Page 119 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
V 0 Vx()

Figure 3.6. Barrier potential and
sketched real part of a harmonic x
0 a
wavefunction hitting the barrier
with energy E = V ⁄ 2 , i.e.,
0
smaller than V 0 .

(
2mU – E)
0
,
λ = ± k = ± ----------------------------- , for x ∈ [ 0 a] (3.46b)
II 2
—
There are no bound states in this case because the region where harmonic
solutions exist is infinite. The states with energies E > U 0 show the
inverse behavior as for the box potential (3.25). This is indicated in
Figure 3.6. Remember that there exist solutions that decay exponentially
inside the barrier and extend to infinity on either side of the barrier for
E < U 0 .

We analyze this situation further. Consider a free electron impinging the
barrier from region I to have a definite wavevector k I . Thus our trial
functions read

ψ x() = a exp ( ik x) + a exp – ( ik x) (3.47a)
1
2
I
I
I
ψ () = b exp ( k x) + b exp k – ( x) (3.47b)
x
II 1 II 2 II
ψ () = c exp ( ik x) + c exp – ( ik x) (3.47c)
x
III 1 I 2 I
We assume the impinging wave to have unit amplitude (a = 1 ) and fur-
1
ther that there is no wave travelling from region III towards the barrier
(c = 0 ). Note that k = k . On either side of the barrier the condi-
2 I III
tions for the continuity of the wavefunction ψ and its spatial derivative

116 Semiconductors for Micro and Nanosystem Technology