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 one-dimensional crystal simulated by the chain of the balls with the masses m1 and m2. 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 one-dimensional 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 g2=4m1m2/(m1+m2)2; w02=2c (m1+m2)/m1m2. 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 long-wave 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 infra-red 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 Un =displacement of
atom n from its equilibrium position
Un-1=displacement of atom n-1 from its equilibrium position
Un+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 Uno = Uo 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 long-wavelength 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 10-100 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:
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 far-infrared 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 far-infrared, 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.Multi-phonon 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 one-phonon 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 temperature-dependent 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 multi-phonon absorption harmonics are present.
Multi-phonon 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 far-infrared. 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 in-phase, the optical branch being formed by out-of-phase vibrations. Generally, for N atoms per unit cell there will be 3 acoustic branches (1 longitudinal and 2 transverse) and 3N-3 optical branches
(N-1 longitudinal and 2N-2 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). Multi-phonon 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 one-phonon reststrahl absorption.