Infrared Material Absorption Theory

coatingli · 发表于 2004-4-2 19:21:00

Introduction

The design of any infrared filtering system requires the selection of materials based upon knowledge of the optical, mechanical and thermal properties available. Frequently, the selection of suitable materials results from compromises between these various properties as no single material will possess the ideal characteristics required to suit the wide variety of applications.

All of the observed intrinsic absorption characteristics present in the spectrum of an infrared optical material can be classified by three fundamental processes involving interaction between the material and the incident electromagnetic radiation, namely; electronic absorption, lattice or phonon absorption and free-carrier absorption.

The electronic absorption characteristics observed towards the higher frequency end of the infrared spectrum are the result of interaction between the incident radiation and the motions of electrons or holes within the material. Only electromagnetic radiation with sufficient energy to cause an electron to transfer between the valence band and conduction band (hf) will be absorbed by this mechanism. The various transitions of these electrons define the position of the short wavelength absorption edge. The resulting spectrum provides information on the width of the energy band-gap of the material, and through spectral anomalies, can indicate the presence of impurities.

The lattice absorption characteristics observed at the lower frequency regions, 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. Hence, all materials are bounded by limiting regions of absorption caused by atomic vibrations in the far-infrared (10µm), and motions of electrons and/or holes in the short-wave visible regions. In the interband region, the frequency of the incident radiation has insufficient energy (E=hf) to transfer electrons to the conduction band and cause absorption; here the material is essentially loss-free.

In addition to the fundamental electronic and lattice absorption process, free-carrier absorption in semiconductors can be present. This involves electronic transitions between initial and final states within the same energy band. The absorption or emission of the resulting photons is accompanied by scattering by optical or acoustic-mode phonon vibrations or by charged impurities.

These intrinsic absorption properties of semiconductors and insulators define the transparency of the material. To be transmitted in the region between the electronic and lattice absorptions, the incident radiation must have a lower frequency than the band-gap (E_g) of the material. This is defined by the short wavelength semiconductor edge at λ = hc/eE_g, preventing electrons transferring to the conduction band. The generalised profile of the electronic absorption edge is known as the Urbach tail where the exponentially increasing absorption coefficient follows the general relationship:

Electronic Absorption

In conductors, absorption due to the presence of a cloud of free electrons or holes is continuous, with a magnitude that increases approximately as the square of the incident wavelength. Overlapping valence and conduction bands provide high reflectivity but prohibit transparency throughout the entire infrared. As the electron energy bands become separated in semiconductors however, the extent to which free electron carriers cause absorption becomes dependent upon the size of the energy gap at any given temperature, and the absence of impurities.

The electronic absorption processes at the higher frequency end of the infrared spectrum caused by band-to-band or band-to-exciton (electron in an electrostatically attracted combination with a hole) transitions can be divided into four main categories for semiconductor materials;

Intrinsic absorption, where in a pure semiconductor transitions between full valence bands and empty conduction bands are free to occur.
Extrinsic absorption, where transitions occur between the valence or conduction band and donor or acceptor sites in the band gap.
Free carrier absorption in which transitions occur within any one energy band, and
Localised energy states caused by defects or impurities, where electrons or holes may be excited into a higher energy state.

At frequencies close to the electronic absorption edge, a change in the bandgap (E_g) of the crystal by a fraction of an electron volt can change the absorption coefficient () by nearly four orders of magnitude. By measurement of the spectral position and profile of the absorption edge, values for the energy band-gap (E_g) can be determined, together with other general information about the energy states either side of the forbidden band responsible for electrical conduction. However, the estimation of the energy gap from the absorption edge is not a straightforward for the following reason;

As the momentum of a photon (h/λ) is very small compared to the crystal momentum (h/a), where a is the lattice constant, the photon-absorption process should conserve the momentum of the electron. However, the absorption coefficient () for a given photon energy hf is proportional to the probability (P_i,f) for the transition from the initial state (n_i) to the final state (n_f) , the density of electrons in the initial state, and also to the density of available final states. These processes must be summed for all the possible transitions between states that are separated by an energy difference equal to hf,

Therefore both the exact positioning and shape of the electronic absorption edge cannot easily be predicted or modelled through solid state theory without detailed knowledge of all the allowed forbidden, direct indirect transitions available, together with knowledge of the density of electrons as given by the product of the density of states and the Fermi-Dirac function, with the probability of an electron level possessing an electron given by:

where, E is the energy level, E_f is the Fermi-energy level, k is Boltzmann' s constant and T is the absolute temperature. The following figure illustrates the various electronic absorption energy band transitions available in a typical semiconductor material.

[Electronic absorption energy band transitions]

(A) Direct valence to conduction band transitions (constant k vector)
(B) Indirect valence to conduction band transitions aided by photon/phonon coupling interactions
(C) Inter-valence band transitions
(D) Valence band free-carrier transitions aided by impurities or photon/phonon interactions
(E) Conduction band free-carrier transitions aided by impurities or photon/phonon interactions

Optical materials that are opaque in the visible, because of comparatively small bandgaps (≤ 1.25eV), are arbitrarily classified as infrared semiconductors whilst materials of larger bandgap, and whose lattice absorption is present in the far-infrared are insulators.

Lattice Absorption

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;

the conservation of energy is maintained,
the conservation of momentum is maintained, and
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/λ. The phonon momentum in the crystal is given by P = h/a, where is the lattice constant for the unit cell. When λ=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;

Reststrahl absorption, this only occurs in ionic crystals and is caused by the creation of single phonons in the lattice.
Multi-phonon absorption which occurs when two or more phonons simultaneously interact and produce an electric moment with which the incident radiation may couple.
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 Absorption

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 = 2π/λ 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

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).

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.

Thermal Vibrations

The concepts of temperature and thermal equilibrium associated with crystal solids are based on individual atoms in the system possessing vibrational motion. The classical theory of thermal energy by atomic vibrations, though providing suitable explanations at elevated temperatures, has proved unsatisfactory at reduced or cryogenic temperatures. Quantum mechanics has subsequently provided theories based upon statistical probability that have provided possible mechanisms to explain some of the observed phenomena. A system of vibrating atoms in a crystal is highly complicated, and beyond the realm of any realisable theoretical method of analysis or calculations to verify spectral measurements from the total thermal energy of a crystalline substrate.

When a particle is bound to a crystal, the energy can only have discrete values as defined by the energy band structure. The quantum-mechanics of a one-dimensional simple harmonic oscillator gives permitted energies of (n+½) where ω is the angular frequency and n is the permitted energy integer. At a position of minimum energy (0K) the energy can never be zero, but has energy of ½(zero-point energy) and as such will still provide crystal vibration.

As an atom can vibrate independently in three dimensions it is equivalent to three separate oscillators. The total thermal energy for N atoms will then be 3NkT, ignoring the ½ term, the specific heat required to change the temperature by one degree will then be 3Nk where the specific heat of a solid for a given number of atoms is independent of temperature if N is the Avagadro number (6.02x10²³). A detailed calculation of this form would require a knowledge of the number of atoms vibrating with frequencies ω₁ ... ω_n, which would depend on the density of states, and integration over the whole range of atomic vibrational frequencies would be required.

The thermal vibrations in a solid produce atomic displacements, which in a three dimensional lattice can be resolved into different states of polarization such that vibrations parallel to the wave vector are longitudinal waves and the two directions at right angles to the wave vector are transverse waves. As the rules of quantum mechanics apply to all the different atomic vibrations in the crystal, the lattice pulsates as a complete assembly in discrete energy steps of (phonons). The phonon is related to both the frequency of vibration and the temperature. If the temperature is raised, the amplitude of atomic vibration increases, and in quantum terms this is considered as an increase in the number of phonons in the system.

The concept of the phonon is therefore considered as the quantum of lattice vibrational energy onto which is superimposed a complex pattern of standing and/or travelling waves that represent changes in temperature. If the crystal is at a uniform temperature the standing wave concept is adequate as the phonon vibrations are uniformly distributed.

ttl · 发表于 2004-4-3 00:11:00

看来还是需要费点力气呀

fredsarah · 发表于 2004-4-9 03:53:00

too hard

jacobcheng · 发表于 2004-4-12 00:50:00

3Q very much

Infrared Material Absorption Theory

Introduction

Electronic Absorption

Lattice Absorption

Single Phonon Absorption

Multi-Phonon Absorption

Thermal Vibrations

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