When a metal is joined by a semiconductor, and thermodynamic
equilibrium is reached, chemical bonding
has to take place.
At an ordinary, polycrystalline MS interface, the bonding
geometry likely changes from place to place, leading to a
locally varying interface dipole.
The measured SBH then reflects some weighted average of this
Because of the randomness of the interface structure, there is the
expectation that the interface dipole can perhaps be estimated
using bulk-derived properties.
It therefore makes some sense to analyze the electric dipole
at a MS interface using established techniques in chemical
physics, developed largely for molecular systems.
The total energy of a multi-atomic molecule can be written,
neglecting higher order terms, as
Etot( QA, ... QN) = S
[ EAo + UAQA + (1/2) YAQA2 ]
+ S (1/2) QAQBJAB},
where EAo is the energy of atom A in the uncharged state,
-eQA is the net charge on atom A,
UA (= cA /2 + IA /2 ) is the
Mulliken potential, YA (= IA - cA )
is the idempotential, and
JAB (= e2 / eo dAB )
is the Coulombic interaction between two
charges which are situated at atomic positions A and B.
In the above, c is the
electron affinity, I is the ionization potential,
and dAB is the distance between atoms A and B.
In thermal equilibrium, one can apply the condition
that the chemical potential
is a constant for all the atoms in the same molecule,
and be able to estimate the charge transfer between atoms.
Because this procedure predicts finite charge transfers
even for atoms placed far apart, it is only physically
meaningful when the interatomic distances are near their
Potential faults notwithstanding, this approach has yielded
dipoles for small heteronuclear molecules in good
quantitative agreement with experiment.
To apply the above method to an MS interface,
the conglomerate of the entire metal-semiconductor
region can be regarded as a giant ``molecule''.
The truncated lattices of a metal and a semiconductor are
assumed to form bonds on an atomically flat interface plane.
A density of chemical bonds, NB,
is assumed to form across the MS interface.
In general, NB needs not equal, and is likely less
than, the total number of semiconductor (or metal) atoms
per unit area on each plane parallel to the interface.
Lattice mismatch, structure incompatibility, interface mixing,
the formation of tilted bonds, etc. all tend to reduce the number
of effective bonds formed across an MS interface.
Without loss of generality, we assume chemical bonds to form in
a square array
with a lateral dimension of dB ( = NB-1/2 ).
We can write a total energy equation according to (4) for
the entire metal-semiconductor system.
For simplicity, we assume Q's to be non-zero only for those metal
and semiconductor atoms on the immediate interface planes which are
involved in the bonding.
Furthermore, we keep only the interactions between
charged atoms which are nearest neighbors.
EMS = S
+ UM QMi + (1/2) YMQMi
2 ) + S( ESo +
US QSi + (1/2) YS QSi2 )
EMS = S
QSi JMS + S
[ JMM QMi
QMj + JSS QSi QSj ] .
Now we require that the chemical potential
of a metal atom at the interface,
mMi = dEMS/ dQMi,
to equal that of a semiconductor atom,
mSi = dEMS/ dQSi.
Because of symmetry, every bonded metal atom at the interface has the
same net charge -eQM and every semiconductor atom has a net
charge of +eQM.
One thus obtains,
QM = [US - UM] / [YM + YS +
4JMM -2JMS +4JSS ] , (7)
where we have used the fact that there are four in-plane nearest
In the spirit of analyzing charge transfer between two crystals, rather
than between atoms, we can let the atoms acquire bulk characteristics.
For a bulk metal, the ionization potential and the electron affinity
are both identified as the work function of the metal, fM, and YM =0.
For a semiconductor, the ionization potential and the
electron affinity, cS, differ by its band gap, Eg.
Therefore, US = cS + Eg/2 and YS
To account for the fact that the Coulombic interactions take place
inside a solid, screening by the respective dielectric medium is
Eq. (7) becomes
QM = [ fM - cS
- Eg /2 ] / [
Eg + k ] , (8)
where k is the sum of all the hopping interactions,
i.e. k = 4e2 / (
eS dB ) - 2e2 /
( eit dMS ), and dMS
is the distance
between metal and semiconductor atoms at the interface.
The voltage drop across this interfacial dipole layer can now be
calculated to be
V"int" = [ -edMS NBQM /
= ( -edMS NB / eit
) x [ fM - cS
- Eg /2 ] / ( Eg + k ) . (9)
Combining Eqs. (1) and (9), one gets
[ 1 - e2 dMSNB
eit-1 ( Eg + k )-1 ]
( fM - cS ) +
eit-1 (Eg + k )-1 /2 ,
which simplifies to
- cS ) + (1 - gB )
Eg/2 , (10)
gB = 1 - e2 dMS NB
eit-1 ( Eg + k )-1 .
Obviously, Eq. (10) predicts a dependence of the SBH on the metal
work function which is similar to that predicted by gap state
models, Eq. (3).
In Eq. (11), the dielectric constant of the interface region,
eit, should have a value somewhere between the
dielectric constant of the semiconductor, einf,
and that of the metal, eM ( = infinity ) .
A simple average eit-1
= eM-1 /2 + einf-1 /2 )
leads to an estimate of 2einf.
Here the optical dielectric constant is used
because the charge transfer occurs between the semiconductor
and the metal, so there should be little ionic contribution
to the screening.
To test the validity of Eqs. (10) and (11), we plot in
Fig. 2 the
experimentally observed SF in the form of
[ einf ( 1 - SF ) ]-1
against the band gap of the semiconductor.
According to Eq. (11), the quantity plotted in Fig. 2 is
2 eo ( Eg + k ) /
( e dMS NB ),
which should display a linear behavior if dMS, NB,
and k do not vary appreciably with semiconductors.
Indeed, a roughly linear relationship is observed, which
approximately extrapolates through the origin, suggesting that,
due to screening, k is small compared with typical band gaps.
A slope of \(ap 0.13 eV-1 is deduced from a linear
fit, which yields a ( dMS NB ) of
~ 9 x 106 cm-1.
Taking dMS to be 0.25 nm, one gets an NB
of ~ 4 x 1014 cm-2 which is a very
reasonable estimate of the number of available bonds on a
typical semiconductor surface.
Overall, Fig. 2 provides strong
support for the chemical bonding picture of FL pinning.
The polarization of the chemical bonds at MS interfaces
leads to a weak dependence of the SBH on the work function
and a natural tendency for
the SBHs to converge toward one half of the band gap,
both of which are in agreement with experimental results and
have long been thought to be attributable only to
the presence of interface gap states.
The excellent agreement of experimental data
with the present theory suggest that
interface bonding is a primary mechanism of SBH formation.
Even though the present theory seems to have captured
the essence of the SBH formation mechanism, it is only
an approximation and should not
be regarded as numerically accurate.
For numerical comparison with SBH experiments, one should
always rely on first principles calculations for a more
accurate treatment of the interface dipole.
The chemical bonding scheme described here is most
applicable for abrupt interfaces and for semiconductors
with large band gaps.
In the present approach, charge is allowed to transfer
only between atoms situated at the interface plane.
In reality, this charge transfer likely takes place over
several atomic planes and the sign of the charge transfer
may even oscillate with the number of planes.
A more sophisticated model of the interface can perhaps
be attempted, but doing so is not advisable given the
simplicity of the present molecular approach and because
precise knowledge on the atomic structure would then be required.
In deriving Eq. 8, we have used quantities characteristic
of isolated crystals to represent the ability of
individual atoms to attract electrons.
Since the metal work function and the semiconductor
electron affinity both contain a surface term which depends
on the structure of the free surface and does not appear
relevant to the interface charge transfer, it may be argued
that only the bulk contributions to the work function
and electron affinity should be used.
This modification shifts the apparent ``pinning'' position
slightly from midgap of the semiconductor.
Since reliable data for the surface dipoles are generally lacking,
the impact of this possible modification is not clear.
One may comment in passing that the work function and electron affinity
in Eq. 1 presumably should not be modified in the same way,
because without surface terms the
discrepancy between the charge densities on metal and semiconductor
Wigner-Seitz cell boundaries would remain.
There are some similarities between Eqs. (4) and (11), especially
if NB / Eg can be identified with Dgs.
Since gap states undoubtedly accommodate some
of the charge transfer in chemical bonding,
the question may arise as to whether the two approaches
are in fact equivalent.
They are not.
The charge in the gap state models is assumed to be
controlled by the FL position.
So, the gap state models artificially build in a negative-feedback
mechanism, with the charge neutrality level as the back drop.
If one imagines slowly "turning on" (or off) the MIGS
mechanism by increasing (or decreasing) Dgs, the interface FL
cannot cross over the neutrality level ( ggs
in Eq. (4) cannot become negative).
In the present theory,
both the pinning and the pinning position (around midgap) come
as a result of the minimization of the total energy.
The FL is not explicitly involved in the evaluation of
the interface dipole, and therefore the contradiction associated
with the bias-dependence of the SBH does not arise.
The distribution of gap states occurs as a result of bonding
and will vary with the metal and the interface structure,
which is in good agreement with the structure dependence of the
SBH seen at epitaxial interfaces, and with the observation
of SBH inhomogeneity at polycrystalline interfaces.
When the experimentally measured interface behavior parameters,
plotted in a fashion suggested by the division-by-epsilon model of
Schottky barrier height formation, the agreement is poor.
The agreement is also poor between experiment and
of the metal-induced gap-state model of Schottky barrier height
To read more about applying chemical concept at
metal-semiconductor interfaces, consult
Physical Review B64, 205310 (2001).