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* Condensed Matter in a Nutshell (Princeton University Press, 2011)
1. Lattice dynamics Crystal Hamiltonian
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
pn2
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
Recall the Born-Oppenheimer approximation, treating the cores as fixed(providing a static periodic potential in which the electrons move).We shall now remove this simplifying assumption, and study the effect of including also the dynamics of the cores.
Introduce bosonic annihilation and creation operators (1D monatomic lattice)
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =X
q
✓1
2Mpqp
†q +
1
2M!2
qyqy†q
◆, !q =
qVq/M
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =X
q
✓1
2Mpqp
†q +
1
2M!2
qyqy†q
◆, !q =
qVq/M
X
q
~!q(a†qaq +
1
2)
a 2a = b
M1 M2
�
aq =1p
2M~!q(M!qyq + ip†q) (3)
a†q =1p
2M~!q(M!qy
†q � ipq) (4)
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =X
q
✓1
2Mpqp
†q +
1
2M!2
qyqy†q
◆, !q =
qVq/M
X
q
~!q(a†qaq +
1
2)
a 2a = b
M1 M2
�
aq =1p
2M~!q(M!qyq + ip†q) (3)
a†q =1p
2M~!q(M!qy
†q � ipq) (4)
state with nq phonons with wave number q
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =X
q
✓1
2Mpqp
†q +
1
2M!2
qyqy†q
◆, !q =
qVq/M
X
q
~!q(a†qaq +
1
2)
a 2a = b
M1 M2
�
aq =1p
2M~!q(M!qyq + ip†q) (3)
a†q =1p
2M~!q(M!qy
†q � ipq) (4)
(a†q)nq |0i = p
nq|nqi
Eq = ~!q(nq +1
2)
1
H = He +Hc +Hec
Hc = K + U (1)
=
X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =
X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =
X
q
✓1
2Mpqp
†q +
1
2
M!2qyqy
†q
◆, !q =
qVq/M
X
q
~!q(a†qaq +
1
2
)
a 2a = b
M1 M2
�
aq =
1p2M~!q
(M!qyq + ip†q) (3)
a†q =
1p2M~!q
(M!qy†q � ipq) (4)
(a†q)nq |0i =
pnq!|nqi
3. Phonon bandstructure
1
H = He +Hc +Hec
Hc = K + U (1)
=
X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =
X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =
X
q
✓1
2Mpqp
†q +
1
2
M!2qyqy
†q
◆, !q =
qVq/M
X
q
~!q(a†qaq +
1
2
)
a 2a = b
M1 M2
�
aq =
1p2M~!q
(M!qyq + ip†q) (3)
a†q =
1p2M~!q
(M!qy†q � ipq) (4)
(a†q)nq |0i = p
nq|nqi
1D phonon bands
1D phonon bands
Diatomic chain with different masses (and ”spring constant” )
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
yq, pq
H =X
q
✓1
2Mpqp
†q +
1
2M!2
qyqy†q
◆, !q =
qVq/M
a 2a = b
M1 M2
�
Phonons in 2D, 3D lattices
2
Eq = ~!q(nq +1
2)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
2
Eq = ~!q(nq +1
2)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
3D Bravais lattice
2
Eq = ~!q(nq +1
2)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
Harmonic approximation: (monatomic lattice)
Dynamical matrix elements:
2
Eq = ~!q(nq +1
2)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
Diagonalize the dynamical matrix !(2x2 in D=2, 3x3 in D=3)
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
Phonons in 2D, 3D lattices with a p-basis
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
D acoustic branches
D(p-1) optical branches
2
Eq = ~!q(nq +1
2)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
2
Eq = ~!q(nq +1
2)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D=3, p=2
4. Phonon density of states
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
DOS for Al
DOS, diatomic chain
2
Eq = ~!q(nq +1
2
)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D(!) ⌘X
�,q�(! � !�,q)
D2D,⌘(!) =!
2⇡v2⌘, ⌘ = LA, TA
!D(⌘) = v⌘qD, ⇡q2D = area of 1BZ
2
Eq = ~!q(nq +1
2
)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D(!) ⌘X
�,q�(! � !�,q)
D2D,⌘(!) =!
2⇡v2⌘, ⌘ = LA, TA
!D(⌘) = v⌘qD, ⇡q2D = area of 1BZ
2
Eq = ~!q(nq +1
2
)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D(!) ⌘X
�,q�(! � !�,q)
D2D,⌘(!) =!
2⇡v2⌘, ⌘ = LA, TA
!D(⌘) = v⌘qD, ⇡q2D = area of 1BZ
2
Eq = ~!q(nq +1
2
)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D(!) ⌘X
�,q�(! � !�,q)
D2D,⌘(!) =!
2⇡v2⌘, ⌘ = LA, TA
!D(⌘) = v⌘qD, ⇡q2D = area of 1BZ
Debye approximation (2D)
van Hove singularities
5. Phonon heat capacity
1
H = He +Hc +Hec
Hc = K + U (1)
=X
n
p2n
2Mn+
1
2
X
n,n0
ZnZn0e2
|rn � rn0 | = K + U (2)
xj = x0,j + yj
1 N j
H =X
j
p2j2M
+ V (y1, y2, ..., yN )
⇤
2
Eq = ~!q(nq +1
2
)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D(!) ⌘X
�,q�(! � !�,q)
D2D,⌘(!) =!
2⇡v2⌘, ⌘ = LA, TA
!D(⌘) = v⌘qD, ⇡q2D = area of 1BZ
CV (T ) ⇡ 3R, kBT � ~!
CV (T ) ⇡ constT 3
2
Eq = ~!q(nq +1
2
)
l = n1l1 + n2l2 + n3l3
V ⇡1
2
X
l,l0;i,j
yilyj
l0V ij
l,l0
V ijq =
X
l,l0eiq·(l�l0)V ij
ll0
Vq
q
!(q)q
D(!) ⌘X
�,q�(! � !�,q)
D2D,⌘(!) =!
2⇡v2⌘, ⌘ = LA, TA
!D(⌘) = v⌘qD, ⇡q2D = area of 1BZ
CV (T ) ⇡ 3R, kBT � ~!
CV (T ) ⇡ constT 3
(Dulong-Petit)
(Debye)
6. Phonon interactions
To correctly predict the thermal expansion or heat conductance of a solid, one must take into account the interaction between phonons. For this, one has to go beyond the harmonic approximation (somewhat cumbersome…).