Page 137 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
(
(
ux() =   ( Ae – ik – κ)x + Be – ik + κ)x ), for 0<x<a (3.96)
 ( Ce ( – ik – λ)x + De ( – ik + λ)x ), for -b<x<0
⁄
⁄
(
where κ = 2mE — and λ = 2mV – E) — . To solve for the coef-
0
ficients A, B, C and D in (3.96) we apply the boundary. The wavefunc-
tions and their derivatives must be equal at x = 0 (u 0() = u 0() and
1
2
u′ 0() = u′ 0() ), and the wavefunctions and their derivatives at
1
2
(
x = – b must equal those at x = a (u a() = u – b) and
1
2
(
u′ a() = u′ – b) ). This gives us a homogeneous system of four linear
2
1
equations, which has only non-trivial solutions if its determinant is zero,
which yields
2
λ – κ 2
-----------------sinh ( λb)sin ( κa) + cosh ( λb)cos ( κa) = cos ( kL) (3.97)
2κλ
We see that already for the simple periodic potential barrier model (3.97)
cannot be solved analytically. A graphical solution is given in
Figure 3.11.

1
κa

-1
i = 1 2 3 4 5 6 ...
Figure 3.11. Graphical solution
for equation (3.97) that shows the
emergence of allowable energy
bands.

134 Semiconductors for Micro and Nanosystem Technology