Consider a p-type semiconductor that is homogeneous andinfinite in extent. Assume a zero applied electric field. For a one-dimensional crystal, assume that excess carriers are beinggenerated at x = 0 only, as indicated in Figure. The excess carriers beinggenerated at x = 0 will begin diffusing in both the – x and +x directions.Calculate the steady-state excess carrier concentration as a function of x.
Question
Consider a p-type semiconductor that is homogeneous andinfinite in extent. Assume a zero applied electric field. For a one-dimensional crystal, assume that excess carriers are beinggenerated at x = 0 only, as indicated in Figure. The excess carriers beinggenerated at x = 0 will begin diffusing in both the – x and +x directions.Calculate the steady-state excess carrier concentration as a function of x.
Solution
The steady-state excess carrier concentration in a semiconductor can be described by the diffusion equation. In one dimension, with generation at x=0 and no applied electric field, the diffusion equation simplifies to:
d²n/dx² = G/D
where:
- n is the excess carrier concentration,
- G is the generation rate of the excess carriers,
- D is the diffusion coefficient.
The general solution to this equation is:
n(x) = Ax + B + (G/D)x²
where A and B are constants.
Since the excess carriers are being generated at x=0, the boundary conditions are:
n(0) = Gτ, dn/dx(0) = 0
where τ is the carrier lifetime.
Applying these boundary conditions gives:
B = Gτ, A = 0
So the steady-state excess carrier concentration as a function of x is:
n(x) = Gτ + (G/D)x²
This equation describes how the excess carrier concentration varies with position in the semiconductor.
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