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Ferroelectric properties of RbNbO and RbTaO 3 3 A. I. Lebedev∗ Physics Department, Moscow State University, Moscow, 119991 Russia (Dated: January 6, 2015) 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 the phonon spectra, the structures of possible distorted phases are determined, theirenergiesarecalculated,anditisshownthatR3mistheground-statestructureofRbNbO3. In 5 RbTaO3,theferroelectric instabilityissuppressedbyzero-pointlatticevibrations. Forferroelectric 1 phases of RbNbO3, spontaneous polarization, piezoelectric, nonlinear optical, electro-optical, and 0 otherpropertiesaswellastheenergybandgapintheLDAandGW approximationsarecalculated. 2 The properties of the rhombohedral RbNbO3 are compared with those of rhombohedral KNbO3, LiNbO3,and BaTiO3. n DOI:10.1134/S1063783415020237 a J PACSnumbers: 61.50.Ah,63.20.dk,77.84.Ek 4 ] i The possibility of ferroelectricity in rubidium niobate ture. At 300 K, the structure of RbNbO3 is similar to c andrubidiumtantalatewiththeperovskitestructurewas thatoftheorthorhombicBaTiO ,andthe dataofdiffer- s 3 - discussedby Smolenskiiand Kozhevnikova1 andthen by entialthermalanalysisindicate phase transitions init at rl Megaw2 in the early 1950s. In Ref. 1, the authors re- 15, 155, and 300◦C.6 mt ferred to unpublished data by V.G. Prokhvatilov who In this work, the equilibrium structures of RbNbO3 t. cde=tec4te.5d1th˚Ae etexthriabgiotinnagl RabpThaaOse3 tprhaansseitwiointhnaea=r 35.2902 K˚A,; caunldatRiobnTs,aOan3dwseproendtaetneeromuisnpedolafrroizmatifiorns,t-dpireilneccitprliecsccoanl-- a m in Ref. 2 these data were simply cited. However, fur- stant, piezoelectric and elastic moduli, nonlinear optical therstudieshaveshownthat,unlikelithium,sodium,and and electro-opticalproperties as well as the energy band - d potassium niobates, RbNbO3 and RbTaO3 crystallize in gaps in the LDA and GW approximations were calcu- n individual crystal structures with triclinic P¯1 symmetry lated for these crystals. o for RbNbO3 and monoclinic C2/m one for RbTaO33–5 Thefirst-principlescalculationswereperformedwithin c when preparedat atmosphericpressure. To obtainthese the density functional theory using the ABINIT soft- [ materials with the perovskite structure, they should be ware.13 The exchange–correlation interaction was de- 1 preparedathighpressures(65–90kbar).6 Duetothedif- scribed in the local density approximation (LDA). The v ficulties in synthesis of RbNbO and RbTaO with the 3 3 optimized norm-conservingpseudopotentials for Nb, Ta, 0 perovskitestructure,thepropertiesofthesecrystalshave and O atoms used in these calculations were taken from 7 been studied very little. 6 Ref. 14. The non-relativistic pseudopotential for the 0 The phase diagrams of Rb2O–Nb2O5 and Rb2O– Rb atom (electronic configuration 4s24p65s0) was con- 0 Ta2O5 systems were studied in Ref. 7 and 8. RbNbO3 is structed according to the scheme of Ref. 15 using the . formed by the peritectic reaction and decomposes above OPIUM program16 with the following parameters: r = 1 s 0 964◦C.7 RbTaO3 decomposes above600◦C probablydue 1.68, rp = 1.72, rd = 1.68, qs = 7.07, qp = 7.27, 5 to the peritectic reactiontoo.8 Rubidium-containing fer- qd =7.07, rmin =0.01, rmax =1.52, and Vloc =1.58 a.u. 1 roelectric materials in the BaNb2O6–NaNbO3–RbNbO3 (for notations, see Ref. 17). The testing of the Rb pseu- : system with the tungsten-bronze structure have high dopotential on the P¯1 phase of RbNbO and the C2/m v 3 i electro-opticalpropertiesthat substantiallyexceedthose phase of RbTaO3, which are stable at atmospheric pres- X of lithium niobate.9,10 In Ref. 11, the possibility of us- sure, showed its sufficiently high quality: the calculated r ing rubidium niobate and rubidium tantalate for pho- latticeparametersandatomiccoordinatesinthesephases a toelectrochemical decomposition of water was discussed. (see Tables I and II) are in good agreement with the In Ref. 12, it was proposed to use the delamination of experimental data;3,5 small underestimates of the calcu- RbTaO3 structure to produce porous TaO3 nanomem- lated lattice parameters are characteristic of the LDA branes with pore sizes of 1.3×0.6 and 1.1×1.1 ˚A, which approximationused in this work. can be used for selective filtration of lithium ions. The lattice parameters and equilibrium atomic posi- Thelackofknowledgeonthepropertiesofcompounds tionsintheunitcellsweredeterminedfromthecondition under consideration appears, in particular, in contradic- when the residual forces acting on the atoms were be- tory data on the ferroelectric properties of RbTaO . For low 5·10−6 Ha/Bohr (0.25 meV/˚A) in the self-consistent 3 example, the existence of the phase transition at 520 K calculation of the total energy with an accuracy better in the tetragonal phase was reported in Ref. 1, whereas than 10−10 Ha. The maximum energy of plane waves the dataofRef.6showedthatRbTaO preparedathigh was 30 Ha for RbNbO and 40 Ha for RbTaO . Inte- 3 3 3 pressure has the cubic perovskite (or close to it) struc- gration over the Brillouin zone was performed using a 2 TABLE I. Calculated lattice parameters and atomic coordi- TABLE II.Calculated lattice parameters and atomic coordinates in RbNbO3 structures. nates in RbTaO3 structures. Atom Position x y z Atom Position x y z Phase P¯1 Phase C2/m a=5.0816, b=8.3047, c=8.7916 ˚A, a=b=6.3396, c=8.0171, α=114.0625, β =93.3891, γ =95.1160◦ α=86.1031, β=93.8969, γ =96.8997◦ Rb2 1a +0.00000 +0.00000 +0.00000 Rb1 4i +0.16000 −0.16000 +0.73758 Rb1 1b +0.00000 +0.00000 +0.50000 Rb2 4g +0.26494 +0.26494 +0.00000 Rb3 2i +0.41251 +0.70257 +0.09488 Ta1 4h +0.31104 +0.31104 +0.50000 Nb1 2i +0.49674 +0.28138 +0.35602 Ta2 4i +0.23924 −0.23924 +0.30214 Nb2 2i +0.02746 +0.51037 +0.30988 O1 8j +0.27303 +0.04639 +0.39643 O1 2i +0.10125 +0.39078 +0.82160 O2 8j +0.54749 −0.21209 +0.28704 O2 2i +0.23664 +0.42747 +0.50994 O3 4i +0.16752 −0.16752 +0.08653 O3 2i +0.28650 +0.71922 +0.45293 O4 4i +0.37705 −0.37705 +0.55235 O4 2i +0.29777 +0.37205 +0.19910 Phase Pm3m O5 2i +0.33951 +0.05579 +0.27137 a=3.9846 ˚A O6 2i +0.78420 +0.27571 +0.22313 Rb 1a +0.00000 +0.00000 +0.00000 Phase Pm3m Ta 1b +0.50000 +0.50000 +0.50000 a=4.0291 ˚A O 3c +0.00000 +0.50000 +0.50000 Rb 1a +0.00000 +0.00000 +0.00000 Nb 1b +0.50000 +0.50000 +0.50000 O 3c +0.00000 +0.50000 +0.50000 Phase P4mm a=4.0037, c=4.1592 ˚A Rb 1a +0.00000 +0.00000 −0.00336 Nb 1b +0.50000 +0.50000 +0.51818 O1 2c +0.50000 +0.00000 +0.47446 O2 1b +0.50000 +0.50000 −0.03654 Phase Amm2 a=3.9928, b=5.7742, c=5.7960 ˚A Rb 2a +0.00000 +0.00000 −0.00341 Nb 2b +0.50000 +0.00000 +0.51447 O1 4e +0.50000 +0.25496 +0.22842 O2 2a +0.00000 +0.00000 +0.47735 FIG. 1. Phonon spectrum of RbNbO3 in the cubic Pm3m Phase R3m phase. Labels near curvesindicate thesymmetry of unstable a=4.0571 ˚A, α=89.8945◦ modes. Rb 1a −0.00308 −0.00308 −0.00308 Nb 1a +0.51193 +0.51193 +0.51193 O 3b −0.02115 +0.48415 +0.48415 ofunstablemodes characteristicofferroelectricchainin- stability which was first observed in KNbO .19 At the 3 center of the Brillouin zone, this mode has the Γ sym- 15 metry,istriplydegenerate,anddescribestheferroelectric 8×8×8 Monkhorst–Pack mesh. The spontaneous polar- distortion of structure. The structures appearing upon izationinferroelectricphaseswascalculatedbytheBerry condensationof the X and M′ modes are characterized 5 3 phasemethod. Thephononspectra,dielectricconstants, by antiparallelorientationof polarizationin neighboring piezoelectric and elastic moduli were calculated within ...–O–Nb–O–... chains. the density-functional perturbation theory similarly to The energies of all RbNbO phases formed upon con- 3 Ref. 17. Nonlinear optical and electro-optical properties densation of the above unstable modes are given in Ta- werecalculatedusing the technique describedin Ref.18. ble III. Among these phases, the R3m phase has the All physical properties presented in this work were cal- lowest energy. The phonon spectrum calculations for culated for the theoretical lattice parameter. the R3m phase show that the frequencies of all optical The phononspectrum ofRbNbO in the cubic Pm3m phonons at the center of the Brillouin zone and at high- 3 phase is shownin Fig. 1. This spectrum containsa band symmetry points at its boundary are positive; the deter- 3 TABLE III. Relative energies of low-symmetry RbNbO3 phasesformed from thecubicperovskitephaseuponconden- sationofunstablephonons,phaseswith6H,4H,9R,and2H structures, and the P¯1 phase prepared at atmospheric pressure (themost stable phase energy is in boldface). Phase Unstable mode Energy, meV Pm3m — 0 P4/nmm M3′ −31.3 Pmma X5 −34.8 Cmcm X5 −38.7 P4mm Γ15 −46.5 Amm2 Γ15 −57.0 R3m Γ15 −58.6 P¯1 — +27.4 FIG. 2. Phonon spectrum of RbTaO3 in the cubic Pm3m phase. Labels near curvesindicate thesymmetry of unstable P63/mmc (6H) — +121.4 modes. P63/mmc (4H) — +334.0 R¯3m (9R) — +568.1 P63/mmc (2H) — +1752 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 energy gainresulting fromthe transitionfromthe Pm3m phase TABLE IV. Relative energies of low-symmetry RbTaO3 to the R3m phase is E = 1.90 meV and the unstable phasesformed from thecubicperovskitephaseuponconden- 0 sation of unstable phonons, phases with 6H, 4H, 9R, and phonon frequency at the Γ point in the Pm3m phase is 2H structures,andtheC2/mphasepreparedatatmospheric ν =84 cm−1. As the energy ratio hν/E0 =5.51 exceeds pressure (themost stable phaseenergy is in boldface). the criticalvalueof2.419obtainedinRef.20,the energy ofthe lowestvibrationalstateinatwo-wellpotentialap- Phase Unstable mode Energy, meV pears above the upper point of the energy barrier sepa- Pm3m — 0 rating the potential wells, and the ferroelectric ordering P4mm Γ15 −1.80 is suppressed by zero-point vibrations. Therefore, the Amm2 Γ15 −1.87 only stable phase of RbTaO3 with the perovskite struc- R3m Γ15 −1.90 tureisthecubic phase. Thecalculatedlattice parameter of this phase is given in Table II; its value is in satisfac- C2/m — +38.5 toryagreementwiththeexperimentaldata(a=4.035˚A, P63/mmc (6H) — +113.3 Ref. 6). P63/mmc (4H) — +352.2 It is known that the formation of phases with hexag- R¯3m (9R) — +589.0 onal BaNiO (polytype 2H), hexagonal BaMnO (poly- 3 3 P63/mmc (2H) — +1839 type 4H), hexagonal BaTiO (polytype 6H), and rhom- 3 bohedralBaRuO (polytype 9R)structures is character- 3 isticofABO perovskiteswiththetolerancefactort>1, 3 minant and all leading principal minors constructed of and the studied compounds belong to this class. Our elastic moduli tensor components are also positive. This calculations showed that the energies of these phases for means that the R3m phase is the ground-state struc- both rubidium compounds are appreciably higher than ture of RbNbO . The calculated lattice parameters and thecubicphaseenergy(TablesIIIandIV).Theseresults 3 atomic coordinates in this phase aregiven in Table I. As explain why it was impossible to observe the transition the samesequenceofphasesasinBaTiO3 issupposedin ofRbNbO3 tothehexagonalstructure6 uponheating,by rubidiumniobatewiththeperovskitestructure,6 thelat- analogy to that occurring in BaTiO3. The high energies tice parameters and atomic coordinates in two other fer- ofthesephases,inparticular,the2H phase,areprobably roelectric phases are also given in this table. The lattice causedbylargersizesandstrongelectrostaticrepulsionof parameterscalculatedfortheorthorhombicRbNbO are Nb5+ ions which occupy face-sharing octahedra in these 3 in good agreement with the experimental data obtained structures. at 300 K (a = 3.9965, b = 5.8360, and c = 5.8698 ˚A, We consider now some properties of ferroelectric Ref. 6). RbNbO . The calculated polarization in RbNbO is 3 3 In RbTaO , the frequency of unstable Γ phonon in 0.46 C/m2 in the P4mm phase and 0.50 C/m2 in the 3 15 the phonon spectrum (Fig. 2) and the energy gain re- Amm2 and R3m phases; these values slightly exceed sulting from the transition to ferroelectric phases (Ta- thecalculatedpolarizationinthesamephasesofKNbO 3 ble IV) are rather low; so it is necessary to additionally (0.37, 0.42, and 0.42 C/m2, respectively). The static di- 4 Thisresultisprobablycausedbyaneffectivelatticecon- TABLEV.Nonzerocomponentsofthepiezoelectrictensoreiν traction which always exists in the LDA calculations. (C/m2)andtensorsofthesecond-ordernonlinearopticalsus- The fact that the specific volume of the Pm3m phase ceptibility diν and the linear electro-optic effect riν (pm/V) is noticeably smallerthan that of P¯1, C2/m, P6 /mmc, in rhombohedral phases of RbNbO3, KNbO3, LiNbO3, and 3 and R¯3m phases suggests that under pressure the cubic BaTiO3. perovskite phase will be the most stable one. To esti- Coefficient RbNbO3 KNbO3 LiNbO3 BaTiO3 matethemaximumvalueoftheactualeffectivepressure, e11 −3.0 −4.2 −2.4 −4.0 thelatticeparametersandatomicpositionsinthe C2/m e15 +4.8 +6.8 +3.5 +7.3 structureofrubidiumtantalatewerecalculatedfordiffer- e31 +2.4 +2.3 +0.1 +3.5 ent pressuresand it was shownthat the unit cell volume equal to the experimental one at 300 K can be obtained e33 +2.9 +3.1 +1.1 +5.1 at an isotropic pressure of −24.7 kbar. At this pressure, d11 +12.7 +11.9 +2.3 +4.4 theenthalpyoftheC2/mphasebecomeslowerthanthat d15 −23.6 −21.9 −11.5 −16.1 of Pm3m by ∼230 meV, i.e., becomes consistent with d31 −23.6 −21.9 −11.5 −16.1 the experimental data. At the above-mentioned nega- d33 −29.4 −27.3 −37.4 −31.1 tive pressure, the ratio hν/E determining the stability 0 r11 −12.8 −17.7 −5.6 −13.7 of the ferroelectric phase in RbTaO3 becomes equal to r15 +27.6 +39.2 +17.1 +43.3 1.90, i.e., it is slightly lower than the critical value of r31 +18.0 +23.9 +10.1 +25.3 2.419. However, if we take into account that the above negative pressure is obviously overestimated because it r33 +30.1 +40.6 +27.3 +48.9 includes the thermal expansion effect, we can suppose that, even taking into account the systematic error in theLDAlatticeparameterdetermination,rubidiumtan- electric tensor in the R3m phase is characterizedby two talate will remain cubic up to the lowest temperatures. eigenvalues: ε0 =21.1 and ε0 =35.8; the optical dielec- k ⊥ The conclusion that RbTaO3 is an incipient ferroelectric tensor eigenvalues are ε∞ = 5.31 and ε∞ = 5.91. tric in which the ferroelectric ordering is suppressed by k ⊥ In the cubic phase, the elastic moduli are C = 412, zero-point vibrations agrees with the data of Ref. 6, but 11 C12 = 84, and C44 = 102 GPa; the bulk modulus is contradicts the data of Ref. 1 in which the phase transi- B =193.5 GPa. The nonzero components of the tensors tionnear520Kwasreported. Wesupposethattantalum- of piezoelectric effect e , second-order nonlinear optical enriched phases (in particular, with the tungsten-bronze iν susceptibilityd ,andlinearelectro-optic(Pockels)effect structure8) could be formed in rubidium tantalate sam- iν r in the R3m phase of rubidium niobate are compared ples discussed in Ref. 1 because of the low temperature iν withthecorrespondingpropertiesofotherrhombohedral of the peritectic reaction, and this could result in the ferroelectrics in Table V. We can see that the piezoelec- observed anomaly. tric moduli in rhombohedral RbNbO3 (as well as in its In Ref. 11, the possibility of using various oxides other polar phases) are slightly lower than in KNbO3. withtheperovskitestructure,inparticularRbNbO3 and The nonlinear optical coefficients in RbNbO3 exceed the RbTaO3, for development of photoelectrochemical solar corresponding values in KNbO3, although the d33 value cells was discussed. We calculated the band gap Eg in in rubidiumniobate is slightly lowerthanin lithium nio- these compounds both in the LDA approximation and bate. As for the electro-optical properties, in rhombo- inthe GW approximationthattakesintoaccountmany- hedral RbNbO3 they are slightly lower than in KNbO3, body effects (the technique of the latter calculationswas but are notably superior to those of lithium niobate. In analogous to that used in Refs. 21–23). In the LDA ap- theorthorhombicphase(stableat300K),nonlinearopti- proximation, ELDA = 1.275 eV in cubic RbNbO when g 3 calpropertiesofRbNbO3 arecomparabletothose ofthe the spin–orbit coupling is neglected; in P4mm, Amm2, same phase of potassium niobate: for example, the d33 and R3m phases, ELDA is 1.314, 1.869, and 2.137 eV, modulus is −30.8pm/Vin RbNbO3 and−30.4pm/Vin respectively. In cubigc RbTaO , ELDA = 2.175 eV when KNbO . 3 g 3 the spin–orbit coupling is neglected. The valence band In cubic RbTaO3, the optical dielectric constant is extremainthecubicphaseofbothcompoundsareatthe ε∞ = 5.58. The static dielectric constant can be es- R point of the Brillouin zone, whereas the conduction timated only in the rhombohedral phase as ∼140. The band extrema are at the Γ point. The calculations using elasticmoduliincubicrubidiumtantalateareC11 =466, the technique of Ref. 23 yield the spin–orbit splitting of C12 = 91.5, and C44 = 120 GPa; B = 216 GPa. The the conduction band edge ∆SO =0.111 eV for RbNbO3 piezoelectric moduli, second-order nonlinear optical sus- and∆ =0.400eVforRbTaO ;thespin-orbitsplitting SO 3 ceptibility, and electro-optical coefficients in the cubic of the valence band edge is absent. After correction for phase are zero. theconductionbandedgeshift(∆ /3),theLDAvalues SO An unexpected result of our calculations is that in of E that take into account the spin–orbit coupling are g bothstudiedcompoundstheP¯1andC2/mphaseswhich 1.238,1.277,1.832,2.100,and2.042eVforfourRbNbO 3 canbe preparedatatmosphericpressureare metastable. phases and for cubic RbTaO , respectively. 3 5 In the GW approximation, the band gap when the the effect is due to intercalation of water molecules into spin–orbit coupling is neglected is EGW = 2.403, 2.616, the loose structures, rather than to hydrolysis of these g 3.291, and 3.609 eV, respectively in cubic, tetragonal, compounds. ThepossibilityofpreparingRbTaO byhy- 3 orthorhombic,and rhombohedralRbNbO and3.302eV drothermal synthesis24 and the low rate of RbTaO ion 3 3 in cubic RbTaO . If the spin–orbit coupling is taken exchange in HCl during its delamination12 support this 3 intoaccount,thesevaluesdecreaseto2.366,2.579,3.254, idea. This suggests that the consideredcompounds with 3.572, and 3.169 eV, respectively. The obtained values the perovskite structure canbe quite stable to humidity. of E are appreciably smaller than those calculated in g In summary, the present calculations of RbNbO and 3 Ref. 11 for cubic phases (3.4 eV for RbNbO and 4.3 eV 3 RbTaO properties and their comparisonwith the prop- 3 for RbTaO ). 3 erties of other ferroelectrics show that rubidium niobate Some authors who studied rubidium niobate and ru- is an interesting ferroelectric material with high nonlin- bidiumtantalatehavenoticedtheirsensitivitytohumid- ear optical and electro-optical properties, and rubidium ity. Evidently, this can be a serious obstacle for practi- tantalate is an incipient ferroelectric. cal applications of these materials. 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