Channeling of acoustic phonons in silicon: the polarization dependence in elastic scattering

Ballistic phonon flux in crystals at low temperatures is highly directional due to phonon focusing. In this paper, a phonon-imaging experiment is reported which shows that scattered phonons, too, can retain a highly directed flux. Basically, the combination of phonon focusing and the elastic-scattering selection rule act to channel phonons along the ballistic focusing directions. Together with Monte Carlo simulations, the experiments show that the scattered phonon flux is indeed due to elastic scattering processes, which depend on the polarizations of the scattering phonons. Isotropic scattering is inconsistent with the data. The experimental technique shows promise for quantifying the microscopic scattering processes and revealing the dynamics of a phonon hot spot.     

Characterization of single-crystalline GaAs by imaging with ballistic phonons

Ballistic phonon propagation in single-crystalline [001]-oriented gallium arsenide has been studied using low-temperature scanning electron microscopy for imaging. Deviations in the phonon focusing pattern due to dispersion effects were found by comparing the phonon images to theoretical calculations of the long-wavelength limit. The phonon propagation behavior in, samples cut from differently prepared wafers has been investigated. For highly impure crystals we found a pronounced increase of the diffusive signal component at the expense of the ballistic one. Samples with varying dislocation densities also showed a sensitive dependence, of the ballistic phonon propagation on these crystal defects. For focusing calculations considering elastic scattering processes the diffusivity of the phonons could be determined as a function of the mean scattering length. We have found phonon mean free paths of 0.35 mm to 0.80 mm for the various GaAs crystals.     

Phonon-pulses propagation and phonon focusing in hexagonal-crystal rods in the boundary-scattering regime

The influence of the sample surface on the propagation of ballistic phonons in cylindrical samples of hexagonal crystals is studied. Our approach is based on the solution of the Boltzmann-Peierls equation with an external phonon source. The phonon irradiation of a detector face is calculated for⁴He and Zn crystals. It is shown how the strong phonon focusing, occurring in the slow modes of these solids, affects on the shape of energy flux falling upon the detector area. For an appropriately chosen lengthto-radius sample ratio, phonons reflected from the sample surface dominate in the detected signal.     

Observation of the quasidiffusive propagation of phonons in Mg<Subscript>2</Subscript>SiO<Subscript>4</Subscript>:Ho<Superscript>3+</Superscript> using time-resolved photo-phonon spectroscopy

The energy flux of phonons produced due to the nonradiative laser-induced transitions of Ho³⁺ impurity ions in forsterite from the ⁵F₅ states has been measured using a superconductor bolometer at a temperature of 2 K. The dependence of the flux on the laser wavelength, the time elapsed after the action of a laser pulse, and the phonon propagation path length is analyzed. It is found that the excitation of Ho³⁺ to some states leads to the diffusive propagation of emitted phonons in the spontaneous frequency decay mode (quasidiffusive mode of propagation): the time of arrival of a phonon pulse is almost a linear function of the path length, but it is several times longer than the longest ballistic time of flight (for transverse phonons). The diffusion coefficient and the nonradiative relaxation time are determined from the best fit to the experiment.     

Ballistic–diffusive phonon transport and size induced anisotropy of thermal conductivity of silicon nanofilms

The effective thermal conductivity of nanofilms is size dependent due to the diffusive–ballistic transport of phonons. In this paper, we investigate the cross-plane phonon transport from the viewpoint of the phonon Boltzmann equation. A predictive model for the size dependent thermal conductivity is proposed and agrees well with the results of molecular dynamics simulation for silicon nanofilms. The ballistic transport has different effects on the heat conduction in the in-plane or cross-plane directions, which causes the anisotropy of thermal conductivity of nanofilms. Such anisotropy is also size dependent and vanishes with the increase of film thickness.     

Phonon scattering at siliconcrystal surfaces

Our observations of the reflection or backscattering of high-frequency phonons (v =280 GHz to 1 THz) at silicon-solid interfaces disagree significantly with predictions from the acoustic mismatch model. Interfaces composed of materials theoretically wellmatched, show high scattering experimentally. In contrast, interfaces theoretically poorly matched, show less phonon scattering than expected. Generally, this is best expressed by the fact that the interface scattering ranges from roughly 30–60% for different phonon modes with little dependence on the material covering the silicon crystal and different techniques of interface preparations. Thus, our experiments indicate that the well-known Kapitza anomaly of the phonon scattering at solid-liquid helium interfaces is not a special case; the same anomaly appears to be present at all tested interfaces. Our experiments are compared with detailed calculations which either assume pure specular or pure diffusive scattering. In these calculations the influence of the crystal anisotropy for the phonon propagation (phonon focussing) is included. This comparison shows, especially for the free silicon surface, that phonons are completely diffuse scattered. Hence, the acoustic mismatched model relying on specular reflection cannot be applied to the real silicon interface. The frequency dependence of phonon scattering at a free silicon interface indicates the existence of at least two different diffusive scattering mechanisms. Within our experimental limits in these two scattering processes the phonons are elastically scattered.     

Scattering of thermal phonon pulses by isotopes in silicon

The pure ballistic propagation of acoustic phonons in crystals at low temperatures can be described within anisotropic continuum acoustics. One needs only the elastic constants and mass densities to calculate the time-dependent spectral phonon irradition of the bolometer for a given radiator pulse power and detector/radiator geometry. We extend this treatment by including single isotope-scattering events for the phonons in a (111)-cut silicon disk on their flight from the radiator to the detector. Using the earlier experimentally determined polarization and frequency dependent phonon absorption in the bolometer metal, the instantaneous temperature of the bolometer can be calculated from this irradiation. This allows a direct comparison with measured bolometer temperatures using exactly the same transmission or reflection arrangement as in calculation. A very satisfying agreement is observed in the expected range of single phonon scattering.     

Propagation of heat pulses in single-crystalline sapphire

We have studied heat-pulse propagation in single-crystalline sapphire using granular aluminum films for the heater and for the superconducting bolometer. The specimen surface carrying the bolometer was in direct contact with the liquid helium bath kept in the temperature range between 2.04 and 2.08 K. By varying the power during the heat pulse in the range between l mW and 2.8 W, we have observed the transition from purely ballistic pulse propagation at low power to diffusive propagation at higher power of the heat pulse. In the diffusive regime the phonon mean-free pathl has been experimentally determined and, using the Stefan-Boltzmann law, the variation ofl with the dominant phonon frequency has been found.     

Evolution of phonon nonequilibrium in single-crystal ZnSe at liquid-helium temperatures

The photoluminescence and nonequilibrium heat transfer observed in ZnSe single crystals grown by the free growth method on an oriented single-crystal ZnSe substrate in a hydrogen atmosphere were investigated. The nonequilibrium heat transfer (or propagation of nonequilibrium phonons) was studied by the heat pulse method. A region of local thermal equilibrium or “a hot phonon spot” was found to form in the material, and the threshold of its formation was determined. The constant of spontaneous anharmonic phonon decay in ZnSe was estimated from an analysis of the propagation of nonequilibrium phonons via comparison of the experimental responses with those calculated by the Monte Carlo method.     

Lattice dynamics of InSb from phonon imaging

The ballistic heat flux radiating from a point source of heat in a cold crystal displays a complex pattern of caustics due to phonon focusing. The caustics correspond to folds in the elastic-wave surface of the crystal. The pattern of caustics is independent of phonon frequency unless the phonon wavelength is comparable to the lattice spacing; i.e., for values of wavevectork approaching /a. We have measured both the shift in the caustic pattern (angular dispersion) and the increasing time-of-flight (velocity dispersion) for ballistic phonons in InSb with wavevectors up to 40% of the Brillouin-zone boundary. Comparison with existing lattice-dynamics models favors the Bond Charge Model (BCM). The phonon-imaging method gives information about theshapes of the wave surfaces which is complementary to the dispersion curves measured only along symmetry directions by neutron scattering.     

Anomalous Hall effect for the phonon heat conductivity in paramagnetic dielectrics

The theory of the anomalous Hall effect for the heat transfer in a parmagnetic dielectric, discovered by Strohm, Rikken, and Wyder [Phys. Rev. Lett. 95, 155901 (2005)]10.1103/PhysRevLett.95.155901, is developed. The appearance of the phonon heat flux normal to both the temperature gradient and the magnetic field is connected with the interaction of magnetic ions with the crystal field oscillations. In crystals with an arbitrary phonon spectrum this interaction creates the elliptical polarization of phonons. The kinetics related to phonon scattering induced by the spin-phonon interaction determines an origin of the off-diagonal phonon density matrix. The combination of both factors is decisive for the phenomenon under consideration.     

Focusing of dispersive phonons in GaAs

The ballistic propagation of highly dispersive phonons in GaAs in analyzed with both a bond-charge model (BCM) and a shell model, and compared to imaging experiments. When scattering is neglected, both models predict dispersive focusing pattern which depend strongly on the selected group velocities. In contrast, velocity-selected phonon images obtained with a PbBi tunnel-junction detector on a 0.7-mm thick crystal, display well-defined caustic lines whose angular positions do not shift at longer delay times. When isotope scattering is considered in the calculation, however, the shell model reproduces the experimental focusing structures remarkably well.For highly dispersive phonons, the bond-charge and shell models predict markedly different phonon-focusing patterns. In particular, a concentration of heat flux along [100] is predicted by BCM forv=1.7–2.0 THz slow-transverse (ST) phonons., but this effect is absent in the shell model. Both models predict new focusing structures in the fasttransverse (FT) mode at high frequencies. Our calculations which include isotope scattering indicate that to observe these new structures considerable technical advances are required in the phonon-imaging experiments.     

Quantitative investigation of thermal phonon pulses in sapphire,silicon, and quartz

Using pulse-heated constantan films as a thermal phonon radiator and superconducting tin bolometer as a phonon detector, we present for the first time a full quantitative comparison between observed bolometer signals and adequate rigorous model calculations for transmission experiments ina-cut sapphire, [111]-cut silicon, as well asX-cut quartz andZ-cut quartz. Details of the observed phonon signals are explained and understood. From these experiments, we are also able to extract information about the phonon absorption coefficient in the normal state of the polycrystalline tin bolometer for longitudinal and transverse polarized phonons in quantitative agreement with an earlier experiment ina-cut sapphire which was performed with a superconducting tunnel junction as a detector. The observed transmission signals can be explained for sapphire and silicon by ballistic propagation with additional small angle scattering, but for quartz strong frequency downconversion occurs for phonons with frequencies above half a Terahertz.In a succeding paper (Part II) the parameter deduced from the transmission experiment are applied to the analysis of the observed phonon signals in reflection experiments in the same crystals under the same conditions.Supported by Deutsche Forschungsgemeinschaft     

Hybrid discrete ordinates—spherical harmonics solution to the Boltzmann Transport Equation for phonons for non-equilibrium heat conduction

The Boltzmann Transport Equation (BTE) for phonons has found prolific use for the prediction of non-equilibrium heat conduction phenomena in semiconductor materials. This article presents a new hybrid formulation and associated numerical procedures for solution of the BTE for phonons. In this formulation, the phonon intensity is first split into two components: ballistic and diffusive. The governing equation for the ballistic component is solved using two different established methods that are appropriate for use in complex geometries, namely the discrete ordinates method (DOM), and the control angle discrete ordinates method (CADOM). The diffusive component, on the other hand, is determined by invoking the first-order spherical harmonics (or P₁) approximation, which results in a Helmholtz equation with Robin boundary conditions. Both governing equations, referred to commonly as the ballistic-diffusive equations (BDE), are solved using the unstructured finite-volume procedure. Results of the hybrid method are compared against benchmark Monte Carlo results, as well as solutions of the BTE using standalone DOM and CADOM for two two-dimensional transient heat conduction problems at various Knudsen numbers. Subsequently, the method is explored for a large-scale three-dimensional geometry in order to assess convergence and computational cost. It is found that the proposed hybrid method is accurate at all Knudsen numbers. From an efficiency standpoint, the hybrid method is found to be superior to direct solution of the BTE both for steady state as well as for unsteady non-equilibrium heat conduction calculations with the computational gains increasing with increase in problem size.     

Measurement of a phonon hot spot in photoexcited Si

High frequency phonons are produced by the thermalization of photoexcited electronhole pairs in a semiconductor. Inelastic and elastic scattering processes determine the frequency down-conversion and diffusion of this thermal energy. Simple estimates of the anharmonic and isotope scattering processes suggest that the acoustic phonons will undergo a quasi-diffusive propagation process. In particular, non-equilibrium phonons high-resolution phonon-imaging experiments presented here show well defined ballistic pulses and sharp phonon-focusing caustics. We explain this discrepancy in terms of a phonon hot spot which acts to efficiently down-convert the high-frequency phonons very near the excitation spot. We present the first measurements of the size of a phonon hot spot, which depends upon excitation power.     

A calculation of nonlocal phonon heat transfer

An evaluation is performed of Levinson's theory of nonlocal phonon heat transfer, which applies to the propagation of phonons in a diffusive medium in the presence of anharmonic processes. The particular case described is that of a semi-infinite substrate, initially at temperature T_o, subjected to an imposed small change in the temperature of its surface. Appropriate formulae are both derived and numerically evaluated to illustrate their content and relevance to experimental data.     

Anomalous Hall effect in the phonon thermal conductivity of paramagnetic dielectrics

A theory of the phonon Hall effect during heat transfer in a paramagnetic dielectric discovered by Strohm et al. [Phys. Rev. 95, 155901 (2005)] is developed. The heat flux emerging in the direction perpendicular to the magnetic field and to the temperature gradient is associated with the interaction of magnetic ions with the oscillating crystal field. In crystals with an arbitrary phonon spectrum, this interaction induces elliptic polarization of phonons. On the other hand, for any type of scattering, the temperature gradient forms part of the phonon density matrix, which is nondiagonal in modes. The combined action of these factors leads to the anomalous Hall effect.     

Effect of phonon decay processes on the formation of a phonon nonequilibrium signal in crystals with two two-level subsystems

The effect of phonon decay on the characteristic propagation time and shape of a phonon nonequilibrium signal in disordered crystals, including crystals containing inelastic phonon scattering centers, is studied theoretically. Attention is focused on slow processes, which are typical of yttrium-aluminum garnet solid solutions and erbium-doped aluminates. It is shown that the temperature dependence of the arrival time of a phonon nonequilibrium signal in these systems can be governed, to a considerable extent, by phonon-phonon interactions. The results of the theoretical studies are compared with experimental data on the propagation of weakly nonequilibrium thermal phonons in solid solutions of rare-earth yttrium-aluminum garnets and aluminates.     

Coherent X-ray scattering by phonons

Interference effects arising from coherently coupled X-ray beams inelastically scattered from phonons have been used to determine phonon eigenvectors in Si and GaAs crystals. An outline of the method, experimental requirements, and theoretical background is presented. The experimental results are analyzed with the help of different lattice dynamical model calculations. Best overall agreement is found with phonon eigendata based on the bond charge model.     

Thermal transport in heavily deformed and γ -irradiated LiF at temperatures near 1 k

The thermal conductivity κ of heavily deformed LiF crystals has been measured at temperatures T ? 0.5 K following exposure of the samples to γ irradiation. The results are in agreement with recent measurements of ballistic phonon propagation in similar samples at an equivalent temperature of ≈ 4 K. A fraction of the phonons have a mean free path of order 1 cm in the heavily deformed crystal, and γ-irradiation increases the fraction having a long mean free path. The measurements support a dynamic (as opposed to static) model of phonon-dislocation interaction.