Page 133 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
When the impact of the switched on interaction is over we obtain
2π
(
Γ mk = ------ V mk 2 δω ) (3.82)
mk
2
—
(3.82) is a very important result, called the Fermi golden rule. The transi-
k
tion rate from state to m is determined by the squared matrix element
of the V mk of the perturbing potential, where the delta-function δω( mk )
ensures that the energy is conserved.
The approach of constant perturbation in this section can be extended to
potentials which vary arbitrarily in time. We will not go into detail but
rather explain the result. Let V ω t,( ) = V exp ± ( ωt) be a perturbation
0
ω
varying with a fixed frequency and an amplitude V 0 . Then the transi-
tion rate becomes

2π
(
Γ = ------ V 2 δω ± ω) (3.83)
mk 2 mk mk
—
The difference between (3.83) and (3.82) is that the energies E and E
m k
of the initial and final states differ now by —ω , which is the energy of the
ω
special perturbation mode with frequency . We already know that elec-
trons couple to periodically oscillating phenomena such as phonons and
electromagnetic waves. Hence, (3.83) will be the basis for describing the
scattering events that electrons in a conduction band of a semiconductor
experience.

3.2.8 Fermion number operators and number states

In the same manner as for the harmonic oscillator we represent the quan-
+
tum state of a fermion by means of a creation operator c and a destruc-
tion operator using the Dirac notation. Defining a fermion vacuum 0|〉
c
we have

+
c 0|〉 = λ |〉 (3.84)
λ

130 Semiconductors for Micro and Nanosystem Technology