Page 114 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
here: simply because also the best simulation program is useless if there
is no understanding of how to obtain a rough estimate of what the result
has to be. We suggest implementing a simple numerical model for the
calculation of 1D eigenvalue problems as is shown in Figure 3.8 for the
harmonic oscillator potential.

Infinite Box We continue our discussion with the case where U → ∞ . A simple
0
Potential physical argument tells us that for the energy of the system to be finite
(the same holds for many other observable quantities), the wavefunction
must vanish in the region where Ux() → ∞ , i.e., ψ x() = 0 for x ≥ la
and x ≤ 0 and for all . The trial solution for the wavefunction
t
ψ x() = Bexp ( ikx) + Cexp – ( ikx) must fulfill the boundary condition
ψ 0() = 0 and ψ a() = 0 . This leads to ψ x() = Bsin ( kx) , with
⁄
k = ( nπ) a and integer and positive starting at n = 1 . Inserting this
n
into the time–independent Schrödinger equation (3.17) gives us the
energy spectrum of the infinite box potential:

2
π —
,,,
2
E = -------------n , with n = 123 … (3.34)
n 2
2ma
The normalized wavefunction reads
2  nπx 
ψ x() = ---sin  --------- (3.35)
n
a a 
We give another explanation to approach the infinite box potential. Sup-
2
2
pose that U » — ⁄ ( ma ) holds. Then there will be a part of the spec-
0
trum with low energies for which the wavefunction decays on a very
short length scale outside the box, i.e., the electron is completely trapped
inside the box.

Three– The above results lead us immediately to the case of an electron in a
Dimensional three-dimensional potential box. Suppose the box is a cuboid region with
Potential 0 ≤≤ a , 0 ≤≤ a , and 0 ≤≤ a . Inside the box Ux y z,,( ) = , 0
z
y
x
Boxes
while outside Ux y z,,( ) → ∞ . The energy spectrum looks like

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