Chapter 4 : Phonons and Crystal vibration
Types of Vibrations in
Crystals
Many solid materials, including all metals, are composed of atoms arranged in a lattice arrangement called crystals. There are a variety of crystal structures like cubic, hexagonal, cubic with an atom in the center of the cube, called body centered cubic, cubic with an atom in the center of each face of the cube, called face centered cubic, and others. The particular structure depends on the relative sizes of the atoms that are nestled together to form the crystal. The reason that materials take crystal form is that these neat geometrical structures represent the lowest energy configuration of the collection of atoms making up the material. To dislodge an atom from the crystal structure requires the addition of energy.
Theoretically, at a temperature of absolute zero, the atoms of a crystal lie at their lowest energy position without moving at all. As thermal energy is added to the crystal it is manifest by vibration of the atoms about this equilibrium location. Within the limits of fairly small vibrations the electric forces bonding the atoms together stretch or compress a bit to a higher energy configuration. Each atom acts as though it were connected to its neighbors by little springs. The added energy is stored in the crystal as the kinetic energy of the atoms in motion and the potential energy of the compressed or stretched springs.
Let us consider the oscillation of the atoms in onedimensional crystal simulated by the chain of the balls with the masses m_{1} and m_{2}. The force applied to every atom depends upon the relative displacement of the nearby atoms and stiffness c of the virtual springs. Therefore, the displacement u of every atom in onedimensional crystal is described by the equations:
These equations determine the oscillatory motion of the atoms:
where_{ }a/2 is the distance between the nearest atoms; k=2p/l is the wave factor, l is the wavelength of the wave in crystal. The equations mentioned above have two solutions for w :
where g^{2}=4m_{1}m_{2}/(m_{1}+m_{2})^{2}_{; }w_{0}^{2}=2c (m_{1}+m_{2})/m_{1}m_{2}. These equations determine two branches of the dispersion curve (so called acoustic and optical branches).
Let us consider the types of the crystal oscillation for these two branches. In the case of the longwave approximation (ak << 1) in acoustic branch the atoms move synchronously and deviation of every atom is about the same at any moment of the time; in optical branch the atoms move in antiphase (it is shown below of the animation). For the shortest waves in acoustic branch the lighter atoms are still and more heavy atoms oscillate; in the optical branch the situation is inverse (see the figure overhand of the animation). Oscillation of atoms in optical branch polarizes the matter electrically and this type of oscillation can be excited by infrared optical radiation. This is the reason why this branch was called "optical".
We can see that the modes differ in the details of how the atoms move, but they both represent energy stored in the crystal, being passed back and forth between the kinetic energy of the atoms and the potential energy of the "springs". The temperature of the crystal is proportional to the average kinetic energy of the atoms.
There are some limitations to this mechanical description of what is going on in crystal vibrations. Because we are dealing with objects as small as individual atoms, quantum mechanical effects may not be neglected. For example, in a metal sample large enough to work with in a laboratory, there may be millions of individual crystals each with millions of individual atoms. On a laboratory size scale, it appears that we can add energy to the sample in any amount, as if the crystal were really composed of weights and springs. In fact, energy may only be added in multiples of some minimum amount. We call that minimum amount of vibrational energy a "phonon" analogous to the photon energy packet familiar in electromagnetic radiation. Energy addition appears continuous on the laboratory scale because the phonon is so small. On the scale of the atoms in the crystals however, the phonon magnitude is significant, and only vibration modes and amplitudes which differ in energy by whole phonon multiples are allowed.
Lets
consider a linear chain of identical atoms of mass M spaced at a distance , the
lattice constant, connected by invisible Hook's law springs. For simplicity we
will consider longitudinal deformations  that is, displacements of
atoms are parallel to the chain.
Let U_{n }=displacement of
atom n from its equilibrium position
U_{n1}=displacement of atom
n1 from its equilibrium position
U_{n+1}=displacement of atom
n+1 from its equilibrium position
The force on atom n will be given by its displacement and the displacement of its nearest neighbors :
The equation of motion is:
where is a spring constant.
The above equation is not obviously a wave equation, but let us assume a traveling wave solution, namely,
Let U_{no} = U_{o} since if it is a wave, it has to have a definite amplitude.
If we substitute our wave solution into equation of motion we find a phonon's dispersion relation for linear monatomic chain as follows:
The dispersion curve is shown below.
One important feature of the dispersion curve is the periodicity of the function. For unit cell length , the repeat period is , which is equal to the unit cell length in the reciprocal lattice. Therefore the useful information is contained in the waves with wave vectors lying between the limits
This range of wave vectors is called the first Brillouin zone. At the Brillouin zone boundaries the nearest atoms of the chain vibrate in the opposite directions and the wave becomes a standing wave .
As k approaches zero (the longwavelength limit) and we have
where is a phase velocity, which is equivalent to the velocity of a sound in the crystal. Phonons with frequency which goes to zero in the limit of small k are known as acoustical phonons.
We should look upon simple crystal structures
with monatomic basis. We want to understand how atom planes can vibrate in
relation to each other. There are three modes of vibrations, one longitudinal
and two of transverse polarization.
Q1. Which modes of vibration are described in figure 2.1.1 to figure 2.1.4?
In
this case we will concentrate to a simple picture with a lattice of primitive
basis containing two different atoms. In this case we have the same modes as in
the previus case. In addition the planes with different atoms can oscillate in
phase or in opposite phase (180 degrees phase difference). An illustration of
this situation can be found in the animation under figure below. If the two
atoms carry opposite charges, we may excite a motion of this type with the
electric field of a light wave, so that the branch is called the optical
branch. The branch discussed in previous section is called the accsustic
branch.
Crystal vibration with two atoms per basis
Q How many modes of vibration is possible in a diamond lattice (include both accustic and optical branches)?
The
band structure of a solid material depends on the location of each atom in the
lattice. Crystal vibration will be transformed into the reciprocal space and as
a consequence the band structure will vibrate. The band structure vibration
will interact with the electron in a strong way.
Crystal vibrations are usually called phonons. A phonon can be regarded as a particle that can interact with holes and electrons. In room temperature a crystal is full of lattice vibrations and in the phonon representation it exits a phonon gas within the crystal.
Different types of vibrations will have different effect on the band structure and therefore the phonon mode will interact differently with the electron. The interaction between an electron and a phonon can be described in terms of transistion rates between one state to an other state in the reciprocal space. The transistion rate the number of transistions from a inital k0 to final k0' that will take place per unit time. A typical scattering rate is in the range of 10100 scatterings per picosecond. The transistion rate can be calculated quantum mechanically. The input data for this type of calculation is a so called deformation potential and the phonon dispersion relation (relation between phonon energy and wave vector).
The deformation potential describes how much the band structure will change due to a change in lattice constant. There are methods that can be used in order to extracted the deformation potentials from experiments.
The dispersion relation can be looked upon as the band structure for the
phonon since it describes the relation between energy and wave vector for the
phonon. Phonon energy is often given in terms of phonon frequency. Figure below
shows a typical 1D dispersion relation for optical and accustical phonons.

Plot of phonon energy vs. phonon wave vector for a typical semiconductor
The dispersion relation for optical phonons are often considered to be a
constant value and is often given as a phonon temperature (see figure 2.3.2).
Phonon frequency, phonon temperature and phonon energy is different
representation of the same physical quantity.

Figure 2.3.2. Plot of the approximated phonon energy vs. phonon wave vector
The accustic phonon dispersion relation is often approximated by the following relation:

In
both cases the correctness of the approximation depends on weather the phonon
wave vector is small or large. In semiconductor modelling the phonon wave
vector is considered to be small and therefore the approximations can be used
without significant errors.
Lattice Photon Absorption
The
lattice absorption characteristics observed at the lower frequency regions of
photons, in the middle to farinfrared wavelength range, define the long
wavelength transparency limit of the material, and are the result of the interactive coupling between
the motions of thermally induced vibrations of the constituent atoms of the
substrate crystal lattice and the incident radiation.
The
conductive properties of many materials that are suitable for use as optical
substrates can provide a good indication of the expected spectral performance,
as the systematic tendencies in the electrical properties tend to parallel the
optical behaviour. Insulator materials show some regions of transparency, either in the near or farinfrared, whilst
good electrical and thermal conductors exhibit a continuous background of electronic
absorption over the whole infrared region.
All
of the resonant absorption processes involved in an infrared material can be
explained by the same common principal. At particular frequencies the incident
radiation is allowed to propagate through the crystal lattice producing the
observed transparency, other frequencies however, are forbidden when the
incident radiation is at resonance with any of the properties of the lattice
material, and as such are transferred as thermal energy, exciting the atoms or
electrons. The resonant vibrational absorption characteristics created by the
lattice are highly complex, consisting of several types of fundamental
vibrations. In order that a mode of
vibration can absorb, a mechanism for coupling the vibrational motion to the
electromagnetic radiation must exist.
Transfer of electromagnetic radiation
from the incident medium to the material is in the form of a couple, where the
lattice vibration produces an oscillating dipole moment which can be driven by
the oscillating electric field (E) of the radiation. In order for the total
transfer of energy to be complete, the following three conditions must be
satisfied;
1.the conservation of energy is
maintained,
2.the conservation of momentum is
maintained, and
3.a coupling mechanism between the
material and the incident medium is present.
The conservation of momentum is
governed by the relationship between de Broglie's particle/wave duality, from
the photon and phonon momenta, where the photon momentum is P = h/l;. The phonon momentum in the crystal is given by
P = h/a, where is the lattice constant
for the unit cell. When l = a, the conservation of
momentum is preserved between the incident photon and thermal phonon, resulting
in complete absorption of the incident radiation by the lattice. However,
the photon has a low momentum when
compared to the momentum of a phonon, therefore two or more photons are required to satisfy the
conservation of momentum and produce total absorption.
The coupling mechanism between the
incident photon and the lattice phonon is produced by a change of state in the
electric dipole moment (M) of the crystal. A dipole moment arises when two
equal and opposite charges are situated a very short distance apart, and is the
product of either of the charges with the distance between them. Thus energy
absorbed from the radiation will be converted into vibrational motion of the
atoms. In simple gas molecules this gives rise to a characteristic spectral
absorption band, as the many molecules form a large number of coupled dipole
moments.
In more complex lattice structures, in
order for a mode of vibration to absorb any incident radiation, the basic
mechanism for coupling must be present. Three different coupled absorption
mechanisms exist;
1.Reststrahl
absorption, this only occurs in ionic crystals and is caused by the creation of
single phonons in the lattice.
2.Multiphonon
absorption which occurs when two or more phonons simultaneously interact
and produce an electric moment with
which the incident radiation may couple.
3.Defect induced one phonon absorption, which
in a pure crystal is where the creation of a single phonon is not accompanied
by a transitional change of state in dipole moment that can act as a couple,
but is induced by the existence of a crystal defect or impurity to aid the
coupling mechanism.
Single
phonon Reststrahl absorption can occur in any material possessing an ionic
character with an alternating pattern of positive and negative ions. This
fundamental onephonon absorption process is associated with the electrostatic
motions of opposite charges which produce an oscillating electric field with
which the incident radiation can couple.
The wave vectors associated with this absorption only follow the longitudinal and transverse optical branches of the phonon dispersion curves as there exists two or more atoms per unit cell. In diatomic ionic crystals, when the interaction between the photon and phonon conserve the wave vector momentum, such that k = 2/l 0, the theory predicts the strongest absorption will be present, such that the crystal becomes totally reflecting, between the transverse and longitudinal optical vibration frequencies at a resonant frequency that corresponds to the following equation;
where m and M are the masses of the two ions. If one ion is much heavier than the other, the smaller of the two masses will determine the value of the bond strength (F). Therefore to achieve transparency to the longest wavelength, requires both ions to be as heavy as possible.
The behaviour of this type of absorption is most suitably described as a damped Lorentz classical oscillator. This is based on the assumption that the material contains charged particles which are bound to equilibrium positions by Hooke's law forces (i.e. for a certain range of atomic stresses (vibrations), the strain produced is proportional to the stress applied). If the magnitude of the force is assumed to be inversely proportional to the square of the distance between the atoms (Coulombic), the resonant frequencies for materials with different atomic masses can be predicted from empirical estimations of F.
In general, ionic crystals exhibit good transmission with constant refractive index and low absorption coefficient up to the lattice absorption band (typically beyond 6µm) at which point the single phonon produces a heavily absorbing mode of vibration and subsequent strong reflection coefficient. The refractive index undergoes a rapid change forcing the Fresnel reflection coefficient to become quiet high. The extinction coefficient also rises rapidly. At wavelengths longer than the resonant Reststrahl frequency, the absorption coefficient decreases, and the refractive index falls to a level slightly higher than on the short wavelength side of the absorption band. The difference in refractive index is characteristic of this absorption mechanism in ionic crystals. The long wavelength limit of transparency is therefore set by the Reststrahl frequency with the absorption falling rapidly at higher frequencies. For most ionic materials more than one absorption peak is present. As the temperature of the material is reduced, the Reststrahl frequency moves slightly towards shorter wavelengths and the peak reflection increases. The refractive index however is unaffected, other than by the characteristic change defined by the temperaturedependent dispersion coefficients.
In homopolar crystals (Ge, Si) where there is an absence of polar electric field interactions, the atomic motions are determined only by the local elastic restoring forces, and as such there is no single phonon interactive coupling and the longitudinal vibration then equals the transverse vibration mode. Hence only weak multiphonon absorption harmonics are present.
MultiPhonon
Absorption
Multiphonon
absorption occurs when two or more phonons simultaneously interact to produce
electric dipole moments with which the incident radiation may couple. These
dipoles can absorb energy from the incident radiation, reaching a maximum
coupling with the radiation when the frequency is equal to the vibrational mode
of the dipole in the farinfrared. The different vibration modes are complex, comprising
several different types of vibrations. There are two modes of vibrations of
atoms in crystals, longitudinal and transverse. In the longitudinal mode the
displacement of atoms from their positions of equilibrium coincides with the
propagation direction of the wave, for transverse modes, atoms move perpendicular
to the propagation of the wave. Where
there is only one atom per unit cell, the phonon dispersion curves are represented
only by acoustic branches. If there is
more than one atom per unit cell both acoustic and optical branches appear. The
difference between acoustic and optical branches being the greater number of
vibration modes available. In a diatomic cell the acoustic branch is formed
when both atoms move together inphase, the optical branch being formed by
outofphase vibrations. Generally, for N atoms per unit cell there will be 3
acoustic branches (1 longitudinal and 2 transverse) and 3N3 optical branches
(N1 longitudinal and 2N2 transverse).
Phonons
in Semiconductors
Compound semiconductors have two transverse optical modes(TO), two transverse acoustic modes(TA), one longitudinal optical mode(LO), and a longitudinal acoustic mode(LA). The two transverse modes can exhibit similar dispersion characteristics on the energy / wave vector diagrams. As phonon emission is quantized, selectivity forbids certain combinations of phonon absorption modes, however the varied combination of all the modes available produces a highly complex absorption structure. In single compound (homopolar) covalently bonded semiconductors such as Silicon and Germanium where there is no bonding dipole, the incident radiation induces a dipole moment with a stronger couple, producing more phonons (usually <4). Multiphonon absorption also occurs in ionic crystals in a form similar to that in homopolar crystals. Its strength is usually greater than in the homopolar case but is substantially weaker than onephonon reststrahl absorption.