Physics 361
Problem Set 7
due November 25, 2009
P7.1 Anisotropic mass
(Cf. Problem 12.1) As we have seen, the maxima and minima of
an energy band need not be isotropic. Thus the general form of
the energy near a minimum is
ε
(
k
) = constant +
¯
h
2
2
(
k

k
0
)
·
ˆ
M

1
(
k

k
0
)
,
where
ˆ
M
is a 3
×
3 (symmetric) matrix called the mass matrix, and
k
0
is the position of
the minimum. If
k
0
is an energy maximum, the second term is negative. For some metals
and most semiconductors all the occupied or empty states are close enough to maxima or
minima that
ε
(
k
) is well described by this massmatrix formula.
a) For such a material find the density of states
g
(
ε
) and find the specific heat in terms
of
ˆ
M
. That is, find the specific heat effective mass
m
*
of p. 48in terms of
ˆ
M
.
b) For such materials the crystal momentum
k
is proportional to the velocity
v
in the
sense
v
=
ˆ
A
k
for some matrix
ˆ
A
. Using Hamilton’s equations (
i.e.,
the semiclassical
formalism of Chapter 12), find the proportionality matrix
ˆ
A
in terms of
ˆ
M
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Pucker
 Energy, Mass, Solid State Physics, Optimization, mass matrix, steadystate average velocity, Drude formula

Click to edit the document details