Minority Carrier Diffusion Equations
We make assumptions when modeling a semiconductor in various applications:
To take for granted; suppose.
A statement accepted or supposed true without proof or demonstration.
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
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
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
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:
The semiconductor is one dimensional, all derivatives are with respect
to one coordinate; usually called the x-coordinate.
We are only working with minority carriers.
An electric field does not exist in the semiconductor region we are analyzing.
Equilibrium minority carrier concentrations are not a function of position,
meaning the doping is uniform.
We assume low-level injection conditions exist.
Thermal recombination-generation occurs indirectly through traps in the
The only other process that may occur would be generation caused by shining
light on the semiconductor.
Assumptions made when analyzing pn-junction
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:
Operation is under steady state conditions.
The doping profile is a nondegenerately doped step junction.
The diode is one dimensional, all derivatives are with respect to one coordinate;
usually the x-coordinate.
Assume low-level injection in quasineutral regions.
Only drift, diffusion, and thermal recombination-generation occur within
The diode is in the dark, GL = 0.
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:
Thermal recombination-generation is insignificant everywhere in both the
emitter-base (EB) and collector-base (CB) depletion regions.
The widths of the emitter and collector are much greater than LN