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Thermal properties of solids ● Lattice specific heat ● Lattice vibrations – classical theory ● Dulong-Petit law ● Phonons ● Quantum theory of specific heat ● Einstein model ● Debye model Ashcroft & Mermin: chapters 22 & 23 + appendix L Lattice vibrations – classical theory ● Two assumptions ● ● regular arrangement of mean equilibrium positions of atoms small displacements around these positions ● Potential energy ● Classical Hamiltonian Harmonic approximation ● ● Taylor expansion of potential Note: in equilibrium positions forces on atoms vanish, i.e. Harmonic approximation ● Final form where ● Adiabatic approximation: electrons move much faster than ions, i.e., from ions' point of view they instantaneously redistribute in accord with displacement of ions ● this contribution is also “hidden” in the matrix D ● matrix D can be calculated by DFT Dulong-Petit law ● Thermal energy (classically) i.e., temperature independent specific heat, determined only by number of degrees of freedom Normal modes ● Bravais lattice (no basis) ● we search for solutions of type ● equation of motion ● dynamical matrix D(k) is even in k and hermitean, i.e., it has 3 real eigenvalues and 3 corresponding eigenvectors per each k-point → normal modes Linear 1D chain with basis ● spring constants K and G ● two solutions Dispersion relation ● Acoustic branch ● Optical branch (at Γ and BZ boundary) Quantum theory - Phonons ● Classical theory predicts constant specific heat ● Reality at low temperatures: ● ● ● Insulators: ● Metals: The T^3 can be explained by quantum theory of lattice vibrations The AT term is due to electronic degrees of freedom Phonons Lattice Hamiltonian in harmonic approximation: Phonon annihilation and creation operators: allow to rewrite the hamiltonian into Phonons specific heat We rewrote the Hamiltonian into a sum of 3N independent harmonic oscillators Phonons are bosons, i.e., we can have any number of them in particular state k,s Energy density Specific heat High-temperature limit For First correction term to Dulong-Petit law: Low-temperature limit Simplifying assumptions: ● ● ● we consider only acoustical modes (optical are too high in energy to be populated) for acoustic branches at lowest energies we can integrate over whole reciprocal space because unless k is close to Γ-point Low-temperature limit where the integral can be evaluated: finally we obtain low-temperature lattice specific heat Einstein model All branches are assumed to have the same and constant frequency Energy density Einstein specific heat Einstein's model works OK for optical branches Debye model Based on acoustic modes instead Model dispersion relation: Specific heat in Debye model For low temperatures the integral evaluates to Einstein vs Debye models Density of states ● ● ● Often we need to evaluate the following type of expressions: It would be useful to turn them into integral over frequency using suitable “density of states” function Density of states can be calculated by