Chapter 7
Thermal Effects
In this chapter we study thermal effects in piezoelectric semiconductor structures
through pyroelectric and thermoelastic couplings. Both homogeneous and composite structures are considered.
7.1 Extension of Homogeneous Rods
Consider the free piezoelectric semiconductor rod shown in Fig. 7.1. The rod is
within |x 3 | < L. The c-axis of the (6mm) crystal is along the x 3 axis. There are no
electromechanical loads. We study the electromechanical fields in the rod associated
with the extensional deformation under a uniform temperature change [1].
Consider a uniform and known temperature change denoted by θ ¼ Θ À Θ 0 from
a uniform reference temperature Θ 0 . At Θ 0 , the carrier concentrations are p 0 ¼ N
À
A
and n 0 ¼ N
þ
D which are assumed to be constants. Under the given temperature
change, the perturbations of the carrier concentrations are denoted by Δp and Δn.
For convenience we collect the relevant equations from Sect. 1.5. They are valid for
the rest of the chapter. Since the temperature change is known, the heat equation in
Eq. (1.43) 3 is not needed. The electromechanical fields are governed by
T ji,j ¼ ρ€ u i ,
ð7:1Þ
D i,i ¼ q Δp À Δn
ð
Þ ,
ð7:2Þ
J
p
i,i ¼ Àq
∂ Δp
ð Þ
∂t
,
J
n
i,i ¼ q
∂ Δn
ð Þ
∂t
,
ð7:3Þ
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_7
177