Download Homework for week 11 File - QMplus

QUEEN MARY, UNIVERSITY OF LONDON
SCHOOL OF PHYSICS AND ASTRONOMY
Condensed Matter A
Homework Set 11
To be handed in by 4 p.m. on Thursday, 30 March 2017
Problem 1: Heat capacity revisited (10 marks)
(a) Explain why only some electrons contribute to the heat capacity of a free electron gas. Which ones do,
and why?
(2)
(b) I claimed in class that the heat capacity of a free electron gas is
kB T
1 2
.
cV,unit volume = 2 π nkB
EF
(2)
Evaluating the dimensions of this shows that it is the heat capacity per unit volume. Starting from this
equation, show that the heat capacity per mole of electrons in the gas is given by
kB T
1 2
cV,mole of electrons = 2 π R
.
EF
(c) At room temperature, silver metal has a face-centred cubic structure with lattice parameter a = 4.09 Å.
Assuming each atom contributes one electron to a free electron gas, calculate the Fermi level EF of this
gas.
(3)
Useful equations:
kF3 = 3π 2 n
E=
h̄2 k2
2m
(d) In the week 9 homework set, you estimated the heat capacity of silver, considering only the phonon
contribution and using the high-temperature limit. Use your results from the previous parts of this question
to recalculate this value, taking into account the electron contribution at 300 K as well. Does this improve
the agreement between experiment and theory?
(3)
Hint: Consider both the magnitude and the sign of the difference between the values: that is, which is
higher, and is this as expected?
Problem 2: The nearly-free electron gas on a 2D hexagonal lattice (20 marks)
Consider a two-dimensional, hexagonal material (again, like the one that turned up in the week 9 homework):
(a) Sketch the first, second, and third Brillouin zones.
(b) Estimate the two lowest energy levels at k
= 0.1a∗ ,
(6)
giving your answer in terms of
h2 /2ma2 .
(c) Assume that each atom contributes two electrons to the electron gas. Sketch the Fermi surface, and shade
the occupied states, in each of the following cases:
i. A free electron gas with no band gap;
(6)
(3)
ii. A nearly-free electron gas with a small band gap;
iii. A band gap big enough that this system becomes an insulator.
(d) Now assume that each atom contributes three electrons to the electron gas. Sketch the Fermi surface, and
shade the occupied states, in each of the following cases:
(3)
i. A free electron gas with no band gap;
ii. A nearly-free electron gas with a small band gap;
iii. A nearly-free electron gas with a very large band gap.
(e) In case (d) iii. above, is this system a metal or an insulator?
Data:
Electronic charge
e = 1.6022 × 10−19 C
Planck constant
h = 6.626 × 10−34 Js
h̄ = h/2π = 1.055 × 10−34 Js
Boltzmann constant
kB = 1.3807 × 10−23 JK−1
Electron mass
m = 9.109 × 10−31 kg
NA = 6.022 × 1023 mol−1
Avogadro number
Page 2
(2)