Metal-semiconductor junctions can be classified into two kinds:
(i) Rectifying Schottky barrier diodes
(ii) Ohmic Contacts

Schottky barrier diode is an important semiconductor device in itself with applications including high speed rectifiers, Photodetectors, etc
Important part of other devices such as MESFETs, HEMTs etcMetal semiconductor Schottky barrier diodes are used as test structures for measuring doping, defect properties etc.
Metal-semiconductor Ohmic contacts are an essential part of all semiconductor devices.
We begin our discussions with an analysis of the junction in equilibrium. As a first step,we will determine the energy band diagram of the junction.

We will first sketch energy band diagram for an ideal metal-semiconductor junction having the following characteristics:

Semiconductor surface same as bulk
No interfacial oxides etc
Semiconductor: uniformly doped N-type Silicon
Metal: Aluminum

The energy band diagram of the junction is determined by first drawing the band diagrams of metal and the semiconductor separately and then suitably combining them.

Energy-band diagram before equilibrium
After equilibrium, Fermi levels will align. This will be accompanied with transfer of electrons from semiconductor(higher Fermi level) to metal (lower Fermi level)
Far from junction: band diagram of semiconductor same as before can be easily drawn.
Metal unaffected by addition of small number of electrons: same as before so that the following diagram can be easily sketched.

Energy-band diagram after equilibrium

As we approach the junction, the semiconductor gets progressively depleted of electrons: bands must bend upwards.

Barrier Height

The barrier height can be expressed as:

Whenever two materials are brought into contact, a contact potential develops upon attainment of equilibrium. The contact potential is the difference in their work functions.

Example 1.1 Determine the contact potential for the following sets of metals deposited on N-type Silicon and compare it with experimentally measured values given below:

Solution : Using Eq.(1), we obtain the following set of values:

Metal	Work Function	Calculated	Experimental
Aluminium	4.28	0.27	0.7
Gold	5.1	1.09	0.79
Titanium	4.33	0.32	0.5
Tungsten	4.55	0.54	0.67

It can be seen from the table that the theoretically predicted barrier heights do not match well with the experimentally measured values at all. Despite change in workfunctions, the barrier heights do not change as much.

Reason for lack of agreement between theory and experiment:
Wrong Assumption: Surface has characteristics same as the bulk semiconductor

Surface has a large number of energy levels within the bandgap unlike the bulk semiconductor. These energy levels are almost distributed in a continuous manner and are described by surface state density

Simple model of a Semiconductor surface:
Surface states either donor or acceptor-like, with each kind distributed within the bandgap.

Assumption: States above Fermi energy are unoccupied and those below it are occupied

Donor states above Fermi energy will be positively charged.

Acceptor states below Fermi energy will be negatively charged.

Semiconductor surface in general will be charged.

The net charge at the surface can be expressed as :

Eq. (3) shows that when

is zero (at the valence band), then the surface would be positively charged and when

is equal to

, the surface would be negatively charged.

Concept of a Neutral Level

We can find a position for the surface Fermi-level somewhere within the bandgap such that net charge at the surface is zero.

One can define now an energy level called neutral Level N E such that if Fermi energy at the surface were equal to it, the net charge at the surface would be zero.

The first tem is zero by the definition of neutral level and the second term can be written as

Implications of Neutral Level:

If the Fermi energy at the surface(

) coincides with the neutral level

, net charge at the surface is zero.

Explanation: Compared to the case where

, there are now some additional states lying between

and

which are occupied(shaded area)

The consequences of a large surface state density is that Fermi energy at the surface is pinned to the neutral level irrespective of other factors such as doping in the bulk etc

Explanation: For simplicity, assume surface state density to be constant

This charge must be balanced by charge in the bulk

For an N-type semiconductor, positive depletion charge will balance the negative surface charge with energy band diagram as shown:

Example 1.2 : For an N-type semiconductor of doping

, determine the difference

for surface state densities of

Solution : This can be obtained using Eq.(13) provided Vbi is known. However, Vbi itself requires knowledge of

. We can get around this circularity by assuming some value of Vbi, calculate

using Eq. (13) and then with the revise value of built-in voltage evaluate Eq.(13) again. The solution is obtained after a few iterations and is shown below:

The results show that for

the Fermi level is almost pinned to the neutral level. It can be verified that if the doping is changed to say

, the Fermi level at the surface remains pinned for high surface state densities. The table above shows that for surface state densities less than

, there is virtually no band bending and Fermi level at the surface is same as that in the bulk.

If surface state density is large(often taken as

for most surfaces), then for all practical doping levels, Fermi energy at the surface is pinned to the neutral level.

Consequences for Schottky barrier height

Pinning of surface Fermi level at the neutral level implies:

The value given by Eq. (14) also does not agree all the time with experimental values if neutral level is assumed to remain at 0.33 eV

A more sophisticated model i that takes into account presence of interfacial oxide(removal of second assumption) gives a better match with experimental results

All the results derived for N-type Schottky barrier also apply to Schottky barriers on P-type semiconductors as well.

Schottky barrier on P-Type Si

Example 1.3 : Determine the contact potential for the following sets of metals deposited on P-type Silicon using the ideal work function difference theory and compare it with experimentally measured values given below:

Solution : Using Eq.(15), we obtain the following set of values:

Metal	Work Function(eV)	Calculated	Experimental
Aluminium	4.28	0.85	0.58
Gold	5.1	0.03	0.34
Titanium	4.33	0.8	0.61
Tungsten	4.55	0.58	0.45

It can be seen that the workfunction theory does not explain the experimental results at all. According to the workfunction theory the sum of N-type Schottky barrier height and P-type Schottky barrier height should add up to the bandgap. This is roughly true for the experimentally measured values for Gold Titanium and Tungsten but not for Aluminum.

Neither the simple work function difference theory nor the simple Fermi level pinning theory adequately explains the experimental values. A more complicated model that takes into account voltage drop across interfacial oxide gives a better match.
Almost all the metals form a junction with Silicon such that a barrier exists for the flow of electrons in N-type material and for holes in P-type material.
A consequence of this is that semiconductor near the junction is depleted of carriers and a space charge region exists.
The detailed nature of potential variation, electric field, space charge region’s width etc can be obtained through the solution of Poisson Equation.

Depletion Approximation

Assumptions:
(i) Neglect

(ii) All donors are ionized
(iii) Neglect

within the space charge region

The neglect of electron and hole density within the space charge region is known as thedepletion approximation. The validity of this assumption will be discussed in detail during the study of PN junctions.

Charge Profile after depletion approx.
With the depletion approximation, the Poisson equation can be easily solved to obtain the following important results:
Although derived for equilibrium, the equations remain valid for non zero values of applied voltage also, provided depletion approximation is assumed to hold. The only change that needs to be made is to substitute Vbi by Vbi - V , where V is the voltage applied between the metal and the semiconductor.

Example 1.4 : Determine the value of maximum electric field and depletion width for a Schottky barrier on N-type Silicon of doping

. Assume that

=0.7eV.

Solution :