Finite Difference Charge Transport Solver (FDCharge)

The FD Charge Transport Solver is based on the drift-diffusion model and self-consistently solves the electrostatic potential and carrier distributions by coupling with Poisson’s equation, enabling accurate reproduction of charge transport processes in semiconductor devices. This model represents a commonly used simplified approach in semiconductor device simulations; its basic physical principles are outlined as follows:

(1) Current density equations for electrons and holes (The drift-diffusion model)

Phenomenologically, three primary factors drive carrier motion in semiconductors: electric potential gradients (drift under an electric field), carrier concentration gradients (diffusion due to non-uniform carrier distribution), and thermal gradients (Seebeck effect). Carrier motion under electromagnetic fields is referred to as drift, whereas motion driven by carrier concentration and temperature gradients is called diffusion. Consequently, the current density equations for electrons (n-type) and holes (p-type) can be expressed as:

$\begin{aligned} \mathbf{J}_n &= q\mu_n n \mathbf{E} + q\mathbf{D}_n \nabla n \\ \mathbf{J}_p &= q\mu_p p \mathbf{E} - q\mathbf{D}_p \nabla p \end{aligned}$

Here,

• $J_n, J_p$: the electron and hole current densities;
• $q$: the elementary charge;
• $\mu_n, \mu_p$: the mobilities of electrons and holes;
• $D_n, D_p$: the diffusivities of electrons and holes;
• $\mathbf{E} = -\nabla \phi$: the electric field derived from the electrostatic potential $\phi$;
• $n, p$: the electron and hole densities.

(2) Poisson equation

To solve the drift-diffusion model, the electric field must be known. Introducing the electric displacement and potential:

$\nabla\cdot\mathbf{D} = \rho, \qquad \mathbf{D} = \varepsilon \mathbf{E}$

Poisson's equation then reads:

$\nabla\cdot(\varepsilon \nabla \phi) = -q\left(p - n + N_D^+ - N_A^-\right)$

where $\varepsilon$ is the dielectric permittivity, $p,n$ are the hole and electron densities, and $N_D^+, N_A^-$ denote ionized donor and acceptor concentrations, respectively.

(3) Carrier continuity equations

Carriers are depleted through radiative (including spontaneous and stimulated emission) and non-radiative mechanisms. Introducing the net electron–hole recombination rate $R$, the continuity equations describing the time evolution of carrier concentrations to keep charge conservation can be obtained as follows:

$\begin{aligned} \frac{\partial n}{\partial t} &= \frac{1}{q} \nabla \cdot \mathbf{J}_n - R \\ \frac{\partial p}{\partial t} &= -\frac{1}{q} \nabla \cdot \mathbf{J}_p - R \end{aligned}$

Here, $R$ denotes the net recombination rate (e.g., Shockley–Read–Hall, Auger, radiative recombination). The recombination rate depends on temperature, doping, carrier concentration, and electric field; the dominant mechanisms should be chosen based on the specific application.

These three sets of equations form a nonlinear, coupled system of partial differential equations, which SimWorks solves numerically using finite-difference discretization.

(4) Numerical methodology

• Multiscale issues: Due to large spatial variations in device dimensions and wide ranges of carrier concentrations, appropriate scaling is necessary to avoid numerical overflow/underflow and to improve stability.
• Convergence difficulties: Standard central differencing may fail to converge under strong nonlinear coupling; therefore, the Scharfetter–Gummel (SG) scheme is employed to discretize the semiconductor current density J.
• Strong nonlinear coupling: The variables form a tightly coupled nonlinear loop— $n \leftrightarrow p \leftrightarrow \phi$ —requiring a self-consistent iterative solution, such as Gummel iteration or the Newton–Raphson method.