u Àa
À
ð
Þ¼u Àa
þ
ð
Þ,
T Àa
þ
ð
ÞÀT Àa
À
ð
Þ¼F=A,
φ Àa
À
ð
Þ¼φ Àa
þ
ð
Þ, D Àa
À
ð
Þ¼D Àa
þ
ð
Þ,
n Àa
À
ð
Þ¼n Àa
þ
ð
Þ, J
n
Àa
À
ð
Þ¼J
n
Àa
þ
ð
Þ,
ð3:126Þ
u a
À
ð Þ ¼ u a
þ
ð Þ, T a
þ
ð Þ À T a
À
ð Þ ¼ ÀF=A,
φ a
À
ð Þ ¼ φ a
þ
ð Þ, D a
À
ð Þ ¼ D a
þ
ð Þ,
n a
À
ð Þ ¼ n a
þ
ð Þ, J
n a
À
ð Þ ¼ J
n a
þ
ð Þ,
ð3:127Þ
T þ1
ð
Þ ¼0, D þ1
ð
Þ ¼ 0, J
n
þ1
ð
Þ ¼ 0:
ð3:128Þ
Equations (3.125), (3.126), (3.127), and (3.128) are 18 conditions, but some of them
are not independent. For example, J
n (À1) ¼ 0, J
n is continuous at Æa, and ∂J
n /
∂x ¼ 0 imply that J
n (1) ¼ 0. Hence, to determine the displacement and potential
fields uniquely, we need to choose À1 as a reference and set
u À1
ð
Þ ¼ 0, φ À1
ð
Þ ¼ 0:
ð3:129Þ
The substitution of Eq. (3.124) for different regions into Eqs. (3.125), (3.126),
(3.127), (3.128), and (3.129) results in a system of linear equations for the
undetermined constants. They are solved on a computer using MATLAB. Then
various electromechanical fields can be calculated.
(a)–(c) of Fig. 3.13 show the effective polarization charge density as defined by
Eq. (3.21), the electron concentration perturbation, and the electric potential produced by two different values of F when n 0 ¼ 10
21
m
À3 and a ¼ 600 nm. There also
exist concentrated polarization charges at x ¼ Æ a. For the purpose of mechanically
manipulating the electrical behavior of the rod, the most basic effect of F is that it
produces a potential well followed by a potential barrier as shown in (c). With the
presence of the potential well and barrier, a mobile charge cannot travel through the
rod unless it has sufficient velocity or kinetic energy. Therefore, a local pair of
concentrated forces forbids the passing of low-energy mobile charges through the
rod. We note the resemblance of the central part of (b) to Fig. 3.8 for the electron
distribution in the extension of a rod by end forces. (d) shows the effect of F on the
current-voltage relation from a nonlinear numerical analysis by COMSOL [7]. It can
be seen that when the applied voltage V is low, no current can flow through the rod in
either direction. When a positive V exceeds a stress-dependent critical value, currents
begin to flow in one direction of the rod but not in the other direction if the sign of the
voltage is changed. When V exceeds a second stress-dependent critical value,
currents can flow in both directions when the sign of V is changed. Therefore the
local stress acts like a switch that has different behaviors in opposite directions. This
provides a basic means for mechanically manipulating currents in a piezoelectric
semiconductor rod.
If the lateral surface of the rod is not traction-free, then the lateral stress T 11 and
T 22 can affect the axial distribution of the electric fields in a similar way through the
piezoelectric constant e 31 .
3.8 Local Extension
57
À
ð
Þ¼u Àa
þ
ð
Þ,
T Àa
þ
ð
ÞÀT Àa
À
ð
Þ¼F=A,
φ Àa
À
ð
Þ¼φ Àa
þ
ð
Þ, D Àa
À
ð
Þ¼D Àa
þ
ð
Þ,
n Àa
À
ð
Þ¼n Àa
þ
ð
Þ, J
n
Àa
À
ð
Þ¼J
n
Àa
þ
ð
Þ,
ð3:126Þ
u a
À
ð Þ ¼ u a
þ
ð Þ, T a
þ
ð Þ À T a
À
ð Þ ¼ ÀF=A,
φ a
À
ð Þ ¼ φ a
þ
ð Þ, D a
À
ð Þ ¼ D a
þ
ð Þ,
n a
À
ð Þ ¼ n a
þ
ð Þ, J
n a
À
ð Þ ¼ J
n a
þ
ð Þ,
ð3:127Þ
T þ1
ð
Þ ¼0, D þ1
ð
Þ ¼ 0, J
n
þ1
ð
Þ ¼ 0:
ð3:128Þ
Equations (3.125), (3.126), (3.127), and (3.128) are 18 conditions, but some of them
are not independent. For example, J
n (À1) ¼ 0, J
n is continuous at Æa, and ∂J
n /
∂x ¼ 0 imply that J
n (1) ¼ 0. Hence, to determine the displacement and potential
fields uniquely, we need to choose À1 as a reference and set
u À1
ð
Þ ¼ 0, φ À1
ð
Þ ¼ 0:
ð3:129Þ
The substitution of Eq. (3.124) for different regions into Eqs. (3.125), (3.126),
(3.127), (3.128), and (3.129) results in a system of linear equations for the
undetermined constants. They are solved on a computer using MATLAB. Then
various electromechanical fields can be calculated.
(a)–(c) of Fig. 3.13 show the effective polarization charge density as defined by
Eq. (3.21), the electron concentration perturbation, and the electric potential produced by two different values of F when n 0 ¼ 10
21
m
À3 and a ¼ 600 nm. There also
exist concentrated polarization charges at x ¼ Æ a. For the purpose of mechanically
manipulating the electrical behavior of the rod, the most basic effect of F is that it
produces a potential well followed by a potential barrier as shown in (c). With the
presence of the potential well and barrier, a mobile charge cannot travel through the
rod unless it has sufficient velocity or kinetic energy. Therefore, a local pair of
concentrated forces forbids the passing of low-energy mobile charges through the
rod. We note the resemblance of the central part of (b) to Fig. 3.8 for the electron
distribution in the extension of a rod by end forces. (d) shows the effect of F on the
current-voltage relation from a nonlinear numerical analysis by COMSOL [7]. It can
be seen that when the applied voltage V is low, no current can flow through the rod in
either direction. When a positive V exceeds a stress-dependent critical value, currents
begin to flow in one direction of the rod but not in the other direction if the sign of the
voltage is changed. When V exceeds a second stress-dependent critical value,
currents can flow in both directions when the sign of V is changed. Therefore the
local stress acts like a switch that has different behaviors in opposite directions. This
provides a basic means for mechanically manipulating currents in a piezoelectric
semiconductor rod.
If the lateral surface of the rod is not traction-free, then the lateral stress T 11 and
T 22 can affect the axial distribution of the electric fields in a similar way through the
piezoelectric constant e 31 .
3.8 Local Extension
57