Linearized Boltzmann Transport Equation Research Articles

First-principles method for electron-phonon coupling and electron mobility: Applications to two-dimensional materials

We present density functional theory calculations of the phonon-limited mobility in n-type monolayer graphene, silicene and MoS$_2$. The material properties, including the electron-phonon interaction, are calculated from first-principles. We provide a detailed description of the normalized full-band relaxation time approximation for the linearized Boltzmann transport equation (BTE) that includes inelastic scattering processes. The bulk electron-phonon coupling is evaluated by a supercell method. The method employed is fully numerical and does therefore not require a semi-analytic treatment of part of the problem and, importantly, it keeps the anisotropy information stored in the coupling as well as the band structure. In addition, we perform calculations of the low-field mobility and its dependence on carrier density and temperature to obtain a better understanding of transport in graphene, silicene and monolayer MoS$_2$. Unlike graphene, the carriers in silicene show strong interaction with the out-of-plane modes. We find that graphene has more than an order of magnitude higher mobility compared to silicene. For MoS$_2$, we obtain several orders of magnitude lower mobilities in agreement with other recent theoretical results. The simulations illustrate the predictive capabilities of the newly implemented BTE solver applied in simulation tools based on first-principles and localized basis sets.

Dosimetric Verification around High-density Materials for External Beam Radiotherapy.

It is generally known that the dose distribution around the high-density materials is not accurate with commercially available radiation treatment planning systems (RTPS). Recently, Acuros XB (AXB) has been clinically available for dose calculation algorithm. The AXB is based on the linear Boltzmann transport equation - the governing equation - that describes the distribution of radiation particles resulting from their interactions with matter. The purpose of this study was to evaluate the dose calculation accuracy around high-density materials for AXB under three X-rays energy on the basis of measured values with EBT3 and compare AXB with various dose calculation algorithms (AAA, XVMC) in RTPS and Monte Carlo. First, two different metals, including titanium and stainless steel, were inserted at the center of a water-equivalent phantom, and the depth dose was measured with EBT3. Next, after a phantom which reproduced the geometry of measurement was virtually created in RTPS, dose distributions were calculated with three commercially available algorithms (AXB, AAA, and XVMC) and MC. The calculated doses were then compared with the measured ones. As a result, compared to other algorithms, it was found that the dose calculation accuracy of AXB at the exit side of high-density materials was comparable to that of MC and measured value with EBT3. However, note that AXB underestimated the dose up to approximately 30% at the plane of incidence because it cannot exactly estimate the impact of the backscatter.

Thermal Properties of β-Ga2O3 from First Principles

ABSTRACTThe thermal conductivity, bulk modulus, thermal expansion and heat capacity for bulk β-Ga2O3 are calculated from lattice dynamics using both a quasi-harmonic approximation and an anharmonic force-constant approach involving a solution of the linearized Boltzmann transport equation. The role of anharmonicity in β-Ga2O3 is determined to be small, which leads to the weak volume dependence of the calculated thermal conductivity. The negligible anharmonic contribution to the overall thermal conductivity is consistent with both thermal expansion measurements and also with comparisons between the quasi-harmonic and anharmonic methods. Phonon-mode-dependent Grüneisen parameters are found to be weakly dependent on temperature. Negative values of the mode Grüneisen parameters are found for certain low energy optical modes, but their net effect on the overall thermal expansion is insignificant. Bulk modulus as well as heat capacity are also given and found to be in agreement with experimental results.

Discontinuous finite element space-angle treatment of the first order linear Boltzmann transport equation with magnetic fields: Application to MRI-guided radiotherapy.

The advent of magnetic resonance imaging (MRI) guided radiotherapy systems demands the incorporation of the magnetic field into dose calculation algorithms of treatment planning systems. This is due to the fact that the Lorentz force of the magnetic field perturbs the path of the relativistic electrons, hence altering the dose deposited by them. Building on the previous work, the authors have developed a discontinuous finite element space-angle treatment of the linear Boltzmann transport equation to accurately account for the effects of magnetic fields on radiotherapy doses. The authors present a detailed description of their new formalism and compare its accuracy to geant4 Monte Carlo calculations for magnetic fields parallel and perpendicular to the radiation beam at field strengths of 0.5 and 3 T for an inhomogeneous 3D slab geometry phantom comprising water, bone, and air or lung. The accuracy of the authors' new formalism was determined using a gamma analysis with a 2%/2 mm criterion. Greater than 98.9% of all points analyzed passed the 2%/2 mm gamma criterion for the field strengths and orientations tested. The authors have benchmarked their new formalism against Monte Carlo in a challenging radiation transport problem with a high density material (bone) directly adjacent to a very low density material (dry air at STP) where the effects of the magnetic field dominate collisions. A discontinuous finite element space-angle approach has been proven to be an accurate method for solving the linear Boltzmann transport equation with magnetic fields for cases relevant to MRI guided radiotherapy. The authors have validated the accuracy of this novel technique against geant4, even in cases of strong magnetic field strengths and low density air.

Strain-Induced Modulation of Electron Mobility in Single-Layer Transition Metal Dichalcogenides MX2 (<inline-formula> <tex-math notation="LaTeX">$M = {\rm Mo}$ </tex-math> </inline-formula>, W; <inline-formula> <tex-math notation="LaTeX">$X = {\rm S}$ </tex-math> </inline-formula>, Se)

In this paper, the effect of biaxial strain on the mobility of single-layer transition metal dichalcogenides (MoS2, MoSe2, WS2, and WSe2) is investigated by accounting for the scattering from intrinsic phonon modes, remote phonons, and charged impurities. $Ab~initio$ simulations are employed to study a strain-induced effect on the electronic bandstructure, and the linearized Boltzmann transport equation is used to evaluate the low-field mobility. The results indicate that tensile strain increases the mobility. In particular, a significant increase in the mobility of single-layer MoSe2 andWSe2 with a relatively small tensile strain is observed. Under a compressive strain, however, the mobility exhibits a nonmonotonic behavior. With a relatively small compressive strain, the mobility decreases and then it partially recovers with a further increase in the compressive strain.

Strain induced mobility modulation in single-layer MoS2

In this paper the effect of biaxial and uniaxial strain on the mobility of single-layer MoS2 for temperatures T > 100 K is investigated. Scattering from intrinsic phonon modes, remote phonon and charged impurities are considered along with static screening. Ab initio simulations are utilized to investigate the strain induced effects on the electronic bandstructure and the linearized Boltzmann transport equation is used to evaluate the low-field mobility under various strain conditions. The results indicate that the mobility increases with tensile biaxial and tensile uniaxial strain along the armchair direction. Under compressive strain, however, the mobility exhibits a non-monotonic behavior when the strain magnitude is varied. In particular, with a relatively small compressive strain of 1% the mobility is reduced by about a factor of two compared to the unstrained condition, but with a larger compressive strain the mobility partly recovers such a degradation.

Adjoint-based deviational Monte Carlo methods for phonon transport calculations

In the field of linear transport, adjoint formulations exploit linearity to derive powerful reciprocity relations between a variety of quantities of interest. In this paper, we develop an adjoint formulation of the linearized Boltzmann transport equation for phonon transport. We use this formulation for accelerating deviational Monte Carlo simulations of complex, multiscale problems. Benefits include significant computational savings via direct variance reduction, or by enabling formulations which allow more efficient use of computational resources, such as formulations which provide high resolution in a particular phase-space dimension (e.g., spectral). We show that the proposed adjoint-based methods are particularly well suited to problems involving a wide range of length scales (e.g., nanometers to hundreds of microns) and lead to computational methods that can calculate quantities of interest with a cost that is independent of the system characteristic length scale, thus removing the traditional stiffness of kinetic descriptions. Applications to problems of current interest, such as simulation of transient thermoreflectance experiments or spectrally resolved calculation of the effective thermal conductivity of nanostructured materials, are presented and discussed in detail.

Ab initioelectronic transport model with explicit solution to the linearized Boltzmann transport equation

Accurate models of carrier transport are essential for describing the electronic properties of semiconductor materials. To the best of our knowledge, the current models following the framework of the Boltzmann transport equation (BTE) either rely heavily on experimental data (i.e., semi-empirical), or utilize simplifying assumptions, such as the constant relaxation time approximation (BTE-cRTA). While these models offer valuable physical insights and accurate calculations of transport properties in some cases, they often lack sufficient accuracy -- particularly in capturing the correct trends with temperature and carrier concentration. We present here a general transport model for calculating low-field electrical drift mobility and Seebeck coefficient of n-type semiconductors, by explicitly considering all relevant physical phenomena (i.e. elastic and inelastic scattering mechanisms). We first rewrite expressions for the rates of elastic scattering mechanisms, in terms of ab initio properties, such as the band structure, density of states, and polar optical phonon frequency. We then solve the linear BTE to obtain the perturbation to the electron distribution -- resulting from the dominant scattering mechanisms -- and use this to calculate the overall mobility and Seebeck coefficient. Using our model, we accurately calculate electrical transport properties of the compound n-type semiconductors, GaAs and InN, over various ranges of temperature and carrier concentration. Our fully predictive model provides high accuracy when compared to experimental measurements on both GaAs and InN, and vastly outperforms both semi-empirical models and the BTE-cRTA. Therefore, we assert that this approach represents a first step towards a fully ab initio carrier transport model that is valid in all compound semiconductors.

SU-E-T-270: Diffusion Synthetic Acceleration for Linear Boltzmann Transport Equation

Purpose: Linear Boltzmann transport equation (LBTE) is as accurate as the Monte Carlo method (MC) for dose calculation in photon/particle therapy (LBTE is a deterministic and Eulerian formulation and MC is a statistical and Lagrangian description). An advantage of LBTE is that numerous acceleration techniques can be utilized for acceleration. This work is to explore the acceleration of LBTE via diffusion synthetic acceleration (DSA). Methods: For simplicity, two-dimensional, steady-state, and within-group LBTE is considered with two angular dimensions and two spatial dimensions. The discrete ordinate method is developed for solving this integro-differential equation. The angular variables are discretized using a level-symmetric quadrature set on the unit sphere. The spatial variables are discretized on the structured grid based on the diamond scheme. The source-iteration method (SI) is used to solve the discretized system.Since SI is slow in optically thick and highly scattering regime. DSA is developed to accelerate SI. The motivation for DSA is that diffusion equation (DE) is a good approximation of LBTE in the above regime. However, DE is much cheaper than LBTE computationally since DE only involves spatial variables. Thus, in each DSA iteration, DSA adds to the SI step a computationally-negligible DE step, i.e., to first solve DE with the SI residual as source term, and then compensate the SI solution with DE solution. Results: DSA was benchmarked and compared with SI. The difference between two methods was within 0.12% which verifies the accuracy of DSA, while DSA demonstrated the great advantage in speed, e.g., the reduction of iteration number to 6% and 4% respectively for cases with 100 and 1,000 scattering-absorption ratio that commonly occur in clinical dose calculation. Conclusion: DSA has been developed as one of many possible means for accelerating the numerical solver of LBTE for dose calculation. The authors were partially supported by the NSFC (#11405105), the 973 Program (#2015CB856000) and the Shanghai Pujiang Talent Program (#14PJ1404500).

TU-F-CAMPUS-I-02: Validation of a CT X-Ray Source Characterization Technique for Dose Computation Using An Anthropomorphic Thorax Phantom

Purpose: To experimentally validate a rotational kV x-ray source characterization technique by computing CT dose in an anthropomorphic thorax phantom using an in-house dose computation algorithm (kVDoseCalc). Methods: The lateral variation in incident energy spectra of a GE Optima big bore CT scanner was found by measuring the HVL along the internal, full bow-tie filter axis. The HVL and kVp were used to generate the x-ray spectra using Spektr software, while beam fluence was derived by dividing the integral product of the spectra and in-air mass-energy absorption coefficients by in-air dose measurements along the bow-tie filter axis. Beams produced by the GE Optima scanner were modeled at 80 and 140 kVp tube settings. kVDoseCalc calculates dose by solving the linear Boltzmann transport equation using a combination of deterministic and stochastic methods. Relative doses in an anthropomorphic thorax phantom (E2E SBRT Phantom) irradiated by the GE Optima scanner were measured using a (0.015 cc) PTW Freiburg ionization chamber, and compared to computations from kVDoseCalc. Results: The agreement in relative dose between dose computation and measurement for points of interest (POIs) within the primary path of the beam was within experimental uncertainty for both energies, however points outside the primary beam were not. The average absolute percent difference for POIs within the primary path of the beam was 1.37% and 5.16% for 80 and 140 kVp, respectively. The minimum and maximum absolute percent difference for both energies and all POIs within the primary path of the beam was 0.151% and 6.41%, respectively. Conclusion: The CT x-ray source characterization technique based on HVL measurements and kVp can be used to accurately compute CT dose in an anthropomorphic thorax phantom.

SU‐E‐T‐317: Dosimetric Evaluation of Acuros XB Advanced Dose Calculation Algorithm in Head and Neck Patients

Purpose:The Acuros XB photon dose calculation algorithm is a newly implemented calculation technique within the Eclipse treatment planning system using deterministic solutions to the linear Boltzmann transport equations. The goal of this study is to assess the clinical impact of dose differences arising from a retrospective comparison of calculations performed using the Analytical Anisotropic Algorithm (AAA) and Acuros XB on patients.Methods:Ten head and neck patients receiving intensity modulated radiation therapy were selected as a pilot study. Initial evaluation was based on the percentage of the planning target volume (PTV) covered by the prescription dose, minimum dose within the PTV, and dose differences in critical structures. For patients receiving boost plans, dosimetric evaluations were performed on the plan sum of the primary and boost plans.Results:Among the ten patients there were a total of 21 PTVs corresponding to primary and boost volumes. Using the same normalization within Eclipse, the average percentage of the PTVs receiving the prescription dose were 95.6% for AAA and Acuros XB. The average minimum doses within the PTVs, expressed as a percentage of the prescription to the volume, were 82.3% and 83.6% for AAA and Acuros XB respectively. Neither comparison showed differences with statistical significance when subjected to a paired t‐test. Statistical significance was found in the average difference of the maximum dose for the mandible (242.5cGy, p=0.0005) and cord with a 5mm radial expansion (105.0cGy, p=0.0005) and in the median dose for the left parotid (25.0cGy, p=0.0423) and oral cavity (36.3cGy, p=0.002).Conclusion:The Acuros XB dose calculation algorithm did not exhibit significant differences in PTV coverage when compared to the AAA algorithm. Significant differences in critical structures are likely attributed to the structures proximity to high atomic number materials or air cavities, regions of known difficulty for the AAA algorithm.

PD-0447: Clinical evaluation of a dose calculation algorithm based on the linear Boltzmann transport equation

PD-0447: Clinical evaluation of a dose calculation algorithm based on the linear Boltzmann transport equation

A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields.

Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today's treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf_macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm(2) field as well as a smaller 2 × 2 cm(2) field. Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm's high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization. The feasibility of including magnetic field effects in a deterministic solution to the first order linear Boltzmann transport equation is shown. The results show a high degree of accuracy when compared against Monte Carlo calculations in all magnetic field strengths and orientations tested.

Size and edge roughness dependence of thermal conductivity for vacancy-defective graphene ribbons.

By incorporating the phonon-phonon scattering, phonon-boundary scattering and phonon-vacancy scattering into the linearized Boltzmann transport equation, we theoretically investigate the effects of size and edge roughness on thermal conductivity of single vacancy-defective graphene ribbons. Due to the severe suppression of high-frequency phonons by phonon-vacancy scattering which includes the impacts of missing mass and linkages, as well as the variation of the force constant of bonds associated with vacancies, the low-frequency ballistic phonons have a higher contribution to the thermal conductivity, which results in the stronger length, weaker width and weaker edge roughness dependence on thermal conductivity of vacancy-defective graphene ribbons than that of pristine ones. Our findings are helpful to understand and manipulate thermal conductivity of graphene by phononic engineering.

Effect of nitrogen and vacancy defects on the thermal conductivity of diamond: Anab initioGreen's function approach

We show that impurities and vacancies affect the thermal conductivity much more strongly than what is predicted by widely accepted models. When local distortions around point defects are strong, standard perturbative approaches fail, and phonon scattering can only be accounted for by an exact Green's function calculation. We apply the theory to the study, from first-principles, of nitrogen and vacancy defects in diamond. The thermal conductivity is computed by solving the linearized Boltzmann transport equation. The Born approximation underestimates the phonon scattering cross sections of nitrogen and vacancies by factors of 3 and 10, respectively. Thermal conductivity calculations are in good agreement with experiment.

A medical image-based graphical platform—Features, applications and relevance for brachytherapy

Brachytherapy dose calculation is commonly performed using the Task Group-No 43 Report-Updated protocol (TG-43U1) formalism. Recently, a more accurate approach has been proposed that can handle tissue composition, tissue density, body shape, applicator geometry, and dose reporting either in media or water. Some model-based dose calculation algorithms are based on Monte Carlo (MC) simulations. This work presents a software platform capable of processing medical images and treatment plans, and preparing the required input data for MC simulations. The A Medical Image-based Graphical platfOrm-Brachytherapy module (AMIGOBrachy) is a user interface, coupled to the MCNP6 MC code, for absorbed dosecalculations. The AMIGOBrachy was first validated in water for a high-dose-rate (192)Ir source. Next, dose distributions were validated in uniform phantoms consisting of different materials. Finally, dose distributions were obtained in patient geometries. Results were compared against a treatment planning system including a linear Boltzmann transport equation (LBTE) solver capable of handling nonwater heterogeneities. The TG-43U1 source parameters are in good agreement with literature with more than 90% of anisotropy values within 1%. No significant dependence on the tissue composition was observed comparing MC results against an LBTE solver. Clinical cases showed differences up to 25%, when comparing MC results against TG-43U1. About 92% of the voxels exhibited dose differences lower than 2% when comparing MC results against an LBTE solver. The AMIGOBrachy can improve the accuracy of the TG-43U1 dose calculation by using a more accurate MC dose calculation algorithm. The AMIGOBrachy can be incorporated in clinical practice via a user-friendly graphical interface.

Gated Si nanowires for large thermoelectric power factors

We investigate the effect of electrostatic gating on the thermoelectric power factor of p-type Si nanowires (NWs) of up to 20 nm in diameter in the [100], [110], and [111] crystallographic transport orientations. We use atomistic tight-binding simulations for the calculation of the NW electronic structure, coupled to linearized Boltzmann transport equation for the calculation of the thermoelectric coefficients. We show that gated NW structures can provide ∼5× larger thermoelectric power factor compared to doped channels, attributed to their high hole phonon-limited mobility, as well as gating induced bandstructure modifications which further improve mobility. Despite the fact that gating shifts the charge carriers near the NW surface, surface roughness scattering is not strong enough to degrade the transport properties of the accumulated hole layer. The highest power factor is achieved for the [111] NW, followed by the [110], and finally by the [100] NW. As the NW diameter increases, the advantage of the gated channel is reduced. We show, however, that even at 20 nm diameters (the largest ones that we were able to simulate), a ∼3× higher power factor for gated channels is observed. Our simulations suggest that the advantage of gating could still be present in NWs with diameters of up to ∼40 nm.

Atomistic calculation of the thermoelectric properties of Si nanowires

The thermoelectric properties of 1.6 nm-thick Si square nanowires with [100] crystalline orientation are calculated over a wide temperature range from 0 K to 1000 K, taking into account atomistic electron-phonon interaction. In our model, the [010] and [001] facets are passivated by hydrogen and there are Si-Si dimers on the nanowire surface. The electronic structure was calculated by using the sp^3 spin-orbit-coupled atomistic second-nearest-neighbor tight-binding model. The phonon dispersion was calculated from a valence force field model of the Brenner type. A scheme for calculating electron-phonon matrix elements from a second-nearest neighbor tight-binding model is presented. Based on Fermi's golden rule, the electron-phonon transition rate was obtained by combining the electron and phonon eigenstates. Both elastic and inelastic scattering processes are taken into consideration. The temperature dependence of transport characteristics was calculated by using a solution of linearized Boltzmann transport equation obtained by means of the iterative Orthomin method. At room temperature, the electron mobility is 195 cm^2/(Vs) and increases with temperature, while a figure-of-mertit ZT=0.38 is reached for n-type doping with a concentration of n=10^19 cm^-3.

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.

TH‐E‐BRE‐01: A 3D Solver of Linear Boltzmann Transport Equation Based On a New Angular Discretization Method with Positivity for Photon Dose Calculation Benchmarked with Geant4

Purpose:The Linear Boltzmann Transport Equation (LBTE) solved through statistical Monte Carlo (MC) method provides the accurate dose calculation in radiotherapy. This work is to investigate the alternative way for accurately solving LBTE using deterministic numerical method due to its possible advantage in computational speed from MC.Methods:Instead of using traditional spherical harmonics to approximate angular scattering kernel, our deterministic numerical method directly computes angular scattering weights, based on a new angular discretization method that utilizes linear finite element method on the local triangulation of unit angular sphere. As a Result, our angular discretization method has the unique advantage in positivity, i.e., to maintain all scattering weights nonnegative all the time, which is physically correct. Moreover, our method is local in angular space, and therefore handles the anisotropic scattering well, such as the forward‐peaking scattering. To be compatible with image‐guided radiotherapy, the spatial variables are discretized on the structured grid with the standard diamond scheme. After discretization, the improved sourceiteration method is utilized for solving the linear system without saving the linear system to memory. The accuracy of our 3D solver is validated using analytic solutions and benchmarked with Geant4, a popular MC solver.Results:The differences between Geant4 solutions and our solutions were less than 1.5% for various testing cases that mimic the practical cases. More details are available in the supporting document.Conclusion:We have developed a 3D LBTE solver based on a new angular discretization method that guarantees the positivity of scattering weights for physical correctness, and it has been benchmarked with Geant4 for photon dose calculation.