Page 130 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
• the excited electron bounces back to its ground state, emitting a pho-
ton.
All those processes require the transitions of electrons between different
electronic states. The scenario described above is what happens in a gas
discharge lamp. The analogous process can be found in the conduction
band of a semiconductor where an electron is hitting a defect atom that
has a spectrum of localized electronic states below the conduction band-
edge. The energy scale is of course one order of magnitude lower and
there are additional degrees of freedom to which the excess energy of the
electron may be transferred, e.g., phonons.

The quantum mechanical description of the electron as given above
focused on the stationary states of the electrons. Now the time depen-
dence of the Schrödinger equation has must be exploited.

3.2.7 Transitions Between Electronic States
The superposition of the time–dependent solutions of the Schrödinger
equation (3.16) gives the general form of the wavefunction with arbitrary
initial conditions

Ψ x t,( ) = ∑ c exp – ( iω t)ψ x() (3.70)
0
n
n
n
n
where ω = E ⁄ — and the ψ x() in (3.70) are the solution of the sta-
n
n
n
tionary eigen-value problem
ˆ
H 0 ψ = E ψ (3.71)
n n n
The coefficients c are determined by the initial condition, i.e.,
n
|
(
,
0
c = 〈 ψ x() Ψ x 0)〉 . The superscript 0 of the wavefunction and the
n n
subscript 0 of the Hamiltonian indicate the unperturbed stationary prob-
lem.

Semiconductors for Micro and Nanosystem Technology 127