7/31/2019 Lecture 11 s

1/25

Solid State Physics

PHONON HEAT CAPACITYLecture 11

A.H. HarkerPhysics and Astronomy

UCL

7/31/2019 Lecture 11 s

2/25

4.5 Experimental Specific Heats

Element Z A Cp Element Z A CpJ K1mol1 J K1mol1

Lithium 3 6.94 24.77 Rhenium 75 186.2 25.48

Beryllium 4 9.01 16.44 Osmium 76 190.2 24.70

Boron 5 10.81 11.06 Iridium 77 192.2 25.10

Carbon 6 12.01 8.53 Platinum 78 195.1 25.86

Sodium 11 22.99 28.24 Gold 79 197.0 25.42Magnesium 12 24.31 24.89 Mercury 80 200.6 27.98

Aluminium 13 26.98 24.35 Thallium 81 204.4 26.32

Silicon 14 28.09 20.00 Lead 82 207.2 26.44

Phosphorus 15 30.97 23.84 Bismuth 83 209.0 25.52

Sulphur 16 32.06 22.64 Polonium 84 209.0 25.75

2

7/31/2019 Lecture 11 s

3/25

Classical equipartition of energy gives specific heat of 3pR per mole,where p is the number of atoms in the chemical formula unit. Forelements, 3R = 24.94 J K1 mol1. Experiments by James Dewar

showed that specific heat tended to decrease with temperature.

3

7/31/2019 Lecture 11 s

4/25

Einstein (1907): If Plancks 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 experi-

ence in other areas of heat theory, contradictions which can be re-moved in the same way.

4

7/31/2019 Lecture 11 s

5/25

Einsteins model If there are N atoms in the solid, assume that eachvibrates with frequency in a potential well. Then

E = Nn =N

e

kBT 1

,

and

CV =

E

T

V

Now

T

kBT=

kBT2

,

so

T e

kBT =

kBT2

e

kBT,

and

CV = N kB

kBT

2 e kBTe

kBT 1

2.

5

7/31/2019 Lecture 11 s

6/25

Limits

CV

= N kB

kBT

2 e

kBTe

kBT 1

2

.

T . Then kBT 0, so e

kBT 1 and e

kBT 1 kBT

, and

CV NkB

kBT

2 1

kBT

2= N kB.This is the expected classical limit.

6

7/31/2019 Lecture 11 s

7/25

Limits

CV

= N kB

kBT

2 e

kBTe

kBT 1

2.

T 0. Then e

kBT >> 1 and

CV N kB

kBT

2 e

kBTe

kBT

2 T2e

kBT.

Convenient to define Einstein temperature, E = /kB.

7

7/31/2019 Lecture 11 s

8/258

7/31/2019 Lecture 11 s

9/25

Einstein theory shows correct trends with temperature.

For simple harmonic oscillator, spring constant , mass m, =

/m. So light, tightly-bonded materials (e.g. diamond) have high fre-

quencies.

But higher lower specific heat.

Hence Einstein theory explains low specific heats of some elements.

9

7/31/2019 Lecture 11 s

10/25

Walther Nernst, working towards the Third Law of Thermodynam-

ics (As we approach absolute zero the entropy change in any process

tends to zero), measured specific heats at very low temperature.

10

7/31/2019 Lecture 11 s

11/25

Systematic deviations from Einstein model at low T. Nernst and Lin-demann 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

7/31/2019 Lecture 11 s

12/25

4.6 Debye Theory

Based on classical elasticity theory (pre-dated the detailed theory of

lattice dynamics).

12

7/31/2019 Lecture 11 s

13/25

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-offD determined by the number ofdegrees of freedom

13

7/31/2019 Lecture 11 s

14/25

4.6.1 The Debye Frequency

The cut-offD is, frankly, a fudge factor.If we use the correct dispersion relation, we get g() by integrating

over the Brillouin zone, and we know the number of allowed valuesofk in the Brillouin zone is the number of unit cells in the crystal, sowe automatically have the right number of degrees of freedom.

In the Debye model, define a cutoffD by

N =D

0g()d,

where N is the number of unit cells in the crystal, and g() is thedensity of states in one phonon branch.

14

7/31/2019 Lecture 11 s

15/25

Taking, as in Lecture 10, an average sound speed v we have for eachmode

g() =V

222

v3,

so

N =

D0

V

222

v3d

=V

62

D3

v3

D3 =

6N2

Vv3

Equivalent to Debye frequency D is D = D/kB, the Debye tem-

perature.

15

7/31/2019 Lecture 11 s

16/25

4.6.2 Debye specific heat

Combine the Debye density of states with the Bose-Einstein distri-

bution, and account for the number of branches S of the phonon

spectrum, to obtain

CV = SD

0V

222

v3kB

kBT

2e

kBT

e

kBT1

2d.

Simplify this by writing

x =

kBT, so =

kBT x

, xD =

DkBT

,

and

CV = S

xD0

V

22k2BT

2x2

2v3kBx

2 ex

(ex 1)2kBT

dx

= SkBV

22

k3BT3

3

v3

xD

0

x4ex

(ex

1)2

dx

16

7/31/2019 Lecture 11 s

17/25

CV = SkBV

22k3BT

3

3v3

xD0

x4ex

(ex 1)2dx.

Remember that

D3 =

6N2

Vv3,

soV

22v3=

3N

D3

=3N3

k3

B

D

3

and

CV = SkB3N3

k3BD3

k3BT3

3

xD0

x4ex

(ex1)2dx,

= 3N SkBT3

D3xD0 x4ex

(ex

1)2dx.

As with the Einstein model, there is only one parameter in this case

D.

17

7/31/2019 Lecture 11 s

18/25

Improvement over Einstein model.

Debye and Einstein models compared with experimental data for Sil-

ver. Inset shows details of behaviour at low temperature.

18

7/31/2019 Lecture 11 s

19/25

4.6.3 Debye model: high T

CV = 3N SkBT3

D3

xD

0

x4ex

(ex 1)2dx.

At high T, xD = D/kBT is small. Thus we can expand the inte-grand for small x:

ex 1,

and

(ex 1) xso xD

0

x4ex

(ex 1)2dx

xD0

x2dx =x3D3

.

The specific heat, then, is

CV 3NSkBT3

D3

x3D3

,

19

7/31/2019 Lecture 11 s

20/25

CV 3NSkBT3

D3

x3D3

,

but

xD = D

kBT= D

Tso

CV N SkB.

This is just the classical limit, 3R = 3NA

kB

per mole. We should have

expected this: as T , CV kB for each mode, and the Debyefrequency was chosen to give the right total number of oscillators.

20

7/31/2019 Lecture 11 s

21/25

4.6.4 Debye model: low T

CV = 3N SkBT3

D3

xD

0

x4ex

(ex 1)2dx.

At low, xD = D/kBT is large. Thus we may let the upper limit ofthe integral tend to infinity.

0

x4ex

(ex 1)2dx =

44

15

so

CV 3NSkBT3

D3

44

15

For a monatomic crystal in three dimensions S = 3, and N, the num-

ber of unit cells, is equal to the number of atoms. We can rewrite thisas

CV 1944

T

D

3which is accurate for T < D/10.

21

7/31/2019 Lecture 11 s

22/25

22

4 6 5 S d h t i

7/31/2019 Lecture 11 s

23/25

4.6.5 Successes and shortcomings

Debye theory works well for a wide range of materials.

But we know it cant be perfect.

23

7/31/2019 Lecture 11 s

24/25

Roughly: only excite oscillators at T for which kBT /. So weexpect:

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 than Debye, so reaches clas-sical limit more slowly.24

U D b t t fitti t t

7/31/2019 Lecture 11 s

25/25

Use Debye temperature D as a fitting parameter: expect:

Very low T: good result with D from classical sound speed;

Low T: rather lower D;

High T: need higher D.

25