Low-energy electrons are for surface structure what x-rays are for
the bulk. We already know the two reasons for this: (1) the mean free
path for low energy electrons
in solids is short and therefore
any technique based on such electrons is rather surface sensitive
and (2) the electron de Broglie wavelength
fits very
well with the typical distances in crystals and thus diffraction
phenomena are to be expected. The discovery of the fact that the electron
has indeed a wave nature was a milestone in the development of modern physics:
The first LEED experiment from Ni single-crystals by Davisson and Germer
was published in 1927. However, the quantitative structure
determination with electrons instead of x-rays also leads to some
difficult problems: the electrons interact with the solid much more
strongly than x-rays. This results in a refraction of the electron wave
at the crystal-vacuum boundary and, even worse, it leads to a high
degree of multiple scattering
.
As we shall see below there are two major applications for LEED. The first one to learn something from the pure inspection of the surface diffraction pattern. One short LEED experiment gives immediate and direct information about the surface order and quality. When the surface is reconstructed or covered with adsorbates, the LEED images can quickly give some information about the surface symmetry and periodicities. The second application of LEED is the quantitative structure determination. This is much more difficult. One has to measure the diffraction intensities as a function of the incidence electron energy and compare them to sophisticated multiple-scattering calculations for a model system. This model system has to be changed until good agreement between calculations and experimental intensities is achieved. Despite of this complicated procedure, LEED is the most important tool for quantitative surface structure determination.
Fig. 
shows a typical LEED apparatus which can be found in
almost every surface science vacuum chamber. The LEED
system has two major components: (1) an electron gun producing
monochromatic electrons and (2) a detector system which detects only
the elastically scattered electrons.
We know already how the electron gun works. The detector consists of four metal grids at different voltages and a fluorescent screen. The first grid (counted from the sample) is on ground potential to ensure a field free region around the sample. The next two grids are set to the so-called retarding voltage. This voltage is slightly lower than the kinetic energy of the electrons produced by the gun. It repels almost all the inelastically scattered electrons. The elastically scattered electrons pass the next grid which is set to ground voltage again and are then accelerated towards the fluorescent screen which is set to a high positive voltage. Behind the screen there is a window in the vacuum system so that the LEED pattern can be observed directly or recorded with a video camera.
The discussion of diffraction from a two-dimensional lattice is very similar to that of a three dimensional crystal. We will therefore only give a very short overview. The diffraction conditions for a two dimensional lattice are given by the two Laue conditions
where h and k are arbitrary integers. We know of course that this condition is fulfilled by any vector of the reciprocal lattice. This gives us the diffraction condition associated with the momentum transfer parallel to the surface
The vertical momentum transfer did so far not enter the discussion at
all. This makes sense since for a two-dimensional lattice 
is not a good quantum number and does not have to be conserved. This
is also true for the semi-infinite solid when electrons cross the
vacuum-solid interface. However, the energy conservation imposes a
restriction on because we have to require that
These two conditions can be made visible by changing the Ewald
construction
known from x-ray scattering to the surface case as shown
in Fig.
. Instead of a three dimensional reciprocal lattice we
have our two dimensional lattice. In the third dimension the
real-space periodicity is infinite which means that in reciprocal
space the lattice points have to be infinitely close to each other.
This leads to reciprocal lattice rods instead of points. We know draw
a 
-vector which ends at the origin of the reciprocal lattice and
has the right length and direction corresponding to our experimental
setup. Then we draw a circle of radius 
around the
starting point of the vector. The intersection of this circle and
the lattice rods gives the possible final 
vectors for
which we will observe scattering maxima. It is evident that we will
see many more spots in the two-dimensional case than in the three
dimensional case because the circle does not have to hit points in
k-space, it just has to intersect with the rods.
Figure: The Ewald construction for the surface case.
We now apply the concepts from the last section to the real LEED
experiment. In most cases, the sample in the LEED setup shown
in Fig. is adjusted such that the electron beam hits the
surface at normal incidence, i.e. such that 
is 0 for the incident electrons. This greatly simplifies the analysis of
the resulting diffraction patterns because (1) the resulting
diffraction maxima can be directly associated with the reciprocal
lattice and (2) the diffraction pattern represents the symmetry of the
surface. In fact, for such a system the diffraction pattern will be
an image of the surface reciprocal lattice. According to equ.
we will find high intensities at
. At the same time we
know the magnitude 
of the outgoing electrons and this
gives us the emission angle
. Now we consider
the imaging by the LEED apparatus (Fig. ). The position of
the intensity maxima on the window is given by
Figure: Linear imaging of the reciprocal lattice by LEED.
From equ.
it is also obvious what happens when one changes the
primary energy E of the electrons. When increasing the energy we
will still see the same spots but they will move closer to the centre
of the window. New spots will move in on the sides of the screen which
have not been visible before. It is obvious that for every reciprocal
lattice point (except the origin) there is a smallest energy of the
primary electrons which is required to see its image on the screen.
The effect is illustrated in Fig. which shows two LEED
images taken at different energies E for the W(100) surface. The
surface unit cell of W(100) is a square and hence the reciprocal
lattice is also a square.
Figure: LEED patterns of W(100) taken at a electron kinetic
energy of 45 eV (left) and 145 eV (right), respectively.
On the right-hand side of Fig. we also give the indexing
of the first LEED spots. This nomenclature refers to the reciprocal
net of the bulk-terminated surface. This means for example that if a
reconstruction or an overlayer with double periodicity is present,
then we will have (1/2,0) spots and so on. The (0,0) spot is invisible
in a normal normal-incidence LEED image because of the electron gun
which is in the way.
When considering the diffraction from the surface instead of a
perfect two-dimensional lattice we have to take into account the
three-dimensional nature of the solid. Our picture of the Ewald-sphere
with rods giving the same intensity in every LEED spot and at every
kinetic energy is not quite correct because the electrons penetrate
into the solid and ``feel'' the third Laue condition
as well. This
leads to very strong intensity variations in the LEED spots as a
function of energy. Fig. shows the intensity of the (0,0)
spot from Ni(100) as a function of electron kinetic energy.
Measurements like this are called an I-V curves (intensity vs.
accelerating voltage of the electrons). A
substantial intensity variation is visible and some of the highest
maxima lie close to the energies which are calculated by application
of the third Laue condition (indicated as arrows on the figure).
The first thing one notes is that the observed intensity maxima are at
a lower kinetic energy than the calculated maxima. This can be
explained by the fact that the electrons have a higher kinetic energy
in the solid than outside due to an ``inner potential''
.
This difference in energy is related to
the bandwidth of the material (which gives a new possible lowest
energy) and the workfunction. The value of the inner potential is
about 10-15 eV, the sum of the bandwidth and the workfunction (see
section ). The
inner potential is also responsible for a refraction effect of the
electrons at the surface. The electron beam which leaves the crystal
will be refracted away from the surface normal as it passes through
the surface (see Fig. ).
Another point worth noticing is the large width of the peaks in Fig.
which is due to small penetration depth of the electrons:
a finite penetration depth means an effective localization in the
first layers, corresponding to a broad k and energy interval. This
is intuitively clear from what we have discussed above. In the case
of zero penetration, equivalent to a purely two dimensional lattice,
the peaks would be infinitely broad and the third Laue condition would
be unimportant. In a real crystal, however, the electrons do
penetrate but their penetration depth is limited for two reasons. The
first is that the peaks actually correspond to a Bragg back-reflection
and therefore the penetration can not be very deep. The second is the
limited electron mean free path
.
The last thing one notices in Fig.
is the
presence of additional peaks apart from the shifted Bragg peaks. This
is due to the multiple scattering
of the electrons in the solid.
Indeed, intensity curves such as Fig. can not be
described by singe-scattering (kinematic theory) like for the
interaction of x-rays with matter. A sophisticated
multiple-scattering formalism is needed to quantitatively describe
the I-V curves. We will come back to this in a
later section.
Figure: Intensity of the (0,0) spot from Ni(100) as a function
of electron kinetic energy. The arrows indicate the positions
where maxima would be expected if the third Laue condition would
be valid. After Ref. [29].
Another new point when going from a two-dimensional lattice to a
surface is the possibility of overlayer structures, i.e. we may have
an overlayer (or a reconstructed layer) on the surface with a
reciprocal lattice which is different from that of the substrate.
What will the LEED pattern look like? One would guess that the LEED
pattern is just the sum of the two reciprocal lattices but this is
only partly true: due to multiple scattering one does not only get
the spots of both reciprocal lattices (truncated bulk and
surface/overlayer) but also all possible combinations between them.
If we adopt the point of view that the lattice of the surface is made
up by the adsorbate and the first few layers of the substrate, these
additional spots enter in a natural way into the reciprocal lattice.
This is illustrated in Fig.
.
Figure: The LEED pattern shows the sum of the reciprocal lattices
from substrate and overlayer plus all possible combinations
between them.
For a simple overlayer structure as in (a), this combination does
not lead to any new spots. For a
coincidence structure
(b) it does (grey spots).
The arrows indicate the size of the surface unit cell as a
whole. When this unit cell is taken to calculate the reciprocal
lattice, the ``extra'' spots appear in a natural way.
A lot about the surface structure can be learned simply by the inspection of the LEED pattern without considering the quantitative I-V behaviour of the spots. Basically, we see the reciprocal lattice of the surface and from this we can construct models for the real lattice. There are, however, several effects complicating this analysis.
The first question which arises when inspecting a LEED pattern of a clean surface is if the surface is reconstructed or not. This can be found out by comparing the position of the spots to the positions one would expect for the (1x1) unreconstructed surface. The simplest way is to estimate the energy at which the (1x1) spots would first appear on the fluorescent screen and compare this to the measured energies. Once the (1x1) spots are identified one can describe any overstructure referring to them.
Now consider the LEED patterns of simple overlayer structures and
coincidence structures. Since we know which spots are the original
(1x1) substrate spots we can now deduce the reciprocal lattice of the
overstructure from the LEED pattern. Fig.
gives a few
examples.
Figure: Three examples for overlayer structures and the LEED
patterns produced by them. (a) a (4x2) structure, (b) a c(4x2)
structure. In the LEED patterns the open circles are the (1x1)
spots. The (1x1) unit cell in reciprocal
space is also given.
From such LEED patterns the surface periodicity and the point-group of the surface may be deduced.
There is, however, one essential problem
which is the existence of domains
. Consider for example the structures
shown in Fig
. For (a) and (b) there are completely
equivalent structures rotated by 
. It is very likely the
an almost equal number of both types of domains exists in the (huge) area
which is sampled by the electron beam. If we neglect the coherent
interference between electrons scattered from different domains then
we will just have to sum up the LEED patterns from the two possible
domains incoherently. Fig. shows the incoherent sum from
the two possible domains in Fig.. It is obvious that the
existence of domains gives the LEED pattern a four-fold symmetry while
the local symmetry of the adsorbate structure is only two-fold.
Figure: LEED patterns resulting from two different domains of
the structures shown in Fig. .
The last point which we only mention very briefly is that there are also lots of imperfections on the surface. The electrons do not scatter from a perfect periodic structure but from a ``real'' surface at finite temperature, with steps, point defects and ``dirt'' in form of unwanted adsorbates. These imperfections cause an intensity loss and a broadening of the diffraction spots and an increase of the background in between the spots. One can turn this problem into an advantage and use the spot profile of the diffraction maxima in order to learn something about the surface imperfections. This technique is called spot profile analysis LEED (SPA-LEED ).
As we have seen, the inspection of the LEED pattern gives
information about the surface periodicities and to some degree also
about the surface symmetry. But there are still many important things
one would like to know. One is the site of the adsorbed atoms.
Consider again Fig. . If we shift the whole overlayer
such that the atoms adsorb in bridge sites instead of on-top sites the
diffraction pattern will be unchanged. Also, one could put more
adsorbate atoms into the same unit cell and still keep the pattern the
same. Clearly, a more quantitative analysis of LEED is needed.
This is achieved by analysing the I-V curves. The I-V curves for a particular model structure can be calculated by a computer program. Then they are compared to the measured I-V curves. If the agreement is not good, the model structure is changed and the structure is re-calculated. This process is repeated until eventually a good agreement between experiment and theory is achieved. The degree of agreement is quantified by a so-called reliability or R-factor . The lower the R-factor, the better the agreement.
In a LEED calculation the crystal is described by a so-called muffin-tin potential . This consists of spherical potentials for the ion cores and a constant potential everywhere else. The spherical potentials are characteristic for the element of the scatterer and depend somewhat also on its environment (but not very much). They can be described by a set of scattering phase shifts. This choice of potential has the advantage that reduces the problem basically to scattering from spherical potentials which can be treated very efficiently. The program must now explicitly solve the Schrödinger equation in the muffin-tin potential including all possibilities of multiple scattering.
There are two additional effects which also have to be taken into consideration by the program. The first is the inelastic scattering of the electrons. This is handled by making the constant part of the potential in the solid complex. The imaginary part corresponds to the energy-dependent mean free path of the electrons and takes care of the inelastic scattering. The other effect is finite temperature. It reduces the scattering coherence in otherwise periodic structures and thereby reduces the intensity in the I-V curves. Finite temperature is taken into account by temperature-dependent scattering phase shifts .
The level of agreement which can be obtained between experiment and
theory is remarkable, at least for many metal surfaces,
and gives a high confidence into the LEED
technique. Fig.
gives an example.
Figure: Agreement between measured and calculated LEED I-V data
for different spots in case of the Al(111) surface. The full line
is experimental data, the dashed line calculation [32].
The possibility of high-quality calculations also means that the structural parameters are determined very precisely by LEED. The atomic positions are given within a tenth of an Angstrom or even better. But there are also some problems associated with the analysis approach employed for LEED. The success depends on the researcher's ability to come up with the right structural model which can then be refined in a trial and error iterative analysis. This is not too difficult for unreconstructed metal surfaces where the truncated bulk can be taken as a starting model. But it is a major problem for semiconductor surfaces where reconstructions with large surface unit cells are possible as we shall see below. Such large unit cells have the further disadvantage that they are extremely expensive in terms of computer time needed for the calculations. Many atoms in the unit cell mean many structural parameters which one has to get all right in order to obtain satisfactory agreement. A great danger is that there are cases where the agreement between theory and experiment is rather good but the model structure is not the right one. If one wants to have confidence in the result, it is important to have a very good agreement, or, in other words, one has to get everything right before one knows that one has anything right.
