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9 Phonons 9.1 Infrared active phonons 9.2 Infrared reflectivity and absorption in polar solids 9.3 Polaritons 9.4 Polarons 9.5 Inelastic light scattering 9.6 Phonon lifetimes 10.1 Infrared active phonons The resonant frequencies of the phonons occur in the infrared spectral region, and the modes that interact directly with light are called infrared active (IR active). The phonon modes of a crystal are subdivided into two general categories: • acoustic or optical; • transverse or longitudinal. It is the “optical” rather than the acoustic modes that are directly IR active. These optically active phonons are able to absorb light at their resonant frequency. Conservation laws require that the photon and the phonon must have the same energy and momentum. Dispersion curves for the acoustic and optical phonon branches in a typical crystal with a lattice constant of a. The dispersion of the photon modes is shown by the dotted line. At small wave vectors the slope of the acoustic branch is equal to vs, the velocity of sound, while the optical modes are essentially dispersionless near q=1. At the intersection , which corresponds to the resonance, the frequency is equal to that of the optical mode at q=0. Photons couple to TO phonons through the driving force exerted on the atoms by the AC electric field of the light wave. This can only happen if the atoms are charged, or the crystal must have some ionic character. The crystals with ionicity contain the ionic crystals and the crystals with partly covalent bonds (polar bonds) 10.2 Infrared reflectivity and absorption in polar solids d2 K q ( x x ) ( x x ) E (t ), 2 dt 10.2.1 The classical oscillator model where 1 1 1 . m m By putting x = x+ -- x- for the relative displacement of positive and negative ions with their unit cell, the equation of motion in the simpler form: d 2x q 2 TO x E (t ), 2 dt Interaction of a TO phonon mode propagating in the z direction with an electromagnetric wave of the same wave vector. The black circles represent positive ions, while the grey circles represent the negative ions. The solid line represents the spatial dependence of the electric field of the electromagnetric wave. When the wave vectors and frequencies match, the strong interaction at resonance between a TO phonon mode and an infrared light occurs at q 0 (~10 m). There will be thousands of unit cells (usually less than 10-9m) within one period of the wave. The equation of motion for the positive and negative displaced ions: d 2 x m K ( x x ) qE(t ) dt 2 d 2 x m K ( x x ) qE(t ) dt 2 where 2TO for K/. represents the natural vibrational frequency of the TO mode at q = 0 in the absence of the extend light field. The equation represents the equation of motion for undamped oscillations of the lattice driven by the forces exerted by AC electric field of the light wave. In reality, a damping term to account for the finite lifetime of the phonon mode should be incorporated into the eqn by introducing a damping rate : d 2x dx q 2 x E (t ). TO dt 2 dt The response of the medium to a light field of angular frequency with E (t) = E0 e it: Nq 2 1 r () 1 , 2 0 (TO 2 i) where represents the non-resonant susceptibility of the medium, and N is the number of unit cells per unit volume. 10.2.1 The classical oscillator model In the limits of low frequency: Nq 2 st r (0) 1 , 2 0TO In the limits of high frequency: r () 1 . Thus we can write: 2 TO r () ( st ) 2 (TO 2 i). where " " in relative sense here. The waves are transverse. If r = 0, the equation can be satisfied with waves in which k E 0, that is, with longitudinal waves. The dielectric can support longitudinal electric field waves at frequencies which satisfy r(’) = 0. The waves at = ’ correspond to LO phonon wave, namely LO, thus 2LO st . 2 TO This is the Lyddane-Sachs-Teller (LST) relationship. 10.2.2 The Lyddane-Sachs-Teller relationship For a lightly damped system with =0, assume r = 0 at ’: 2 r ( ' ) 0 ( st ) TO . ( '2 ) 2 TO This can be solved to obtain: 1 2 ' st TO . What does r=0 mean physically? When = 0, D (r 0 E) 0, For r 0 and wave solutions with the form: i ( kr t ) E (r , t ) E0e , we have k E 0. The data in the table imply that the LO phonon and TO phonon modes of non-polar crystals are degenerate. There is no infrared resonance, and therefore r = . This is indeed the case for the purely covalent crystals of the group IV element namely diamond (c), silicon and germanium. 10.2.3 Restrahlen Set = 0, the dielectric constant has the following frequency dependence: 2 vTO r (v) ( st ) 2 . (vTO v 2 ) A wave number of 1 cm-1 is equivalent to a frequency of 2.9981010 Hz. 1THz = 1012 Hz. 1ps = 10-12 s III-V semiconductor At low frequency the dielectric constant is just equal to r. As v increases from 0, r (v) gradually increases until it diverges when the resonance at vTO is reached. Between vTO and vLO , r is negative. Precisely at v = vLO, r = 0. thereafter, r is positive, and gradually increases towards the value of . The most important optical property of a polar solid in the infrared spectral region is the reflectivity. 2 n~ 1 R ~ n 1 r 1 r 1 2 . The reflectivity is equal to 100 % in the frequency region between vTO and vLO. This frequency region is called the restralen band. Light cannot propagate in the restrahlen band. Infrared reflectivity of InAs and GaAs at 4.2 K. there is a sharp dip in the reflectivity just above the LO phonon resonance. This reduction in reflectivity from 100 % is caused by ignoring the damping term. The damping also broadens the edge so that there is only a minimum in R just above vLO rather than a zero. The magnitude of obtained in this way are around 1011-1012 s-1, which implies that the optical phonons have a lifetime of about 1-10 ps. 10.2.4 Lattice absorption The dispersion of the polaritons as q 0 and q : The absorption coefficients expected at the resonance with the TO phonon can be calculated from the imaginary part of the dielectric constant. At = TO we have r (TO ) i( st ) TO . The extinction coefficient and the absorption coefficient (typically in the range of 106 – 107 m-1) can be work out from r. The absorption is very high at the TO phonon resonance frequency. as q 0, vq c q, st as q vq c q, st TO LO TO LO > > TO, r < 0, restrahlen band; > LO, r > 0, v c / ( )1/2.. 10.3 Polariton 1 {( st 02 c 2 q 2 ) ( st 02 c 2 q 2 ) 2 402c 2 q 2 } 2 Polariton dispersion predicted for a crystal with vTO = 10 THz, st = 12.1 and = 10. the asymptotic velocities are vst and v. Dispersion of the TO and LO phonons in GaP measured by Raman scattering. The solid lines are the predictions of the polariton model with hvTO = 45.5 meV, = 9.1 and st 11.0. 10.4 Polarons The polaron theory cab be applied to holes or electrons in the formula. In a non-polar crystal such as silicon, = st , and ep = 0, there is no polaron effect. In the table the coupling constant increases from the III-V, which is predominantly covalent, to the I-VII, which is highly ionic. The effective mass can be measured using cyclotron resonance. 1 The electron energy: En n C , 2 Where This is a schematic representation of a polaron. A free electron moving through an ionic lattice attracts the positive ions, and repel the negative ones. This produces a local distortion of lattice within the polaron radius that is a new elementary excitation of the crystal. The optical properties of the whole host can depend on the polaron effect of the electron-phonon coupling. The strength of the electron-phonon interaction in a polar solid can be quantified by the dimensionless coupling constant ep, which is given by: ep 1 mc 137 2 LO * 2 1 2 1 1 , st The effective mass: 1 d 2E m 2 . dk 2 C eB . m* Optical transition (absorption) with n = 1 can take place at a wavelength ( in the far-IR) given by: hc eB m The effective mass can be deduced from and B at resonance, however, it is usually not the value determined by the curvature of the bands given by m*. For the small ep: m 1 1 1 ep . m 1 ep / 6 6 For III-V material like GaAs with ep<0.1, m** only differs from m* by about 1%. The polaron effect is thus only a small correction. The correction become more significant for II-VI (e.g. ~7% for ZnSe). With highly ionic crystals like AgCl, this is 0.43 (about 50% larger than the rigid lattice value). The small ep approximation is not valid. The additional energy correction: Eg epLO , This means a reduction in the band gap by the amount. The radius, rp, which specifies how far the lattice distortion extends. If ep is small: rp 2m LO 10.5.1 General principles of inelastic light scattering Inelastic light scattering can be subdivided into two types: • Stokes scattering: emitting a phonon or other excitation; • Anti-Stokes scattering: absorbing a phonon. 1 2 . The gives rp = 4.0 nm for GaAs and 3.1 nm for ZnSe. Large polaron: rp >> a (unit cell size, ~ 0.5 nm) and ep is small; Small polaron: rp ~ a and ep is not small. Small polar existing in highly ionic crystals leads to selftrapping of the charge carriers. The local lattice distortion is so strong that the carrier can get completely trapped in its own lattice distortion. The only way they move is by hopping to a new site. The electrical conductivity of most alkali halide crystals is limited by this thermally activated hopping process at RT. The effects are also improtant in the conduction processes in organic semiconductors . 10.6 Inelastic light scattering Inelastic light scattering can be medicated by many different types of elementary excitations in a crystal such as phonons, magnons or plasmons. Inelastic light scattering from phonons is subdivided as: • Raman scattering: scattered from optical phonons; • Brillouin scattering: scattered from acoustic phonons Conservation of energy: Conservation of energy: 1 2 , k1 k2 q. The light is shifted down in frequency – Stokes; The light is shifted up in frequency – anti-stokes. The ratio of anti-stokes to stokes scattering is given by: I anti Stokes exp( / k BT ). I stokes The probability for anti-Stokes scattering is very low at cryogenic temperatures. The maximum phonon frequency in a typical crystal is about 1012 – 1013 Hz. This is almost two orders of magnitude smaller than the frequency of a photon in the visible spectral region. Thus the maximum frequency shift for photon will be around 1%. n k 2 k1 , c and q k1 k 2 . The maximum possible value of q thus occurs for the back-scattering geometry: n q k (k ) 2 . c The maximum value of q is of order 107 m-1. This is very small compared to the size of the Brillouin zone in a typical crystal (~ 1010 m-1). Inelastic scattering is thus only able to probe small wave vector phonons. 10.5.2 Raman scattering Raman scattering is mainly used to determine the frequency of LO and TO modes near the Brillouin zone centre, and gives little information about the dispersion of the phonons (optical ponons are essentially dispersionless near q = 0). IR reflectivity and light scattering is of complementarity. The full selection rule requires the use of group theory. However, a simple rule can be given for crystals that posses inversion symmetry: • The odd parity modes are IR active; • The even parity modes are Raman active. This is called the rule of mutual exclusion. In non-centrosymmetric crystals, some modes may be simultaneously IR and Raman active. Experimental apparatus used to record Raman spectra. Raman spectra obtained from four III-V crystals at 300 K using a Nd:YAG laser at 1.064 m. the spectra are plotted against the wave number shift: 1 cm-1 is equivalent to an energy shift of 0.124 meV. These correspond to Stokesshifted signals from the TO phonons and LO phonons. The LO mode is the one at higher frequency. 10.5.3 Brillouin scattering It refers to inelastic light scattering from acoustic phonons. Its main purpose is to determine the dispersion of these acoustic modes. The frequency shift of the phonon in the scattering is given by s One phonon is annihilated and two new phonons are created. 2 n sin , c 2 where s is the velocity of the acoustic waves, and is the angle through which is light is scattered. This can be used to determined the velocity of the sound waves. 10.6 Phonon lifetimes Analysis of the experimental data on the reflectivity in the restrahlen band show that damping constant is typically in the range 1011—10 12 s-1. Since is equal to lifetime -1, the data implies that is in the range 1 – 10 ps. The very short lifetime of the optical phonons is caused by anharmonicity in the crystal. In general, the atoms sit in a potential well of the form: Decay of an optical phonon into two acoustic phonons by a three -phonon interaction . U ( x) C2 x2 C3 x3 C4 x4 . The term in x2 is the harmonic term. The term in x3 and higher are the anharmonic terms. These anharmonic terms allow phonon – phonon scattering processes. For example, the term in x3 allows interactions involving three phonons. The lifetime of the optical phonons can be deduced from Raman data in three different ways. Firstly, the spectral width of the Raman line is affected by lifetime broadening, which in frequency unit is expected to be (2)-1; Secondly, can be measured directly by time-resolved Raman spectroscopy using short pulse lasers; thirdly, the lifetime of the optical phonons can be deduced from reflectivity measurements. Examination: 1. The static and high frequency dielectric constants of NaCl are st = 5.9 and = 2.25 respectively, and the TO phonon frequency vTO is 4.9 THz. i) Calculate the upper and lower wavelengths of the restrahlen band. ii) Estimate the reflectivity at 50 m, if the damping constant of the phonons is 1012s-1. 1 iii) Calculate the absorption coefficient at 50 m. 1 2 2. Show that the reflectivity of an undamped polar solid falls to zero at a frequency given by st TO , 1 where st and are the low and high frequency dielectric constants, and the frequency of the TO phonon mode at the Brillouin zone centre. 3. When light from an argon ion laser operating at 514.5 nm is scattered by optical phonons in a sample of AlAs, two peaks are observed at 524.2 nm and 525.4 nm. What are the values of the TO phonon and LO phonon energies? 4. Figure .1 show the measured infrared reflectivity of AlSb crystals. Use this data to estimate: i) the frequencies of the TO and LO phonons of AlSb near the Brillouin zone centre; ii) the static and high frequency dielectric constants, st and ; iii) the lifetime of the TO phonons. Are the experimental values found in i) and ii) consistent with the LST relationship?