Boltzmann Transport Equation Model Research Articles

Towards energy filtering in Mg2X-based composites: Investigating local carrier concentration and band alignment via SEM/EDX and transient Seebeck microprobe analysis

A comprehensive understanding of multi-phase materials requires the characterization of transport properties at the micro/nano-scale. Optimized alignments of the conduction bands and valence bands at the interfaces of multi-phase materials can enhance the properties of thermoelectric materials by filtering out undesired charge carriers and is a promising path towards high performance thermoelectric devices. Here, a micro-scale characterization technique using transient Seebeck microprobe analysis, scanning electron microscopy with energy-dispersive X-ray spectroscopy, and electronic transport modelling based on the Boltzmann transport equation modeling is introduced to assess the band diagram and estimate the band offset in multi-phase materials. This characterization technique is applied to a composite of magnesium silicide-based materials. This material class is prone to form self-assembling nano-structured composites driven by a miscibility gap in the solid solutions series. Our analysis reveals changes in carrier concentration upon composite formation and considerable band offsets for both conduction and valence band when synthesizing nominally undoped composites. Employing Bi as dopant we show that Bi exhibits a preference for the Sn-rich phase, changing the carrier concentration differently in the Sn-rich matrix phase compared to the Si-rich secondary phase, effectively altering the band alignment of the composite. This demonstrates that our approach can be utilized to measure and manipulate the band offset within the composite to achieve optimal thermoelectric performance.

A combined theoretical-experimental investigation of thermal transport in low-dose irradiated thorium dioxide

During reactor operation, nuclear fuels are subject to extreme temperature and irradiation conditions which can significantly degrade the fuel's thermal transport properties. The reduction in thermal conductivity of the fuel as a result of irradiation-induced lattice defects is arguably the most important fuel performance metric in regard to reactor efficiency and safety. Because thorium dioxide (ThO2) is suitable as a model system for more complex materials such as UO2 and its mixed oxides, we present a theoretical investigation of thermal conductivity reduction seen in defect-bearing thorium dioxide and compare directly to experimental measurements. Phonon-mediated thermal transport of the fuel is modeled by a solution to the Boltzmann transport equation (BTE) for phonons. A cluster dynamics (CD) model for lattice defect evolution during irradiation predicts defect densities which are used as input to the BTE for modeling phonon-defect scatterings. Phonon scatterings by lattice defects include those from point defects and vacancy clusters and interstitial clusters of various sizes. The CD model is benchmarked against structural defect characterization of irradiated thorium dioxide using electron microscopy. Thermal conductivity predicted by the BTE model is compared to measured values for irradiated thorium dioxide specimens below room temperature to isolate effects of phonon-defect scattering from intrinsic 3-phonon processes, which dominate at higher temperatures. The computed conductivity values are in partial agreement at temperatures close to room temperature while slight deviations are observed at the lowest measured temperatures, suggesting that implemented phonon-defect scattering cross-section expressions may not be adequate for low temperatures. The presented work provides a necessary investigation of the influence of irradiation induced defects on fuel performance and represents a first step toward a full characterization of phonon mediated thermal transport in irradiated materials with complex defect microstructure.

Modeling ballistic phonon transport from a cylindrical electron beam heat source

Recent electron microscopy experiments have used focused electron beams as nanoscale heat sources or thermometers to enable high spatial resolution studies of heat transfer in nanostructures. When the electron beam radius is smaller than the heat carrier mean free path, Fourier’s law will underpredict the temperature rise due to electron beam-induced heating, motivating the development of subcontinuum models to interpret thermal electron microscopy measurements. Here, electron beam-induced heating of nonmetallic samples is modeled by applying a recently developed general solution of the governing Boltzmann transport equation (BTE) under the relaxation time approximation. The analytical BTE solution describes thermal phonon transport from a time-periodically heated cylindrical region in a homogeneous infinite medium. The BTE results show that ballistic phonon effects in this radial heat spreading scenario are more conveniently represented using a ballistic thermal resistance rather than an effective thermal conductivity. Calculations of this ballistic resistance for three semiconductors (Si, GaAs, and 3C-SiC) show that ballistic effects dominate the total thermal resistance to radial heat flow for typical STEM or SEM beam radii (<10 nm), indicating that the ballistic resistance could potentially be measured using thin-film electron beam heating experiments. However, combining the BTE solution with recent calorimetric measurements shows that the magnitude of the temperature rise remains negligibly small (<1 K) under typical electron microscopy conditions, even when considering these ballistic effects. These BTE modeling results can be used to quantify electron beam-induced heating or to design experiments probing ballistic phonon transport using electron beam heat sources.

Coupled electron-phonon transport and heat transfer pathways in graphene nanostructures

A hybrid Boltzmann transport equation model coupled with a two-temperature model is developed to simulate the heat dissipation in single layer graphene (SLG) suspended or supported on SiO2 and with/without Cu contacts. This hybrid model includes the phonon transport in SLG, electron-phonon interaction (EPI) in SLG, and phonon/electron transmission at the interface between SLG and its surroundings. The EPI strength between electrons and various phonon modes is obtained at different temperatures using first-principle calculations. Simulations are performed for SLG with different graphene lengths, contact configurations and heating powers. For the SLG suspended on SiO2, adding Cu contact at ends can remarkably improve the heat dissipation and lower the maximum temperature in SLG, especially at small lengths of SLG. The SLG/Cu interface can still be the major heat dissipation pathway when its contact area is similar with that of SLG/SiO2 interface, which can be attributed to the higher thermal conductivity of Cu. For the SLG fully supported on SiO2, adding Cu contact can effectively lower the temperature in SLG of small length, but the dominating heat dissipation pathway will be shifted from SLG/Cu interface to SLG/SiO2 interface as the SLG length increases.

Solving Nongray Boltzmann Transport Equation in Gallium Nitride

Several studies have validated that diffusive Fourier model is inadequate to model thermal transport at submicron length scales. Hence, Boltzmann transport equation (BTE) is being utilized to improve thermal predictions in electronic devices, where ballistic effects dominate. In this work, we investigated the steady-state thermal transport in a gallium nitride (GaN) film using the BTE. The phonon properties of GaN for BTE simulations are calculated from first principles—density functional theory (DFT). Despite parallelization, solving the BTE is quite expensive and requires significant computational resources. Here, we propose two methods to accelerate the process of solving the BTE without significant loss of accuracy in temperature prediction. The first one is to use the Fourier model away from the hot-spot in the device where ballistic effects can be neglected and then couple it with a BTE model for the region close to hot-spot. The second method is to accelerate the BTE model itself by using an adaptive model which is faster to solve as BTE for phonon modes with low Knudsen number is replaced with a Fourier like equation. Both these methods involve choosing a cutoff parameter based on the phonon mean free path (mfp). For a GaN-based device considered in the present work, the first method decreases the computational time by about 70%, whereas the adaptive method reduces it by 60% compared to the case where full BTE is solved across the entire domain. Using both the methods together reduces the overall computational time by more than 85%. The methods proposed here are general and can be used for any material. These approaches are quite valuable for multiscale thermal modeling in solving device level problems at a faster pace without a significant loss of accuracy.

A Monte Carlo Study on the Effect of Energy Barriers on the Thermoelectric Properties of Si

Abstract Energy filtering by energy barriers has been proposed to interpret observations on large thermoelectric power factor (TPF) enhancement in highly doped nanocrystalline Si (nc-Si). Previous Boltzmann transport equation (BTE) modeling indicated that high TPFs could be explained as the result of the presence of energy barriers at the grain boundaries, the high Fermi energy due to the high doping level, and the formation of a low thermal conductivity second phase. To test the assumptions of the BTE modeling and provide more realistic simulations, we have performed Monte Carlo (MC) simulations on the transport properties of composite nc-Si structures. Here, we report on (i) the effect of an energy barrier, and (ii) the effect of multiple barriers on the conductivity and the Seebeck coefficient. In short structures, a TPF enhancement was found and it has been attributed to energy filtering by the energy barrier. The MC indicated that the TE performance can be improved by multiple barriers in close separation. It has been shown that TPF enhancement is possible even when the condition for thermal conductivity non-uniformity across the composite structure is not-fulfilled.

Solution of the Bethe–Salpeter equation in a nondiffusive random medium having large scatterers

We present a formalism for solving the scalar Bethe–Salpeter equation (BSE) in the nondiffusive regime under the ladder approximation and for an infinite randomly scattering medium having scatterers of size on the order of or larger than the wavelength. We compare the information content in a wave transport model (the BSE) with that in energy-based transport, the Boltzmann transport equation (BTE), in the spatial frequency domain. Our results suggest that when absorption dominates scatter, the intensity Green’s function from a BTE model is similar to the field correlation Green’s function from a BSE solution. When scatter dominates loss, there are significant differences between the BTE and BSE representations, and the BTE solutions appear to be smoothed versions of those from the BSE. Therefore, field correlation measurements, perhaps extracted from intensity correlations over frequency and space, offer significantly more information than a mean-intensity measurement in the weakly scattering and nondiffusive regime. Our work provides a mathematical framework for electric field correlation-based imaging methods based on the BSE that hold promise in, for example, near-surface tissue imaging.

Multiscale thermal device modeling using diffusion in the Boltzmann Transport Equation

Thermal modeling of some nano-scale devices requires attention to multiple length scales and physical phenomena, ranging from continuum level heat diffusion to atomic-scale interactions and phonon confinement. At the nanometer-scale, thermal phenomena such as ballistic phonon transport may be important. The present paper uses a multiscale thermal modeling approach including diffusion in the Boltzmann Transport Equation (BTE) to study heat transport in several applications like Silicon on Insulator transistors, Si thin films with heat pulse, a Si thin film sandwiched between two bulk layers of Silicon dioxide as in solar cells. Analytical solutions of both a multilayered Fourier model and a simple gray BTE with and without a diffusion term in the steady state are computed. Thermal modeling and analysis of some of these applications is performed using the COMSOL Multiphysics finite element software package. The BTE model with and without diffusion for multilayers is discussed for interface conditions involving reflectance and transmittance of phonons. The results obtained from the models are compared to existing published results for Si/Si, Si/Diamond layers and are found to be in good agreement.

Optical photon transport in powdered‐phosphor scintillators. Part 1. Multiple‐scattering and validity of the Boltzmann transport equation

In Part 1 of this two-part work, predictions for light transport in powdered-phosphor screens are made, based on three distinct approaches. Predictions of geometrical optics-based ray tracing through an explicit microscopic model (EMM) for screen structure are compared to a Monte Carlo program based on the Boltzmann transport equation (BTE) and Swank's diffusion equation solution. The purpose is to: (I) highlight the additional assumptions of the BTE Monte Carlo method and Swank's model (both previously used in the literature) with respect to the EMM approach; (II) demonstrate the equivalences of the approaches under well-defined conditions and; (III) identify the onset and severity of any discrepancies between the models. A package of computer code (called phsphr) is supplied which can be used to reproduce the BTE Monte Carlo results presented in this work. The EMM geometrical optics ray-tracing model is implemented for hypothesized microstructures of phosphor grains in a binder. The BTE model is implemented as a Monte Carlo program with transport parameters, derived from geometrical optics, as inputs. The analytical solution of Swank to the diffusion equation is compared to the EMM and BTE predictions. Absorbed fractions and MTFs are calculated for a range of binder-to-phosphor relative refractive indices (n = 1.1-5.0), screen thicknesses (t = 50-200 μm), and packing fill factors (pf = 0.04-0.54). Disagreement between the BTE and EMM approaches increased with n and pf. For the largest relative refractive index (n = 5) and highest packing fill (pf = 0.5), the BTE model underestimated the absorbed fraction and MTF50, by up to 40% and 20%, respectively. However, for relative refractive indices typical of real phosphor screens (n ≤ 2), such as Gd2O2S:Tb, the BTE and EMM predictions agreed well at all simulated packing densities. In addition, Swank's model agreed closely with the BTE predictions when the screen was thick enough to be considered turbid. Although some assumptions of the BTE are violated in realistic powdered-phosphor screens, these appear to lead to negligible effects in the modeling of optical transport for typical phosphor and binder refractive indices. Therefore it is reasonable to use Monte Carlo codes based on the BTE to treat this problem. Furthermore, Swank's diffusion equation solution is an adequate approximation if a turbidity condition, presented here, is satisfied.

Thermal conductivity modeling of periodic porous silicon with aligned cylindrical pores

We present a frequency-dependent phonon Boltzmann transport equation (BTE) solver to study phonon transport in arbitrary geometries. For composite and porous structures, most simulations adopted either gray-medium approximation or geometric simplification in phonon BTE model. To show the importance of considering the frequency-dependent phonon transport, transverse thermal transport in periodic porous silicon (PS) with aligned square-cylindrical pores is investigated by the present frequency-dependent phonon BTE solver and gray-medium phonon BTE solver. It is found that phonon size effect is underestimated by adopting the gray-medium approximation in sub-micron scale. To demonstrate geometry effect, the frequency-dependent phonon BTE solver is applied to study transverse thermal transport in the PS with square-cylindrical and circular-cylindrical pores for various characteristic sizes and porosities. The pore shape is found to make great difference to the thermal conductivity of the PS when the characteristic size is decreased or the porosity is increased. Our results indicate the importance of considering the frequency dependence of phonon transport as well as the exact geometry of material structure in the analysis of micro- and nanostructured materials.

Phonon Transport Modeling Using Boltzmann Transport Equation With Anisotropic Relaxation Times

A sub-micron thermal transport model based on the phonon Boltzmann transport equation (BTE) is developed using anisotropic relaxation times. A previously-published model, the full-scattering model, developed by Wang, directly computes three-phonon scattering interactions by enforcing energy and momentum conservation. However, it is computationally very expensive because it requires the evaluation of millions of scattering interactions during the iterative numerical solution procedure. The anisotropic relaxation time model employs a single-mode relaxation time, but the relaxation time is derived from detailed consideration of three-phonon interactions satisfying conservation rules, and is a function of wave vector. The resulting model is significantly less expensive than the full-scattering model, but incorporates directional and dispersion behavior. A critical issue in the model development is the role of three-phonon normal (N) scattering processes. Following Callaway, the overall relaxation rate is modified to include the shift in the phonon distribution function due to N processes. The relaxation times so obtained are compared with the data extracted from equilibrium molecular dynamics simulations by Henry and Chen. The anisotropic relaxation time phonon BTE model is validated by comparing the predicted thermal conductivities of bulk silicon and silicon thin films with experimental measurements. The model is then used for simulating thermal transport in a silicon metal-oxide-semiconductor field effect transistor (MOSFET) and leads to results close to the full-scattering model, but uses much less computation time.

Influence of Interface Arrangement on Phonon Heat Transfer in Nanocomposites

The influence of interface arrangement in two- and three-dimensional silicon–germanium nanostructures on heat transfer is investigated by phonon Boltzmann transport equation model. For all nanocomposites studied here, the thermal conductivity decreases monotonically with increasing interface density and decreases monotonically with decreasing characteristic size. The interface density plays an important role to correlate the size effect of phonon transport. Another factor called structure efficiency, which taking into account both the structure dimensions and the interface arrangement, is also found to affect significantly the thermal conductivities of the nanocomposites. For the structure efficiency of a composite structure, it is found that the superlattice structure has the best structure efficiency in transverse transport and the compacted-wire structure has the best structure efficiency in longitudinal transport. High-dimensional (wire and particle) structures contain large interface density which is beneficial for reducing thermal conductivity. At the same characteristic size, the compacted-particle structure can yield the lowest thermal conductivity in transverse transport and the compacted-wire structure can yield the lowest thermal conductivity in longitudinal transport. The obtained results may provide essential information for the development of bulk-nanostructured thermoelectric devices.

Influence of Interface Arrangement on Phonon Heat Transfer in Nanocomposites

The influence of interface arrangement in two- and three-dimensional silicon–germanium nanostructures on heat transfer is investigated by phonon Boltzmann transport equation model. For all nanocomposites studied here, the thermal conductivity decreases monotonically with increasing interface density and decreases monotonically with decreasing characteristic size. The interface density plays an important role to correlate the size effect of phonon transport. Another factor called structure efficiency, which taking into account both the structure dimensions and the interface arrangement, is also found to affect significantly the thermal conductivities of the nanocomposites. For the structure efficiency of a composite structure, it is found that the superlattice structure has the best structure efficiency in transverse transport and the compacted-wire structure has the best structure efficiency in longitudinal transport. High-dimensional (wire and particle) structures contain large interface density which is beneficial for reducing thermal conductivity. At the same characteristic size, the compacted-particle structure can yield the lowest thermal conductivity in transverse transport and the compacted-wire structure can yield the lowest thermal conductivity in longitudinal transport. The obtained results may provide essential information for the development of bulk-nanostructured thermoelectric devices.

Parallel Computation of the Phonon Boltzmann Transport Equation

The performance of a parallel implementation of the phonon Boltzmann transport equation (BTE) is tested. Two three-phonon scattering models, the full-scattering model and the anisotropic relaxation-time model, are considered. The finite-volume method is used to solve the BTE numerically. Two strategies are explored: spatial domain decomposition and phonon band decomposition. The parallel computations are validated. Overall parallel performance, as measured by speed-up, is presented for the two BTE models using the two decomposition strategies on two architectures: the IBM Power 4+, and a cluster of 8-core Dell 1950 systems with Gigabit Ethernet and Infiniband.

Spin Accumulation From the Spin Hall Effect Studied Using the Effective Mean-Free-Path Model

This paper investigates following Zhang's procedure the study the spin accumulation resulted from the spin Hall effect. The procedure is that firstly the electron distribution is in the absence of the spin-orbital coupling and then the spin-orbit coupling by an anomalous term in current is included. Zhang started with the spin-diffusion equation derived from the Boltzmann transport equation, but this paper begins with the effective mean-free-path model equivalent to the Boltzmann transport equation model. In the calculations, the scattering from the boundary effect is considered, the spin-flip scattering is neglected, and the assumption is unnecessary that the anomalous Hall conductivity is much smaller than the Drude conductivity.

Comparison of Different Phonon Transport Models for Predicting Heat Conduction in Silicon-on-Insulator Transistors

The problem of self-heating in microelectronic devices has begun to emerge as a bottleneck to device performance. Published models for phonon transport in microelectronics have used a gray Boltzmann transport equation (BTE) and do not account adequately for phonon dispersion or polarization. In this study, the problem of a hot spot in a submicron silicon-on-insulator transistor is addressed. A model based on the BTE incorporating full phonon dispersion effects is used. A structured finite volume approach is used to solve the BTE. The results from the full phonon dispersion model are compared to those obtained using a Fourier diffusion model. Comparisons are also made to previously published BTE models employing gray and semi-gray approximations. Significant differences are found in the maximum hot spot temperature predicted by the different models. Fourier diffusion underpredicts the hot spot temperature by as much as 350% with respect to predictions from the full phonon dispersion model. For the full phonon dispersion model, the longitudinal acoustic modes are found to carry a majority of the energy flux. The importance of accounting for phonon dispersion and polarization effects is clearly demonstrated.