Thermal Properties of Crystal Lattices

1. Thermal Properties of Crystal Lattices Introduction to Solid State Physicshttp://www.physics.udel.edu/~bnikolic/teaching/phys624/phys624.html

2. Statistical mechanics of Phonons (=bosons) →Motion of harmonic crystal is described by a set of decoupled harmonic oscillators (phonons): →Average excitation level of normal mode at temperature can also be interpreted as the number of indistinguishable quanta, i.e., phonons, per oscillator: →Crystal in equilibrium with a heath bath at temperature → justified by dividing infinite system into a finite number of subsystems which interact weakly with the remaining system (acting as a heat bath for the give subsystem):

3. Partition function for Phonons

4. Evaluating : Virial theorem →For harmonic oscillator:

5. Evaluating : Phonon density of states

6. Example: Phonon DoS in 1D chain

7. Average RMS displacement of ions from equilibrium positions →One atom per unit cell: →Use virial theorem and phonon DOS:

8. Models of lattice dispersion: Debye • →For thermodynamic properties optical modes are irrelevant: • Retain only acoustic modes, while replacing them with a purely linear mode with the same initial dispersion. →Since the total DoS is finite, we have to introduce a cutoff at Debye frequency.

9. Models of lattice dispersion: Einstein →Each atom oscillates independently of other atoms — model is dispersionless: →Helium absorbed on atomically perfect surface – each atom is attracted weakly to the surface by van der Walls forces and sits in the local minimum of the surface lattice potential: it oscillates with a frequency without interacting with its neighbors.

10. Long-range order →Long-range order (which is an initial assumption for introducing phonons!) in the lattice at low temperature (at high temperature all lattices melt) exists if and only if : →Example of the Mermin-Wagner theorem!

11. One-dimensional systems: no long-range order →Random fluctuations of atoms in 1D lattice can accumulate to produce a very large average RMS displacement of the atoms out of small interatomic displacements. • In higher dimensional systems: the displacements in any directions are constrained by the neighbors in orthogonal directions. • “Real” 2D systems (monolayer of gas deposited on atomically perfect surface) do have long-range order due to the surface potential (corrugation of surface).

12. Specific heat of Phonon gas →Debye approximation:Elastic isotropic medium where cutoff frequency is the same for all three acoustic modes (this crude approximation better fits experiments than introducing separate cutoff for longitudinal and transverse branches!).

13. Specific heat of Phonon gas: low vs. high temperature limits Characteristic signature of low-energy phonon excitations! → Debye temperature is a measure of the stiffness of the crystal: above all modes are getting excited, and below modes begin to be “frozen out”.

14. Anharmonic effects: Thermal expansion →An unconstrained cubic system of linear dimension L will change its length with temperature — described by a coefficient of free (p=0) expansion: →Harmonic crystals do not expand when heated! →Introduce anharmonicity in the potential — quasi-harmonic approximation:

15. Cubic terms generates thermal expansion →As the average energy (temperature) of particle trapped in a cubic potential increases, its mean positions shifts! →Cubic potential is not any more exactly solvable — use mean-field approximation:

16. Grüneissen number →If free energy is expressed as a function of the volume, than the condition of zero stress for every temperature yields the relation between V and T: →Generalized to “real” solid:

17. Anharmonicity: Three-phonon processes →Physically: Phonon can decay into two other phonons while conserving the crystal momentum. →Graphically: Three phonon processes arising from cubic terms in the inter-ion potential (six other three phonon-processes are also possible).

18. Thermal conduction →Thermal current density = heat conductivity times temperature gradient = energy density times the velocity: →Metals carry heat via free electrons, and are good conductors of both heat and electricity. →Insulators lack free electrons and, therefore, carry heat via lattice vibrations ↔phonons. Although most insulators are not good thermal conductors, some very stiff insulating crystals have very high thermal conductivities (often highly temperature dependent):

19. Linear response theory of transport coefficients →If temperature gradient is small, will deviate only slightly from its equilibrium value : →Thus, to first order, heat current density is given by: →Since we already know , the calculation of and heat conductivity reduces to finding linear change in the phonon density due to the transport of energy.

20. Changing phonon number: decay →Change of phonon density occurs either by phonon decay or by phonon diffusion in and out of the region.

21. Changing phonon number: diffusion

22. Properties of thermal conductivity →Phonons near the BZ boundary or optical modes with small velocity contribute very little to thermal conduction. →Stiff materials, with very fast velocity of the acoustic modes will have a large thermal conductivity. →Thermal conductivity is small for material with short mean free path — affected by defects, anharmonic umklapp processes, …

23. Umklapp processes →At low temperature, only low-energy acoustic modes are excited for which and crystal momentum , so that one has to worry about anharmonic processes which do not involve a reciprocal lattice vector in requirement for momentum conservation. →At high temperatures, conservation of momentum in an anharmonic process may involve a reciprocal lattice vector if of an excited mode is large enough, and there exists sufficiently small reciprocal vector , so that momentum reversal occurs when: Umklapp process: Involves a very large change in the heat current (almost a reversal) — therefore and can become small at high temperatures.