1. Condensed Matter Physics: review
Homework solutions posted today
Exam 2 solutions will be posted by the end of
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2. Free Electron Gas in Metals
Solid metals are bonded by the metallic bond
One or two of the valence electrons from each atom are free to move throughout the solid
All atoms share all the electrons. A metal is a lattice of positive ions immersed in a gas of electrons. The binding between the electrons and the lattice is what holds the solid together

3. 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:

4. Free Electron Gas in Metals
Since we are dealing with fermions, the
electron gas obeys Fermi-Dirac statistics.
The number of electrons with
energy in the range E to E+dE is given by
where, the density of states g(E) is given by

5. Free Electron Gas in Metals
The electron has two spin states, so W = 2.
If the electron’s speed << c, we can take its
energy to be
So the number of states in (E, E+dE) is given by

6. Free Electron Gas in Metals
The number of electrons in the interval
E to E+dE is therefore
The first term is the Fermi-Dirac distribution
and the second is the density of states g(E)dE

7. Free Electron Gas in Metals
From n(E) dE we can calculate many global characteristics of the electron gas. Here are just a few
The Fermi energy – the maximum energy level occupied by the free electrons at absolute zero
The average energy
The total number of electrons in the electron gas

8. Free Electron Gas in Metals
The total number of electrons N is given by
the integral
At T=0, the Fermi-Dirac distribution is equal
to 1 if E < EF and 0 if E > EF.
At T=0, all energy levels are filled up to
the energy EF, called the Fermi energy.

9. Free Electron Gas in Metals
The total number of electrons N is given by
The average energy of a free electron is given by

10. Free Electron Gas in Metals
At T = 0, the integrals are easy to do.
For example, the total number of electrons is

11. Free Electron Gas in Metals
The average energy of an electron is
This implies
EF = k TF defines
the Fermi
temperature

12. Summary of metallic state
The ions in solids form regular lattices
A metal is a lattice of positive ions immersed in a gas of electrons. All ions share all electrons
The attraction between the electrons and the lattice is called a metallic bond
At T = 0, all energy levels up to the Fermi energy are filled

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

14. 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

15. 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

16. 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

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

18. 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

19. 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