Recall, general results:
m = number in basis
LO
ω
Optical: 3m-3 branches;
ω ≠ 0 at k = 0.
TO
LA
TA
k
1st Brillouin Zone: N k-points
(N = # primitive cells)
3m modes/point (m = # basis atoms);
3Nm = 3*number of atoms in crystal
Acoustic – always 3 (in 3D);
ω → 0 as k → 0.
Longidudinal & Transverse
modes:
§ Same as compression &
shear waves for isotropic
elastic media
§ May work best in large
λ limit; L & T not
always an appropriate
designation.
General results:
LO
ω
TO
Optical modes: always
designated “optical”
even if not optically
active
LA
TA
k
1st Brillouin Zone: N k-points
(N = # primitive cells)
3m modes/point (m = # basis atoms);
3Nm = 3*number of atoms in crystal
∂ω
• Group velocity (
):
∂k
Generally zero at high
symmetry points on Z.B.
(“standing waves”)
!
!
• 3D case: v g = ∇ k ω
Ge:
Weber, Phys. Rev. B 15 (1977) 4789
FCC examples
Pd: Stewart, New J. Phys. 10, 043025 (2008).
Typical plots: dispersion curves
along selected paths in k-space
Mg2Ca and Al2Ca
(more complex hexagonal structure)
(Zhang et al., Intermetallics 22, 17, 2012)
(note density of modes g(ω) at
bottom.)
54 atoms / cell
(x-ray, same structure;
I showed this before)
Ba8Al16Ge30
Nenghabi and Myles, J Phys.
Cond. Mat. 20, 415214 (2008)
Y. Li et al., Phys. Rev. B 75, 054513 (2007).
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)
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)
(theoretical result, material = Si)
(from a talk posted by A. Kirk, McGill Univ.)
!
!
vg = ∇ kω
Quantized Modes (phonons):
⎛ pˆ i2
⎞
⎛ pˆ i2
⎞
H = ∑i , j ⎜
+ uˆi ⋅ Dij ⋅ uˆ j ⎟ ⇒ ∑i , j ⎜
+ K (uˆi − uˆ j ) 2 ⎟
2
⎝ 2M
⎠
⎝ 2M
⎠ (1D)
Canonical operators
N atoms
pˆ i uˆi = −i! + uˆi pˆ i
Convert to sum over N wave-vectors k (appendix L)
1
Qk =
N
∑ uˆ e
i
−ikxi
1
Pk =
N
mω
1
ak =
Qk + i
Pk
2!
2!mω
∑ pˆ e
i
−ikxi
(note N k vectors in 1st BZ make complete set)
mω t
1
t
ak =
Qk − i
Pk
2!
2!mω
t
• Analogs of lowering, raising operators; “remove/add a phonon”
• ω(k) is classical lattice solution
Quantized Modes (phonons):
⎛ pˆ i2
⎞
⎛ pˆ i2
⎞
H = ∑i , j ⎜
+ uˆi ⋅ Dij ⋅ uˆ j ⎟ ⇒ ∑i , j ⎜
+ K (uˆi − uˆ j ) 2 ⎟
2
⎝ 2M
⎠
⎝ 2M
⎠ (1D)
Canonical operators
N atoms
pˆ i uˆi = −i! + uˆi pˆ i
Convert to sum over N wave-vectors k (appendix L)
1
Qk =
N
∑ uˆ e
i
−ikxi
1
Pk =
N
mω
1
ak =
Qk + i
Pk
2!
2!mω
1
t
∑k !ωak ak = N
⎛ mω 2
∑k ⎜⎜ 2
⎝
i, j
i
(note N k vectors in 1st BZ make complete set)
mω t
1
t
ak =
Qk − i
Pk
2!
2!mω
t
∑ uˆ uˆ e
i
∑ pˆ e
−ikxi
j
ik ( Ri − R j )
1
!ω
ik ( Ri − R j ) ⎞
⎟
+
pi p j e
−∑
∑
⎟
2m i , j
⎠ k 2
Quantized Modes (phonons):
⎛ pˆ i2
⎞
⎛ pˆ i2
⎞
H = ∑i , j ⎜
+ uˆi ⋅ Dij ⋅ uˆ j ⎟ ⇒ ∑i , j ⎜
+ K (uˆi − uˆ j ) 2 ⎟
2
⎝ 2M
⎠
⎝ 2M
⎠ (1D)
N atoms
1
t
∑k !ωak ak = N
Canonical operators
⎛ mω 2
∑k ⎜⎜ 2
⎝
∑ uˆ uˆ e
i
pˆ i uˆi = −i! + uˆi pˆ i
ik ( Ri − R j )
j
i, j
∑ (uˆ ⋅ D
i, j
⇒ H = ∑k ,ν !ωkν
(
1
2
i
ij
1
!ω
ik ( Ri − R j ) ⎞
⎟−∑
+
pi p j e
∑
⎟
2m i , j
⎠ k 2
⋅ uˆ j )
+ 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.
g(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