Phonon focusing of elastically scattered phonons in GaAs

The propagation of high-frequency phonons through crystals at low temperatures is characterized by both ballistic and diffusive processes. Ballistic propagation of heat pulses is highly anisotropic due to phonon focusing, while diffusive propagation is expected to be nearly isotropic in cubic crystals. By using phonon imaging techniques, we have attempted to identify the heat flux from ballistic and scattered phonons in GaAs. Comparison of this data to Monte Carlo calculations which incorporate elastic scattering shows that the flux from phonons scattered a few times in the bulk retains a significant degree of anisotropy. In particular, a sharp feature discovered by Stock, Ulbrich, and Fieseler and attributed to ballistic propagation of phonons with frequencies up to 1.5 THz is now identified with the scattering of sub-THz phonons. Our analysis provides insights into the evolution of heat propagation from the ballistic to diffusive regimes.     

Relaxation times and mean free paths of phonons in the boundary scattering regime for silicon single crystals

The phonon focusing in cubic dielectric crystals and its influence on the heat transfer in the boundary phonon scattering regime at low temperatures have been investigated. The mean free paths of phonons of different polarizations in samples of infinite and finite lengths with circular and square cross sections have been calculated in the anisotropic continuum model. For samples of infinite length with circular and square cross sections in the case of the equality of the cross-sectional areas, the angular dependences of the mean free paths normalized by the Casimir length almost completely coincide. It has been shown that the anisotropy of the mean free paths decreases significantly upon changing over from infinite samples to samples of finite length. For silicon crystals, the anisotropy of the phonon mean free paths has been analyzed for each of the branches of the phonon spectrum. It has been found that the mean free paths for phonons of each vibrational mode reach maximum values in the directions of focusing, and, in these directions, they exceed the mean free paths for phonons of the other vibrational modes.     

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.     

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.     

Phonon focusing in quartz and sapphire imaged by electron beam scanning

Anisotropic phonon propagation and phonon focusing in single-crystalline -quartz and sapphire has been investigated using electron beam scanning for imaging the ballistic phonon propagation. The samples were circular disks of 20 mm diameter and 2.0 mm thickness. The phonons generated at the upper flat sample surface by the electron beam were detected with a small-size (43 m×41m) bolometer at the center of the opposite surface. The experiments were performed using a scanning electron microscope with a low-termperature stage. During electron irradiation of the upper sample surface the bottom of the sample was in direct contact with the liquid-He bath. For -quartz, also time-resolved imaging of the phonon intensity has been performed in addition to imaging of the time-integrated bolometer signal. Our results agree remarkably well with calculations by Rösch and Weis.Supported by a grant of the Deutsche Forschungsgemeinschaft     

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.     

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.     

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     

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.     

Phonon focusing in [001] germanium

Phonon focusing in [001]-oriented Ge crystals has been studied at temperatures near 2.0K using electron-beam scanning for phonon generation and thin-film bolometers with an effective area of 10 m×10 m or smaller for phonon detection. The thickness of the Ge crystals was 2 or 3 mm. In our experiments the angular resolution appeared to be dominated by effective diameter of the local region heated by the beam and acting as phonon source. Whereas metallic overlay films did not appreciably effect the angular resolution, a distinct increase in resolution due to high-frequency beam modulation has been observed which can be understood in terms of the thermal skin effect. From a comparison between theoretical calculations and our experimental results the effective diameter of the phonon source has been found to be 20–30 m. Our focusing images show appreciable deviations from theoretical calculations in the low-frequency limit. However, excellent agreement is obtained, if dispersion is included and if the dominant phonon frequencies are place in the range 300–400 GHz.     

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.     

Acoustic waves and phonon focus in Li crystal: the first principle study

The acoustic waves in Li are investigated by solving the Christoffel equations.Theoretical calculations of the three dimensional slowness surfaces give insights into themixing of longitudinal and transverse modes and show the origin of the phonon caustics.The results are explained in terms of phonon focusing due to the anisotropy of crystals.The propagation of elastic waves in Li is affected by the elastic anisotropy of thelattice.     

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.     

Reduction of phonon mean free path: From low-temperature physics to room temperature applications in thermoelectricity

It has been proposed for a long time now that the reduction of the thermal conductivity by reducing the phonon mean free path is one of the best way to improve the current performance of thermoelectrics. By measuring the thermal conductance and thermal conductivity of nanowires and thin films, we show different ways of increasing the phonon scattering from low-temperature up to room-temperature experiments. It is shown that playing with the geometry (constriction, periodic structures, nano-inclusions), from the ballistic to the diffusive limit, the phonon thermal transport can be severely altered in single crystalline semiconducting structures; the phonon mean free path is in consequence reduced. The diverse implications on thermoelectric properties will be eventually discussed.     

Phonon propagation through photonic crystals—media with spatially modulated acoustic properties

The thermal conductivity κ of photonic crystals differing in degree of optical homogeneity (single crystals of synthetic opals) was measured in the 4.2–300 K temperature range. The thermal conductivity revealed, in addition to the conventional decrease in comparison with solid amorphous SiO₂ characteristic of porous solids, a noticeable decrease for T<20 K, the range wherein the phonon wavelength in amorphous SiO₂ approaches the diameters of the contact areas between the opal spheres. This effect is enhanced in the case of phonon propagation along the SiO₂ sphere chains (six directions in the cubic opal lattice). The propagation of light waves (photons) through a medium with spatially modulated optical properties (photonic crystals) is presently well studied. The propagation of acoustic waves through a medium with spatially modulated acoustic properties (phononic crystals) may also reveal specific effects, which are discussed in this paper; among them are, e.g., the ballistic mode of phonon propagation and waveguide effects.     

Phonon focusing catastrophes

We present an elegant application of formal catastrope theory to phonon focusing phenomena in crystals. The standard analysis of phonon focusing breaks down near singular directions known as caustics. The theory we present, in addition to being numerically accurate, provides a qualitative picture of caustic surfaces which yields important new insights into the results of phonon propagation experiments. Supporting experimental evidence is also presented.     

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.     

Electron transport and hot phonons in carbon nanotubes

We demonstrate the key role of phonon occupation in limiting the high-field ballistic transport in metallic carbon nanotubes. In particular, we provide a simple analytic formula for the electron transport scattering length, which we validate by accurate first principles calculations on (6, 6) and (11, 11) nanotubes. The comparison of our results with the scattering lengths fitted from experimental I-V curves indicates the presence of a nonequilibrium optical phonon heating induced by electron transport. We predict an effective temperature for optical phonons of thousands Kelvin.     

Mode dependent scattering of phonons by domain walls in ferroelectric KDP

Thermal conductivity and ballistic phonon imaging measurements in KH₂PO₄ (KDP) at low temperature (T<3K) indicate that scattering from domain walls has a large effect on phonon transport. kDP has a ferroelectric phase transition from tetragonal to orthorhombic structure atT _c =122 K. BelowT _c domains of opposite electric polarization and crystal orientation form unless the sample is colled in an electric field. Thermal conductivity measured along the [100] (tetragonal) axis drops 30% when domain walls are present, which is independent of sample size and temperature. We attribute this decrease to phonon polarization-dependent scattering at the domain boundaries. This is verified by measurements of ballistic transport, using phonon imaging techniques, which reveal the phonon polarization and mode dependence of the scattering. The scattering is successfully modelled using continuum acoustics with simple acoustic mismatch at the domainwall. The interface scattering is found to be mode dependent: Caustic structures in the phonon images due to slow transverse phonons are most affected by the domain wall scattering, which channels these phonons along parallel planes by multiple reflections without mode conversion. Mode conversion scattering, though possible for a number of phonons, has little effect on the overall phonon transmission.     

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.