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Pyroelectricity and the strength of many-body forces
Volume 47A, number 4
PHYSICS LETTERS
8 April 1974
P Y R O E L E C T R I C I T Y AND T H E S T R E N G T H O F M A N Y - B O D Y F O R C E S P.J. GROUT and N.H. MARCH Department o f Physics, Imperial College, South Kensington, London, S. IV. 7., UK Received 19 February 1974 The linear relation between pyroelectric coefficient and specific heat for ZnO is interpreted in terms of rigid ions. Some other materials behave differently and exhibit stronger pyroelectrieity. This is interpreted as due to many-body forces. Conditions favourable for pyroelectricity are thereby suggested.
Pyroelectric materials are currently of technological interest [ 1] and it is of some importance therefore to attempt to understand pyroelectricity in fundamental terms. The purpose of this work is firstly to propose an interpretation of the experimental result of Heiland and Ibach [2] that the pyroelectric coefficient of ZnO, with wurtzite structure, is accurately proportional to the specific heat, whereas such a relation is not obeyed for materials having different structures and larger pyroelectric coefficients. Secondly, we are led to suggest conditions which should favour strong pyroelectricity. Finally, some experiments are proposed to test further the interpretation put forward here. By considering the thermodynamics of pyreelectric crystals, it can be shown [3] that the total pyroelectric coefficient p = dPs/dT, where Ps is the spontaneous polarization, consists of the sum of two terms, a true or primary coefficient Pt and an apparent or secondary term Pa" Furthermore, the term Pa can be expressed in terms of the elastic constants c, piezoelectric constants d and the thermal expansion ot as
pa= diik ~klm 'Vlm'
(1) summation over repeated indices being implied. Unfortunately, few measurements of c, d and a are available over a wide range of temperature. However, it is already clear from (1) that, at low temperatures, the thermal expansion a dominates the temperature dependence o f p a, leading to Pa proportional to T 3. The major problem is therefore to understand the behaviour of the true coefficient Pt" In this connection, we note that for ZnO, measurements exist over 288
a substantial temperature range, p itself varies as T 3 at low temperatures and there is a linear relation between measured specific heat and pyroelectric coefficient over a wide temperature range, as can be seen from figs. 1 and 2 of Heiland and Ibach [2]. In the course of treating non-linear optical effects, Garrett [4] noted that a one-dimensional shell model, with of course appropriate anharmonicity, can explain Pt varying linearly with specific heat. But it might seem from his argument that such a relation should apply generally in pyroelectrics, whereas subsequent experimental work has shown that this is not the case. To clarify this situation, with particular reference to ZnO, some properties of the wurtzite structure are summarized below. Following Bedincourt, Jaffe and Shiozawa [5] we start from a model which assumes: a) Rigid ions b) That the ideal wurtzite structure is preserved independently of temperature and pressure. The consequences of this model are readily derived [5] and the results are: i) The two thermal expartsion constants must be equal. ii) The elastic compliances satisfy the relation S l l + S 1 2 = $33 + S 1 3
(2)
iii) The resultant electric displacement is zero, hence the pyroelectric effect disappears and the piezoelectric constants obey the relation d33 + 2d31 = 0.
(3)
Values of four materials with the wurtzite structure
Volume 47A, number 4
PHYSICS LETTERS
are given below BeO CdS CdSe ZnO
d33 2d31 X 1012 0.24 -0.24 10.32 -10.36 7.84 -7.84 10.60 - 10.42
c/N
The relation (3) i~-~een to be strikingly well obeyed. We stress that (3), derived from a rigid-ion model [5], will not be expected to hold when many body forces are important. Actually, of course, the pyroelectric constants of these materials are not zero, but from experiment they are an order of magnitude smaller than for other pyroelectrics. For the wurtzite structure, Berlincourt et al. interpret the non-zero values as due to the temperature dependence of the deviation from the ideal wurtzite structure, which is also revealed by some observed anisotropy of the thermal expansion. It should also be noted that the elastic isotropy condition (2) is satisfied within 5% for CdS and CdSe. Our conclusion, therefore, is that, allowing for nonideality in the c/a ratio, the data available on the wurtzite structure is usefully interpreted on the basis of a rigid ion model. There is one obvious difficulty though, which must be referred to: such a model will not correctly relate the long wavelength limit of the transverse optic mode to the longitudinal optic-mode, since the generalised Lyddane-Sachs-Teller relation [6] would apply with e** = 1. For the above wurtzite materials co, >3. We are currently investigating this point further, but from the point of view of relating the various macroscopic properties discussed above, the relevant phonon modes appear to be usefully described by a rigid-ion model. Further support for such a model in the wurtzite structure is offered by Keffer and Portis [7] who show that the rather subtle deviations of the hexagonal-close-packed lattice c/a ratio from its ideal value can be understood on the rigid-ion picture. The interpretation then that we propose for the result that in ZnO p is proportional to the specific heat is that it is a direct consequence of rigid-ion behaviour and that the departures from it in other structures are due to the inapplicability of rigid-ion model of the lattice dynamics in such materials. To support this on theoretical grounds, it is perhaps worth noting that, if we switch off the shell term in
8 April 1974
Garrett's model [4], we are indeed still left with Pt proportional to the specific heat. Secondly, a much earlier paper of Born [8] also leads to such a result for a rigid-ion model. The cases known for which p is not proportional to the specific heat are characterized by pyroelectric moments an order of magnitude larger than for the wurtzite materials and suggest that departures from a rigid-ion model, which are equivalent to many-body forces being involved in the dynamical motion of the atoms, lead to markedly enhanced pyroelectricity. This conclusion from the above data is also supported by Born's work [8]. While the details of this probably should not be pressed, it leads to the conclusion that the mean thermal energy of an oscillator, which enters linearly the expression for the polarization, has to be replaced, when the rigid-ion model is inapplicable, by the mean square amplitude of the oscillators, in suitable units. This then leads to pcx T, not T 3 , at low temperatures. This prompts us to suggest that conditions for strong pyroelectricity, obtained by maximizing manybody forces, may include some or all of the following: i) An appreciable element of covalency should be present in the bonding. Materials that are polar, but can be prepared in amorphous as well as crystalline form, often have a substantial component of covalency. ii) Strong electric fields should be generated when ions are moved off the lattice sites by freezing in a phonon, the electric fields at the equilibrium sites being zero by the Hellmann - Feynman theorem. Presumably, this is likely to be most often the case in the lower symmetry structures. iii) Related to ii), polarizable ions should be involved. Mate'rials with heavy anions would seem to be favourable. iv) The Debye temperature should be low, to get an appreciable pyroelectric moment at a given temperature, say 300°K. v) Deviations from Cauchy relations between elastic constants should be large. Finally, it would be of interest to carry out some experiments to test the interpretation given here as to the role of many-body forces: a) Measurement of the pyroelectric coefficient in ZnO and similar wurtzite materials down to the lowest possible temperatures. The temperature at which the pyroelectric coefficient deviates from a T 3 law 289
Volume 47A, number 4
PHYSICS LETTERS.
should be a measure of residual many-body forces. b) The study of a system like I Br, with crystal class C2v, and particularly a comparison o f the temperature dependence of its apparent coefficient Pa with the total coefficient p over a wide temperature range. We are particularly grateful to Drs. T.P. McLean and E.H. Putley for interesting us in this problem and to them and their colleagues at RRE for much help and encouragement. Discussions with Professors C.W. McCombie, M.P. Tosi, G.S. Handler and Dr. M.J. Norgett have also been valuable. Support from Mullard's and from the Ministry of Defence during the course of the work is acknowledged.
290
8 April 1974
References [ 1] H. Boot and--R~.Taylor, Contemp. Phys., 14 (1973) 55. [2] G. Heiland and H. Ibach, Solid State. Commun. 4 (1966) 353. [3l J. Nye, Physical properties of crystals (Oxford University Press) 1957. [4] C.G.B. Garrett, IEEE J. Quantum Electronics QE4 (1968) 70. [5] D. Berlincourt, D., H. Jaffe and I.R. Shiozawa, Phys. Rev. 129 (1963) 1009. [6] W. Cochran and R.A. Cowley, Phys. Chem. Solids 23 (1962) 447. [71 F. Keffer and A.M. Portis, J. Chem. Phys. 27 (1957) 675. [81 M. Born, Rev. Mod. Phys. 17 (1945) 245.
PHYSICS LETTERS
8 April 1974
P Y R O E L E C T R I C I T Y AND T H E S T R E N G T H O F M A N Y - B O D Y F O R C E S P.J. GROUT and N.H. MARCH Department o f Physics, Imperial College, South Kensington, London, S. IV. 7., UK Received 19 February 1974 The linear relation between pyroelectric coefficient and specific heat for ZnO is interpreted in terms of rigid ions. Some other materials behave differently and exhibit stronger pyroelectrieity. This is interpreted as due to many-body forces. Conditions favourable for pyroelectricity are thereby suggested.
Pyroelectric materials are currently of technological interest [ 1] and it is of some importance therefore to attempt to understand pyroelectricity in fundamental terms. The purpose of this work is firstly to propose an interpretation of the experimental result of Heiland and Ibach [2] that the pyroelectric coefficient of ZnO, with wurtzite structure, is accurately proportional to the specific heat, whereas such a relation is not obeyed for materials having different structures and larger pyroelectric coefficients. Secondly, we are led to suggest conditions which should favour strong pyroelectricity. Finally, some experiments are proposed to test further the interpretation put forward here. By considering the thermodynamics of pyreelectric crystals, it can be shown [3] that the total pyroelectric coefficient p = dPs/dT, where Ps is the spontaneous polarization, consists of the sum of two terms, a true or primary coefficient Pt and an apparent or secondary term Pa" Furthermore, the term Pa can be expressed in terms of the elastic constants c, piezoelectric constants d and the thermal expansion ot as
pa= diik ~klm 'Vlm'
(1) summation over repeated indices being implied. Unfortunately, few measurements of c, d and a are available over a wide range of temperature. However, it is already clear from (1) that, at low temperatures, the thermal expansion a dominates the temperature dependence o f p a, leading to Pa proportional to T 3. The major problem is therefore to understand the behaviour of the true coefficient Pt" In this connection, we note that for ZnO, measurements exist over 288
a substantial temperature range, p itself varies as T 3 at low temperatures and there is a linear relation between measured specific heat and pyroelectric coefficient over a wide temperature range, as can be seen from figs. 1 and 2 of Heiland and Ibach [2]. In the course of treating non-linear optical effects, Garrett [4] noted that a one-dimensional shell model, with of course appropriate anharmonicity, can explain Pt varying linearly with specific heat. But it might seem from his argument that such a relation should apply generally in pyroelectrics, whereas subsequent experimental work has shown that this is not the case. To clarify this situation, with particular reference to ZnO, some properties of the wurtzite structure are summarized below. Following Bedincourt, Jaffe and Shiozawa [5] we start from a model which assumes: a) Rigid ions b) That the ideal wurtzite structure is preserved independently of temperature and pressure. The consequences of this model are readily derived [5] and the results are: i) The two thermal expartsion constants must be equal. ii) The elastic compliances satisfy the relation S l l + S 1 2 = $33 + S 1 3
(2)
iii) The resultant electric displacement is zero, hence the pyroelectric effect disappears and the piezoelectric constants obey the relation d33 + 2d31 = 0.
(3)
Values of four materials with the wurtzite structure
Volume 47A, number 4
PHYSICS LETTERS
are given below BeO CdS CdSe ZnO
d33 2d31 X 1012 0.24 -0.24 10.32 -10.36 7.84 -7.84 10.60 - 10.42
c/N
The relation (3) i~-~een to be strikingly well obeyed. We stress that (3), derived from a rigid-ion model [5], will not be expected to hold when many body forces are important. Actually, of course, the pyroelectric constants of these materials are not zero, but from experiment they are an order of magnitude smaller than for other pyroelectrics. For the wurtzite structure, Berlincourt et al. interpret the non-zero values as due to the temperature dependence of the deviation from the ideal wurtzite structure, which is also revealed by some observed anisotropy of the thermal expansion. It should also be noted that the elastic isotropy condition (2) is satisfied within 5% for CdS and CdSe. Our conclusion, therefore, is that, allowing for nonideality in the c/a ratio, the data available on the wurtzite structure is usefully interpreted on the basis of a rigid ion model. There is one obvious difficulty though, which must be referred to: such a model will not correctly relate the long wavelength limit of the transverse optic mode to the longitudinal optic-mode, since the generalised Lyddane-Sachs-Teller relation [6] would apply with e** = 1. For the above wurtzite materials co, >3. We are currently investigating this point further, but from the point of view of relating the various macroscopic properties discussed above, the relevant phonon modes appear to be usefully described by a rigid-ion model. Further support for such a model in the wurtzite structure is offered by Keffer and Portis [7] who show that the rather subtle deviations of the hexagonal-close-packed lattice c/a ratio from its ideal value can be understood on the rigid-ion picture. The interpretation then that we propose for the result that in ZnO p is proportional to the specific heat is that it is a direct consequence of rigid-ion behaviour and that the departures from it in other structures are due to the inapplicability of rigid-ion model of the lattice dynamics in such materials. To support this on theoretical grounds, it is perhaps worth noting that, if we switch off the shell term in
8 April 1974
Garrett's model [4], we are indeed still left with Pt proportional to the specific heat. Secondly, a much earlier paper of Born [8] also leads to such a result for a rigid-ion model. The cases known for which p is not proportional to the specific heat are characterized by pyroelectric moments an order of magnitude larger than for the wurtzite materials and suggest that departures from a rigid-ion model, which are equivalent to many-body forces being involved in the dynamical motion of the atoms, lead to markedly enhanced pyroelectricity. This conclusion from the above data is also supported by Born's work [8]. While the details of this probably should not be pressed, it leads to the conclusion that the mean thermal energy of an oscillator, which enters linearly the expression for the polarization, has to be replaced, when the rigid-ion model is inapplicable, by the mean square amplitude of the oscillators, in suitable units. This then leads to pcx T, not T 3 , at low temperatures. This prompts us to suggest that conditions for strong pyroelectricity, obtained by maximizing manybody forces, may include some or all of the following: i) An appreciable element of covalency should be present in the bonding. Materials that are polar, but can be prepared in amorphous as well as crystalline form, often have a substantial component of covalency. ii) Strong electric fields should be generated when ions are moved off the lattice sites by freezing in a phonon, the electric fields at the equilibrium sites being zero by the Hellmann - Feynman theorem. Presumably, this is likely to be most often the case in the lower symmetry structures. iii) Related to ii), polarizable ions should be involved. Mate'rials with heavy anions would seem to be favourable. iv) The Debye temperature should be low, to get an appreciable pyroelectric moment at a given temperature, say 300°K. v) Deviations from Cauchy relations between elastic constants should be large. Finally, it would be of interest to carry out some experiments to test the interpretation given here as to the role of many-body forces: a) Measurement of the pyroelectric coefficient in ZnO and similar wurtzite materials down to the lowest possible temperatures. The temperature at which the pyroelectric coefficient deviates from a T 3 law 289
Volume 47A, number 4
PHYSICS LETTERS.
should be a measure of residual many-body forces. b) The study of a system like I Br, with crystal class C2v, and particularly a comparison o f the temperature dependence of its apparent coefficient Pa with the total coefficient p over a wide temperature range. We are particularly grateful to Drs. T.P. McLean and E.H. Putley for interesting us in this problem and to them and their colleagues at RRE for much help and encouragement. Discussions with Professors C.W. McCombie, M.P. Tosi, G.S. Handler and Dr. M.J. Norgett have also been valuable. Support from Mullard's and from the Ministry of Defence during the course of the work is acknowledged.
290
8 April 1974
References [ 1] H. Boot and--R~.Taylor, Contemp. Phys., 14 (1973) 55. [2] G. Heiland and H. Ibach, Solid State. Commun. 4 (1966) 353. [3l J. Nye, Physical properties of crystals (Oxford University Press) 1957. [4] C.G.B. Garrett, IEEE J. Quantum Electronics QE4 (1968) 70. [5] D. Berlincourt, D., H. Jaffe and I.R. Shiozawa, Phys. Rev. 129 (1963) 1009. [6] W. Cochran and R.A. Cowley, Phys. Chem. Solids 23 (1962) 447. [71 F. Keffer and A.M. Portis, J. Chem. Phys. 27 (1957) 675. [81 M. Born, Rev. Mod. Phys. 17 (1945) 245.