aa r X i v : . [ c ond - m a t . m t r l - s c i ] J a n Ferroelectric properties of RbNbO and RbTaO A. I. Lebedev ∗ Physics Department, Moscow State University, Moscow, 119991 Russia (Dated: July 9, 2018)Phonon spectra of cubic rubidium niobate and rubidium tantalate with the perovskite structureare calculated from first principles within the density functional theory. Based on the analysis ofunstable modes in the phonon spectra, the structures of possible distorted phases are determined,their energies are calculated, and it is shown that R m is the ground-state structure of RbNbO . InRbTaO , the ferroelectric instability is suppressed by zero-point lattice vibrations. For ferroelectricphases of RbNbO , spontaneous polarization, piezoelectric, nonlinear optical, electro-optical, andother properties as well as the energy band gap in the LDA and GW approximations are calculated.The properties of the rhombohedral RbNbO are compared with those of rhombohedral KNbO ,LiNbO , and BaTiO .DOI: 10.1134/S1063783415020237 PACS numbers: 61.50.Ah, 63.20.dk, 77.84.Ek
The possibility of ferroelectricity in rubidium niobateand rubidium tantalate with the perovskite structure wasdiscussed by Smolenskii and Kozhevnikova and then byMegaw in the early 1950s. In Ref. 1, the authors re-ferred to unpublished data by V.G. Prokhvatilov whodetected the tetragonal RbTaO phase with a = 3 .
92 ˚A, c = 4 .
51 ˚A exhibiting a phase transition near 520 K;in Ref. 2 these data were simply cited. However, fur-ther studies have shown that, unlike lithium, sodium, andpotassium niobates, RbNbO and RbTaO crystallize inindividual crystal structures with triclinic P ¯1 symmetryfor RbNbO and monoclinic C /m one for RbTaO when prepared at atmospheric pressure. To obtain thesematerials with the perovskite structure, they should beprepared at high pressures (65–90 kbar). Due to the dif-ficulties in synthesis of RbNbO and RbTaO with theperovskite structure, the properties of these crystals havebeen studied very little.The phase diagrams of Rb O–Nb O and Rb O–Ta O systems were studied in Ref. 7 and 8. RbNbO isformed by the peritectic reaction and decomposes above964 ◦ C. RbTaO decomposes above 600 ◦ C probably dueto the peritectic reaction too. Rubidium-containing fer-roelectric materials in the BaNb O –NaNbO –RbNbO system with the tungsten-bronze structure have highelectro-optical properties that substantially exceed thoseof lithium niobate. In Ref. 11, the possibility of us-ing rubidium niobate and rubidium tantalate for pho-toelectrochemical decomposition of water was discussed.In Ref. 12, it was proposed to use the delamination ofRbTaO structure to produce porous TaO nanomem-branes with pore sizes of 1.3 × × . Forexample, the existence of the phase transition at 520 Kin the tetragonal phase was reported in Ref. 1, whereasthe data of Ref. 6 showed that RbTaO prepared at highpressure has the cubic perovskite (or close to it) struc- ture. At 300 K, the structure of RbNbO is similar tothat of the orthorhombic BaTiO , and the data of differ-ential thermal analysis indicate phase transitions in it at15, 155, and 300 ◦ C. In this work, the equilibrium structures of RbNbO and RbTaO were determined from first-principles cal-culations, and spontaneous polarization, dielectric con-stant, piezoelectric and elastic moduli, nonlinear opticaland electro-optical properties as well as the energy bandgaps in the LDA and GW approximations were calcu-lated for these crystals.The first-principles calculations were performed withinthe density functional theory using the ABINIT soft-ware. The exchange–correlation interaction was de-scribed in the local density approximation (LDA). Theoptimized norm-conserving pseudopotentials for Nb, Ta,and O atoms used in these calculations were taken fromRef. 14. The non-relativistic pseudopotential for theRb atom (electronic configuration 4 s p s ) was con-structed according to the scheme of Ref. 15 using the OPIUM program with the following parameters: r s =1 . r p = 1 . r d = 1 . q s = 7 . q p = 7 . q d = 7 . r min = 0 . r max = 1 .
52, and V loc = 1 .
58 a.u.(for notations, see Ref. 17). The testing of the Rb pseu-dopotential on the P ¯1 phase of RbNbO and the C /m phase of RbTaO , which are stable at atmospheric pres-sure, showed its sufficiently high quality: the calculatedlattice parameters and atomic coordinates in these phases(see Tables I and II) are in good agreement with theexperimental data; small underestimates of the calcu-lated lattice parameters are characteristic of the LDAapproximation used in this work.The lattice parameters and equilibrium atomic posi-tions in the unit cells were determined from the conditionwhen the residual forces acting on the atoms were be-low 5 · − Ha/Bohr (0.25 meV/˚A) in the self-consistentcalculation of the total energy with an accuracy betterthan 10 − Ha. The maximum energy of plane waveswas 30 Ha for RbNbO and 40 Ha for RbTaO . Inte-gration over the Brillouin zone was performed using a TABLE I. Calculated lattice parameters and atomic coordi-nates in RbNbO structures.Atom Position x y z Phase P ¯1 a = 5 . b = 8 . c = 8 . α = 114 . β = 93 . γ = 95 . ◦ Rb2 1 a +0.00000 +0.00000 +0.00000Rb1 1 b +0.00000 +0.00000 +0.50000Rb3 2 i +0.41251 +0.70257 +0.09488Nb1 2 i +0.49674 +0.28138 +0.35602Nb2 2 i +0.02746 +0.51037 +0.30988O1 2 i +0.10125 +0.39078 +0.82160O2 2 i +0.23664 +0.42747 +0.50994O3 2 i +0.28650 +0.71922 +0.45293O4 2 i +0.29777 +0.37205 +0.19910O5 2 i +0.33951 +0.05579 +0.27137O6 2 i +0.78420 +0.27571 +0.22313Phase P m ma = 4 . a +0.00000 +0.00000 +0.00000Nb 1 b +0.50000 +0.50000 +0.50000O 3 c +0.00000 +0.50000 +0.50000Phase P mma = 4 . c = 4 . a +0.00000 +0.00000 − b +0.50000 +0.50000 +0.51818O1 2 c +0.50000 +0.00000 +0.47446O2 1 b +0.50000 +0.50000 − Amm a = 3 . b = 5 . c = 5 . a +0.00000 +0.00000 − b +0.50000 +0.00000 +0.51447O1 4 e +0.50000 +0.25496 +0.22842O2 2 a +0.00000 +0.00000 +0.47735Phase R ma = 4 . α = 89 . ◦ Rb 1 a − − − a +0.51193 +0.51193 +0.51193O 3 b − × × in the cubic P m m phase is shown in Fig. 1. This spectrum contains a band TABLE II. Calculated lattice parameters and atomic coordi-nates in RbTaO structures.Atom Position x y z Phase C /ma = b = 6 . c = 8 . α = 86 . β = 93 . γ = 96 . ◦ Rb1 4 i +0.16000 − g +0.26494 +0.26494 +0.00000Ta1 4 h +0.31104 +0.31104 +0.50000Ta2 4 i +0.23924 − j +0.27303 +0.04639 +0.39643O2 8 j +0.54749 − i +0.16752 − i +0.37705 − P m ma = 3 . a +0.00000 +0.00000 +0.00000Ta 1 b +0.50000 +0.50000 +0.50000O 3 c +0.00000 +0.50000 +0.50000FIG. 1. Phonon spectrum of RbNbO in the cubic P m m phase. Labels near curves indicate the symmetry of unstablemodes. of unstable modes characteristic of ferroelectric chain in-stability which was first observed in KNbO . At thecenter of the Brillouin zone, this mode has the Γ sym-metry, is triply degenerate, and describes the ferroelectricdistortion of structure. The structures appearing uponcondensation of the X and M ′ modes are characterizedby antiparallel orientation of polarization in neighboring...–O–Nb–O–... chains.The energies of all RbNbO phases formed upon con-densation of the above unstable modes are given in Ta-ble III. Among these phases, the R m phase has thelowest energy. The phonon spectrum calculations forthe R m phase show that the frequencies of all opticalphonons at the center of the Brillouin zone and at high-symmetry points at its boundary are positive; the deter- TABLE III. Relative energies of low-symmetry RbNbO phases formed from the cubic perovskite phase upon conden-sation of unstable phonons, phases with 6 H , 4 H , 9 R , and 2 H structures, and the P ¯1 phase prepared at atmospheric pres-sure (the most stable phase energy is in boldface).Phase Unstable mode Energy, meV P m m — 0 P /nmm M ′ − P mma X − Cmcm X − P mm Γ − Amm − R m Γ − P ¯1 — +27.4 P /mmc (6 H ) — +121.4 P /mmc (4 H ) — +334.0 R ¯3 m (9 R ) — +568.1 P /mmc (2 H ) — +1752TABLE IV. Relative energies of low-symmetry RbTaO phases formed from the cubic perovskite phase upon conden-sation of unstable phonons, phases with 6 H , 4 H , 9 R , and2 H structures, and the C /m phase prepared at atmosphericpressure (the most stable phase energy is in boldface).Phase Unstable mode Energy, meV P m m — 0 P mm Γ − Amm − R m Γ − C /m — +38.5 P /mmc (6 H ) — +113.3 P /mmc (4 H ) — +352.2 R ¯3 m (9 R ) — +589.0 P /mmc (2 H ) — +1839 minant and all leading principal minors constructed ofelastic moduli tensor components are also positive. Thismeans that the R m phase is the ground-state struc-ture of RbNbO . The calculated lattice parameters andatomic coordinates in this phase are given in Table I. Asthe same sequence of phases as in BaTiO is supposed inrubidium niobate with the perovskite structure, the lat-tice parameters and atomic coordinates in two other fer-roelectric phases are also given in this table. The latticeparameters calculated for the orthorhombic RbNbO arein good agreement with the experimental data obtainedat 300 K ( a = 3 . b = 5 . c = 5 . , the frequency of unstable Γ phonon inthe phonon spectrum (Fig. 2) and the energy gain re-sulting from the transition to ferroelectric phases (Ta-ble IV) are rather low; so it is necessary to additionally FIG. 2. Phonon spectrum of RbTaO in the cubic P m m phase. Labels near curves indicate the symmetry of unstablemodes. test the stability of the ferroelectric distortion with re-spect to zero-point lattice vibrations. For this purpose,we used the technique proposed in Ref. 20. The energygain resulting from the transition from the P m m phaseto the R m phase is E = 1 .
90 meV and the unstablephonon frequency at the Γ point in the
P m m phase is ν = 84 cm − . As the energy ratio hν/E = 5 .
51 exceedsthe critical value of 2.419 obtained in Ref. 20, the energyof the lowest vibrational state in a two-well potential ap-pears above the upper point of the energy barrier sepa-rating the potential wells, and the ferroelectric orderingis suppressed by zero-point vibrations. Therefore, theonly stable phase of RbTaO with the perovskite struc-ture is the cubic phase. The calculated lattice parameterof this phase is given in Table II; its value is in satisfac-tory agreement with the experimental data ( a = 4 .
035 ˚A,Ref. 6).It is known that the formation of phases with hexag-onal BaNiO (polytype 2 H ), hexagonal BaMnO (poly-type 4 H ), hexagonal BaTiO (polytype 6 H ), and rhom-bohedral BaRuO (polytype 9 R ) structures is character-istic of AB O perovskites with the tolerance factor t > to the hexagonal structure upon heating, byanalogy to that occurring in BaTiO . The high energiesof these phases, in particular, the 2 H phase, are probablycaused by larger sizes and strong electrostatic repulsion ofNb ions which occupy face-sharing octahedra in thesestructures.We consider now some properties of ferroelectricRbNbO . The calculated polarization in RbNbO is0.46 C/m in the P mm phase and 0.50 C/m in the Amm R m phases; these values slightly exceedthe calculated polarization in the same phases of KNbO (0.37, 0.42, and 0.42 C/m , respectively). The static di- TABLE V. Nonzero components of the piezoelectric tensor e iν (C/m ) and tensors of the second-order nonlinear optical sus-ceptibility d iν and the linear electro-optic effect r iν (pm/V)in rhombohedral phases of RbNbO , KNbO , LiNbO , andBaTiO .Coefficient RbNbO KNbO LiNbO BaTiO e − − − − e +4.8 +6.8 +3.5 +7.3 e +2.4 +2.3 +0.1 +3.5 e +2.9 +3.1 +1.1 +5.1 d +12.7 +11.9 +2.3 +4.4 d − − − − d − − − − d − − − − r − − − − r +27.6 +39.2 +17.1 +43.3 r +18.0 +23.9 +10.1 +25.3 r +30.1 +40.6 +27.3 +48.9 electric tensor in the R m phase is characterized by twoeigenvalues: ε k = 21 . ε ⊥ = 35 .
8; the optical dielec-tric tensor eigenvalues are ε ∞k = 5 .
31 and ε ∞⊥ = 5 . C = 412, C = 84, and C = 102 GPa; the bulk modulus is B = 193 . e iν , second-order nonlinear opticalsusceptibility d iν , and linear electro-optic (Pockels) effect r iν in the R m phase of rubidium niobate are comparedwith the corresponding properties of other rhombohedralferroelectrics in Table V. We can see that the piezoelec-tric moduli in rhombohedral RbNbO (as well as in itsother polar phases) are slightly lower than in KNbO .The nonlinear optical coefficients in RbNbO exceed thecorresponding values in KNbO , although the d valuein rubidium niobate is slightly lower than in lithium nio-bate. As for the electro-optical properties, in rhombo-hedral RbNbO they are slightly lower than in KNbO ,but are notably superior to those of lithium niobate. Inthe orthorhombic phase (stable at 300 K), nonlinear opti-cal properties of RbNbO are comparable to those of thesame phase of potassium niobate: for example, the d modulus is − and − .In cubic RbTaO , the optical dielectric constant is ε ∞ = 5 .
58. The static dielectric constant can be es-timated only in the rhombohedral phase as ∼ C = 466, C = 91 .
5, and C = 120 GPa; B = 216 GPa. Thepiezoelectric moduli, second-order nonlinear optical sus-ceptibility, and electro-optical coefficients in the cubicphase are zero.An unexpected result of our calculations is that inboth studied compounds the P ¯1 and C /m phases whichcan be prepared at atmospheric pressure are metastable. This result is probably caused by an effective lattice con-traction which always exists in the LDA calculations.The fact that the specific volume of the P m m phaseis noticeably smaller than that of P ¯1, C /m , P /mmc ,and R ¯3 m phases suggests that under pressure the cubicperovskite phase will be the most stable one. To esti-mate the maximum value of the actual effective pressure,the lattice parameters and atomic positions in the C /m structure of rubidium tantalate were calculated for differ-ent pressures and it was shown that the unit cell volumeequal to the experimental one at 300 K can be obtainedat an isotropic pressure of − C /m phase becomes lower than thatof P m m by ∼
230 meV, i.e., becomes consistent withthe experimental data. At the above-mentioned nega-tive pressure, the ratio hν/E determining the stabilityof the ferroelectric phase in RbTaO becomes equal to1.90, i.e., it is slightly lower than the critical value of2.419. However, if we take into account that the abovenegative pressure is obviously overestimated because itincludes the thermal expansion effect, we can supposethat, even taking into account the systematic error inthe LDA lattice parameter determination, rubidium tan-talate will remain cubic up to the lowest temperatures.The conclusion that RbTaO is an incipient ferroelec-tric in which the ferroelectric ordering is suppressed byzero-point vibrations agrees with the data of Ref. 6, butcontradicts the data of Ref. 1 in which the phase transi-tion near 520 K was reported. We suppose that tantalum-enriched phases (in particular, with the tungsten-bronzestructure ) could be formed in rubidium tantalate sam-ples discussed in Ref. 1 because of the low temperatureof the peritectic reaction, and this could result in theobserved anomaly.In Ref. 11, the possibility of using various oxideswith the perovskite structure, in particular RbNbO andRbTaO , for development of photoelectrochemical solarcells was discussed. We calculated the band gap E g inthese compounds both in the LDA approximation andin the GW approximation that takes into account many-body effects (the technique of the latter calculations wasanalogous to that used in Refs. 21–23). In the LDA ap-proximation, E LDA g = 1 .
275 eV in cubic RbNbO whenthe spin–orbit coupling is neglected; in P mm , Amm R m phases, E LDA g is 1.314, 1.869, and 2.137 eV,respectively. In cubic RbTaO , E LDA g = 2 .
175 eV whenthe spin–orbit coupling is neglected. The valence bandextrema in the cubic phase of both compounds are at the R point of the Brillouin zone, whereas the conductionband extrema are at the Γ point. The calculations usingthe technique of Ref. 23 yield the spin–orbit splitting ofthe conduction band edge ∆ SO = 0 .
111 eV for RbNbO and ∆ SO = 0 .
400 eV for RbTaO ; the spin-orbit splittingof the valence band edge is absent. After correction forthe conduction band edge shift (∆ SO / E g that take into account the spin–orbit coupling are1.238, 1.277, 1.832, 2.100, and 2.042 eV for four RbNbO phases and for cubic RbTaO , respectively.In the GW approximation, the band gap when thespin–orbit coupling is neglected is E GWg = 2 . and 3.302 eVin cubic RbTaO . If the spin–orbit coupling is takeninto account, these values decrease to 2.366, 2.579, 3.254,3.572, and 3.169 eV, respectively. The obtained valuesof E g are appreciably smaller than those calculated inRef. 11 for cubic phases (3.4 eV for RbNbO and 4.3 eVfor RbTaO ).Some authors who studied rubidium niobate and ru-bidium tantalate have noticed their sensitivity to humid-ity. Evidently, this can be a serious obstacle for practi-cal applications of these materials. However, we wouldlike to note that this property is inherent to phases pre-pared at atmospheric pressure and having loose struc-tures whose specific volume is 26–28% larger than thatof the perovskite phase. In Ref. 8, it was suggested that the effect is due to intercalation of water molecules intothe loose structures, rather than to hydrolysis of thesecompounds. The possibility of preparing RbTaO by hy-drothermal synthesis and the low rate of RbTaO ionexchange in HCl during its delamination support thisidea. This suggests that the considered compounds withthe perovskite structure can be quite stable to humidity.In summary, the present calculations of RbNbO andRbTaO properties and their comparison with the prop-erties of other ferroelectrics show that rubidium niobateis an interesting ferroelectric material with high nonlin-ear optical and electro-optical properties, and rubidiumtantalate is an incipient ferroelectric.The calculations presented in this work were performedon the laboratory computer cluster (16 cores).This work was supported by the Russian Foundationfor Basic Research, project no. 13-02-00724. ∗ [email protected] G. A. Smolenskii and N. V. Kozhevnikova, Dokl. Akad.Nauk SSSR , 519 (1951). H. D. Megaw, Acta Cryst. , 739 (1952). M. Serafin and R. Hoppe,J. Less-Common Metals , 299 (1980). M. Serafin and R. Hoppe,Angewandte Chem. , 387 (1978). M. Serafin and R. Hoppe,Z. Anorg. Allg. Chem. , 240 (1980). J. A. Kafalas, in
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