The Debye Model Professor Stephen Sweeney

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

Solid State Physics - Lecture 8 Recap from Lecture 7

• Concepts of “temperature” and thermal Dulong-Petit 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 Summary of Dulong-Petit and Einstein models of heat capacity

Dulong-Petit model (1819) Einstein model (1907)

• Atoms on lattice vibrate • Atoms on lattice vibrate independently of each independently of each other other • Completely classical • Quantum mechanical • Heat capacity (vibrations are quantised) independent of • Agreement with

temperature (3NkB) experiment good at very • Poor agreement with high (~3NkB) and very low experiment, except at (~0) temperatures, but high temperatures not inbetween

Solid State Physics - Lecture 8 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 3D case

Java applet: http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html

Solid State Physics - Lecture 8 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

The number and type of Einstein: number of modes = number of atoms modes are the key difference

Debye: each mode has its own k value (and hence frequency)

Solid State Physics - Lecture 8 Modes: lattice vibrations

Modes exist in various areas of physics/nature

Butterfly wing-beat Water Molecules

Guitar modes

Solid State Physics - Lecture 8 Modes: Quantum mechanics

Modes are quantised in units of   where the fundamental frequency of each mode is 

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)

Solid State Physics - Lecture 8 Standing waves: revision

n = 4 Consider a vibrating string  Lowest (fundamental) frequency  L n = 3 2 n = 2  2L More generally n  L  λ  n = 1 2 n

L v vn vn Other results follow: f    ω  2πf   2L L 2π n k    L

Solid State Physics - Lecture 8 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 1D array of atoms: k    L=Na λ L Na

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

 Highest order mode λ  2a  k  (n=N) min max a nmax  N Therefore we get N modes for N atoms

Solid State Physics - Lecture 8 Standing waves in 2D crystals

Fundamental mode (2D)

Each component of the wave is quantised separately and added in quadrature π x k  ky  x L y L L Magnitude of k-vector for mode

2 2 π k  k  ky  2 x L Corresponding angular frequency  v ω  v 2 ω  2πf  2  vk L 

Solid State Physics - Lecture 8 Standing waves in 2D crystals: Degeneracy 2π π k  k  π y L 2π y L k  k  x L x L

y L L L L

2 2 2 2    2  π  2    π k        5 k        5 L L L  L   L  L      In both cases ω  v 5 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

kl2 gkdk  dk 2

Solid State Physics - Lecture 8 Number of States in 3D

In 3D we consider the number of states k within a sphere of radius k z

k 4 3 Sphere “volume” = k 3 ky  3 “volume” of k-state = kx l 3

 l Vk 2 gkdk  2 dk 2 k-state

Solid State Physics - Lecture 8 Number of States in 3D

2 k Vk z gkdk  2 dk 2 k

We know that ω  vk  dω  vdk ky 2 V k Hence gd  d x 2 2v3

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

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

Solid State Physics - Lecture 8 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

max 3N  3gd 0 1 3  2 N  ωmax 2 3V V So, ωmax  v6π   dω  ω3  V   2 3 2 3 max 0 2π v 2π v

Solid State Physics - Lecture 8 Some crystal modes of vibration

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

Solid State Physics - Lecture 8 Debye model: Total average energy of System

Energy of Phonon Average no. No. of From earlier: = x x system energy of phonons modes

max 1 max 3g E     3gd   d 0    0    exp  1 exp  1  kBT   kBT 

Integrate over all modes

(NB: Ignoring zero-point energy) Solid State Physics - Lecture 8 Debye model: Total average energy of System

max 3g E   d     0 exp  1 3V max  3  k T  E  d  B  2 2v3     0 exp  1 2   V  kBT  From before: gd  d 2 2v3

ω ωmax Make substitution: x  and define Debye temperature: θD  kBT kB

D 3Vk 4T 4 T x3 E  B dx From which (finally) we can 2 3 3  extract the heat capacity, C 2 v  0 expx1

Solid State Physics - Lecture 8 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

1 3 ω  2 N  From earlier, we know that θ  max and ω  v6π  D max V kB  

1  3 max DkB  2 N  Therefore, v  1  6   N  3   V  6 2   V  So if we know N/V then we can predict the speed of sound in a solid

Solid State Physics - Lecture 8 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

Solid State Physics - Lecture 8 Debye model: Heat Capacity

D 3Vk 4T 4 T x3 ω E  B dx where x  2 3 3  2 v  0 expx1 kBT

At high T: x is small  exp(x) 1 x ... so…

D D D T x3 T x3 T  3 dx  dx  x2dx  D    3 0 expx1 0 1 x ...1 0 3T

1 4 3  3 Vk T  k  2 N  E  B D and since v  D B 6  2 2v33   V  Dulong-Petit ! dE dE E  3Nk T Heat capacity, C   3Nk  C   3N k B dT B molar dT A B

Solid State Physics - Lecture 8 Debye model: Heat Capacity

D 3Vk 4T 4 T x3 ω E  B dx where x  2 3 3  2 v  0 expx1 kBT

At low T:  x3  4 Take limit that  /T   and use identity dx  D  0 expx1 15

3Vk 4T 4  4 3 4 Nk T 4 so… E  B  B 2 3 3 3 Debye T3 law 2 v  15 5D

3 dE 12 4  T    Heat capacity, C   NkB    1 dT 5 D   k  N  3 v  D B 6 2    V 

Solid State Physics - Lecture 8 Debye T3 law

Heat capacity for solid Argon (from Kittel)

)

1 1

- HeatCapacity (mJ mol

T3 (K3)

Solid State Physics - Lecture 8 Comparison of Dulong-Petit, Einstein and Debye models of heat capacity

Dulong-Petit

Solid State Physics - Lecture 8 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

T T high low 1 dE dT Area, A Q    A dt dx 1 dE dx 1 dE dx      Energy flow along x A dt dT A dT dt

C v i.e. thermal conductivity scales with heat capacity

Solid State Physics - Lecture 8