# Minority Carrier Diffusion Equations

### Assumptions:

Assume:
To take for granted; suppose.
Assumption:
A statement accepted or supposed true without proof or demonstration.
We make assumptions when modeling a semiconductor in various applications:

#### Why do we make assumptions?

We use the continuity equations to model the processes that are taking place inside a semiconductor. They are complicated equations containing partial differentials, one for each carrier action (drift, diffusion, thermal R-G, and other processes). Making assumptions about the area(s) we are studying allows us to simplify the mathematical equations into something we can solve by hand. However, all assumptions must be tested.
Assumptions made when using the continuity equations
We do not make any assumptions when using the continuity equations because they take into account all the processes that are occurring within a semiconductor.  The continuity equations can be solved only by computer since they are too mathematically complicated to allow a nice analytical solution.
Assumptions are needed to obtain the minority carrier diffusion equations
The continuity equations are mathematically complicated equations containing partial differentials that describe the change in carrier concentration due to individual processes; i.e., drift, diffusion, recombination, generation, and other processes.  These equations can be solved only by computer.  The minority carrier diffusion equations are derived from the continuity equations and are used to model semiconductors by obtaining the rate of change of the minority carrier concentration with respect to space and/or time.
The following assumptions are made in order to derive the minority carrier diffusion equations from the continuity equations, and these conditions must exist in order to use the MCDEs:
1. The semiconductor is one dimensional, all derivatives are with respect to one coordinate; usually called the x-coordinate.
2. We are only working with minority carriers.
3. An electric field does not exist in the semiconductor region we are analyzing.
4. Equilibrium minority carrier concentrations are not a function of position, meaning the doping is uniform.
5. We assume low-level injection conditions exist.
6. Thermal recombination-generation occurs indirectly through traps in the bandgap.
7. The only other process that may occur would be generation caused by shining light on the semiconductor.
Assumptions made when analyzing pn-junction diodes
When modeling a pn-junction diode, we are solving for the minority carrier current densities in the diode as a function of position and the applied voltage. In order to do this we must first solve the MCDE to obtain Dnp and Dpn. Since we use the MCDEs to analyze diodes, all those assumptions are made and must be valid.   We also assume:
1. Operation is under steady state conditions.
2. The doping profile is a nondegenerately doped step junction.
3. The diode is one dimensional, all derivatives are with respect to one coordinate; usually the x-coordinate.
4. Assume low-level injection in quasineutral regions.
5. Only drift, diffusion, and thermal recombination-generation occur within the diode.
6. The diode is in the dark, GL = 0.
7. No R-G in space charge region (depletion region).