Page 345 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

ACOUSTIC WAVE PROPAGATION 325

2
where e kij is the piezoelectric constant in units of C/m , E k is the kth component of the
E
electric field, and c ijkl is measured either under a zero or a constant electric field.
In nonpiezoelectric materials, the electrical displacement D is related to the electric
field applied by
D=s re 0E (10.13)

where e r is the relative permittivity, formerly called the dielectric constant, and e 0 is the
permittivity of vacuum, now known as the electric constant. For piezoelectric materials,
the electrical displacement is extended to:

S
Di = e iklS kl+e ikE k (10.14)
S
where e ik is measured at constant or zero strain.
Equations (10.12) and (10.14) are often referred to as piezoelectric constitutive
equations. In matrix notation, Equations (10.12) and (10.14) can be written as (Auld
1973b):

T
[T] = [c][S]–[e ]E
D = [e][S ] + [e]E (10.15)
where, [e] is a 3 x 6 matrix with its elements depending on the symmetry of the piezo-
T
electric crystal and [e ] is the transpose of the matrix [e]. For quartz having a trigonal
crystal classification, the [e] matrices are

/ e {[ -e n 0 e\4 0 0 \
[e] = 0 0 0 0 -e 14 -e 11 (10.16)
V 0 0 0 0 0 0
The difference between poled and naturally piezoelectric materials is that in the former,
the presence of a large number of grain boundaries and its anisotropic nature would lead
to a loss of acoustic signal fidelity at high frequencies. This is one of the reasons SAW
devices are, usually, only fabricated out of single-crystal piezoelectrics.

10.5 ACOUSTIC WAVE PROPAGATION

10.5.1 Uniform Plane Waves in a Piezoelectric Solid:
Quasi-Static Approximation

For the numerical calculations of acoustic wave propagation, the starting point is the
equation of motion in a piezoelectric material (Auld 1973a)

pu i = T ij.j i, j = 1,2, 3 (10.17)

where, p is the mass density, and u i is the particle displacement.
2
In tensor notation, the two dots over a symbol denotes a /at 2 and a subscript i
preceded by a comma denotes a/ax i. The piezoelectric constitutive equations in (10.15)
are rewritten in tensor notation: