[3.1.29]
The diffusion current in a metal-semiconductor diode is derived based on the assumption that the depletion layer is large compared to the mean free path, so that the concepts of drift and diffusion are valid. We start from the expression for the total current and then integrate it over the width of the depletion region:
[3.1.29]
which can be rewritten by using E = -df/dx and multiplying both sides of the equation with exp(-f/Vt), yielding:
[3.1.30]
Integration of both sides of the equation over the depletion region yields:
[3.1.31]
Where the following values were used for the electron density and the potential:
|
x |
n(x) |
f (x) |
|
0 |
Nc exp(-fB/V t) |
- f i + Va |
|
x d |
N d = N c exp(-fB/V t) exp(fi/V t) |
0 |
and f* = f + fi - Va. The integral in the denominator can be solved using the potential obtained from the full depletion approximation [3.1.5], or
[3.1.32]
so that f* can be written as:
[3.1.33]
The second term was dropped since the linear term is dominant because x << xd. Using this approximation one can solve the integral as:
[3.1.34]
for (fi – Va) > Vt. This yields the final expression for the current due to diffusion:
[3.1.35]
This expression indicates that the current depends exponentially on the applied voltage, Va, and the barrier height, fB. The prefactor can be understood physically if one rewrites that term as a function of the electric field at the metal-semiconductor interface, Emax:
[3.1.35a]
yielding:
[3.1.35b]
so that the prefactor equals the drift current at the metal-semiconductor interface, which for zero applied voltage exactly balances the diffusion current.
© Bart Van Zeghbroeck 1997