ψ ¼ ξx 3 À ωt, ω
2
¼
b c 33
ρ
ξ
2 ,
b c 33 ¼ c 33 þ
e
2
33
ε 33
, B ¼
e 33
ε 33
A:
ð3:183Þ
The electric field accompanying the wave in Eq. (3.182) is given by
E ¼ B
0 sin ψ, B
0
¼
e 33
ε 33
Aξ:
ð3:184Þ
With the electric potential in Eq. (3.182) considered as known, Eq. (3.181)
becomes linear in n. However, it has variable coefficients depending on both x 3
and t through φ and is still a mathematically challenging problem. We consider
uniform doping and denote n ¼ N
þ
D . We write n as the sum of a uniform static part n
which is a constant and a dynamic part b n x 3 , t
ð
Þ caused by the waves in Eq. (3.182),
i.e.,
n x 3 , t
ð
Þ ¼ n þ b n x 3 , t
ð
Þ:
ð3:185Þ
Substituting Eq. (3.185) into Eq. (3.181), we obtain the following equation for b n:
_
b n ¼ μ 33 E 3,3 n þ μ 33 E 3,3 b n þ μ 33 E 3 b n ,3 þ D 33 b n ,33 ,
ð3:186Þ
where the electric field E is given by Eq. (3.184) which, when substituted into
Eq. (3.186), results in
_
b n ¼ μ 33 nB
0
ξ cos ψ þ μ 33 B
0
ξ cos ψ b n þ μ 33 B
0 sin ψ b n ,3 þ D 33 b n ,33 :
ð3:187Þ
The first three terms on the right-hand side of Eq. (3.187) are from the nonlinear drift
current, and the last term is from the diffusion current. The first term does not depend
on b n and is effectively a driving term from the wave in Eq. (3.182).
Consider a special case of Eq. (3.187) first. If b n is very small and only the first
term on the right-hand side of Eq. (3.187) is kept, the equation reduces to
_
b n ¼ μ 33 nB
0
ξ cos ψ,
ð3:188Þ
which admits the following solution
b n ¼ Àμ 33 nB
0 ξ
ω
sin ψ:
ð3:189Þ
Equation (3.189) describes a simple sinusoidal wave motion of the carriers driven by
the electromechanical wave in Eq. (3.182).
74
3 Extension of Rods
2
¼
b c 33
ρ
ξ
2 ,
b c 33 ¼ c 33 þ
e
2
33
ε 33
, B ¼
e 33
ε 33
A:
ð3:183Þ
The electric field accompanying the wave in Eq. (3.182) is given by
E ¼ B
0 sin ψ, B
0
¼
e 33
ε 33
Aξ:
ð3:184Þ
With the electric potential in Eq. (3.182) considered as known, Eq. (3.181)
becomes linear in n. However, it has variable coefficients depending on both x 3
and t through φ and is still a mathematically challenging problem. We consider
uniform doping and denote n ¼ N
þ
D . We write n as the sum of a uniform static part n
which is a constant and a dynamic part b n x 3 , t
ð
Þ caused by the waves in Eq. (3.182),
i.e.,
n x 3 , t
ð
Þ ¼ n þ b n x 3 , t
ð
Þ:
ð3:185Þ
Substituting Eq. (3.185) into Eq. (3.181), we obtain the following equation for b n:
_
b n ¼ μ 33 E 3,3 n þ μ 33 E 3,3 b n þ μ 33 E 3 b n ,3 þ D 33 b n ,33 ,
ð3:186Þ
where the electric field E is given by Eq. (3.184) which, when substituted into
Eq. (3.186), results in
_
b n ¼ μ 33 nB
0
ξ cos ψ þ μ 33 B
0
ξ cos ψ b n þ μ 33 B
0 sin ψ b n ,3 þ D 33 b n ,33 :
ð3:187Þ
The first three terms on the right-hand side of Eq. (3.187) are from the nonlinear drift
current, and the last term is from the diffusion current. The first term does not depend
on b n and is effectively a driving term from the wave in Eq. (3.182).
Consider a special case of Eq. (3.187) first. If b n is very small and only the first
term on the right-hand side of Eq. (3.187) is kept, the equation reduces to
_
b n ¼ μ 33 nB
0
ξ cos ψ,
ð3:188Þ
which admits the following solution
b n ¼ Àμ 33 nB
0 ξ
ω
sin ψ:
ð3:189Þ
Equation (3.189) describes a simple sinusoidal wave motion of the carriers driven by
the electromechanical wave in Eq. (3.182).
74
3 Extension of Rods