Download Exam 2 from Spring 2005

Second Exam Physics 555
Introduction to Solid State Physics
You may use your textbook, a calculator and a ruler during the exam but no other
material/notes are allowed.
Name:
SSN:
, March-31-2004, 11:10 AM till 12:25
PM (If students need more time, please be flexible and give them more time).
Note: Show all steps and identify all symbols that you use. Write in ink on both sides of this
paper and extra papers if necessary. Choose four problems out of the total of six problems.
1. (25 points) The Free and Independent Electron Gas in Two dimensions
i) What is the relation between n and kF in two dimensions? (5 points)
3 1/ 3
j) What is the relation kF and rs in two dimensions, where rs [ = (
) )]is the radius of
4π n
the free electron sphere? (5 points)
k) Prove that in two dimensions the free electron density of state D (ε ) is a constant
independent of ε for ε >0, and 0 for ε <0. What is the constant? (7 points)
l) Show that because D(ε ) is constant, every term in the Sommerfeld expansion for n
vanishes except the T=0 term. Deduce that µ = ε F at any temperature (8 points).
2. (25 points) Try the jigsaw puzzle task of fitting the second through fifth zones of a hexagonal
lattice into the hexagonal size and shape of the reduced zone.
3. (25 points) Consider a set of energy bands given by
Ek = ± = 2 k 2 ∆ / m * + ∆ 2
with all the positive-energy states empty, and all negative-energy states full.
a.
At
time
zero
add
an
electron
with
k x = k0 , k y = k z = 0, and a field ε x = ε y = 0; ε z = ε .
Obtain the current as a
function of time. Note particularly the limit as t goes to infinity. (Do exactly
rather than by making the band parabolic).
b.
At time zero empty instead a state in the lower band at the same k as in part a, and
obtain the current as a function of time with the same field applied.
4. (25 points) Consider a one-dimensional crystal, for which energy varies with wavevector in
accordance with
ε = ε1 + (ε 2 − ε1 ) sin 2 ( ak x / 2)
Populating this band with a single electron, which is not scattered. Discuss the behavior of
the effective mass, the electron velocity, and the electron position in real space under the
influence of a time-independent electric field, assuming that Bragg reflection occurs at the
zone boundaries as expected by Houston. If a = 1 angstrom, then for how long must a field of
100 V/m be applied to make this electron execute one complete oscillation in space? If the
band is 1eV wide, what is the range of distance covered in this oscillation?
5. (25 points) Consider the problem of an extremal cyclotron orbit in the k x k y plane of the
Fermi surface for a non-isotropic solid.
The equation for the Fermi energy is
2
2
ε = (α k x + β k y ) . Apply a magnetic induction Bz, and describe the rate at which k changes
during an orbit when α and β are unequal. Describe the orbit performed in real space for
this situation.
6. (25 points) Consider a one-dimensional crystal with primitive lattice translation a, and let the
energy of an electron be given by
1
=2  7

− cos ka + cos 2ka 
2 
ma  8
8

Determine the effective mass at the bottom of the band and at the top from
quadratic expansion of E in the departure of k from these points.
Determine an effective inertial mass for electrons by applying a force F and writing
dv 1
=
F
dt mi
1
Determine an effective velocity mass such that v =
=k (The velocity mass enters
mv
expressions for the conductivity).
Note that the number of states per unit length of crystal in the wavenumber interval
dk is dk/2π, Write the density of states per unit energy in terms of k and a densityof-states mass, md. (The formula should give the correct value for free electrons in
one dimension if md is taken equal to m).
E (k ) =
a.
b.
c.
d.