The questions of surface structure can be roughly divided into the “large scale” and the “small scale” structure. The two are divided by an arbitrary and unclear borderline. Some interesting and more macroscopic phenomena still lack a detailed understanding on the microscopic level, for example catalysis or friction. Here we discuss the “large scale” structure only very briefly in connection with surface free energies and crystal shape. We also mention some very interesting phenomena such as the roughening transition and surface melting . This discussion is done in the framework of thermodynamics. All the rest of the chapter is concerned with the microscopic structure. We are interested in questions like: What is the structure of clean and adsorbate-covered surfaces? Why is the structure like this? How does one measure the structure?
We start this section off with the expression for the internal energy of a homogeneous thermodynamic system, having in mind the bulk crystal. This is given by the Euler equation
| U=TS−PV+µ N. (7.1) |
Now suppose that this crystal is cleaved and a surface area is created. Suppose further that this process is carried out reversibly at a constant temperature, system volume and chemical potential. The creation of the surface adds a term to the energy which must be proportional to the amount of surface area created and positive because otherwise bulk crystals would cleave spontaneously. The energy is then
| U=TS−PV+µ N+γ A, (7.2) |
where the constant γ has the dimension of energy per surface area and is called the surface tension . It would be very desirable to measure the surface tension (or energy) of a solid/vacuum interface because it is the most fundamental quantity emerging from a modern total-energy calculation of the surface properties. The theory would greatly benefit from a comparison to reliable experimental numbers. Unfortunately, it is very difficult to measure the surface tension. Some hint about the magnitude of the quantity can be taken from measurements of the liquid gas interface but this is not sufficient to test the interesting details in the theory.
When we think about the microscopic origin of γ it is obvious that it will be different for the various possible surface planes of a crystal. As we shall see below, the number of bonds which have to be broken to generate a certain surface plane depends on the orientation of the plane. A surface with many steps might be particularly unfavourable because several of bonds have to be broken to create a low-coordination step atom. Therefore we write γ with a directional dependence γ(n).
These considerations have important consequences on the macroscopic shape of crystals. While liquids are always found in the shape of smallest surface area (a sphere) this is not the case for solids. The solid crystal wants to have a shape where a large fraction of the surface area is given by crystal planes with a low γ. More formally, if we look for the minimum in the free energy (but at zero temperature) we have to require that
| ∫ |
| γ(n)=minimum. (7.3) |
This requirement leads to quite complex equilibrium shapes of crystals.
If γ(n) is known, the problem can be solved graphically by
the so-called Wulff construction
shown in Fig. 7.1. The
procedure for finding the equilibrium crystal shape at 0K is as
follows: Draw γ(n). Draw a plane perpendicular to the
radius vector which intersects γ(n) at each point. The
inner envelope of all these planes is the equilibrium crystal shape.
Figure 7.1: The Wulff construction for the determination of equilibrium crystal shape.
It is interesting to consider this crystal shape if we want to prepare a surface by cleaving a crystal. The cleaving process will only work (if at all) for surfaces which also appear on the equilibrium crystal. If we attempt a cleavage in another direction, the result will be a surface with facets of low-energy planes.
At finite temperature the situation changes. Now excitations which do not cost much energy but increase the entropy of the system become important. Such excitations are steps and other defects. As the temperature is raised, the free energy for the steps decreases. The sharp edges of the Wulff-crystal disappear and the facets decrease. Eventually, the facets will vanish completely and the crystal will have a round shape. This behaviour can be described by a phase transition, the so-called roughening transition. It can also be observed on certain crystal surfaces, well below the actual melting temperature of the crystal.
Another process worth mentioning is the phenomenon of surface melting. This is the formation of a thin, liquid-like phase on the surface below the melting temperature of the bulk. The most important example of this phenomenon is the surface melting of ice which permits for example skating. The existence of surface melting can be made plausible by considering the Lindemann formula for bulk melting. It states that
| Tm ∝ ΘD2 (7.4) |
where Tm is the melting temperature and ΘD is the Debye temperature. The latter is often found to be much smaller at the surface. A rule of thumb is that the surface Debye temperature is about a factor of √2 smaller than its bulk counterpart. This simply means that the surface atoms vibrate more strongly than the bulk atoms at a given temperature and this eventually leads to melting. But one has to keep in mind that this point of view is much too simple. Note that also the completely counter-intuitive phenomenon of surface overheating is found in which the bulk melts at a lower temperature than the surface.
The simplest picture of a microscopic surface structure is that of the truncated bulk (the so-called ideal surface). Suppose the crystal is cleaved along a plane specified by its Miller indices (hkl) . In the truncated bulk model, all the atoms on the cleaved crystal’s surface stay exactly where they have been in the bulk crystal. This means that one can immediately draw the surface geometry of the crystal. Fig. 7.2 shows the surface geometry for the fcc(111) surface.
When looking at a surface, we think of the bulk as being made of
planes parallel to the surface plane. We define a unit cell in the
first plane and, if required, a basis. We define furthermore a
vector r
connecting the atoms in successive planes. A “plane" does of course
not necessarily mean that all the atoms have the same z value (z
being the distance perpendicular to the surface). We can extend the
unit cell over several layers if we want to. In Fig. 7.2,
however, we have chosen a primitive unit cell and all the atoms in
one layer are at the same height. We call the z component of r
the distance between the planes. In the side view of the surface in Fig. 7.2
we can see the familiar ABCABC…stacking sequence of the fcc
lattice.
Figure 7.2: The (111) surface of an fcc crystal.
In Fig. 7.3 the truncated bulk surfaces of a few important
cases are shown.
Figure 7.3: A few important truncated bulk surface structures.
The most severe problem with the truncated bulk model is that it
completely neglects the dramatic change in the coordination and
potential due to the abrupt termination of the crystal in the
direction normal to the surface. This change will for almost all
surfaces lead to a phenomenon called relaxation
(see Fig.
7.4). A relaxation
is a change in the distances between the first few layers with
respect to the bulk values. For most surfaces the distance d1 is
smaller than the corresponding bulk value. This can be made plausible
by the model of Finnis and Heine
shown in Fig. 7.5 [28].
In the
bulk (of a metal) the ion cores are screened by the conduction
electrons around them. If we divide up the crystal in Wigner-Seitz
cells, it is easy to see what happens in the surface case: the original
distribution of electrons in the Wigner-Seitz cells would lead to a
highly corrugated electron distribution at the surface. This is,
however, very unfavourable because of the high kinetic energy of
“bent” wave functions. The electrons at the surface will
re-distribute themselves leading to a smooth charge density at the
surface. This creates an asymmetric screening of the ion cores in the
first layer and a net electrostatic force which pushes them “into”
the crystal and thus reduces d1. The charge smoothing at the
surface is called the Smoluchowski effect
.
Figure 7.4: Relaxation (left) and reconstruction (middle) and adsorbate superstructures (right) on surfaces.
Figure 7.5: The Finnis Heine model for inward relaxations on metal surfaces [28].
A more severe change of structure is the phenomenon of surface reconstruction (see Fig. 7.4). In a reconstruction the periodicity parallel to the surface is changed with respect to that of the bulk. Surface reconstructions are the rule in the case of semiconductors. There the bonds are highly directional. Cleaving the crystal leaves the structure in a unfavourable elastic state and also gives rise to half-occupied “dangling” bonds. Reconstructions give a considerable gain in energy and reduce the number of dangling bonds. The resulting structures can be rather complicated. Most metals surfaces do not reconstruct but some do. In most cases, this happens for metals were localized d or f electrons take part in the bonding for reasons similar as on the semiconductor surfaces. But there are also a few simple metal surfaces which reconstruct.
Another phenomenon, somewhat similar to reconstruction, happens when atoms or molecules are adsorbed on a surface. The adsorbates often form ordered structures (due to their mutual interaction) which have unit cells larger than the substrate unit cell (see Fig. 7.4). In most cases, however, there is still a simple ratio between the substrate and adsorbate unit cell (due to the adsorbate - substrate interaction). Adsorbates will in general change the structure of the underlying substrate. In particular, they can induce a lift of the reconstruction of the clean surface.
Apart from these simple phenomena there are many things which can make life much more complicated: adsorbate structures have domains and domain boundaries, the surface may have many imperfections such as steps and terraces, the atoms and molecules which are adsorbed on the surface do not show any long-range order and so on.
The concepts for lattice and reciprocal lattice on surfaces are very similar to what we know from the bulk. In any case, it is worth repeating them!
The 14 possible Bravais lattices
of the bulk are reduced to 5
two-dimensional surface Bravais lattices which are shown in Fig. 7.6.
Figure 7.6: The 5 possible two-dimensional Bravais lattices.
The two-dimensional lattice is then the combination of one of the Bravais lattices and a basis. It is important to notice that since a surface is not two-dimensional, the basis atoms need not be in one plane. Once the basis is assigned one can find out the two-dimensional point group of the lattice. The point group will be some sub-group of the highest possible symmetry compatible with the Bravais lattice under consideration. The final symmetry of the system, the space group, is then formed by a combination of the translation group (i.e. the Bravais lattice) and the point group. Like in the three dimensional case this combination can lead to entirely new symmetry elements which are glide-lines in the two-dimensional case. In total, there are 17 possible two-dimensional space groups.
The phenomena of reconstruction and ordered overlayers make it necessary to have a nomenclature which describes the periodicity and symmetry of the surface with respect to that of the bulk. Suppose the two-dimensional lattice vectors of the bulk are a1 and a2. By “two-dimensional lattice vectors of the bulk” we mean the lattice vectors for the bulk-truncated crystal or, equivalently, the vectors which represent the lattice of the bulk projected onto the surface. Let the lattice vectors of the surface including possible adsorbate overlayers be b1 and b2. A simple nomenclature of surface structures is that of Woods . The surface structure is described by
| N | ⎛ ⎜ ⎜ ⎝ |
| × |
| ⎞ ⎟ ⎟ ⎠ | RΘ (7.5) |
where N=”p” or “c” for primitive or centred cells,
respectively, and Θ is the
angle by which the surface vectors have to be rotated with respect to
those of the bulk (see Fig 7.7). The nomenclature of Woods has
the advantage of simplicity. It is, however, not possible to describe
all surface structures because the rotation angle might not be the
same for both vectors.
Figure 7.7: The Woods terminology for surface lattices.
Some examples for the application of the Woods nomenclature are given
in Fig. 7.8. Note that despite of its lack of generality the
Woods nomenclature is still useful because many structures can be
described by it.
Figure 7.8: Examples for structures described by the Woods terminology.
A more general description of the surface structure is the so-called matrix notation . One writes
| b1=m11a1+m12a2, (7.6) |
| b2=m21a1+m22a2. (7.7) |
or, in other words
| ⎛ ⎜ ⎜ ⎝ |
| ⎞ ⎟ ⎟ ⎠ | = | ⎛ ⎜ ⎜ ⎝ |
| ⎞ ⎟ ⎟ ⎠ | ⎛ ⎜ ⎜ ⎝ |
| ⎞ ⎟ ⎟ ⎠ | . (7.8) |
The inspection of the matrix directly allows the classification of the overlayer structures into three types which are illustrated in Fig. 7.9:
It is obvious that the relative strength of the substrate-adsorbate
and adsorbate-adsorbate interactions will favour one type of structure
over the others.
Figure 7.9: Three types of overlayers: (a) simply related to the substrate, (b) rationally related and (c) a incommensurate structure with no common periodicity between substrate and adsorbate lattice.
The reciprocal lattice of the surface is defined in the same way as that of the three-dimensional crystal:
| (7.9) |
This means that
| (7.10) |
and
| (7.11) |
The two last equations also give a simple recipe to construct the reciprocal lattice.
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 λ = h/p 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. 7.10 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.
Figure 7.10: A LEED system.
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
| (7.12) |
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
| (7.13) |
The vertical momentum transfer did so far not enter the discussion at all. This makes sense since for a two-dimensional lattice k⊥ 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 k⊥ because we have to require that
| (7.14) |
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. 7.11. 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 now draw
a ki-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 |ki| around the
starting point of the vector. The intersection of this circle and
the lattice rods gives the possible final kf 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 7.11: 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. 7.10 is adjusted such that the electron beam hits the surface at normal incidence, i.e. such that k∥ 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. 7.13 we will find high intensities at k∥=hg1+kg2. At the same time we know the magnitude |k| of the outgoing electrons and this gives us the emission angle sinΘhk=|k∥|/|k|. Now we consider the imaging by the LEED apparatus (Fig. 7.12). The position of the intensity maxima on the window is given by
| (7.15) |
Figure 7.12: Linear imaging of the reciprocal lattice by LEED.
From equ. 7.15 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. 7.13 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 7.13: 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. 7.13 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. 7.14 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 8.1). 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. 8.10).
Another point worth noticing is the large width of the peaks in Fig. 7.14 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. 7.14 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. 7.14 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 7.14: 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.7.15.
Figure 7.15: 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.7.16 gives a few
examples.
Figure 7.16: 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 7.16. For (a) and (b) there are completely
equivalent structures rotated by 90∘. It is very likely that
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. 7.17 shows the incoherent sum from
the two possible domains in Fig.7.16. 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 7.17: LEED patterns resulting from two different domains of the structures shown in Fig. 7.16.
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. 7.16. 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. 7.18 gives an example.
Figure 7.18: 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.
As we have already discussed above, most metal surfaces do not
reconstruct. The only change in the geometry upon the creation of the
surface are relaxations of the layer distances. In most cases, the
first to second layer distance contracts like in the simple Finnis and
Heine model
. The charge-smoothing effect which leads to this
relaxation should be more important for more “open” surfaces (like
fcc(110)) and less for “closed” surfaces (like fcc(111)). This is
indeed the case. Fig. 7.19 shows the change in the first layer
distance as a function of the bulk value of this distance (both
normalized to the nearest neighbour distance in the bulk). The bulk
value of the layer distance is a measure for the openness of the
surface. Closed packed surfaces are to the right side of the plot,
open surfaces to the left. The plot does not only show the
experimental values for the distances (which were obtained using
quantitative LEED) but also the result from modern first-principles
calculations. The agreement between experiment and theory is rather
good. We further note that all the points lie more or less on one
line, the “universal curve” for the interlayer distance of simple
metals. The only remaining problem is that the qualitative
Finnis-Heine model gets into trouble for the closed packed surfaces.
They relax into the other direction! The first layer distance at the
surface is even bigger than that in the bulk.
Figure 7.19: Relative change in the distance between first and second layer of simple metal surfaces as a function of bulk interlayer distance (both normalized by the nearest neighbour distance in the bulk). After Ref. [30].
Very many ordered adsorbate systems have been solved using LEED. Actually, it is quite fortunate that so many adsorbates form ordered structures such that they can be studied by this technique! We give just two examples here.
Alkali atoms have always been of great
interest as particularly simple adsorbate systems. We have already
looked at alkalis in the lecture about adsorption. Basically, the
picture is very simple: the alkali s level broadens when the atom
gets close to the surface and gets more or less emptied upon adsorption.
The result is a fairly ionic bond. If the substrate for alkali
adsorption is a stable, closed packed surface of a simple metal one
would be expecting simple adsorption sites for the alkali atoms
without a severe perturbation of the substrate, let alone a
reconstruction. A recent LEED investigation of K adsorbed on Al(111)
has shown that this simple picture is not necessarily correct.
[31].
K forms
a (√3 × √3)R30∘) for adsorption at low
temperature and at room temperature. At low temperature the K atoms
are adsorbed in on-top sites on the substrate. However, when adsorbed
at (or heated to)
room temperature they lead to a severe reconstruction of the
surface and are found in substitutional sites (see Fig. 7.20).
Figure 7.20: Adsorption geometry for K on Al(111) at 100 K (top) and 300 K (bottom). Both structures give rise to a (√3 × √3)R30∘) LEED pattern. After Ref. [31].
Another example for an adsorbate-induced reconstruction is the (2x1)
oxygen structure found on Cu(110) at half a monolayer coverage. This
structure is shown if Fig. 7.21. It consists of oxygen-copper
chains in the [001] direction. Evidently, half a monolayer of the top
copper atoms has to be removed or added to form such a structure but
LEED can not give information about the actual mechanism.
Figure 7.21: Adsorption geometry for (2x1)-O on Cu(110). Left: top view of the clean surface, Right: top view of the reconstruction. After Ref. [33].
As mentioned above, semiconductor surfaces tend to reconstruct, sometimes in rather complicated ways. These reconstructions are very closely related to bonding in semiconductors and thus to the electronic structure. We will come back to this in a later lecture. The reconstructions present a major problem to the LEED analysis because of the complicated structure and the large unit cell. Apart from the reconstructions there is the general feeling that the present LEED theory does not work as good for semiconductors as it does for metals. Why this is so is a tricky question and definitely beyond the scope of these lectures. Anyway, the LEED structure determination of semiconductor surfaces is an important present research area.
We just want to show two examples of semiconductor reconstructions in
order to illustrate how complicated they can be. The first is the
famous (7x7) reconstruction of Si(111)
. When Si is cleaved in the
(111) plane several reconstructions can be obtained depending on the
temperature at which the cleaving is performed: (1x1), (2x1) and
(7x7). Annealing the cleaved surface to high temperature always
results in the formation of the (7x7) structure which remains also
when cooling the crystal down again. Therefore, it is thought that this
reconstruction is the one with the lowest total energy. Several
structural models have been proposed for the (7x7) reconstruction. The
model which is thought to be the right one is shown in Fig. 7.22.
It has to be pointed out, though, that this structure has not been
found by LEED.
Figure 7.22: Left: ideal Si(111) surface. Right: the Takayanagi model for the Si(111) (7x7) reconstruction. After Ref. [34].
Another example is the (2x1) reconstruction of the technologically
important Si(100)
surface. This surface has been studied with a
variety of techniques including quantitative LEED. The model favoured
at present is that of a dimer reconstruction of the surface as shown
in Fig 7.23. There is, however, still a considerable dispute
about the details of the reconstruction, in particular on the
question if the dimer is symmetric or not.
Figure 7.23: The unreconstructed Si(100) surface and the asymmetric dimer model.
Studying insulator surfaces with electron spectroscopy immediately brings up the problem that the surface charges up when it is bombarded with electrons or when it emits electrons. No meaningful experiment can be done. If we want to look at insulators with LEED we have to overcome this problem somehow. There are two main strategies to do this. The first is to have a sufficient number of defects in the insulator which provide carriers such that a small conductivity results. In the case of insulating oxides this is mostly done by heating the sample in vacuum which results in oxygen vacancies. The disadvantage is that one studies a modified version of the oxide. The other possibility is to prepare a thin film of the oxide on a metallic substrate. This also gives enough conductivity to do electron spectroscopy but one has to worry about the fact that one may not really study the surface properties of the real bulk material.
Surfaces of oxides where used a lot in the early days of surface science because
some are so inert that they can be studied for several days even under
rather poor vacuum conditions. An example for an early LEED
investigation of an insulator is the (001) surface of NiO
. NiO has a
rock-salt structure shown in Fig. 7.24. It is a
prototype material for a highly correlated electron system. The
electronic configuration of the Ni2+ ions is 3d8 and this
partially filled band would correspond to a metallic ground state in
a band structure picture. Yet, the strong on-site d−d electron
repulsion causes the material to be an insulator with a band gap of
more than 4 eV! The LEED pattern from NiO(001) was found to be (1x1).
For this surface there are two possibilities of structural changes
consistent with such a LEED pattern: an interlayer relaxation as we
know from metal surfaces and a buckling in the surface layer (i.e. a
structural change where the Ni ions and the oxygen ions are not in the
same plane any more). The latter geometry change is not possible with
only one atom per unit cell.
The LEED result is that the surface shows a small inward relaxation
but no buckling.
Figure 7.24: Structure and surface unit cell of NiO (top view). Top-layer Ni ions are red, second layer Ni ions are pink, oxygen ions are black.
There are a couple of other techniques which we should mention. The first one is x-ray scattering from surface. This technique is very similar to bulk x-ray scattering with the major advantage over LEED that the theory is purely kinematic. The problem of the long mean free path of the x-rays is overcome by using a grazing incidence angle. In such a geometry total external reflection occurs because the refractive index for x-rays is smaller than one. It is, however, only very little smaller than one (in the order of 10−5 such that the incidence angle below which total reflection occurs is very small, a few tenth of a degree. Therefore, surface x-ray scattering requires a highly collimated x-ray beam which can only be produced at a synchrotron radiation source delivering high-brilliance light.
It is interesting to compare the advantages and disadvantages of LEED and surface x-ray scattering. One is that the momentum transfer, and hence the sensitivity to structural parameters, is nearly perpendicular to the surface in case of LEED and nearly parallel to the surface in the case of x-ray scattering. The penetration depth is a little higher for x-ray scattering but it has the same order of magnitude as for LEED.
There are several interesting experiments involving the scattering of atoms and ions but we are not going to discuss them here (apart from He scattering in the section about vibrational properties, section 10.4). If you are interested in these techniques, consult the book by Woodruff and Delchar, section 1.3.
Two techniques based on electron scattering are discussed here in further detail: (S)EXAFS and photoelectron diffraction. Both do not rely on long-range order on the surface (like LEED). Instead, they can be used to determine the structure locally around an atom of interest. This is achieved by using specific atoms as “electron sources” instead of an external source like in LEED.
The techniques of EXAFS and SEXAFS have been made possible by the construction of synchrotron radiation (SR) sources. SR has a continuous energy spectrum. In combination with a monochromator it provides a tunable x-ray source. The (S)EXAFS measurements involve scanning the photon energy around the absorption edges of the atoms in a material or on a surface. The fine structure in the absorption cross section gives information about the neighbours of the emitting atoms. The main advantages of these x-ray absorption techniques are that they work for materials where long-range order is not present and that at least the nearest neighbour distances can be obtained with rather high precision.
Fig. 7.25 shows an example for an EXAFS spectrum.
Such a spectrum can be taken
by exposing a thin film of material to x-rays and simply measuring the
transmission through the film.
The x-ray
absorption of Cu is plotted vs the photon energy. As the energy
reaches the K-edge the absorption increases steeply because of the the
possibility to excite the K-electrons. Above the edge the absorption
shows a slow decrease due to the matrix element. On this slow
decrease rapid oscillations in the cross section are superimposed.
These are the EXAFS oscillations.
Figure 7.25: X-ray absorption of Cu in the vicinity of the K-edge.
The physical origin of these oscillations is quite easy to understand.
The absorption cross section is given by Fermi’s golden rule and thus by
a matrix element of the type <f|H′|i>. The
initial state is simply the localized core state. The final state is
the extended state of the outgoing electron wave, including all the
multiple scattering
processes. One can now think about the EXAFS
oscillations in the following way. At low kinetic energies, from zero
to a few hundred electron volts, the cross section for the
back-scattering of the electrons from the neighbour atoms is rather
high. These back-scattered waves have to be added coherently to the
outgoing wave and this directly influences the final state at the
emitter and thus the matrix element. The interference from outgoing
and back-scattered waves changes with a periodicity given by the
nearest neighbour distance. The effect is illustrated in Fig.
7.26.
Figure 7.26: Schematic illustration of the interference leading to the EXAFS oscillations
This simple picture works only for electron energies which are not too close to the edge. Below kinetic energies of 50 eV or so the oscillations contain resonant absorption from the valence states.
It is convenient to extract the EXAFS oscillations from the slowly varying background. This is done by the definition of a “fine structure function” χ
| χ(k)= |
| . (7.16) |
where σ is the measured absorption and σ0 is the absorption due to the free atom.
If we consider a simple singe-scattering picture the fine structure function is given by
| χ(k)=−k−1 |
| Ai(k)sin | ⎡ ⎣ | 2kRi+φi(180∘,k) | ⎤ ⎦ | . (7.17) |
where k is the electron wave number, Ai is an amplitude function defined below and φi is a phase shift. The sum runs over different “shells” of neighbours, a shell being defined as a set of neighbours having the same distance from the emitter, Ri. In principle, the desired values for Ri can be extracted from χ by a Fourier transformation. The phase shift would not be a problem in such an analysis but its energy dependence is one. It leads to wrong values for Ri. Thus the phase shifts have to be included in the analysis. One can use calculated phase shifts or one can, in contrast to LEED and Photoelectron Diffraction, use “experimental” phase shifts. The reason is that only the phase shift for 180∘ back scattering is of interest, not the phase shifts for all the other scattering angles. The 180∘ phase shift can be obtained from a material which contains the scatterer of interest and has a known structure, e.g. a single crystal.
Let’s look again at the amplitude function for the different shells. It is given by
| Ai(k)=(Ni/Ri2)|fi(180∘,k)|W(T,K)exp(−2Ri/λ). (7.18) |
Ni is the number of atoms in the shell. The 1/Ri2 factor leads to an effective localization of EXAFS explaining the success of a single-scattering treatment. It is caused by the fact that both emitter and scatterer are point sources. The next factor is the modulus of the scattering amplitude. W(T,K) is a Debye-Waller factor which takes the thermal vibrations into account and the last factor describes the inelastic scattering of the electrons in the solid. As mentioned above, the single scattering approach works quite well, at least in order to determine the distance of the nearest neighbour shell.
The surface version of EXAFS is called SEXAFS. One can for example use the technique for determining the bond distances of adsorbate atoms to the substrate. However, what one actually wants to know is the adsorption site and this is difficult to get from just the bondlength. Help could come from considering the absolute amplitude of the modulations because this should give the number of nearest neighbours (equation 7.17). But due to experimental difficulties (see below) it is very dangerous to use the absolute modulation strength. One can also play the following trick. The synchrotron radiation one uses for the experiment is polarized. This means that the electrons from a core level will have a certain angular distribution, depending on the direction of the polarization vector. One can take SEXAFS data for different directions of the polarization vector such that different possible neighbours would be “hit” by a high intensity of photoelectrons. Comparing the different SEXAFS spectra one can work out where the nearest neighbours are.
One has to add a few words of caution, though. SEXAFS is a very difficult experiment. The first problem is what to measure in order to get the absorption of a particular atomic species on the surface. One possibility is the intensity of the Auger signal which is emitted by the decay of the core electron one creates. Such Auger peaks are usually positioned on a high background of inelastically scattered electrons, leading to a bad signal-to-noise ration. To make matters worse, the SEXAFS modulations are only a tiny fraction of the absorption cross section (one percent or so) and there are only a few adsorbate atoms compared to a bulk EXAFS experiment.
We just give one example here which demonstrates how powerful SEXAFS is as a technique for structural investigations. SEXAFS has been used to determine the structure of the co-adsorption system SO2 + O on Cu(111) which is formed upon SO2 adsorption [36]. Fig. 7.27 shows data taken at the oxygen K-edge for two different polarizations of the incident photons. The spectra are quite different, illustrating the usefulness of the “trick” mentioned above. The Fourier transforms of the data are shown together with the result of a simulation for a particular geometry. The agreement for the closer distances is very good.
Figure 7.27: SEXAFS data from the oxygen K-edge of the co-adsorption system SO2 + O on Cu(111). After ref. [36].
Fig. 7.28 shows the final result of this structural
determination showing the position of the adsorbates and
a very complex surface reconstruction.
Figure 7.28: Structure SO2 + O on Cu(111) as determined by SEXAFS. After ref. [36].
The principle in photoelectron diffraction is given in the Fig.
7.29. One
measures the intensity of a core level line as a function of energy and
emission angle. The electron can reach the detector on a direct path or it
can be scattered at the atoms surrounding the emitter and then reach the
detector. The final state is given by interference between the direct and
the scattered components. This interference depends on the path-length
difference and on the scattering phase shifts
. This, in turn, depends on
the electron wavelength / kinetic energy, on the position of the detector
and on the position of the emitter with respect to the scatterers.
Figure 7.29: Principle of photoelectron diffraction
By varying the position of the detector or the kinetic energy of the emitted electrons the interference conditions are changed. Normally the photoemission intensity is recorded as a function of kinetic energy or emission angle. The modulation in the intensity which results from the change in the interference conditions can be used to extract geometrical information.
The technique is related to SEXAFS and LEED. SEXAFS is similar to PhD in that the electron source is also a core level from an atom at the surface. Here the interference between the outgoing and the 180∘ backscattered waves is of major importance. The backscattering changes the final state intensity at the core and this in turn changes the absorption cross section. SEXAFS is very similar to PhD with the difference that the emitter itself is the detector. In a simple picture, it can be viewed as an angle-integrated PhD experiment. The PhD modulations do, of course, also contain the SEXAFS part, i.e. the modulations in the absolute cross section. This is, however, not a problem because the SEXAFS modulations are much weaker (1-3%) than the PhD modulations (30-50%).
The big difference between these two techniques and LEED is that the electron source in the latter is not in the system itself but in the far-field. It is not possible to pick an atom in the adsorbate layer and make it special by considering just the core-level intensity from that atom. All the atoms are of equal importance as scatterers. The information about the adsorbate layer relative to the substrate is only contained in the wave-part scattered by the adsorbate and its interference with the wave-part scattered by the substrate. In PhD and SEXAFS the position of the adsorbate relative to the substrate is contained in ALL scattering pathways. A further difference is that LEED is a true diffraction technique which relies on the long-range order in the adsorbate layer. PhD and SEXAFS only probe the local structure around the adsorbate. This is a great advantage for disordered systems. Multiple scattering is very important in LEED, less important in PhD and least important in SEXAFS. The reason is the 1/r fall-off of the spherical wave from the electron source and, in SEXAFS, also from the electron scatterer. PhD contains the 1/r effect once, SEXAFS twice. Finally, SEXAFS and PhD have chemical sensitivity: it is possible to pick out one specific atom in the adsorbate. Indeed, if there are two, say, carbon atoms in a different chemical environment it is even possible to pick one of them as an emitter because of the chemical shift in the XPS lines (only in PhD, see below).
The basis of the theory is the scattering of electrons by ion cores. For
the simple case of plane-wave scattering this can be described in terms of
a scattering amplitude and a phase shift. Both are angle and energy
dependent. Fig. 7.30 shows the modulus of the scattering factor.
Figure 7.30: Modulus of the electron scattering factor.
The strongest scattering always occurs in the forward scattering direction, i.e. for a scattering angle of 0∘, especially for high electron energies. At low energies there is also some scattering in the back scattering direction. For LEED the back scattering direction is of particular importance (scattering angles between 90∘ and 180∘. For SEXAFS only 180∘ scattering is relevant. In photoelectron diffraction both back scattering and forward scattering has to be considered.
The most important point in the theory is to realise that the intensity at the detector is the coherent sum of the direct wave and all the waves reaching the detector on however complicated scattering paths:
| I(k,Θ,φ)∝|Ψ0+ |
| Ψsj|2 (7.19) |
The Ψ’s describe the amplitude of the different components of the wave field. Ψ0 is reaching the detector directly. The Ψs are the amplitudes after several scattering processes.
The first question is how the wave function of the outgoing electron can be described without any scatters. The standard optical selection rules apply for the angular distribution, i.e.
| l′=l ± 1 and m′=m,m ± 1. (7.20) |
The intensity can be written as an intuitive formula when the following assumptions are made:
The result for the intensity looks like this (see also Fig. 7.31):
| I(k)∝ | ⎪ ⎪ ⎪ ⎪ | cosΘk+ |
|
| |f(Θj,k)|exp(ikrj(+ cosΘj)+φ(Θj,k)) | ⎪ ⎪ ⎪ ⎪ |
| (7.21) |
Figure 7.31: Symbols in single scattering theory.
The first term corresponds to the direct wave. The sum is over all relevant scatterers. The first term in the sum describes the intensity which arrives at the scatterer. f is the complex scattering factor. Its magnitude describes how much of this intensity is scattered in the direction of the detector. Its phase φ(Θj,k) describes the scattering phase shift. The exponential term describes the pathlength difference with the direct wave.
There are two very important ingredients missing in the above formula. The inelastic scattering of the electrons attenuates the propagation of the waves in the crystal. It also makes the calculation easier because it reduces the number of important scatterers. It can be taken into account via the usual mean-free path concept. The finite temperature of the crystal makes the atoms vibrate. This means that the pathlength differences are modulated. This effect tends to wash out the photoelectron diffraction modulations. It can be taken into account via a Debye-Waller factor like in x-ray scattering. Adding this to the above formula gives:
|
Note that the inelastic mean free path λ is twice as big as the value normally taken in electron spectroscopy since it is an amplitude attenuation factor and not an intensity attenuation factor.
In the angle-scan mode of PhD the core level intensity is measured as a function of emission angle keeping the photon energy fixed. An obvious advantage of this approach is that a laboratory x-ray source can be used instead of a synchrotron radiation facility.
Although such an experiment is dominated by high-energy forward scattering,
it can be used to learn something about adsorbates. The intensity from an
adsorbate core level can be observed at grazing
emission (see Fig. 7.32. The intensity can be expected to be strongest for directions
where substrate atoms lie close to the emitter. One can learn something
about the symmetry of the adsorption site.
Figure 7.32: A grazing emission forwards scattering experiment.
Angle-scan photoelectron diffraction can also be used to determine the orientation of molecules on the surfaces. The intensity of the atom closer to the surface is measured. At a certain angle a forward scattering condition is met where another atom of the molecule is between the detector and the emitter. There the intensity will be increased. This allows the determination of the molecular axis orientation. Note that at high energies only a very small fraction from the core level electrons from an adsorbate will be scattered back by the substrate.
Fig. 7.33 shows the situation for a CO molecule adsorbed on a surface. Consider the emission from the carbon 1s core level (we know that the carbon atom is going to the atom which forms the the chemical link to the surface, hence it has to be “under” the oxygen atom) [37]. At high energies forward scattering dominates and the only important scattering process is along the molecular axis. Moving the analyser over the surface (in angle) will give the right tilt angle of the molecule. Fig. 7.34 shows the result of such angular scans for CO adsorbed on Ni(100). Evidently at higher coverage tilted CO species can be found.
Figure 7.33: A forward scattering experiment to determine molecular orientation.
Figure 7.34: A forward scattering experiment to determine molecular orientation of CO on Ni(100). After Ref. [37].
Note: this simple interpretation is not only based on the assumption of strong forward scattering but also on a small scattering phase shift. Otherwise the interference in the direction of the molecular bond might turn out to be destructive. This has lead to some confusion and has been wrongly termed "shadowing".
In scanned energy mode photoelectron diffraction the intensity at the detector is measured as a function of photoelectron kinetic energy in a fixed emission direction. Since the binding energy of the core level under consideration is constant, synchrotron radiation has to be used to vary the kinetic energy of the photoelectron. This mode of PhD is particularly interesting and it has close ties to both LEED and SEXAFS.
To illustrate this consider a typical XPS spectrum of a clean and
adsorbate-covered (C2H2) Ni(111) surface (Fig. 7.35).
Figure 7.35: XPS spectra of clean and C2H2 covered Ni(111).
As the photon energy is changed the kinetic energy of all the XPS peaks
also changes. Fig. 7.36 shows a group plot of very many XPS spectra
taken around the O 1s region for the system CO on Cu(110). One of the
peaks is magnified.
Figure 7.36: Group plot of many XPS spectra of the O 1s peak from CO on Cu(110). All the spectra are taken with different photon energies. One spectrum is magnified.
Already in this raw data plot a clear modulation of the intensity is visible. The function I(E) describes now the integrated intensity of each peak. Like in SEXAFS a modulation function is defined by I(E) and the slowly varying part I0(E)
| χ(E)= |
| . (7.23) |
The integrated I(E) and the modulation function for the system above
are shown in Fig. 7.37.
The modulation functions for various emission directions can be viewed as the analogue to the LEED I/V beams. The interpretation of the modulation functions is not as obvious as in the angle-scan case. Multiple scattering is far more important and this also means that there is more structural information. The various modulation functions for different emission directions have to be compared to theory to obtain structural information.
Let us illustrate this by a simple example, the adsorption of the acetate
species on Cu(110) [38]. A O 1s diffraction spectrum is compared to theoretical
calculations for different adsorption sites and distances from the
surfaces. The best agreement is found for a bridge site (see Fig.
7.38).
Figure 7.38: O 1s diffraction data to determine the adsorption site of the acetate species on Cu(110). After Ref. [38]. The R-factor has the same meaning as in a quantitative LEED analysis.
The big advantage of PhD with respect to LEED lies in the dependence of local order only. However, if a molecule is adsorbed on more than one site at the same time more information is needed to obtain the right structural parameters. In such a case many modulation functions are measured and compared to theory.
A further advantage of PhD is its chemical sensitivity. The positions of O and C
in the above example can be determined separately considering the O 1s and
C 1s emission. Moreover, the two C atoms are in a chemically
different environment. This means that the binding energies are also
different (chemical shift, see 5.3.4
). Hence the PhD from both C atoms can be
measured independently and the modulations are quite different.
This is illustrated in Fig. 7.39.
Figure 7.39: Modulation functions for the two chemically different C atoms in the acetate species on Cu(110). After Ref. [38].
D. P. Woodruff in Angle-Resolved Photoemission, Theory and Current Applications (eds. S. D. Kevan) Elsevier, Amsterdam (1992).
A. M. Bradshaw and D. P. Woodruff in Applications of Synchrotron Radiation: High Resolution Studies of Molecules and Molecular Adsorbates on Surfaces (eds. W. Eberhardt) Springer, Berlin
It is clearly desirable to have a real-space microscopic technique which can image the structure of surfaces on a truly atomic scale. Field emission microscopy is a possibility with a rather limited range of possible applications. First the advent of scanning tunnelling microscopy has made a real space atomic scale view on most surfaces possible. But the section should start with a clear warning. STM does not measure the structure of surfaces but, as we shall see below, the electronic structure. It is essential to keep this in mind, in particular when working with semiconductors and insulators.
Scanning tunnelling microscopy is based on the quantum mechanical effect of tunnelling illustrated in Fig. 7.40. The wavefunctions at the Fermi level exponentially leak out of the metal with an inverse decay length of
| (7.24) |
where m is the mass and φ is the local workfunction. If now two metals are brought in close contact and a small voltage is applied between them, a tunnelling current can be measured which is
| (7.25) |
where d is the distance between the conductors. The important
message here, and the reason why STM works, is the exponential
dependence of the tunnelling current on the distance between the
conductors. We will come back to this several times in the following.
Figure 7.40: (a) Exponential leakage of the wavefunctions from a conductor into the vacuum. (b) Application of a voltage and tunnelling between two conductors because of the overlap of the wavefunction tails. Φ is the workfunction which is discussed in section 8.1.
Fig. 7.41 shows the principle setup for an STM. It consists of a sharp tip, very close to the sample, which can be moved with high precision using three mutually orthogonal piezoelectric transducers (PET). A small voltage is applied between the tip and the sample and the current is measured. Typical values for the tunnelling voltage are from a few mV to several V or so and for the current from 0.5 to 5 nA. The tip-sample distance is a few Angstrom. The tunnelling current depends very strongly on this distance. A change of 1 Åcauses a change in the tunnelling current by a factor of ten.
Most STM
topography studies are performed in the so-called constant current
mode. The tip is scanned across the surface by the X and Y PETs. The
Z PET is in a feedback loop which applies a correction voltage to the
Z voltage in order to maintain a constant tunnelling current as the
XY position of the tip is changed. Since the piezo extension is
proportional to the applied voltage, this correction voltage is a
direct measure for the change in Z the tip has to perform in order to
follow the contours of the sample and it can be used as an “image”
of the sample when displayed as a function of X and Y voltages.
The exponential decay of the tunnelling current with distance from the
sample is crucial for this operation: even if we can only keep the
current stable by 10 percent or so, this will still mean that we have
a very high precision in the Z measurement because a small
uncertainty in the current means virtually no uncertainty in the
distance.
Figure 7.41: Schematic construction principle of an STM.
While the principle of STM operation is simple, its practical realization faces some formidable difficulties. The first is that the tip has to be brought at a distance of a few Angstrom from the surface and has to be stabilized there with sub-Angstrom stability. This process has to be performed in the presence of mechanical vibrations and thermal drift. The vibrational problem is solved by vibrational insulation, like suspensions by springs or the use of a support frame which has a resonance frequency very different from the usual noise frequencies of the environment. The only way to get completely rid of the thermal drift problem is to stabilize the whole microscope and sample at the same (low) temperature. The most stable STMs today are working in a UHV vessel which is placed inside a dewar filled with liquid Helium. The next practical problem is to make an atomically sharp tip. Different techniques are used like cutting and etching and most tips are made off W or Ir. Again, the exponential decay of the tunnelling current helps: one can hope that there is one single atom sticking out a little further than all the others on an otherwise rather blunt tip. Then most of the tunnelling will happen through this atom and it will be possible to obtain images with atomic resolution. Unfortunately, it is not possible to prepare a tip in a controlled and reproducible way. In particular, one has no control about the chemical nature of the outermost atom of the tip but the images depend a lot on this.
The calculation of STM images is very difficult. For the sample one can make reasonable total energy calculations for a perfect surface. For the tip such calculations are not so easy because of the unknown shape. Even if one could calculate the wavefunctions for each system, sample and tip, separately bringing them together would render the problem hopelessly complicated because now one has a new system which totally lacks the translational invariance which was needed to solve the surface problem. The last hurdle is circumvented by Bardeen’s suggestion to basically use the unperturbed wavefunctions of tip and sample and to consider just the tunnelling between them. Tersoff and Hamann have developed a simple and useful STM theory based on this approximation. They furthermore assume that the tip can be approximated by an s-wave state (i.e. that its end has spherical symmetry) and that temperature and tunnelling voltage are very low. This results in
| (7.26) |
where R is the radius of the tip, Dtip is the tip density of states at the Fermi level, rtip is the centre position of the tip and
| (7.27) |
is the local density of sample states at the Fermi level. The only z dependence in these equations is in the wavefunctions of 7.27 such that we recover our exponential decay into the vacuum, as we must. These equations give a simple interpretation for constant-current mode STM images. If we scan across the surface at constant current or, more precisely, at constant It/Vt, adjusting just the distance of the tip over the surface, we basically follow the contours of sample density of states at the Fermi-level. Note, however, that this statement is only correct if we assume that there are no lateral changes in the workfunction of the sample.
We add an additional rather crude approximation. We write ρsamp(rtip,EF) as a superposition of charge densities of the free atoms
| (7.28) |
where the sum extends over all the atoms making up the solid. Then we say that the interesting density of states at the Fermi level is just a fraction of the total charge
| (7.29) |
This approximation clearly ignores the fact that there is any bonding in the solid. While the approximation of no bonding at all obviously is not a good one, we may expect it to work best for delocalized bonding because this does at least avoid the presence of strongly oriented bonds which might show up as structure in the STM images. It is evident that in this approximation the STM sees “atoms” but we have to be aware of the limits of this view!
In practice, the tunnelling voltage is not always very small. Especially for semiconducting materials a small tunnelling voltage can be impossible because there are no carriers in the gap which can be involved in the tunnelling. The tunnelling voltage can be chosen to be both negative or positive. This means that STM can look at both, occupied and unoccupied states of the sample depending on the bias voltage.
Taking STM data with atomic resolution on metal surfaces is much harder and historically later than on semiconductor surfaces. The reason is the small electronic corrugation of metal surfaces. We have already encountered this phenomenon as Smoluchowski effect / Finnis-Heine model for the metal relaxation. Nevertheless, STM investigations of metal surfaces are now possible and we give two examples.
The first is the (1x2) missing row reconstruction
on the (110) surface
of the fcc metal Au. Fig. 7.42. In the large scale image we
can clearly see the remaining rows as well as several terraces. In
the small-scale image we can again see the rows and even the atomic
corrugation parallel to the rows.
Figure 7.42: STM topographs of the (1x2) missing row reconstruction on Au(110). (a) (800 x 800) Å2 with the presence of atomic steps in the upper part of the image [39] and (b) (84x 84) Å2 with atomic resolution even along the rows. After [40].
Au(111) shows a more complicated reconstruction . The unit cell of the reconstructed surface is much bigger than the (1x1) unit cell, as observed by LEED. In fact, a correct description of the overlayer structure is (
| 22 | 0 |
| −1 | 2 |
). The reconstruction is caused by a denser layer
of Au atoms in the first layer than in all other layers. The first
layer contains 4.5 percent more atoms. The reconstruction is a
compression of a hexagonal layer to accommodate these atoms.
It was suggested that the atoms in the top layer are arranged such that
there are regions of fcc stacking sequence (ABCABC…, like in the
bulk) and hcp stacking sequence (ABAB...). These regions have to be
separated by transition regions where the atoms are in less symmetric
sites.
It is impossible to solve such a structure
quantitatively by LEED. It was just clear that the unit cell is very
large but the position of the atoms in the unit cell were not
known. STM was able to shed light on the reconstruction. The
images in Fig. 7.43 show the reconstructed surface on different
scales. Fig. 7.43 (a) shows a pattern with pairs of bright lines which
are due to the reconstruction. The dark area between a pair of stripes
was identified as the region of hcp stacking and the area between two
stripe pairs as fcc stacking. This means that there is a higher area
of fcc stacking which also makes sense energetically. It is consistent
with intuition, that the transition regions should appear brighter.
The first-layer atoms are not placed in hollow sites and stick out
further than the three-fold hollow site atoms.
An atomically resolved image (b) allows
to identify the unit cell and to state that the reconstruction is
caused by a uniform contraction in the [110] direction. In (c),
finally, a bigger overview is given which shows rotational domains of
the reconstruction and transitions between them. The accommodation of
more atoms in the first layer than in the other
combined with the uniaxial contraction causes an elastic
strain which
leads to a long range “herringbone”
pattern formed by the reconstruction.
Figure 7.43: STM topographs of the reconstruction on Au(111). (a) (360 x 420), (b) (57 x 89) Å2 and (b) (1230 x 1280) Å2. Reprinted figure with permission from [41]. Copyright (1990) by the American Physical Society.
How do adsorbates show up in STM images? According to our theory section the appearance of adsorbates will depend on the change they induce in the local density of states at the Fermi level. If they increase this LDOS, they will appear as protrusion, if they decrease it, they will appear as hole in the images, if they do not change the LDOS we will not be able to see them at all! Consider for example the case of Li, Si and Cl shown in Fig. 6.5. Si atoms would be well visible on jellium , Li less and Cl not at all. In a real STM experiment, however, there is still the possibility to work at a higher tunnelling voltage to improve things.
Unfortunately, it is not possible to chemically identify individual adsorbates by the appearance in the STM images. Conclusions about the chemical identity can only be drawn rather indirectly. One can, for example dose some gas on a surface which was clean before. If one can then see adsorbates, these must be related to the gas. In the case of a monoatomic gas, the adsorbates can be identified as the gas atoms. For a molecular adsorbate one might identify the whole molecules or the fragments. Information about possible dissociation products can be taken from some other technique. There is a very recent development in STM spectroscopy which could lead to some chemical sensitivity. We will discuss this in the lecture on surface vibrations, section 10.5.
Suppose we know the chemical nature of the adsorbate. Can we then learn something about the adsorption geometry, or at least about the adsorption site? This is sometimes possible and done in the following way. Suppose we have a small adsorbate like an atom ar a diatomic molecule. In the case of a small coverage we might be able to image the adsorbates and the substrate lattice at the same time. Then one can infer the adsorption site by comparing the adsorbate’s position to those of the substrate atoms. Fig. 7.44 illustrates this idea. It is an STM images taken from the adsorption system S on Ni(100). One can clearly identify the lattice and the adsorbates and one would guess that the carbon atoms are adsorbed in a four-fold hollow site. However, one has to be careful since this strategy can go wrong very easily: suppose the Ni(100) surface is imaged such that the holes and not the bright spots correspond to the Ni atoms. Then carbon would be adsorbed in an on-top or even in a substitutional site.
Yet another aspect of adsorbates is the high sensitivity of STM towards surface contamination and defects. Note that a low contamination level of 1 percent, almost invisible in AES and XPS, at least for certain atoms, would show up rather drastically in an STM image if you take into account that it would mean one adsorbate atom in 10x10 substrate atoms.
Figure 7.44: STM image of S adsorbed on Ni(100). Reprinted figure with permission from [42]. Copyright (1993) by the American Physical Society.
STM can contribute a lot to the understanding of adsorbate induced
reconstructions. Consider our example of the
adsorbate system (2x1) oxygen on Cu(110). LEED has shown that this
reconstruction consists of copper-oxygen chains in the first layer
(see Fig. 7.21). STM can also reveal how the transport of
substrate atoms needed for the reconstruction
takes place. Fig. 7.45 shows a series of STM images taken as
the reconstruction proceeds.
Figure 7.45: STM images taken during the oxygen adsorption on Cu(110). On the upper left corner of the first image two steps are seen. The grey lines perpendicular to the steps are due to the reconstruction. As more and more oxygen is adsorbed the reconstructed area increases but the steps are “eaten” away. This is clear evidence for the fact the the reconstruction is of added-row type with copper atoms stemming from the steps [43].
Atomic resolution is much easier to get on semiconductor than on metal surfaces. The reason is the higher corrugation of the latter. STM can also be helpful to get some starting ideas about the complex structures in semiconductor reconstructions which can then be refined by a “real” structural analysis like LEED or surface x-ray diffraction. On the other hand, the more directional bonding in semiconductors should make the interpretation of STM images as pure atomic structure even more problematic than on metal surfaces. We give two examples with which we come back to the two Si structures discussed in connection with LEED.
The Si(111)(7x7) reconstruction
was among the first surfaces studied
with STM. STM has contributed a lot to the understanding of this
structure. Fig. 7.46 shows calculated STM images based on the
superposition of atomic charges (equ. 7.28) together with the
experimental data. Despite of all the shortcomings of this
approximation it gives an almost quantitative agreement with
Takayanagi’s model of the reconstruction.
Figure 7.46: Calculated STM images for various models of the Si(111)(7x7) reconstruction and comparison to the experimental data. Images and two linescans through the images. Solid lines are calculations, dashed lines the data. Reprinted figure with permission from [44]. Copyright (1986) by the American Physical Society.
A similar insight was gained for the (2x1) reconstruction
of Si(100).
First, it was found that the reconstruction consists of rows of
dimers like in Fig. 7.21. Later, it was established that the dimers
are buckled but this buckling might disappear at sufficiently high
temperature. Fig. 7.47 shows an example of the
atomically-resolved Si(100)-(2x1) structure. The asymmetry of the
dimers is clearly evident. Unfortunately, the authors of the paper
did not mention the temperature at which the experiment was done!
Figure 7.47: STM image of the Si(100)-(2x1) structure. The dimer rows are found along the (110) structure. A linescan across the dimers in the bottom of the figure shows their asymmetry. Reprinted figure with permission from [45]. Copyright (1995) by the American Physical Society.
It is obvious that STM is not the right tool to study the surfaces of
insulators. A tunnelling current through the sample is not possible
and the tip crashes into the sample upon approach. Like in all
electron spectroscopy techniques, it helps to heat the sample in order
to have (thermally activated) carriers. If a sufficient density of
carriers is present in the bulk, one still has to choose the
tunnelling voltage such that one can either tunnel into unoccupied
states above the gap or out of occupied states below the gap. As an
example, we discuss an STM investigation of the NiO(100)
surface[46].
The sample
was heated to about 200∘C during the measurements, at room
temperature no imaging was possible1.
Fig. 7.48 (a) shows an atomically-resolved empty state
image of the NiO(100) surface, i.e. an image obtained with a positive
bias voltage on the sample. It was argued, that the bright
spots on the image correspond to the Ni ions. Fig. 7.48 (b)
also shows an empty state image with a point defect in the middle. The
previous Ni ions are still barely visible. A mesh is laid above
the Ni ions. It permits the defect to be localized on a Ni site.
The defect changes the contrast in the Ni ions surrounding it. This
change is strong in the next-nearest neighbours, not in the nearest
neighbours. This view is consistent with the present picture of
covalent bonding in NiO shown in Fig. 7.48 (c). The bonding
between the d states of the Ni atoms is achieved only through the
p orbitals of the oxygen atoms in between. Since the p states on
each oxygen atom are orthogonal, two simple cubic sub-lattices are
formed which are displayed in different shades in the figure. If a
defect on a Ni site is created, it is obvious that the nearest
neighbour Ni ions will not notice this very much but the next-nearest
neighbours will. The STM image therefore confirms this picture of
bonding in NiO.
Figure 7.48: STM images taken on NiO(100). (a) atomically-resolved empty state image (b)atomically-resolved empty state image with a defect (c) sketch of the bonding in NiO. Reprinted figure with permission from [46]. Copyright (1997) by the American Physical Society.
STM can be applied to many other problems which can not be treated in this context. An interesting field is the dynamics of surfaces. One can learn a lot about diffusion and similar phenomena. With a fast STM, “movies” of changes on the surface can be taken.
A possible problem in the interpretation of STM images is the strong tip surface interaction due to the high electric field. One has to keep in mind that one does not necessarily observe the surface but the surface in presence of the tip. This disadvantage can also be turned into an advantage: one can use the tip to manipulate surface and/or adsorbate atoms and build artificial structures on the nanometre scale. One can for example write letters with the STM tip or create patterns of atoms to store information. It is, however, rather questionable if such techniques will ever be applied to something really useful because it simply takes too much time to build such structures.
Another field which has taken off considerably in the last few years is the use of scanning tunnelling spectroscopy (STS) where the tip is parked over a particular point on the surface and basically the tunnelling current is measured as a function of voltage. This opens the possibility to measure electronic effects on a very small scale but it is also difficult, since the STM has to be very stable. In addition to this, one really has to worry about the electronic structure of the tip when doing such an experiment. We will come back to STS in the lectures about surface electronic structure and surface vibrations.
Last but not least, there are interesting technical improvements. One
is the development of scanning force microscopy.
A scanning force
microscope does not rely on a tunnelling current and can therefore also
be used on insulators. Fig. 7.49 shows a sketch of such an
instrument. The tip of the microscope is mounted on the end of a
cantilever. As the tip is brought into close contact with the sample,
a force is exerted on the tip which leads to a small deflection of
the cantilever. This deflection can be measured as a function of x
and y coordinates in the same way as for the STM. Recent progress in
this technique has yielded measurements with truly atomic resolution
on some surfaces.
Figure 7.49: Principle of the scanning force microscope.
For a discussion of surface thermodynamics, consult [3]. The basics of surface structure is discussed in [2]. A detailed discussion on LEED can be found in the same book but for more detailed information see [47][49][48]. A good review of photoelectron diffraction is [50]. There are many books about scanning tunnelling probe techniques by now. See for example [51]