Lecture 13 Phonons: thermal properties

(1)

Lecture 13

Phonons: thermal properties

Lattice contribution to the thermal

properties of solids, in 3

-D

-

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 3

T

U

c

V V

∂

∝

∂

=

Debye, T3 _Law Debye, T3 _Law

(2)

Lattice heat

-

capacity

Heat capacity

Follows from differentiating the internal energy (as usual).

Internal energy

Density of modes g(

ω

).

Einstein Approximation: all modes have the same frequency, ω_E. (See lecture 5)

Debye approximation: In the low temperature limit acoustic modes, with small q, dominate. So assume ω = v_s q.

Exact: calculate g(ω), numerically, from the phonon dispersion curves

Einstein approximation gave the correct high-temperature 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)

(

)

( )

∞

−

=

0

exp

ω

1

ω

g

d

kT

U

No. of phonons in dω at ω

Energy per phonon (Planck formula, Lecture 5)

(3)

Vol, V = a3_.

Debye

Model

Density of states

Assume

ω

= v_s 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.

where

ω

= v_s q has been substituted in RHS. Energy becomes:

Formula gives the full T dependence. We are interested in the behaviour at low T.

3 acoustic modes for each q 3 acoustic modes for each q

ω

π

ω

π

d

3

d

)

(

d

2

.

3

d

)

(

2 3 2 3 2 2 3 s

v

a

g

q

a

q

g

=

ω ω)d ( d ) (q q g g =

(

)

(

)

−

=

−

=

D D

d

kT

v

V

d

kT

v

V

U

s s ω ω

ω

π

ω

π

ω

0 3 3 2 0 2 3 2

1

exp

1

2

3

1

exp

1

2

3

Upper limit on integral guaran-tees the correct, total number of modes (3N). It is known as the

Debye frequency.

Upper limit on integral guaran-tees the correct, total number of modes (3N). It is known as the

(4)

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

(~

ω

_D) can be ignored and

ω

_D replaced by ∞.

Differentiating gives the heat capacity as

gives the correct, observed dependence at low temperatures. Recall the Einstein model gave an exponential dependence at low T.

(

)

( )

(

)

( )

3 4 4 2 0 1 3 4 3 2 0 3 3 2

10

1

exp

2

3

1

exp

1

2

3

T

v

k

V

U

d

x

kT

v

V

d

kT

v

V

U

s s s

π

ω

π

ω

π

=

−

=

−

=

∞ ₋ ∞ Integral = π4_/15 Integral = π4_/15 3

T

U

c

V V

∂

∝

∂

(5)

Measured density of states

Example: Aluminium

(shows common features)(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)

(6)

Thermal conductivity

Phonons and thermal conductivity

Phonons have energy and momentum and, therefore, can conduct heat.

Kinetic theory gives the thermal conductivity

Excess temperature of phonons crossing plane

Excess energy of each phonon

θ

cos

l

dz

dT

z

dz

dT

T

=

∆

=

−

∆

θ

cos

l

dz

dT

c

T

c

_ph

∆

=

−

_ph l θ z ∆z = - cos l θ heat capacity heat capacity

(7)

Definition of thermal conductivity

conductivity...

Number density of phonons, n

Heat flux across plane

Thermal conductivity

l

c

C

3

1

=

κ

Mean free pathMean free path

Average speed

Heat capacity per unit vol

θ dθ 2 sin dπ θ θ

( )

c

dc

f

n

( )

[

][

]

[

]

( )

(

)

3 3 1 cos cos 2 1 cos sin 2 1 cos cos 2 sin 0 2 0 0 2 0 0 c nl c dz dT dz dT c nl c H d c dz dT nl c H dc c f c d dz dT nl c H dz dT l c c d dc c nf H ph ph ph ph ph = = + = + = − = − = ∞ ∞

κ

θ

π π π number with speed c to c+dc number with speed c to c+dc fraction with angles θ to θ+dθ fraction with angles θ to θ+dθ

2

sin

4

sin

2

π

θ

d

θ

π

=

θ

d

θ

c

speed normal to plane

net heat per phonon

(8)

Temperature dependence of

thermal conductivity

Mean free path

– limited by scattering processes

With many scattering processes:

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).

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

+

=

1

₁

1

₂

1

l

10 100 T/K K /(W m K ) 1 1 0 -1 -1 Ge purified κ C, and hence κ~T3