Page 283 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
7.6.1 Lattice Defects
Suppose that at specific lattice sites R there might be a different atom
j
sitting, an atom missing, or even a built in distortion of the lattice be
present. This will lead to the fact that the periodic potential of the crystal
lattice that the electrons suffer V x –( R ) will be different from the
j
unperturbed periodic potential V x –( R ) . Let us define the deviation
0 j
v x –( R ) = V x – R ) V x – R ) . The additional potential that
(
(
–
j j 0 j
enters the Schrödinger equation as a perturbation is built by the sum over
all impurity sites j. So the transition matrix element between two Bloch
waves as defined in (3.92) will yield
n'k' nk ∑ ( ( ( , ( ( , 3
V , = ∫ exp i kk')x)u k' x)v x – R )u kx)d x (7.97)
–
j
n'
n
j
where and denote the different bands of the electronic states and k
n'
n
and k' their respective wavevectors.
Let us assume that the impurities are distributed statistically so that we
can identify an impurity density, i.e.
N
∑ v x –( R ) = ------ I ∑ v x –( R ) (7.98)
j
i
j N 0 i
where in (7.98) the averaged sum over j counts all the impurity sites and
the sum over i counts all the lattice sites. For a periodic perturbation
potential thus the wavevector is a “good” quantum number, i.e., k will be
preserved in this scattering, at least in lowest order perturbation theory.
0 ()
Therefore, we have V = N v δ , where the superscript (0) indi-
,
n'k' nk I n'n k'k
cates the lowest order approximation. Inside a band nothing will change
due to that. This only affects interband scattering of electron.

We take a quick look at terms of higher order, i.e., averages of the form

280 Semiconductors for Micro and Nanosystem Technology