Thomas Ouldridge
Keble College - Hillary 2011
Section VI: Condensed Matter Physics
Tutorial 4 - Nearly Free Electrons, Tight Binding Model and Semiconductors
Please leave your work in my Keble pigeon hole by 5pm on Wednesday of 7th week.
Suggested reading: Hook & Hall 4, 5; Kittel 7,8; Ashcroft/Mermin 8-12,28.
1
The Tight Binding Model
1. Refer back to the tutorial sheet on chemical bonding, Qn. 2. There we took a two-nucleus
system, and constructed approximate eigenstates from combining the lowest-lying states of the
isolated nuclei. We now consider a one-dimensional monatomic chain with a single atomic
orbital per atom. Assuming, as usual, independent electrons, the Hamiltonian for a single
electron is given by:
H=
N
X
ε|nihn|
−
n
t|n + 1ihn|
−
t|n − 1ihn| .
(1)
Here, N is the number of atoms in the system, |ni is the orbital around the nth atom, ε is
the energy of the atomic orbital and t is a hopping integral, with the simplifying assumption
hn|mi = δn,m .
Define |ki states by
|ni =
√1
N
X
k
e−ikna |ki
,
hn| =
√1
N
X
k
eikna hk|
(2)
with a as the lattice parameter. By substituting Eqn. 2 into Eqn. 1, show that the Hamiltonian
diagonalises to
X
H=
ε − 2t cos(ka) |kihk|.
(3)
k
Sketch the dispersion relation, ε(k), and illustrate the filling in the monovalent case. What
happens to the “band” as the hopping term t gets stronger or weaker?
2
Nearly Free Electrons
2. We have just approached the electron in a periodic potential problem by perturbing about
electrons which are tightly bound to atoms. We now consider the opposite limit – electrons
which are only slightly perturbed by a periodic potential. The good thing is that we will draw
some very similar conclusions.
Consider an electron in a 1-dimensional system with periodic potential V (x). Argue that the
potential can be written as:
X
(4)
V (x) =
VK exp(iKx).
K
Which values of K are summed over? We will consider the effect of one component of the
periodic potential. Assuming V (x) is real and symmetric, show that VK = V−K = VK∗ . Thus
our perturbation becomes V (x) = VK (exp(iKx) + exp(−iKx)).
1
We can write the Schrödinger equation in matrix from as
hΨ|H0 |Ψi + hΨ|V (x)|Ψi = E
(5)
Our unperturbed eigenstates are plane waves, |Ψk (x)i. Argue that
hΨk |V (x)|Ψ′k i = VK δk−k′ ,±K
(6)
The Hamiltonian only acts to connect states with k − k′ = ±K. Can you see tat this means the
true eigenstates are superpositions of states from different Brillouin zones? This is known as
Bloch’s theorem, more on this in class. Unfortunately, this is still an infinite number of states.
Ideally, we would like to restrict ourselves to pairs of states with |k| ≈ |k′ | ≈ |K|/2, as this
makes the problem easy to solve. Why might this be a reasonable assumption? Think about
the second order term in perturbation theory: is it large if the unperturbed states have very
different energies?
Hence show that a state given by |Ψ(x)i = a|Ψk (x)i + b|Ψk−K (x)i obeys the Schrödinger
equation:
! h̄2 k 2
V
a
a
K
2m
=E
(7)
h̄2 (k−K)2
b
b
VK
2m
Solve for the eigenvalues, expanding to quadratic order in δ = k − K/2. What are the eigenvectors if k lies on the Brillouin zone boundary at K/2? What sort of waves are these?
3
Fermi Surface vs Brillouin Zones
3. Consider a 2-dimensional crystal with a simple cubic structure and a monoatomic basis. Using
the free electron model, draw pictures in k-space showing the first Brillouin zone and the Fermi
surface when the atoms are
(a) monovalent
(b) divalent
Show how the Fermi surfaces are modified when the crystal potential is taken into account.
With reference to band structure, explain briefly why:
(a) Sodium (BCC crystal with trivial basis) is a metal.
(b) Calcium (FCC crystal with trivial basis) is a metal.
(c) Diamond (FCC lattice with 2 atom basis) is an insulator whereas silicon and germanium,
which have similar structures, are semiconductors.
4
Electrons and Holes
4. Explain the terms group velocity and effective mass as applied to an electron in a crystal. How
can a negative effective mass be reconciled with the fact that the mass of a free electron is
positive?
Consider an electron in a one dimensional energy band E(kx ), where kx is the wavevector along
the x-direction. An electric field F is applied along the x-direction. Show that the equation
describing the motion of the electron along the x-direction is
h̄
dkx
= −eF
dt
(8)
Derive an expression for the effective mass in terms of the band structure E(kx ).
What is a hole? Why is the concept useful? How are the properties of holes related to those of
the electron state to which they correspond?
2
5. Briefly summarize (in a couple of sentences for each) how to find the following properties of a
semiconductor experimentally:
(a) The sign of the majority charge carrier.
(b) The concentration of charge carriers.
(c) The band gap.
(d) The effective mass.
(e) The mobility of the majority carrier.
5
Semiconductors: Doping and the Law of Mass Action
6. Explain the principle of doping a semiconductor with impurities to produce an excess of electrons or holes. Give an example of a suitable material with which to dope silicon to produce
an excess of electrons.
[7]
Give expressions for the density of free electrons (n) and holes (p) in a semiconductor in terms
of the density of states in the conduction and valence bands and the Fermi-Dirac distribution
function.
[4]
Using the free-electron form of the density of states and approximating the Fermi-Dirac distribution function in each case by an exponential, derive an expression for the hole and electron
densities as a function of temperature and Fermi energy in a semiconductor with a band gap
Eg and electron and hole effective masses of m∗e and m∗h . Show that the results give a product
np which is independent of the Fermi energy EF .
[9]
A material with Eg = 1 eV and m∗e = m∗h = 0.1 me is doped with 1 × 1023 m−3 shallow donor
impurities. Calculate the density of holes present at room temperature.
[5]
Z
∞
1/2
dx x
0
−x
e
√
π
=
2
(9)
[1996 A4 question 5 (old course)]
7. Explain what the terms direct gap and indirect gap mean when used to describe semiconductors.
[5]
Show that the number of electrons per unit volume in the conduction band of an intrinsic gap
semiconductor at a temperature T is given by
−Eg
n = AT 3/2 exp
(10)
2kB T
where Eg is the energy gap and A is a constant which need not be evaluated.
[10]
m3 C −1
The Hall coefficient RH for a sample of undoped silicon is measured to be 625
at 300K
3
−1
and 0.016 m C
at 500K. Assuming only one type of carrier contributes to RH deduce a
value for Eg from these data.
[6]
The optical absorption of silicon as a function of photon energy exhibits large increases at 1.2
eV and 3.3 eV . Comment on your value for Eg in relation to these optical data.
[4]
[2000 A4 question 4 (old course)]
3