γ ¼
q
k B T
e
b εe ck
b λ þ b pb c
A
1
ð Þ
þ A
2
ð Þ
Θ 0 ,
e c ¼
b c þ e
2
=b ε
A
1
ð Þ
þ A
2
ð Þ
:
ð7:106Þ
e c is an averaged, piezoelectrically stiffened elastic constant. At the end of a long rod
where kL is large, we have sinhkL ffi cosh kL. Therefore γ describes the relative
change of carrier concentration near the ends of the rod per unit relative temperature
change. It can be used as a measure of the strength of the coupling effect of the
thermally induced carrier redistribution in the rod. We are interested in the change of
number of carriers in the differential element in Fig. 7.14. From Eq. (7.105), for a
θ=0.1K
θ=0.5K
θ=1.0K
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
-1.0
-0.5
0.0
0.5
1.0
Δn
(m
-3
)
×10
22
z (µm)
Fig. 7.15 Electron
concentration perturbation
in the rod for different
temperature change.
n 0 ¼ 10
23
/m
3
n 0 =10
21
/m
3
n 0 =10
22
/m
3
n 0 =10
23
/m
3
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
-1.0
-0.5
0.0
0.5
1.0
Δn (m
-3
)
×10
21
z (µm)
Fig. 7.16 Electron
concentration perturbation
in the rod for different n 0 .
θ¼0.1 K
202
7 Thermal Effects