Lecture 13
Phonons: thermal properties
Lattice contribution to the thermal
properties of solids, in 3-D
Aims:
Thermal properties of a crystalline solid:
Heat capacity:
Debye treatment
T3 law for low temperature heat capacity
Thermal conductivity:
Phonon scattering
Mean-free path
Debye,
Debye, TT33 Law
Law
cV =
December 06
Lecture 13
∂U
∝T3
∂T V
1
Lattice heat-capacity
Heat capacity
Follows from differentiating the internal energy
(as usual).
No. of phonons
Internal energy
in dω at ω
U=
∞
0
ω
exp( ω kT ) − 1
g (ω )dω
Energy per phonon
(Planck formula, Lecture 5)
Density of modes g(ω).
Einstein Approximation: all modes have the
same frequency, ωE. (See lecture 5)
Debye approximation: In the low temperature
limit acoustic modes, with small q, dominate. So
assume ω = vs q.
Exact: calculate g(ω), numerically, from the
phonon dispersion curves
Einstein approximation gave the correct hightemperature behaviour (C = 3NkT) and gave
C 0 as T 0 though the exact temperature
dependence was inaccurate (the reason, we
can now understand from the phonon
dispersion curves)
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Lecture 13
2
Debye Model
Density of states
Assume ω = vs q. i.e. dispersionless waves
Result is therefore similar to that for photons in
a 3-D cavity (black body radiation), except for a
numerical constant.
3
a
2
g (q) d q = 3 .
q
dq
2
2π
g (q ) d q = g (ω ) d ω
33 acoustic
acoustic modes
modes
a3 2
g (ω ) d ω = 3 2 3 ω d ω
for
for each
each qq
π vs
where ω = vs q has been substituted in RHS.
Energy becomes: Upper
Upper limit
limit on
on integral
integral guaranguaran-
tees
tees the
the correct,
correct, total
total number
number
of
of modes
modes (3N).
(3N). ItIt isis known
known as
as the
the
Debye
Debye frequency.
frequency.
U=
Vol,
Vol, VV==aa33..
ωD
0
3Vω 2
1
ω
dω
2 3
2π vs
exp( ω kT ) − 1
3V
= 2 3
2π vs
ωD
0
ω3
1
exp( ω kT ) − 1
dω
Formula gives the full T dependence. We are
interested in the behaviour at low T.
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Lecture 13
3
heat capacity at
low-temperture
Limiting behaviour as T
0.
At low temperature the higher frequency
modes are not excited. Thus contributions to
the integral for large ω (~ωD) can be ignored
and ωD replaced by ∞.
3V
U= 2 3
2π vs
∞
0
ω
3V
kT
= 2 3
2π vs
3
1
exp( ω kT ) − 1
4
∞
0
dω
x 3 (exp( x ) − 1) dω
Vπ 2 k 4 4
U=
T
3
10(vs )
−1
Integral
Integral == ππ44/15
/15
Differentiating gives the heat capacity as
cV =
∂U
∝T3
∂T V
3
Debye,
Debye, TT3 Law
Law
gives the correct, observed dependence at low
temperatures. Recall the Einstein model gave
an exponential dependence at low T.
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Lecture 13
4
Measured density of states
Example: Aluminium (shows common features)
Measured density of states compared with
Debye approximation.
Both measured and Debye density of states
are similar at low ω, as expected (ω∝q).
Debye frequency chosen to give same total
number of modes (i.e. equal area under both
curves)
Largest deviations where phonon modes
approach zone boundary.
Measured curve is complex because the 3-D
zone has a relatively complicated shape, and
the transverse and longitudinal modes have
different dispersions (as we have seen earlier)
December 06
Lecture 13
5
Thermal conductivity
Phonons and thermal conductivity
Phonons have energy and momentum and,
therefore, can conduct heat.
Kinetic theory gives the thermal conductivity
l
z
θ
∆ z = -l cos θ
Excess temperature of phonons crossing plane
dT
dT
∆T =
∆z = − l cos θ
dz
dz
Excess energy of each phonon
dT
c ph ∆T = −c ph
l cos θ
dz
heat
heat capacity
capacity
of
of aa phonon
phonon
December 06
Lecture 13
6
conductivity..........
Number density of phonons, n
number
number with
with
speed
speed cc to
to c+dc
c+dc
n f (c )dc
fraction
fraction with
with
angles
angles θθ to
to θθ+d
+dθθ
2π sin θ dθ
θ
dθ
2π sin θ dθ 4π = sin θ dθ 2
speed
speed normal
normal to
to plane
plane
net
net heat
heat per
per phonon
phonon
Heat flux across plane
H=
π
∞
0
0
[nf (c )dc sin θ dθ 2][c cosθ ][− c phl cosθ dT
dz ]
∞
1
dT π
2
H = − c ph nl
sin θ cos θ dθ c f (c )dc
0
2
dz 0
π
1
dT
c
H = + c ph nl
c cos 2 θ d (cos θ )
0
2
dz
1
dT
dT
H = + c ph nl c
=κ
3
dz
dz
κ = c ph nl c 3
Definition
Definition of
of thermal
thermal conductivity
conductivity
Thermal conductivity
1
κ= C c l
3
Mean
Mean free
free path
path
Average
Average speed
speed
Heat
Heat capacity
capacity per
per unit
unit vol
vol
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Lecture 13
7
Temperature dependence of
thermal conductivity
Mean free path – limited by scattering processes
With many scattering processes:
1 l = 1 l 1 + 1 l2 +
Thus, the shortest mean free path dominates
“Geometric” scattering:
Sample boundaries (only significant for purest
samples at low temperatures).
Impurities: scattering rate ~ independent of T.
Phonon-phonon scattering:
Phonons, being normal modes, should not affect
each other. However, in an anharmonic lattice,
they can scatter. Free path ~ 1/T.
Insulators (no contribution from electrons).
-1
-1
CC,, and
and hence
hence κκ~T
~T33
December 06
Lecture 13
Ge
purified
K/(Wm K )
1
10
In pure crystalline form
the conductivity can be
very high (larger than
metals)
N.B. Non-crystalline
systems (eg glass) have
much lower conductivity.
l ~ local order
10
100
T/K
8