SOLID STATE PHYSICS
Homework problem paper 0041/3
Answers should be submitted by 5pm GMT on Wednesday, 9th March 2022.
1. In tight-binding theory, the electron energy Ek as a function of wavevector k =
(kx , ky , kz ) may be written as
Ek = −α − γ
X
exp(−ik · rm ),
m
where the sum is over the nearest neighbours connected by the vector rm , α is
the on-site overlap integral and γ is the nearest neighbour overlap integral.
Consider a monoatomic crystal with a body-centred cubic (bcc) structure.
(a) Write down all the vectors rm joining a given atom to all its nearest neighbours. Hence use the tightbinding model to show that the dispersion relation for this crystal is
Ek = α − 8γ cos
kx a
2
cos
ky a
2
cos
kz a
2
.
[8]
(b) Sketch the reciprocal lattice of this crystal, indicating the relevant dimesions.
[2]
(c) Deduce the distances from the Brillouin zone centre to the Brillouin zone
boundary along [100] and [110]?
[2]
(d) Plot the dispersion relation within the first Brillouin zone along wavevectors
parallel to the [100] and [110] directions.
[4]
CONTINUED
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2. Consider a monoatomic 2-dimensional crystal with a centred rectangular lattice.
The conventional unit cell for this structure is shown below, where b > a.
(a) Sketch a primitive unit cell for this structure and write down the two primitive vectors.
[4]
(b) Find the corresponding primitive vectors of the reciprocal lattice and sketch
the reciprocal lattice.
[6]
(c) Draw the first Brillouin zone, clearly indicating the positions of the Brillouin
zone boundaries relative to the nearby reciprocal lattice points.
√
(d) If 1 < ba < 3, show that the shortest and longest distances between
q
2
π
the Brillouin zone centre and Brillouin zone boundaries are b 1 + ba2 and
π a
b
+
respectively.
b b
a
[4]
[6]
(e) Assuming that the atoms are divalent and that the magnitude of the bandgap
is the same across the Brillouin zone boundary, use the nearly free electron model to derive an expression for the minimum bandgap required for
this material to be an insulator at zero temperature.
[4]
CONTINUED
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3. Silicon has a band gap of 1.1 eV. Given that the reverse-bias current of a siliconbased p-n junction at 300 K saturates at −1 µA, stating any assumptions you
make, calculate the following:
(a) the forward bias current at 300 K and +0.1 V;
[4]
(b) the forward bias current at 310 K and +0.1 V. [You may assume that the recombination rate does not change significantly between 300 K and 310 K.]
END OF PAPER
0041/3/2022
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[6]
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