Page 118 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
3.2.3 Periodic Boundary Conditions
A practical approach to set boundary conditions would be to have an infi-
nite box potential with a large extension from x = 0 to x = L forcing
the wavefunction to vanish on the boundaries. This seems to be artificial:
because of the vanishing wavefunction it is hard to imagine how elec-
trons get into or out of the sample. There is another way of introducing
boundary conditions for free electrons in one dimension that requires the
wavefunction to be periodic, i.e., ψ 0() = ψ L() . This allows us to have
plane wave solutions for the Schrödinger equation, that are normalized to
L
the length :
/
ψ x() = L () – 12 exp ( ikx) (3.43)
Applying periodic boundary conditions the wavevector is restricted to

2πn
k = ---------- (3.44)
L
⋅
Taking L = N a , i.e., in multiples of the atomic spacings, there are N
k
discrete values for to fulfill the periodic boundary condition.

3.2.4 Potential Barriers and Tunneling

Let us discuss the situation inverse to that given in Figure 3.3, i.e., a
potential of the form

 U , x ∈ [ 0 a]
,
Ux() =  0 (3.45)
 0, x ∉ [ 0 a]
,
(see also Figure 3.6). We insert the potential (3.45) into the Hamiltonian.
⋅
Again we choose ψ = const. exp ( λx) as the trial solution. Thus we
obtain

– 2mE
,
λ = i ± k = ± --------------- , for x ∉ [ 0 a] (3.46a)
I 2
—

Semiconductors for Micro and Nanosystem Technology 115