3.2 Electrostatic analysis of an M-S junction

Electrostatic analysis

1. Full depletion analysis
2. Barrier lowering due to image charges in the metal.
3. Linearized Poisson equation
4. Exact analysis for non-degenerate M-S junctions.

3.2 Electrostatic analysis of an M-S junction

We define the barrier height, f_B, for a junction containing n-type material as the difference between the metal work function, F_M, and the electron affinity, c. For p-type material it is given by the difference between the valence band edge, E_C, and the Fermi energy, E_M in the metal:

(msj1)

(msj2)

In addition we define the built-in potential, f_i, as the difference between the Fermi energy of the metal and that of the semiconductor.

(msj3)

(msj4)

msfig2.gif

We first present the electrostatic analysis based on the full depletion approximation. This approximation assumes that the charge density, r, equals q N_d in the "depletion region" and equals zero outside the depletion region. This analysis provides a series of useful analytic expressions which relate the applied voltage to the depletion layer width, the electric field and the potential in the semiconductor.

Next we derive an expression for the barrier lowering due to the potential caused by the image charge in the metal. This correction to the barrier height is especially of interest if the semiconductor is highly doped and causes the reverse bias current to be voltage dependent.

This derivation is followed by a small potential analysis of the M-S junction which can be adequately described with a linearized Poisson equation. We introduce the Debye length which is the characteristic length related to the equation.

Finally we present an exact solution to the M-S junction with a non-degenerate carrier density. An analytic expression for the charge, electric field and capacitance is obtained as a function of the applied voltage. A numeric solution for the charge density, electric field and potential as a function of position is also provided.