Chapter 2
Exact Solutions
This chapter gathers a few exact solutions satisfying the three-dimensional equations
of piezoelectric semiconductors. Solutions like these are possible only in a few
relatively special cases. Most of them are based on the linearized theory.
Section 2.1 is simple but nontrivial. Section 2.2 is a one-dimensional problem
mathematically in the sense that there is only one spatial variable. Sections 2.3,
2.4, and 2.5 are antiplane problems. Section 2.3 is axisymmetric and
one-dimensional. The other two are two-dimensional.
2.1 Uniform Doping
N
À
A and N
þ
D in Eq. (1.1) 2 may depend on spatial variables representing nonuniform
doping. In the special case of uniform doping, N
À
A and N
þ
D are constants. In this case,
it can be verified that
p ¼ N
À
A , n ¼ N
þ
D ,
E ¼ 0, D ¼ 0,
J
p
¼ 0, J
n
¼ 0,
S ¼ 0, T ¼ 0,
ð2:1Þ
satisfy the static form of Eqs. (1.1) and (1.2) when there is no body force, recombination, and generation. Hence, for a finite body without surface traction, charge,
and currents, Eq. (2.1) is the solution of the problem. From Eq. (1.5), the displacement field can still have an undetermined rigid-body displacement. Similarly, the
electric potential can still have an arbitrary constant.
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_2
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