Diffusing phonons

Joshi, A.A. and A. Majumdar, Transient Ballistic and Diffusive Phonon Transport in Thin Films. Journal of Applied Physics, 1993. 74(1) p. 31-39. 

The rate of heat transfer in a thermal gradient is lower than might be expected for transport of vibrational energy heat transport by conductivity is much slower than sound propagation. The accepted physical model for heat transfer is a process of diffusing phonons (wave packets in the vibrating lattice) heat conduction in solids and in fluids is observed to be diffusive. 

The heat transfer via the solid backbone of aerogels depends on the backbone stmcture and connectivity (Chap. 21), and its chemical composition. For a given temperature gradient within an aerogel, heat is transferred by diffusing phonons via the chains of the aerogel backbone, where the mean free path of the phonons is far below the dimensions of the mostly amorphous, dielectric primary particles. Within the primary particles, the thermal conductivity is a property of the backbone material and is described in terms of the phonon diffusion model by Debye [7], i.e., ... 

Sph and are the contributions to the thermopower due to electron diffusion, phonon drag and magnon drag (or paramagnon drag), respectively. 

The situation is very different in indirect gap materials where phonons must be involved to conserve momentum. Radiative recombination is inefficient, resulting in long lifetimes. The minority carrier lifetimes in Si reach many ms, again in tire absence of defects. It should be noted tliat long minority carrier lifetimes imply long diffusion lengtlis. Minority carrier lifetime can be used as a convenient quality benchmark of a semiconductor. 

Charge carriers in a semiconductor are always in random thermal motion with an average thermal speed, given by the equipartion relation of classical thermodynamics as m v /2 = 3KT/2. As a result of this random thermal motion, carriers diffuse from regions of higher concentration. Applying an electric field superposes a drift of carriers on this random thermal motion. Carriers are accelerated by the electric field but lose momentum to collisions with impurities or phonons, ie, quantized lattice vibrations. This results in a drift speed, which is proportional to the electric field = p E where E is the electric field in volts per cm and is the electron s mobility in units of cm /Vs. 

For local deviations from random atomic distribution electrical resistivity is affected just by the diffuse scattering of conduction electrons LRO in addition will contribute to resistivity by superlattice Bragg scattering, thus changing the effective number of conduction electrons. When measuring resistivity at a low and constant temperature no phonon scattering need be considered ar a rather simple formula results ... 

These carriers of heat do not move balistically from the hotter part of the material to the colder one. They are scattered by other electrons, phonons, defects of the lattice and impurities. The result is a diffusive process which, in the simplest form, can be described as a gas diffusing through the material. Hence, the thermal conductivity k can be written as ... 

The different behaviour of contact resistance in the two cases can be examined through the two models the just described acoustic mismatch model and the diffuse mismatch model which suppose that all the phonons are scattered at the interface. Hence these two models define two limits in the behaviour of phonons at a discontinuity. 

Some experiments outlined the frequency dependence of phonon scattering on surfaces [74]. Thus, Swartz made the hypothesis that a similar phenomenon could take place at the interface between solids and proposed the diffuse mismatch model [72]. The latter model represents the theoretic limit in which all phonons are heavily scattered at the interface, whereas the basic assumption in the acoustic mismatch model is that no scattering phenomenon takes place at the interface of the two materials. In the reality, phonons may be scattered at the interface with a clear reduction of the contact resistance value as calculated by the acoustic model. 

In the diffuse mismatch model, the scattering destroys the correlation between the wave vector of the impinging phonon and that of the diffused one. In other words, the scattering probability is the same independent of which of the two materials the phonon comes from. This probability is proportional to the phonon state density in the material (Fermi golden rule). 

The linear and nonlinear optical properties of the conjugated polymeric crystals are reviewed. It is shown that the dimensionality of the rr-electron distribution and electron-phonon interaction drastically influence the order of magnitude and time response of these properties. The one-dimensional conjugated crystals show the strongest nonlinearities their response time is determined by the diffusion time of the intrinsic conjugation defects whose dynamics are described within the soliton picture. 

The polydiacetylene crystals (1-4) most strikingly corroborate these conjectures. Along this line of thought is also shown that this electron-phonon interaction is intimately interwoven with the polymerisation process in these materials and plays a profound role there. We make the conjecture that this occurs through the motion of an unpaired electron in a non-bonding p-orbital dressed with a bending mode and guided by a classical intermolecular mode. Such a polaron type diffusion combined with the theory of non radiative transitions explains the essentials of the spectral characteristics of the materials as well as their polymerisation dynamics. ... 

The linear and nonlinear optical properties of one-dimensional conjugated polymers contain a wealth of information closely related to the structure and dynamics of the ir-electron distribution and to their interaction with the lattice distorsions. The existing values of the nonlinear susceptibilities indicate that these materials are strong candidates for nonlinear optical devices in different applications. However their time response may be limited by the diffusion time of intrinsic conjugation defects and the electron-phonon coupling. Since these defects arise from competition of resonant chemical structures the possible remedy is to control this competition without affecting the delocalization. The understanding of the polymerisation process is consequently essential. 

The third term describes the polarization set up by ultrafast drift-diffusion currents, which can excite coherent phonons via TDFS (or via the buildup of electric Dember fields [9,10]). The first two terms represent the second- and the third-order nonlinear susceptibilities, respectively [31]. The fourth term describes the polarization associated with coherent electronic wavefunctions, which becomes important in semiconductor heterostructures. 

In combination with DFT calculations, the time- and depth-dependent phonon frequency allows to estimate the effective diffusion rate of 2.3 cm2 s 1 and the electron-hole thermalization time of 260 fs for highly excited carriers. A recent experiment by the same group looked at the (101) and (112) diffractions in search of the coherent Eg phonons. They observed a periodic modulation at 1.3 THz, which was much slower than that expected for the Eg mode, and attributed the oscillation to the squeezed phonon states [9]. 

The connection between anomalous conductivity and anomalous diffusion has been also established(Li and Wang, 2003 Li et al, 2005), which implies in particular that a subdiffusive system is an insulator in the thermodynamic limit and a ballistic system is a perfect thermal conductor, the Fourier law being therefore valid only when phonons undergo a normal diffusive motion. More profoundly, it has been clarified that exponential dynamical instability is a sufRcient(Casati et al, 2005 Alonso et al, 2005) but not a necessary condition for the validity of Fourier law (Li et al, 2005 Alonso et al, 2002 Li et al, 2003 Li et al, 2004). These basic studies not only enrich our knowledge of the fundamental transport laws in statistical mechanics, but also open the way for applications such as designing novel thermal materials and/or... 

Despite the difficulty cited, the study of the vibrational spectrum of a liquid is useful to the extent that it is possible to separate intramolecular and inter-molecular modes of motion. It is now well established that the presence of disorder in a system can lead to localization of vibrational modes 28-34>, and that this localization is more pronounced the higher the vibrational frequency. It is also well established that there are low frequency coherent (phonon-like) excitations in a disordered material 35,36) These excitations are, however, heavily damped by virtue of the structural irregularities and the coupling between single molecule diffusive motion and collective motion of groups of atoms. 

A further generalization is to write down a multi-dimensional GLE, in which the system is described in terms of a finite munber of degrees of freedom, each of which feels a frictional and random force. For example, an atom diffusing on a surface, moves in three degrees of freedom, two in the plane of the surface and a third which is perpendicular to the siuface. Each of these degrees of Ifeedom feels a phonon friction. Multi-dimensional generalizations and considerations may be foimd in Refs. 72-82. 

From an experimental point of view, a quantity of major interest is the hopping probability distribution Pj. A major source of friction for surface diffusion of metal atoms on metal surfaces is phonon friction. As shown in Refs. 164-167, the typical phonon friction is expected to be Ohmic (although there are claims... 

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