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Electrostriction in a uniaxial ferroelectric with second order phase transition
Volume 90A, number 6
PHYSICS LETTERS
19 July 1982
ELECTROSTRICTION IN A UNIAXIAL FERROELECTRIC WITH SECOND ORDER PHASE TRANSITION B.J. LUYMES Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands Received 18 February 1982
The electrostriction constant ~(m2 v2) of a uniaxial ferroelectric with a second order phase transition has been calculated as a function of the dielectric constant (aP/aE) 0. The results have been experimentally verified on triglycine sulphate (TGS) by measuring~ and (aP/aE)0. A sign reversal of ~ above the Curie temperature is presented.
Predicting electrostrictive constants is difficult. For instance we used a first order Raylor expansion of the polarisation with respect to the electric field and predicted a positive electrostrictive constant in TGS. However we measured a negative one. We therefore introduce a second order approximation below, which explains this discrepancy well. Electrostriction in uniaxial ferroelectrics is described by 2 the following equation [11 L~.L/L= np ~
‘ ‘
‘
where I~.L/Lis the relative change in thickness L of a sample in the direction of the polarisation P (C m2) along the ferroelectricexperiments, axis. Q is a constant, In electrostriction an electric field E is applied to a dielectric sample. The field F consists of an ac field L~Esuperimposed on a much larger poling field F 0. The minimum electric field strength has to be several times higher than the coercive field Ec in order to avoid hysteresis effects. In general, electrostriction is described by = (z~L/L)0 + (d0 + y~E+ ...) LiE (2) where d0 (mV~)describes the linear converse piezoelectric effect. The electrostrictive coefficient is y. The index 0 refers to the situation in which E = E0. The coefficient ‘y can be expressed as a function of Q by expandmg Pm the second order Taylor expansion P—F ~0
,
0~
i ~0
“
2
o 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
where P0 is the polarisation atE = E0 and P~jand P~ are first and second order partial derivatives ofF with respect to F. Combining eqs. (1), (2) and (3) leads to ‘2
y=Q(P0 +Pç~P0)
(L~L/L)0=
d0=2QF0P0
Qr~.
(4,
Our experiments on a uniaxial ferroelectric crystal TGS resulted in a negative valueTc(49.4°C, for y/Q below transition (Curie) temperature [3]). the According to eq. (4) such a negative value of ‘y/Q can be explained by a negative value of ~ The sign ofP 1~can be deduced from thea uniaxial expression 3) for of the elastic Gibbs energy G (J m ferroelectric crystal [4] G= EP + ~I3(T Tc) P2 + ~6F4, (5) —
—
where IG1 I, ~ and 6 are positive constants. T stands for temperature. Differentiating G with respect toP and then twice more with respect to F leads to =
(6)
.
Thermal equilibrium has been assumed, so that ~G/1JP= 0. The positive signs of P0 and P~5result in an expected negative value ofP~jin eq. (6). Combining eqs. (4) and (6) leads to ~ 6b), b = 6P~P6, 0
The coefficient ‘y will now be analysed as a func313
Volume 90A, number 6
PHYSICS LETTERS
tion of T. At temperatures below T~the factors P?j and P6 are equal to O(T~ fl/ö and ~13~1 (T~ —
1
tors P0 and P6 are roughly proportional to (T T~Y [4], leading to b~O.The result is 0
Tc+ [(1 3b)/f3] [6E~/b(l _2b)2]~3.
a
~
—
respectively [4]. This leads to bt4. Above T~the fac
T
19 July 1982
(8)
A relation between ~ and ~6can be derived from calculating ~2G/~ThP and ~2G/~E~P. This leads first to
T~~(T
p
2) y•
~s 2-2 ~omV 40 50 20
0
45
50
T~o
T~°C)
20 -25 i,rj ~
-so 80 ~lOO
EJP 0/6i T = —IIP0P6.
(9)
Differentiating eq. (9) with respect to F and supplementing with eq. (4) gives =
—(Q/13) aPt/aT.
(10)
All calculations are done under isothermal conditions. Differentiating eq. (7) with respect to T and eliminating aF0/a T and ?1P6/a T according to eqs. (9) and (10), respectively, leads to eq. (11), if7 is eliminated finally as given in eq. (7) 3 13(1 3b)(l 1 8b). (11) = —2QP6 According to eq. (7), y shows a sign reversal if b = 1/6. —
—
The temperature T1 at which this happens can be calculated eq. (8). According extrema from at temperatures Tc and to T2 eq. ifb(11)7 = 1/3 has and b = 1/18, respectively. T~and T 2 can also be calculated with the aid of eq. (8). The extrema of 7, i.e. 7(Tc) and 7(T2), are calculated by analogy with the calculations leading to eq. (8): 7(Tc)
=
—~Q(oE~)213,
(12)
Fig. Both the electrostriction constant ‘y and the temperature derivative of the dielectric constant P~of the electrical poled TGS are shown as a function of temperature. An ac measurement determines P/,. The broken line shows aP/~/aT. The tnangles show calculated points of y according to eqs. (8), (12) and (13).
dence of 7 on E0, do not show a sign reversal of y as a function of temperature, since these experiments have been performed without a poling field. In our experiments F0 = 500 V/mm and LiE = 80 V/mm. Hysteresis effects are avoided since Ec(TGS) ~ 40 V/mm above 20°C[8]. ‘y(T~),T 1 and ‘y(T2) are calculated for TGS. The results are marked in fig. 1. We used 2m4 [91,6by= triangles +3.5 X lO~F~K1m [10]Q =and +2.35 C 6 = +6.5 X 1011 F1C2m5 [10]. The value of Q for TGS is constant [11]. Calculating 7~aP~/a~r for each temperature interval of 1 K from 22°Cto 60°Cleads to a value 7 J m3K1 according to of eq.j3/Q (10). of(1.5 ± 0.25) X i0 The results of ç3/Q are shown in fig. 2 and are in good agreement with the figures for 13 and Q mentioned above.
(13) The above described theory of 7 as a function of temperature has been verified on TGS. Fig. 1 shows y~2hasbeenmeasuredbya1ock-intechniqueinan both ~ and aPbIaT. The electrostrictive dilatation electronically stabilized Michelson hiterferometer [5]. P6 has been measured by a Sawyer—Tower circuit [6]. An arbitrary frequency of 67 Hz was used in both cases. Earlier experiments on TGS [7] showing a depen314
20r
(o~~3K~1)1~
30
40~ 50 T(°C)
60
Fig. 2. The coefficient ~t/Q = —(iF/ia fl/~yhas been calculated for TGS for different temperatures. The broken line gives the expected value of~3/Qaccording to refs. [9,10].
Volume 90A, number 6
PHYSICS LETTERS
An inaccuracy in the temperature measurement of 0.5 K caused by a temperature gradient between thermocouple and TGS crystal led to scattering when measuring T~,T1 and T2. The agreement between the calculated and measured values of the extrema of 7 is good, as shown in fig. 1. Strictly correction [1 (10) + 216Cp(Tcspeaking 7)] inan theadiabatic right-hand side of eq. b13 should be introduced. This first order approximation has been calculated, since 7, Q and P6 are experimentally determined adiabatically. C~,is the heat capacity at constant polarisation and is minimal 2.6 X 106 ~ m3K1 for TGS [8]. In calculating the adiabatic correction we used the method given by Devonshire [1]. The correction is less than 1% in TGS and may be neglected in view of the experimental uncertainties in 7, Q and Pb. —
19 July 1982
References [1] A.F. Devonshire,
Adv. Phys. 3 (1954) 99.
[2] J.F. Nye, Physical properties of crystals (Clarendon, [3] Oxford, T. Mitsui,1969). An introduction to the physics of ferroelectnics (Gordon and Breach, New York, 1976). [4] physics, E. Fatuzzo and Merz, SelectedAmsterdam, topics in solid state VoL VIIW.J. (North-Holland, 1967). [5] Th. Kwaaitaal, B.J. Luymes and G.A. van der Pyll, J. Phys. D13 (1980) 1005. [6] C.B. Sawyer and C.H. Tower, Phys. Rev. 35 (1930) 269. [7] A.A. Fotchenkov and M.P. Zaitsewa, Soy. Phys. Crystallogr. 8(1964)579. [8] Landolt—Bornstein, Vol. III, no. 3 (Springer, Berlin, pp. 495, [9] 1969) G. Schmidt and496. P. Pfannschmidt, Phys. Stat. Sol. 3 (1963) 2215. [10] 5. Tniebwasser, IBM J. Res. July (1958) 212. [11] K.H. Ehses and H. Meister, Ferroelectrics 25 (1980) 573.
315
PHYSICS LETTERS
19 July 1982
ELECTROSTRICTION IN A UNIAXIAL FERROELECTRIC WITH SECOND ORDER PHASE TRANSITION B.J. LUYMES Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands Received 18 February 1982
The electrostriction constant ~(m2 v2) of a uniaxial ferroelectric with a second order phase transition has been calculated as a function of the dielectric constant (aP/aE) 0. The results have been experimentally verified on triglycine sulphate (TGS) by measuring~ and (aP/aE)0. A sign reversal of ~ above the Curie temperature is presented.
Predicting electrostrictive constants is difficult. For instance we used a first order Raylor expansion of the polarisation with respect to the electric field and predicted a positive electrostrictive constant in TGS. However we measured a negative one. We therefore introduce a second order approximation below, which explains this discrepancy well. Electrostriction in uniaxial ferroelectrics is described by 2 the following equation [11 L~.L/L= np ~
‘ ‘
‘
where I~.L/Lis the relative change in thickness L of a sample in the direction of the polarisation P (C m2) along the ferroelectricexperiments, axis. Q is a constant, In electrostriction an electric field E is applied to a dielectric sample. The field F consists of an ac field L~Esuperimposed on a much larger poling field F 0. The minimum electric field strength has to be several times higher than the coercive field Ec in order to avoid hysteresis effects. In general, electrostriction is described by = (z~L/L)0 + (d0 + y~E+ ...) LiE (2) where d0 (mV~)describes the linear converse piezoelectric effect. The electrostrictive coefficient is y. The index 0 refers to the situation in which E = E0. The coefficient ‘y can be expressed as a function of Q by expandmg Pm the second order Taylor expansion P—F ~0
,
0~
i ~0
“
2
o 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
where P0 is the polarisation atE = E0 and P~jand P~ are first and second order partial derivatives ofF with respect to F. Combining eqs. (1), (2) and (3) leads to ‘2
y=Q(P0 +Pç~P0)
(L~L/L)0=
d0=2QF0P0
Qr~.
(4,
Our experiments on a uniaxial ferroelectric crystal TGS resulted in a negative valueTc(49.4°C, for y/Q below transition (Curie) temperature [3]). the According to eq. (4) such a negative value of ‘y/Q can be explained by a negative value of ~ The sign ofP 1~can be deduced from thea uniaxial expression 3) for of the elastic Gibbs energy G (J m ferroelectric crystal [4] G= EP + ~I3(T Tc) P2 + ~6F4, (5) —
—
where IG1 I, ~ and 6 are positive constants. T stands for temperature. Differentiating G with respect toP and then twice more with respect to F leads to =
(6)
.
Thermal equilibrium has been assumed, so that ~G/1JP= 0. The positive signs of P0 and P~5result in an expected negative value ofP~jin eq. (6). Combining eqs. (4) and (6) leads to ~ 6b), b = 6P~P6, 0
The coefficient ‘y will now be analysed as a func313
Volume 90A, number 6
PHYSICS LETTERS
tion of T. At temperatures below T~the factors P?j and P6 are equal to O(T~ fl/ö and ~13~1 (T~ —
1
tors P0 and P6 are roughly proportional to (T T~Y [4], leading to b~O.The result is 0
Tc+ [(1 3b)/f3] [6E~/b(l _2b)2]~3.
a
~
—
respectively [4]. This leads to bt4. Above T~the fac
T
19 July 1982
(8)
A relation between ~ and ~6can be derived from calculating ~2G/~ThP and ~2G/~E~P. This leads first to
T~~(T
p
2) y•
~s 2-2 ~omV 40 50 20
0
45
50
T~o
T~°C)
20 -25 i,rj ~
-so 80 ~lOO
EJP 0/6i T = —IIP0P6.
(9)
Differentiating eq. (9) with respect to F and supplementing with eq. (4) gives =
—(Q/13) aPt/aT.
(10)
All calculations are done under isothermal conditions. Differentiating eq. (7) with respect to T and eliminating aF0/a T and ?1P6/a T according to eqs. (9) and (10), respectively, leads to eq. (11), if7 is eliminated finally as given in eq. (7) 3 13(1 3b)(l 1 8b). (11) = —2QP6 According to eq. (7), y shows a sign reversal if b = 1/6. —
—
The temperature T1 at which this happens can be calculated eq. (8). According extrema from at temperatures Tc and to T2 eq. ifb(11)7 = 1/3 has and b = 1/18, respectively. T~and T 2 can also be calculated with the aid of eq. (8). The extrema of 7, i.e. 7(Tc) and 7(T2), are calculated by analogy with the calculations leading to eq. (8): 7(Tc)
=
—~Q(oE~)213,
(12)
Fig. Both the electrostriction constant ‘y and the temperature derivative of the dielectric constant P~of the electrical poled TGS are shown as a function of temperature. An ac measurement determines P/,. The broken line shows aP/~/aT. The tnangles show calculated points of y according to eqs. (8), (12) and (13).
dence of 7 on E0, do not show a sign reversal of y as a function of temperature, since these experiments have been performed without a poling field. In our experiments F0 = 500 V/mm and LiE = 80 V/mm. Hysteresis effects are avoided since Ec(TGS) ~ 40 V/mm above 20°C[8]. ‘y(T~),T 1 and ‘y(T2) are calculated for TGS. The results are marked in fig. 1. We used 2m4 [91,6by= triangles +3.5 X lO~F~K1m [10]Q =and +2.35 C 6 = +6.5 X 1011 F1C2m5 [10]. The value of Q for TGS is constant [11]. Calculating 7~aP~/a~r for each temperature interval of 1 K from 22°Cto 60°Cleads to a value 7 J m3K1 according to of eq.j3/Q (10). of(1.5 ± 0.25) X i0 The results of ç3/Q are shown in fig. 2 and are in good agreement with the figures for 13 and Q mentioned above.
(13) The above described theory of 7 as a function of temperature has been verified on TGS. Fig. 1 shows y~2hasbeenmeasuredbya1ock-intechniqueinan both ~ and aPbIaT. The electrostrictive dilatation electronically stabilized Michelson hiterferometer [5]. P6 has been measured by a Sawyer—Tower circuit [6]. An arbitrary frequency of 67 Hz was used in both cases. Earlier experiments on TGS [7] showing a depen314
20r
(o~~3K~1)1~
30
40~ 50 T(°C)
60
Fig. 2. The coefficient ~t/Q = —(iF/ia fl/~yhas been calculated for TGS for different temperatures. The broken line gives the expected value of~3/Qaccording to refs. [9,10].
Volume 90A, number 6
PHYSICS LETTERS
An inaccuracy in the temperature measurement of 0.5 K caused by a temperature gradient between thermocouple and TGS crystal led to scattering when measuring T~,T1 and T2. The agreement between the calculated and measured values of the extrema of 7 is good, as shown in fig. 1. Strictly correction [1 (10) + 216Cp(Tcspeaking 7)] inan theadiabatic right-hand side of eq. b13 should be introduced. This first order approximation has been calculated, since 7, Q and P6 are experimentally determined adiabatically. C~,is the heat capacity at constant polarisation and is minimal 2.6 X 106 ~ m3K1 for TGS [8]. In calculating the adiabatic correction we used the method given by Devonshire [1]. The correction is less than 1% in TGS and may be neglected in view of the experimental uncertainties in 7, Q and Pb. —
19 July 1982
References [1] A.F. Devonshire,
Adv. Phys. 3 (1954) 99.
[2] J.F. Nye, Physical properties of crystals (Clarendon, [3] Oxford, T. Mitsui,1969). An introduction to the physics of ferroelectnics (Gordon and Breach, New York, 1976). [4] physics, E. Fatuzzo and Merz, SelectedAmsterdam, topics in solid state VoL VIIW.J. (North-Holland, 1967). [5] Th. Kwaaitaal, B.J. Luymes and G.A. van der Pyll, J. Phys. D13 (1980) 1005. [6] C.B. Sawyer and C.H. Tower, Phys. Rev. 35 (1930) 269. [7] A.A. Fotchenkov and M.P. Zaitsewa, Soy. Phys. Crystallogr. 8(1964)579. [8] Landolt—Bornstein, Vol. III, no. 3 (Springer, Berlin, pp. 495, [9] 1969) G. Schmidt and496. P. Pfannschmidt, Phys. Stat. Sol. 3 (1963) 2215. [10] 5. Tniebwasser, IBM J. Res. July (1958) 212. [11] K.H. Ehses and H. Meister, Ferroelectrics 25 (1980) 573.
315