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The usual way to explain the formation of a Schottky barrier between a metal and semiconductor is the following:

Suppose a metal and a semiconductor with different work functions are put into contact. To compensate for the work function difference ΔW of the two materials a charge layer will build up at the interface creating an electrostatic potential φ(z) s.th.

-eφ(z=0) = ΔW

Where z=0 is at the interface. Now what I don't understand is this. Since the metal is a perfect conductor we must have that φ(0) = 0 at the interface between the metal or at least that φ tends to a constant. The usual way to solve this problem is to employ the method of images, i.e. we mirror the charge distribution in the semiconductor at the interface, since this effectively is the same as solving the problem of having a charge distribution above an infinitely grounded plane at z=0, in which case (V(z=0)=0, V(∞)=0) are the appropriate boundary conditions.

But are these two requirements compatible? For me it seems that the latter requires that there can be no correction to the conduction band energy at the metal-semiconductor interface, which is exactly what we want when ΔW≠0.

Suppose a metal and a semiconductor with different work functions are put into contact. To compensate for the work function difference ΔW of the two materials a charge layer will build up at the interface creating an electrostatic potential φ(z) s.th.

-eφ(z=0) = ΔW

Where z=0 is at the interface. Now what I don't understand is this. Since the metal is a perfect conductor we must have that φ(0) = 0 at the interface between the metal or at least that φ tends to a constant. The usual way to solve this problem is to employ the method of images, i.e. we mirror the charge distribution in the semiconductor at the interface, since this effectively is the same as solving the problem of having a charge distribution above an infinitely grounded plane at z=0, in which case (V(z=0)=0, V(∞)=0) are the appropriate boundary conditions.

But are these two requirements compatible? For me it seems that the latter requires that there can be no correction to the conduction band energy at the metal-semiconductor interface, which is exactly what we want when ΔW≠0.

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