1. Band Theory of Solids
So far we have neglected the lattice of positively charged ions
Moreover, we have ignored the Coulomb repulsion between the electrons and the attraction between the lattice and the electrons
The band theory of solids takes into account the interaction between the electrons and the lattice ions

2. Band Theory of Solids Consider the potential energy of a
1-dimensional solid
which we approximate by the Kronig-Penney Model

3. Band Theory of Solids The task is to compute the quantum states and
associated energy levels of this simplified model
by solving the Schrödinger equation 1 2 3

4. Band Theory of Solids For periodic potentials, Felix Bloch showed that
the solution of the Schrödinger equation must
be of the form
and the wavefunction must
reflect the periodicity of
the lattice: 1 2 3

5. Band Theory of Solids By requiring the wavefunction and its derivative
to be continuous everywhere, one finds energy
levels that are grouped into bands separated by
energy gaps. The gaps occur at
The energy gaps
are basically energy
levels that cannot
occur in the solid 1 2 3

6. Band Theory of Solids
Completely free
electron
electron in a
lattice

7. Band Theory of Solids When, the wavefunctions become
standing waves. One wave peaks at the lattice
sites, and another peaks between them. Ψ2, has
lower energy
than Ψ1. Moreover, there is a jump in energy
between these states, hence the energy gap

8. Band Theory of Solids The allowed ranges of the wave vector k are
called Brillouin zones.
zone 1: -p/a < k < p/a; zone 2: -2p/a < - p/a;
zone 3: p/a < k < 2p/a etc.
The theory can explain why some
substances are conductors, some
insulators and
others
semi
conductors

9. Conductors, Insulators, Semiconductors
Sodium (Na) has one electron in the 3s state, so
the 3s energy level is half-filled. Consequently, the
3s band, the valence band, of solid
sodium is also half-filled. Moreover,
the 3p band, which for Na is
the conduction band, overlaps with
the 3s band.
So valence electrons can easily be
raised to higher energy states.
Therefore, sodium is a good
conductor

10. Conductors, Insulators, Semiconductors
NaCl is an insulator, with a band gap of 2 eV,
which is much larger than the thermal energy at
T=300K
Therefore, only a tiny fraction of electrons are
in the conduction band

11. Conductors, Insulators, Semiconductors
Silicon and germanium have band gaps of 1 eV and
0.7 eV, respectively. At room temperature, a small
fraction of the electrons are in the conduction
band. Si and Ge are intrinsic semiconductors