1. EEE539 Solid State Electronics 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. EEE539 Solid State Electronics  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 is the occupancy factor of a mode K frm branch p.. 5.1 Phonon heat capacity

3. EEE539 Solid State Electronics  The Planck distribution is found from the considerati- on of a set of identical harmonic oscillators in the (n+1)- 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. EEE539 Solid State Electronics  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. EEE539 Solid State Electronics  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 (N+1) par- ticles. The particles at s=0 and at s=N are held fixed. From the harmonic approximation and the condition that u N =0, 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. EEE539 Solid State Electronics  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  K=2  /L between successive values of K.  DOS function The DOS function in 1D and in analogy, in 3D, is given by:

7. EEE539 Solid State Electronics  The Debye and the Einstein models for C V  Debye model The Debye model is valid for acoustic phonons near the zone center, for which  = v s K 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. EEE539 Solid State Electronics At very low temperatures, the heat capacity is found from the assumption that  /T  , when  Therefore, at very low temperatures, the T3 approximati- on is quite good model for the acoustic modes.  Note also that out of the allowed K-space, the volume occupied equals (  T /  D ) 3 where h  T =k B T. CVCV 3Nk B T (T/  ) 2 T/ 

9. EEE539 Solid State Electronics  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 C V describes exponential decay with T, whereas the experiments show T 3 behavior.

10. EEE539 Solid State Electronics  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. EEE539 Solid State Electronics 5.2 Anharmonic Crystal Interactions  Harmonic Theory Assumptions The following are the assumptions made in the harmonic theory: 1.Two lattice waves do not interact with each other 2.There is no thermal expansion 3.Adiabatic and the isothermal lattice constants are the same 4.The elastic constants are independent of preasure and temperature  The heat capacity C V becomes constant at T>  None of the above assumptions is satisfied in real crystals in the case of three phonon interactions.

12. EEE539 Solid State Electronics  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)=cx 2 - gx 3 - fx 4 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. EEE539 Solid State Electronics Note that the slope of with T is proportional to the thermal expansion coefficient g. T

14. EEE539 Solid State Electronics 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. EEE539 Solid State Electronics In this last expression, C=nc and l=v  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 l~1/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: K 1 + K 2 = K 3 - umklapp phonon process: K 1 + K 2 = K 3 + G