condensed matter physics 2016physics.gu.se/~tfkhj/161124.pdfcondensed matter physics 2016...
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Condensed Matter Physics 2016 preliminary planning 24/11 – 15/12
24/11 Phonons 25/11 Electron-phonon interaction 29/11 Temperature-dependent transport 2/12 Superconductivity: BCS theory 6/12 Superconductivity: Ginzburg-Landau theory 8/12 Quasicrystals (Stellan Östlund) 9/12 Topological superconductors (Jan Budich) 13/12 Phase transitions and broken symmetries 15/12 Magnetism (note: 4 hours)
All lectures start 13:15. See TIME EDIT for lecture room.
Condensed Matter Physics 2016 Lecture 24/11: Phonons
1. Lattice dynamics 2. Quantization: Phonons 3. Phonon band structure 4. Density of states 5. Heat capacity 6. Phonon interactions
References: Ashcroft & Mermin, 22, 23 Taylor & Heinonen, 3.5-3.9 Mahan*, 7.1-7.4
* 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.
Template: 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
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
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
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 )
⇤harmonic approximation, Fourier transformed collective variables
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
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 )
⇤ blackboard
2. Quantization: phonons
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)
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…).