Coupled Boltzmann Transport Equations Research Articles

Panoramic Mapping of Phonon Transport from Ultrafast Electron Diffraction and Scientific Machine Learning.

One central challenge in understanding phonon thermal transport is a lack of experimental tools to investigate frequency-resolved phonon transport. Although recent advances in computation lead to frequency-resolved information, it is hindered by unknown defects in bulk regions and at interfaces. Here, a framework that can uncover microscopic phonon transport information in heterostructures is presented, integrating state-of-the-art ultrafast electron diffraction (UED) with advanced scientific machine learning (SciML). Taking advantage of the dual temporal and reciprocal-space resolution in UED, and the ability of SciML to solve inverse problems involving coupled Boltzmann transport equations, the frequency-dependent interfacial transmittance and frequency-dependent relaxation times of the heterostructure from the diffraction patterns are reliably recovered. The framework is applied to experimental Au/Si UED data, and a transport pattern beyond the diffuse mismatch model is revealed, which further enables a direct reconstruction of real-space, real-time, frequency-resolved phonon dynamics across the interface. The work provides a new pathway to probe interfacial phonon transport mechanisms with unprecedenteddetails.

Coupled Boltzmann transport equations of heavy quarks and quarkonia in quark-gluon plasma

We develop a framework of coupled transport equations for open heavy flavor and quarkonium states, in order to describe their transport inside the quark-gluon plasma. Our framework is capable of studying simultaneously both open and hidden heavy flavor observables in heavy-ion collision experiments and can account for both, uncorrelated and correlated recombination. Our recombination implementation depends on real-time open heavy quark and antiquark distributions. We carry out consistency tests to show how the interplay among open heavy flavor transport, quarkonium dissociation and recombination drives the system to equilibrium. We then apply our framework to study bottomonium production in heavy-ion collisions. We include ϒ(1S), ϒ(2S), ϒ(3S), χb(1P) and χb(2P) in the framework and take feed-down contributions during the hadronic gas stage into account. Cold nuclear matter effects are included by using nuclear parton distribution functions for the initial primordial heavy flavor production. A calibrated 2 + 1 dimensional viscous hydrodynamics is used to describe the bulk QCD medium. We calculate both the nuclear modification factor RAA of all bottomonia states and the azimuthal angular anisotropy coefficient v2 of the ϒ(1S) state and find that our results agree reasonably with experimental measurements. Our calculations indicate that correlated cross-talk recombination is an important production mechanism of bottomonium in current heavy-ion experiments. The importance of correlated recombination can be tested experimentally by measuring the ratio of RAA(χb(1P)) and RAA(ϒ(2S)).

Nondiffusive electron transport in metals: A two-temperature Boltzmann transport equation analysis of thermoreflectance experiments

We develop a theoretical framework based on the electron-phonon coupled Boltzmann transport equations (BTEs) for the interpretation of nondiffusive thermal conductivity measurements in metals made via frequency domain thermoreflectance (FDTR). The thermal conductivity of a bulk gold crystal was measured over a temperature range of 23--304 K as a function of FDTR's laser spot size. Our interpretation of these measurements by a two-temperature heat diffusion model finds that the thermal conductivity is suppressed when the laser spot size is comparable to electron mean free paths. Using a simplified spherical geometry that enables analytical solutions, we compare the two-temperature diffusion model with the coupled BTEs to identify a thermal conductivity suppression function. We also conclude that over the timescales of our experiment electron-phonon nonequilibrium is negligible, but that the length scale of heat deposition by hot electrons critically influences our interpretation.

Quarkonium production in heavy ion collisions: coupled Boltzmann transport equations

We develop a set of coupled Boltzmann equations to describe the dynamical evolution of heavy quarks and quarkonia inside the quark-gluon plasma. The quarkonium dissociation and recombination terms are calculated from pNRQCD. Their interplay drives the system to a detailed balance. The heavy quark energy loss term is necessary for the system to reach kinematic thermalization. By coupling the transport equations with initial particles' momenta generated by Pythia and hydrodynamic medium evolutions, we can describe the RAA of ϒ family at both RHIC and LHC energies. The transverse momentum azimuthal anisotropy of ϒ(1S) in 2.76 TeV peripheral Pb-Pb collisions is also studied.

An Energy Transport Model Describing Electro-Thermal Transport in Silicon Carbide Semiconductors

ABSTRACTBy taking the moments of the coupled Boltzmann transport equations for electrons and phonons, an extended hydrodynamic model describing the electron transport in silicon carbide semiconductors has been obtained. This model is coupled with the heating of the crystal lattice. All the transport coefficients and the constitutive equations are determined by using the maximum entropy principle of extended thermodynamics. By using this model with a suitable limit, thermoelectric effects in silicon carbide devices are investigated.

Electro-thermal simulation based on coupled Boltzmann transport equations for electrons and phonons

To study the thermal effect in nano-transistors, a simulator solving self-consistently the Boltzmann transport equations for both electrons and phonons has been developed. It has been used to investigate the self-heating effects in a 20 nm-long double-gate MOSFET (Fig. 1). A Monte Carlo solver for electrons is coupled with a direct solver for the steady-state phonon transport. The latter is based on the relaxation time approximation. This method is particularly efficient to provide a deep insight of the out-of-equilibrium thermal dissipation occurring at the nanometer scale when the device length is smaller than the mean free path of both charge and thermal carriers. It allows us to evaluate accurately the phonon emission and absorption spectra in both real and energy spaces.

TH‐E‐BRE‐02: A Forward Scattering Approximation to Dose Calculation Using the Linear Boltzmann Transport Equation

Purpose:To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy.Methods:Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium.Results:Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm.Conclusion:Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not required, such as in certain optimization algorithms.

SU‐C‐500‐06: Development of a Hybrid Stochastic‐Deterministic Method for Dose Calculation in Radiotherapy

Purpose: To develop a hybrid stochastic‐deterministic method, COMET‐PE, for dose calculation in external beam photon radiotherapy. Methods: To calculate dose, COMET‐PE solves the coupled Boltzmann Transport Equations for photons and electrons. The method uses a deterministic iteration to compose response functions that are pre‐computed using Monte Carlo. Thus, COMET‐PE takes advantage of Monte Carlo physics without incurring the computational costs typically required for statistical convergence. Dose distributions are calculated for a heterogeneous benchmark problem using both COMET‐PE and DOSXYZnrc (Monte Carlo) methods. The benchmark consists of a CT‐based lung phantom, composed of air, lung, soft tissue, and bone, irradiated by a 2 cm × 2 cm photon field. The 6 MV source spectrum comes from Monte Carlo simulation of a Varian Clinac 2100. The COMET‐PE solution is computed at resolution of 1 mm (27,648,000 voxels) before being reduced to a resolution of 5 mm (221,184 voxels) for compatibility with the DOSXYZnrc reference solution. Results: The agreement between dose distributions calculated with COMET‐PE and Monte Carlo is excellent. Of voxels receiving greater than 10% of the maximum dose, 98.73% pass the 2% (point‐wise relative difference) or 2 mm (distance‐to‐agreement) criterion and 99.38% pass the 3% / 3 mm criterion. Localized discrepancies are observed at the beam corners; these are caused by the difficulty of using a continuous representation for the discontinuous primary fluence. Most of the failures, however, occur where the beam exits the phantom and where Monte Carlo uncertainties are the highest. The COMET‐PE calculation is over 10 times faster than the Monte Carlo reference solution. Conclusion: The COMET‐PE method calculates dose with accuracy comparable to Monte Carlo while using only a fraction of the time and providing a solution with orders of magnitude more detail.

Validation of a new grid-based Boltzmann equation solver for dose calculation in radiotherapy with photon beams

A new grid-based Boltzmann equation solver, Acuros™, was developed specifically for performing accurate and rapid radiotherapy dose calculations. In this study we benchmarked its performance against Monte Carlo for 6 and 18 MV photon beams in heterogeneous media. Acuros solves the coupled Boltzmann transport equations for neutral and charged particles on a locally adaptive Cartesian grid. The Acuros solver is an optimized rewrite of the general purpose Attila© software, and for comparable accuracy levels, it is roughly an order of magnitude faster than Attila. Comparisons were made between Monte Carlo (EGSnrc) and Acuros for 6 and 18 MV photon beams impinging on a slab phantom comprising tissue, bone and lung materials. To provide an accurate reference solution, Monte Carlo simulations were run to a tight statistical uncertainty (σ ≈ 0.1%) and fine resolution (1–2 mm). Acuros results were output on a 2 mm cubic voxel grid encompassing the entire phantom. Comparisons were also made for a breast treatment plan on an anthropomorphic phantom. For the slab phantom in regions where the dose exceeded 10% of the maximum dose, agreement between Acuros and Monte Carlo was within 2% of the local dose or 1 mm distance to agreement. For the breast case, agreement was within 2% of local dose or 2 mm distance to agreement in 99.9% of voxels where the dose exceeded 10% of the prescription dose. Elsewhere, in low dose regions, agreement for all cases was within 1% of the maximum dose. Since all Acuros calculations required less than 5 min on a dual-core two-processor workstation, it is efficient enough for routine clinical use. Additionally, since Acuros calculation times are only weakly dependent on the number of beams, Acuros may ideally be suited to arc therapies, where current clinical algorithms may incur long calculation times.

Boltzmann transport study of bulk and interfacial spin depolarization effects in spin valves

A theoretical model is proposed to analyze both bulk and interfacial spin depolarization effects on the magnetoresistance (MR) of nanoscale spin valves (SVs). The model is based on the spin coupled Boltzmann transport equations where the momentum spin relaxation arising from spin flip and non-spin-flip scattering are considered. In the boundary conditions we include the parameter q which denotes the interfacial spin flip probability, while bulk spin depolarization is characterized by the ratio r of spin flip to non-spin-flip scattering times. We consider a typical Fe∕Cr∕Fe pseudo-SV trilayer, and calculate the current for parallel and antiparallel alignments, to deduce the MR. A decreasing trend in MR is observed with an increase in either r or q, with the dependence on q being more pronounced. We also studied the combined effect of interfacial diffusive scattering (described by parameters Ns and D↑) and spin flip scattering. We found that although diffusive scattering generally results in an improvement in MR, this is more than offset by the MR suppression effect arising from spin flip scattering.

Coupled dynamics of electrons and phonons in metallic nanotubes: Current saturation from hot-phonon generation

We show that the self-consistent dynamics of both phonons and electrons is the necessary ingredient for the reliable description of the hot phonons generation during electron transport in metallic single-wall carbon nanotubes (SWNTs). We solve the coupled Boltzmann transport equations to determine in a consistent way the current vs. voltage (IV) curve and the phonon occupation in metallic SWNTs which are lying on a substrate. We find a good agreement with measured IV curves and we determine an optical phonon occupation which corresponds to an effective temperature of several thousands K (hot phonons), for the voltages typically used in experiments. We show that the high-bias resistivity strongly depends on the optical phonon thermalization time. This implies that a drastic improvement of metallic nanotubes performances can be achieved by increasing the coupling of the optical phonons with a thermalization source.

Modelling the transport of ionizing radiation using the finite element method

Radiation therapy treatment planning is based on the calculation of the absorbed dose in the patient domain. For exact dose calculations, the solution of three coupled Boltzmann transport equations (BTEs) is needed to cover the transport of photons, electrons and positrons. In many situations, however, two coupled systems for photons and electrons are enough. The use of numerical methods in finding the exact solution of the unknown particle fluxes is necessary. In the stationary case, the BTE has six variables, three spatial, two directional and one energy variable. In this paper, we describe an approach in which the finite element method (FEM) is used to solve the six-dimensional problem. For the coupled photon–electron system, the variational formulation and the existence and uniqueness of the solution are derived. We simulate the solution of two coupled BTEs describing the travelling of photons and electrons in two spatial dimensions. The results are compared to Monte Carlo calculations with good agreement.

Deterministic modeling of impact ionization with a random-k approximation and the multiband Boltzmann equation

We present here an approach for determining impact ionization coefficients for the spherical multiband model in silicon. Using random-k approximation, the impact ionization rate is determined to reflect the multiband density of states in silicon. To account for the actual density of states, we have solved four coupled Boltzmann transport equations by combining a generalized Legendre polynomial expansion method with numerical techniques using finite differences and sparse matrices. Calculated values for the impact ionization coefficients agree with experiments for electrons in silicon, while being obtained in significantly less CPU time than required by analogous Monte Carlo calculations. Different multiband transport parameter sets are also compared.

Ultrafast carrier relaxation and vertical-transport phenomena in semiconductor superlattices: A Monte Carlo analysis.

The ultrafast dynamics of photoexcited carriers in semiconductor superlattices is studied theoretically on the basis of a Monte Carlo solution of the coupled Boltzmann transport equations for electrons and holes. The approach allows a kinetic description of the relevant interaction mechanisms such as intra- miniband and interminiband carrier-phonon scattering processes. The energy relaxation of photoexcited carriers, as well as their vertical transport, is investigated in detail. The effects of the multiminiband nature of the superlattice spectrum on the energy relaxation process are discussed with particular emphasis on the presence of Bloch oscillations induced by an external electric field. The analysis is performed for different superlattice structures and excitation conditions. It shows the dominant role of carrier\char21{}polar-optical-phonon interaction in determining the nature of the carrier dynamics in the low-density limit. In particular, the miniband width, compared to the phonon energy, turns out to be a relevant quantity in predicting the existence of Bloch oscillations.

AN EFFICIENT SOLUTION OF THE MULTI‐BAND BOLTZMANN TRANSPORT EQUATION IN SILICON

A computationally very efficient solution of the multi‐band homogeneous Boltzmann transport equation (MB‐HBTE) for Silicon is presented. We used a Legendre polynomial (LP) expansion approach to solve coupled Boltzmann transport equations. To account for the multi‐band model, a Boltzmann transport equation is formulated for each of four conduction energy bands. The respective Boltzmann equations are coupled through the interband phonon scatterings. Analytical methods using Legendre polynomials are combined with numerical techniques using matrices to solve the coupled system of MB‐HBTE's. The multi‐band calculation requires approximately 8 GPU seconds on a SUN SPARC workstation. The efficiency of this method proves to be appropriate for use in computer‐aided design (CAD) tools for semiconductor devices, and results for the distribution function agree with Monte Carlo (MC) calculations.

On Electro- and Thermomigration and Thermoelectric Transport Coefficients in Metals

Abstract The system of coupled Boltzmann transport equations for electrons, phonons and impurities in metals is set up and solved with special interest for the effect of the cross-coupling terms. General expressions for the effective charge in electromigration and heat of transport in thermomigration are given. The cross-coupling terms in the transport equations lead to a number of corrections in the thermoelectric transport coefficients. Noteworthy is the result that the resistance by impurities depends now on the temperature. The investigation is restricted to high temperatures, isotropic dependence of the energies of electrons and phonons on wave number vector, and to normal scattering processes of phonons.

Electrical Resistivity of Metals at Low Temperatures

Abstract For the system of the coupled Boltzmann transport equations for electrons, phonons and impurities in metals the influence of the crosscoupling terms is investigated. It is shown that this results at low temperatures in a contribution to the electrical resistivity proportional to 1/√T. This is due to the fact that by phonon-impurity scattering the phonon-drag contribution is influenced by the presence of the impurities. A comparison with recently published experimental work is made.

Intersubband scattering effect on the mobility of a Si (100) inversion layer at low temperatures

Effects of the occupation of higher subbands on the mobility limited by Coulomb scattering and surface-roughness scattering are studied in an $n$-channel inversion layer on the silicon (100) surface. The first excited subband associated with the two valleys which give the ground subband is assumed to be populated first with increasing electron concentration. Coupled Boltzmann transport equations are solved and intersubband scattering is shown to be very important and to greatly reduce the mobility. Effects of the level broadening are also considered within a simple model of the surface-roughness scattering. A discontinuous drop of the mobility when the excited subband becomes occupied by electrons is almost completely smeared out by the broadening effect. The results explain experimental results of Hartstein et al. quite well, if parameters characterizing the surface roughness are chosen reasonably.

The role of the Boltzmann transport equation in radiation damage calculations

The role of the Boltzmann transport equation in radiation damage calculations

Der Einfluß des Phonon -Mitzieheffektes auf die thermoelektrischen Koeffizienten in Metallen / Influence of Phonon Drag on the Thermoelectric Coefficients in Metals

Abstract The system of coupled Boltzmann transport equations for electrons and phonons in a metal is considered in the absence of a magnetic field. Usually these equations are decoupled by “Bloch′s assumption”, that is the phonon gas is considered to stay in equilibrium with respect to the trans­ port equation for the electrons. It is shown that the coupled system of equations has finite solutions only if some other relaxation process than the electron-phonon interaction is present, for instance the phonon-phonon-interaction. With some simplifying assumptions the full system of equations is solved for high temperatures. The contributions of the phonon-drag to the thermoelectric transport coefficients are derived and discussed.