Chapter 1
Macroscopic Theory
This chapter presents a concise summary of the basic equations of the phenomenological theory of piezoelectric semiconductors. The Cartesian tensor notation is used,
along with the summation convention for repeated tensor indices and the convention
that a comma followed by an index denotes partial differentiation with respect to the
coordinate associated with the index. A superimposed dot represents a time
derivative.
1.1 Basic Theory
Piezoelectric materials may be dielectrics or semiconductors. Mechanical fields and
mobile charges in piezoelectric semiconductors can interact through electric fields,
which is called the acoustoelectric effect. The basic behaviors of piezoelectric
semiconductors can be described by coupling the linear theory of piezoelectricity
[1–4] and the macroscopic theory of nonpiezoelectric semiconductors [5, 6]. The
three-dimensional phenomenological theory consists of the equation of motion
(Newton’s law), the charge equation of electrostatics (Gauss’s law), and the conservation of charge for holes and electrons (continuity equations):
T ji,j þ f i ¼ ρ€ u i ,
D i,i ¼ q p À n þ N
þ
D À N
À
A
À
Á
,
_
p ¼ À
1
q
J
p
i,i þ _
pj thermal R‐G þ _
pj other processes ,
_
n ¼
1
q
J
n
i,i þ _
nj thermal R‐G þ _
nj other processes ,
ð1:1Þ
where T is the stress tensor, ρ the mass density, f the body force which is usually
zero, u the mechanical displacement vector, and D the electric displacement vector.
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_1
1
Macroscopic Theory
This chapter presents a concise summary of the basic equations of the phenomenological theory of piezoelectric semiconductors. The Cartesian tensor notation is used,
along with the summation convention for repeated tensor indices and the convention
that a comma followed by an index denotes partial differentiation with respect to the
coordinate associated with the index. A superimposed dot represents a time
derivative.
1.1 Basic Theory
Piezoelectric materials may be dielectrics or semiconductors. Mechanical fields and
mobile charges in piezoelectric semiconductors can interact through electric fields,
which is called the acoustoelectric effect. The basic behaviors of piezoelectric
semiconductors can be described by coupling the linear theory of piezoelectricity
[1–4] and the macroscopic theory of nonpiezoelectric semiconductors [5, 6]. The
three-dimensional phenomenological theory consists of the equation of motion
(Newton’s law), the charge equation of electrostatics (Gauss’s law), and the conservation of charge for holes and electrons (continuity equations):
T ji,j þ f i ¼ ρ€ u i ,
D i,i ¼ q p À n þ N
þ
D À N
À
A
À
Á
,
_
p ¼ À
1
q
J
p
i,i þ _
pj thermal R‐G þ _
pj other processes ,
_
n ¼
1
q
J
n
i,i þ _
nj thermal R‐G þ _
nj other processes ,
ð1:1Þ
where T is the stress tensor, ρ the mass density, f the body force which is usually
zero, u the mechanical displacement vector, and D the electric displacement vector.
© Springer Nature Switzerland AG 2020
J. Yang, Analysis of Piezoelectric Semiconductor Structures,
https://doi.org/10.1007/978-3-030-48206-0_1
1