JAISWAL and SHARMA : Atomic Displacements in Perovskite Strontium Titanate 321 Atomic Displacements in Perovrkite Strontium Titanate Ti. K. JAISWAL and P. K. SHARMA With 5 Figures Abstract The relative atomic displacements of the sublattices of perovskita strontium titenate associated with optical frequency mode are calculated using all poaaible interionic couplings. The couplings are estimated from the observed optical frequencies of SrTiO,. The results of the calculation are plotted as a function of a parameter a(-T - T,J. It is found that and LAST. two sets of solutions exist which correspond to the proposals of SLATER 1. Introduction Several interesting paraelectric properties of ferroelectric materials have been interpreted in terms of BORNVON KLRMLNlattice dynamics (SILVERMAN [l-41, VINOGRADOV [j],D V ~ Rand ~ XGLOGAR [ S ] ) . The theoretical basis for such a discussion goes back to an observation of COCHRAN that the ferroelectric transition is associated with the temperature dependence of the low-lying transverse optic mode frequency. I n recent years there have been several attempts to calculate the optic mode frequencies and to assign motions associated with these frequencies. RAJAQOPAL and SRINIVASAN  have obtained infrared active frequencies of SrTiO, from a nearest neighbour general force model using experimental elastic constants to determine the couplings between the ions. DV~RAK and JANOVEC  have used FOWLER'S [lo] short range forces to compute the coupling coefficients and the frequencies of the transverse optical modes. However, the results of these calculations do not agree well with the experimental values obtained from neutron scattering and infrared spectrometry. Recently JOSEPH and SILVERMAN [ll]have carried out calculations of the atomic displacements associated with infrared active modes using three couplings between the ions. The optic modes of the perovskite ferroelectrics are characterized by the motions of the five inequivalent sublattices. If the vibrations are considered along one of the Cartesian axes, the motion of these sublattices is completely determined by seven sublattice couplings. Of these strontium-titanium coupling is very small and can be neglected altogether. If the other three couplings are neglected, the remaining three couplings can be obtained from the three observed optical mode frequencies of SrTiO,. These couplings can then be used to calculate the relative atomic displacements for each mode. Such an approximation has been used by JOSEPH and SILVERMAN [ll] to determine the atomic 21 Ann. Physik. ?.Fo&c, Bd. 24 322 ,4nndenderPhysik * 7.Folge * Band24,Heft7/8 1970 displacements in SrTiO,. These workers, however, do not take into account the Ti-0,, Ti-0, and Sr-0, couplings. If we assume them to be comparable with Ti-0, and Sr-0, couplings, the equations determining the couplings are greatly simplified. One is then concerned with the solution of a second degree equation rather than a sixth degree equation which is somewhat difficult to solve. I n the present paper we have calculated the relative atomic displacements of the sublattices of SrTiO, using all possible couplings. It is shown t h a t the solutions consist of two branches which correspond t o the ferroelectric modes propounded by SLATER and LAST. 2. Numerical computation The equations of motion for ions are written here in a manner similar to t,hose of JOSEPH and SILVERMAN [ll].However, in the present work we have also considered the couplings Ti--,, Ti- 0, and Sr-OO,. The perovskite structure 8Sr 00 Ti Fig. 1. A unit cell of SrTiO,. The springs depict the coupling considered in the calculation. Displacements y are along the [loo] direction 5 in cubic phase has five atoms in the unit cell as shown in Fig. 1. For these sublattices we denote the various couplings as follows: k,: Ti-02, Ti-0, and Ti-0, couplings, k2: 0,-0, and 0,-0, couplings, k,: Sr-OO,, Sr-0, and Sr-0, couplings. We assume that Sr-Ti coupling is negligible as shown by D V ~ Rand ~ RJANOVEC [S]. Since 0,-0, coupling determines the torsional mode frequency, it can also be neglected. Assuming the displacements t o be along the [loo] direction, the equations of motion for the sublattices labelled in Fig. 1 can be written as + - + + + + + + + + + Gkiy, 2kiyz 2kiy3 2kiY4, 2kiyI - (2k1 8kz 4k3) yz 4kZy3 m3Ys= 2k,yl 4kzyZ- (2k, 4k2 4kJ Y, m4Y4= 2k1y1 4kzy2- (2kl 4k2 4k3)y4 nGi6 = 4kqz 4k3~3 4k3~4- 12k3~5n ~ &= ntzyz = + + + + + 442~4+ 4Qci9 + 44% + 4ky5, (1) Here mi and yi are the mass and displacement of the atom a t the site i. The solution of these equations can be obtained by writing yt = gieimt, where the eigen frequencies w are given by setting the determinant of the coefficients of the displacements equal to zero. The three oxygen atoms a t the sites 2, 3 and 4 are identical, hence we can write in2 = m3 = m4 = nz. The 5 x 5 determinant JAISWAL and SHARMA : Atomic Displacenients in Perovskite Strontium Titanate 323 of the coefficients when evaluated gives Q[Q where - P k l + 4k2 + 4k3)J x [Q - (2k1 + 12k2 + 4k3)l[a2- Q(& + @3) + 3P), p = 4(1 + 391, y = 24(P + 9 + 3 P d , 01 + (2) ~ k l k 3 l= 0, = 2(1 (3) Q = mwa, p = mlm,, q = mlm5. The solution 52 = 0 corresponds to the uniform translation motion of all the atoms and hence represents the acoustical mode. The solution represented by the second factor has the displacement y1 = ya = y5 = 0, y3 y4 = 0 and therefore corresponds to torsional mode. The three positive frequencies obtained from the last two factors of Eq. ( 2 ) correspond to the transverse optical modes. From the last second degree equation for Q, the couplings k, and k3 can be calculated. The optic mode frequencies of SrTiO, a t room temperature have been measured by several workers (BARKERand TINEHAM , Cowley , SPITZER et al. [lG], LEFEOWITZ). It is now well known that several interesting paraelectric properties of ferroelectrics result from the temperature dependence of the low-lying transverse optic mode of vibrations. In order to study the effect of this temperature dependence on couplings and hence on displacements, we assume that the two high optical mode frequencies are temperature independent and are given by their room temperature values while the low frequency mode is temperature dependent governed by a parameter Q, proportional to T - T,, T, being the CURIE temperature. We denote the three measured optic mode frequencies by Q,, $2, and Q3. These frequencies can be written as (SPITZER et al. [lG], LEFKOWITZ ): Ql = 28.30 . lo4 g rad2 s-,, Q, = 3.00 104 g rad2 s-,, Q, = 0.95 Q, lo4 g rad2 s-,. If these frequencies represent the optically active modes of Eq. (e), we obtain + - 0,= Zkl Q, + Q3 + 12k2 + 4k3, =& + pk3, (4) Q@3 = Yklk3. One can easily solve these equations for'k,, k, and k3. I n the present work we are concerned with the dispalcements associated with optical modes and therefore we take Qias given above. COWLEY has measured the torsional mode frequency of SrTiO, using inelastic neutron scattering technique. If his measured value is taken for the torsional mode frequency, this will change only the value of k,. The other two couplings k, and k8 will remain unaffected. This will slightly modify the values of relative displacements ya/yland y3/yl, while ~ 5 / remains ~ 1 unaffected. 21' 324 Annalen der Physik * 7.Folge * Band 24, Heft 718 * 1970 The couplings k, and k3 can now be evaluated as a function of 52, and D3by solving the second degree equation resulting from the simplification of Eq. (4). It was found that for each value of Q, two solutions for the couplings are obtained. One of them corresponds to LAST'S[^^] proposal and the other to SLATER’S  description of the ferroelectric mode. From Eq. (1)the relative atomic displacements are given by and 3. Results and discussion The numerical values of the relevant constants for SrTiO, used in the calculation are listed below : , . rn = 26.75 10-24 g, rnl = 80.09 * rn5 = 146.50 - g, g. The couplings k,, k, and k, were calculated from Eq. (4) using experimental values of sZi as given above. It was found that for @ > 0.872 no real solutions pr- J-rc) Fig. 2. Fig. 3. Fig. 2. The coupling coefficients as a function of the parameter 0 (NT - T,) Fig. 3. Relative atomic displacements yz/yl, yJyl and ys/yl as a function of 0 (-T - T,) for low optical frequency mode. The solid and dashed curves show our calculations and the dotted and dash-dotted curves represent JOSEPH and SILVERM.4N’S calculations JAISWAL and SHARNA: Atomic Displacements in Perovskite Strontium Titsnate 325 existed for them. The calculated values of the couplings as a function of the parameter @(-T - T,)are plotted in Fig. 2. The superscript “S” shows one type of solution and “L” the other solution. In Fig. 3 we have plotted the relative atomic displacements y2/yl,y3/y1 and y5/y1for both the solutions corresponding to low optical frequency mode. We notice that as @ --f 0, the displacements for “S” solutions, viz., y, = y3 = y4 = y5 $: y, are in accordance with SLATER’S  proposal for ferroelectric mode, -22; 0 07 02 ! 03 ’ 04 @ I -T-TC) ’ 05 ’ 08 ’ 07 ’ J 080872 L L 07 02 03 04 0 5 06 07 080872 -O60 4 1- T- Tc I Fig. 4. Fig. 5. Fig. 4. Relative atomic dispalcements y2/y1,y3/yland y5/y1for “L” solution as a function of @ (NT - T,)for intermediate optical frequency mode. The smooth solid curves depict our calculations and the dashed curves represent JOSEPH and SILVERMAN’S caloulations Fig. 5. Relative atomic displacements yl/ya,ya/yzand y5/yzfor “S” solution as a function of @ (NT - T,)for intermediate optical frequency mode. The solid curves represent our calculations and the dashed curves show JOSEPH and SILVERMAN’S calculations * i. e., the titanium ion vibrates against all the remaining ions. The displacements for the couplings labelled “L” are: y1 = y, = y3 = y4 y5. These are in accordance with the suggestion of LAST, that is, the vibration of the strontium ion against all the remaining ions plays dominant role in the ferroelectric mode of SrTiO,. Fig. 4 shows the relative atomic displacements y2!y1, y31yl and y5/ylfor the “L” solutions of the couplings corresponding to the mtermediate frequency of the optical mode. I n Fig. 5 we have plotted the calculated values of yl/y2,yJy, and y5/y2 for the “S” values of couplings corresponding to the intermediate frequency. The displacements corresponding to high frequency optical mode frequency 9,have not been plotted as the present treatment does not hold good for them. This is due to the fact that the calculated k, for a given value of Q, makes the factor (2k1 + 12k, + 4k3 - Q) zero for every value of k, and k3. 326 Annalen der P h p i k * 7.Folge * Band 24, Heft 7/8 1970 It would be worthwhile to compare our present calculations with those of JOSEPH and SILVERMAN [ll] which neglect certain interionic couplings. For this purpose we have also shown in Figs. 3-5 the JOSEPHand SILVERMAN’B calculations. It will be seen from these figures that our calculations show striking similarity with them. However, in the present calculation we have introduced certain couplings left by JOSEPH and SILVERMAN [ll]without increasing the number of couplings. This innovation makes the problem very simple. I n this approximation, the 5 x 5 determinant of the coefficients of the displacements gives a simple second degree equation for the determination of couplings rather than a sixth degree equations as obtained by JOSEPHand SILVERMAN. The sixth degree equation has to be solved by a numerical technique which is definitely a cumbersome procedure. I n the present treatment, we are concerned with a second degree equation which can be solved easily and without much labour we are sure that there will be only two real solutions of k,, k, and k, for each value of @. Acknowledgments We are thankful to the Department of At,omic Energy, Bombay and the Council of Scientific and Industrial Research, New Delhi for financial assistance. References [l] SILVERMAN, B. D., Phys. Rev. 125 (1962) 1921.  SILVERMAN, B. D., Phys. Rev.. 131 (1963) 2478. B. D., Technical Memorandum T-584, Ra-ytheon Company, Waltham,  SILVERMAN, Massachusetts 1964 (unpublished).  SILVERMAN, B. D., Phys. Rev. 135 (1964) A 1596.  VINOGRADOV, V. S., Proceedings of International Conference on Lattice Dynamics, Copenhagen, Pergamon Prees Ltd., New York 1965, p. 421. 161 D v ~ R ~V., K ,and P. GLOGAR, Phys. Rev. 143 (1966) 344.  COCHRAN,W., Adv. Phys. 9 (1960) 387.  RAJAGOPAL, A. K., and R. SRINIVASAN, J. Phys. Chem. Solids 23 (1962) 633.  Dv~RAK, V., and V. JANOVEC, Czech. J. Phys. B 12 (1962) 461. [lo] FOWLER, R. H., Statistical Mechanics, Cambridge University Press, Cambridge 1936. [ll] JOSEPH, R. I., and B. D. SILVERMAN, J. Phys. Chem. Solids 24 (1963) 1349.  SLATER,J. C., Phys. Rev. 78 (1950) 748.  LAST,J. T., Phys. Rev. 106 (1957) 1740.  BARKER,A. S., and M. TINKHAM, Phys. Rev. 125 (1962) 1527.  COWLEY,R. A., Phys. Rev. 184(1964) A 981. and L. E. HOWORTH.Phys. Rev. [l6] SPITZER,W. G., R. C. MILLER, D. A. KLEIMANN 126 (1962) 1710.  LEFKOWITZ, I.. Proc. phys. Soc. London 80 (1962) 868. A l l a h a b a d - B/India, University of Allahabad, Department of Physics. Bei der Redaktion eingegangen am 28. Juli 1969.