Transient Energy and Heat Transport in Metals

A recently developed Shastry formalism for energy transport is used to analyze the temporal behavior of the energy and heat transport in metals. Comparison with Cattaneo’s equation is performed. Both models show the transition between ballistic and diffusive regimes. Furthermore, because the new model considers the discrete character of the lattice, it highlights some new phenomena such as oscillations in the energy transport at very short time scales. The energy relaxation of the conduction band electrons in metals is considered to be governed by the electron-phonon scattering, and the scattering time is taken to be averaged over the Fermi surface. Using the new formalism, one can quantify the transfer from ballistic modes to diffusive ones as energy propagates in the material and it is transformed into heat. While the diffusive contribution shows an almost exponentially decaying behavior with time, the non-diffusive part shows a damped oscillating behavior. The origin of this oscillation will be discussed as well as the effect of temperature on the dynamics of the energy modes transport. INTRODUCTION Development of high-power short-pulse laser sources with a pulse width in the sub-ps range has provided an opportunity to study the propagation of energy and heat at very short time scales; it has also created many applications in thin film analysis or in material processing. Ultrafast laser sources have made possible the study of many interesting fundamental physical phenomena in condensed matter such as electrons transitions in semiconductors [1, 2], electron-phonon coupling in metals [3-7], and electron dynamics in semiconductor superlattices [8, 9]. Shorter laser pulses in the sub-fs range have recently opened the field to explore electronic interactions within the atom itself [10]. Energy and heat transport during short-pulse laser heating of solid materials is an important phenomenon that needs to be fully understood to better control the abundant applications in which short-pulse laser sources are used. The question of energy and heat transport mechanisms at short time scales is the basis of numerous theoretical and experimental papers. From the microscopic point of view, energy deposits into and propagates through a material in different ways, depending on the excitation, the structure of the material, and the nature of the energy carriers. At short time scales, Fourier’s law becomes invalid and many Non-Fourier energy conduction models have been developed to overcome problems associated with the Fourier model (e.g. infinite speed of propagation of heat) [11-14]. Most importantly, the distinction between diffusive and non-diffusive (ballistic) regimes of energy transport becomes very relevant at these short time scales [15]. Recently, B. S. Shastry has developed a new formalism based on linear theory to describe coupled charge and energy transport in solid material systems [16]. The formalism is general and gives a set of equations for the electro-thermal transport coefficients in the frequency-wave vector domain. One of the most important results of this formalism is the introduction of new response functions describing the change in energy density, charge density, and the currents arising from the input excitation (coefficients M1, M2, N1, and N2 as defined in the reference article [16]). Among these response functions, N2 is of particular interest since this new function gives a measure of the change in energy density and hence temperature at various points in the system in response to the applied excitation at the top free surface of the system, and as such represents the energy (heat) Green’s function of the system. The objective of the current paper is to analyze the transient energy and heat transport in a metal occurring after application of a heating source at its top free surface in the frame work of the above mentioned Shastry’s formalism. In addition, the analysis will consider a comparison with Cattaneo’s model [17], both in time and frequency domains.