1. Physics 342 Lecture 28 Band Theory of Electronic Structure in Solids
Problems from Chapter 10: 1, 6, 7, 9, 10, 14, 18, 19, 20 and 23
Due next Wednesday the 20th
Read Chapter 11
We will be discussing a lot about semiconductors
and metals

2. Band Theory of Solids
To understand the electronic states of a solid we must consider the presence of many N~1023 atoms
The band theory of solids describes the interaction between the electrons and the lattice ions that comprise a solid

3. Band Theory: “Bound” Electron Approach
For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width E.
Leads to a band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level).
Two atoms
Six atoms
Solid of N atoms
Electrons must occupy different energies due to Pauli Exclusion principle.
Phys Baski

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

5. 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

6. 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

7. 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

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

9. 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

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

11. Fermi-Dirac “Filling” Function
Probability of electrons to be found at various energy levels.
For E – EF = 0.05 eV  f(E) = 0.12 For E – EF = 7.5 eV  f(E) = 10 –129
Exponential dependence has HUGE effect!
Temperature dependence of Fermi-Dirac function shown as follows:

12. 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

13. 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

14. 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

15. Summary
In solids, the discrete energy levels of the individual atoms merge to form energy bands
Energy gaps arise in solids because they contain standing wave states
The size of the energy gap between the valence and conduction bands determines whether a substance is a conductor, an insulator or a semiconductor