Page 111 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
Bound
Electrons, There are harmonic solutions for E > U 0 , because the radicands in
(3.28a) and (3.28b) are always negative in this case and thus the solution
Free
λ
for is imaginary. Electrons in these states are called unbound electrons
Electrons
or free electrons. Their λ is the well known wavevector and their
k
wavefunctions do not necessarily vanish for x → ± ∞ . We shall discuss
their properties together with the free electron states. For now we are
interested in the solutions with energy eigenvalues E < U , called bound
0
states. Their wavefunction decays exponentially in regions I and III (see
Figure 3.3). The plus sign in (3.28b) must hold for x < 0 , while the
Vx()

V
0

x
0 a
Figure 3.3. Finite box potential in
one dimension.

minus sign must hold for x > a . In the regions I and III we call λ = κ .
Inside the box we have a harmonic solution, with wavevector . Thus we
k
obtain

ψ x() = Aexp ( κx) (3.29a)
I
x
ψ () = Bexp ( ikx) + Cexp – ( ikx) (3.29b)
II
x
ψ () = Dexp – ( κx) (3.29c)
III
The boundary conditions read

108 Semiconductors for Micro and Nanosystem Technology