Solid State Physics
PHONON HEAT CAPACITY
Lecture 11
A.H. Harker
Physics and Astronomy
UCL
4.5
Experimental Specific Heats
Element
Z
Lithium
Beryllium
Boron
Carbon
Sodium
Magnesium
Aluminium
Silicon
Phosphorus
Sulphur
3
4
5
6
11
12
13
14
15
16
A
6.94
9.01
10.81
12.01
22.99
24.31
26.98
28.09
30.97
32.06
Cp
J K−1mol−1
24.77
16.44
11.06
8.53
28.24
24.89
24.35
20.00
23.84
22.64
Element
Z
Rhenium
Osmium
Iridium
Platinum
Gold
Mercury
Thallium
Lead
Bismuth
Polonium
75
76
77
78
79
80
81
82
83
84
A
186.2
190.2
192.2
195.1
197.0
200.6
204.4
207.2
209.0
209.0
Cp
J K−1mol−1
25.48
24.70
25.10
25.86
25.42
27.98
26.32
26.44
25.52
25.75
2
Classical equipartition of energy gives specific heat of 3pR per mole,
where p is the number of atoms in the chemical formula unit. For
elements, 3R = 24.94 J K−1 mol−1. Experiments by James Dewar
showed that specific heat tended to decrease with temperature.
3
Einstein (1907): “If Planck’s theory of radiation has hit upon the
heart of the matter, then we must also expect to find contradictions
between the present kinetic molecular theory and practical experience in other areas of heat theory, contradictions which can be removed in the same way.”
4
Einstein’s model If there are N atoms in the solid, assume that each
vibrates with frequency ω in a potential well. Then
N ~ω
,
E = N hni~ω = ~ω
e kB T − 1
and
∂E
CV =
∂T V
Now
~ω
~ω
∂
,
=−
2
∂T kBT
kB T
so
~ω
~ω
∂
~ω
kB T ,
e kB T = −
e
∂T
kB T 2
and
~ω
2
~ω
e kB T
CV = N kB
~ω
2 .
kB T
e kB T − 1
5
Limits
~ω
CV = N kB
kB T
T → ∞. Then k~ωT → 0, so
B
~ω
e kB T
2
~ω
e kB T
~ω
e kB T
→ 1 and
2
CV → N kB k~ωT
B
2 .
−1
~ω
e kB T
− 1 → k~ωT , and
B
1 2 = N kB.
~ω
kB T
This is the expected classical limit.
6
Limits
~ω
CV = N kB
kB T
T → 0. Then
~ω
e kB T
2
~ω
e kB T
2 .
−1
~ω
e kB T
>> 1 and
2
CV → N kB k~ωT
B
~ω
e kBT! →
~ω 2
e kB T
− k~ωT
−2
T e B .
Convenient to define Einstein temperature, ΘE = ~ω/kB.
7
8
• Einstein theory shows correct trends with temperature.
• For
p simple harmonic oscillator, spring constant α, mass m, ω =
α/m.
• So light, tightly-bonded materials (e.g. diamond) have high frequencies.
• But higher ω → lower specific heat.
• Hence Einstein theory explains low specific heats of some elements.
9
Walther Nernst, working towards the Third Law of Thermodynamics (As we approach absolute zero the entropy change in any process
tends to zero), measured specific heats at very low temperature.
10
Systematic deviations from Einstein model at low T. Nernst and Lindemann fitted data with two Einstein-like terms.Einstein realised that
the oscillations of a solid were complex, far from single-frequency.
Key point is that however low the temperature there are always some
modes with low enough frequencies to be excited.
11
4.6
Debye Theory
Based on classical elasticity theory (pre-dated the detailed theory of
lattice dynamics).
12
The assumptions of Debye theory are
• the crystal is harmonic
• elastic waves in the crystal are non-dispersive
• the crystal is isotropic (no directional dependence)
• there is a high-frequency cut-off ωD determined by the number of
degrees of freedom
13
4.6.1
The Debye Frequency
The cut-off ωD is, frankly, a fudge factor.
If we use the correct dispersion relation, we get g(ω) by integrating
over the Brillouin zone, and we know the number of allowed values
of k in the Brillouin zone is the number of unit cells in the crystal, so
we automatically have the right number of degrees of freedom.
In the Debye model, define a cutoff ωD by
Z ω
D
N=
g(ω)dω,
0
where N is the number of unit cells in the crystal, and g(ω) is the
density of states in one phonon branch.
14
Taking, as in Lecture 10, an average sound speed v we have for each
mode
V ω2
g(ω) = 2 3 ,
2π v
so
Z ω
D V ω2
N =
dω
2
3
2π v
0
V ωD3
= 2 3
6π v
2
6N
π
ωD3 =
v3
V
Equivalent to Debye frequency ωD is ΘD = ~ωD/kB, the Debye temperature.
15
4.6.2
Debye specific heat
Combine the Debye density of states with the Bose-Einstein distribution, and account for the number of branches S of the phonon
spectrum, to obtain
~ω
R ωD V ω2 ~ω 2
k
BT
e
CV = S 0 2π 2 v 3 kB k T
!2 dω.
~ω
B
e kBT −1
Simplify this by writing
~ω
kB T x
~ωD
x=
, so ω =
, xD =
,
kB T
~
kB T
and
Z x
x
D V k 2 T 2 x2
e
kB T
2
B
CV = S
kB x
dx
2
2
2
3
2π ~ v
(ex − 1) ~
0
Z x
3
3
D
V kB T
x4ex
= SkB 2 3 3
dx
2
x
2π ~ v 0
(e − 1)
16
Z x
3
3
D
x4ex
V kB T
CV = SkB 2 3 3
dx.
2
x
2π ~ v 0
(e − 1)
Remember that
so
2
6N
π
v 3,
ωD3 =
V
V
3N
3N ~3
=
= 3 3
2
3
3
2π v
ωD
kBΘD
and
3 k3 T 3 R x
4ex
3N
~
x
D
B
CV = SkB 3 3 ~3 0
2 dx,
x
kBΘD
(e −1)
3 Rx
4ex
T
x
D
= 3N SkB 3 0
dx.
2
x
ΘD
(e −1)
As with the Einstein model, there is only one parameter – in this case
ΘD.
17
Improvement over Einstein model.
Debye and Einstein models compared with experimental data for Silver. Inset shows details of behaviour at low temperature.
18
4.6.3
Debye model: high T
CV = 3N SkB
Z x
3
D
T
ΘD3 0
x4ex
(ex − 1)2
dx.
At high T, xD = ~ωD/kBT is small. Thus we can expand the integrand for small x:
ex ≈ 1,
and
(ex − 1) ≈ x
so
Z x
Z x
3
4
x
D
D
x
x e
2dx = D .
dx
≈
x
3
(ex − 1)2
0
0
The specific heat, then, is
T 3 x3D
CV ≈ 3N SkB 3 ,
ΘD 3
19
T 3 x3D
CV ≈ 3N SkB 3 ,
ΘD 3
but
so
~ωD ΘD
xD =
=
kB T
T
CV ≈ N SkB.
This is just the classical limit, 3R = 3NAkB per mole. We should have
expected this: as T → ∞, CV → kB for each mode, and the Debye
frequency was chosen to give the right total number of oscillators.
20
4.6.4
Debye model: low T
CV = 3N SkB
Z x
3
D
T
ΘD3 0
x4ex
(ex − 1)2
dx.
At low, xD = ~ωD/kBT is large. Thus we may let the upper limit of
the integral tend to infinity.
Z ∞
4π 4
x4ex
dx =
2
15
(ex − 1)
0
so
T 3 4π 4
CV ≈ 3N SkB 3
ΘD 15
For a monatomic crystal in three dimensions S = 3, and N , the number of unit cells, is equal to the number of atoms. We can rewrite this
as
3
T
CV ≈ 1944
ΘD
which is accurate for T < ΘD/10.
21
22
4.6.5
Successes and shortcomings
Debye theory works well for a wide range of materials.
But we know it can’t be perfect.
23
Roughly: only excite oscillators at T for which ω ≤ kBT /~. So we
expect:
• Very low T: OK
• Low T: real DOS has more low-frequency oscillators than Debye,
so CV higher than Debye approximation.
• High T: real DOS extends to higher ω than Debye, so reaches classical limit more slowly.
24
Use Debye temperature ΘD as a fitting parameter: expect:
• Very low T: good result with ΘD from classical sound speed;
• Low T: rather lower ΘD;
• High T: need higher ΘD.
25