Page 271 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
and ψ
electrons had two possible states ψ
where they may be found
1
2
in, e.g., the valence or conduction band edge. As we know from electro-
statics and from the polar optical phonon interaction, the energy is given
by the product of the incident electric field E and the dipole moment p
of the electronic system. In classical terms the dipole moment is given as
charge times distance p = er . Assume that the electric field varies little
in space with respect to the extension of the electronic states. Since the
charge of the electron has a density distribution according to the wave-
function Ψ r t,( ) , which for the unperturbed problem is a superposition
,
(
Ψ r t) = j ∑ c t()exp – ( iω t)ψ r() . The matrix elements read
j
j
j
Dipole V = ∫ ψ ∗ r()erE R t,( )ψ r() Vd (7.80)
mn m n
approxi-
mation
where R is the position of the center of mass of the dipole. We may inter-
pret the integral in (7.80) as a dipole moment that couples to an external
field. In most cases the wavefunctions are of this kind that V mm = , 0
i.e., there is no static dipole moment present in the system. The only pos-
sibility to have a non vanishing matrix element is for m ≠ n . We return to
a harmonic wave and assume the electronic system to be rather localized,
e.g., atomic electron states. In addition we deal with wavelengths of the
electric field that are much larger than the extension of the atom. So that
the dipole approximation holds. Then we have

⁄
Transition V 12 = V ∗ = µµ µ µ ( exp ( iωt) + exp ( i – ωt))ER() 2 (7.81)
12
21
Dipole Matrix
Element with µµ µµ = ∫ ψ ∗ r()erψ r() V , which usually are called the transition
d
12 1 2
dipole matrix elements.
7.4.1 Intra- and Interband Effects

There are three basic processes involved in the interaction: interband
transition, intra-band transitions, and free carrier absorption (see
Figure 7.9).

268 Semiconductors for Micro and Nanosystem Technology