1
Lecture 19: Phonons
We discuss the longitudinal sound wave of jellium. We view it as a charged plasma where
the oscillation of the positively charged jellium is screened by the electrons. The plasma
2
frequency ωpl
= 4πn0 e2 /M where M , which is the nuclear mass, is screened by replacing e2
by e2 /ε(q, ω = 0) and we use the Thomas Fermi screening
q2
1
= 2
,
ε(q)
q + κ2
(1)
where κ2 = 4πe2 (2N (0)), and N (0) is the single spin electronic density of states at the Fermi
level. This gives
2
ω 2 (q) = ωpl
/ε(q)
= c2 q 2
(2)
where
c2 =
=
4πno e2
M κ2
1m 2
v .
3M F
Then the sound velocity is reduced from the Fermi velocity by a factor
(3)
�
m
.
3M
This is a
useful small parameter when discussing phonons and electrons.
Next we introduce a local displacement vector d(r, t) for the jellium. A modulation of
d(r, t) causes density modulation given by
δn(r, t) = −n0 � · d(r, t) .
Then we can write the Lagrangian as
(4)
2
1
L =
2
�
1
=
2
�
�
d3 r M n
�
d3 r M n
�
�
�2
d
B
d − 2 (δn)2
dt
n0
�
�
�2
d
2
d − B (� · d)
dt
(5)
where B is the bulk modulus. This gives the usual classical wave equation with sound
velocity c2 = B/M n0 .
We introduce the canonical momentum density
∂d(r, t)
∂t
(6)
�
1 �
π · π + B(� · d)2 .
M n0
(7)
π(r, t) = M n0
and the Hamiltonian
1
H=
2
�
d3 r
The problem is quantized by demanding the canonical commutation relation between d and
π. This is satisfied by the Fourier expansion
i
1 �ˆ
√
d(r, t) = −√
k
M n0 V k
�
�
� � ik·r−iωk t
† −ik·r+iωk t
bk e
− bk e
.
2ωk
(8)
Then
�
�
1�
1
†
�ωk bk bk +
H=
2 k
2
(9)
and
�
�
bk , b†k� = δk,k� .
(10)
In real solids there are transverse acoustic modes in addition to the longitudinal mode
discussed above. If there are two or more atoms per unit cell, there are optical modes as
well. All these can be quantized by extending the above procedure.
Reading: Marder 13.2, 13.3