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First of all, the functional relation between current and voltage drop
across a diode is an exponential, not a power.
An exponential increases much fast that a quadratic.
Second, the actual f(V) function is not simply an exponential, since there
is turn-on point, below which the diode hardly conducts at all.
The function that closely approximates the I-V relation of a diode is I =
Irs(exp(Vd/Vt - 1), where Irs is called the reverse saturation current
(or current for reverse bias,
a few nA for Si, micro-A for Ge),