The famous Schottky-Mott relationship,
where fM is the work function
of the metal and cS is the
electron affinity of the semiconductor,
can be viewed as a zeroth-order theory on the formation of the SBH.
Even though the barrier now bears his name, the theory that
Schottky advanced has not demonstrated much relevance for real
Schottky's main contribution, instead, lies in correctly predicting
the electric potential and field distribution in the space charge region.
A common perception of the Schottky-Mott relationship is that this
rule predicts the barrier height in the absence of interface states.
Perhaps the best way to see the rationale behind this perception
is to derive the Schottky-Mott condition in a
In this exercise, large flat
surfaces of metal and semiconductor are assumed to be placed parallel
to each other and with a small gap
dgap separating them.
An electrical connection is made externally, thus equating the
Fermi level of the two crystals ( Va = 0 ).
For a uniformly doped, non-degenerate semiconductor, the total charge
per unit area arising from charges in the depletion region is
QSC = (2eeS ND
Vbb )1/2. If the semiconductor has no surface states,
as assumed, the space charge, QSC, is the only source of
charge on the semiconductor surface. The charge on the semiconductor
is balanced by a charge of equal magnitude, but of opposite sign,
on the surface of the metal. These charges lead to a constant electric
field in the gap between the semiconductor and the metal, much like that
in a parallel plate capacitor, and a total potential drop of
- dgap QSC /
Since the electric potential is continuous everywhere, the electric field
in the gap is also required to be, according to this figure,
cS + eVbb + eVN -
Combining Eqs. (2) and (3) allows the total band bending to be determined
as a function of the gap width,
as plotted in (b) of the this figure.
As expected, the band bending is zero when the separation is large.
And when the gap collapses ( dgap -> 0 ),
one gets what amounts to the Schottky-Mott condition.
So, the Schottky-Mott relationship can be viewed as the asymptotic
result of the band lineup when a semiconductor without surface states
approaches a metal. There is thus some perceived connection between
the Schottky-Mott relationship and the absence of surface states.
This perception, even though expressed profusely in the literature,
Some may be quick to point out that the Schottky-Mott relationship is
also trivially obtained if the vacuum level outside the metal is lined
up with the vacuum level outside the semiconductor. What this entails
is for the electric field to vanish in the gap between the metal and semiconductor,
which can be accomplished by applying an external bias of (
cS - eVbb - eVN )
to the semiconductor, as illustrated in
a second Gedanken experiment.
With such a bias applied, the semiconductor band bending will be independent
of the width of gap and the semiconductor can approach the metal with its CBM
already positioned at fM -
cS above the EF of the metal.
Again, the Schottky-Mott relationship can be inferred as the asymptotic result when
the gap becomes zero. Note that this result is always obtained, independent
of the density of surface states on the semiconductor.
Therefore, the Schottky-Mott relationship has nothing to do with the
density of surface states.
Rather, it is simply related to the absence of a gap between the metal
and the semiconductor. This somewhat disturbing result is of course just
an artifact of the analysis model that has been used. The analysis model
assumes that the interface dipole arises from a transfer of charge between
the metal and some pre-existing states on the semiconductor surface.
So when the gap vanishes, the charge is not transferred over any
distance and there can be no interface dipole!
With a grasp of the concept of the interface specific region (ISR),
it is plain to see what the Schottky-Mott relationship really implies about the
charge distribution at such a MS interface. It can be shown that the Schottky-Mott
relationship is equivalent to
DISR = DhklM -
namely, the interface dipole is the difference between the surface dipole
of the metal and that of the semiconductor.
This condition is trivially satisfied if the charge distributions on the two original,
isolated surfaces are maintained, or frozen-in, at the eventual MS interface.
In other words, the Schottky-Mott relationship results from an absence of
charge rearrangement or interaction. Since significant charge redistribution
at an MS interface usually occurs when new bonds are formed and/or when old bonds
are broken, the Schottky-Mott relationship is not expected to give accurate
predictions of the SBH at MS interfaces where bonding is unavoidable.
And, as we all know, bonding is unavoidable at all stable MS interfaces.