5. Phonons

1
5. Phonons Thermal Properties

• Issues that are addressed in this chapter
  include
• Phonon heat capacity with explanation to the
  phonon occupancy number
• Anharmonic crystal interactions
• Thermal conductivity

2
5.1 Phonon heat capacity

• Heat capacity and Planck distribution function
• By heat capacity we mean change in the inthernal
  energy of the system for unit change in
  temperature for fixed volume,i.e.
• The total energy of the phonon bath at a
  temperature T equals the sum of all branches p
  and all phonon modes K,
• Where ltnK,pgt is the occupancy factor of a mode K
  frm branch p..

3

• The Planck distribution is found from the
  considerati-on of a set of identical harmonic
  oscillators in the (n1)-st and n-th state
• The fraction of the total of oscillators in the
  n-th quantum state is
• The occupancy factor, i.e. the average excitation
  number of the oscillator, is then given by
• In deriving the above result we have used

4

• With the above, the energy of a collection of
  oscillators of frequency ?K,p is
• By conserving the of states, I.e. going from
  sum over K to an integral over ?, we can express
  the lattice heat capacity as

5

• The Density of States (DOS) Function
• When calculating the DOS function, one can use
  either vanishing or periodic boundary conditions.
  For simplicity, we start with a 1D derivation
  and then go to a 3D case.
• Vanishing boundary conditions
• We consider a 1D line of length L that carries
  (N1) par- ticles. The particles at s0 and at
  sN are held fixed. From the harmonic
  approximation
• and the condition that uN0, we get
• i.e. there are (N-1) allowed independent values
  of K. Hence, the number of modes per unit range
  of K is L/?.

6

• ? Periodic boundary conditions
• Another way of looking at this problem is to
  assume that the medium is unbound, but the
  solutions have to be periodic over a large
  distance L, so that
• Again, for harmonic displacements this leads to
• This method gives the same number of modes, one
  per atom, but we have or values and ?K2?/L
  between successive values of K.
• DOS function
• The DOS function in 1D and in analogy, in 3D, is
  given by

7

• The Debye and the Einstein models for CV
• Debye model
• The Debye model is valid for acoustic phonons
  near the zone center, for which ?? vsK and the
  DOS function and the cut-off, or the Debye
  frequencies are given by
• The thermal energy is represented with
• And the heat capacity equals to

8
(No Transcript)
9
? Einstein model In the Einstein model, all N
oscillators oscillate with the same frequency w,
and in that case The thermal energy of these
optical modes of vibration is given by and the
heat capacity equals to Note that at low T,
the expression for CV describes exponential decay
with T, whereas the experiments show T3 behavior.
10

• Derivation of the DOS Function
• The last thing we want to consider is to find a
  general expression for the DOS function
• To evaluate the volume of the shell, we take dS?
  to be an element of the surface in k-space and
  dK? to be the perpendicular distance between the
  surface ?const. and the surface ?d?const.
  Then
• that gives
• and

11
5.2 Anharmonic Crystal Interactions

• Harmonic Theory Assumptions
• The following are the assumptions made in the
  harmonic theory
• Two lattice waves do not interact with each other
• There is no thermal expansion
• Adiabatic and the isothermal lattice constants
  are the same
• The elastic constants are independent of preasure
  and temperature
• The heat capacity CV becomes constant at Tgt?
• None of the above assumptions is satisfied in
  real crystals in the case of three phonon
  interactions.

12

• Thermal Expansion
• The thermal expansion coefficients describe the
  increase in the lattice constant in the crystal
  with increa- sing temperature.
• Let us denote the potential energy of the atoms
  at a displacement x from the equilibrium position
  as
• U(x)cx2 - gx3 - fx4
• The average displacement of atoms is calculated
  by using Boltzmann distribution, i.e.

Asymmetry of the mutual repulsion
Softening of the vibration at large amplitude
13
Note that the slope of ltxgt with T is
proportional to the thermal expansion coefficient
g.
ltxgt
T
14
5.3 Thermal Conductivity
The thermal conductivity coefficient is defined
with respect to the steady-state flow of heat
down a temperature gradient, i.e. To arrive at
the expression for the thermal conductivity, we
start from where Substituting back the above
results leads to
15

• In this last expression, Cnc and lv? is the
  mean free time between collisions that is defined
  by the following
• Geometric scattering crystal boundaries and
  lattice imperfections
• Scattering by other phonons due to anharmonic
  coupling that predicts that l1/T, and therefore
  ?1/T. Here we are interested in the phonon
  processes that contribute to the thermal
  conductivity and limit its value. Examples are
  normal and umklapp three phonon processes, out of
  which the umklapp processes make the largest
  contribution.
• - normal phonon process K1 K2 K3
• - umklapp phonon process K1 K2 K3 G