Tunneling at metal-semiconductor contact

We consider next a contact between a metal and a semiconductor which has a band gap Cgap. For an intrinsic semiconductor without any impurities, the Fermi level will be in the middle of the band gap (see chapter 9). At equilibrium the Fermi level on both sides of the contact must be the same. We will assume that the gap is sufficiently 

Subtracting the two expressions to obtain the total current flowing through the contact we find 

What is usually measured experimentally is the so called differential conductance, given by the derivative of the current with respect to applied voltage  

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