Time-dependent Boltzmann Transport Equation Research Articles

Physics-informed neural networks for solving time-dependent mode-resolved phonon Boltzmann transport equation

The phonon Boltzmann transport equation (BTE) is a powerful tool for modeling and understanding micro-/nanoscale thermal transport in solids, where Fourier’s law can fail due to non-diffusive effect when the characteristic length/time is comparable to the phonon mean free path/relaxation time. However, numerically solving phonon BTE can be computationally costly due to its high dimensionality, especially when considering mode-resolved phonon properties and time dependency. In this work, we demonstrate the effectiveness of physics-informed neural networks (PINNs) in solving time-dependent mode-resolved phonon BTE. The PINNs are trained by minimizing the residual of the governing equations, and boundary/initial conditions to predict phonon energy distributions, without the need for any labeled training data. The results obtained using the PINN framework demonstrate excellent agreement with analytical and numerical solutions. Moreover, after offline training, the PINNs can be utilized for online evaluation of transient heat conduction, providing instantaneous results, such as temperature distribution. It is worth noting that the training can be carried out in a parametric setting, allowing the trained model to predict phonon transport in arbitrary values in the parameter space, such as the characteristic length. This efficient and accurate method makes it a promising tool for practical applications such as the thermal management design of microelectronics.

J–V characteristics of dark current in truncated conical quantum dot infrared photodetectors (QDIPs)

Quantum Dot Infrared Photodetector (QDIP) is one of the promising candidates for infrared photodetection due to its controllable heterojunction bandgap and sensitivity to normal incident radiation. It is expected to be superior to infrared photodetectors of mature technologies such as Mercury Cadmium Telluride (HgCdTe) or a quantum well infrared photodetector. In the presented paper, we have developed a theoretical model for the dark current in truncated conical QDIP as the truncated conical shaped QD structure is more appropriate to describe the fabricated dots. The dark current model is based on the drift diffusion model solving the main governing Poisson’s and continuity equations. In this model, the carrier mobility is calculated by solving time-dependent Boltzmann transport equation in the photodetector material with embedded truncated conical QDs using finite difference technique. The results of the developed model have been compared with the dark current characteristics with published experimental results of Indium Arsenide/Gallium Arsenide (InAs/GaAs) truncated QDIP. The effects of QD volume, QD aspect ratio and QD density and the operating temperature on the dark current characteristics have also been investigated.

The Time-Dependent Asymptotic PN Approximation for the Transport Equation

In this study, a spatio-temporal approach for the solution of the time-dependent Boltzmann (transport) equation is derived. Finding the exact solution using the Boltzmann equation for the general case is generally an open problem and approximate methods are usually used. One of the most common methods is the spherical harmonics method (the approximation), when the exact transport equation is replaced with a closed set of equations for the moments of the density with some closure assumption. Unfortunately, the classic closure yields poor results with low-order N in highly anisotropic problems. Specifically, the tails of the particles’ positional distribution as attained by the approximation are inaccurate compared to the true behavior. In this work, we present a derivation of a linear closure that even for low-order approximation yields a solution that is superior to the classical approximation. This closure is based on an asymptotic derivation both for space and time of the exact Boltzmann equation in infinite homogeneous media. We test this approximation with respect to the one-dimensional benchmark of the full Green function in infinite media. The convergence of the proposed approximation is also faster when compared to (classic or modified) approximation.

Validation of the SHNC time-dependent transport code based on the spherical harmonics method for complex nuclear fuel assemblies

The diffusion approximation to the time-dependent Boltzmann transport equation gives accurate results for traditional nuclear reactor designs, but new reactor designs and new fuel elements require neutron transport methods. We develop a numerical approximation to the time-dependent transport equation coupled to delayed neutron precursors based on the spherical harmonics PL equations, for odd L, and on the Backward Euler finite difference discretization of time. The resulting scheme can be written as a stationary form of diffusive second order PL equations. This allows a reduction by half to the number of unknowns and also to apply a nodal collocation method to the spatial discretization of the problem, using coarse spatial grids to further reduce memory requirements. This scheme is validated with several transient benchmarks, where the convergence properties are established and compared with the simplified PL approximation. A more realistic transient benchmark, based on the two-group C5 MOX problem, is finally introduced, showing the need of high order PL approximation for complex fuel geometries.

Solution of a new model of fractional telegraph point reactor kinetics using differential transformation method

A new model of fractional telegraph point reactor kinetics FTPRK is introduced to approximate the time dependent Boltzmann transport equation considering new terms that contain time derivative of the reactivity and fractional integral of the neutron density. Caputo fractional derivatives and fractional Leibniz rule are used for such derivation. Cattaneoequation is applied to overcome the flaw of infinite neutron velocity and to describe the anomalous transport. Effect of the new term on the neutron behaviour is discussed. The new model is applied to both TRIGA reactor and to commercial pressured water reactor of a Three Mile Island type reactor, TMI-type PWR. Results for step, ramp and sinusoidal excess reactivities with thermal hydraulic feedback are presented and discussed for different values of anomalous sub-diffusion exponent, the fractional order, 0 < µ ≤ 1. To maintain the reactor safe at start-up after insertion of step reactivity and based on the concept of prompt jump approximation, the FTPRK model is simplified and solved analytically by Mittag–Liffler function. Physical interpretations of the fractional order µ and relaxation time τ and their effects on the behaviour of the neutron population are discussed. Also, the effect of a small perturbation in the geometric buckling on the neutron behaviour is discussed for finite reactor core. The new model is solved numerically using the fractional order multi-step differential transform method MDTM. The MDTM constitutes an easy algorithm based on Taylor's formula and Caputo fractional derivative. Two theorems with their proofs are introduced to solve the fractional system. Two major disadvantages of the method about the choice of the fractional order values and the step size length are addressed. We present a procedure which enables us to solve the system with appropriate values of fraction orders.

Enhanced geometric capabilities for the transient analysis code T-ReX and its application to simulating TREAT experiments

Advances in computational architecture have prompted a resurgence in the simulation of reactor transients from first principles. Most codes are unable to simulate transient events with complex models, and require numerous approximations. The code T-ReX (Transient-Reactor eXperiment simulator), an extensive update to TDKENO, has been developed as a transient analysis tool with few geometric limitations, and minimal theoretical approximations. T-ReX achieves this by employing the Improved Quasi-Static (IQS) method to solve the time-dependent Boltzmann transport equation with explicit representation of delayed neutrons. The primary change in T-ReX relative to TDKENO is the incorporation of a modified version of the Monte Carlo code KENO-VI to calculate the flux shape and model the geometry of a problem. Using KENO-VI to model systems allows exact representation of the geometry. The changes to T-ReX are verified by comparison of solutions to computational benchmark problems found with a previous version of TDKENO that made use of KENO V.a, and several other codes with time-dependent capabilities. In addition, a three-dimensional KENO-VI model of the Transient Reactor Test Facility (TREAT) core is used in simulations of several temperature-limited transient experiments from the M8 Calibration series. T-ReX produces results that agree with benchmark problems and are in better agreement with TREAT experimental data than TDKENO.

AlmaBTE : A solver of the space–time dependent Boltzmann transport equation for phonons in structured materials

almaBTE is a software package that solves the space- and time-dependent Boltzmann transport equation for phonons, using only ab-initio calculated quantities as inputs. The program can predictively tackle phonon transport in bulk crystals and alloys, thin films, superlattices, and multiscale structures with size features in the nm–μm range. Among many other quantities, the program can output thermal conductances and effective thermal conductivities, space-resolved average temperature profiles, and heat-current distributions resolved in frequency and space. Its first-principles character makes almaBTE especially well suited to investigate novel materials and structures. This article gives an overview of the program structure and presents illustrative examples for some of its uses.PROGRAM SUMMARYProgram Title:almaBTEProgram Files doi:http://dx.doi.org/10.17632/8tfzwgtp73.1Licensing provisions: Apache License, version 2.0Programming language: C++External routines/libraries: BOOST, MPI, Eigen, HDF5, spglibNature of problem: Calculation of temperature profiles, thermal flux distributions and effective thermal conductivities in structured systems where heat is carried by phononsSolution method: Solution of linearized phonon Boltzmann transport equation, Variance-reduced Monte Carlo

Performance comparison of Si, III–V double-gate n-type MOSFETs by deterministic Boltzmann transport equation solver

A deterministic time-dependent Boltzmann transport equation (BTE) solver is employed to carry out a comparison work among 10 nm double-gate n-type MOSFETs with channel materials of Si, In0.53Ga0.47As, and GaSb in different surface orientations. Results show that the GaSb device has the highest drive current, while scattering affects carrier transport in the Si device the most. The InGaAs device exhibits the highest injection velocity but suffers from the density of state (DOS) bottleneck seriously.

Fourier Convergence Analysis of the Infinite Homogenous Multigroup Time-Dependent Boltzmann Transport Equation Using Discrete Ordinates Formulation

Fourier analysis of the continuous infinite homogenous multigroup (MG) formulation is investigated in this paper for the time-dependent Boltzmann transport equation using discrete ordinates formulation. In addition, a continuous two-group (2G) and one-group (1G) Fourier analysis is performed to generate an analytical spectral radius and provide the basis for a theoretical analysis of the convergence. The discrete 1G formulation is then presented, and the theoretical analysis is found to predict the same spectral radius as the continuous 1G formulation. A typical pressurized water reactor pin cell problem with 47-group library is then homogenized with reflective boundary conditions, and the numerical spectral radius is calculated using the MPACT code. The theoretical predictions and the numerical results from the pin cell case agree very well and are found to have the following behavior: (1) The spectral radius is usually very close to unity for standard parameters for realistic transient application, (2) the spectral radius generally decreases as a function of inners per outer M, (3) the spectral radius generally decreases as a function of time-step size and then increases beyond unity for extremely small time steps, and (4) the spectral radius is almost constant as a function of the inserted reactivity. Good agreement is observed with the MG Fourier analysis. Finally, it is shown that the group sweeping coarse mesh finite difference method is theoretically and numerically very slow to converge the time-dependent neutron transport equation and that it is necessary to move the right-hand-side fission and transient source to the left-hand side and to solve the entire matrix form of the system.

Investigation of the surface orientation influence on 10-nm double gate GaSb nMOSFETs**Project supported by the National Natural Science Foundation of China (Grant Nos. 61674008, 61421005, and 61404005).

The performance of double gate GaSb nMOSFETs with surface orientations of (100) and (111) are compared by deterministically solving the time-dependent Boltzmann transport equation (BTE). Results show that the on-state current of the device with (111) surface orientation is almost three times larger than the (100) case due to the higher injection velocity. Moreover, the scattering rate of the (111) device is slightly lower than that of the (100) device.

Vectorization of the time-dependent Boltzmann transport equation: Application to deep penetration problems

We introduce an alternative method to calculate the steady state solution of the angular photon flux after a numerical evolution of the time-dependent Boltzmann transport equation (BTE). After a proper discretization the transport equation was converted into an ordinary system of differential equations that can be iterated as a weighted Richardson algorithm. As a different approach, in this work the time variable regulates the iteration process and convergence criteria is based on physical parameters.Positivity and convergence was assessed from first principles and a modified Courant-Friedrichs-Lewy condition was devised to guarantee convergence. The Penelope Monte Carlo method was used to test the convergence and accuracy of our approach for different phase space discretizations. Benchmarking was performed by calculation of total fluence and photon spectra in different one-dimensional geometries irradiated with 60Co and 6MV photon beams and radiological applications were devised.

Investigation of transient responses of nanoscale transistors by deterministic solution of the time-dependent BTE

In this work, the transient characteristics of nanoscale field-effect transistors (FETs) have been investigated using a deterministic solver based on the time-dependent multi-subband Boltzmann transport equation (BTE). The response to a step signal superimposed on the gate or drain electrode is simulated. The transient process can be understood as a combination of electrostatic and transport relaxation. The extracted transient relaxation time for the drain current, which is unrelated to the direct-current (DC) shift, is important for transient device modeling.

Impact of the Substrate Material on the RF Performance of Carbon-Nanotube Transistors

We examine the effect of the substrate material on the radio-frequency (RF) behavior of carbon-nanotube transistors by considering the impact of substrate polar phonons (SPPs). We consider SPP scattering from AlN, SiO <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$_{2}$</tex></formula> , HfO <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$_{2}$</tex></formula> , and ZrO <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$_{2}$</tex></formula> substrates within a semiclassical approach by solving the time-dependent Boltzmann transport equation self-consistently with the Poisson equation. Various RF figures of merit, such as the unity-current gain frequency <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$f_T $</tex></formula> , the unity-power-gain frequency <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$f_{{\rm max}} $</tex></formula> , the transconductance <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$g_m $</tex></formula> , and the two-port <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$y$</tex></formula> -parameters, are determined in order to characterize the impact of SPP scattering. We first consider the impact of SPP scattering on the RF behavior of an intrinsic single-tube carbon nanotube field-effect transistor (CNFET). These single-tube results are then combined with the external parasitic elements to analyze the pitch-dependent, RF behavior of an extrinsic array-based CNFET. It is shown that AlN substrates have the least impact in degrading the RF performance of a CNFET, while the more polar substrates (HfO <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$_{2}$</tex></formula> or ZrO <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$_{2}$</tex> </formula> ) have a greater impact. This varying behavior can be attributed to the SPP energies, which are higher in AlN compared to the other materials, making CNFETs with AlN substrates less susceptible to SPP scattering even at room temperature. Our results suggest that substrate engineering will become an important component in the design process of emerging devices to achieve an optimized RF performance.

Electrical tunability of soft parametric resonance by hot electrons in graphene

We show that the onset of soft parametric resonance (SPR) at ω=ηωF, where ωF=eFovf/ℏωOP for hot carriers interacting with optic phonons ℏωOP in graphene is independent of temperature, whereas η ∼ 0.56 is practically independent of the DC field Fo, which makes SPR tunable over a sizeable range of Fo. Our model based on the linearization of the time-dependent Boltzmann transport equation (BTE) predicts a secondary resonance at ω∼ωF, emerging from a weak shoulder in the ac current at T = 300 K, when the temperature decreases to T = 77 K and lower. The two resonances behave differently as scattering at low energy strengthens.

A Finite Volume Method for the Multi Subband Boltzmann Equation with Realistic 2D Scattering in Double Gate MOSFETs

AbstractWe propose a deterministic solver for the time-dependent multi-subband Boltzmann transport equation (MSBTE) for the two dimensional (2D) electron gas in double gate metal oxide semiconductor field effect transistors (MOSFETs) with flared out source/drain contacts. A realistic model with six-valleys of the conduction band of silicon and both intra-valley and inter-valley phonon-electron scattering is solved. We propose a second order finite volume method based on the positive and flux conservative (PFC) method to discretize the Boltzmann transport equations (BTEs). The transport part of the BTEs is split into two problems. One is a 1D transport problem in the position space, and the other is a 2D transport problem in the wavevector space. In order to reduce the splitting error, the 2D transport problem in the wavevector space is solved directly by using the PFC method instead of splitting into two 1D problems. The solver is applied to a nanoscale double gate MOSFET and the current-voltage characteristic is investigated. Comparison of the numerical results with ballistic solutions show that the scattering influence is not ignorable even when the size of a nanoscale semiconductor device goes to the scale of the electron mean free path.

RF Performance Potential of Array-Based Carbon-Nanotube Transistors—Part I: Intrinsic Results

A comprehensive study, which is presented in two parts, is performed to assess the radio-frequency (RF) performance potential of array-based carbon-nanotube field-effect transistors (CNFETs). In Part I, which is presented in this paper, the time-dependent Boltzmann transport equation is solved self-consistently with the Poisson equation to study the impact of nanotube phonon scattering on different aspects of intrinsic (single-tube, contact-independent) CNFET operation, including the attainable drive current and transconductance per tube, the intrinsic cutoff frequency, the intrinsic y-parameters, and the small-signal equivalent circuit for the intrinsic transistor. These intrinsic results are used to assess the tube-to-tube distance (pitch) that would be required in a multitube array-based structure to meet the drive current and transconductance requirements of the International Technology Roadmap for Semiconductors for the year 2015, which we use as a benchmark for CNFET technology going forward. In Part II of our paper, we elaborate on the results of Part I by adding the effects of extrinsic (contact-dependent) parasitics, thus providing an overall performance assessment of array-based structures.

Understanding the Frequency- and Time-Dependent Behavior of Ballistic Carbon-Nanotube Transistors

The high-frequency and time-dependent behavior of carbon-nanotube (CN) transistors is examined by numerically solving the time-dependent Boltzmann transport equation self-consistently with the Poisson equation. The two-port admittance matrix, containing the transistor's y-parameters, is extracted. At frequencies below the transistor's unity-current-gain frequency fT, the y-parameters are shown to agree with those predicted from a quasi-static description of transistor operation, provided that the partitioning factor for the device charge is extracted through application of an appropriate time-dependent ramp voltage to the gate. The physics of time-dependent transport is described, and by examining the positive- and negative-going components of electron charge in the nanotube, it is shown for an nin device structure that the n regions can add a time delay to the device response, even though these regions do not affect the transistor's extrapolated fT. For very high frequencies, or for very fast transients, it is pointed out that the conventional ldquofloating boundary conditionrdquo approach, which was originally suggested for dc simulations of ballistic Mosfets, becomes questionable when applied to time-dependent simulations of nanotubes. While this paper omits collisions and focuses on an intrinsic transistor structure that excludes external parasitics, it provides a first useful step toward the full frequency- and time-dependent characterization of CN transistors.

Space–time streamline upwind Petrov–Galerkin methods for the Boltzmann transport equation

This paper presents a space–time streamline upwind Petrov–Galerkin formulation for the time-dependent Boltzmann transport equation. Conventional linear space–time formulations typically take the distance across space–time elements as the representative distance across which dissipation is introduced. The proposed method however, calculates the representative distances based on the gradient of the solution in space–time. Therefore, the method in its most general form is non-linear. The calculation of distances based on the solution gradient means that even if small time-steps are used the method will attempt to optimise the numerical accuracy while eliminating any spurious oscillations.Unlike conventional non-linear Petrov–Galerkin formulations the proposed class of methods introduces dissipation only in the streamline direction (in space–time). Conventional non-linear discontinuity capturing schemes add dissipation in the steepest gradient direction. In order to demonstrate the accuracy and robustness (in terms of suppressing spurious oscillations), the method is applied to a series of one-group, 2-D, fixed source, steady-state and time-dependent radiation transport problems.

Analytical form of the particle distribution based on the cumulant solution of the elastic Boltzmann transport equation

An analytical expression of the particle distribution based on an analytical cumulant solution of the time-dependent elastic Boltzmann transport equation (BTE) is presented. This expression improves upon the previous second order cumulant solution of the BTE described by a Gaussian distribution in two aspects: (1) separating the ballistic component from the scattered component to ensure that the summation in expressions is convergent; and (2) enforcing the causality condition to ensure that no particle travels faster than the free speed of the particles. Time-resolved profiles obtained using the analytical form are compared with those obtained by the Monte Carlo simulation, for both transmission and backscattering. The calculating time using our analytical form is much faster than that using the Monte Carlo approach.

Cumulant solution of the elastic boltzmann transport equation in an infinite uniform medium

We consider an analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrary phase function. We obtain (1) the exact distribution in angle, (2) the exact first and second spatial cumulants at any angle, and (3) an approximate combined distribution in position and angle and a spatial distribution whose central position and half-width of spread are always exact. The resulting Gaussian distribution has a center that advances in time, and an ellipsoidal contour that grows and changes shape providing a clear picture of the time evolution of the particle migration from near ballistic, through snakelike and into the final diffusive regime.