Forward bias in a silicon p-n junction diode

[Whistlekins]

Homework Statement

The forward bias current I in the diode described (silicon p-n junction) varies as a function of the voltage V and temperature T described by the formula I = I_s*|e^[(|e|*V]/[n*k_b*T]) - 1| where |e| is the fundamental charge, k_b is the Boltzmann constant, I_s is the reverse saturation current. n is the ideality factor. For all intents and purposes we will assume that n = 1.

Also, I_s is related to temperature with I_s = A*T^3 * e^(-E_g/[n*k_b*T]), A and E_g are constant.


The questions: (1) Under forward bias conditions, the exponent |e|V/nk_bT >> 1. Why is this so? (2) And why does the current I vary markedly with the voltage V and the temperature T?

Homework Equations

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The Attempt at a Solution

My guess is that for (1), unless V-T ratio is very very small (10^-5), the exponent will turn out magnitudes higher than 1, since |e|/nk_b ~ 11594. The question is in relation to an experiment in which the values of T vary from 0K - 300K, and the smallest value of V is 0.4V. The smallest V-T ratio would then be on order 10^-3, which when multiplied by the exponent, still gives a value of 15.

As for (2), I'm not quite sure what it's asking. Obviously increasing the voltage would increase the current exponentially, but I think they want the underlying priciples as to why it's exponential rather than not. Similarly for T, since it has a cubed term in I_s, it would produce a great variance in I.

I apologise for poor equation formatting, I'm not too familiar with latex.

 

[mfb]

V-T ratio

This ratio has units V/K - do not ignore units, it will lead to many problems.
For T<=300K, k_bT is of the order of 1/40 eV or less, which directly gives an exponent larger than 1 for a voltage of more than 25mV.

Similarly for T, since it has a cubed term in I_s, it would produce a great variance in I.

Exponential functions usually dominate over polynomials (like T^3).
The exponent is the ratio of electric energy of an electron to its thermal energy (up to a prefactor).