Saad Bin Mansoor

Influence of Heat Source Size on Phonon Transport in Thin Silicon Film

Energy transfer is mainly governed by the phonon transport in dielectric films. The polarization and dispersion of the phonons alter the thermal resistance of the film as the film size becomes comparable to the mean path of the substrate material. This is because of the quasi-ballistic behavior of the transport characteristics. In this case, the ballistic phonons do not undergo scattering in the film while suppressing the thermal resistance increase across the film. In the present study, the quasi-ballistic phonon transport and the effect of heat source size on the phonon transport characteristics are investigated in the two-dimensional silicon film. The heat source is located at one edge of the film while other edges assumed to be at uniform temperature. Since the Knudsen number is small (∼1), the Boltzmann transport equation is solved numerically, incorporating the polarization and dispersion of phonons, to obtain phonon intensity distribution in the film. Equivalent equilibrium temperature is introduced to assess the phonon intensity distribution in the film. The transient behavior of the phonon transport is incorporated in the analysis to predict the time to reach steady state value of equivalent temperature in the film. It is found that the size of the heat source has a significant effect on the phonon transport in the film. The effective thermal conductivity reduces significantly as the heat source size reduces.

Phonon Transport Characteristics in a Thin Silicon Film

Phonon transport characteristics in thin silicon film are investigated due to temperature disturbance at the film edges. Phonon intensity distribution ratio , where Iomax is the maximum equilibrium intensity, I is the phonon intensity corresponding to acoustics branch at any location and time in the film) and phonon wavelength ratio , where k is the wavenumber, , and kcutoff is the cutoff wavenumber based on the film length, are introduced to assess the effects of quasi-ballistic and ballistic phonons on the film thermal conductivity and specific heat. It is found that phonon intensity ratio due to longitudinal acoustic branch is considerably higher than that of transverse acoustics branch. The influence of phonon wavelength ratio is not significantly high on the intensity ratio due to longitudinal acoustics branch. Reducing the wavelength ratio creates the size effect in the film while lowering thermal conductivity and specific heat of the film, which is more pronounced when the reduction becomes more than 20%.

Thermal transport across a thin film composite due to laser short-pulse heating

Abstract Laser short-pule irradiation of silicon-diamond-aluminum thin films is considered and energy transport across the films is modelled using the Boltzmann transport equation. Electron-phonon coupling is adopted to formulate energy transfer across the electron and lattice sub-systems in the aluminum film. Thermal boundary resistance is incorporated at the film interfaces. The transfer matrix method is used to account for the transmittance, reflection and absorption of the incident laser radiation across the films. Equivalent equilibrium temperature is introduced to access energy transport in the films, which represents the average energy of all phonons around a local point when they redistribute adiabatically to an equilibrium state. It is found that equivalent equilibrium temperature increases sharply in the electron sub-system due to electron excess energy gain from the irradiated field. Equivalent equilibrium temperature decays gradually in the lattice sub-system with increasing period due to phonon scattering in the film.

Phonon Transport in Silicon Thin Film: Effect of Temperature Oscillation on Effective Thermal Conductivity

In this article, we consider phonon transport in thin silicon film and examine the influence of temporal oscillation of the temperature heat source on the phonon transport. We solve the frequency-dependent transient two-dimensional Boltzmann equation numerically to observe the quasi-ballistic effects of the phonons on the film effective thermal conductivity. Temperature oscillation at different periods is incorporated at the bottom face of the film as a heat source. The size of the temperature source is varied by introducing the spatial Gaussian distribution along this face. In order to assess the phonon distribution in the film, we introduce the equivalent equilibrium temperature. We determine the effective thermal conductivity of the film and analyze its variation due to the different periods of the temperature oscillation and the Gaussian parameters. It is found that reducing the period of the temperature oscillation lowers the effective thermal conductivity of the film, which is more pronounced for low Gaussian parameters. In this case, the quasi-ballistic behavior of phonons contributes adversely to the effective thermal conductivity of the film.

Phonon Transport in Curved Thin Film: Effect of Film Curvature and Radius on Transport Characteristics

ABSTRACT Phonon transport inside a curved film is considered and phonon intensity distribution is instigated for various film arc angles and film radii. A thermal disturbance is applied at the edges of the curved film and resulting phonon transport characteristics are simulated by incorporating the radiative phonon transport equation, which is derived for the general coordinate system to fit any film geometry. Equivalent equilibrium temperature is introduced to quantify the phonon intensity distribution inside the film. In the analysis, the total area and the thickness of the film are kept constant for various film arc angles and radii. It is found that equivalent equilibrium temperature decay is sharp in the close region of the high temperature edge inside the curved film. Normalized equivalent equilibrium temperature reduces with increasing arc angle; however, this decrease is not considerable, i.e., it is in order of 4%. Similar arguments are applied for the temperature distribution along the film thickness in the radial direction.

A New Approach for Semi-Analytical Solution of Cross-plane Phonon Transport in Silicon–Diamond Thin Films

Abstract Transient analysis of phonon cross-plane transport across two consecutively placed thin films is considered, and a new approach is introduced to obtain the semi-analytical solution for the equation of phonon radiative transport. The orthogonality properties of trigonometric functions are used in the mathematical analysis. Silicon and diamond thin films are used to resemble the consecutively placed thin films. The films are thermally disturbed from its edges to initiate the phonon transport, and thermal boundary resistance is introduced at the films interface. Equivalent equilibrium temperature is incorporated to quantify the phonon intensity distribution in the films. It is found that the results of the analytical solution agree well with their counterparts obtained from the numerical simulations. Phonon intensity at the film edges and interface reduces significantly due to boundary scattering. The analytical solution captures phonon scattering at boundaries and interface correctly, and provides considerable simplification of the numerical treatment of the equation for phonon radiative transport. It also reduces significantly the numerical efforts required for solving the transient phonon radiative transport equation pertinent to the cross-plan transport across the thin films in terms of program size and run-time.

Some Aspects of Statistical Thermodynamics

This chapter provides useful information to familiarize the readers with some understanding of statistical mechanics. The information provided bridges the gap between the microscopic and macroscopic picture of the matter. In the beginning, some useful definitions and the basic understanding of statistical thermodynamics are provided such as microstates and distributions, phase space, and so on. The probability of energy states and energy distribution is given and some applications are provided in the later part of this chapter. In addition, the concepts of ensemble and statistical distribution function are provided.

Chapter 5 – Analytical Treatment of Phonon Transport in Thin Films

The radiative phonon transport equation is critically important to assess the energy transport characteristics in a thin film. In this chapter, analytical treatment of the radiative transport equation is presented, and the formulation of heat transport due to thermal excitation of the thin film is introduced. The closed form solution for the radiative transport equation is presented with the appropriate boundary conditions for the one-dimensional heating situation. In addition, the closed form solution of the hyperbolic heat equation derived from the electron kinetic theory approach is presented, incorporating the thermal disturbance due to laser short-pulse irradiation of the metallic substrates. Since the metallic materials thermally separate under the nonequilibrium heating situations, thermal coupling in terms of the electron–phonon coupling parameter is introduced for the energy transport across the electron and lattice subsystems. The radiative transport equation is modified for thin metallic films, and thermal coupling across the electron and lattice subsystem is incorporated. The limitations of the two-equation model in micro/nanoscale heating are also discussed.

Phonon Radiative Transfer in Curvilinear Coordinate Systems

The mathematical formulation of the energy transport equation in terms of the radiative phonon transport is introduced, incorporating the curvilinear coordinate system. The new formulation provided the general solution for the phonon radiative transport equation without the coordinate transformation, in particular for the curved thin films. The equation for radiation transport can be specialized for two common systems, namely the cylindrical and the spherical coordinate systems. A mathematical procedure outlining the numerical solution of the frequency-independent, steady-state equation of phonon radiative transfer (EPRT) in an irregular, two-dimensional domain is included by introducing the curvilinear grid system. Since the development of the EPRT in this chapter pertains only to Euclidean spaces and not to Riemannian spaces, the curvature tensor vanishes in the formulations. An application of the EPRT for two-dimensional curved thin film is also included, and the findings are discussed at the end of this chapter.

Chapter 6 – Heat Transfer Applications in One- and Two-Dimensional Thin Films

One- and two-dimensional treatments of the equation for phonon transport are introduced, and case studies associated with these treatments are provided. Since the resulting equation of phonon radiative transport is in the form of an integrodifferential equation, the numerical solution of the transport equation is provided in line with the boundary and initial conditions. The transient analysis for the phonon transport in dielectric thin films is provided, and the thermal disturbance due to the temperature difference across the thin film is adopted to demonstrate the phonon transport characteristics across the thin film. The case studies are extended to include two-dimensional films, and transient analysis for the two-dimensional thin films is provided. Thin film materials such as silicon and diamond are the primary focus in the case studies. To quantify the phonon intensity distribution in the film, the equivalent equilibrium temperature is introduced, which provides the physical quantities of transport characteristics in the window of engineering understanding. The case studies related to the metallic thin films are also provided, and transport characteristics inside the film due to laser short-pulse heating are included.