When the periodic structure of a crystal lattice is terminated
at a surface, electronic states particular to the surface are created.
These are states that have no equivalent in the band structure
of the bulk crystal.
Surface-specific states can be true surface states with wave functions
which are peaked near the surface plane and which decay in amplitude
away from the surface, both toward vacuum and toward the bulk crystal.
Surface-specific states can also be surface resonant states that
have enhanced amplitudes at the surface but are coupled to bulk states.
Of course, the distribution of surface-specific states depends on
the atomic structure of the surface, and conversely, the atomic
structure of the surface is determined more or less as a result
of the minimization of the surface energy, to which the
surface-specific states are a major contribution.
Surface-specific states are present at the surfaces of all matters.
On metallic surfaces, they are known to lead to a surface dipole
which contributes to the work function of the metal surface.
On semiconductors, the presence of surface states in the band gap
is known to "pin" the Fermi level position of the semiconductor.
Pinning does not happen on every semiconductor surface,
however, because surface states are not positioned inside
the band gap of some semiconductor surfaces, such as the
non-polar (110) surfaces
of III-V semiconductors.
So, on some cleaved non-polar surfaces, there is little band bending.
Central to the concept of Fermi level pinning by gap states is
the charge neutrality level (CNL) concept,
which can be roughly
defined as follows. At zero temperature, the surface states
are populated from the lowest energy states up to the Fermi level.
If electrons are filled to a point short of the CNL, i.e. when the
Fermi level is lower than the CNL, the very surface region,
meaning the first few atomic planes of the surface, has
net charge. If the Fermi level is above the CNL, the surface has
an excess of electrons and is
negatively charged. Since the Fermi
level is a constant throughout all regions of a semiconductor, the
CNL allows a unique correspondence to be made between the band
bending at the semiconductor surface and the population of surface
states. To calculate the exact pinning position of the surface
Fermi level, one can assume the density of surface states, DGS,
to be roughly constant near the CNL.
The net charge per unit area of
the surface, QGS, is then simply
QGS = e DGS (
fCNL - EG ) ,
where fCNL is the CNL
position with respect to the valence band maximum, and
FoB,n is the
difference between the FL and the CBM at the surface.
Note that DGS has the unit of # of states per area
per energy (cm-2 eV-1). Here the subscript
"GS" (gap states) is used to represent surface states, to be
consistent with later notations. As
is analogous to the n-type SBH at an MS interface, here it
is allowed to retain its usual symbol even though there
is no Schottky barrier at a semiconductor surface.
One should also note that the definition of the CNL
at a surface involves a tacit assumption: the energy levels
of surface states are assumed to be referenceable to the
bulk bands of the semiconductor. Since the bulk bands are
considerably modified near a surface, strictly speaking
there is no CBM right on the atomic plane of a semiconductor
surface. The location closest to the surface where bulk
semiconductor bands can be identified is a few lattice
spacings away and it is the CBM position at that location
which provides the reference point. Since the definition
of a CNL involves states that are spatially separated, it
may be subject to variation due to changes in the relative
electric potential (chemical shift). While this definition
does not lead to any ambiguities on the determination of the
FL position on a semiconductor surface, it can render the
concept of CNL fuzzy for MS interfaces.
For an intrinsic (meaning undoped) semiconductor with homogeneous surfaces,
the Fermi level should coincide with the CNL of the surface states.
For uniformly doped semiconductors, the surface Fermi level
position deviates slightly from the CNL to yield the necessary
surface net charge which balances the charge due to exposed
dopants in the space charge region:
e DGS (
fCNL - EG ) +
ND ( FoB,n
-e VN ) ]1/2 = 0
This equation can be used to solve for the Fermi level position
at the surface of a semiconductor.
Alternatively, one notes that FoB,n
can be obtained graphically, as shown in
this figure, by plotting both terms of the above equation on the
same graph. The graphical method is of some value as its results
can be more transparent than analytic expressions, especially
when the system gets more complex.
As shown by the above equation, even a relatively low density
of surface states, say 1x1013 cm-2 eV-1,
can quite effectively fixate the surface Fermi level to within 0.05eV
of the CNL independent of the doping type or the doping level of
a moderately doped (ND < 1x1017
cm-3) semiconductor. The surface states are said to
pin the Fermi level near the CNL under these conditions.
Often, the density of surface states is thought to have peaks
at certain energetic positions, such as due to states associated
with defects or dangling bonds. With a non-uniform density
of surface states, the net charge due to surface states,
the first term of the above equation is replaced by an integral.
For a distribution of gap states with discrete peak(s),
as schematically illustrated in
this figure, the surface Fermi level can be pinned tightly
at the position of the peak. A review of the figures
makes it clear that only those gap states with energy above
the CNL, i.e. acceptors, affect the surface Fermi level position
on an uncharged, n-type semiconductor; and only gap states with
energy below the CNL (donors) can affect the Fermi level position
on a p-type semiconductor surface.
Perhaps because the net surface charge is the only relevant parameter
for the determination of surface Fermi level, the surface pinning
concept is clear and straightforward. An important aspect of
the surface states is the formation of the surface dipole, which
directly affect the electron affinity
of the semiconductor.
The magnitude of a surface dipole has to do with the entire spatial
distribution of charge. It is not difficult to imagine that
patches of the surface that have the same net charge desnity,
or even the same density of surface states distribution,
can turn out to have different surface dipoles.