Electronic structure of surfaces.
Electron states in solids.
Electrons in solids are in localized or delocalized states, levels, or orbitals. The
inner or core states are localized, and are very similar to the states in free atoms. The
outer or valence electrons are delocalized in the sense that the can extend outside the
region of just one atom. In metals, the delocalization of the valence electron is such
that we can say that they are shared by all atoms in the solid. Two important properties
of electronic states are their binding energy U and width, G = DU.
Electron binding energies
The binding energy is the minimum energy needed to remove a particular electron from
the atom. There are two basic ways of measuring this energy, depending on how we remove
the electron. We can place it in the first available empty state (at T=0 K) that is
delocalized, the bottom of the conduction. Empty states can exist in the band gap band of
non-metals, but they are localized and so the electron is not removed from the atom by
placing it in a band-gap state. Another way to remove an electron is to place it at
infinity, that is, outside the solid. This is the definition of binding energy used for
free atoms and molecules. An operational definition of binding energy in electron
spectroscopy is the energy that would be needed to put the electron at the Fermi level.
The Fermi level of the material where the atom is may not contain available delocalized
empty levels (case of non-metals) but still the energy can be referred to that of a metal
in electrical contact and in thermal equilibrium with the material, which shares the same
Fermi level.
Widths of electronic states
The width of the electronic state is related to t,
the lifetime of a hole produced in the state when we remove the electron. This is the
Heisenberg relationship: G = h/2pt.
The broadening results from the fact that there is only a finite time t to measure the
binding energy of the state before the hole is replenished with another electron. A hole
can be filled by an electron from a higher (less bound) state of the same atom, or by
electron transfer from another atom. In the case of core level, there is very little
overlap with the electron cloud of a similar level in a neighbor atom, so the lifetimes
are very close to those of free atoms (or ions in the case of some insulators). Typical
core lifetimes are 1 fs to 100 fs. In contrast, a hole in the valence band will be more or
less delocalized (depending on the type of material), and lifetimes can be as low as 0.1
fs. Since h/2p = 6.582 x 10-16 eV.sec, the widths of core
levels are 0.001 - 0.1 eV, whereas the width of valence band states are 0.1 eV to several
eV.
Band-gap states
The band-gap in semiconductors and insulators results from the interplay between the
electron wavelengths and the symmetric arrangement of atoms in the solid. At the surface,
the symmetry is broken which relaxes the condition that inhibits electrons to propagate
inside the lattice. The symmetry is also broken by defects and impurities. Broken symmetry
results in electronic states which can exist in the band gap. The existence of states in
the band have that pertain to the surface, or surface states, is the general rule.
Population of valence states
The occupation probability P(E) of valence electrons of energy E is given by the
Fermi-Dirac distribution:
P(E) = 1/[1+exp(-(E-EF)/kT) ]
Notice the very small tail above the Fermi level, even at very high temperatures.
However, the small tail can have some profound effects, like thermoionic emission of
electrons outside the solid or transfer of electrons at a metal-semiconductor interface.
Electron energies near surfaces
Vacuum level
The potential energy of an electron in vacuum is taken as zero. This is called the
vacuum level. In practice, it represents the potential outside the surface at a distance
larger than grain sizes but smaller than the size of the sample. If these conditions are
not met, the electron will find a patch of electric field and what one needs to use
depends on the problem at hand.
Work Function f
The energy difference between the Fermi level and the vacuum level. For a metal, this
is the minimum energy required to eject an electron into vacuum (at T = 0 K), since the
electrons with the minimum binding energy are at the Fermi level. At T > 0 K it is
found that the work function varies slightly with temperature. Also, and independently
from the previous statement, the minimum energy to emit an electron from the solid depends
on the sensitivity of the measurement, since the tail of the Fermi-Dirac distribution
extends beyond the vacuum level. For most metals, f is about
4-5 eV, the minimum values are around 1.5 eV for metal surfaces with an adsorbed layer of
cesium.
In a semiconductor and insulator, the work function is defined in
the same way as in the metal: f = Evacuum - EFermi,
but now there are no electrons at the Fermi level.
Electron Affinity c and Inner Potential I
The energy gained by an electron when it enters a solid is the difference in energy
between the vacuum level and the bottom of the conduction band. This is the electron
affinity in the case of a non-metal. Typical values are below 1 eV. In some materials like
solid argon, the electron affinity is negative, that means that the bottom of the
conduction band is above the vacuum level. Hence, an electron that is excited to the
conduction band will spill out the solid, unless it is trapped at a band-gap state.
For a free electron metal, I is the sum of the Fermi energy
and the work function. The energy gained by the electron when it enters is important, for
instance, in LEED since it will be accompanied by a change in the associated de Broglie
wavelength.
Photoelectric threshold
This is the minimum energy needed to extract an electron from the solid. In the case of
metal, as we have seen, it is the work function. In the case of a semiconductor or an
insulator, it is the energy difference between the vacuum level and the top of the
conduction band, or: c+Egap.
Image potential
From standard electrostatic theory, a charge q in vacuum at a distance z in front of a
perfect conductor (infinite dielectric constant e and zero
internal electric field) induces an image charge -q in the solid. This
image charge is virtual and behaves as if it was inside the solid at a depth -z. The
image force is F = q2/(2z)2 = q2/4z2.
Therefore the image potential (the integral of the field E = F/q) is:
U = q/4z
If the distance is measured in Å,
U(eV) = 3.6/z
For an arbitrary solid of dielectric constant e, the
potential is:
U = [(e-1)/(e+1)]
q/4z = 3.6 eV Å (e-1)/z(e+1)
In reality, what happens is that the charge polarizes the medium. For instance, if an
ion is outside the solid, a cloud of electrons piles up at the surface (not at a distance
-z). When the distance between the external charge and the surface becomes of the order of
the distance between charges in the solid, the expression U ~1/z breaks down. The
image potential merges into the inner potential in the solid.
Realistic surface potential for a Cu(110) surface
Models of solids and surfaces
Solids cannot be described accurately as atoms, because of the large number of
interactions. Several simplifying models exist, notably:
Jellium model. The charge of the ion cores is spread over the solid (jellium) and the
electrons then move in the potential produced by this jellium. Density functional theory
is used where the properties of the electron "gas" depends only on the electron
density. This is sometimes refined by adding non-local corrections to the properties. We
note that a uniform electron gas is not a good approximation at the surface.
Pseudopotentials. Free electrons plus a small correction to take into account the
discrete potentials of the atoms. Harder.
Tight-binding. Electrons are bound to atoms and corrections are added for overlap
of the electron density produced by different atoms. Best applied for directional
bonds, like semiconductors and insulators.
Surface dipole
In the jellium model, the positive background terminates abruptly at the surface
(jellium edge). The electrons are allowed to readjust. The finite wavelength of the
electrons causes Friedel oscillations in the electron density near the surface (this is
analogous to what happens when one tries to express a step function as a sum of sinusoidal
functions up to a maximum frequency). The sharpness of the jellium and the spread of
the electron density (which decays exponentially outside the solid) produces a deficit of
electrons just inside the jellium edge and an excess outside. This produces a dipole
layer. This dipole attracts electrons to the surface and produces a step in the
surface potential.
The total potential seen by the electrons (inner potential) is the
electrostatic potential caused by the distribution of charge density (Poisson equation),
plus the exchange-correlation potential produced by electron-electron correlations.
The exchange-correlation potential evolves into the image potential outside the
solid. The electrostatic potential includes the surface dipole whose value depends
on the roughness of the surface, both at the atomic scale and that produced by steps.
Thus, the work function, which is the inner potential minus the Fermi energy,
depends on the crystallographic orientation of the face of the crystal. For
instance, the work function of Cu (fcc) is 4.94 eV, 4.59 eV and 4.48 eV for the (111),
(100) and (110) surfaces, respectively. The work function will be changed when
permanent or induced dipoles are added during adsorption of gases on the surface.
These additional dipoles can increase or decrease the work function.
It is important to notice that the work function is not measured
from the Fermi level to the potential at infinity, but to the potential at a small
distance to the surface, say 1 micron, where the image potential is already negligible.
The potential in vacuum at distances large compared to the dimension of the solid
will depend on the combined effect of surface patches of different crystallographic
orientation. That is why one cannot gain energy by removing an electron from a patch
of low work function and then putting it back in a patch of high work function. In a
parallel plate capacitor made of different materials, but connected electrically (Fermi
levels at the same potential) there is an electric field between the plates in vacuum (and
thus a potential difference outside) given by the difference in work functions.
The Kelvin Probe
The Kelvin probe is a vibrating metal plate used to determine the work function
of a sample, fs, relative to that of the probe, fp. The probe is located close to the sample which acts as
the second plate of a capacitor. The capacitance is C(t) = kA/d where k is the dielectric
constant of the medium between the plates, A is the area of the plates, and a(t) is the
separation between the plates of the capacitor. Since the sample is not likely to be a
flat plate, A and d should be taken as effective values.
The probe is made to vibrate at a frequency f (angular frequency w=2pf) with an amplitude a1
around a mean spacing a0, that is, a(t) =a0+a1coswt, and therefore the capacitance also varies with time t.
Now a voltage VA is applied between the probe and the sample
(positive side to the probe), causing their Fermi levels to differ by the same amount. The
potential difference between the plates V will not be VA but VA+Df. if the work functions of the plates differ by Df
= fs–fp. Thus, the change in
capacitance produced by the vibration of the plate induces a change in the charge CV at
each plate and therefore a displacement current, I, that can be measured by intercalating
an ammeter in the circuit.
I = d(CV)/dt = (VA+Df)(kAwa1/a02) sinwt
If we now adjust VA so that the current is zero, VA=–Df.
This result is independent of a0, a1, A
and w but in practice, the current can be nullified only
within some noise level. The sensitivity of the method depends directly on the magnitude
of the factor multiplying (VA+Df) is largest. This
means that it is better to have a large area, a small separation, and large amplitude and
frequency of vibration. However, there are practical limits to this. The area of the
sample is typically limited by other considerations and the area of the probe needs to be
smaller to avoid including in the measurement the sample mount. The minimum value of the
initial separation will depend on how well one can place the probe parallel to the sample.
The maximum frequency is limited by the inertia of the vibrating circuit.
Finally, note that the Kelvin probe method does not measure work functions but differences
in work function. This requires the probe to me made of a material whose work function is
known and which is stable when the environment of the sample changes by heating, exposure
to gas, particle bombardment, etc. Useful materials are Au and SnO coated glass.
Experimental details and references can be found in Woodruff and
Delchar (Ch. 7). The Kelvin method has been adapted to
scanning probe microscopy; some details are given here.
Field Electron Emission from Surfaces
Applications
Scanning Tunneling Microscope
Woodruff & Delchar, Ch. 6
The Homebrew STM page (build your own STM)
STM Image Gallery
at IBM-Almaden
Flat Panel Displays
Field emitter arrays for flat panel displays at MCNC.
A single tip. The radius of curvature is 5 nm.
