- Email: [email protected]
Critical behaviour of SrTiO3 near the 105°K phase transition
~)
SolidStateCommunications, Vol, 88, Nos. 11/12,pp. 901-904, 1993.
0038-I098/9356.00+,00
Printed in Great Britain.
Pergamon Press Ltd
C R I T I C A L BEHAVIOUR OF SrTiOs NEAR T H E 105°K PHASE TRANSITION T. R"-ste, E.J. Samuelsen and K. Otnes Institutt for Atomenergi, 2007 KjeIler, Norway and J. Feder Universitetet i Oslo, Oslo 3, Norway
(Received 1 June 1971 by N.1. Meyer)
The critical behaviour of SrTiO s near the 10S°K structural p h a s e transition has been studied by neutron scattering. T h e study r e v e a l s the e x i s t e n c e of a central critical mode in addition to the condensing soft phonons. In our interpretation this mode e x i s t s b e c a u s e adiab a t i c order fluctuations must become isothermal in the critical region.
AT T H E CUBIC to tetragonal transition in SrTi03, neighbouring TiO c octahedra rotate an angle ~ in opposite s e n s e s about < I 0 0 > . t The thermal average <&>, i.e. the order parameter, is measured by the neutron intensity of superlattice points, I "~ <~b>2. Intensity data are shown in Fig. 1. Extinction corrections were found to be l e s s than 1%. Near the transition temperature Tc fluctuations in c~ contribute with a critical component, which comes in addition to the Bragg intensity. Rotating the c r y s t a l - 0 . 9 ° from the Bragg position, the results of Fig. 2 are obtained. The maximum intensity is at Tc = (106.1 ± 0.3)°K. F o r T > Tc the intensity above the incoherent background is critical scattering, similar to that • 2 • . observed in magnetic s u b s t a n c e s . Critical scattering at Tc - A T is set equal to half the intensity at Tc + A T, in analogy with the observation on magnetic s u b s t a n c e s . Using this with A T = 24°K, and invoking Fig. 1 for the shape of the Bragg scattering, we arrive at curve 2, the residual Bragg intensity, and curve 3 for the critical scattering is obtained with an uncertaintly l e s s than 20 per cent for Tc T < 5°K. -
s0o
03( ½½])
"~400
srT,
-z 300
• • thick crystal . = thin crystal
EO=|3.5meY • m e e•
"~20o abe
~ I0c
,e
3
~"
9()"
1
lOO"
('K)
110"
FIG. 1. Neutron intensity vs. temperature for (1/2 1/2 3[2) superlattice reflection. The dotted curve is obtained after correction for critical scattering near Tc. We are then able to correct the data of Fig. 1 for critical scattering, using curve 3. The correction is significant only for Tc - T < 5°K and does not affect the temperature exponent ]3 very much. The resulting dotted curve is proportional to <¢~>2. For the temperature range 79°K < T < Tc we find < 4 > "~, (To - T ) # with ]3 = 0.343 +0.001.
Previouslypublished in:Solid State Commun. Vol. 9, No. 17, pp. 1455-1458 (I971)
902
SrTiO3 NEAR THE 105°K PHASE TRANSITION
.A
'I . . . . . . . . 0J.,!
80"
05"
Voi. 88, Nos. l 1/12
Tc.oOi.lt.3).,(
....... 90"
95"
100"
".. 105"
110"
115"
120"
125"
('K)
130"
FIG. 2. Residual intensityobserved at q ~, 0.04~ -~ near (1/2 1/2 3/2) when the crystal is rotated out of its Bragg position. Below Tc the intensity is decomposed into residual Bragg scattering (curve 2) and critical scattering (curve 3). E P R measurements by Mflller and Berlinger s give /3 = 0.33 ± 0.02. 04 /
We can now turn to the discussion of critical scattering. The simultaneous existence of two modes, a soft phonon mode and a c e n t r a l m o d e , is exhibited by the energy s c a n s at constant q of Fig. 3. As Tc is approached the phonon energy decreases, the phonon line width increases and the intensity of a central mode increases.
d
• 0/ . . . . . . . ~ O.el_-I O
I(q, o~) o [KS + q=]-' • Dq=. [c~=+ D q']-'
(1)
where K and D are parameters to be determined. We have not found a finite energy width of the central peak; from the instrumental resolution we can deduce D E 0 . 2 c m a s e c -I at I15°K. For comparison we mention that the thermal diffusivity of SrTiOs is about 0 . I cm=sec -I. A finite q-width, however, was observed. From the integrated intensity at q = 0 and 0.04/~-I we find g ¢ [ T - T¢Iv , with 7)' = 0.55 ± 0.09 and v ffi 0.76 ± 0.08 below and above T~, respectively. For the data/led data we refer to the full account of the work. s Uncertainties are mostly due to corrections for instrumental resolution.
1
~
J,~,
2
125 K
........
3
o , I.
4.
.
o,F ..,' 0l
The hypothesis which led to the discovery of the central mode was that a diffusive mode seemed necessary in order to establish correlations over loss distances, in analogy with the diffusive modes at magnetic phase transitions. The intensity distribution expected is then 4
!
.
.
',,/. .
,,~2 , ; r !
1, K
.
. ~,
,~
0
1
,
,
2
,
, ,~---~_I
3
4
!,t:r A 0.5 0
-. -I
'
J
0
l
I
I
I
I
2
I , I
Energy meV
3
I
i"
6
I
FIG. 3. Energy s c a n s at q = 0 for (1/2 1/2 3/2) for three different temperatures. At 125 ° and 115°K the full-drawn curves are calculated ones, n o n - a l i z e d to observed intensity, assuming phonons of infinite lifetime and with energy denoted by arrows. The dashed curves indicate instrumental resolution as seen from incoherent background. At 108°K the curve drawn is only a guide to the eye. Another qualitative observation related to the nature of the central mode is that its neutron intensity varies from superlattice point to superlattice point as the Bragg intensity below Tc does. Since a similar variation exists for the
Vol. 88, Nos. 11/12
SrTiO 3 NEAR THE 105"K PHASE TRANSITION
soft mode, this indicates a relation between the central, critical mode and the soft mode. Finally we give a brief interpretation of the central mode. Well above Tc the order parameter fluctuations are short range and can be described in terms of soft phonons. T h e s e fluctuations are adiabatic and the integrated intensity of the phonon doublet is Idoub(q) "~ x s ( q ) ' where X , is the adiabatic susceptibility describing the response of the order parameter to its conjugate force at wave vector q. The neutron cross section is proportional to the isothermal susceptibility X r ( q ) , and in the critical region x r ( q ) differs significantly from x,(q)" To see this, assume t h a t ' s fluctuation ~be decays adiabatically. Then a s y s t e m a t i c temperature change is produced and <~6(0)~b(t)> can disappear only for times long enough to e s t a b l i s h thermal equilibrium, i,e. for t >> (DTq2) -I , where D~. is the thermal diffusivity. Consequently one expects a contribution to l ( q , ~o) of the form in equation (1). More precisely we find ~ at all temperatures for the integrated i n t e n s i t i e s
Inserting the appropriate values for SrTiO 3 we get A "'~ 5.5/~, (k/Cp As) "~ 0.06 and X T / X s I + 3 . 3 - 10-4 A-2/(KZ + q2). Thus we estimate that at l15°K where K "~ 0.05A -1 , X r / X a > ~ l + 0.1 for q = 0. Experimentally Itot/ldoub is roughly 1.5. As K-, 0, /tot//doub diverges for q = 0, and the central mode dominates i(q, oJ) for T "~ Tc. The width in q-space is determined by K. For T >> T~, X r "~ X, and thus the central mode is significant only near Tc . For T << Tc, <~b2> ~" <~>z and our argument reduces essentially to that used by Heller 7 for the antiferromagnetic c a s e . However, his approach is limited to T < T¢. The general behaviour described here is expected in any second order phase transition. For instance replacing ~( with compressibilities, equation (2) is the total (light scattering) intensity of the Rayleigh line to the Brillouin doublet intensity, well known in the theory of the g a s liquid critical point. A new feature of our theory is that equation (2) gives X ' r / ~ s > 1, even for c a s e s where, with
itot(q)/Idoub(q) = xr(q)/xs(q) = 1 + x r ( q ) .
.903
(2)
The last relation is obtained by thermodynamic arguments. We have made use of (OT/cgc~) s =. TC~l(d~f/aTOc~) where f is the free energy density and C¢ the s p e c i f i c heat at constant ~b. The occurrence of the expection value in equation (2) is essential. It is evaluated using the probability distribution for temperature and order parameter fluctuation. In order to obtain an estimate of the average in equation (2) one may u s e a Landau form for [,
any order parameter ¢, <(aT/ac~)s> is zero. In these c a s e s we have only a second order effect, which , however, becomes all important in the critical region, where order parameter fluctuations become large. We now have the following picture of a soft mode transition. The soft mode itself drives the transition. However, the fluctuation spectrum is in the critical region dominated by the central mode, whose width is related to the thermal diffusivity. .
f = fo(T) + t 2 a ( T ) ~ 2 + t4bcb'÷ z~C(V6) 2. 1
Introducing a ( T ) = a' • ( T - Tc ), ~ = ( C / 2 a ' T c )', and C ¢ " Cp, we obtain xr(q)/xs(q)
= i + TC~a'2<&z> xr(q)
+ ~
[ X2(K 2 + q2)]-'
5
Complete accounts of the experimental and the theoretical part 6 of this work will be given elsewhere.
"~ 1
(3)
where we have used X r ( q ) "~ (C[K 2 + q2])-,, k is the Boltzmann constant.
A c k n o w l e d g e m e n t s - The authors are indebted to Dr. J e n s Lothe for valuable discussions of the theoretical interpretation, and to Dr. G. Shirane and Mr. B. Berre for the loan of the SrTiO 3 samples. Financial support through NATO Research Grant No. 455 is gratefully acknowledged.
904
SrTiO NEAR THE 105°K PHASE TRANSITION
Vol. 88, Nos. ll/12
REFERENCES 1.
SHIRANEG. and YAMADA Y., Phys. Rev. 177, 858 (1969).
2.
MCREYNOLDS A.W. and RISTE T., Phys. Rev. 95, 1161 (1954).
3.
MOLLER K.A. and BERLINGER W., Phys. Rev. Lett. 26, 13 (1971).
4.
VAN HOVE L., Phys. Rev. 95, 249 (1954); 1374 (1954).
5.
RISTE T., SAMUELSEN E.J. and OTNES K., Proc. NATO Adv. Study inst. on 'Structural Phase Transitions and So/t Modes', Geilo, Norway, April (1971). (To be published).
6.
FEDER J., to be published.
7.
HELLER P., Int. J. Magnetism 1, 53 (1971).
Kritische P h e n o m ~ e in SrTiO bei der 105°K stmkturellen PhasenUmwandlung wurden mit Neutronenstreuexperimenten studiert. Zus~tzlich den 'soft modes' wurde eine 'central mode' gefunden. Diese 'central mode' existiert nach unserer Interpretation, well die adiabatischen Ordnungsfluktuationen in dem kritischen Bereich in isothermen Fluktuationen fibergehen m~issen.
SolidStateCommunications, Vol, 88, Nos. 11/12,pp. 901-904, 1993.
0038-I098/9356.00+,00
Printed in Great Britain.
Pergamon Press Ltd
C R I T I C A L BEHAVIOUR OF SrTiOs NEAR T H E 105°K PHASE TRANSITION T. R"-ste, E.J. Samuelsen and K. Otnes Institutt for Atomenergi, 2007 KjeIler, Norway and J. Feder Universitetet i Oslo, Oslo 3, Norway
(Received 1 June 1971 by N.1. Meyer)
The critical behaviour of SrTiO s near the 10S°K structural p h a s e transition has been studied by neutron scattering. T h e study r e v e a l s the e x i s t e n c e of a central critical mode in addition to the condensing soft phonons. In our interpretation this mode e x i s t s b e c a u s e adiab a t i c order fluctuations must become isothermal in the critical region.
AT T H E CUBIC to tetragonal transition in SrTi03, neighbouring TiO c octahedra rotate an angle ~ in opposite s e n s e s about < I 0 0 > . t The thermal average <&>, i.e. the order parameter, is measured by the neutron intensity of superlattice points, I "~ <~b>2. Intensity data are shown in Fig. 1. Extinction corrections were found to be l e s s than 1%. Near the transition temperature Tc fluctuations in c~ contribute with a critical component, which comes in addition to the Bragg intensity. Rotating the c r y s t a l - 0 . 9 ° from the Bragg position, the results of Fig. 2 are obtained. The maximum intensity is at Tc = (106.1 ± 0.3)°K. F o r T > Tc the intensity above the incoherent background is critical scattering, similar to that • 2 • . observed in magnetic s u b s t a n c e s . Critical scattering at Tc - A T is set equal to half the intensity at Tc + A T, in analogy with the observation on magnetic s u b s t a n c e s . Using this with A T = 24°K, and invoking Fig. 1 for the shape of the Bragg scattering, we arrive at curve 2, the residual Bragg intensity, and curve 3 for the critical scattering is obtained with an uncertaintly l e s s than 20 per cent for Tc T < 5°K. -
s0o
03( ½½])
"~400
srT,
-z 300
• • thick crystal . = thin crystal
EO=|3.5meY • m e e•
"~20o abe
~ I0c
,e
3
~"
9()"
1
lOO"
('K)
110"
FIG. 1. Neutron intensity vs. temperature for (1/2 1/2 3[2) superlattice reflection. The dotted curve is obtained after correction for critical scattering near Tc. We are then able to correct the data of Fig. 1 for critical scattering, using curve 3. The correction is significant only for Tc - T < 5°K and does not affect the temperature exponent ]3 very much. The resulting dotted curve is proportional to <¢~>2. For the temperature range 79°K < T < Tc we find < 4 > "~, (To - T ) # with ]3 = 0.343 +0.001.
Previouslypublished in:Solid State Commun. Vol. 9, No. 17, pp. 1455-1458 (I971)
902
SrTiO3 NEAR THE 105°K PHASE TRANSITION
.A
'I . . . . . . . . 0J.,!
80"
05"
Voi. 88, Nos. l 1/12
Tc.oOi.lt.3).,(
....... 90"
95"
100"
".. 105"
110"
115"
120"
125"
('K)
130"
FIG. 2. Residual intensityobserved at q ~, 0.04~ -~ near (1/2 1/2 3/2) when the crystal is rotated out of its Bragg position. Below Tc the intensity is decomposed into residual Bragg scattering (curve 2) and critical scattering (curve 3). E P R measurements by Mflller and Berlinger s give /3 = 0.33 ± 0.02. 04 /
We can now turn to the discussion of critical scattering. The simultaneous existence of two modes, a soft phonon mode and a c e n t r a l m o d e , is exhibited by the energy s c a n s at constant q of Fig. 3. As Tc is approached the phonon energy decreases, the phonon line width increases and the intensity of a central mode increases.
d
• 0/ . . . . . . . ~ O.el_-I O
I(q, o~) o [KS + q=]-' • Dq=. [c~=+ D q']-'
(1)
where K and D are parameters to be determined. We have not found a finite energy width of the central peak; from the instrumental resolution we can deduce D E 0 . 2 c m a s e c -I at I15°K. For comparison we mention that the thermal diffusivity of SrTiOs is about 0 . I cm=sec -I. A finite q-width, however, was observed. From the integrated intensity at q = 0 and 0.04/~-I we find g ¢ [ T - T¢Iv , with 7)' = 0.55 ± 0.09 and v ffi 0.76 ± 0.08 below and above T~, respectively. For the data/led data we refer to the full account of the work. s Uncertainties are mostly due to corrections for instrumental resolution.
1
~
J,~,
2
125 K
........
3
o , I.
4.
.
o,F ..,' 0l
The hypothesis which led to the discovery of the central mode was that a diffusive mode seemed necessary in order to establish correlations over loss distances, in analogy with the diffusive modes at magnetic phase transitions. The intensity distribution expected is then 4
!
.
.
',,/. .
,,~2 , ; r !
1, K
.
. ~,
,~
0
1
,
,
2
,
, ,~---~_I
3
4
!,t:r A 0.5 0
-. -I
'
J
0
l
I
I
I
I
2
I , I
Energy meV
3
I
i"
6
I
FIG. 3. Energy s c a n s at q = 0 for (1/2 1/2 3/2) for three different temperatures. At 125 ° and 115°K the full-drawn curves are calculated ones, n o n - a l i z e d to observed intensity, assuming phonons of infinite lifetime and with energy denoted by arrows. The dashed curves indicate instrumental resolution as seen from incoherent background. At 108°K the curve drawn is only a guide to the eye. Another qualitative observation related to the nature of the central mode is that its neutron intensity varies from superlattice point to superlattice point as the Bragg intensity below Tc does. Since a similar variation exists for the
Vol. 88, Nos. 11/12
SrTiO 3 NEAR THE 105"K PHASE TRANSITION
soft mode, this indicates a relation between the central, critical mode and the soft mode. Finally we give a brief interpretation of the central mode. Well above Tc the order parameter fluctuations are short range and can be described in terms of soft phonons. T h e s e fluctuations are adiabatic and the integrated intensity of the phonon doublet is Idoub(q) "~ x s ( q ) ' where X , is the adiabatic susceptibility describing the response of the order parameter to its conjugate force at wave vector q. The neutron cross section is proportional to the isothermal susceptibility X r ( q ) , and in the critical region x r ( q ) differs significantly from x,(q)" To see this, assume t h a t ' s fluctuation ~be decays adiabatically. Then a s y s t e m a t i c temperature change is produced and <~6(0)~b(t)> can disappear only for times long enough to e s t a b l i s h thermal equilibrium, i,e. for t >> (DTq2) -I , where D~. is the thermal diffusivity. Consequently one expects a contribution to l ( q , ~o) of the form in equation (1). More precisely we find ~ at all temperatures for the integrated i n t e n s i t i e s
Inserting the appropriate values for SrTiO 3 we get A "'~ 5.5/~, (k/Cp As) "~ 0.06 and X T / X s I + 3 . 3 - 10-4 A-2/(KZ + q2). Thus we estimate that at l15°K where K "~ 0.05A -1 , X r / X a > ~ l + 0.1 for q = 0. Experimentally Itot/ldoub is roughly 1.5. As K-, 0, /tot//doub diverges for q = 0, and the central mode dominates i(q, oJ) for T "~ Tc. The width in q-space is determined by K. For T >> T~, X r "~ X, and thus the central mode is significant only near Tc . For T << Tc, <~b2> ~" <~>z and our argument reduces essentially to that used by Heller 7 for the antiferromagnetic c a s e . However, his approach is limited to T < T¢. The general behaviour described here is expected in any second order phase transition. For instance replacing ~( with compressibilities, equation (2) is the total (light scattering) intensity of the Rayleigh line to the Brillouin doublet intensity, well known in the theory of the g a s liquid critical point. A new feature of our theory is that equation (2) gives X ' r / ~ s > 1, even for c a s e s where, with
itot(q)/Idoub(q) = xr(q)/xs(q) = 1 +
.903
(2)
The last relation is obtained by thermodynamic arguments. We have made use of (OT/cgc~) s =. TC~l(d~f/aTOc~) where f is the free energy density and C¢ the s p e c i f i c heat at constant ~b. The occurrence of the expection value in equation (2) is essential. It is evaluated using the probability distribution for temperature and order parameter fluctuation. In order to obtain an estimate of the average in equation (2) one may u s e a Landau form for [,
any order parameter ¢, <(aT/ac~)s> is zero. In these c a s e s we have only a second order effect, which , however, becomes all important in the critical region, where order parameter fluctuations become large. We now have the following picture of a soft mode transition. The soft mode itself drives the transition. However, the fluctuation spectrum is in the critical region dominated by the central mode, whose width is related to the thermal diffusivity. .
f = fo(T) + t 2 a ( T ) ~ 2 + t4bcb'÷ z~C(V6) 2. 1
Introducing a ( T ) = a' • ( T - Tc ), ~ = ( C / 2 a ' T c )', and C ¢ " Cp, we obtain xr(q)/xs(q)
= i + TC~a'2<&z> xr(q)
+ ~
[ X2(K 2 + q2)]-'
5
Complete accounts of the experimental and the theoretical part 6 of this work will be given elsewhere.
"~ 1
(3)
where we have used X r ( q ) "~ (C[K 2 + q2])-,, k is the Boltzmann constant.
A c k n o w l e d g e m e n t s - The authors are indebted to Dr. J e n s Lothe for valuable discussions of the theoretical interpretation, and to Dr. G. Shirane and Mr. B. Berre for the loan of the SrTiO 3 samples. Financial support through NATO Research Grant No. 455 is gratefully acknowledged.
904
SrTiO NEAR THE 105°K PHASE TRANSITION
Vol. 88, Nos. ll/12
REFERENCES 1.
SHIRANEG. and YAMADA Y., Phys. Rev. 177, 858 (1969).
2.
MCREYNOLDS A.W. and RISTE T., Phys. Rev. 95, 1161 (1954).
3.
MOLLER K.A. and BERLINGER W., Phys. Rev. Lett. 26, 13 (1971).
4.
VAN HOVE L., Phys. Rev. 95, 249 (1954); 1374 (1954).
5.
RISTE T., SAMUELSEN E.J. and OTNES K., Proc. NATO Adv. Study inst. on 'Structural Phase Transitions and So/t Modes', Geilo, Norway, April (1971). (To be published).
6.
FEDER J., to be published.
7.
HELLER P., Int. J. Magnetism 1, 53 (1971).
Kritische P h e n o m ~ e in SrTiO bei der 105°K stmkturellen PhasenUmwandlung wurden mit Neutronenstreuexperimenten studiert. Zus~tzlich den 'soft modes' wurde eine 'central mode' gefunden. Diese 'central mode' existiert nach unserer Interpretation, well die adiabatischen Ordnungsfluktuationen in dem kritischen Bereich in isothermen Fluktuationen fibergehen m~issen.