Anharmonic processes of scattering and relaxation of slow quasi-transverse phonons in cubic crystals

Relaxation of slow quasi-transverse phonons in anharmonic processes of scattering in cubic crystals with positive (Ge, Si, diamond) and negative (KCl, NaCl) anisotropies of the second-order elastic moduli has been considered. The dependences of the relaxation rates on the direction of the wave vector of phonons in scattering processes with the participation of three quasi-transverse phonons (the TTT relaxation mechanisms) are analyzed within the anisotropic continuum model. It is shown that the TTT relaxation mechanisms in crystals are associated with their cubic anisotropy, which is responsible for the interaction between noncollinear phonons. The dominant contribution to the phonon relaxation comes from large-angle scattering. For crystals with significant anisotropy of the elastic energy (Ge, Si, KCl, NaCl), the total contribution of the TTT relaxation mechanisms to the total relaxation rate exceeds the contribution of the Landau-Rumer mechanism either by several factors or by one to two orders of magnitude depending on the direction. The dominant role of the TTT relaxation mechanisms as compared to the Landau-Rumer mechanism is governed, to a considerable extent, by the second-order elastic moduli. The total relaxation rates of slow quasi-transverse phonons are determined. It is demonstrated that, when the anharmonic processes of scattering play the dominant role, the inclusion of one of the relaxation mechanisms (the Landau-Rumer mechanism or the mechanisms of relaxation of the slow quasi-transverse mode by two slow or two fast modes) is insufficient for describing the anisotropy of the total relaxation rates in cubic crystals.     

The mechanism of Landau-Rumer relaxation of thermal and high-frequency phonons in cubic crystals of germanium,silicon, and diamond

Transverse phonon relaxation according to the Landau-Rumer mechanism is considered for an isotropic medium and crystals of germanium, silicon, and diamond possessing a cubic symmetry. The energy of elastic deformation caused by the anharmonicity of vibrations of the cubic crystal lattice is expressed via the second-and third-order moduli of elasticity. Using the known values of these elastic moduli, parameters determining the frequencies of the transverse phonon relaxation in the Landau-Rumer mechanism are evaluated for the germanium, silicon, and diamond crystals. It is shown that the dependence of the relaxation frequency on the wavevector of thermal and high-frequency phonons sharply differs from the classical Landau-Rumer relationship both in the isotropic medium and in the cubic crystals. It is established that the observed peculiarities in the relaxation frequency are related to the angular dependence of the probability of anharmonic scattering and the anisotropy of elastic properties of the germanium, silicon, and diamond crystals. A new method is proposed for the experimental determination of the relaxation frequency of high-frequency phonons as a function of the wavevector using the temperature dependence of the coefficient of absorption of high-frequency ultrasound.     

Normal phonon-phonon scattering processes and the thermal conductivity of germanium crystals with isotope disorder

The influence of the normal phonon-phonon scattering processes on the thermal conductivity was theoretically studied for germanium crystals with various degrees of the isotope disorder. The theory takes into account redistribution of the phonon momentum in the normal scattering processes both inside each oscillation branch (Simons mechanism) and between various phonon oscillation branches (Herring mechanism). Contributions to the thermal conductivity due to the drift mobility of the longitudinal and transverse phonons are analyzed. It is shown that the momentum redistribution between longitudinal and transverse phonons according to the Herring relaxation mechanism leads to a significant suppression of the drift motions (and to the corresponding drop in contribution to the thermal conductivity) of the longitudinal phonons in isotopically pure germanium crystals. The results of the thermal conductivity calculations involving the Herring relaxation mechanism agree well with the experimental data available for germanium crystals with various degrees of the isotope disorder.     

The effect of normal phonon-phonon scattering processes on the maximum thermal conductivity of isotopically pure silicon crystals

The effect of normal phonon-phonon scattering processes on the thermal conductivity of silicon crystals with various degrees of isotope disorder is considered. The redistribution of phonon momentum in normal scattering processes is taken into account within each oscillation branch (the Callaway generalized model), as well as between different oscillation branches of the phonon spectrum (the Herring mechanism). The values of the parameters are obtained that determine the phonon momentum relaxation in anharmonic scattering processes. The contributions of the drift motion of longitudinal and transverse phonons to the thermal conductivity are analyzed. It is shown that the momentum redistribution between longitudinal and transverse phonons in the Herring relaxation model represents an efficient mechanism that limits the maximum thermal conductivity in isotopically pure silicon crystals. The dependence of the maximum thermal conductivity on the degree of isotope disorder is calculated. The maximum thermal conductivity of isotopically pure silicon crystals is estimated for two variants of phonon momentum relaxation in normal phonon-phonon scattering processes.     

Low-temperature anisotropy of the thermal conductivity of single-crystal nanofilms and nanowires

The effect of phonon focusing on the phonon transport in single-crystal nanofilms and nanowires is studied in the boundary scattering regime. The dependences of the thermal conductivity and the free path of phonons on the geometric parameters of nanostructures with various elastic energy anisotropies are analyzed for diffuse phonon scattering by boundaries. It is shown that the anisotropies of thermal conductivity for nanostructures made of cubic crystals with positive (LiF, GaAs, Ge, Si, diamond, YAG) and negative (CaF₂, NaCl, YIG) anisotropies of the second-order elastic moduli are qualitatively different for both nanofilms and nanowires. The single-crystal film plane orientations and the heat flow directions that ensure the maximum or minimum thermal conductivity in a film plane are determined for the crystals of both types. The thermal conductivity of nanowires with a square cross section mainly depends on a heat flow direction, and the thermal conductivity of sufficiently wide nanofilms is substantially determined by a film plane orientation.     

Elastic waves and the Landau-Rumer mechanism of relaxation of quasi-transverse phonons in GaAs crystals

The dependences of the relaxation rate and the absorption coefficient of ultrasound in GaAs crystals on the direction of the wave vector of quasi-transverse phonons for the Landau-Rumer mechanism are analyzed in the framework of the anisotropic-continuum model. The calculations are performed with two sets of second-order and third-order elastic moduli experimentally measured by different authors. It is demonstrated that the angular dependences of the relaxation rates of quasi-transverse phonons and the ultrasonic absorption coefficients calculated from these data differ qualitatively. The correctness of the determination of the third-order elastic moduli available in the literature can be checked by measuring the ultrasonic absorption coefficients for GaAs crystals.     

The effect of phonon focusing on the electron-phonon relaxation and electron transport in potassium crystals

The influence of anisotropy of elastic energy on electron-phonon relaxation and the role of shear waves in the electrical resistance of potassium crystals are investigated. It is shown that, at temperatures much lower than the Debye temperature (T<< θ_D), the contribution of slow quasi-transverse phonons to the electrical resistance of potassium crystals exceeds that of longitudinal phonons by an order of magnitude. Earlier, the Bloch-Grüneisen theory left aside this component under the above conditions. At the same time, at high temperatures(T>>θ_D), the contribution of longitudinal phonons to the electrical resistance turns out to be 4 times greater than the total contribution of electron relaxation by fast and slow transverse modes. The role of shear waves in the electrical resistance of potassium crystals is analyzed. It is shown that, at low temperatures, this mechanism provides 32% of the total electrical resistance. It is 4 times higher than the contribution of longitudinal phonons to the electrical resistance and should be taken into account when analyzing the electrical resistance of alkali metals. The distribution function of the most effective phonons for electrical resistance is defined, and the inelasticity of electron-phonon scattering is analyzed. It is shown that the calculated results of the electrical resistance of potassium in the temperature range from 40 to 400 K, taking into account the anisotropy of elastic energy, are in good agreement with the experimental data without the use of fitting parameters.     

Absorption coefficient of infrared laser window materials

The absorption coefficient of a number of potential i.r. laser window materials has been measured in the region of multiphonon absorption. Absorptions as low as 0·004 cm^?1 have been detected by using thick samples and a differential technique with a dual-beam spectrophotometer. The absorption coefficients of LiF, CaF₂, BaF₂, SrF₂, MgF₂, Al₂O₃, KCl, NaCl and KBr all decrease exponentially with increasing wavenumber. The slope of the exponential bears a nearly constant relationship to the longitudinal optical phonon frequency, consistent with a multiphonon process whose strength depends on the number of phonons generated. Absorption data are also presented for Si, Ge, GaAs: Cr, GaAs: Fe, ZnSe and CdTe.     

Anisotropy of longitudinal ultrasonic absorption in anharmonic processes of scattering in Ge,Si, InSb,MgO, and KCl cubic crystals: The role of damping of phonon states

Physics of the Solid State - The relaxation of longitudinal phonons and absorption of ultrasound in cubic crystals with positive (Ge, Si, InSb, MgO) and negative (KCl) anisotropies of the...     

Effect of dispersion and damping of thermal phonon states on the longitudinal ultrasonic absorption in germanium crystals

A method has been proposed for approximating a phonon spectrum of cubic crystals, which has been obtained from data on inelastic neutron scattering for symmetric directions, over the entire Brillouin zone in the form appropriate for studying relaxation characteristics of phonon systems. The effect of dispersion and damping of thermal phonon states on the longitudinal ultrasonic absorption in anharmonic processes of scattering with the participation of three longitudinal phonons has been investigated for germanium crystals. It has been shown that the inclusion of the dispersion leads to a decrease in the anisotropy of ultrasonic absorption in the LLL relaxation mechanism and makes it possible to fit the results obtained from calculations of the ultrasonic absorption coefficients to the experimental data in the low-temperature range. The temperature dependence and anisotropy of the relaxation rate of longitudinal thermal phonons in germanium crystals have been determined from experimental data on ultrasonic absorption. The performed analysis has refined values of the relaxation parameters obtained from the interpretation of the data on thermal conductivity of germanium crystals with different isotopic compositions in the isotropic-medium model.     

Phonon focusing and electron–phonon drag in semiconductor crystals with degenerate charge-carrier statistics

We study the effect of anisotropy in elastic properties on the electron–phonon drag and thermoelectric phenomena in gapless semiconductors with degenerate charge-carrier statistics. It is shown that phonon focusing leads to a number of new effects in the drag thermopower at low temperatures, when diffusive phonon scattering from the boundaries is the predominant relaxation mechanism. We analyze the effect of phonon focusing on the dependences of the thermoelectromotive force (thermopower) in HgSe:Fe crystals on geometric parameters and the heat-flow directions relative to the crystal axes in the Knudsen regime of the phonon gas flow. The crystallographic directions that ensure the maximum and minimum values of the thermopower are determined and the role of quasi-longitudinal and quasi-transverse phonons in the drag thermopower in HgSe:Fe crystals at low temperatures is analyzed. It is shown that the main contribution to the drag thermopower comes from slow quasi-transverse phonons in the directions of focusing in long samples.     

Longitudinal-phonon relaxation mechanisms in cubic crystals of germanium,silicon, and diamond

The relaxation rates of thermal and high-frequency longitudinal phonons are calculated using an anisotropic-continuum model. Three-phonon scattering mechanisms (L ? L + L, L ? T + L) for the phonon relaxation are considered. Anisotropic anharmonic phonon scattering in cubic crystals is described in terms of the second-and third-order elastic moduli. The parameters determining the longitudinal-phonon relaxation rates are found for germanium, silicon, and diamond crystals. The long-wavelength limit and the transition to the isotropic-medium model are considered, and the dependences of the relaxation rates of thermal and high-frequency phonons on temperature and phonon wave vector are analyzed for these crystals.     

TE-surface polaritons at the interface of the isotropic media

Spectra of surface polaritons at resonance with a transverse phonon of the macroscopic transition layer (or thin film on a substrate) have been studied both theoretically and experimentally. The existence of TE-SP in the gap of the TM SPs is predicted and observed experimentally for a LiF film on a CaF₂ crystal.     

Temperature dependence of hypersonic attenuation in NaCl and KCl crystals

The usual theories of ultrasonic attenuation (Landau-Rumer and Akhiezer) are in disagreement with experiments involving hypersonic phonons (Brillouin scattering). This is shown here by measurements on NaCl and KCl. We observe a Tⁿ dependence with n approximately equal to 2.8, and an attenuation increasing with temperature in the low and high temperature range respectively. An extension of the Landau-Rumer and Akhiezer theories to hypersonic frequencies is proposed. Our theoretical expressions give a good account of the temperature dependence of the attenuation in the whole temperature range.     

Anisotropic attenuation of transverse ultrasound in cubic crystals of Ge,Si, and diamond with various isotopic compositions

The attenuation of transverse ultrasound in germanium, silicon, and diamond crystals is considered with allowance for competing isotopic and anharmonic scattering processes. The dependence of the attenuation of transverse ultrasound on the direction of the wave vector of quasi-transverse phonons is analyzed within an anisotropic continuum model. The Landau—Rumer mechanism is considered for anharmonic scattering processes. Given the second-and third-order elastic moduli, the parameters are found determining ultrasonic absorption in the above crystals with various degrees of isotopic disorder. The attenuation coefficients of transverse ultrasound associated with isotopic and anharmonic scattering processes are shown to have qualitatively different angular dependences. Therefore, from studying the anisotropic attenuation of ultrasound in cubic crystals, one can determine the dominant mechanism of ultrasonic absorption in isotopically modified crystals.     

Elastic and photoelastic constants of NaCl,KBr and LiF by Brillouin scattering

Elastic and photoelastic constants of NaCl, KBr and LiF were determined from Brillouin frequency shifts and intensities. Reported differences between the hypersonic and ultrasonic elastic constants for KBr and LiF were not confirmed. For each material the constants were in good agreement with the ultrasonic values. Scattering from longitudinal, transverse and mixed phonon modes was clearly observed for each material.     

Generation of coherent terahertz phonons by sharp focusing of a femtosecond laser beam in the bulk of crystalline insulators in a regime of plasma formation

The generation of coherent terahertz phonons in a regime of plasma formation by a femtosecond laser radiation with an intensity of 10¹³ W/cm² in the bulk of crystalline quartz has been detected by the method of probing by a probe pulse of the third harmonic. A smooth increase in the frequency of coherent terahertz phonons from 2.2 to 5.5 THz has been detected, along with its subsequent sharp decrease down to 2.2 THz due to an α-β phase transition in crystalline quartz. The generation of 1-THz coherent phonons has been detected in BaF₂ crystals. A smooth variation of the frequency of coherent phonons from 2 to 2.5 THz has been detected in leucosapphire. The generation of coherent phonons during local laser excitation in CaF₂ and LiF crystals develops at the frequencies of 2.3 and 0.1 THz, respectively.     

Ultrasonic attenuation in insulating crystals

A theory for the dampingΓ of ultrasonic waves due to three-phonon processes is developed by using a Green's function method. The imaginary part of the self-energy of the impressed ultrasound phonons interacting with thermal phonons is calculated. In the limits ofω τ very large and very small the known results are rederived, whereω is the frequency of the ultrasonic wave andτ the thermal phonon relaxation time. The intermediate range ofω τ values is discussed in detail for the case of longitudinal phonon attenuation. It is found, that forω τ>1 a Landau-Rumer type law applies also for longitudinal phonons,Γ～ωT ⁴. But it is shown that dispersion effects and large third-order elastic anisotropy can lead to a stronger temperature dependence thanT ⁴ and a weaker dependence on frequency thanω. These results are compared with recent experiments.     

A study of the structure and scattering mechanisms of subterahertz phonons in lithium fluoride single crystals and optical ceramics

The methods of optical, electron, and atomic force microscopy (AFM) are applied to the study of the real structure of optical lithium fluoride ceramic obtained by hot deformation of single crystals. A comparative analysis is carried out of the scattering mechanisms of weakly nonequilibrium thermal phonons at liquid helium temperatures in LiF single crystals and ceramics. It is demonstrated that the phonon scattering in the original single crystals is determined by the forced vibrations of dislocations in the stress field of an elastic plane wave (a phonon), i.e., by the flutter mechanism. As the degree of deformation of the original material increases, the ceramics exhibit a change in the plastic deformation mechanisms, which leads to a decrease in the average size of grains and to an ordered structure. In this case, the dominant scattering is that by intergrain boundaries. The thickness and the acoustic impedance of these boundaries are evaluated.     

Dispersive effects and correction term in two-mode phonon conduction model for Ge

A complete analysis of the phonon conductivity κ, still lacking in the literature, is presented in the two-mode conduction model for Germanium. First a method is derived from which the correction term κ_c of the Callaway model is separated into its longitudinal and transverse parts and then the effects of strong phonon dispersion and the role of longitudinal and transverse phonons on κ_c are studied. For this purpose we have also proposed some new empirical expressions for the three phonon relaxation rates τ_3ph⁻¹'s which are valid in the entire temperature range. This improvised model, when applied simultaneously to the phonon conductivity data of both normal and enriched Ge, yields some new results. These are (i) κ_c neglected by the earlier workers in the two-mode phonon conduction model, gives a substantial contribution beyond the conductivity maximum and (ii) the longitudinal phonons are the major carriers of heat at high temperature.