Minority Carrier Diffusion Equations

Assume:

To take for granted; suppose.

Assumption:

A statement accepted or supposed true without proof or demonstration.

• In using the continuity equations,
• In deriving the minority carrier diffusion equations,
• In analyzing diodes,
• In analyzing transistors.

Why do we make assumptions?

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 Dn_p and Dp_n. 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, G_L = 0.
7. No R-G in space charge region (depletion region).

Assumptions made when analyzing BJTs

Solving for BJT parameters and terminal currents involves assumptions that parallel those for the pn-junction diode, then we add:

1. Thermal recombination-generation is insignificant everywhere in both the emitter-base (EB) and collector-base (CB) depletion regions.
2. The widths of the emitter and collector are much greater than L_N or L_P.