Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

Chapter A The Basic Semiconductor Equations

The Poisson equation and continuity equations, derived from Maxwell’s equations, form the basis of semiconductor device modeling describing the structure of the field problem. Since the modern semiconductor devices satisfy the criterion for quasi-electrostatic approximation¹, the displacement current ( $\partial D/\partial t$ ) and induction ( $\partial B/\partial t$ ) can be neglected in Maxwell’s equations. Thus, the right hand side in Faraday’s law $\nabla \times E=\partial B/\partial t$ becomes zero and allowing introduction of a scalar potential $\psi$ .

$(A.1) \begin{equation} E=-\nabla \psi , \end{equation}$

when coupled with Gauss’ law ( $\nabla .D=\rho$ )

$(A.2) \begin{equation} \nabla .(\hat {\epsilon }.\nabla \psi )=-\rho . \end{equation}$

Assuming the permittivity to be scalar and spatially dependent,

$\begin{eqnarray*} \epsilon .\nabla ^{2}\psi & = & -\rho , \end{eqnarray*}$

$\begin{eqnarray*} \nabla ^{2}\psi & = & -\frac {\rho }{\epsilon } \end{eqnarray*}$

$(A.3) \begin{equation} \nabla ^{2}\psi =-\frac {q}{\epsilon }\left (n-p-C\right ),\label {eq:poisson's} \end{equation}$

where n is the electron density, p the hole density and C the density of ionized impurities and dopants. Equation A.3 is known as Poisson’s equation. The continuity equations are laid down for the electron and hole current density ( $J_{\mathrm {n}}$ , $J_{\mathrm {p}}$ , respectively).

$(A.4) \begin{equation} \nabla .J\mathrm {_{n}}-q\frac {\partial n}{\partial t}=qR,\label {eq:continuity_n} \end{equation}$

$(A.5) \begin{equation} \nabla .J\mathrm {_{p}}-q\frac {\partial p}{\partial t}=-qR,\label {eq:continuity_p} \end{equation}$

where R denotes the net recombination rate. i.e. rate of electron-hole pair generation minus rate of recombination. Since the total number of charge particles remains constant, the Equations. A.4 and A.5 add up to zero. The unknowns in the three equations (eqns. A.3, A.4, and A.5) are $\psi$ , $n$ , $p$ , $J_{\mathrm {n}}$ , $J_{\mathrm {p}}$ (neglecting $R$ , considering detailed balance for simplification). Thus two more equations are needed to evaluate the unknowns, which are endowed by the transport models.

¹ Quasi-electrostatic approximation is valid when the characteristic length of a system is considerably ( $\sim$ factor of $10$ ) smaller than the shortest electromagnetic wavelenght present in the system.