**Branch :**Computer Science and Engineering

**Subject :**Fundamental of Electronic Devices

## The Continuity Equation

In the discussion of diffusion of excess carriers, we have thus far neglected the important effects of recombination. These effects must be included in a description of conduction processes, however, since recombination can cause a variation in the carrier distribution.

**Continuity Equation for holes:
**

For example, consider a differential length Δx of a semiconductor sample with area A in the yz-plane (Fig given below). The hole current density leaving the volume, J_{p}(x Δx), can be larger or smaller than the current density entering, J_{p}(x), depending on the generation and recombination of carriers taking place within the volume. The net increase in hole concentration per unit time, dp/dt, is the difference between the hole flux per unit volume entering and leaving, minus the recombination rate. We can convert hole current density to hole particle flux density by dividing J_{p} by q. The current densities are already expressed per unit area; thus dividing Jp(x)/q by Δx gives the number of carriers per unit volume entering ΔxA per unit time, and (1/q)J_{p}(x Δx)/Δx is the number leaving per unit volume and time:

As Ax approaches zero, we can write the current change in derivative form:

The expression is called the continuity equation for holes.

**Continuity Equation for electrons:
**

we can write

since the electronic charge is negative. When the current is carried strictly by diffusion (negligible drift), we can replace the currents in above equ by the expressions for diffusion current; for example, for electron diffusion we have

Substituting this into the continuity equ for holes we obtain the diffusion equation for electrons

and similarly for holes,

These equations are useful in solving transient problems of diffusion with recombination.