Solid State Physics

Solid State Physics Lecture 8 – The Debye model Professor Stephen Sweeney Advanced Technology Institute and Department of Physics University of Surrey, Guildford, GU2 7XH, UK

[email protected]

Solid State Physics - Lecture 8

Recap from Lecture 7 • Concepts of “temperature” and thermal equilibrium are based on the idea that individual particles in a system have some form of motion

• Heat capacity can be determined by considering vibrational motion of atoms

• We considered two models: • Dulong-Petit (classical) • Einstein (quantum mechanical) • Both models assume atoms act independently – this is made up for in the Debye model (today) Solid State Physics - Lecture 8

Dulong-Petit

Summary of Dulong-Petit and Einstein models of heat capacity Dulong-Petit model (1819)

Einstein model (1907)

•

•

• • •

Atoms on lattice vibrate independently of each other Completely classical Heat capacity independent of temperature (3NkB) Poor agreement with experiment, except at high temperatures

• •

Atoms on lattice vibrate independently of each other Quantum mechanical (vibrations are quantised) Agreement with experiment good at very high (~3NkB) and very low (~0) temperatures, but not inbetween


A more realistic model… •

Both the Einstein and Dulong-Petit models treat each atom independently. This is not generally true.

•

When an atom vibrates, the force on adjacent atoms changes causing them to vibrate (and vice-versa)

•

Oscillations can be broken down into modes

1D case Nice animations here: http://www.phonon.fc.pl/index.php Java applet: http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html Solid State Physics - Lecture 8

3D case

Debye model •

Basic idea similar to Einstein model, with one key difference:

Einstein: Energy of system = Phonon Energy x Average number of phonons Debye: Energy of system = Phonon Energy x Average number of phonons x number of modes

Einstein: number of modes = number of atoms Debye: each mode has its own k value (and hence frequency)


The number and type of modes are the key difference

Modes: lattice vibrations Modes exist in various areas of physics/nature

Butterfly wing-beat

Water Molecules

Guitar modes


Modes: Quantum mechanics Modes are quantised in units of mode is 



where the fundamental frequency of each

The Einstein model assumed that each oscillator has the same frequency Debye theory accounts for different possible modes (and therefore different



)

Modes with low  will be excited at low temperatures and will contribute to the heat capacity. Therefore heat capacity varies less abruptly at low T compared with Einstein model

Low frequency modes correspond to multiple atoms vibrating together (sound or acoustic waves)


Standing waves: revision Consider a vibrating string

n=4

Lowest (fundamental) frequency

n=3 n=2

More generally n=1

L Other results follow:

 2



2L n L  λ 2 n

vn vn f    ω  2πf   2L L v

k

2π





n L


L

Standing waves in a 1D crystal Consider solid as a continuous elastic medium: N atoms, 3 degrees of freedom  3N standing modes a

2π πn n k   λ L Na

1D array of atoms: L=Na

max  2 L  2 Na  kmin Fundamental mode (n=1)

π  Na

λmin  2a  kmax 

Highest order mode (n=N) Therefore we get N modes for N atoms Solid State Physics - Lecture 8

 nmax  N

 a

Standing waves in 2D crystals Fundamental mode (2D)

Each component of the wave is quantised separately and added in quadrature

k x  ky 

x L

y L

π L

Magnitude of k-vector for mode

k

kx

2

π  ky  2 L 2

Corresponding angular frequency

ω  2πf  2

v



 vk

ωv 2

 L


Standing waves in 2D crystals: Degeneracy ky 

π kx  L

2π L

ky 

2π kx  L

π L

x y L

L

L

L

π     2  k       5 L L  L  2

π  2     k      5 L  L  L 2

2

In both cases

ωv 5



2

so these two modes are degenerate

L

As frequency increases, more and more states share the same frequency & energy (called DEGENERACY) Solid State Physics - Lecture 8

Back to reciprocal space… (2D) •

We can represent each mode as a point in reciprocal (k) space

Q. How many modes are available at a particular k value? A. Need three pieces of information: 1. How “big” is an individual k-state 2. How much of k-space is covered at a particular k 3. Account for degeneracy

kl 2 g k dk  dk 2 Solid State Physics - Lecture 8

Number of States in 3D In 3D we consider the number of states within a sphere of radius k Sphere “volume” =

kz

4 3 k 3

k

ky

“volume” of k-state =

3 l3

kx

2

Vk g k dk  dk 2 2

 l

 l

k-state


 l

Number of States in 3D Vk 2 g k dk  dk 2 2 We know that

Hence

kz

k

ω  vk  dω  vdk

V 2 g  d  d 2 3 2 v

ky kx

i.e. the number of standing waves (modes) increases as 2

 l

Sound can propagate with 2 transverse and 1 longitudinal wave in a solid  total no. of states = 3g()d 


 l

k-state

 l

Debye frequency For any one wavelength of oscillation there are shorter wavelength oscillations that will also have the atoms in the same position on the lattice (c.f. aliasing in electronics) There is a minimum wavelength which can oscillate which corresponds to a maximum frequency, max (Debye frequency) We can calculate max since we know (from earlier) that the maximum number of states = 3N

 3N 

max

 3g  d 0



ωmax

 0

3V V 3 dω  ω max 2 3 2 3 2π v 2π v 2


So,

ωmax

 2 N  v 6π  V 

1 3

Some crystal modes of vibration

Phonon animations here: http://www.phonon.fc.pl/index.php


Debye model: Total average energy of System From earlier: Energy of = system

E

max

 0

Phonon Average no. x energy of phonons

1    3g  d       1 exp   k BT 

Integrate over all modes

(NB: Ignoring zero-point energy)


x

max

 0

No. of modes

3g   d      1 exp   k BT 

Debye model: Total average energy of System E

max

 0

3g   d      1 exp   k BT 

3V E 2 2 v 3

V 2 From before: g  d  d 2 3 2 v

ω x k BT

Make substitution:

4 B

4

3Vk T E  2 2 v 3 3

D

T

 0

max

 0

 3 d      1 exp   k BT 

and define Debye temperature:

x3 dx exp x   1

ωmax θD  kB

From which (finally) we can extract the heat capacity, C


The Debye Temperature D This is perhaps the most useful parameter in the Debye theory • It allows us to predict the heat capacity at any temperature • It provides an indication of the temperature at which we approach the classical limit of the Dulong-Petit theory

ω θ D  max kB

From earlier, we know that

Therefore,

v

max  2 N  6  V 

1

 3

 DkB 

and

2 N  6      V

N  ωmax  v 6π 2  V 

1 3

 13

So if we know N/V then we can predict the speed of sound in a solid


The Debye Temperature D: examples High D corresponds to a large max Large max implies large forces, low max implies weak bonds Diamond D = 2230K

Dulong-Petit poor fit at room temperature. Strongly bonded

Iron

D = 457K

Dulong-Petit reasonable fit at room temperature.

Lead

D = 100K

Dulong-Petit good fit at room temperature. Weakly bonded


Debye model: Heat Capacity 4 B

4

3Vk T E 2 2v 3 3

D

T

 0

x3 dx exp x   1

where

ω x k BT

At high T:

exp( x)  1  x  ...

x is small  D

T

 0

3

x dx  exp x   1

Vk T E  2 v 

4 3 B D 2 3 3

 E  3Nk BT

and since

D

T

 0

so… 3

x dx  1  x  ...  1

v

 DkB 

2 N  6      V

Heat capacity, C 

dE  3Nk B dT


D

T

 0

x 2 dx 

 D3 3T 3

 13

Dulong-Petit !

dE  Cmolar   3N A k B dT

Debye model: Heat Capacity 4 B

4

3Vk T E 2 2v 3 3

D

T

 0

x3 dx exp x   1

where

At low T:



Take limit that D/T   and use identity

so…

x3 4 0 exp x   1dx  15

3Vk B4T 4  4 3 4 Nk BT 4 E  2 3 3 2 v  15 5 D3 Heat capacity, C 

v

ω x k BT

 DkB 

N  6    V

 13

Debye T3 law

T  dE 12  Nk B   dT 5  D 

2


4

3

Debye T3 law

Heat Capacity (mJ mol-1 K-1)

Heat capacity for solid Argon (from Kittel)

T3 (K3)


Comparison of Dulong-Petit, Einstein and Debye models of heat capacity Dulong-Petit


Thermal Conductivity •

Thermal conduction is a measure of how much heat energy is transported through a material per unit time

•

In metals conduction is due to free electrons (a later lecture)

•

In non-metals conduction is largely due to phonons • most hard insulators have a low thermal conductivity

•

Phonons have energy and can therefore conduct heat • Scattering mechanisms limit the thermal conductivity of non-metals, due to • Imperfections (grain boundaries, point defects, dislocations) • Phonons themselves can scatter other phonons (Umklapp processes – we won’t cover that here) Solid State Physics - Lecture 8

Thermal Conductivity

Thermal conductivity

Thigh

Tlow

1 dE dT Q   A dt dx

Area, A

Energy flow along x

 

1 dE dx 1 dE dx  A dt dT A dT dt

i.e. thermal conductivity scales with heat capacity


C

v