Poisson Equation is the fundamental partial differential equation relating the electrostatic potential to the spatial distribution of charge in a semiconductor device — one of the three coupled equations (with electron and hole continuity) that form the complete drift-diffusion TCAD framework, it is the electrostatic backbone of all device simulation.
What Is the Poisson Equation in Semiconductors?
- Definition: The semiconductor Poisson equation is nabla^2(phi) = -rho/epsilon = -(q/epsilon)*(p - n + N_D+ - N_A-), relating the curvature of the electrostatic potential phi to the net charge density from holes (p), electrons (n), ionized donors (N_D+), and ionized acceptors (N_A-).
- Physical Meaning: Positive net charge (excess holes or donors) causes the potential to curve downward (local potential maximum); negative net charge (excess electrons or acceptors) causes upward curvature — the Poisson equation is the mechanism by which charge creates electric fields and bands bend.
- Boundary Conditions: At metal contacts, the potential is specified by the applied voltage plus the contact workfunction difference; at insulating surfaces, the normal component of the electric displacement is continuous across the interface; at semiconductor-dielectric interfaces, charge sheets (interface states) modify the boundary condition.
- Nonlinearity: Because electron and hole concentrations depend exponentially on potential (n ~ exp(q*phi/kT)), the Poisson equation is highly nonlinear and requires iterative numerical methods (Newton-Raphson) for solution.
Why the Poisson Equation Matters
- Electrostatic Foundation: Every device characteristic — threshold voltage, depletion width, junction capacitance, breakdown field, channel charge — is ultimately determined by the solution of the Poisson equation in the device geometry. Without it, none of the principal device design parameters can be calculated.
- TCAD Core Equation: The Poisson equation is solved simultaneously with the electron and hole continuity equations at every mesh point in TCAD simulation — it is the electrostatic solver that converts the charge state of the device into the potential landscape that drives current.
- Short-Channel Effects: In short-channel MOSFETs, drain voltage modifies the two-dimensional Poisson solution in the channel, pulling down the source barrier and causing DIBL, threshold voltage roll-off, and subthreshold slope degradation — effects that cannot be predicted from the one-dimensional analysis used for long channels.
- Gate Control Analysis: The differential of potential in the channel with respect to gate voltage (the body factor m = 1 + C_dep/C_ox) comes directly from the Poisson solution in the channel depletion region and determines how much gate voltage is required to invert the channel.
- Quantum Correction Need: The classical Poisson equation places peak electron density exactly at the oxide interface; quantum mechanics pushes it approximately 1nm away. This discrepancy, visible in the Poisson solution, motivated the development of quantum correction models (Schrodinger-Poisson and density-gradient coupling).
How the Poisson Equation Is Solved in Practice
- Linearization: The Newton-Raphson method linearizes the Poisson equation at each iterate by expanding the nonlinear carrier density terms in a Taylor series around the current potential estimate, solving a linear system at each Newton step.
- Meshing: The accuracy of the Poisson solution depends critically on mesh density — fine mesh spacing (0.1-1nm) is required in the depletion region and inversion layer where potential varies rapidly; coarser mesh is adequate in neutral bulk regions.
- Coupled Iteration: In full device simulation, the Poisson, electron continuity, and hole continuity equations are coupled — the standard approach is either fully coupled (simultaneously solving all three at each Newton step) or decoupled (Gummel iteration, solving each equation sequentially until convergence).
Poisson Equation is the electrostatic law that governs every aspect of semiconductor device potential and charge distribution — its solution defines the band diagram, depletion width, threshold voltage, and electric field profile that determine device behavior, making it the most fundamental equation in device physics and the central computation in every TCAD solver from the simplest 1D diode analyzer to the most advanced 3D FinFET simulation.