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ISSN 1063 7834, Physics of the Solid State, 2015, Vol. 57, No. 2, pp. 331336. © Pleiades Publishing, Ltd., 2015. Original Russian Text © A.I. Lebedev, 2015, published in Fizika Tverdogo Tela, 2015, Vol. 57, No. 2, pp. 316320.

FERROELECTRICITY

Ferroelectric Properties of RbNbO3 and RbTaO3
A. I. Lebedev
Moscow State University, Moscow, 119991 Russia e mail: swan@scon155.phys.msu.ru
Received July 30, 2014

Abstract--Phonon spectra of cubic rubidium niobate and rubidium tantalate with the perovskite structure are calculated from first principles within the density functional theory. Based on the analysis of unstable modes in phonon spectra, symmetries of possible distorted phases are determined, their energies are calcu lated, and it is shown that R3m is the ground state structure of RbNbO3. In RbTaO3, the ferroelectric insta bility is suppressed by zero point lattice vibrations. For ferroelectric phases of RbNbO3, spontaneous polar ization, piezoelectric, nonlinear optical, electro optical, and other properties as well as the energy band gap in the LDA and GW approximations are calculated. The properties of rhombohedral RbNbO3 are compared with those of rhombohedral KNbO3, LiNbO3, and BaTiO3. DOI: 10.1134/S1063783415020237

The possibility of ferroelectricity in rubidium nio bate and rubidium tantalate with the perovskite struc ture was discussed by Smolenskii and Kozhevnikova [1] and then by Megaw [2] in the early 1950s. In [1], the authors referred to unpublished data by V.G. Prokhvatilov who detected the tetragonal RbTaO3 phase with a = 3.92 е, c = 4.51 е exhibiting a phase transition near 520 K; in [2], these data were simply cited. However, further studies have shown that, unlike lithium, sodium, and potassium niobates, RbNbO3 and RbTaO3 crystallize in individual crystal structures with triclinic P 1 symmetry for RbNbO3 and monoclinic C2/m one for RbTaO3 [35] when pre pared at atmospheric pressure. To obtain these materi als with the perovskite structure, they should be pre pared at high pressures (6590 kbar) [6]. Due to the difficulties in synthesis of RbNbO3 and RbTaO3 with the perovskite structure, the properties of these crys tals have been studied very little. The phase diagrams of Rb2ONb2O5 and Rb2O Ta2O5 systems were studied in [7, 8]. RbNbO3 is formed by the peritectic reaction and decomposes above 964°C [7]. RbTaO3 decomposes above 600°C probably due to the peritectic reaction too [8]. Rubid ium containing ferroelectric materials in the BaNb2O6NaNbO3RbNbO3 system with the tung sten bronze structure have high electro optical prop erties that substantially exceed those of lithium nio bate [9, 10]. In [11], the possibility of using rubidium niobate and rubidium tantalate for photoelectrochem ical decomposition of water was discussed. In [12], it was proposed to use the delamination of RbTaO3 structure to produce porous TaO3 nanomembranes

with pore sizes of 1.3 в 0.6 and 1.1 в 1.1 е, which can be used for selective filtration of lithium ions. The lack of knowledge on the properties of com pounds under consideration appears, in particular, in contradictory data on the ferroelectric properties of RbTaO3. For example, the existence of the phase tran sition at 520 K in the tetragonal phase was reported in [1], whereas the data of [6] showed that RbTaO3 pre pared at high pressure has the cubic perovskite (or close to it) structure. At 300 K, the structure of RbNbO3 is similar to that of the orthorhombic BaTiO3, and the data of differential thermal analysis indicate phase transitions in it at 15, 155, and 300°C [6]. In this work, the equilibrium structures of RbNbO3 and RbTaO3 were determined from first principles calculations, and spontaneous polarization, dielectric constant, piezoelectric and elastic moduli, nonlinear optical and electro optical properties as well as the energy band gaps in the LDA and GW approximations were calculated for these crystals. The first principles calculations were performed within the density functional theory using the ABINIT software [13]. The exchangecorrelation interaction was described in the local density approximation (LDA). The optimized norm conserving pseudopo tentials for Nb, Ta, and O atoms used in these calcula tions were taken from [14]. The non relativistic pseudopotential for the Rb atom (electronic configu ration 4s24p65s0) was constructed according to the scheme of [15] using the OPIUM program [16] with the following parameters: rs = 1.68, rp = 1.72, rd = 1.68, qs = 7.07, qp = 7.27, qd = 7.07, rmin = 0.01, rmax = 1.52, and Vloc = 1.58 a.u. (for notations, see [17]). The testing of the Rb pseudopotential on the P 1 phase of

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LEBEDEV

Table 1. Calculated lattice parameters and atomic coordi nates in RbNbO3 structures Atom Position x y z

Phase P 1 a = 5.0816 е, b = 8.3047 е, c = 8.7916 е, = 114.0625°, = 93.3891°, = 95.1160° Rb2 Rb1 Rb3 Nb1 Nb2 O1 O2 O3 O4 O5 O6 1a 1b 2i 2i 2i 2i 2i 2i 2i 2i 2i +0.00000 +0.00000 +0.41251 +0.49674 +0.02746 +0.10125 +0.23664 +0.28650 +0.29777 +0.33951 +0.78420 +0.00000 +0.00000 +0.70257 +0.28138 +0.51037 +0.39078 +0.42747 +0.71922 +0.37205 +0.05579 +0.27571 +0.00000 +0.50000 +0.09488 +0.35602 +0.30988 +0.82160 +0.50994 +0.45293 +0.19910 +0.27137 +0.22313

Phase Pm3m a = 4.0291 е Rb Nb O 1a 1b 3c +0.00000 +0.50000 +0.00000 +0.00000 +0.50000 +0.50000 +0.00000 +0.50000 +0.50000

Phase P4mm a = 4.0037 е, c = 4.1592 е Rb Nb O1 O2 1a 1b 2c 1b +0.00000 +0.50000 +0.50000 +0.50000 +0.00000 +0.50000 +0.00000 +0.50000 0.00336 +0.51818 +0.47446 0.03654

Phase Amm2 a = 3.9928 е, b = 5.7742 е, c = 5.7960 е Rb Nb O1 O2 2a 2b 4e 2a +0.00000 +0.50000 +0.50000 +0.00000 +0.00000 +0.00000 +0.25496 +0.00000 0.00341 +0.51447 +0.22842 +0.47735

Phase R3m a = 4.0571 е, = 89.8945° Rb Nb O 1a 1a 3b 0.00308 +0.51193 0.02115 0.00308 +0.51193 +0.48415 0.00308 +0.51193 +0.48415

RbNbO3 and the C2/m phase of RbTaO3, which are stable at atmospheric pressure, showed its sufficiently high quality: the calculated lattice parameters and atomic coordinates in these phases (see Tables 1 and 2)

are in good agreement with the experimental data [3, 5]; small underestimates of the calculated lattice parameters are characteristic of the LDA approxima tion used in this work. The lattice parameters and equilibrium atomic positions in the unit cells were determined from the condition when the residual forces acting on the atoms were below 5 в 106 Ha/Bohr (0.25 meV/е) in the self consistent calculation of the total energy with an accuracy better than 1010 Ha. The maximum energy of plane waves was 30 Ha for RbNbO3 and 40 Ha for RbTaO3. Integration over the Brillouin zone was per formed using a 8 в 8 в 8 MonkhorstPack mesh. The spontaneous polarization in ferroelectric phases was calculated by the Berry phase method. The phonon spectra, dielectric constants, piezoelectric and elastic moduli were calculated within the density functional perturbation theory similarly to [17]. Nonlinear opti cal and electro optical properties were calculated using the technique described in [18]. All physical properties presented in this work were calculated for the theoretical lattice parameter. The phonon spectrum of RbNbO3 in the cubic Pm3m phase is shown in Fig. 1. This spectrum con tains a band of unstable modes characteristic of ferro electric chain instability which was first observed in KNbO3 [19]. At the center of the Brillouin zone, this mode has the 15 symmetry, is triply degenerate, and describes the ferroelectric distortion of structure. The structures appearing upon condensation of the X5 and M '3 modes are characterized by antiparallel orienta tion of polarization in neighboring ...ONbO... chains. The energies of all RbNbO3 phases formed upon condensation of the above unstable modes are given in Table 3. Among these phases, the R3m phase has the lowest energy. The phonon spectrum calculations for the R3m phase show that the frequencies of all optical phonons at the center of the Brillouin zone and at high symmetry points at its boundary are positive; the determinant and all leading principal minors con structed of elastic moduli tensor components are also positive. This means that the R3m phase is the ground state structure of RbNbO3. The calculated lattice parameters and atomic coordinates in this phase are given in Table 1. As the same sequence of phases as in BaTiO3 is supposed in rubidium niobate with the per ovskite structure [6], the lattice parameters and atomic coordinates in two other ferroelectric phases are also given in this table. The lattice parameters calculated for the orthorhombic RbNbO3 are in good agreement with the experimental data obtained at 300 K (a = 3.9965 е, b = 5.8360 е, and c = 5.8698 е [6]). In RbTaO3, the frequency of unstable 15 phonon in the phonon spectrum (Fig. 2) and the energy gain resulting from the transition to ferroelectric phases (Table 4) are rather low; so it is necessary to addition
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ally test the stability of the ferroelectric distortion with respect to zero point lattice vibrations. For this pur pose, we used the technique proposed in [20]. The energy gain resulting from the transition from the Pm3m phase to the R3m phase is E0 = 1.90 meV and the unstable phonon frequency at the point in the Pm3m phase is = 84 cm1. As the energy ratio h/E0 5.51 exceeds the critical value of 2.419 obtained in [20], the energy of the lowest vibrational state in a two well potential appears above the upper point of the energy barrier separating the potential wells, and the ferroelectric ordering is suppressed by zero point vibrations. Therefore, the only stable phase of RbTaO3 with the perovskite structure is the cubic phase. The calculated lattice parameter of this phase is given in Table 2; its value is in satisfactory agreement with the experimental data (a = 4.035 е [6]). It is known that the formation of phases with hex agonal BaNiO3 (polytype 2H), hexagonal BaMnO3 (polytype 4H), hexagonal BaTiO3 (polytype 6H), and rhombohedral BaRuO3 (polytype 9R) structures is characteristic of ABO3 perovskites with the tolerance factor t > 1, and the studied compounds belong to this class. Our calculations showed that the energies of these phases for both rubidium compounds are appre ciably higher than the cubic phase energy (Tables 3 and 4). These results explain why it was impossible to observe the transition of RbNbO3 to the hexagonal structure [6] upon heating, by analogy to that occur ring in BaTiO3. The high energies of these phases, in particular, the 2H phase, are probably caused by larger sizes and strong electrostatic repulsion of Nb5+ ions which occupy face sharing octahedra in these struc tures. We consider now some properties of ferroelectric RbNbO3. The calculated polarization in RbNbO3 is 0.46 C/m2 in the P4mm phase and 0.50 C/m2 in the

Table 2. Calculated lattice parameters and atomic coordi nates in RbTaO3 structures Atom Position x y z

Phase C 2/m a = b = 6.3396 е, c = 8.0171 е, = 86.1031°, = 93.8969°, = 96.8997° Rb1 Rb2 Ta1 Ta2 O3 O1 O2 O4 4i 4g 4h 4i 8j 8j 4i 4i +0.16000 +0.26494 +0.31104 +0.23924 +0.27303 +0.54749 +0.16752 +0.37705 0.16000 +0.26494 +0.31104 0.23924 +0.04639 0.21209 0.16752 0.37705 +0.73758 +0.00000 +0.50000 +0.30214 +0.39643 +0.28704 +0.08653 +0.55235

Phase Pm3m a = 3.9846 е Rb Ta O 1a 1b 3c +0.00000 +0.50000 +0.00000 +0.00000 +0.50000 +0.50000 +0.00000 +0.50000 +0.50000

Amm2 and R3m phases; these values slightly exceed the calculated polarization in the same phases of KNbO3 (0.37, 0.42, and 0.42 C/m2, respectively). The static dielectric tensor in the R3m phase is character ized by two eigenvalues: || = 21.1 and = 35.8; the optical dielectric tensor eigenvalues are || = 5.31 and = 5.91. In the cubic phase, the elastic moduli are C11 = 412 GPa, C12 = 84 GPa, and C44 = 102 GPa; the bulk modulus is B = 193.5 GPa. The nonzero compo
1000 800 600 , cm1 400 200
0 0

800 600
1

400 200 0 -200 5 X 3' M 15 R M

, cm-

0 5 -200 X M 15 R M

Fig. 1. Phonon spectrum of RbNbO3 in the cubic Pm3m phase. Labels near curves indicate the symmetry of unsta ble modes. PHYSICS OF THE SOLID STATE Vol. 57 No. 2 2015

Fig. 2. Phonon spectrum of RbTaO3 in the cubic Pm3m phase. Labels near curves indicate the symmetry of unsta ble modes.

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LEBEDEV Table 4. Relative energies of low symmetry RbTaO3 phases formed from the cubic perovskite phase upon condensation of unstable phonons, phases with 6H, 4H, 9R, and 2H struc tures, and the C2/m phase prepared at atmospheric pressure (the most stable phase energy is in boldface) Phase Pm3m P4mm Amm2 R3m C2/m P63/mmc (6H) P63/mmc (4H) R3m (9R) P63/mmc (2H) Unstable mode
15 15 15

Table 3. Relative energies of low symmetry RbNbO3 phases formed from the cubic perovskite phase upon condensation of unstable phonons, phases with 6H, 4H, 9R, and 2H struc tures, and the P 1 phase prepared at atmospheric pressure (the most stable phase energy is in boldface) Phase Pm3m P4/nmm Pmma Cmcm P4mm Amm2 R3m P1 P63/mmc (6H) P63/mmc (4H) R3m (9R) P63/mmc (2H) Unstable mode M '3 X X
5 5

Energy, meV 0 31.3 34.8 38.7 46.5 57.0 58.6 +27.4 +121.4 +334.0 +568.1 +1752

Energy, meV 0 1.80 1.87 1.90 +38.5 +113.3 +352.2 +589.0 +1839

15 15 15

tive lattice contraction which always exists in the LDA calculations. The fact that the specific volume of the Pm3m phase is noticeably smaller than that of P 1 , C2/m, P63/mmc, and R 3 m phases suggests that under pressure the cubic perovskite phase will be the most stable one. To estimate the maximum value of the actual effective pressure, the lattice parameters and atomic positions in the C2/m structure of rubidium tantalate were calculated for different pressures and it was shown that the unit cell volume equal to the exper imental one at 300 K can be obtained at an isotropic pressure of 24.7 kbar. At this pressure, the enthalpy of the C2/m phase becomes lower than that of Pm3m by ~230 meV, i.e., becomes consistent with the exper imental data. At the above mentioned negative pres sure, the ratio h/E0 determining the stability of the ferroelectric phase in RbTaO3 becomes equal to 1.90, i.e., slightly less than the critical value of 2.419. How ever, if we take into account that the above negative pressure is obviously overestimated, since it includes the thermal expansion effect, we can suppose that, even taking into account the systematic error in the LDA lattice parameter determination, rubidium tan talate will remain cubic up to the lowest temperatures. The conclusion that RbTaO3 is an incipient ferro electric in which the ferroelectric ordering is sup pressed by zero point vibrations agrees with the data of [6], but contradicts the data of [1] in which the phase transition near 520 K was reported. We suppose that tantalum enriched phases (in particular, with the tungsten bronze structure [8]) could be formed in rubidium tantalate samples discussed in [1] because of the low temperature of the peritectic reaction, and this could result in the observed anomaly. In [11], the possibility of using various oxides with the perovskite structure, in particular RbNbO3 and
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nents of the tensors of piezoelectric effect ei, second order nonlinear optical susceptibility di, and linear electro optic (Pockels) effect ri in the R3m phase of rubidium niobate are compared with the correspond ing properties of other rhombohedral ferroelectrics in Table 5. We can see that the piezoelectric moduli in rhombohedral RbNbO3 (as well as in its other polar phases) are slightly lower than in KNbO3. The nonlin ear optical coefficients in RbNbO3 exceed the corre sponding values in KNbO3, although the d33 value in rubidium niobate is slightly lower than in lithium nio bate. As for the electro optical properties, in rhombo hedral RbNbO3 they are slightly lower than in KNbO3, but are notably superior to those of lithium niobate. In the orthorhombic phase (stable at 300 K), nonlinear optical properties of RbNbO3 are comparable to those of the same phase of potassium niobate: for example, the d33 modulus is 30.8 pm/V in RbNbO3 and 30.4 pm/V in KNbO3. In cubic RbTaO3, the optical dielectric constant is = 5.58. The static dielectric constant can be esti mated only in the rhombohedral phase as ~140. The elastic moduli in cubic rubidium tantalate are C11 = 466 GPa, C12 = 91.5 GPa, and C44 = 120 GPa; B = 216 GPa. The piezoelectric moduli, second order nonlinear optical susceptibility, and electro optical coefficients in the cubic phase are zero. An unexpected result of our calculations is that in both studied compounds the P 1 and C2/m phases which can be prepared at atmospheric pressure are metastable. This result is probably caused by an effec

PHYSICS OF THE SOLID STATE

FERROELECTRIC PROPERTIES Table 5. Nonzero components of the piezoelectric tensor ei (C/m2) and tensors of the second order nonlinear optical susceptibility di and the linear electro optic effect ri (pm/V) in rhombohedral phases of RbNbO3, KNbO3, LiNbO3, and BaTiO3 Coefficient RbNbO3 e e e e
11 15 31 33 11

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KNbO3 4.2 +6.8 +2.3 +3.1 +11.9 21.9 21.9 27.3 17.7 +39.2 +23.9 +40.6

LiNbO3 2.4 +3.5 +0.1 +1.1 +2.3 11.5 11.5 37.4 5.6 +17.1 +10.1 +27.3

BaTiO3 4.0 +7.3 +3.5 +5.1 +4.4 16.1 16.1 31.1 13.7 +43.3 +25.3 +48.9

3.0 +4.8 +2.4 +2.9 +12.7 23.6 23.6 29.4 12.8 +27.6 +18.0 +30.1

d d d d r r r

15 31 33

r11
15 31

33

RbTaO3, for development of photoelectrochemical solar cells was discussed. We calculated the band gap Eg in these compounds both in the LDA approxima tion and in the GW approximation that takes into account many body effects (the technique of the latter calculations was analogous to that used in [2123]). In LDA the LDA approximation, E g = 1.275 eV in cubic RbNbO3 when the spinorbit coupling is neglected; in P4mm, Amm2, and R3m phases, E
LDA g

The obtained values of Eg are appreciably smaller than those calculated in [11] for cubic phases (3.4 eV for RbNbO3 and 4.3 eV for RbTaO3). Some authors who studied rubidium niobate and rubidium tantalate have noticed their sensitivity to humidity. Evidently, this can be a serious obstacle for practical applications of these materials. However, we would like to note that this property is inherent to phases prepared at atmospheric pressure and having "loose" structures whose specific volume is 2628% larger than that of the perovskite phase. In [8], it was suggested that the effect is due to intercalation of water molecules into the "loose" structures, rather than to hydrolysis of these compounds. The possibility of pre paring RbTaO3 by hydrothermal synthesis [24] and the low rate of RbTaO3 ion exchange in HCl during its delamination [12] support this idea. This suggests that the considered compounds with the perovskite struc ture can be quite stable to humidity. Thus, the present calculations of RbNbO3 and RbTaO3 properties and their comparison with the properties of other ferroelectrics show that rubidium niobate is an interesting ferroelectric material with high nonlinear optical and electro optical properties, and rubidium tantalate is an incipient ferroelectric. The calculations presented in this work were per formed on the laboratory computer cluster (16 cores). ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research (project no. 13 02 00724). REFERENCES
1. G. A. Smolenskii and N. V. Kozhevnikova, Dokl. Akad. Nauk SSSR 76, 519 (1951). 2. H. D. Megaw, Acta Crystallogr. 5, 739 (1952). 3. M. Serafin and R. Hoppe, J. Less Common Met. 76, 299 (1980). 4. M. Serafin and R. Hoppe, Angew. Chem. 90, 387 (1978). 5. M. Serafin and R. Hoppe, Z. Anorg. Allg. Chem. 464, 240 (1980). 6. J. A. Kafalas, in Proceedings of the 5th Materials Research Symposium, Gaithersburg, Maryland, United States, October 1821, 1971 (NBS Spec. Publ., No. 364, 287 (1972)). 7. A. Reisman and F. Holtzberg, J. Phys. Chem. 64, 748 (1960). 8. H. Brusset, H. Gillier Pandraud, M. Chubb, and R. MahИ, Mater. Res. Bull. 11, 299 (1976). 9. D. F. O'Kane, G. Burns, E. A. Giess, B. A. Scott, A. W. Smith, and B. Olson, J. Electrochem. Soc. 116, 1555 (1969). 10. G. Burns, E. A. Giess, and D. F. O'Kane, US Patent No. 3 640 865 (1972).

is 1.314, 1.869,
LDA

= and 2.137 eV, respectively. In cubic RbTaO3, E g 2.175 eV when the spinorbit coupling is neglected. The valence band extrema in the cubic phase of both compounds are at the R point of the Brillouin zone, whereas the conduction band extrema are at the point. The calculations using the technique of [23] yield the spinorbit splitting of the conduction band edge SO = 0.111 eV for RbNbO3 and SO = 0.400 eV for RbTaO3; the spin orbit splitting of the valence band edge is absent. After correction for the conduc tion band edge shift (SO/3), the LDA values of Eg that take into account the spinorbit coupling are 1.238, 1.277, 1.832, 2.100, and 2.042 eV for four RbNbO3 phases and for cubic RbTaO3, respectively. In the GW approximation, the band gap when the GW spinorbit coupling is neglected is E g = 2.403, 2.616, 3.291, and 3.609 eV, respectively in cubic, tet ragonal, orthorhombic, and rhombohedral RbNbO3 and 3.302 eV in cubic RbTaO3. If the spinorbit cou pling is taken into account, these values decrease to 2.366, 2.579, 3.254, 3.572, and 3.169 eV, respectively.
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LEBEDEV 17. A. I. Lebedev, Phys. Solid State 51 (2), 362 (2009). 18. M. Veithen, X. Gonze, and P. Ghosez, Phys. Rev. B: Condens. Matter 71, 125107 (2005). 19. R. Yu and H. Krakauer. Phys. Rev. Lett. 74, 4067 (1995). 20. A. I. Lebedev, Phys. Solid State 51 (4), 802 (2009). 21. A. I. Lebedev, Phys. Solid State 54 (8), 1663 (2012). 22. A. I. Lebedev, J. Alloys Compd. 580, 487 (2013). 23. A. I. Lebedev, Phys. Solid State 56 (5), 1039 (2014). 24. D. Gompel, M. N. Tahir, M. Panthofer, E. Mugnaioli, R. Brandscheid, U. Kolb, and W. Tremel, J. Mater. Chem. A 2, 8033 (2014).

11. I. E. Castelli, D. D. Landis, K. S. Thygesen, S. Dahl, I. Chorkendorff, T. F. Jaramillo, and K. W. Jacobsen, Energy Environ. Sci. 5, 9034 (2012). 12. K. Fukuda, I. Nakai, Y. Ebina, R. Ma, and T. Sasaki, Inorg. Chem. 46, 4787 (2007). 13. The ABINIT code is a common project of the UniversitИ Catholique de Louvain, Corning Incorporated and other contributions. http://www.abinit.org/. 14. A. I. Lebedev, Phys. Solid State 52 (7), 1448 (2010). 15. A. M. Rappe, K. M. Rabe, E. Kaxiras, and J. D. Joan nopoulos, Phys. Rev. B: Condens. Matter 41, 1227 (1990). 16. Opium--Pseudopotential Generation Project. http:// opium.sourceforge.net/.

Translated by A. Kazantsev

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