1. Phonons II: Thermal properties specific heat of a crystal
density of states
Einstein model
Debye model
anharmonic effect
thermal conduction
M.C. Chang
Dept of Phys

2. Specific heat: experimental fact
Specific heat approaches a constant 3R (per mole) at high temperature
(Dulong-Petit law)
Specific heat drops to zero at low temperature
After rescaling the temperature by  (Debye temperature), which differs from material to material, a universal behavior emerges:

3. Debye temperature
In general, a harder material has a higher Debye temperature

4. Specific heat: theoretical framework
Internal energy U of a crystal is the summation of vibrational energies (consider an insulator so there’s no electronic energies)
For a crystal in thermal equilibrium, the average phonon number is determined by (see Kittel, p.107)
Therefore, we have
Specific heat is nothing but the change of U(T) w.r.t. to T:

5. Density of states D() (DOS, 態密度)
For example, assume N=16, then there are 22=4 states within the interval d
Flatter (k) curve, higher DOS.
D()d is the number of states within the constant- surfaces  and +d
Once we know the DOS, we can reduce the 3-dim k-integral to a 1-dim  integral.
Connection between summation and integral
f(x) x b a

6. DOS: 3-dim (assume (k)= (k) is isotropic)
a simple change of variable
Ex: Calculate D() for the 1-dim string with (k)=M|sin(ka/2)|
There is no use to memorize the result, just remember the way to derive it.
DOS: 3-dim (assume (k)= (k) is isotropic)

7. Einstein model of calculating the specific heat CV (1907)
Assume that each atom vibrates independently of each other, and every atom has the same vibration frequency 0
3 dim  number of atoms
(Activation behavior)

8. Debye model (1912)
In actual solids, because of the atomic bonds, the atoms vibrate collectively in a wave-like fashion.
Debye assumed a simple dispersion relation :  = vs k. Therefore, Ds()=(L3/22vs3) 2
Actual density of states for silicon

9. Also, the range of integration is slightly changed.
The 1st BZ is approximated by a sphere with the same volume
for 3 identical branches
= 4/15 as T 0

10. Comparison between the two models
At low T, Debye’s curve drops slowly because, in Debye’s model, long wave length vibration can still be excited.
Energy dispersion
solid Argon (=92 K)

11. A simple explanation for the T3 behavior:
Suppose that
1. All the phonons with wave vector k<kT are excited with thermal energy kT.
2. All the modes between kT and kD are not excited. kT kD
Then the fraction of excited modes
= (kT/kD)3 = (T/)3.
energy U  kT3N(T/)3, and the
heat capacity C  12Nk(T/)3

12. More discussion on DOS (general case)
Surface =const
Surface +d =const
dk
dS

13. Thermal properties of crystals specific heat of a crystal
density of states
Einstein model
Debye model
anharmonic effect
thermal conduction

14. Anharmonic effect in crystals
If there is no anharmonic effect, then
V(r) r
There is no thermal expansion
There is no phonon-phonon interaction
A crystal would vibrate forever
Thermal conductivity would be infinite
...

15. Thermal conductivity Thermal current density (Fourier’s law, 1807)
In metals, thermal current is carried by both electrons and phonons. In insulators, only phonons can be carriers.
The collection of phonons are similar to an ideal gas
Ashcroft and Mermin, Chaps 23, 24

16. Phonon-phonon scattering is a result of the anharmonic vibration
Heat conductivity K=1/3 Cv
where C is the specific heat, v is the velocity, and  is the mean free path of the phonon. (Kittel p.122)
A phonon can be scattered by an electron, a defect (including a boundary), and other phonons. Such scattering will shorten the mean free path.
Phonon-phonon scattering is a result of the anharmonic vibration
Modulation of elastic const.

17. Umklapp process (轉向過程, Peierls 1929):
Normal process (正常過程): total momentum of the 2 phonons remains the same before and after scattering, no resistance to thermal current!
Umklapp process (轉向過程, Peierls 1929):
1st BZ

18. T-dependence of the mean free path
At low T, for a crystal with few defects, a phonon does not scatter frequently with other phonons and the defects. The mean free path is limited mainly by the boundary of the sample.
At high T, C is T-independent. The number of phonons are proportional to T, therefore the mean free path ~ 1/T
T-dependence of the thermal conductivity (K=1/3 Cv) :
At low T, K~C~T3
At high T, K~ ~1/T