Thermal Properties of Crystal Lattices

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Thermal Properties of Crystal Lattices

• Introduction to Solid State Physics
  http//www.physics.udel.edu/bnikolic/teaching/phy
  s624/phys624.html

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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)
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Partition function for Phonons
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Evaluating Virial theorem

?For harmonic oscillator
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Evaluating Phonon density of
states
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Example Phonon DoS in 1D chain
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Average RMS displacement of ions from equilibrium
positions
?One atom per unit cell
?Use virial theorem and phonon DOS
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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.
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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.
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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!
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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).

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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!).
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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.
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Anharmonic effects Thermal expansion
?An unconstrained cubic system of linear
dimension L will change its length with
temperature described by a coefficient of free
(p0) expansion
?Harmonic crystals do not expand when heated!
?Introduce anharmonicity in the potential
quasi-harmonic approximation
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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
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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
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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).
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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)
273.2K 298.2K
Carbon 26.2 23.2
Copper 4.03 4.01
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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.
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Changing phonon number decay
?Change of phonon density occurs either by phonon
decay or by phonon diffusion in and out of the
region.
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Changing phonon number diffusion
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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,
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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.