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.     

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.     

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.     

Effect of normal phonon-phonon scattering on the mutual electron-phonon drag and kinetic phenomena in degenerate conductors

The effect of normal phonon-phonon scattering processes on momentum relaxation in a nonequilibrium electron-phonon system is considered. A system of rate equations for the electron and phonon distribution functions has been solved with the inclusion of mutual electron-phonon drag. The kinetic coefficients of conductors have been calculated in a linear approximation in the degeneracy parameter. The effect of normal phonon-phonon scattering processes on the electron-phonon drag and on the kinetic phenomena in conductors with degenerate carrier statistics is analyzed.     

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 quasiparticle scattering and kinetic effects in degenerate semiconductors

The influence of normal processes of electron-electron and phonon-phonon scattering on quasiparticle momentum relaxation in nonequilibrium electron-phonon systems of degenerate semiconductors is investigated. A system of kinetic equations is solved for the electron and phonon distribution functions, and the kinetic coefficients of a semiconductor are calculated in the linear approximation in the degeneracy parameter. The influence of normal scattering of quasiparticles on the electrical conductivity, thermopower, and heat conductivity of a degenerate semiconductor is analyzed. Redistribution of the phonon momentum in N processes within each branch of the vibrational spectrum, as well as among different branches, is taken into account.     

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.     

High field transport properties in a GaN/AlGaN heterostructure

The drift velocity, electron temperature, electron energy and momentum loss rates of a two-dimensional electron gas are calculated in a GaN/AlGaN heterojunction (HJ) at high electric fields employing the energy and momentum balance technique, assuming the drifted Fermi–Dirac (F–D) distribution function for electrons. Besides the conventional scattering mechanisms, roughness induced new scattering mechanisms such as misfit piezoelectric and misfit deformation potential scatterings are considered in momentum relaxation. Energy loss rates due to acoustic phonons and polar optical phonon scattering with hot phonon effect are considered. The calculated drift velocity, electron temperature and energy loss rate are compared with the experimental data and a good agreement is obtained. The hot phonon effect is found to reduce the drift velocity, energy and momentum loss rates, whereas it enhances the electron temperature. Also the effect of using drifted F–D distribution, due to high carrier density in GaN/AlGaN HJs, contrary to the drifted Maxwellian distribution function used in the earlier calculations, is brought out.     

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.     

Theory of Raman light scattering by nanocrystal acoustic vibrations

A theory of Raman scattering of light by acoustic phonons in spherical nanocrystals of zinc-blende and wurtzite semiconductors has been developed with the inclusion of the complex structure of the valence band. The deformation-potential approximation was used to describe the exciton-phonon interaction. It is shown that this approximation allows only Raman scattering processes involving spheroidal acoustic phonons with a total angular momentum F=0 or 2. The effect of phonon quantum confinement on linewidth in Raman scattering spectra and scattered polarization is analyzed. An expression for the shape of the spectral line corresponding to nonresonant scattering from F=0 phonons was obtained. Fiz. Tverd. Tela (St. Petersburg) 41, 1473–1483 (August 1999)     

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.     

Electron diffusion thermopower of alkali metals

The electron diffusion thermopower of K, Rb and Cs is calculated and analysed by using an accurate phonon spectrum generated from neutron scattering data. It is found that electron-phonon normal scattering generates a negative thermopower which is linear in temperature. On the other hand, Umklapp scattering produces a large positive and extremely non-linear thermopower. Since the individual contributions are weighted by the respective thermal resistivities, the positive Umklapp component dominates for Rb and Cs, whereas the negative normal component does so for K. Although the observed sign anomaly of the thermopower is generally attributed to the phonon drag effect, we see that the electron diffusion thermopower also plays a significant role.     

Relaxation of quasi-transverse phonons in the Herring mechanism and ultrasound absorption in cubic crystals with positive and negative anisotropies of the second-order elastic moduli

The phonon relaxation and quasi-transverse ultrasound absorption in the course of Herring and Landau-Rumer anharmonic scattering processes in cubic crystals with positive (Ge, Si, diamond, InSb, LiF, MgO) and negative (KCl, NaCl, CaF₂) anisotropies of the second-order elastic moduli have been investigated. A new mechanism of transverse phonon relaxation, according to which the fusion of a transverse (slow or fast) phonon with a slow phonon generates a fast transverse phonon, has been considered in the long-wave-length approximation. This mechanism is similar to the Herring relaxation mechanism for longitudinal phonons. It has been demonstrated that, for crystals of the first group with a considerable anisotropy of the elastic energy (Ge, Si, InSb, LiF, MgO), “anomalous” relaxation processes in which the fusion of a slow transverse phonon with a fast phonon generates a slow transverse phonon are possible in contrast to the Herring relaxation mechanism for longitudinal phonons. These relaxation processes appear to be impossible for all crystals of the second group (KCl, NaCl, CaF₂), as well as for crystals of the first group with a small anisotropy of the elastic energy, such as diamond. The angular dependences of the ultrasound absorption coefficient for the Herring and Landau-Rumer mechanisms have been analyzed using the anisotropic-continuum model. It has been shown that, for the crystals of the first group under consideration, the contribution of the Herring mechanism to the long-wavelength ultrasound absorption is small compared to the contribution of the Landau-Rumer mechanism. However, for the KCl and NaCl crystals of the second group in directions of the [001] type, the contribution of the Herring mechanism can significantly exceed the contribution of the Landau-Rumer mechanism.     

Transient response of hot electrons in narrow-gap semiconductors at low temperatures in the presence of a longitudinal quantizing magnetic field

Transient response of hot electrons in narrow-gap semiconductors to a step electric field in the presence of a longitudinal quantizing magnetic field has been studied at low temperatures using displaced Maxwellian distribution. The energy and momentum balance equations are used assuming acoustic phonon scattering via deformation potential responsible for the energy relaxation and elastic acoustic phonon scattering together with ionized impurity scattering for momentum relaxation. The calculations for the variation of drift velocity and electron temperature as functions of time are made for n-Hg_0.8Cd_0.2 Te in the extreme quantum limit at 1.5 K and 4.2 K. The momentum and energy relaxation times are found to be of the same order of magnitudes as with the experimental values. The magnetic field and lattice temperature dependences of the relaxation rates have been investigated.One of the authors, Suchandra Bhaumik, acknowledges the Council of Scientific and Industrial Research (New Delhi) for financial support.     

Lattice thermal conductivity of skutterudite compounds

A generalized expression is used on the basis of relaxation time approximation to facilitate calculation of lattice thermal conductivity of dielectric materials as well as skutterudite family consists of compounds of the form AB₃. It is assumed that phonon scattering processes are independent and is represented by frequency dependent relaxation times. The contributions of normal three phonon scattering processes are included explicitly as redistribution of phonon momentum between two oscillation branches is considered. Magnitudes of relaxation times are estimated from the experimental data. The result for CoSb₃ is in reasonably good agreement with the experimental result in the temperature range 1–1000°K. It is observed that redistribution of phonon momentum between two oscillation branches leads to a significant suppression of thermal conductivity maximum and it is observed that for unfilled skutterudite the main dominant mechanism at the thermal conductivity maximum is three phonon normal scattering process.     

Effect of normal processes on thermal conductivity of germanium,silicon and diamond

The effect of normal scattering processes is considered to redistribute the phonon momentum in (a) the same phonon branch — KK-S model and (b) between different phonon branches — KK-H model. Simplified thermal conductivity relations are used to estimate the thermal conductivity of germanium, silicon and diamond with natural isotopes and highly enriched isotopes. It is observed that the consideration of the normal scattering processes involving different phonon branches gives better results for the temperature dependence of the thermal conductivity of germanium, silicon and diamond with natural and highly enriched isotopes. Also, the estimation of the lattice thermal conductivity of germanium and silicon for these models with the consideration of quadratic form of frequency dependences of phonon wave vector leads to the conclusion that the splitting of longitudinal and transverse phonon modes, as suggested by Holland, is not an essential requirement to explain the entire temperature dependence of lattice thermal conductivity whereas KK-H model gives a better estimation of the thermal conductivity without the splitting of the acoustic phonon modes due to the dispersive nature of the phonon dispersion curves.      

Transient magnetoresponse of photoexcited electrons: Negative cyclotron absorption and evolution of Faraday angle

The transient magnetooptical response of electrons with partly inverted initial distribution produced by an ultrashort optical pulse near the optical phonon energy is studied theoretically. Transient cyclotron absorption and Faraday rotation of polarization plane are considered for bulk semiconductors (GaAs, InAs, and InSb) as well as for a GaAs-based quantum well. Damping of the response due to electron momentum relaxation associated with elastic scattering from acoustic phonons is taken into account in calculations, as well as the evolution of the electron distribution due to quasi-elastic energy relaxation at acoustic phonons and effective inelastic transitions accompanied by spontaneous emission of optical phonons. Nonstationary negative absorption in the cyclotron resonance conditions and peculiarities of Faraday rotation of the polarization plane associated with partial inversion of the initial distribution are considered. The possibility of transient enhancement of the probe field under cyclotron resonance conditions is indicated.     

声子角动量与手性声子

在经典的物理学理论中,声子广泛地被认为是线极化的、不具有角动量的.最近的理论研究发现,在具有自旋声子相互作用的磁性体系(时间反演对称性破缺)中,声子可以携带非零的角动量,在零温时声子除了具有零点能以外还带有零点角动量;非零的声子角动量将会修正通过爱因斯坦-德哈斯效应测量的回磁比.在非磁性材料中,总的声子角动量为零,但是在空间反演对称性破缺的六角晶格体系中,其倒格子空间的高对称点上声子具有角动量,并具有确定的手性;三重旋转对称操作给予声子量子化的赝角动量,赝角动量的守恒将决定电子谷间散射的选择定则;此外还理论预测了谷声子霍尔效应.     

Anisotropy of the mean free paths of phonons in single-crystal films of germanium,silicon, and diamond at low temperatures

The physical aspects of the influence of the elastic energy anisotropy of crystals on the anisotropy of the mean free paths of phonons in single-crystal films of germanium, silicon, and diamond in the diffuse scattering of phonons at the boundaries of the samples have been considered. It has been shown that, for sufficiently wide films of germanium, silicon, and diamond with the {100} and {111} orientations and the lengths of less than or equal to their width, the phonon mean free paths are isotropic (independent of the direction of the temperature gradient in the plane of the film). The anisotropy of the phonon mean free paths depends primarily on the orientation of the film plane and is determined by the focusing and defocusing of phonon modes. For single-crystal films of germanium, silicon, and diamond with the {100} and {111} orientations and lengths much larger than their width, the phonon mean free paths are anisotropic.