Download Exercises in Statistical Mechanics

Exercises in Statistical Mechanics
Based on course by Doron Cohen, has to be proofed
Department of Physics, Ben-Gurion University, Beer-Sheva 84105, Israel
This exercises pool is intended for a graduate course in “statistical mechanics”. Some of the
problems are original, while other were assembled from various undocumented sources. In particular some problems originate from exams that were written by B. Horovitz (BGU), S. Fishman
(Technion), and D. Cohen (BGU).
====== [Exercise 3540]
Ideal Fermi gas in semiconductor
Consider a gas of electrons in a semiconductor, the temperature is T , and the chemical potential is µ. The single particle density of states g(E) = gv (E) + gc (E) consists of valence and conduction bands, separated by a gap
Eg = Ec − Ev . In the vicinity of the energy gap, one can use the following approximation:
gc (E) ≈ 2
gv (E) ≈ 2
V
3
1
2
· (2mc ) 2 (E − Ec ) 2
2
· (2mv ) 2 (Ev − E) 2
(2π)
V
3
(2π)
1
The electron has Fermi occupation f (E − µ), optionally it is customary to define an occupations function
f˜(E − µ) = 1 − f for the holes.
(a) What are the occupation functions of the electrons in the conduction band, and of the holes in the valance band,
in the Boltzmann approximation.
(b) What is the condition for the validity of this approximation? Assume that this condition is satisfies in the following
items.
(c) Derive expressions for the number of electrons Nc (β, µ) and for the number of holes Nv (β, µ). in the conductance
and valence band respectively. Explain how the product Nc Nv could be optionally deduced from the law of mass
action.
(d) Consider a closed system, such that at T = 0 the valence band is fully occupied, while the conductance band is
empty. The temperature is raised to T . Find the chemical potential and evaluate Nc (T ) and Nv (T ).
E
E=E r
µ
E=Ev
E=0
gap