Notes:
• Next exam scheduled for April 13; if no conflicts can
move to Friday (15th) same as before.
• I have a meeting trip May 2&3 (last 2 class days). As
noted before, the plan will be to start 25 minutes
early on Fridays (starting April 1).
Recall, lattice modes:
LO
ω
Optical: 3m-3 branches;
ω ≠ 0 at k = 0.
TO
LA
Acoustic – always 3 (in 3D);
ω → 0 as k → 0.
TA
m = number in basis
k
Phonon modes
H = ∑k ,ν !ωkν
(
1
2
)
(
+ akt ν akν ⇒ E = ∑ ω k,ν nk,ν + 1
2
k,ν
§ Each classical oscillator mode amplitude nkv = 0 to ∞ ,
“number of phonons present”
§ Raising and lowering operators add/remove a phonon.
)
Equilibrium phonon occupation
(
E = ∑ ω k,ν nk,ν + 1 2
k,ν
)
nkν = 1,2,3 … = quantum number for each of 3N modes.
Sum over all modes for energy average:
− β ∑ (nk,ν +1/2)ω k,ν
∂
with
Z = ∑ e k,ν
E =−
ln Z
∂β
Same as,
{nkν }
⎛ ∞ − β ( n +1/ 2 ) !ωk ,ν
Z = ∏⎜∑e
k ,ν ⎝ n = 0
− β!ω
⎡1
e k ,ν
E = ∑ !ωk ,ν ⎢ +
− β!ω k ,ν
2
1
−
e
k ,ν
⎣
β = 1 kT
All sets of 3N quantum numbers
1
− β!ω k ,ν
2
e
⎞
=
⎟ ∏
− β!ω k ,ν
1
−
e
⎠ k ,ν
⎤
⎡1
⎤
=
!
ω
+
n
⎥ ∑ k ,ν ⎢
kν ⎥
2
⎣
⎦
⎦ k ,ν
Mean occupation number at equilibrium
nk ,ν =
1
e
β!ω k ,ν
−1
Equilibrium phonon occupation
1
⎡1
⎤
⎡1
⎤
E = ∑ !ωk ,ν ⎢ + β!ωk ,ν ⎥ = ∑ !ωk ,ν ⎢ + nkν ⎥
− 1⎦ k ,ν
⎣2 e
⎣2
⎦
k ,ν
nk ,ν =
1
e
β!ω k ,ν
−1
• Planck formula, sum over all normal modes
• Phonons act as Bose particles; photon statistics
• Note, photons maximum frequency unlimited, unlike phonons
kT << !ω :
E ≈ !ωk ,ν e
kT >> !ω :
E ≈ kT
− β!ω k ,ν
Low-T occupation vanishingly small,
except for lowest acoustic modes.
(Note omitted zero-point term here.)
for each mode: Classical result.
Equilibrium phonon occupation
⎡1
⎤
E = ∑ !ωk ,ν ⎢ + nkν ⎥
⎣2
⎦
k ,ν
nk ,ν =
1
e
β!ω k ,ν
−1
kT << !ω :
E ≈ !ωk ,ν e
kT >> !ω :
E ≈ kT
− β!ω k ,ν
Low-T occupation vanishingly small,
except for lowest acoustic modes.
(Note omitted zero-point term here.)
for each mode: Classical result.
Dulong-Petit law (required by correspondence
principle). 3N total modes: CV = 3Nk B / V
Specific heat & phonon density of modes:
⎡1
⎤
E = ∑ !ωk ,ν ⎢ + nkν ⎥
⎣2
⎦
k ,ν
nk ,ν =
1
e
β!ω k ,ν
−1
General relation:
g (ω ) =
1
(2π )
3
∫ω
shell @
d 2k
∇ kω
E = V ∫ g (ω )!ωk ,ν (12 + nkν )dω
g (ω ) ∝ ω 2
Low T, Debye method works
E = 3Nk BT
High T
Phonon Density of Modes:
• density of k-points = V/(2π)3
• per point, 3Nm branches
• For quantities depending only on
energy use density of modes = g(ω)
Saw this before
g (ω ) =
1
(2π ) 3
∫ω
shell @
d 2k
∇ kω
Same argument as for electrons
Or:
⎡ 1
⎤
×
(
k
−
space
volume
d ⎢
⎥
(2π ) 3
g (ω ) =
⎥
dω ⎢
converted to ω units ) ⎥⎦
⎢⎣
Note, g(ω) is same as
density of quantized
modes (phonons)
Debye approximation:
• Assume ω = kc “average sound
velocity” represents all modes.
• Assume 3 branches, with cut off
sphere containing # k-points = 3N.
Result, Debye wavevector kD = 3 6π 2 n
ω D = c3 6π 2 n
Θ D = !ω D / k B
= (!c / k B )3 6π 2 n
3 ω2
g (ω ) = 2
3
c
2π
Debye
temperature
Einstein approximation:
• Assumes that ω = ωo
fixed value for all
modes.
Specific Heat (Debye model):
E = V ∫ g (ω )!ωk ,ν (12 + nkν )dω
3 ω2
g (ω ) = 2
c3
2π
Debye model
∂
C=
∂T
ωD
∫
0
3ω ω dω
3k T
=
2 3 β ω
2π c e −1 2π c 
kT >> !ω :
kT << !ω :
2
4 3
B
2 3 3
CV = 3Nk B / V
ΘD
∫
0
T
4
3
x dx e
T
= 9kB n 3
x
2
(e −1)
ΘD
ΘD
∫
T
0
x 4 dx e x
(e x −1)2
classical limit, easy to show.
∞
x 4 dxe x 4π 4
1 / T → ∞; ∫ x
=
2
(e − 1)
15
0
Standard notation
x
Debye function
Can look up, solve
numerically, etc.
C = (12 / 5)π 4 nk BT 3 / Θ3D ≡ βT 3
phonons
C ≡ βT + γT
3
(Exact low-temperature
limit for both contributions)
electrons
Debye approximation: ▪ Common measure of phonon behavior ▪ But note g(ω) can be calculated even more precisely (e.g. using accurate electron codes) when needed.
from “The Specific Heat of Matter at Low
Temperatures” [Tari, 2003].
X Zheng et al. Phys.
Rev. B 85, 214304
(2012) [my lab]:
Einstein
Anomalous Specific
heat of thermoelectric
crystal.
Anharmonic lattice effects:
► Real potential not precisely
quadratic; effects can normally be
treated as perturbation.
Results:
• Thermal expansion occurs as higher vibrational
states/numbers become populated.
• Vibrational states not stationary states; finite lifetime.
• Leads to phonon-phonon scattering:
energy and crystal momentum conserved.