Context

With few exceptions this book has concerned itself with the propagation of vibrational waves in the bulk of a crystal. As indicated in Section B of Chapter 12, there are also vibrational modes associated with the surface of the medium. They are known as Rayleigh surface waves (RSWs) and pseudo–surface waves (PSWs). As with bulk waves, the velocities of these surface waves depend on the propagation direction, so one might reasonably expect phonon-focusing effects. Indeed, Tamura and Honjo and Camley and Maradudin have predicted phonon-focusing patterns for Rayleigh waves in a variety of media. An example of the latter results is shown in Figure 1: (a) is the slowness curve for Rayleigh waves on the (111) surface of Ge and (b) shows the corresponding singular directions of energy flux emanating from a point source in this plane. Phonon caustics arise from the zero-curvature regions of the slowness surface, just as discussed with bulk waves in Figure 8 of Chapter 2.

Unfortunately, high-frequency surface phonons are likely to have much shorter mean free paths than bulk phonons, due to the sensitivity of these waves to surface irregularities. The scattering of surface phonons from such defects, plus the potentially huge background of scattered bulk phonons, makes it extremely difficult to observe the ballistic propagation and phonon focusing of surface phonons. A recent study by Höss and Kinder, however, has reported experimental evidence for the focusing of surface phonons on a laser-annealed Si surface.

The physics of nonequilibrium phonons in crystals, as with most scientific phenomena, has been unveiled by a combination of experimental and theoretical work. We have seen that the propagation of acoustic phonons in elastically anisotropic media contains a rich spatial complexity. The advent of imaging techniques has permitted visual insights into elasticity theory and, as shown in later chapters, into the propagation and scattering of high-frequency phonons.

To study phonons one must be able to create and detect them. How does one jiggle the atoms in the crystal and detect their collective motion? Because the nuclei (or atomic cores) are charged, one possibility is the application of oscillating electric fields such as electromagnetic waves. Examples of this include ultrasound generation in piezoelectric materials and phonon generation by absorption of infrared light. Alternatively, because electronic motion produces time-varying local fields, an effective way of creating nonequilibrium phonons is to electrically or optically excite the electrons in the crystal. It comes as no surprise, then, that an efficient means of creating and detecting nonequilibrium phonons is to employ materials with high electron density – namely, metals. Conversely, the macroscopic motion of phonons is studied in materials with low densities of free carriers – namely, insulators and semiconductors. The common approach, therefore, is to attach metallic generators and detectors to the nonmetallic crystals of interest.

This chapter begins with an overview of the measurement techniques involved in a phonon-imaging experiment.

In the previous chapters we saw how continuum elasticity theory leads to a rich variety of phonon focusing patterns. The predicted phonon images are completely determined by the elastic constants, in some cases modified by piezoelectric stiffening. An alternative approach is to model the microscopic forces between atoms in the crystal lattice. Such lattice dynamics models are necessary to explain the propagation of phonons with wavelengths comparable to the lattice spacing. Of course, these microscopic models must also be compatible with the continuum theory in the limit of long wavelengths. The purpose of this chapter is threefold: (1) to introduce the mathematical underpinnings of lattice dynamics, (2) to examine the connection between continuum mechanics and the theories of lattice dynamics, and (3) to summarize the predictions of selected lattice dynamics models for phonon wavelengths approaching the atomic spacings.

A basic question that arises is how the intricate anisotropies in elastic-wave propagation relate to the crystal structure and the forces between atoms. In particular, what arrangements of atoms and types of interatomic forces lead to elastic stability? … to positive or negative anisotropy factor, Δ? … to an elastically isotropic medium? Can one construct a hypothetical lattice that has the same sound velocity in all directions? What kinds of slowness surfaces do the simple force models predict? How sophisticated must the microscopic models be in order to predict the measured elastic properties of real crystals?

Historical overview

Anisotropy is a recurring theme in the physics of crystalline solids. The regular arrangement of atoms in crystals produces direction-dependent physical properties. For example, the response to applied electric and magnetic fields usually depends on the direction of the fields with respect to the symmetry axes of the crystal. Properties such as effective mass, electrical conductivity, magnetic susceptibility, and electric polarizability are second-rank tensors because they relate two vectors, such as current and electric field.

Thermal conductivity is also a second-rank tensor, giving the heat flow produced by an imposed temperature gradient. In nonmetallic crystals, heat is conducted by high-frequency elastic waves, or phonons. The velocity of an elastic wave depends on the direction of propagation with respect to the crystal axes because the elasticity of a crystal is anisotropic. Yet, in cubic crystals at room temperature, the thermal conductivity is nearly isotropic. This isotropy results from the fact that, at normal temperatures, phonons scatter frequently, making heat flow a diffusive rather than ballistic process. At low temperatures, however, the mean free paths of phonons lengthen dramatically, and it is possible to observe the ballistic propagation of heat, i.e., no phonon scattering between source and detector. Ballistic heat flow occurs at or near the velocities of sound and exhibits the anisotropies expected from elasticity theory. Heat flow in the ballistic regime is governed by the fourth-rank elasticity tensor.

For a crystal containing N atoms, there are 3N normal modes of vibration. These normal modes are generally described in terms of plane waves of the form cos(k · r − ωt), where k is the wavevector with magnitude 2π/λ and ω ≡ 2πν is the angular frequency of the wave with wavelength λ and frequency ν. The boundary conditions of the crystal restrict the possible values of k, leading to a uniform grid of discrete wavevectors in k space, bounded by the Brillouin zone of the particular lattice. This quantization of wavevectors, k, is simply a result of classical mechanics.

Models of lattice dynamics seek to determine the vibrational frequency for a given k by considering the elastic forces between atoms. As an example, the dispersion relation, ω(k), for Ge along the axis is reproduced in Figure 1(a). The acoustic and optical branches are shown. Given this dispersion relation, the density of modes per unit frequency, D(ω), can be calculated. The density of modes – essentially a “vibrational spectrum” of the crystal – is basic to the thermal and electrical properties of the particular material.

Now quantum mechanics comes into play. If we think of a vibrational mode with wavevector k and (angular) frequency ω as a plane wave spread uniformly over the entire crystal, then, according to the quantum mechanics of the harmonic oscillator, the energy (and, hence, amplitude) of this wave can have only discrete values.

This chapter attempts to characterize the acoustic phonons generated when a semiconductor crystal is excited by light. These phonons are a byproduct of the energy relaxation of photoexcited carriers. The problem naturally divides into two parts: (a) the distributions (in space, time, and frequency) of acoustic phonons emanating from the excitation region, i.e., the phonon source, and (b) the subsequent propagation and decay (downconversion) of acoustic phonons in the bulk of the sample. Because elastic scattering from mass defects is highly dependent on frequency, the instantaneous diffusion constant increases with time as phonons downconvert – a process commonly referred to as quasidiffusion. In principle, low-frequency phonon sources produce ballistically propagating phonons and high-frequency sources lead to quasidiffusion.

Previous chapters have dealt mainly with heat pulses generated by optical excitation of a metal film that was deposited on the nonmetallic crystal of interest. Nonequilibrium phonons are again produced by the relaxation of the carriers excited by the light. In the metal, however, there is a high density of free carriers (1022–1023 cm−3), and the interactions among carriers and between carriers and phonons tend to establish a single temperature. In that case, the phonons radiated from the metal film into the crystal have an approximately Planckian energy distribution, and the subsequent propagation in the crystal can be modeled by anharmonic decay, elastic scattering, and ballistic propagation (with frequency dispersion).

The Kapitza problem

The transport of thermal energy across solid/solid and solid/liquid interfaces has occupied and perplexed many researchers since Kapitza's famous experiments in 1941. He discovered the existence of a sharp temperature gradient at the surface of a heated solid that was immersed in liquid He. This surprisingly steep temperature drop was observed to occur within a few tens of micrometers of the surface, which was the spatial resolution of the experiment.

It was first suggested that the source of this discontinuity was a thick boundary layer of He that limited the thermal conductivity. However, such mechanisms proved unnecessary, as Khalatnakov realized that the measured thermal boundary resistance between the solid and liquid was actually much smaller than predicted from a simple acoustic-mismatch theory. One defines the thermal boundary resistance as, where ΔT is the temperature drop across the boundary for a given power flow.

Khalatnakov reasoned that the solid and the He have quite disparate acoustic impedances (Z = ρν, with ρ the density and ν the sound velocity), which implies a small phonon transmission coefficient across the interface. Modeling the solid and liquid as classical acoustic media, he predicted that the average transmission coefficient of an acoustic wave should be approximately 1%, whereas the measured thermal boundary resistances corresponded to a transmission coefficient of approximately 50%. This disparity between theory and experiment is known as the “Kapitza anomaly.” Fifty years after Kapitza's measurements, this anomaly is not completely resolved.

This book could have been entitled “Phonon Caustics,” for it is these unusual concentrations of thermal energy resulting from point excitation of crystals that have provided a wealth of new perspectives on the physics of acoustic phonons. The concept of caustics, however, is not common to the parent field of physical acoustics. Why not? In large part, it is because acoustic experiments on solids usually employ planar sources of vibrational energy. Also, the coherence and macroscopic wavelengths of radio-frequency waves are not characteristics of nonequilibrium phonons. This chapter describes some recent ultrasonic-imaging experiments and calculations that help to establish the conceptual links between phonon imaging and physical acoustics. More importantly, we shall see that the extension of phonon-imaging techniques to ultrasound greatly expands the applicability of these methods.

Throughout the text we have used the word “acoustic” to designate the lowest branches of the vibrational spectrum – those waves which, in the limit of low frequency, have the same velocities and polarizations as sound waves in the solid. Indeed, many of the caustic patterns we have seen are predictable from continuum elasticity theory. Yet, we have implicitly assumed that the wavelengths of the phonons are smaller than the source and detector sizes, and certainly much smaller than the sample dimensions. This assumption is true even for the lowest frequencies observed in a typical heat-pulse experiment – approximately 100 GHz, corresponding to λ≃50 nm.

The phonon is an elementary excitation of a crystal. How it couples to other elementary excitations is critical to many areas of condensed-matter physics. Foremost are the interactions of phonons with electrons. In this final chapter, I will touch upon a few emerging technologies and applications of phonon imaging that mainly involve the interactions of electrons and phonons. The topics chosen include the optical properties of semiconductors and heterostructures, mechanisms of superconductivity and development high-energy particle detectors.

Of course, we have already encountered the interaction of nonequilibrium phonons with electrons. Chapter 10 dealt with the spatial and frequency distributions of phonons created by photoexcited carriers in semiconductors. In contrast, an increasing variety of experiments employ ballistic phonons to probe both intrinsic electronic states and those associated with impurities. In addition, it is possible in some cases to detect nonequilibrium phonons by their effect on the optical and electronic properties of crystals. We begin by considering a few crystalline systems that are suitable for time- and space-resolving phonon detectors, while keeping in mind the broader goal of elucidating the physics of new materials or devices.

Optical detection of nonequilibrium phonons

For the most part, the optical properties of crystals in the visible or nearinfrared region are associated with transitions between electronic states. Phonons of sufficient energy can induce transitions between low-lying electronic states, thereby affecting the optical absorption or emission of the solid.

Because matter is composed of light and heavy particles – namely, electrons and nucleons – the properties of solid materials naturally divide into two main categories: electronic and vibrational. For the most part, the rapid motion of electrons governs the binding and elasticity of a solid, its conductivity, and its optical and magnetic properties. In contrast, the more sluggish motion of the atomic nuclei determines the vibrational properties of a solid. The descriptions of electronic and nuclear motion in solids are generally developed in terms of wave motion. Such waves in solids are most easily understood for well-ordered systems such as crystals, where the atoms are arranged in regular arrays.

This book is about vibrational waves in crystals – more specifically, those classified as acoustic waves. These are the waves that depend on the elasticity of the medium and whose frequencies extend from zero to a few terahertz (1 THz = 1012 Hz). Experimental probes of the motion of acoustic waves in solids traditionally range from ultrasonics to thermal conductivity. The major focus of this book is the propagation of phonons – the tiny packets of vibrational energy associated with heat and typically observed in the frequency range from 1010 to 1012 Hz. The propagation of ultrasonic waves in the megahertz range will also be examined here using imaging techniques initially developed for high-frequency phonons.

Phonons govern the thermal properties of nonmetallic solids. At normal temperatures their motion is highly diffusive.

Phonon scattering and thermal conduction

We have seen that the mean free path of high-frequency phonons is greatly limited by scattering in the bulk of the crystal. The experiments discussed in Chapter 7 were concerned with the relatively small fraction of large-k phonons that traverse the crystal ballistically as identified by their sharp focusing pattern. In those experiments, the bulk scattering of high-frequency phonons by mass defects acted as a low-pass filter, which, in combination with a high-pass detector, provided an effective means of frequency selection for the ballistic phonons. In Chapter 8, we developed a theoretical basis for phonon scattering. Now we will describe experiments that measure the phonon scattering processes in detail.

There are several different phonon-scattering processes to be considered. As discussed in Chapter 8, the simplest form of scattering is from atomic mass defects that differ from their neighbors only in isotopic mass. Because most atoms occur naturally with several isotopic masses, this type of defect is ubiquitous. Only a few crystals, such as NaF, occur naturally in an isotopically pure form. Because the natural isotopic abundance of atoms is generally well known, however, phonon scattering from randomly occurring isotopes (i.e., isotope scattering) is very predictable and serves as an important test case for phonon-scattering theories.

Another common form of defect scattering occurs when an impurity atom substitutes for a host atom in the crystal.