Solids can be classified according to the dominant contribution to their
cohesive energy: van der Waals, ionic, hydrogen-bonded, covalent solids,
and metals. In principle, the total
energy of a solid depends on the coordinates and velocities of the atomic
nuclei and on the coordinates and velocities (or, in quantum mechanics, the wavefunctions)
of the electrons. However, in the adiabatic approximation the potential
energy is expressed as 
,
where the 
are the atomic positions, with no
explicit dependence on their electronic states.
When the nuclei are at equilibrium positions ( 
), the energy U
equals the binding energy or cohesive energy of the solid.
For van der Waals solids the potential energy can be quite well calculated by summing up the pair potential of interacting atoms. For two atoms the interaction is described by the the Lennard-Jones potential:
or
The energy of the lattice can be expressed with the lattice sums
and 
such that
For ionic solids the Coulomb potential, supplemented with a strongly repulsive core force, is used:
where e is the electron charge, and B and n are parameters to describe the repulsive atomic core. The Madelung energy is obtained by adding up the Coulomb potentials in the crystal
where N is the number of primitive unit cells in the crystal,
d is the equilibrium nearest-neighbor
distance between the centers of the positive and negative ions, and
is the distance between the ith and jth ion,
measured in units of d.
The 
factor compensates for double counting the ion pairs
as the sum is performed, and the 
represents the attractive and
repulsive contributions.
The Madelung constant is the coefficient M in Eq.
2.5.
In covalent and hydrogen-bonded materials, the calculation of the cohesive energy is much more complicated. In general, the potential energy cannot be calculated as a sum over pair potentials acting between atoms. For simple metals, an estimate of the cohesive energy can be obtained by balancing the kinetic energy of the electrons against the Coulomb attraction between the atomic cores and the conduction electrons. To calculate the Coulomb energy, the electrons are treated as a uniform negative background; in first approximation the kinetic energy is obtained by solving a simple ``particle in a box" quantum mechanics problem. This procedure yields
where r is the average distance between the electrons.
Lattice vibrations are elastic waves propagating
within crystals. Phonons are quantized elastic waves.
The expression can be
expanded around the equilibrium position of the atoms by 
,
where represents the equilibrium position and 
is the
displacement vector. We obtain the harmonic expansion:
where the tensor 
is obtained from the second derivatives of the
potential. Quite often it is enough to keep only the nearest-neighbor terms in
the above summation.
The equation of motion is
where 
is the mass of the ith atom. The solution is searched in
the form of a lattice wave,
where the three 
(s=1,2,3) are the
polarization vectors of the vibration.
With one atom per unit cell (every 
), in the harmonic approximation,
the solution to the equation of motion is reduced to solving a simple linear
set of equations:
where 
(independent of 
) is the dynamical matrix.
In one dimension, for nearest-neighbor-only interactions, Eq. 2.7 simplifies to
and the equation of motion is also much simpler.
The general solution of the equation of motion provides
the phonon dispersion or phonon spectrum
.
A continuous set of 
values is a
phonon branch.
The long wavelength (or small wavenumber,
, where a is a lattice spacing)
vibrations are sound waves. The phonon branches that
start from 
at 
are the acoustic phonons.
It is easy to show that for acoustic phonons at small 
the phonon frequencies are proportional to k (although the constant of
proportionality c, the sound velocity, may depend on the direction
of propagation).
If the wave vector 
is along appropriate symmetry axes of the crystal, then one of the
polarization vectors will point parallel to (corresponding to the
longitudinal mode), while the other two are perpendicular (transverse
modes). For general directions of , the concept of longitudinal and
transverse modes is only approximately valid.
For p;SPMgt;1 atoms in the primitive unit cell, the phonon
spectrum will have more branches, including the p-1 higher-frequency
optical phonons.
The total number of possible values of are fixed by the periodic
boundary condition as 
, where N is the number of
primitive unit cells in the crystal (for simple crystals with one atom per
unit cell, N is the number of atoms).
In three dimensions, for a system of 
atoms the total
number of possible phonon modes is always 
.
At high-symmetry points in the Brillouin zone the calculation of the phonon
mode frequencies is much simpler than finding the general solution.
For example, in one dimension a zone boundary ( 
) mode corresponds
to neighboring atoms oscillating with opposite phases. With this in mind,
the equation of motion for the N atom can be reduced to that of a two-atom
problem.
Since atoms are massive, a fairly accurate picture of phonons can be obtained
without using quantum mechanics. A somewhat oversimplified
transition to the ``quantum world"
is provided by the correspondence principle: The phonon energy
is 
and the momentum is 
.
Note that, loosely speaking, the amplitude of the classical vibrations
with wavenumber 
and frequency 
corresponds to the number of phonons in the , state,
. More accurately, the expectation value of the amplitude,
, and the expectation value of the number of phonons
are related by
where M is the mass of the atoms.
Remember: The classical energy of the oscillation, 
, has nothing to do with the phonon energy!
Two convenient models are frequently used in connection to lattice vibrations.
The Einstein model is appropriate for the optical modes; it consists of
independent oscillators with the same resonance frequency, set equal to the
frequency of a typical optical phonon. The total number of oscillators
equals N, the number of optical phonon modes around that frequency.
The Debye approximation replaces the density of states for
each of the three acoustic modes with
the density of states corresponding to the low- part of the spectrum,
where the phonon frequency is proportional to the wavenumber,
, with c being the sound velocity.
Furthermore, the Brillouin zone is replaced with a sphere of radius 
,
the Debye wavenumber, so that the total number of states within this sphere
is equal to the total number of states in the Brillouin zone, N.
The upper cutoff frequency is the Debye frequency, 
; the corresponding temperature is the Debye temperature
.
As we will discuss later, at temperatures much higher than the Debye
temperature the number of phonons increases as 
.
The Lindemann melting formula uses the temperature dependence of the
phonon number and Eq. 2.12 to estimate the melting point of solids:
is the amplitude of the thermally excited oscillation
at the melting point, 
is the average size of the unit cell, and
M is the mass of the atoms.
Neutron scattering is the best way to explore the phonon spectrum,
but limited information can also be obtained from optical spectroscopy
(see introductory text in Chapter 
) and other spectroscopic
methods.
To tackle the task of presenting a multivalued function of three independent
variables (the three components of ), it is common to show the
phonon dispersion only along a few symmetry directions of the crystal.
In metals the lattice deformations influence the motion of the electrons
and vice versa. The electron-phonon interaction leads to a
damping of the lattice vibration, or finite phonon lifetime.
The finite phonon lifetime appears as a finite energy width in
the scattering experiment. For the electrons, an important contribution
to the finite electrical resistance (see introductory text in Chapter
) is due to phonons; however, the electron-phonon interaction
can lead to
a total destruction of the metallic conductivity, like in the case of
superconductivity or charge density waves.
In the harmonic approximation the compressibility is independent
of the temperature, and the
the linear thermal expansion coefficient is zero.
To understand the thermal expansion, one has to go beyond the harmonic
expansion. Anharmonicity also leads to the volume dependence of
phonon frequencies. The Grüneisen parameter,
, describes this effect for small
changes in volume. For insulators, the approximate formula
points to the intimate relationship between the
two manifestations (i.e., thermal expansion and phonon frequency change)
of the anharmonicity.
With the anharmonic terms in the potential, the lattice waves (Eq. 2.9) are not exact solutions of the equation of motion. However, the lattice waves and the phonon concept can be saved if the anharmonic terms are viewed as a source of interaction between the phonons. In this description, phonons have finite lifetimes, and they can decay into other phonons, as long as their energies and momenta (wavenumbers) are conserved.
Laszlo Mihaly
Thu Oct 31 13:23:11 EST 1996
Cover Page
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