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Atomic Displacements in Perovskite Strontium Titanate.

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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[7] 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
[8] 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
[9] 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[12] and LAST[13].
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
[14], Cowley [15], SPITZER
et al. [lG], LEFEOWITZ[17]). 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
[17]):
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[15] 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
[12] 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
[12] 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[13], 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.
[2] SILVERMAN,
B. D., Phys. Rev.. 131 (1963) 2478.
B. D., Technical Memorandum T-584, Ra-ytheon Company, Waltham,
[3] SILVERMAN,
Massachusetts 1964 (unpublished).
[4] SILVERMAN,
B. D., Phys. Rev. 135 (1964) A 1596.
[5] 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.
[7] COCHRAN,W., Adv. Phys. 9 (1960) 387.
[8] RAJAGOPAL,
A. K., and R. SRINIVASAN,
J. Phys. Chem. Solids 23 (1962) 633.
[9] 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.
[12] SLATER,J. C., Phys. Rev. 78 (1950) 748.
[13] LAST,J. T., Phys. Rev. 106 (1957) 1740.
[14] BARKER,A. S., and M. TINKHAM,
Phys. Rev. 125 (1962) 1527.
[15] 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.
[17] 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.
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