Metallic ferroelectricity induced by anisotropic unscreened Coulomb interaction in LiOsO
3H. M. Liu,1Y. P. Du,1Y. L. Xie,1J.-M. Liu,1Chun-Gang Duan,2and Xiangang Wan1,*
1National Laboratory of Solid State Microstructures, School of Physics, Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
2Key Laboratory of Polar Materials and Devices, Ministry of Education, East China Normal University, Shanghai 200062, China (Received 27 October 2014; revised manuscript received 27 January 2015; published 20 February 2015)
As the first experimentally confirmed ferroelectric metal, LiOsO3 has received extensive research attention recently. Using density-functional calculations, we perform a systematic study on the origin of the metallic ferroelectricity in LiOsO3. We confirm that the ferroelectric transition in this compound is order-disorder-like.
By doing electron screening analysis, we unambiguously demonstrate that the long-range ferroelectric order in LiOsO3 results from the incomplete screening of the dipole-dipole interaction along the nearest-neighboring Li-Li chain direction. We conclude that highly anisotropic screening and local dipole-dipole interactions are the two most important keys to form LiOsO3-type metallic ferroelectricity.
DOI:10.1103/PhysRevB.91.064104 PACS number(s): 77.80.B−,61.50.Ah,71.20.−b,72.80.Ga I. INTRODUCTION
Ferroelectric (FE) instability can be explained by a del- icate balance between short-range elastic restoring forces supporting the undistorted paraelectric (PE) structure and long-range Coulomb interactions favoring the FE phase [1].
Itinerant electrons can screen the electric fields and inhibit the electrostatic forces; metallic systems are thus not expected to exhibit ferroelectric-like structural distortion. Despite the incompatibility, using a phenomenological theory, Anderson and Blount proposed in 1965 that metals can break inversion symmetry [2]. They found that the FE metal is possible through a continuous structural transition accompanied by the appearance of a polar axis and the disappearance of an inversion center [2]. Very recently, Puggioni and Rondinelli propose a microscopic mechanism about how to eliminate the incompatibility between metallicity and acentricity [3]. In 2004, Cd2Re2O7 had been proposed as a rare example of ferroelectric metals [4]; however, it was found that although this compound exhibits a second-order phase transition to a structure that lacks inversion symmetry a unique polar axis could not be identified [5], which does not fit the criteria about the FE metal.
In 2013, the first convincing success was achieved exper- imentally in LiOsO3 [6]. LiOsO3 remains metallic behavior while it undergoes a second-order phase transition from the high-temperature centrosymmetric R3c to a FE-like R3c structure at Ts = 140K [6]. Neutron and x-ray-diffraction studies showed that the structural phase transition involves the displacements of Li ions accompanying also a slight shift of O ions [6]. The electronic structure and lattice instability were studied by several groups [7–9]. It was found that the local polar distortion in LiOsO3is solely due to the instability of the A-site Li ion [7–9]. The importance of the Coulomb interaction among 5d electrons and the hybridization between oxygenp orbitals and Os emptyeg orbitals has also been emphasized by Giovannetti and Capone [9]. Despite these efforts devoted to understanding the origin of the FE-like structural transition in this metallic system, there are still two fundamental issues
*Corresponding author: [email protected]
that have not been clearly clarified. The first is the origin of the ferroelectric instability: is it displacive or order-disorder?
Second, as the FE-like phase transition of LiOsO3 occurs at a relatively high temperature (140 K), how can these local dipoles line up to form long-range order, as if there are no conduction electrons to screen the dipole interactions?
In this paper, based on the density-functional theory (DFT) calculations, we reveal the microscopic mechanism for the FE-like structural transition in LiOsO3. Our study shows that, different from other 5dtransition-metal oxides [10–12,14], for LiOsO3, the effect of spin-orbital coupling (SOC) is small and the electronic correlation is weak. Our comprehensive poten- tial surface calculations suggest that the structural transition is order-disorder-like. The most striking finding is that the electric screening in LiOsO3 is highly anisotropic despite its metallic nature. Consequently, the dipole-dipole interactions are unscreened along certain directions, which results in the long-range FE order at considerably high temperature. This is in sharp contrast to the case in the displacive type FE compounds, where the FE structural transition is usually driven by hybridization or a lone pair [17], and consequently the change of the electric dipole (namely, the atomic motion) will modify the valence band significantly. If such displacive type FE compounds become metallic, the interactions between their electric dipoles will be strongly screened out, and the metallic FE phase is highly unlikely to occur.
Before the formal presentation of the calculated results, we would like to first discuss our strategy to study the electric screening effect. As is well known, the major difference be- tween the insulator and metal is that there are free nonlocalized electrons in metals, whereas in insulators there are only bound electrons. Consequently, electrostatic forces will be strongly screened by the itinerant electron in the metallic system. The screening effect actually can be described as the electron charge difference induced by a perturbation such as the change of the dipole or external electric field [18]. However, seldom efforts have been carried out to study the screening effect in the bulk metal, as people generally believe there is no macroscopic electric field inside metals. In the current study, we try to study the electric response to a local dipole in the bulk metal. This is done by analyzing the charge difference before and after the local dipole is introduced. Such a strategy provides an explicit
III. RESULTS AND DISCUSSIONS
There are ten atoms in the primitive unit cell of LiOsO3. The atomic arrangements are sketched in Fig. 1. In the R3c PE structure, the Os atoms are at the centers of the oxygen octahedrons, while Li atoms are centered between two adjacent Os atoms along the polar axis on average. Using the experimental lattice parameters, we optimize all independent internal atomic coordinates of the FE structure until the Hellman-Feynman forces on every atom are converged to less than 1 meV/ ˚A; the optimized internal atomic coordinates are listed in TableI, and the experimental PE and FE structures have also been presented in Table I for comparison. The calculated results coincide with previous experimental and calculated results [6–9], and the FE structural phase transition mainly involves the displacements of Li atoms: Li atoms shift along the polar axis aboutd∼0.47 ˚A from the mean positions of the PE phase [see gray arrowd in Fig.1(a)] and O atoms slightly displace about 0.056 ˚A [6,7,9].
Based on calculated lattice structure, we first perform standard GGA calculation to see the basic features of the electronic structure of LiOsO3. We show the total and partial density of states (DOS) in Fig.2. Our results are consistent with previous work [6,7]. The energy range−9.0 to−2.4 eV is
FIG. 1. (Color online) Primitive unit cell of (a) PE and (b) FE phases of LiOsO3. The green, blue, and red balls are the Li, Os, and O ions, respectively.dand−dcorrespond to the displacements of Li ions along the polar axis.
dominated by the O-2porbital with an additional contribution from the Os-5d state, indicating hybridization between them.
Notice that the phase transition involves a slight shift of O ions; the hybridization between Os-5d and O-2pstates may have a finite contribution to the ferroelectric-like transition as discussed in Ref. [9]. As shown in Fig.2(c), Li is highly ionic and its bands are far from the Fermi level. The Os atom is octahedrally coordinated by six O atoms, making the Os 5d band split into thet2g andeg states, and thet2g bands are located from−2.2 to 1.2 eV, as shown in Fig.2(c). Due to the extended nature of 5dstates, the crystal splitting between t2g andeg states is large, and theeg states are located about
FIG. 2. (Color online) (a) The total DOS patterns of LiOsO3 in PE (blue) and FE (pink) phases. The partial DOS of (b) Li-1s, (c) Os-5d, and (d) O-2p states in PE (blue) and FE (pink) phases, respectively. The Fermi energy is positioned as zero.
FIG. 3. (Color online) Band structure of LiOsO3, shown along the high-symmetry directions. (a) GGA. (b) GGA+SO.
3.0 eV higher than the Fermi energy and disperse widely.
As shown in the comparison of DOS of PE and FE, the electronic structures almost do not change during the phase transition, which is consistent with the previous theoretical work [7]. It is worth mentioning that this is quite different from prototype FE systems such as BaTiO3, in which hybridization is necessary for the FE phase transitions [22–24]. Thus we think the hybridization is not the major driving force for the structural instability in LiOsO3.
It is well known that the SOC of 5d electrons is very strong [25] and usually changes the 5d band dispersion significantly, as demonstrated in Sr2IrO4 [10], pyrochlore iridates, and spinel osmium [11,12]. In the case of LiOsO3, as shown in Fig.2, the O-2porbitals are almost fully occupied, while the bands of Li are mainly empty; thus Os occurs in its 5+valence state and there are basically three electrons in its t2gband. Since thet2gband is half filled, it is natural to expect the effect of SOC to be small despite the large strength of SOC [13,15,16]. This has been confirmed by the comparison of the band structures obtained in the presence and absence of SOC. The SOC slightly enhances thet2g bandwidth as shown in Fig.3. Besides this, the band-structure difference around the Fermi level is small.
Although the 5d orbitals are spatially extended, it has been found that the electronic correlations are important for 5d transition-metal oxides [10–14]. The values of electronic correlationUobtained in Sr2IrO4/Ba2IrO4are between 1.43 and 2.35 eV [14]. Although the accurate value of U is not known for this system, we generally expect screening to be larger in three-dimensional systems than in two-dimensional systems like Sr2IrO4. Furthermore, the Os-Os bond length of LiOsO3 is shorter than that of NaOsO3; thus we expect that theU in LiOsO3 is even smaller than in NaOsO3, for which U is around 1 eV [13]. Here, we estimate the Sommerfeld coefficient based on the numerical DOS at the Fermi level.
Our numerical result (6.1 mJ mol−1K−2) is just slightly less than that of the experimental one (γ =7.7 mJ mol−1K−2) [6], which indicates that the electronic correlation is indeed weak in LiOsO3.
One fundamental issue about this system is the mechanism for the ferroelectric instability: is it displacive or order- disorder? Using comprehensive total-energy calculations, we
FIG. 4. (Color online) The olive, blue, and red curves represent the potential-energy changes with respect to O displacements only, Li displacements only, and the coupled displacements of the Li and O ions. The total energy and displacements of PE states are set as zero. The displacements of corresponded FE states are set as 100%.
now try to solve this issue. Following the common procedure used in the study of FE structures, we first calculate the potential-energy profile along different displacive soft modes, i.e., the evolution paths from the PE structure to the FE structure. The results, as shown in Fig. 4, suggest that the energy difference between the PE and FE structures is
Li
4 Os O 5 O 2
Li
(a) (b)
1
3 O Os Li 5 O 2
Li Li
1 5
Li Os
6 Os
1)
Li
(0 0
Li
1 3
Li5
Li1 3
Li5
LiO O O
Os Os Os
(c)
Li Os Li Os Os
)
Os Li Os Os Li
(0 0 1)
O Os O Os O Os
FIG. 5. (Color online) Partial electron densities contour maps for PE LiOsO3taken through (a) [1 -1 0] and (b) [2 -1 0] planes. Contour levels shown are between 0 (blue) and 0.3 e/ ˚A3 (red). (c) Charge density difference between FE and PE structures for Li pair 1, 3, and 5 through the [1 -1 0] plane. See text for details. Contour levels shown are between−0.004 (blue) and 0.004 e/ ˚A3(red). The (0 0 1) direction here is the same as the (1 1 1) direction in Fig.1.
charge transport above the transition temperature [6], which is possibly caused by the scattering induced by disorder of Li off-center displacement. For order-disorder transition, the Li atoms oscillate between the double wells, and the potential wells remain basically unchanged throughout the phase transition; thus we expect that there is no softening mode in the Raman spectra of LiOsO3. Therefore, a Raman measurement is useful to clarify this issue.
As mentioned above, the Li ions in LiOsO3 favor an off-center displacement and form local dipoles as shown in Fig.1. Thus it is a puzzle why the local electric dipoles in different unit cells can interact with each other and form a long-range order at 140 K, noticing that the distance between them is far (even the nearest-neighbor dipole distance is larger than 3.5 ˚A) and the DOS at the Fermi level is rather large (Fig.2). We find that the bands located below−10 eV are quite narrow and have negligible hybridization with other bands.
The electrons at these bands are tightly bounded with the ion, and thus almost do not change with the motion of the Li ion;
namely, these electrons almost have no contribution to the electric screening effect. On the other hand, the displacements of Li ions just slightly affect the Os-5d and O-2p electrons, as shown in Fig. 2. To have a straightforward view of the charge distribution, we sketched the electron densities of PE LiOsO3 arising from states between −10 eV and the Fermi level in Figs.5(a)and5(b). There are two distinctive characters in these two figures. One is that the electronic density is relatively high between the Os and O ions, which again indicates the strong hybridization between Os-5p and O-2pstates. The second is that there is almost no conduction charge at all in a relative large space around the Li ions; i.e., the Li ion is literally a bareion. We will demonstrate later that the later character directly results in incomplete electric screening of dipole-dipole interactions and forms long-range dipole ordering.
Following the previously described procedure, we then demonstrate the screening effect by doing charge difference calculation. Since the FE-like transition basically involves displacements of Li ion, the change of local dipole can be approximated by the Li movement from the PE structure and the dipole interactions can be labeled as Li-Li pairs. In Fig.5, we use, e.g., symbols 1, 2, and 3 to denote the Li-Li pairs with the first-, second-, and third-nearest distance between them.
As is clear in Fig.5(a), there is almost no conduction charge distribution between pair 1 (red solid line), and thus it is natural to expect the screening effect for the nearest dipole-dipole
unchanged, indicating that the Os-5dand O-2phybridizations are neither important for the electric dipole interaction nor affected by the Li dipoles. The most interesting thing is, as shown in the left panel of Fig.5(c), that there is almost no modification of charge distribution in pair 1 at all. This clearly demonstrates that dipole interaction in pair 1 is only slightly screened. We then apply the same strategy to study other Li pairs. For pair 2 [Fig.5(b)], as there is an O atom between two Li ions, we observe noticeable screening to prevent the direct dipole interaction. However, for pair 3, the dipole interaction is again not fully screened, as shown in the middle panel of Fig.5(c). Detailed analysis indicates that this is because this Li pair is 0.65 ˚A away from the O ion plane. For other pairs, as there is either an O or Os atom between two Li ions, the electric screening effect is strong. The example of pair 5, which is also at the [1−1 0] plane like pairs 1 and 3, is shown in the right panel of Fig.5(c)for comparison.
Above, we have provided a qualitative picture about why locale dipole interactions are not fully screened in LiOsO3. To give a more quantitative explanation, we try to estimate the interaction strength between the local electric dipole moments. Again using the above adopted supercell, we obtain coupling constant Ji (i=1–6) between the ith Li pairs from the energy difference between the local FE and antiferroelectric (AFE) states (ith Li pairs are AFE ordered), i.e., Ji=[EFE−EAFE]/2. The Li-Li distances di of each pair and the obtained interaction parameterJi are listed in TableII. Consistent with the above screening discussions,J1 andJ3are much larger than all other interactions, indicating that the dipole interactions are highly anisotropic. Despite d6 being longer than d2, d4, and d5, J6 is considerably larger than J2, J4, and J5, as shown in Table II, which also indicates the anisotropic screening effect in this metallic compound.
To show that these dipole interaction parameters obtained in the above procedure are reasonable, we perform Monte Carlo (MC) simulations using an effective Ising-like Hamiltonian:
H=
iJiDmDn, whereJi is the coupling constant between dipole momentsDm andDn. The obtained phase transition temperature is 210 K with only J1 considered and 330 K with bothJ1 andJ3 considered, which is reasonably higher than the experimental Ts (140 K), and the overestimation may come from the rigid dipole model used in our MC simulations. This, again, shows that our explanation on the mechanism of the lineup of local dipoles in metallic LiOsO3is self-consistent.
FIG. 6. (Color online) Potential-energy surface based on experimental displacements [6] using other available exchange-correlation functionals: (a, b) PW91. (c–e) LDA. (f–j) PBE. For each exchange-correlation functional, the calculations are performed using pseudopotentials with different valence electrons, and the used valence states for Li, Os, and O are inserted in related figures. The olive, blue, and red curves represent the potential-energy changes with respect to O displacements only, Li displacements only, and the coupled displacements of the Li and O ions. The total energy and displacements of PE states are set as zero. The displacements of corresponded FE states are set as 100%.
IV. SUMMARY
In summary, by performing DFT calculation, we investigate the microscopic mechanism of ferroelectricity in metallic LiOsO3. We find that, in contrast to other 5d transition-metal oxides, for LiOsO3, the effect of SOC is small and the
electronic correlation is weak. We propose that the structural phase transition is of order-disorder type, and we find that the electronic states at the Fermi level are only weakly coupled to the ferroelectric-like transition, which makes the metallic ferroelectricity become possible, as discussed by Ref. [3].
In addition to that [3], by using a straightforward method,
a rather high temperature. We also want to emphasize that the above picture implied that the ferroelectric-like transition should be of order-disorder instead of displacive type. This is because the displacive-type ferroelectric transition occurring in ABO3 perovskite structures is generally triggered by the hybridization between the B and O electronic states or the lone pair in the A site. In either case the change of the electric dipole will modify the valence band consid- erably. Therefore in a displacive system the dipole-dipole interaction will be strongly screened out in the metallic phase.
We also calculate the potential-energy surface based on experimental displacements [6] by using available exchange-correlation functionals (such as local-density approximation, Perdew-Becke-Erzenhof, and PW91), and the results are presented in Fig.6. In each case, a densek mesh and huge energy cutoff for the basis set are carefully checked for better convergence. Adopting experimental coordinates will obtain unreasonable results, as the well depth caused by the sole Li ion movements is even larger than that of both the Li and O moments.
[1] W. Cochran,Adv. Phys.9,387(1960).
[2] P. W. Anderson and E. I. Blount,Phys. Rev. Lett.14,217(1965).
[3] D. Puggioni and J. M. Rondinelli,Nature Commun.5, 3432 (2014).
[4] I. A. Sergienko, V. Keppens, M. McGuire, R. Jin, J. He, S. H.
Curnoe, B. C. Sales, P. Blaha, D. J. Singh, K. Schwarz, and D.
Mandrus,Phys. Rev. Lett.92,065501(2004).
[5] V. Keppens,Nature Mater.12,952(2013).
[6] Y. Shi, Y. Guo, X. Wang, A. J. Princep, D. Khalyavin, P. Manuel, Y. Michiue, A. Sato, K. Tsuda, S. Yu, M. Arai, Y. Shirako, M.
Akaogi, N. Wang, K. Yamaura, and A. T. Boothroyd,Nature Mat.12,1024(2013).
[7] H. Sim and B. G. Kim,Phys. Rev. B89,201107(R)(2014).
[8] H. J. Xiang,Phys. Rev. B90,094108(2014).
[9] G. Giovannetti and M. Capone,Phys. Rev. B90,195113(2014).
[10] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park, C. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, J.-H. Park, V. Durairaj, G.
Cao, and E. Rotenberg,Phys. Rev. Lett.101,076402(2008);
H. Watanabe, T. Shirakawa, and S. Yunoki,ibid.105,216410 (2010).
[11] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov,Phys.
Rev. B83,205101(2011); B. J. Yang and Y. B. Kim,ibid.82, 085111(2010).
[12] X. Wan, A. Vishwanath, and S. Y. Savrasov,Phys. Rev. Lett.
108,146601(2012).
[13] Y. Du, X. Wan, L. Sheng, J. Dong, and S. Y. Savrasov,Phys.
Rev. B85,174424(2012).
[14] R. Arita, J. Kuneˇs, A. V. Kozhevnikov, A. G. Eguiluz, and M.
Imada,Phys. Rev. Lett.108,086403(2012).
[15] Y. G. Shi, Y. F. Guo, S. Yu, M. Arai, A. A. Belik, A. Sato, K. Yamaura, E. Takayama-Muromachi, H. F. Tian, H. X. Yang, J. Q. Li, T. Varga, J. F. Mitchell, and S. Okamoto,Phys. Rev. B 80,161104(R) (2009).
[16] S. Calder, V. O. Garlea, D. F. McMorrow, M. D. Lumsden, M. B. Stone, J. C. Lang, J.-W. Kim, J. A. Schlueter, Y. G. Shi, K. Yamaura, Y. S. Sun, Y. Tsujimoto, and A. D. Christianson, Phys. Rev. Lett.108,257209(2012).
[17] D. Khomskii,Physics2,20(2009).
[18] C.-G. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S.
Jaswal, and E. Y. Tsymbal,Phys. Rev. Lett.101,137201(2008).
[19] G. Kresse and J. Hafner,Phys. Rev. B47,558(R) (1993).
[20] G. Kresse, and J. Furthm¨uller,Phys. Rev. B54,11169(1996).
[21] J. P. Perdew, K. Burke, and M. Ernzerhof,Phys. Rev. Lett.77, 3865(1996).
[22] R. E. Cohen,Nature (London)358,136(1992).
[23] N. A. Hill,J. Phys. Chem. B104,6694(2000).
[24] P. Ghosez, J.-P. Michenaud, and X. Gonze,Phys. Rev. B58, 6224(1998).
[25] L. F. Mattheiss,Phys. Rev. B13,2433(1976).
[26] I. Inbar and R. E. Cohen,Phys. Rev. B53,1193(1996).
