# Browsing by Subject "Boltzmann transport equation"

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Item Experimental and theoretical investigation of thermal and thermoelectric transport in nanostructures(2010-05) Moore, Arden Lot, 1982-; Shi, Li, Ph. D.; Ezekoye, Ofodike; Ferreira, Paulo; Howell, John; Tutuc, EmanuelShow more 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.Show more Item On study of deterministic conservative solvers for the nonlinear boltzmann and landau transport equations(2014-08) Zhang, Chenglong; Martínez Gamba, Irene, 1957-Show more 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.Show more Item Thermal transport in low-dimensional materials(2015-12) Marepalli, Prabhakar; Murthy, Jayathi; Shi, Li; Akinwande, Deji; Wang, Yaguo; Singh, DhruvShow more Recent years have witnessed a paradigm shift in the world of electronics. Researchers have not only continued to postpone the long dreaded end-of-Moore’s-law, but have also opened up a new world of possibilities with electronics. The future of electronics is widely anticipated to be dominated by wearable and implantable devices, the realization of which will be made possible by the discovery of new materials. Graphene and hexagonal boron nitride (hBN) are two such materials that have shown promising properties to make these devices possible. It has been shown that an energy bandgap can be opened in graphene by patterning it as a narrow ribbon, by applying an electric displacement field to a bilayer configuration, and by other means. The possibility of tuning the bandgap makes graphene an ideal channel material for future electronics. Similarly, hexagonal boron nitride (hBN) and its ribbon configurations have been shown to be excellent dielectric materials. In addition, the similarities in the atomic configurations of graphene and hBN allow them to conform extremely well to each other, achieving atomically smooth interfaces. Graphene devices on hBN substrates have been shown to have mobilities an order of magnitude larger than graphene devices fabricated on silicon dioxide. In addition to their outstanding electrical properties, graphene and hBN have been shown to have excellent thermal properties compared to their traditional counterparts (silicon and silicon dioxide, respectively). More specifically, these materials have been shown to have size dependent thermal properties which may be used to tune device performance. In this thesis, we study the thermal transport of three important classes of materials – graphene nanoribbons, hBN nanoribbons and graphene-hBN heterostructures using the phonon Boltzmann transport equation in a linearized framework. An exact solution of the Boltzmann transport equation is obtained ensuring that normal and umklapp phonon scattering processes are appropriately treated. In the first part of the thesis, we present a computational technique called method of automatic code differentiation to calculate sensitivities in nanoscale thermal transport simulations. Key phonon parameters like force constants, group velocities, the Gruneisen parameter, etc., which can be expressed as sensitivities or derivatives, are computed using this technique. The derivatives computed using this technique are exact and can be generalized to any order with minimal effort. This technique can be unintrusively integrated with existing first-principles simulation codes to obtain the sensitivities of parameters computed therein to chosen inputs. The next focus is to investigate the thermal properties of three main classes of materials – graphene nanoribbons, hBN nanoribbons,and graphene-hBN heterostructures. For nanoribbons, we consider ribbons of varying widths to investigate the transition of key thermal properties with width. The lattice structure of the ribbon structures considered is fully resolved. An efficient parallelization technique is developed to handle the large number of atoms in a unit cell. The thermal conductivity is obtained by an iterative solution of the linearized Boltzmann transport equation. For graphene and hBN ribbons, we find that the thermal conductivity increases with the ribbon width following a power-law trend. The rate of increase of thermal conductivity with width for hBN ribbons is found to be slower compared to graphene. Flexural phonons are found to contribute to the majority of heat conduction in both the materials. Frequency- and polarization-resolved transport is analyzed for ribbon of all widths. The thermal conductivity of single- and few-layer hexagonal boron nitride is also computed and compared with measured data. It is found that the thermal conductivity of hBN based nanostructures (single-layer, few-layer and ribbons) is around 6-8 times smaller than that for the corresponding graphene-based nanostructure. The effect of strain in both these materials is investigated. We find that the thermal conductivity of single-layer hBN is very sensitive to strain whereas graphene shows relatively less sensitivity for the same strains. Finally, thermal transport in graphene-hBN heterostructures is simulated. Two different structures are considered – single-layer graphene on an hBN substrate, and bilayer graphene on an hBN substrate. Substrates of different thickness are considered. Due to the weak interlayer coupling in these heterostructures, it is found that the phonon dispersion remains largely unchanged from the dispersions of the individual layers. The only difference in dispersion is noticed for flexural phonons, which are the only modes affected by interlayer coupling. The addition of an hBN layer underneath the graphene/bilayer graphene layer is found to drastically reduce the thermal conductivity of the heterostructures. This reduction is due to breakdown of the selection rule for flexural phonons which results in increased scattering channels for these phonons. The thermal conductivity gradually decreases, saturating to a bulk value with an increase in the number of hBN layers. The results presented in this thesis are expected to help guide the design of graphene/hBN based flexible electronics.Show more