Thesis advisor: David A. Broido / A theory is presented for describing the lattice thermal conductivities of graphene and single-walled carbon nanotubes. A phonon Boltzmann transport equation approach is employed to describe anharmonic phonon-phonon, crystal boundary, and isotopic impurity scattering. Full quantum mechanical phonon scattering is employed and an exact solution for the linearized Boltzmann transport equation is determined for each system without use of common relaxation time and long-wavelength approximations. The failures of these approximations in describing the thermal transport properties of nanotubes is discussed. An efficient symmetry based dynamical scheme is developed for carbon nanotubes and selection rules for phonon-phonon scattering in both graphene and nanotubes are introduced. The selection rule for scattering in single-walled carbon nanotubes allows for calculations of the thermal conductivities of large-diameter and chiral nanotubes that could not be previously studied due to computational limitations. Also due to this selection rule, no acoustic-only umklapp scattering can occur, thus, acoustic-optic scattering must be included in order to have thermal resistance from three-phonon processes. The graphene selection rule severely restricts phonon-phonon scattering of out-of-plane modes. This restriction leads to large contributions to the total thermal conductivity of graphene from the acoustic, out-of-plane modes which have been previously neglected. Empirical potentials used to model interactions in carbon-based materials are optimized to better describe the lattice dynamics of graphene-derived systems. These potentials are then used to generate the interatomic force constants needed to make calculations of the thermal conductivities of graphene and carbon nanotubes. Calculations of the thermal conductivities of single-walled carbon nanotubes and graphene for different temperatures and lengths are presented. The thermal conductivities of SWCNTs saturate in the diffusive regime when the effects of higher-order scattering processes are estimated and correctly reproduce the ballistic limit for short-length nanotubes at low temperatures. The effects of isotopic impurity scattering on the thermal conductivities of graphene and SWCNTs are explored. Isotopic impurities have little effect in the low (high) temperature regime where boundary (umklapp) scattering dominates the behavior of the thermal conductivities. In the intermediate temperature regime, modest reductions in the thermal conductivities, 15-20%, occur due to impurities. The thermal conductivities of a wide-range of SWCNTs are explored. The thermal conductivities of successively larger-diameter, one-dimensional nanotubes approach the thermal conductivity of two-dimensional graphene. / Thesis (PhD) — Boston College, 2010. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Heat dissipation is a critical bottleneck for microelectronic device performance and longevity. At micrometer and nanometer length scales heat carriers scatter at the boundaries of the material reducing its thermal conductivity. Additionally, thermal boundary conductance across dissimilar material interfaces becomes a dominant factor due to the increase in surface area relative to the volume of device layers. Therefore, techniques for monitoring spatially varying temperature profiles, and methods to improve thermal performance are critical to future device design and optimization. The first half of this thesis focused on frequency domain thermoreflectance (FDTR) to measure thermal transport in nanometer-thick polymer films and across an organic-inorganic interface. Hybrid structures of organic and inorganic materials are widely used in devices such as batteries, solar cells, transistors, and flexible electronics. The Langmuir-Blodgett (LB) technique was used to fabricate nanometer-thick polymer films ranging from 2 - 30 nm. FDTR was then used to experimentally determine the thermal boundary conductance between the polymer and solid substrates. The second half of the thesis focused on developing a fundamental understanding of thermal transport in wide-bandgap (WBG) materials, such as GaN, and ultrawide-bandgap (UWBG) materials, such as diamond, to improve thermal dissipation in power electronic devices. Improvements in WBG materials and device technologies have slowed as thermal properties limit their performance. UWBG materials can provide a dramatic leap in power electronics technologies while temporarily sidestepping the problems associated with their WBG cousins. However, for power electronic devices based on WBG- and UWBG-materials to reach their full potential the thermal dissipation issues in these hard-driven devices must be understood and solved. FDTR provides a comprehensive pathway towards fully understanding the physics governing phonon transport in WBG- and UWBG-based devices. By leveraging FDTR imaging and measuring samples as a function of temperature, defect concentration, and thickness, in conjunction with transport models, a well-founded understanding of the dominant thermal-carrier scattering mechanisms in these devices was achieved. With this knowledge we developed pathways for their mitigation.

Thesis advisor: David Broido / We present fundamental studies of phonon and electron transport in semiconductors. First principles density functional theory (DFT) is combined with exact numerical solutions of the Boltzmann transport equation (BTE) for phonons and electrons to calculate various transport coefficients. The approach is used to determine the lattice thermal conductivity of three hexagonal polytypes of silicon carbide. The calculated results show excellent agreement with recent experiments. Next, using the infinite orders T-matrix approach, we calculate the effect of various neutral and charged substitution defects on the thermal conductivity of boron arsenide. Finally, we present a general coupled electron-phonon BTEs scheme designed to capture the mutual drag of the two interacting systems. By combining first principles calculations of anharmonic phonon interactions with phenomenological models of electron-phonon interactions, we apply our implementation of the coupled BTEs to calculate the thermal conductivity, mobility, Seebeck and Peltier coefficients of n-doped gallium arsenide. The measured low temperature enhancement in the Seebeck coefficient is captured using the solution of the fully coupled electron-phonon BTEs, while the uncoupled electron BTE fails to do so. This work gives insights into the fundamental nature of charge and heat transport in semiconductors and advances predictive ab initio computational approaches. We discuss possible extensions of our work. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

This investigation is a theoretical study of the influence of defects of finite volume on the electrical conductivity in the quantum size effect regime. Correction terms to existing equations are derived, and a physical explanation of the results is given. Many macroscopic properties of films exhibit an oscillatory dependence on thickness when the thickness is comparable to the de Broglie wavelength of an electron at the Fermi surface. This behavior is called the quantum size effect. In very thin films, scattering from surfaces, phonons, and crystal defects plays an increasingly important role. In this investigation the influence of scattering centers (defects) in semimetal films on the electrical conductivity is explored by extending existing work to include scattering centers of finite range. The purpose of this study is to determine the overall change in the conductivity and the alteration of the amplitude of the oscillations. The Boltzmann transport equation is the starting point for the calculation. An equation for the vector mean free path is derived, and a solution is obtained by the iterative process. The relaxation approximation need not be made since the vector mean free path is determined. The sample is a thin slab that is infinite in two dimensions. The assumption is made that the electron wave function is zero at the walls of the sample. It is further assumed that there is a known number of randomly located defects within the slab. The noninteracting electrons are considered free except in the vicinity of the scattering centers. The defects are characterized by a potential that is constant within a small cube and zero outside of it. This approach allows the potential matrix elements to be evaluated by expanding in a power series. The electrical conductivity is calculated for three defect sizes, and a comparison is made to 3-function (infinitely small) scattering centers. An overall decrease in the conductivity is found in each case, and the absolute magnitude of the oscillations is decreased. The percentage of oscillation, however, is increased. The general conductivity decrease is attributed to the increase in the scattering range. The change in the amplitude of the oscillations is explained by analyzing the transition probabilities to available energy states at critical film thicknesses. The oscillations are found to be a result of transitions from states with large energies in the plane of the film to states with small energies in the plane of the film. The number of electrons occupying the various states is determined at critical film thicknesses, and a comparison with the conductivity equation shows excellent agreement.

The calculation of the electron–phonon coupling from first principles is computationally very challenging and remains mostly out of reach for systems with a large number of atoms. Semi-empirical methods, like density functional tight binding (DFTB), provide a framework for obtaining quantitative results at moderate computational costs. Herein, we present a new method based on the DFTB approach for computing electron–phonon couplings and relaxation times. It interfaces with PHONOPY for vibrational modes and DFTB+ to calculate transport properties. We derive the electron–phonon coupling within a non-orthogonal tight-binding framework and apply them to graphene as a test case.

This work presents the development and application of analytical, numerical, and experimental methods for the study of thermal and electrical transport in nanoscale systems, with special emphasis on those materials and phenomena which can be important in thermoelectric and semiconductor device applications. Analytical solutions to the Boltzmann transport equation (BTE) using the relaxation time approximation (RTA) are presented and used to study the thermal and electrical transport properties of indium antimonide (InSb), indium arsenide (InAs), bismuth telluride (Bi₂Te₃), and chromium disilicide (CrSi₂) nanowires. Experimental results for the thermal conductivity of single layer graphene supported by SiO₂ were analyzed using an RTA-based model and compared to a full quantum mechanical numerical BTE solution which does not rely on the RTA. The ability of these models to explain the measurement results as well as differences between the two approaches are discussed. Alternatively, numerical solutions to the BTE may be obtained statistically through Monte Carlo simulation for complex geometries which may prove intractable for analytical methods. Following this approach, phonon transport in silicon (Si) sawtooth nanowires was studied, revealing that thermal conductivity suppression below the diffuse surface limit is possible. The experimental investigation of energy transport in nanostructures typically involved the use of microfabricated devices or non-contact optical methods. In this work, two such approaches were analyzed to ascertain their thermal behavior and overall accuracy as well as areas for possible improvement. A Raman spectroscopy-based measurement design for investigating the thermal properties of suspended and supported graphene was examined analytically. The resulting analysis provided a means of determining from measurement results the thermal interface conductance, thermal contact resistance, and thermal conductivity of the suspended and supported graphene regions. Previously, microfabricated devices of several different designs have been used to experimentally measure the thermal transport characteristics of nanostructures such as carbon nanotubes, nanowires, and thin films. To ascertain the accuracy and limitations of various microdevice designs and their associated conduction analyses, finite element models were constructed using ANSYS and measurements of samples of known thermal conductance were simulated. It was found that designs with the sample suspended were generally more accurate than those for which the sample is supported on a bridge whose conductance is measured separately. The effects of radiation loss to the environment of certain device designs were also studied, demonstrating the need for radiation shielding to be at temperatures close to that of the device substrate in order to accurately calibrate the resistance thermometers. Using a suspended microdevice like those analyzed using finite element analysis, the thermal conductivities of individual bismuth (Bi) nanowires were measured. The results were correlated with the crystal structure and growth direction obtained by transmission electron microscopy on the same nanowires. Compared to bulk Bi in the same crystal direction, the thermal conductivity of a single-crystal Bi nanowires of 232 nm diameter was found to be 3 - 6 times smaller than bulk between 100 K and 300 K. For polycrystalline Bi nanowires of 74 nm to 255 nm diameter the thermal conductivity was reduced by a factor of 18 - 78 over the same temperature range. Comparable thermal conductivity values were measured for polycrystalline nanowires of varying diameters, suggesting a grain boundary scattering mean free path for all heat carriers in the range of 15 - 40 nm which is smaller than the nanowire diameters. An RTA-based transport model for both charge carriers and phonons was developed which explains the thermal conductivity suppression in the single-crystal nanowire by considering diffuse phonon-surface scattering, partially diffuse surface scattering of electrons and holes, and scattering of phonons and charge carriers by ionized impurities such as oxygen and carbon of a concentration on the order of 10¹⁹ cm⁻³. Using a similar experimental setup, the thermoelectric properties (Seebeck coefficient, electrical conductivity, and thermal conductivity) of higher manganese silicide (HMS) nanostructures were investigated. Bulk HMS is a passable high temperature thermoelectric material which possesses a complex crystal structure that could lead to very interesting and useful nanoscale transport properties. The thermal conductivities of HMS nanowires and nanoribbons were found to be reduced by 50 - 60 % compared to bulk values in the same crystal direction for both nanoribbons and nanowires. The measured Seebeck coefficient data was comparable or below that of bulk, suggesting unintentional doping of the samples either during growth or sample preparation. Difficulty in determining the amorphous oxide layer thickness for nanoribbons samples necessitated using the total, oxide-included cross section in the thermal and electrical conductivity calculation. This in turn led to the determined electrical conductivity values representing the lower bound on the actual electrical conductivity of the HMS core. From this approach, the measured electrical conductivity values were comparable or slightly below the lower end of bulk electrical conductivity values. This oxide thickness issue affects the determination of the HMS nanostructure thermoelectric figure of merit ZT as well, though the lower bound values obtained here were found to still be comparable to or slightly smaller than the expected bulk values in the same crystal direction. Analytical modeling also indicates higher doping than in bulk. Overall, HMS nanostructures appear to have the potential to demonstrate measurable size-induced ZT enhancement, especially if optimal doping and control over the crystallographic growth direction can be achieved. However, experimental methods to achieve reliable electrical contact to quality four-probe samples needs to be improved in order to fully investigate the thermoelectric potential of HMS nanostructures. / text

The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided. / text

Heat transport by phonons and magnons in crystals due to a local perturbation in temperature is described by the Boltzmann transport equation. In this project a one phonon one magnon system was studied in a one dimensional rod with reflective boundaries. Using the splitting algorithm the problem was reduced to a transport and collision term. The resulting stabilization time for a initial phonon and magnon distribution and the respective temperatures at equilibrium was calculated. This study shows how energy propagates in crysials and gives further understanding of how the coupling of phonons and magnons affect heat transport.

Temperature rise due to Joule heating of on-chip interconnects can severely affect performance and reliability of next generation microprocessors. Thermal predictions become difficult due to large number of features, and the impact of electron size effects on electrical and thermal transport. It is thus necessary to develop efficient numerical approaches, and accurate metal and dielectric thermal characterization techniques. In this research, analytical, numerical, and experimental techniques were developed to enable accurate and efficient predictions of interconnect temperature rise. A finite element based compact thermal model was developed to obtain temperature rise with fewer elements and acceptable accuracy. Temperature drop across the interconnect cross-section was ignored. The compact model performed better than standard finite element model in two and three-dimensional case studies, and the predictions for a real world structure agreed closely with experimentally measured temperature rise. A numerical solution was developed for electron transport based on the Boltzmann Transport Equation (BTE). This deterministic technique, based on the path integral solution of BTE within the relaxation time approximation, free electron model, and linear response, was applied to a constriction in a finite size thin metallic film. A correlation for effective conductance was obtained for different constriction sizes. The Atomic Force Microscope (AFM) based Scanning Joule Expansion Microscopy (SJEM) was used to develop a new technique to measure thermal conductivity of thin metallic films in the size effect regime. This technique does not require suspended metal structures, and thus preserves the original electron interface scattering characteristics. The thermal conductivities of 43 nm and 131 nm gold films were extracted to be 82 W/mK and 162 W/mK respectively. These measurements were close to Wiedemann-Franz Law predictions and are significantly smaller than the bulk value of 318 W/mK due to electron size effects. The technique can potentially be applied to interconnects in the sub-100 nm regime. A semi-analytical solution for the 3-omega method was derived to account for thermal conduction within the metallic heater. It is shown that significant errors can result when the previous solution is applied for anisotropic thermal conductivity measurements.

Electronics industry has been developing at a tremendous rate for last five decades and currently is one of the biggest industries in the world. The key to the rapid growth of electronics industry is innovation that made possible the constant scaling of transistors with reduced cost and improved performance. Scaling transistors were simpler at the beginning, but currently as the gate length of transistors has reached few nanometers, different short channel effects have emerged and power density of transistors has also increased drastically, which made further scaling much more challenging. To study electro-thermal transport in these reduced dimensionality devices, continuum models are no longer sufficient. In this work, the electrical and thermal transport properties have been modeled by solving Boltzmann Transport Equation (BTE) for electrons and phonons, respectively, using the Monte Carlo (MC) technique. To solve BTE for the phonons, a coupled Molecular Mechanics-Monte Carlo approach is employed where phonon band-structure is obtained using the atomistic modified Valence Force Field (VFF) model and is coupled with a Monte Carlo Phonon Transport kernel which solves the BTE for phonons. The phonon-phonon scattering is modeled in relaxation time approximation (RTA) using Holland’s formalism. Diffusive boundary scattering for phonons has been modeled using the Beckmann-Kirchhoff (B-K) surface roughness scattering model taking into account the effects of phonon wavelength, incident angles and degree of surface roughness. The effect of rough surface on longitudinal acoustic (LA) and transverse acoustic (TA) phonon branches has been studied with the help of the B-K model and it has been found that, at elevated temperatures, there is less backscattering to the LA branch due to rough surface. Effort has been made then to couple the developed phonon Monte Carlo transport simulator with an electron Monte Carlo transport simulator to study the origin and effects of self-heating in a nanoscale field-effect transistor (FET). In contrast to the widely used continuum model, where Fourier heat diffusion equation is usually solved to describe the thermal transport, the simulator developed in this dissertation treats both the electrons and the phonons at the particle level. Acoustic and intervalley g and f type electron-phonon scattering mechanisms are considered and the resulting local temperature modification has been used to bridge the electron and phonon transport paths. Phonon transport at the oxide-silicon interface has been modeled using the Diffuse Mismatch (DM) model, whereas, the phonons in the oxide have been described using the Debye model and temperature and frequency dependent relaxation time. The simulator is then benchmarked and used to study the electron-phonon transport processes in a FinFET device with a gate length of 18 nm, channel width of 4 nm, and a fin height of 8 nm. Preliminary results show that there can be a current degradation of as high as ~9.56% due to self-heating effect. Also, temperature in the entire channel region could rise due to self-heating. The maximum temperature rise in the channel region is found to be ~30K.