With successive substitutions from Eqs. (1.2) and (1.5), we can write Eq. (1.1) as six
equations for u, φ, p, and n. This book is based on the macroscopic theory in
Eqs. (1.1), (1.2), and (1.5) without the use of any information from the microscopic
theory of semiconductors except Eq. (1.4) which serves as a relationship between the
mobility and the diffusion constants. The right-hand side of Eq. (1.1) 2 is in terms of
p-n. Therefore, working with p-n and p+n instead of p and n may have some
mathematical advantages. N
À
A ÀN
þ
D appears as a load in Eq. (1.1) 2 . When N
À
A ÀN
þ
D
and f are zero, Eq. (1.1) becomes homogeneous. Then the state with u, φ, p, and n all
being zero satisfies Eq. (1.1), which may be used as the reference state. For a finite
piezoelectric semiconductor body, the electric field in the surrounding free space is
usually neglected as an approximation. Then, on the surface of a finite body, the
mechanical displacement vector or the traction vector, the electric potential or the
normal component of D, and the carrier concentrations or the normal components of
the current density vectors may be prescribed as boundary conditions. Integrating
Eq. (1.1) 2 over a volume V with boundary surface S and unit outward normal n, we
obtain
Z
S
n i D i dS ¼
Z
V
q p À n þ N
þ
D À N
À
A
À
Á
dV,
ð1:6Þ
which is a relationship among the boundary normal value of D and doping as well as
carrier concentrations. Similar relationships can also be derived from the stress
equation of motion and the continuity equations, in particular their static form.
1.2 Linearization
Because of the nonlinearity in Eq. (1.2) 3,4 associated with the drift currents of holes
and electrons which are the products of the unknown carrier concentrations and the
unknown electric field, theoretical analyses of piezoelectric semiconductor devices
may present considerable mathematical challenges. For many purposes, a linearized
version of Eq. (1.2) 3,4 is sufficient. Consider the following change of unknown
variables from p and n to Δp and Δn:
p ¼ p 0 þ Δp,
n ¼ n 0 þ Δn,
ð1:7Þ
where, for simplicity, we have denoted
p 0 ¼ N
À
A , n 0 ¼ N
þ
D :
ð1:8Þ
1.2 Linearization
3
equations for u, φ, p, and n. This book is based on the macroscopic theory in
Eqs. (1.1), (1.2), and (1.5) without the use of any information from the microscopic
theory of semiconductors except Eq. (1.4) which serves as a relationship between the
mobility and the diffusion constants. The right-hand side of Eq. (1.1) 2 is in terms of
p-n. Therefore, working with p-n and p+n instead of p and n may have some
mathematical advantages. N
À
A ÀN
þ
D appears as a load in Eq. (1.1) 2 . When N
À
A ÀN
þ
D
and f are zero, Eq. (1.1) becomes homogeneous. Then the state with u, φ, p, and n all
being zero satisfies Eq. (1.1), which may be used as the reference state. For a finite
piezoelectric semiconductor body, the electric field in the surrounding free space is
usually neglected as an approximation. Then, on the surface of a finite body, the
mechanical displacement vector or the traction vector, the electric potential or the
normal component of D, and the carrier concentrations or the normal components of
the current density vectors may be prescribed as boundary conditions. Integrating
Eq. (1.1) 2 over a volume V with boundary surface S and unit outward normal n, we
obtain
Z
S
n i D i dS ¼
Z
V
q p À n þ N
þ
D À N
À
A
À
Á
dV,
ð1:6Þ
which is a relationship among the boundary normal value of D and doping as well as
carrier concentrations. Similar relationships can also be derived from the stress
equation of motion and the continuity equations, in particular their static form.
1.2 Linearization
Because of the nonlinearity in Eq. (1.2) 3,4 associated with the drift currents of holes
and electrons which are the products of the unknown carrier concentrations and the
unknown electric field, theoretical analyses of piezoelectric semiconductor devices
may present considerable mathematical challenges. For many purposes, a linearized
version of Eq. (1.2) 3,4 is sufficient. Consider the following change of unknown
variables from p and n to Δp and Δn:
p ¼ p 0 þ Δp,
n ¼ n 0 þ Δn,
ð1:7Þ
where, for simplicity, we have denoted
p 0 ¼ N
À
A , n 0 ¼ N
þ
D :
ð1:8Þ
1.2 Linearization
3