Analytical Simulation of MOS Capacitor Electrostatics

A MATLAB-based numerical simulation tool engineered to model and analyze the 1D electrostatic behavior and charge dynamics of a Metal-Oxide-Semiconductor (MOS) structure. By solving the 1D Poisson equation exactly, this tool maps the transitions across accumulation, depletion, and inversion regimes.

Simulation Profiles

The plots below illustrate the continuous physical transitions of the semiconductor interface under varying gate bias. The top profile visualizes the exponential scaling of the space-charge density ($Q_S$) relative to band bending, while the bottom profile characterizes the pinning of the surface potential ($\phi_s$) upon entering the strong inversion regime.

Core Analytical Focus

• Surface Potential Tracking: Evaluates the continuous relationship between the applied Gate Voltage ($V_G$) and interface band bending ($\phi_s$).
• Charge Density Analysis: Maps total semiconductor charge density ($Q_S$) behavior across all operational regimes, highlighting charge control mechanics essential for MOSFET device design.

Technical Specifications & Parameters

The numerical solver is configured using standard physical constants and the following device layout definitions:

• Substrate Type: p-type Silicon ($N_A = 10^{14} \text{ cm}^{-3}$)
• Gate Dielectric: Silicon Dioxide ($t_{ox} = 1 \mu\text{m}$)
• Thermal Environment: $300 \text{ K}$ (Room Temperature)
• Dielectric Constants: $\epsilon_{SiO_2} = 3.9$, $\epsilon_{Si} = 11.7$

Mathematical Framework

The framework evaluates the total semiconductor charge profile ($Q_S$) utilizing the exact solution of the 1D Poisson equation:

$$Q_S = \pm \sqrt{2\epsilon_{Si}kT N_A} \cdot \sqrt{\left(e^{-\frac{q\phi_s}{kT}} + \frac{q\phi_s}{kT} - 1\right) + \frac{n_i^2}{N_A^2}\left(e^{\frac{q\phi_s}{kT}} - \frac{q\phi_s}{kT} - 1\right)}$$

The corresponding operational Gate Bias ($V_G$) is calculated by mapping the ideal voltage balance across the gate stack, factoring in the potential drop across the oxide buffer layer ($V_{ox} = Q_S / C_{ox}$):

$$V_G = \frac{Q_S}{C_{ox}} + \phi_s$$