Editing Continuity equation (section)

==Semiconductor==
The total current flow in the semiconductor consists of [[drift current]] and [[diffusion current]] of both the electrons in the conduction band and holes in the valence band.

General form for electrons in one-dimension:
<math display="block">\frac{\partial n}{\partial t} = n \mu_n \frac{\partial E}{\partial x} + \mu_n E \frac{\partial n}{\partial x} + D_n \frac{\partial^2 n}{\partial x^2} + (G_n - R_n)</math>
where:
* ''n'' is the local concentration of electrons
* <math>\mu_n</math> is [[electron mobility]]
* ''E'' is the electric field across the [[depletion region]]
* ''D<sub>n</sub>'' is the [[diffusion coefficient]] for electrons
* ''G<sub>n</sub>'' is the rate of generation of electrons
* ''R<sub>n</sub>'' is the rate of recombination of electrons

Similarly, for holes:
<math display="block">\frac{\partial p}{\partial t} = -p \mu_p \frac{\partial E}{\partial x} - \mu_p E \frac{\partial p}{\partial x} + D_p \frac{\partial^2 p}{\partial x^2} + (G_p - R_p)</math>
where:
* ''p'' is the local concentration of holes
* <math>\mu_p</math> is hole mobility
* ''E'' is the electric field across the [[depletion region]]
* ''D<sub>p</sub>'' is the [[diffusion coefficient]] for holes
* ''G<sub>p</sub>'' is the rate of generation of holes
* ''R<sub>p</sub>'' is the rate of recombination of holes

===Derivation===

This section presents a derivation of the equation above for electrons. A similar derivation can be found for the equation for holes.

Consider the fact that the number of electrons is conserved across a volume of semiconductor material with cross-sectional area, ''A'', and length, ''dx'', along the ''x''-axis. More precisely, one can say:
<math display="block">\text{Rate of change of electron density} = (\text{Electron flux in} - \text{Electron flux out}) + \text{Net generation inside a volume}</math>

Mathematically, this equality can be written:
<math display="block">\begin{align}
 \frac{dn}{dt} A \, dx &= \left[J(x+dx)-J(x)\right]\frac{A}{e} + (G_n - R_n)A \, dx \\
 &= \left[J(x)+\frac{dJ}{dx}dx-J(x)\right]\frac{A}{e} + (G_n - R_n)A \, dx \\[1.2ex]
 \frac{dn}{dt}         &= \frac{1}{e}\frac{dJ}{dx} + (G_n - R_n)
\end{align}</math>Here ''J'' denotes current density(whose direction is against electron flow by convention) due to electron flow within the considered volume of the semiconductor. It is also called electron current density.

Total electron current density is the sum of drift current and diffusion current densities:
<math display="block">J_n = en\mu_nE + eD_n\frac{dn}{dx}</math>

Therefore, we have
<math display="block">\frac{dn}{dt} = \frac{1}{e}\frac{d}{dx}\left(en\mu_n E + eD_n\frac{dn}{dx}\right) + (G_n - R_n)</math>

Applying the product rule results in the final expression:
<math display="block">\frac{dn}{dt} = \mu_n E\frac{dn}{dx} + \mu_n n\frac{dE}{dx} + D_n\frac{d^2 n}{dx^2} + (G_n - R_n)</math>

===Solution===
The key to solving these equations in real devices is whenever possible to select regions in which most of the mechanisms are negligible so that the equations reduce to a much simpler form.