O ions. elastic energy change ( ∼ Ku2 2 ) are of the second order Our numerical results show that regardless the values of the off-center displacement. Therefore, when the of U and O-displacement, the ground state of MnO is anion located between magnetic ions is shifted away always AFM. One important parameter relevant to the from the center-symmetric point, the increase in elastic magnetic ordering energy is the energy difference between AFM and FM state (ΔE=EFM-EAFM). For U energy may be compensated by the decrease of 29 magnetic exchange energy, consequently, forming an = 3 eV , the results are shown in Fig. 2a. It is clear electric dipole and resulting in ferroelectricity, as Fig. 1 from the figure that ΔE enhances with increasing O shows. displacement, which directly supports our conclusion: The ferroelectricity is long believed to originate off-center motion of the anion between two magnetic from a delicate balance between the short-range cations would enhance the magnetic exchange favoring the undistorted paraelectric structure and the interaction. As clearly shown in Fig.2b, at strain-free long range Coulomb interactions favoring the state (0%), the curve for total energy change with the O ferroelectric phase26,27. Now we see that indirect off-center displacement is parabolic, which indicates a magnetic exchange, which is short-range in nature, paraelectric state. When the compressive strain applies, provide another driving force for the off-center atomic the bottom of the curve becomes flat. Then, at a motion. To numerically prove the above conclusion, reasonable strain, e. g., -3%, ferroelectricity is induced we then carried out a series of first-principles in the ground state of MnO. At even larger calculations (see Methods). For the purpose of compressive strain (-5%), the energy curve is clearly in avoiding the complication due to complex magnetic the shape of double-well － the symbol of order and achieving an unambiguous result, we choose ferroelectricity. The calculated polarization is about 2 the simple insulator MnO with rock-salt structure, 0.38 C/m , which is comparative to that of typical which is AFM at low temperature (~ 120 K) and perovskite ferroelectrics. Furthermore, our numerical paramagnetic (PM) under normal conditions, as the results show that enlarging U will suppress the demonstrating system. magnetic ordering energy, and consequently the critical Although cannot deal with the high-temperature strain for the occurrence of ferroelectricity will PM state of MnO, DFT+U scheme is adequate for the increase. This is expected from our theory [see Eq. (4)], zero-temperature magnetically ordered insulating and again demonstrates the importance of magnetic

3 ordering energy for the ferroelectricity. For the zero-temperature magnetically ordered To confirm that the above ferroelectric instability is insulating state, we use the projector augmented wave of magnetic origin instead of other mechanism30, (PAW) method implemented in the Vienna Ab-Initio LDA+DMFT calculations are then carried out. We use Simulation Package (VASP)32. The exchange-correla- the highly accurate continue time quantum Monte tion potential is treated in the generalized gradient Carlo (CT-QMC) as the impurity solver31. For low approximation (GGA). We use the energy cut-off of temperature, CT-QMC would be very demanding on 500 eV for the plane wave expansion of the PAWs and the computer resource, we thus only consider the high a 10 x 10 x 10 Monkhorst-Pack grid for k-point temperature PM phase of MnO by LDA+DMFT. All of sampling in the self-consistent calculations. We vary our calculations, regardless the temperature (T=200, the onsite Coulomb energy U from 1 to 6 eV and 300 and 400 K) and U, give the same qualitative confirm its value do not change our qualitative conclusion: losing the magnetic ordering will suppress conclusions. The epitaxial strain is defined as (a − the off-center displacement, and even a large strain a0)/a0, where a is the in-plane lattice parameter and a0 (-5%) can no longer induce the ferroelectric instability is the theoretical equilibrium lattice constant in cubic for PM phase (see line with star symbols in Fig. 2b, symmetry. The out-of-plane lattice parameter c is which is the result of T=300 K and U=3.0 eV). Thus optimized at every strain. The Berry phase technique is we unambiguously demonstrate that here the magnetic used to calculate ferroelectric polarizations. interaction is essential for the onset of ferroelectricity. We use the LDA+DMFT method28,31 to calculate the Speaking solely from the point of view of magnetic high temperature PM state. We use continue time exchange interaction, antiferroelectric phases are also quantum Monte Carlo as the impurity solver31, and possible states to lower the magnetic energy of the crossing check our results by the non-crossing system. We show here that these states are not approximation. Calculations are fully self-consistent in energetically favored by considering the electrostatic charge density, chemical potential, impurity level and energy. This is confirmed by our phonon frequency total energy. The line with star symbol in Fig.2b shows calculation on the paraelectric phase of a 2×2×2 the LDA+DMFT results for -5% strain with supercell of MnO (64 atoms) under different in-plane temperature equal to 300K and U equal to 3.0 eV. strains. We find that only the Au mode (vibration of Mn and O ion along z-axis) has been gradually Acknowledgements XW thanks Kristjan Haule for softened when the compressive epitaxial strain useful discussion. The work was supported by the increases. No lattice instability corresponds to an National Key Project for Basic Research of China antiferroelectric phase. Therefore, the off-center O (Grant no. 2011CB922101 and 2010CB923404), movement will result in ferroelectricity instead of NSFC (Grant No. 91122035, 61125403, 50832003); antiferroelectricity. PCSIRT, NCET, Shanghai ShuGuang. Computations Our finding, that magnetic exchange interaction were performed at the ECNU computing center. S.Y.S could induce ferroelectricity even in simple magnetic was supported by DOE Computational Material oxide insulators, is unexpected and exciting. As shown Science Network (CMSN) and DOE SciDAC Grant in the above analysis, the bonus coming with the No. SE-FC02-06ER25793. induced ferroelectricity would be the enhancing of the magnetic transition temperature. In addition, this Correspondence and requests for materials should be mechanism is ubiquitous, i.e., not confined in addressed to X. W or C.G. D. antiferromagnets. Therefore various magnetic systems could be potential multiferroics. References In summary, we have confirmed both analytically 1. Kimura, T. et al. Magnetic control of ferroelectric and numerically that indirect magnetic exchange, polarization. Nature 426, 55-58 (2003). contrary to what people previously thought, favors 2. Efremov, D. V., Van den Brink, J. & Khomskii, D. ferroelectricity even in collinear magnetic systems. Our I. Bond-versus site-centred ordering and possible research provides a new way to explain the coexistence ferroelectricity in manganites. Nat. Mater. 3, of ferroelectricity and magnetism and might be useful 853-856 (2004). to the search of novel multiferroics suitable of practical 3. Fiebig, M. Revival of the magnetoelectric effect. J. use. Phys. D: Appl. Phys. 38, R123-R152 (2005). 4. Eerenstein, W., Mathur, N. D. & Scott, J. F. Multiferroic and magnetoelectric materials. Nature Methods 442, 759-765 (2006).

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