18 On Converse Piezoelectricity
343
18.3.3 Measuring Surface-Dependent Converse Piezoelectricity
In order to measure the converse piezoelectric coefficient we propose the following
approach using, for illustrative purposes, the linear chain shown in Fig. 18.1. Our
approach guarantees that surface effects, other than those discussed above, are eliminated as soon as the samples are large enough to reach the thermodynamic limit.
We also do not need to assume that the surface regions have an almost vanishing
spatial extent. Finally, this approach can be used to separate the coefficients d 0 and
d 1 as well as to measure the direct piezoelectric effect.
Let us consider a set of samples that have the same terminations but different
lengths. Thus, they differ only in the size of the central region (at least, to a good
approximation for a large system). When these samples are exposed to an electrostatic field their length, l, will change. Since the termination regions may respond
differently than the central region
l = l R + l L + K C · a
= l R0 (1 + d R E) + l L0 (1 + d L E) + K C a 0 (1 + d C E),
(18.34)
where d X is the converse piezoelectric coefficient for region X and the subscript 0
again indicates the field-free value. We seek the converse piezoelectric coefficient d C . To that end it is convenient to re-write Eq. (18.34) as
l = l 0 (1 + d C E) + l R0 (d R − d C )E + l L0 (d L − d C )E,
(18.35)
whereas l R0 and l L0 are independent of the size of the sample, l 0 is not. Thus, a plot
of l as a function of l 0 for samples of different lengths, but subject to the same field,
will give a straight line with slope (1 + d C E). Ultimately, by repeating the whole
process for another set of samples with different terminal charges, it is possible
to separate the bulk converse piezoelectric coefficient into a reference value and a
surface-dependent term as done in Eq. (18.32).
18.4 Metals
So far we have assumed that the occupied and unoccupied orbitals are separated by
a finite gap, i.e., that the system is insulating or semiconducting. It is natural to ask
whether it is possible to treat a metallic system with the approaches presented here.
As it turns out, there are aspects in which a metal is fundamentally different from a
semiconductor, or an insulator, that make it impossible to do so.
First, we discuss the infinite periodic case. In principle, for a metal the electronic
part of the dipole moment per unit could be formulated using an expression like
that of Eq. (18.8). The only difference would be that the sum over occupied orbitals
would not include the complete band, i.e., the j summation in Eq. (18.8) would have
an upper limit that is k-dependent. Although the substitution of Eq. (18.16) is still
possible, this means that the conditions of Eqs. (18.17) and (18.18) cannot be used
343
18.3.3 Measuring Surface-Dependent Converse Piezoelectricity
In order to measure the converse piezoelectric coefficient we propose the following
approach using, for illustrative purposes, the linear chain shown in Fig. 18.1. Our
approach guarantees that surface effects, other than those discussed above, are eliminated as soon as the samples are large enough to reach the thermodynamic limit.
We also do not need to assume that the surface regions have an almost vanishing
spatial extent. Finally, this approach can be used to separate the coefficients d 0 and
d 1 as well as to measure the direct piezoelectric effect.
Let us consider a set of samples that have the same terminations but different
lengths. Thus, they differ only in the size of the central region (at least, to a good
approximation for a large system). When these samples are exposed to an electrostatic field their length, l, will change. Since the termination regions may respond
differently than the central region
l = l R + l L + K C · a
= l R0 (1 + d R E) + l L0 (1 + d L E) + K C a 0 (1 + d C E),
(18.34)
where d X is the converse piezoelectric coefficient for region X and the subscript 0
again indicates the field-free value. We seek the converse piezoelectric coefficient d C . To that end it is convenient to re-write Eq. (18.34) as
l = l 0 (1 + d C E) + l R0 (d R − d C )E + l L0 (d L − d C )E,
(18.35)
whereas l R0 and l L0 are independent of the size of the sample, l 0 is not. Thus, a plot
of l as a function of l 0 for samples of different lengths, but subject to the same field,
will give a straight line with slope (1 + d C E). Ultimately, by repeating the whole
process for another set of samples with different terminal charges, it is possible
to separate the bulk converse piezoelectric coefficient into a reference value and a
surface-dependent term as done in Eq. (18.32).
18.4 Metals
So far we have assumed that the occupied and unoccupied orbitals are separated by
a finite gap, i.e., that the system is insulating or semiconducting. It is natural to ask
whether it is possible to treat a metallic system with the approaches presented here.
As it turns out, there are aspects in which a metal is fundamentally different from a
semiconductor, or an insulator, that make it impossible to do so.
First, we discuss the infinite periodic case. In principle, for a metal the electronic
part of the dipole moment per unit could be formulated using an expression like
that of Eq. (18.8). The only difference would be that the sum over occupied orbitals
would not include the complete band, i.e., the j summation in Eq. (18.8) would have
an upper limit that is k-dependent. Although the substitution of Eq. (18.16) is still
possible, this means that the conditions of Eqs. (18.17) and (18.18) cannot be used