Stochastic Galerkin Methods for the Boltzmann-Poisson system (1910.13525v3)

Abstract: 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.

• The paper introduces stochastic Galerkin methods to quantify uncertainty in semiconductor electron transport using the Boltzmann-Poisson system.
• Numerical simulations with discontinuous Galerkin solvers reveal significant impacts on electron density, mean energy, and electric current.
• The study offers insights into integrating stochastic modeling for improved semiconductor device simulation and design in nanotechnology.

Stochastic Galerkin Methods for the Boltzmann-Poisson System: An Analytical Overview

The paper "Stochastic Galerkin Methods for the Boltzmann-Poisson system" by Jos A. Morales Escalante and Clemens Heitzinger presents an in-depth paper on uncertainty quantification within the Boltzmann-Poisson (BP) system, an important model for electron transport in semiconductors. The authors apply the stochastic Galerkin method to handle the inherent randomness associated with electron transport problems, focusing on the uncertainties linked to initial conditions, quantum collision operators, energy bands, lattice temperature, phonon energy, and boundary conditions.

Deterministic and Stochastic Aspects of the Boltzmann-Poisson System

In the deterministic framework, the BP system is a semiclassical model that strives to approximate quantum behavior using a statistical mechanics approach, given the immense number of particles involved, which are primarily electrons. The BP model consists of a coupling between the Boltzmann equation, articulating the statistical distribution of electrons in phase space, and the Poisson equation, which accounts for the electrostatic field.

Uncertainties in this system are multifaceted:

• Initial Conditions and Large Particle Number: A probability density function (PDF) in phase space represents the probabilistic nature of initial conditions.
• Quantum Mechanical Features: Particularly manifest in the collision operator, relying on quantum scattering matrices, these influence electron transitions.
• Energy Band Structure: This affects both the transport behavior and electron-phonon interactions.
• Environmental Parameters: Such as the lattice temperature, which affects electron-phonon scattering probabilities.
• Boundary Conditions and Material Properties: Including uncertainties in doping concentrations and material permittivity.

These complexities necessitate the employment of stochastic approaches, such as the stochastic Galerkin method, to statistically characterize the model's behavior under uncertainty.

Stochastic Galerkin Approach

The stochastic Galerkin method is employed to incorporate random variables where uncertainties typically occur. Noteworthy variables include the phonon energy and the lattice temperature, which directly affect the collision operator and the scattering processes.

1. Phonon Energy as a Random Variable: The authors explore scenarios where phonon energy is treated as a non-constant entity due to variances evidenced in practice. This is initially handled through linear approximation via distributional derivatives, followed by a comprehensive non-linear treatment.
2. Lattice Temperature as a Random Variable: By treating the lattice temperature as a scalar random variable, the authors delineate how variations influence scattering coefficients in collision integrals without affecting Dirac delta arguments traditionally utilized in Fermi's Golden Rule.

Numerical Simulations and Results

Numerical experiments are executed using a diode setup where disturbance in the lattice environment and variations in the stochastic models are analyzed. The stochastic Galerkin system captures the uncertainty propagation through numerical simulations performed using discontinuous Galerkin (DG) solvers.

The numerical results underscore the impact of these uncertainties on electron density, mean energy, electric current, and other kinetic moments. A comparative analysis between stochastic and deterministic models reveals the nuanced predictions in current distribution attributed to the stochastic model's resolution capabilities.

Implications and Future Directions

This paper elucidates the impact of stochastic modeling on the reliability and accuracy of semiconductor simulation. By quantifying the uncertainties inherent in real-world scenarios, this work facilitates the prediction and control of semiconductor device behavior under variable conditions, potentially informing design in nanotechnology applications.

Further research can extend these methodologies to more complex band structures and broader classes of uncertainty, emphasizing the integration of stochastic methods with real-time measurement data to enhance predictive accuracy. As the semiconductor industry continuously evolves, such predictive models will prove invaluable in innovating novel materials and device architectures.