Imperial College London
BSc/MSci EXAMINATION 2023
This paper is also taken for the relevant Examination for the Associateship
SOLID STATE PHYSICS
For Third & Fourth-Year Physics Students
Monday, May 22nd, 2023: 10:00 to 12:00
Answer ALL questions.
Marks shown on this paper are indicative of those the Examiners anticipate assigning.
General Instructions
Complete the front cover of each of the THREE answer books provided.
If an electronic calculator is used, write its serial number at the top of the front cover of
each answer book.
USE ONE ANSWER BOOK FOR EACH QUESTION.
Enter the number of each question attempted in the box on the front cover of its corresponding answer book.
Hand in THREE answer books even if they have not all been used.
You are reminded that Examiners attach great importance to legibility, accuracy and
clarity of expression.
© Imperial College London 2023
PHYS60003
1
Go to the next page for questions
1.
(i) Consider the longitudinal vibrations of a chain of particles of individual mass m.
Let un (t) denote the displacement of mass n along the direction of the chain.
The masses in the chain obey the equation of motion
κ
d 2 un
=
(un+1 + un−1 − 2un )
dt 2
m
for all n.
(a) Write down a plane-wave solution for un (t) with wavevector q and frequency
ω(q). Hence, show that the vibrations follow the dispersion relation,
ω2 (q) =
4κ
qa
sin2
m
2
for −π/a < q ≤ π/a.
[5 marks]
(b) Explain why we restrict attention to a finite range of wavevectors.
[5 marks]
(c) Sketch the dispersion relation ω(q) as a function of q over the range
−2π/a < q < 2π/a. Indicate on your plot the relevant solution zones and
the Debye frequency.
[5 marks]
(ii) Consider electrons in a 3D metal described by free-electron theory, with an
energy dispersion, E = ~2 k 2 /2m, where k is the electron wavevetor and m is
the electron mass.
(a) Describe the Fermi surface of this system. By considering the volume enclosed by the Fermi surface, show that the electron density, n, in the conduction band at zero temperature is a function of the Fermi energy, EF ,
1 2mEF
n=
3π2
~2
!3 / 2
.
You may use without proof the electron density of states in reciprocal space
(k-space), (L /2π)3 (per spin state), for a cubic sample with side length L .
[4 marks]
(b) Write down an expression relating the metal’s electron density and the density of states, g(E) (states per unit energy and per unit volume including
spin degeneracy) at zero temperature. Thus show that,
1 2m
g(E) =
2π2 ~2
!3 / 2
√
E
[3 marks]
(c) Thus show that the system’s internal energy density, u = (3n/5)EF , and
explain why it is non-zero at absolute zero?
[3 marks]
(iii) Electrons within a solid have 1D energy dispersion, E(k ) = α + β cos(ka), where
k is the carrier wavenumber. In the semi-classical model of carrier transport,
classical equations of motion are assigned quantum mechanical parameters of
carrier velocity, v = ~−1 dE(k )/dk , and effective mass, m∗ .
PHYS60003
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[This question continues on the
next page . . . ]
(a) Considering the incremental work done on an electron at velocity, v, by a
force, F, show that
~
dk
= F.
dt
[4 marks]
(b) Use Newton’s second law to show that the effective carrier mass is
" 2
∗
m =~
2
d E(k )
dk 2
# −1
.
[2 marks]
(c) Derive an expression for the velocity of electrons subject to a constant external electric field, E, and explain the physical interpretation of their motion.
[4 marks]
(d) Explain the Drude-Sommerfeld model of electrical conduction in solids making sure to describe the underlying physical processes involved.
[5 marks]
[Total 40 marks]
PHYS60003
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Please go to the next page
2.
(i) Consider a 2D monoatomic crystal with a triangular
lattice described by primitive
√
lattice vectors a1 = (a, 0) and a2 = (a /2, 3a /2). Plot atomic positions in real
space as a function of x and y over a few unit cells to illustrate the crystal
structure. Identify the primitive unit cell and calculate its area.
[Hint: you may find it useful to assume the third primitive lattice vector, a3 = ẑ.]
[5 marks]
(ii) (a) Reciprocal lattice vectors are defined by G = h1 b1 + h2 b2 where h1,2 are
integers. Identify the reciprocal lattice basis vectors b1,2 of this crystal.
(b) Plot reciprocal lattice points as a function of kx and ky in 2D k-space, and
identify the First Brillouin Zone (FBZ). Write down the ranges of the FBZ
along the kx and ky directions.
[9 marks]
(iii) The electronic properties of the 2D crystal are described by the tight-binding
model. The corresponding discrete Schrödinger equation, describing the energy spectrum of band α, is
Eαk = Eα − tα
X
e ik.R ,
R
where k = (kx , ky ) is the electron wavevector, Eα are atomic orbital energies, tα
are hopping energies, and R = n1 a1 + n2 a2 , for integer n1,2 , are lattice vectors.
(a) Explain the main assumptions of the tight binding model.
(b) Considering nearest neighbour interactions for the triangular lattice, show
that the electron states for band α are described by
Eαk = Eα − 2tα cos(kx a) + cos
a
2
√
(kx +
3ky ) + cos
a
2
√
(kx −
3ky ) .
[8 marks]
(iv) The Fermi energy, EF , lies within band α = 3, with parameters E3 = 6 eV and
t3 = 1 eV. It is determined that this band contains 0.2 electrons per unit cell.
(a) Use the parabolic band
p approximation to show that the Fermi surface is a
circle of radius, kr ≈ 2EF /(3a 2 ).
(b) Calculate the Fermi energy for this electronic system.
[8 marks]
[Total 30 marks]
PHYS60003
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Please go to the next page
3. The densities of electrons, n, in the conduction band and holes, p, in the valence
band of a direct band gap semiconductor are given by,
n = NC e −(EC −EF )/kB T and p = NV e (EV −EF )/kB T ,
where NC and NV NC are effective carrier densities of the conduction and valence
bands, respectively, EC and EV are the conduction and valence band edge energies,
respectively, T is temperature, and EF is the Fermi level. Note that NC /V ∝ (mc∗/v T )3/2 ,
where mc∗ mv∗ are effective masses for conduction and valence bands.
(i) (a) Sketch the semiconductor band energies near the conduction and valence
band edges as a function of the electron wavevector magnitude, k . Indicate
EC , EV , the energy gap, Eg , as well as the Fermi level position for T > 0.
(b) Explain what is meant by the intrinsic carrier density, ni , of a semiconductor.
Being careful to explain all assumptions, derive an expression for ni .
[8 marks]
(ii) (a) Explain how the conductivity of a semiconductor can be controlled by
adding impurities, paying particular attention to the role of temperature. Explain the saturation and freeze-out regimes of extrinsic semiconductors.
(b) The semiconductor is doped with a density, ND , of donor atoms, where
ni ND < NC . In the saturation regime, show that the Fermi level position
of the n-type semiconductor is,
(n)
EF = EC − kB T ln
NC
ND
!
(c) For a p-type semiconductor, with acceptor density NA , where ni NA < NV ,
also in the saturation regime, show that the Fermi level position is,
(p)
EF = EV + kB T ln
NV
NA
!
[8 marks]
(iii) Sketch the band edge energy versus space diagram for a pn-Junction in equilibrium, identifying the Fermi level, depletion region and built in potential. Explaining your working, show that the potential across the junction is,
ND NA
kB T
ln
V0 =
e
ni2
!
[6 marks]
(iv) (a) Sketch the pn-Junction’s band edge energy versus space diagram under
forward bias, Va . Indicate the quasi-Fermi levels in the depletion region
assuming negligible recombination near the junction.
(b) Use your result from part (iii) to determine the excess carrier densities injected across the pn-Junction.
Thus show
that the pn-Junction current follows Shockley’s result, I ∝ e eVa /kB T − 1 .
[8 marks]
[Total 30 marks]
PHYS60003
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End of examination paper