# Maxwell’s Equations¶

The Maxwell’s equations are: and the Lorentz force is: where: This corresponds to: ## Four Potential¶

The four potential is defined by: this corresponds to: The Maxwell’s equations can then be written as (note that the two eq. without sources are automatically satisfied by the four potential): where we have employed the Lorentz gauge .

# Semiconductor Device Physics¶

In general, the task is to find the five quantities: where ( ) is the electron (hole) concentration, ( ) is the electron (hole) current density, is the electric field.

And we have five equations that relate them. We start with the continuity equation: where the current density is composed of electron and hole current densities: and the charge density is composed of mobile (electrons and holes) and fixed charges (ionized donors and acceptors): where and is the electron and hole concetration, is the net doping concetration ( where is the concentration of ionized donors, charged positive, and is the concentration of ionized acceptors, charged negative) and is the electron charge (positive). We get: Assuming the fixed charges are time invariant, we get: where is the net recombination rate for electrons and holes (a positive value means recombination, a negative value generation of carriers). We get the carrier continuity equations:

(1) Then we need material relations that express how the current is generated using and and . A drift-diffusion model is to assume a drift current ( ) and a diffusion ( ), which gives:

(2) where , , , are the carrier mobilities and diffusivities.

Final equation is the Gauss’s law: (3) ## Equations¶

Combining (2) and (1) we get the following three equations for three unknowns , and : And it is usually assumed that the magnetic field is time independent, so and we get:

(4) These are three nonlinear (due to the terms and ) equations for three unknown functions , and .

### Example 1¶

We can substract the first two equations and we get: and using and , we get: So far we didn’t make any assumptions. Most of the times the net doping concetration is time independent, which gives: Assuming further , we just get the equation of continuity and the Gauss law: Finally, assuming also that that doesn’t depend on time, we get: ### Example 2¶

As a simple model, assume , , , and are position independent and , : Using we get: ## Example 3¶

Let’s calculate the 1D pn-junction. We take the equations (4) and write them in 1D for the stationary state ( ): We expand the derivatives and assume that and is constant: and we put the second derivatives on the left hand side:

(5) now we introduce the variables : and rewrite (5): So we are solving the following six nonlinear first order ODE:

(6) 