Notes:
• Exam: Tues, Oct 13.
§
§
§
In class.
Covers through ch. 5
Open book = Kittel only.
• This week: Read chapter 5, also appendix C.
(theoretical result, material = Si)
(from a talk posted by A. Kirk, McGill Univ.)
Recall last time:
∂ω
• Group velocity (
):
∂k
Generally zero at high
symmetry points on Z.B.
(“standing waves”)
!
!
• 3D case: v g = ∇ k ω
Phonon Density of Modes:
• Recall density of k-points = V/(2π)3
• Per point, 3Nm branches
(3 × # cells × number in basis)
• For quantities depending only on
energy use density of modes [D(ω)]
we’ll show:
L
D(ω ) =
dω
2
π
1D
In Debye
approximation:
For later use:
dk
V
D(ω ) =
3
(
2
π
)
3D,
∫ω
shell @
d 2k
∇ kω
per branch
3V ω 2
D(ω ) = 2
3
c
2π
⎡ V
⎤
×
(
k
−
space
volume
d ⎢
⎥
(2π ) 3
D(ω ) =
⎥
dω ⎢
converted to ω units ) ⎥⎦
⎢⎣
Note, D(ω) is same as
density of quantized
modes (phonons)
Debye approximation:
• Assume ω = kc for all modes.
• Assume 3 branches, cut off at a
sphere containing # k-points same
as number of atoms.
• “Debye wavevector” etc kD = 3 6π 2 n
ω D = c3 6π 2 n
Θ D = ω D / kB = (c / kB ) 3 6π 2 n
3V ω 2
D(ω ) = 2
3
c
2π
Debye approximation: Commonly used as measure of phonon behavior (even when “real” behavior can be obtained)
from “The Specific Heat of Matter at Low
Temperatures” [Tari, 2003].
X Zheng et al. Phys.
Rev. B 85, 214304
(2012) [my lab]:
Specific heat of
thermoelectric crystal.
Quantized Modes (phonons):
!
⎛ pi2
⎞
⎛ pi2
⎞
H = ∑i , j ⎜
+ uˆi ⋅ Φ ij ⋅ uˆ j ⎟ ⇒ ∑i , j ⎜
+ K (uˆi − uˆ j ) 2 ⎟
2
⎝ 2M
⎠
⎝ 2M
⎠ (1D)
N atoms
Convert to sum over N wave-vectors k (appendix C)
1
Qk =
N
∑ uˆ e
i
−ikxi
1
Pk =
N
∑pe
i
+ ikxi
⎛ Pk P− k
Mωk2ν Qk Q− k ⎞
⇒ H = ∑k ,ν ⎜
+
⎟ = ∑k ,ν !ωkν
2
M
2
⎝
⎠
(note N k vectors in 1st BZ make complete set)
(
1
2
+ akt ν akν
)
(branch)
Equivalent to 3N harmonic oscillators = “phonon modes”
Frequency parameter in Hamiltonian same as classical solutions.
Raising and lowering operators at right: add/remove a phonon.
Quantized Modes (phonons):
H = ∑k ,ν !ωkν
(
1
2
+ akt ν akν
)
Solution:
(
E = ∑ ω k,ν nk,ν + 1 2
k,ν
)
nkν = 1,2,3 … = quantum number for each of the 3N modes.
D(ω) obtained from the classical solutions can be used for
energy averages:
− β ∑ (nk,ν +1/2)ω k,ν
∂
with
Z = ∑ e k,ν
E =−
ln Z
∂β
Same as,
{nkν }
All sets of 3N quantum numbers
1
− β!ω k ,ν
2
e
⎛ ∞ − β ( n +1/ 2 ) !ωk ,ν ⎞
Z = ∏⎜∑e
⎟=∏
− β!ω k ,ν
1
−
e
k ,ν ⎝ n = 0
⎠ k ,ν
− β!ω
⎡1
e k ,ν
E = ∑ !ωk ,ν ⎢ +
− β!ω k ,ν
2
1
−
e
k ,ν
⎣
⎤
⎡1
⎤
=
!
ω
+
n
⎥ ∑ k ,ν ⎢
kν ⎥
2
⎣
⎦
⎦ k ,ν
nk ,ν =
1
e
β!ω k ,ν
−1
Quantized Modes (phonons):
− β!ω
⎡1
e k ,ν
E = ∑ !ωk ,ν ⎢ +
− β!ω k ,ν
2
1
−
e
k ,ν
⎣
⎤
⎡1
⎤
=
!
ω
+
n
⎥ ∑ k ,ν ⎢
kν ⎥
2
⎣
⎦
⎦ k ,ν
nk ,ν =
1
e
β!ω k ,ν
−1
• Planck formula
• But sum is over lattice normal modes
kT << !ω :
kT >> !ω :
− β!ω k ,ν
E ≈ e kT << !ω :
E ≈ !ωk ,ν / kT
(correction)
Vanishing phonon number except
lowest acoustic modes
for each mode