Page 129 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
value. This multiple eigenvalue phenomenon is well known from linear
algebra, and is called degeneracy. All energies of the hydrogen spectrum
with equal quantum number are degenerate. Note that the Pauli princi-
n
ple is still valid, because the wavefunctions are different. This degener-
acy can be lifted by adding a potential that acts on the respective
wavefunctions differently for different quantum numbers. In fact, nature
provides us with such additional interaction potentials and an exact mea-
surement of the spectrum a hydrogen atom does not show theses degener-
acies. The conclusion is that our model is too simple. Nevertheless, it
explains the principles that the electronic system of atoms follows. One
major incompleteness is that for many-electron systems the Coulomb
interaction between the electrons must be taken into account. This makes
the Schrödinger equation highly non-linear and thus other techniques
including numerics must be used. Moreover, if we want to understand the
formation of crystal symmetry by just putting atoms together, this inter-
action is responsible for the details in the electronic band structure.

Scattering So far we have only dealt with the bound states of the atom and an elec-
tron occupying one of them. There is no dynamics in this picture,
because there is no process represented by an interacting potential that
might change this static situation. Suppose a free electron collides with
the atom, then there are several possibilities:

• the electron gets trapped and subsequently occupies a quantum state
of the atom, it emits a photon carrying away the energy difference
between the free state and the bound state;

• it is scattered and moves on with a different wave-vector, there is a
momentum transfer and an energy transfer between the electron and
the atom;

• the transferred energy excites a bound electron already occupying a
quantum state, the free electron moves on with a lower kinetic energy,
the bound electron either occupies an excited state higher in energy,
or even leaves behind an ionized atom moving freely;

126 Semiconductors for Micro and Nanosystem Technology