Download Thermal properties of solids

Thermal properties of solids
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Lattice specific heat
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Lattice vibrations – classical theory
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Dulong-Petit law
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Phonons
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Quantum theory of specific heat
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Einstein model
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Debye model
Ashcroft & Mermin:
chapters 22 & 23 + appendix L
Lattice vibrations – classical theory
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Two assumptions
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regular arrangement of mean equilibrium positions
of atoms
small displacements around these positions
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Potential energy
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Classical Hamiltonian
Harmonic approximation
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Taylor expansion of potential
Note: in equilibrium positions forces on atoms
vanish, i.e.
Harmonic approximation
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Final form
where
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Adiabatic approximation:
electrons move much faster than ions, i.e., from
ions' point of view they instantaneously
redistribute in accord with displacement of ions
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this contribution is also “hidden” in the matrix D
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matrix D can be calculated by DFT
Dulong-Petit law
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Thermal energy (classically)
i.e., temperature independent specific heat,
determined only by number of degrees of freedom
Normal modes
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Bravais lattice (no basis)
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we search for solutions of type
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equation of motion
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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
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spring constants K and G
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two solutions
Dispersion relation
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Acoustic branch
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Optical branch
(at Γ and BZ boundary)
Quantum theory - Phonons
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Classical theory predicts constant specific heat
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Reality at low temperatures:
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Insulators:
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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:
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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
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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