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Stochastic Galerkin Methods for the Boltzmann-Poisson system
Jose A. Morales Escalante, Clemens Heitzinger
TL;DR
The paper addresses uncertainty quantification for a probabilistic, quantum-informed Boltzmann-Poisson model of semiconductor electron transport. It applies stochastic Galerkin methods to random phonon energy and lattice temperature in electron-phonon collisions, including a distributional-derivative treatment for phonon energy. Numerically, temperature randomness produces coefficient variations in phase-space regions similar to the mean coefficient, while recombination changes current only slightly relative to the average.
Problem
The Boltzmann-Poisson model contains uncertainty from many-particle initial conditions and quantum collision mechanisms, with additional uncertainty in quantities such as phonon energy and temperature.
Method
The paper uses stochastic Galerkin methods to propagate uncertainty in phonon energy and lattice temperature through the electron-phonon collision operator and probability density.
Results
Temperature randomness produces coefficient α1 variations in phase-space regions similar to α0, while including recombination changes the current moment by two orders of magnitude below its average value.
Takeaways & Limitations
The methods quantify how physically varying temperature and nonconstant phonon energy propagate into electron probability densities and kinetic moments such as density, energy, and current.
Takeaways & Limitations
The lattice-temperature example assumes a uniform distribution and uses a first-order expansion, while excluding Gaussian modeling because zero temperature causes singular integrals.
Abstract
from arXiv · showhide
We study uncertainty quantification for a Boltzmann-Poisson system that models electron transport in semiconductors and the physical collision mechanisms over the charges. We use the stochastic Galerkin method in order to handle the randomness associated with the problem. The main uncertainty in the Boltzmann equation concerns the initial conditions for a large number of particles, which is why the problem is formulated in terms of a probability density in phase space. The second source of uncertainty, directly related to the quantum nature of the problem, is the collision operator, as its structure in this semiclassical model comes from the quantum scattering matrices operating on the wave function associated to the electron probability density. Additional sources of uncertainty are transport, boundary data, etc. In this study we choose first the phonon energy as a random variable, since its value influences the energy jump appearing in the collision integral for electron-phonon scattering. Then we choose the lattice temperature as a random variable, since it defines the value of the collision operator terms in the case of electron-phonon scattering by being a parameter of the phonon distribution. The random variable for this case is a scalar then. Finally, we present our numerical simulations.
1 Introduction
The paper frames semiconductor electron transport with the Boltzmann-Poisson system, whose probabilistic and quantum-mechanical structure creates multiple uncertainty sources. It applies stochastic Galerkin methods primarily to uncertainties in electron-phonon collisions, focusing on phonon energy and lattice temperature.
- Motivation: The Boltzmann-Poisson model uses a phase-space probability density because the system contains very many charge carriers with uncertain initial positions and momenta.Quantum considerations also make exact knowledge of these initial conditions impossible.
- Motivation: The collision operator models quantum-mechanical electron scattering, with electron-phonon transitions represented through scattering kernels and energy-conserving Dirac delta distributions.In the low-density regime, the collision operator can be approximated as linear in the probability density.
- Uncertainty sources: Uncertainty may enter the energy band, lattice temperature, phonon energy, Poisson parameters, boundary conditions, initial data, and quantum collision modeling.The paper identifies lattice-temperature fluctuations and experimentally nonconstant phonon energy as key collision-related uncertainties.
- Stochastic formulation: The stochastic formulation augments the probability density with random parameters associated with these uncertainty sources.The Boltzmann-Poisson equations retain the Poisson coupling while allowing f to depend on an additional random vector.
- Approach: The study uses stochastic Galerkin methods for electron-phonon collision variables, beginning with scalar lattice temperature because it enters the phonon distribution with minimal dimensional cost.The paper positions this approach relative to earlier stochastic Galerkin work on kinetic equations and semiconductor Boltzmann models.
2 Stochastic Galerkin Method for the Boltzmann-Poisson System
The stochastic Galerkin formulation represents identified uncertainties with random variables and propagates them through the Boltzmann collision operator. The paper treats random phonon energy both with a distributional-derivative approximation and without that approximation, then treats random lattice temperature.
- Random-variable formulation: The stochastic Galerkin method assigns random variables to uncertainties in initial data, quantum collisions, energy bands, lattice temperature, phonon energy, and Poisson parameters.These variables enter the probability density, dispersion relation, collision terms, or Poisson coefficients.
- Random phonon energy: Random phonon energy replaces ¯hωp by ¯hωp + z, making both phonon occupation and collision energy transitions random.The paper studies a distributional-derivative approximation and a fully general treatment without that approximation.
- Random phonon energy: The distributional-derivative approximation linearizes the scattering cross section around z = 0 using σ0 and the derivative coefficient ˜σ1.The derivative coefficient is defined as the z-derivative of the cross section evaluated at zero.
- Random phonon energy: For the unapproximated random phonon-energy case, the formulation uses n = 1, P = 1, and a two-coefficient density expansion.The collision equation is then written using the random collision term directly.
- Random lattice temperature: For random lattice temperature, randomness enters collision coefficients through the phonon distribution rather than the arguments of the Dirac delta distributions.The resulting operator separates deterministic collision action on each band from temperature-related recombination and diagonal terms.
- Random lattice temperature: The lattice-temperature model uses a uniform random distribution and a first-order Legendre expansion, with f approximated by α1 + wα2.The paper excludes a Gaussian distribution because temperature reaching zero would create singular integrals.
3 Stochastic Galerkin Method for the Boltzmann-Poisson System Using Deterministic Discontinuous Galerkin Solvers
The deterministic discontinuous Galerkin solver provides the numerical foundation for the stochastic Galerkin Boltzmann-Poisson system. It transforms momentum into spherical coordinates, formulates the weighted phase-space equation, and advances transport, collisions, and electrostatics iteratively.
- Solver framework: The solver uses deterministic DG methods for two bands, representing the stochastic Galerkin coefficient vector and its matrix-integral collision operator.Earlier DG studies treated a single PDF without randomness.
- k-spherical coordinates: A spherical transformation places a conduction-band minimum at the momentum-space origin, making the radial coordinate proportional to energy at low bias.The transformed PDF must be weighted by the Jacobian when computing momentum-space moments.
- Transformed equations: The transformed formulation produces a divergence-form Boltzmann equation with transport coefficients derived from the energy gradient and a transformed linear collision operator.The scattering term represents electron-phonon interactions in the two-band system.
- Coupling: The electron density is obtained from the first PDF coefficient, which represents the mean of the PDF.This coefficient supplies the density used in the coupled Poisson calculation.
- Time advancement: Each time step computes charge density, solves the Poisson equation for potential and electric field, evaluates transport coefficients, advances Boltzmann transport by DG, and time-steps the resulting ODE system.Partial time steps repeat the charge, Poisson, and transport stages as needed.
- DG discretization: The DG approximation uses piecewise linear polynomials on a rectangular Cartesian grid in transformed phase space.The coordinates include physical position and spherical momentum variables r, µ, and ϕ.
4 SDG-BP: Stochastic Discontinuous Galerkin Method for the Boltzmann-Poisson System
The study applies stochastic discontinuous Galerkin methods to a one-dimensional silicon diode, solving for truncated PDF coefficients under specified numerical conditions. Results compare deterministic and recombination cases and examine stochastic effects on phase-space coefficients and current.
- Computational setup: The simulations model a 1 µm one-dimensional n+–n–n+ silicon diode with a 400 nm central n-channel and azimuthal symmetry in momentum space.Only radial and polar momentum coordinates are retained under the symmetry assumptions.
- Computational setup: The computational domain uses x ∈ [0, 1], r ∈ [0, rmax], and µ ∈ [−1, 1], with rmax ≈ 36 for a 0.5 V bias in a 400 nm channel.The initial condition is specified through the zeroth and first PDF coefficients, with a normalization constant enforcing initial charge neutrality.
- Boundary and time discretization: The numerical scheme imposes charge-neutral endpoints, fixed source and drain potentials, a vanishing distribution at r = rmax, and analytically zero transport boundary integrals at point boundaries.Time evolution is computed with an RK2 method.
- Numerical results: In the deterministic benchmark, α1 = 0, so α0 contains all information about the PDF; −log α0 is shown for a 1 µm diode at 0.5 V bias and t0 = 10.0 ps.The vanishing first coefficient is associated with the absence of random effects and recombination in this case.
- Numerical results: With recombination, variations in α1 occupy similar phase-space regions, whereas variations in α0 appear finer and more pronounced.The reported coefficient fields include −log α0 and −log α1 at the same diode size, bias, and final time conditions.
- Numerical results: The main difference between recombination and no-recombination cases appears in the predicted current, at two orders of magnitude below its mean value.The authors interpret this as indicating finer momentum resolution from the stochastic Galerkin method and also compute expectation, variance, and standard deviation.
5 Conclusions
The study applies stochastic Galerkin methods to propagate uncertainty in phonon energy and lattice temperature through the Boltzmann-Poisson system and its kinetic moments. Numerical results show localized random effects, small current differences between recombination settings, and a new distributional-derivative treatment for phonon-energy uncertainty.
- Numerical results: Temperature randomness produces α1 variations in regions similar to α0, while α0 exhibits finer and more pronounced phase-space variations.This concerns simulations of the stochastic Galerkin Boltzmann-Poisson system with a random temperature in the electron-phonon collision operator.
- Numerical results: The recombination and no-recombination simulations differ mainly in current, with the difference two orders of magnitude below the average current.Using α0 to calculate moments, the comparison attributes the slight momentum difference to a nonzero averaged random coefficient and the product structure of density and energy.
- Uncertainty propagation: The methods quantify phonon-energy uncertainty using distributional derivatives with respect to the random variable.The paper identifies this as a departure from the usual stochastic Galerkin collision-term structure for kinetic equations.
- Uncertainty propagation: The study propagates Gaussian or uniform phonon-energy uncertainty and uniformly distributed lattice-temperature uncertainty into the probability density and associated kinetic moments.The phonon-energy treatment includes both a first-order approximation and a full calculation, while temperature variation is modeled as a scalar random variable.
- Implications: The computed uncertainty targets measurable observables including electric current, average energy, and density, with future work aimed at scalar energy-band structures ε(k).The stated application is predicting how physically variable or idealized model parameters affect electronic-device behavior.