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Distortion of charge distribution due to internal electric fields described by the drift-diffusion semiconductor model

Masakazu Yamamoto

TL;DR

We address the large-time behavior of the Debye–Hueckel drift-diffusion model in $\\mathbb{R}^3$, given by $\\partial_t u - \\Delta u = \\nabla\\cdot(u\\nabla\\psi)$ and $-\\Delta\\psi = u$, with initial data $u_0$. The approach uses a renormalized mild formulation and Escobedo–Zuazua analysis to obtain an explicit asymptotic expansion for $u(t)$ in terms of the heat kernel $G(t)$, its gradient, and nonlinear corrections. The main contributions are (i) a radially symmetric nonlinear correction $U_1^{\\mathrm{rad}}$ and a logarithmic shift $K_2\\log t$, both determined by the initial moments $M_0=\\int u_0$ and $M_1=-\\int x u_0(x)\\,dx$, and (ii) an explicit expression for $U_1^{\\mathrm{rad}}$ via a double integral and a precise decay estimate $\\|u(t)-U_0(t)-U_1(t)\\|_{L^q}=O(t^{-\\gamma_q-1})$; the log-term coefficient is $(2\\pi-3\\sqrt{3})M_0^3/(2^{7}\\cdot 3^2 \\pi^3)$. These results reveal symmetry distortion and internal-field–driven scale shifts that persist beyond linear diffusion, with implications for MOSFET simulations and plasma diffusion modeling.

Abstract

In this paper, the initial value problem for the Debye--Hueckel drift-diffusion equation is studied. This equation was introduced as a model describing plasma behavior and is also known as a simulation model of MOSFET, and so its solution describes charge density. It is well-known that, if the initial density is localized, then the density is adjusted to be radially symmetric due to the linear diffusion. Consequently, the electric field is also governed by a radially symmetric potential, and its effects are expected to act radially symmetrically. The main result express the electric field and its effect on the charge density as concrete functions. It also denotes the distortion of symmetry and the shift of scale on the density due to the internal electric field. Unlike the historical paper via Escobedo and Zuazua and the followers, the main result captures stronger nonlinearity than the logarithmic shift.

Distortion of charge distribution due to internal electric fields described by the drift-diffusion semiconductor model

TL;DR

We address the large-time behavior of the Debye–Hueckel drift-diffusion model in , given by and , with initial data . The approach uses a renormalized mild formulation and Escobedo–Zuazua analysis to obtain an explicit asymptotic expansion for in terms of the heat kernel , its gradient, and nonlinear corrections. The main contributions are (i) a radially symmetric nonlinear correction and a logarithmic shift , both determined by the initial moments and , and (ii) an explicit expression for via a double integral and a precise decay estimate ; the log-term coefficient is . These results reveal symmetry distortion and internal-field–driven scale shifts that persist beyond linear diffusion, with implications for MOSFET simulations and plasma diffusion modeling.

Abstract

In this paper, the initial value problem for the Debye--Hueckel drift-diffusion equation is studied. This equation was introduced as a model describing plasma behavior and is also known as a simulation model of MOSFET, and so its solution describes charge density. It is well-known that, if the initial density is localized, then the density is adjusted to be radially symmetric due to the linear diffusion. Consequently, the electric field is also governed by a radially symmetric potential, and its effects are expected to act radially symmetrically. The main result express the electric field and its effect on the charge density as concrete functions. It also denotes the distortion of symmetry and the shift of scale on the density due to the internal electric field. Unlike the historical paper via Escobedo and Zuazua and the followers, the main result captures stronger nonlinearity than the logarithmic shift.

Paper Structure

This paper contains 3 sections, 1 theorem, 40 equations.

Table of Contents

  1. Introduction
  2. Notations.
  3. Proof of main result

Key Result

Theorem 1.1

Let $u_0 \in L^1 (\mathbb{R}^3) \cap L^\infty (\mathbb{R}^3)$ be positive and $x u_0 \in L^1 (\mathbb{R}^3)$. Then as $t\to +\infty$ for $M_0 = \int_{\mathbb{R}^3} u_0 (x) dx$ and $M_1 = - \int_{\mathbb{R}^3} x u_0 (x) dx$, and $1 \le q \le \infty$ and $\gamma_q = \frac{3}{2} (1-\frac{1}{q})$. In addition, if $|x|^2 u_0 \in L^1 (\mathbb{R}^3)$, then as $t\to +\infty$.

Theorems & Definitions (3)

  • Theorem 1.1
  • Remark 2.1
  • Remark 2.2