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).
Anharmonic lattice effects: (ch 25)
Real potential not precisely
quadratic; effects can normally be
treated as perturbation.
► Model as 3rd and higher terms:
►
ε=
∑µ ν u µ D
1
2
Ri , R j , ,
i
ij , µν
u jν ⇒
∑µ νDλ
u u jν ukλ + ...
1
ijk , µνλ iµ
2
Ri , R j , Rk , , ,
Results:
• Thermal expansion occurs as higher vibrational
states/numbers become populated.
See text; Grüneisen parameter.
• Vibrational states not stationary states; finite lifetime.
• Leads to phonon-phonon scattering:
energy and crystal momentum conserved.
Anharmonic lattice effects:
Umklapp dominates thermal conductivity
Other processes may be considered…
Lattice thermal conductivity:
κ = 13 C!v
Thermal conductivity by phonon “gas”:
Heat
capacity
(more generally a
summation over all modes)
n = kT / !ω
→ const.
Heat
capacity
boundary
Mean free path for
scattering that impedes
heat flow
scattering
!
T3
0
•
Mean
free
path
Group ∂ω
velocity ∂k
Umklapp 1/T
(3-phonon
processes)
ΘD
T
→
Non-Umklapp phonon scattering:
generally preserves heat flow
Lattice thermal conductivity:
κ = 13 C!v
→ const.
Heat
capacity
T3
T3
Mean free path
boundary
scattering
!
0
Umklapp 1/T
(3-phonon
processes)
ΘD
T
→
Note for metals,
κ = κ! + κe
1/ T
Lattice thermal conductivity:
T3
•
•
Large ΘD generally large κL peak.
Anharmonics decrease high-T κL.
1/ T
Scattering from point defects
analogous to Rayleigh scattering
(long-wave case)
Negligible at low T leads to
boundary scattering limit.
Glasses: “minimum thermal
conductivity”, mean free path ~λ.
Boron nitride
isotope effect
S. Barman,
Europhys. Letters
2011
Electron term (metals):
1
Thermal conductivity, κ el = C!v
3
• v = Fermi velocity for good metals.
• Specific heat we have seen: C = γT
• Alloy or very strong disorder, peak may disappear.
κ el
2
κ el π ⎛ k B ⎞
=
⎜ ⎟T
σ
3 ⎝ e ⎠
2
Wiedemann-Franz law
(see ch. 1, different
result for classical
case)
⎛π 2 ⎞
1
c ≅ k BTg (ε F )⎜⎜ ⎟⎟ ≡ γT
V
⎝ 3 ⎠
g (ε F )vF2 e 2τ
σ=
3
Electrical resistivity of metals (ch. 26):
Matthiessen’s rule:
1
τ
=
1
τ impurity
+
1
τ phonons
+ ...
• Uncorrelated processes, metal resistivities add.
• Impurities always decrease mobility (increase resistivity)
• Alloy resistivity may be large & constant vs. T.
• Semiconductors resistivity normally decrease vs. T.
With impurities
eτ
µ=
m
σ = neµ ≡ 1 / ρ
mobility
ρ = ρ impurity + ρ phonons + ...
1
τ
∝ n phonon
High T,
Good metal
ρ ~ T.
Electrical resistivity of metals (ch. 26):
With impurities
1
τ
∝ n phonon
High T,
Good metal
ρ ~ T.
ΘD
• Low-T “Bloch T5 law”: multiple phonon scattering required
to affect electron direction. Scattering rate in electrical
conductivity relationship not same as e-ph scattering rate.
Electrical resistivity of metals (ch. 26):
Cu nanowires
Phys. Rev. B 74, 035426 (2006)
1
τ
∝ n phonon
High T,
Good metal
ρ ~ T.
ΘD
• More general relation: Bloch-Grüneisen eqn. (won’t show
derivation), based on Debye model.
• Other factors affecting resistivity: magnetic disorder,
Umklapp terms (see text)…
Inelastic Neutron scattering:
•
Neutron energy vs. k comparable to phonon (&
magnetic) excitations
•
Observe elastic and inelastic scattering.
•
Crystal momentum, energy conserved
Triple-axis: analyze for momentum/energy of outgoing
neutrons (NMI3 website)
Example, scanned
energy transfer,
giving phonon
dispersion curves.