For all the following considerations, it will be relevant that materials have a work function, i.e. that one has to pay some energy in order to extract an electron from a solid. In a photoemission process, for example, the electrons loose kinetic energy as they cross the surface. Let’s examine this a little further.
The workfunction of a metal is defined as the smallest energy needed to extract an electron at 0 K. Formally, this definition is made for an infinitely large crystal plane. One takes an electron from infinitely deep inside the crystal and brings it through the surface, infinitely far away into the vacuum. In practice, one wishes to avoid external fields and fields set up by the edges of the crystal. The definition is modified such that one brings the electron far away from the surface compared to atomic dimensions but not far compared to crystal dimensions. The energetics involved in this is displayed in Fig. 8.1. There are two potentials displayed. The first is the electrostatic potential φ(z). This potential changes a little when going into the crystal. The change is due to the surface dipole layer which is caused by the spill-out of the conduction electrons (see Fig. 3.5). The potential difference between inside and outside the crystal is called Δ φ. The other potential is the full one-electron potential νeff(z). This is obtained from the electrostatic potential by adding the exchange-correlation potential µxc. νeff(z) shows that the electron reaches a much lower total potential energy inside the crystal than that caused by the electrostatic part. This must be so, because the exchange and correlation will cause the electrons to go out of each others way and therefore their potential energy will decrease. A lowering of potential energy goes along with an increase of kinetic energy. The zero for the kinetic energy in the bulk is different from the zero outside. The difference is the so-called inner potential V0=µ+Φ, i.e. the occupied band width plus the work function.
The chemical potential can now be referenced to the νeff(z) or to φ(z). It is called µ or µ, respectively. The workfunction, finally, is the difference between the Fermi level and the vacuum level. It is called Φ. We can write down an expression for the workfunction:
| Φ = φ(+∞)−µ = Δ φ − µ. (8.1) |
The right hand side of this equation now allows us to think about the
workfunction as being made of two parts. A surface-part Δ
φ and a bulk-part µ. Workfunctions of metals have values
between about 1.5 eV and 5.5 eV.
Figure 8.1: Definition of the energies contributing to the workfunction.
The surface part of the workfunction is of interest here because any change of the surface in terms of morphology or adsorption will be changing the workfunction. The workfunction (change) can be used as a fingerprint of the state the surface is in. Workfunction changes upon adsorption can be in the range of 100 meV to 1.5 eV for a full monolayer. One can measure workfunction changes within about 1 meV, so that this technique is very sensitive towards the state of the surface. We do not want to describe the technical methods used for measuring workfunctions here. We just give two examples for workfunction changes.
First we illustrate how the workfunction for a specific material
depends on the surface orientation. Consider a closed packed and an
“open” surface of some material. On the open surface we find
the Smoluchowski effect
of charge smoothing [28].
This smoothing leads to
a dipole moment which opposes the dipole created by the flow-out of
the electrons (see Fig. 8.2 and also Fig. 7.5).
Hence, the work function of a
closed packed surface will be higher than that of an open surface.
This is illustrated in Fig. 8.3 for a number of W surfaces. The
closed packed (110) surface has the highest workfunction.
Figure 8.2: Charge distribution at a closed packed (left) and at an open (right) surface.
Figure 8.3: Workfunction of W for different surface orientations. After Ref. [52].
Another important example is the change of the workfunction upon adsorption. A famous case is the adsorption of alkali metals which drastically lowers the workfunction. We can immediately see why, considering the charge distribution of an alkali atom on a jellium surface (see Fig. 6.6). In very simple terms, the bond is ionic and the alkali metal gives an electron to the surface. This does also lead to a dipole moment which also opposes the spill-out of the electrons and reduces the workfunction. At low coverage, the dipole-dipole interaction between the alkali atoms will keep them far apart and the workfunction decreases linearly as a function of coverage. At high coverage, the same interaction causes a depolarization of the dipoles and leads to a metallic bond. This increases the workfunction again by a small amount. Fig. 8.4 shows the workfunction change upon the adsorption of potassium on tungsten.
The workfunction change upon alkali
metal adsorption can be very useful to calibrate the coverage.
In the real world, adsorption can also be used to lower the
workfunction of tungsten filaments such that they do not have to be heated so
much to produce the same amount of electrons and therefore live
longer.
Figure 8.4: Workfunction change upon the adsorption of K on W(110). After Ref. [53].
In section 3.3.6 we have seen that one can obtain new solutions to the Schrödinger equation caused by the introduction of the surface. Inside the crystal they have the form
| ψk(r)=uk∥(r∥) ei k∥r∥ e−κ r⊥ (8.2) |
with a complex wave-vector κ perpendicular to the surface. The new solutions decay exponentially both into the vacuum and into the bulk and are thus located at the surface and called surface states. They are characterized by the quantum number k∥ and an energy E(k∥). k⊥ is not a good quantum number any more because of the broken periodicity in the direction perpendicular to the surface. If we wish to relate surface states to bulk states in the reciprocal space we can think about the k⊥ of the surface state as a rod like in the Ewald construction known from LEED (Fig. 7.11).
It is important to realize that a true surface state can not be degenerate with any bulk state. By this we mean the following: For a true surface state with k∥ and E(k∥) there can not be any bulk state with the same energy and k∥ for any value of k⊥, i.e. on the whole k⊥-rod of the surface state. If there was such a state, the surface state could couple to it and penetrate infinitely into the bulk. It would not be a surface state any more.
This requirement gives a necessary condition for the existence of a
surface state. We can illustrate it by introducing the concept of the
projected bulk band structure.
Fig. 8.5 shows the structure and Brillouin zone of the hcp metal Be
and the surface Brilloin zone for the closed packed Be(0001) surface.
Every point in the surface Brillouin zone is characterized by a
k∥. For every point we can ask: at which binding
energies are there bulk electronic states with this particular
k∥ and an arbitrary k⊥ somewhere in the
bulk Brilloin zone. The answer to this question for many points along
high-symmetry lines of the surface Brilloin zone is the so-called
projected band
structure.
For Be(0001), it is shown in Fig. 8.6.
Figure 8.5: Real space structure and Brillouin zone of Be. The Be(0001) surface is the closed packed surface on top of the real-space hexagon. The surface Brillouin zone of Be(0001) is also shown. It is the projection of the bulk Brillouin zone in the (0001) direction.
Figure 8.6: Projected bulk band structure and electronic surface states for Be(0001). For the shaded areas there are bulk states with the same k∥ and energy for a k⊥ somewhere in the bulk Brilloin zone. After Ref. [54].
In order to illustrate again how the projected band structure is
formed, we calculate it for just one point,
k∥=(0,0)=Γ, i.e. the centre of the surface Brillouin
zone. For this point we have to consider all the k-points in the bulk
Brilloin zone with the same k∥. These points lie
all along the Γ−A direction of the bulk Brillouin zone. Now
let us look at the bulk band structure in this direction in Fig.
8.7.
There is a free-electron like band going from Γ to A, being
folded back to Γ. For all energies
between the bottom of the valence band and a binding energy of about 4
eV it is possible
to find a value of k⊥ such that
there is a bulk state with k∥=(0,0) and that energy. So
there can not be any surface states. This energy range is also shaded
for the Γ point, i.e. for k∥=(0,0)
in Fig. 8.6. At higher
energies, up to the Fermi level, there are no bulk states in the Γ−A
direction.
There is a gap in the projected bulk band structure where a surface
state could “live”. Fig. 8.6 does
indeed show one as a dashed line.
Figure 8.7: Bulk band structure of Beryllium. After [55].
A close inspection of Fig. 8.6 shows that there are also dashed lines (meaning surface states) which do go into the projected bulk band continuum, in contrast to what we have said above. These are bulk states with a high amplitude at the surface. They are called surface resonances.
There is also a completely different type of surface states which we should mention in a few words. Consider an electron in front of a metal surface. The screening properties of the metal can be described by a positive image charge in the metal which has an attractive interaction with the electron. One can describe this as if the electron moves inside an attractive Coulomb potential in front of the surface. Such a potential can actually support unoccupied bound states, so-called image potential states. These states lie above the Fermi energy of the solid but below the vacuum energy. This means that image potential states can be populated but the electrons in these states can not leave the solid.
In angle-resolved photoemission spectroscopy (ARPES) one illuminates a surface with UV light and detects the photoemitted electrons. The spectrometer is designed such that it accepts just a small solid angle and analyses the energy-distribution of the electrons emitted in that solid angle. As we shall see below, angle-resolved photoemission is the technique for the determination of surface (and bulk) band structures. It was developed in the mid 70s. Before, the photoelectron distribution from a sample was only measured in an angle- integrated way, i.e. I(Ekin,hν). It was then that people realized that angle-resolved photoelectron spectroscopy I(Ekin,hν,Θ,Φ) could be used to determine the occupied band structure of a crystal. The basic idea is very simple: one determines the energy and (all three components) of the k vector of the emitted electrons. From this one tries to infer E(k) inside the solid, i.e. the occupied band structure.
A necessary ingredient for the success of ARPES was the development of synchrotron radiation. As we shall see later, the ability to change the photon energy continuously is of high importance.
It is trivial but worth mentioning that photoelectron spectroscopy can only be used to study the occupied electronic states of the sample up to the Fermi level. The unoccupied states can be measured by a technique called inverse photoemission which is based on shooting electrons at a surface and detecting the emitted photons. This technique is not discussed in any detail here.
In photoemission we are concerned with transitions from (occupied) initial states to free final states. The transition rate is given by Fermi’s golden rule:
| = |<ΨfH′Ψi>|2δ(Ef−Ei−hν), (8.3) |
where H′ is the perturbation Hamiltonian. For a system irradiated by light H′ can be written down by replacing P in the original Hamiltonian H0 by P + (e/c) A and adding the electrostatic energy due to the scalar potential. So the full Hamiltonian is
| H=H0+ |
| (AP+PA) −eΦ+ |
| |A|2. (8.4) |
The last term is always small and can be neglected. Furthermore, one can always find a gauge where the scalar potential vanishes. So the differential photoionization cross section can be written as:
| ∝ |<Ψf|AP+PA|Ψi>|2 δ(Ef−Ei−hν), (8.5) |
or, using the commutator [P,A]=−ih∇A,
| ∝ |<Ψf|2AP−ih∇A|Ψi>|2 δ(Ef−Ei−hν). (8.6) |
For not too high energies (< 1 keV or so), we can assume that ∇A is small and can be neglected. This is true in the crystal and in free space but not necessarily at the surface. If we neglect this surface contribution for the moment, we can make the so-called dipole-approximation and use a constant vector potential A0. Using commutation relations the transition rate can be written:
| ∝ |<Ψf|A0P|Ψi>|2 δ(Ef−Ei−hν), (8.7) |
| ∝ |<ΨfA0|r|Ψi>|2 δ(Ef−Ei−hν), (8.8) |
| ∝ |<Ψf|A0∇ V|Ψi>|2 δ(Ef−Ei−hν), (8.9) |
where V is the potential in the original Hamiltonian H0. From the last equation it can already be seen that for a constant potential (free space, for instance) there is not photoemission. This is also obvious when we think about the requirement to conserve energy and momentum at the same time.
So far, we did not talk about the nature of the wave functions in the transition rate. They should be the exact many body wave functions of the system but it is more convenient to work in a single- particle picture. Let us write the ground state many particle wave-function as
| Ψ0N=φiΨiN−1, (8.10) |
where φi is the single-particle wave-function of the electron to be removed and ΨfN−1 is the properly anti-symmetrized determinant of the remaining N−1 electrons. Similarly we write the final state as
| ukΦi,jN−1, (8.11) |
where uk is the wave-function of the free electron and Φi,jN−1 are the possible states for the ionic system, i.e. the system with one electron missing. The notation is that there is an electron missing from orbital i and the ion is in the excited state j. Now the transition matrix element can be split up into a single and many-particle part:
| ∝ |<uk|AP|Φi><Φi,jΨi>|2 δ(Ef−Ei−hν). (8.12) |
This turns into the single-particle picture if <Φi,jΨi>=δi,0 , i.e. when the electrons in the ionic state just remain where they have been before in the non-ionized state. But this is not the case. So when we project ΨiN−1 onto the eigenstates of the ionic system there will not only be overlap with the ground state but also with some of the excited states with j not equal to zero. If the system is left in an excited state there will be less energy available for the photoelectron. So-called satellite or shake- up lines will appear at higher binding energy (lower kinetic energy). We have already encountered this, less formally, in the section about XPS 5.3.
Photoemission from molecules can illustrate how simple symmetry considerations can be employed to learn something about the initial states. Consider a linear diatomic molecule like CO. The molecular orbitals can be classified into σ and π orbitals by their symmetry (see Fig. 6.7). The σ orbitals are rotationally symmetric around the main axis. The π orbital wave function changes its sign upon a 180∘ rotation around the main axis. They must thus have a nodal plane containing the axis.
Fig. 8.8 shows the calculated photoemission intensity for
CO for photoemission from the 4σ initial state. The polarization vector
and the detector are along the axis of the molecule which means that
a final state of an electron reaching the detector must
also have σ symmetry. This is consistent with the plotted
intensity. A π initial state would lead to a zero photoemission
intensity in this geometry.
This method can be used to determine the orientation of molecules on a surface. We
come back to this point later.
Figure 8.8: Photoemission from the CO 4σ orbital with polarized light. After Ref. [56].
Let us discuss the photoemission process in the presence of a potential, like in a real crystal. We ignore the effect of the surface, i.e. the fact that the potential ends at some point.
The simplest case of a potential is a constant potential. The electronic states are free-electron like,
i.e. E=(ℏ2k2/2m). The problem as illustrated in Fig.
8.9 is that energy and momentum can not be
simultaneously conserved in this process because the momentum of the photons
is so small (ℏ ω =cℏ k).
Figure 8.9: Momentum conservation in angle-resolved photoemission.
For a periodic potential, band gaps open at the zone boundary (extended zone scheme). Because of the periodicity in k, all the bands can be folded back into the first zone by adding a reciprocal lattice vector. In the reduced zone scheme vertical transitions are possible. Formally the momentum conservation has changed from kf=ki to kf=ki+G where G is a reciprocal lattice vector. Note, that even the most simple vertical transitions are Umklapp processes in that they involve a reciprocal lattice vector. Formally this can be seen by putting the appropriate (Bloch) initial and final states in the photoemission matrix element:
| |<Ψf|∇ V|Ψi>A0|2 δ(Ef−Ei−hν), (8.13) |
It is the gradient of the potential which provides the possibility of photoemission. A periodic potential has, of course, a non-vanishing gradient.
In this context it is interesting to mention the effect of the potential associated with the termination of the crystal, i.e. with the surface. In the most simple case this is a step. Then
| ∇ V= |
| ez, (8.14) |
and the matrix element becomes
| |<Ψf| |
| Az|Ψi>|2 δ(Ef−Ei−hν). (8.15) |
The step perpendicular to the surface chooses only the perpendicular component of A. The step can provide any momentum k in the direction perpendicular to the surface since its Fourier transform contains all possible frequencies. Therefore the k-vector component perpendicular to the surface k⊥ is not conserved but the component parallel to the surface k∥ still is.
There are two commonly used light sources in UV photoemission. One is a gas-discharge lamp in which a noble gas discharge is taking place in a high electric field. The light is guided into the UHV chamber by a quartz-glass tube with a tiny hole (to avoid a high noble-gas pressure in the chamber). The lamp is differentially pumped. The other light source is synchrotron radiation. Its most important advantages are tunability and polarization.
The most common electron analysers are hemispherical analysers like discussed in an earlier lecture. The angular resolving capability of such analysers can be achieved by a small hole (aperture) in the entrance or by an electrostatic lens. For ARPES it is important to change the angle of emission in order to change k∥. This can be done by either moving the sample or the analyser. Moving the sample with the required precision is very difficult. A common way to change the angles is to mount the analyser on a goniometer in the chamber.
An intuitive and often-used way to look on photoemission is the so-called three step model.
1. A photon is absorbed in the crystal by an electron. The electron is excited from an occupied initial to an unoccupied final state.
2. The excited electron is brought to the surface taking into account the direction of propagation of the final state and the mean free path.
3. The electronic wave function of the final state is matched to a free-electron wave function outside the solid.
In the last step the electron has to overcome the potential barrier at the surface. For the kinetic energy of the electron this means that it will be lower in the vacuum since the potential energy is higher.
It is important to notice that the whole system of semi-infinite solid and vacuum has a translational symmetry only parallel to the surface. Therefore the k-vector parallel to the surface k∥ is a good quantum number and is conserved during the photoemission process.
This is not true for the perpendicular
component of k. k⊥ is not determined by the photoemission
experiment. The most severe change in k⊥ happens because of
the kinetic energy change which leads to a
refraction at the surface barrier (see Fig. 8.10).
We have encountered this
process already in our discussion about LEED.
The picture may imply that
k⊥ is merely changed during the transmission of an electron through the
surface but that it is still well defined. Unfortunately, this is not
the case. The symmetry breaking normal to the surface means that
k⊥ is not a well-defined quantity any more.
Figure 8.10: Refraction at the surface potential barrier.
How is angle resolved photoemission now used to determine band structure, i.e. to determine Ei(ki)? Let’s follow the three step model.
1. An electron is excited from an initial state with energy Ei to a final state with energy Ef.
In the reduced zone scheme this excitation is vertical with kf=ki. In the (more appropriate) extended zone scheme we have
| kf=ki+G (8.16) |
2. The final state electron is transmitted through the crystal. It may get lost on the way but at this point we don’t really care about those electrons which get lost.
3. The electron is transmitted through the surface. This costs kinetic energy due to the work function.
We measure k and Ekin of the electron in the vacuum. If k⊥ was conserved then we would directly obtain the energy and k of the initial state. This would solve the experimental task of band-structure determination. Unfortunately, we just get information about the initial state energy and the wave vector component parallel to the surface.
The problem can safely be ignored in the case of a two-dimensional sample. There, only k∥ is of interest anyway. The most important example for this in our context is the dispersion of an electronic surface state E(k∥). But there are also other important cases of two-dimensional or quasi two-dimensional systems, namely layered materials. The most common is graphite. Others are transition metal dichalcogenides like MoS2. Other quasi two-dimensional materials of great current interest are transition metal perovskite oxides. The cuprate high TC superconductors are members of this family.
A classical example for band mapping in two dimensions
is the free-electron surface state on Cu(111). It exists in the necks of the bulk
Fermi surfaces. Fig. 8.11 shows
this surface state seen with ARPES as the angle of emission is
varied. The data shown in the left-hand part of
Fig. 8.11 consists of
so-called energy distribution curves (EDCs). In an EDC the photoemission
intensity is measured as a function of electron kinetic energy
at a fixed emission direction and photon energy. The electronic
states show up as peaks in an EDC. As the emission angle is varied,
the binding energy of the peak changes. In the present case the
binding energy decreases for higher off-normal angles. Eventually,
the peak disappears because it crosses the Fermi level and turns into
an empty band. On the right-hand side of 8.11 the same
data is displayed as a grey-scale plot. Such plots are common today
and are made possible by technical progress which permits taking
large data sets with many angles.
Figure 8.11: Surface state dispersion on Cu(111). Left: EDCs close to normal emission. The dispersion of the surface state is clearly evident. Right: Grey-scale image of the same data.
It is easy to infer the band structure:
| k∥ i=k∥ f = sin(Θ) |
|
| √ |
| = sin(Θ) |
|
| √ |
| (8.17) |
Θ is the emission angle as defined in Fig. 8.11. Ebin is the binding energy, measured with respect to the Fermi level. Ekin is the kinetic energy of the electrons outside the solid, measured with respect to the vacuum level. This definition of the energy zero is of practical importance. As we shall see later, the Fermi edge shows up in many photoemission spectra such that the binding energy of a peak can be read directly from the spectrum as the difference between the peak position and the position of the Fermi edge. From this binding energy, the kinetic energy outside the solid can be calculated by simply subtracting the work function Φ (see Fig. 8.1). The work-function Φ must be obtained by some other technique (this can also be done by photoemission).
After having read the binding energy from the spectrum,
equ. 8.17 can be used to calculate k∥
i and one obtains the desired dispersion
Ebin(k∥
i) which is shown in Fig. 8.12.
Figure 8.12: Surface state dispersion on Cu(111). After Ref. [57].
Let us return to the problem of determining the band-structure of a three-dimensional solid. How can we circumvent the problem with k⊥? We restrict ourselves to normal emission geometry. This makes things easy, mainly because the component of the wave-vector parallel to the surface is 0:
| k∥ f∝ sin(Θ) | √ |
| , (8.18) |
independent of the kinetic energy of the electrons. Note that if we take an EDC in any other geometry, the k∥-vector is different for every point in the spectrum because the kinetic energy is different but this is not so for normal emission.
Fig. 8.13 shows normal emission data from Be(0001) for
various photon energies. There are two
peaks. One changes its binding energy as the photon energy is
changed, the other does not. The latter is a surface
state which we will discuss later.
The first state is a bulk state. We do not know the exact value of
k⊥ but we do see that the dispersion reaches two
extrema, at about 33 eV and 99 eV photon energy. These
extrema correspond to high-symmetry extrema
in the band structure.
Figure 8.13: Normal emission spectra from Be(0001) taken with different photon energies. After Ref. [58].
Consider again the Brillouin zone of Be in Fig. 8.5. For normal emission from the (0001) closed packed surface the initial states lie on the Γ−A−Γ rod. The moving peak has a smallest binding energy of about 5 eV at 99 eV photon energy. This extremum corresponds to the band with the lowest binding energy at the Γ-point in the band structure of Fig. 8.7. The other extremum at 33 eV corresponds to the band with the highest binding energy, also at Γ.
This example illustrates how ARPES can be used to determine the
energies of extrema in the dispersion of the three dimensional bulk band
structure. If further assumptions about the final states are made, it
is also possible to figure out the actual dispersion of the bands in
between these extrema. We do not wish to go into the details
of this here but an example is given in Fig. 8.14. It
is a
band structure determination of Cu in two bulk high symmetry
directions.
The left part is taken in normal emission from Cu(111) and the right part from Cu(110). The
triangles and circles are obtained with different polarizations.
Figure 8.14: Band structure of Cu determined using photoemission and free electron final states. The lines are theory. After Ref. [59].
Before we start giving specific examples, here are a few important general practical considerations. Suppose we have measured an EDC from a surface and we find peaks in that spectrum. How do we know if a peak is due to a surface state? We give some criteria and discuss them:
Figure 8.15: When taking a spectrum with two different photon energies but for the same k∥, bulk-related peaks will in general show a dispersion in the spectrum while surface state peaks stay at a fixed binding energy. One can think of the surface state as being present for all values of k⊥, similar as in the Ewald construction in Fig. 7.11. See also Fig. 8.13.
The reason for (1) was that the surface state must not mix with a bulk state. Otherwise it would not be a genuine surface state. This requires some more discussion. Actually the surface state does not have to lie in a real projected bulk band gap. It is sufficient that it lies apart from states it can mix with. This subtle difference means that it can lie in a region of bulk bands with another symmetry. Sometimes strong and obviously surface-related features can be found in regions of degenerate bulk states with the same symmetry. These features could be either surface resonances , i.e. states which penetrate deeply into the bulk but have a strong amplitude at the surface, or the investigator could be mislead by band structure calculations which places the bulk band gap edges at positions which are actually not right.
A nice demonstration of (2) was the set of Be(0001) spectra taken at different photon energies shown in Fig. 8.13. It is evident that the binding energy of the surface state is independent of the photon energy and k⊥ while the binding energy of the bulk state is not.
Point (3) is a little tricky. It should be possible to severely change the appearance of a surface state
in a spectrum by putting impurities on the surface or by creating disorder. This is indeed a nice test:
many surface states can be "killed" by adsorbing small amounts of contaminating atoms. However,
some care is needed: A change in the surface cleanliness or order can also affect the intensity of
bulk transitions. Fig. 8.16 shows an example:
a Bi surface which supports a surface state is measured twice:
once well-ordered and once after sputtering
without annealing. The alleged surface state peaks
decreases a lot in intensity. But the other peaks change, too.
Figure 8.16: Spectra from well-ordered and sputtered Bi surface. After Ref. [60].
The last point (4) is not a real criterion: it is often found that surface states are very sharp peaks in the photoemission spectrum. The reason is related to the fact that surface states have no dispersion with k⊥. Such a dispersion, combined with the uncertainty in k⊥ leads to a considerable broadening of the bulk peaks which is not present in the case of surface states.
In the foregoing discussion we have already seen three examples for surface states on metals
(Be(0001), Cu(111) and Al(001)). In all three cases the surface state band was derived from the s-p
band of the metal. The states are fairly delocalized and can be described in a nearly-free electron
model. Historically these states are known as
Shockley states. One can also, of course, approach the
question of surface states from the opposite viewpoint, namely in a tight-binding picture. There the
atomic orbitals which stick into the vacuum because the atoms’ neighbours
have been cut off, have
very different electronic properties than the equivalent bulk orbitals and represent surface localized
states. This kind of more localized surface state is called Tamm state. Examples of Tamm states can
also be found on metals. One is a surface state which is derived from
the Cu(100) d-band and shown in Fig. 8.17.
Figure 8.17: Tamm-surface state on Cu(001). After Ref. [61].
We should also mention that s−p surface states do not necessarily have to have a free electron like dispersion around the Γ point like in all our previous examples. We will discuss an example in the exercises.
Angle resolved photoemission can of course also be used to probe the modification to the surface electronic structure which is induced by adsorption. The electronic states of the atoms and molecules on the surface are changed from the gas phase due to the bonding. Theses adsorbate states may be viewed as extrinsic surface states.
Fig. 8.18 shows two spectra taken from CO
on Ni(100).
Both are taken in normal emission but the
direction of the polarization vector is different. There are peaks found in the spectrum which are
not present on the clean surface and which are related to the CO. A comparison with the energy of
the peaks from the gas phase CO can be used to identify the peaks. When the polarization vector is
parallel to the surface no σ peaks are found. This means that the molecular axis is
perpendicular to
the surface: then the light has π symmetry, the final state has
σ symmetry and only π symmetry
states are allowed as initial states. In this way the orientation of adsorbed molecules can be
determined.
Figure 8.18: Photoemission from CO on Ni(100). After Ref. [62].
Another example involving CO adsorption illustrates what one can learn about intermolecular interactions. We have already mentioned that these interactions can (and do) often lead to ordered adsorbate structures. In an ordered structure the interactions between the molecular orbitals will lead to a two dimensional overlayer band structure which can be investigated by ARPES. A particularly interesting orbital for studying these interactions is the 4σ orbital because it often has a higher binding energy than all the electronic states in the substrate and does therefore not mix strongly with the substrate states. Fig. 8.19 shows the measured and calculated (by a tight-binding scheme) dispersion of the band derived by the 4σ state in the adsorption systems (2√3 × 2√3)R30∘CO-Co(0001) and (√3 × √3)R30∘CO-Co(0001). The sizes of the two Brillouin zones have been artificially made equal for a more convenient comparison. There is indeed a dispersion of the electronic states due to the intermolecular interaction. Furthermore, the close-packed structure shows a considerably larger bandwidth caused by the stronger interaction. Note, that the (2√3 × 2√3)R30∘CO-Co(0001) structure has a higher coverage (Θ = 7/12 ML) and thus closer intermolecular distance than the (√3 × √3)R30∘CO-Co(0001) structure (Θ = 1/3 ML).
Figure 8.19: Dispersion of the CO 4σ state for (2√3 × 2√3)R30∘CO-Co(0001) and (√3 × √3)R30∘CO-Co(0001). After Ref. [63].
In the discussion of surface structure we have already seen that semiconductor surfaces tend to reconstruct in complicated ways rather than staying bulk terminated. This fact is very closely related to the electronic structure of the semiconductor surfaces as we shall see below. This close relation, combined with the difficulties of solving the semiconductor structures with LEED has lead to a very different aspect in the application of photoemission. Many important semiconductor structures have, in fact, been solved, at least approximately, by comparing the measured dispersion of the electronic surface states to various models.
For understanding the surface states on semiconductors, let us look at the bonding in these materials first. The elemental semiconductors Si and Ge and many of the common compounds such as GaAs have a tetrahedral bonding geometry in common, i.e. each atom is bound to four neighbours with sp3 hybrid orbitals. Although the bonding can be partly ionic in some materials, this strongly directional view is very instructive. When a surface is formed, the sp3 hybrid bonds are cut and dangle into the vacuum. What about the electronic states derived from these dangling bonds? Suppose we have one dangling bond per unit cell. In a band structure picture we can describe the dangling bond state by a band. If the band is fully occupied it can accommodate two electrons. But the dangling bond state has only one electron and hence the band has to cross the Fermi level such that only one half of the surface Brillouin zone has occupied states due to the dangling bond. In other words, the semiconductor’s surface is metallic! In the real world this does not happen very often. The dangling bonds don’t like to dangle alone and the semiconductor surfaces reconstruct in complicated ways to get rid of the dangling bonds.
This concept can be illustrated nicely by the (2x1) reconstruction
of
Si(111). This reconstruction is obtained when Si(111) is cleaved at
room temperature. It is a meta-stable structure since annealing the
crystal to high temperature causes an irreversible phase transition
into the (7x7)
structure. Let’s consider the bulk terminated structure
of Si(111). There is one dangling bond in the surface Brillouin zone
(see Fig. 7.22) so
that we have precisely the scenario discussed above, a metallic
surface with an energetically unfavourable dangling bond. What
happens if we introduce a (2x1) reconstruction is shown in
Fig. 8.20). The larger surface unit cell leads to a
surface Brillouin zone of half the original size. The new Brillouin
zone boundary causes the familiar band splitting. If he original
dangling bond state has a Fermi level crossing at the new boundary, the
reconstruction divides the band up in one which is completely occupied
and one which is completely unoccupied. It is immediately evident that
energy is gained by this reconstruction because the occupied states in
the vicinity of the zone boundary have lowered their kinetic energy.
The surface is now semiconducting.
Figure 8.20: (2x1) reconstruction on Si(001).
The actual surface geometry of the Si(111)-(2x1) reconstruction
has been disputed for a
long time. The model favoured now is the so-called π-bonded chain
model. A previous favourite has been a buckling model. Both are shown
with the calculated of surface states in Fig. 8.21. Both models
lead to a semiconducting surface behaviour as expected. But there are
qualitative differences. The band-width for the π-bonded chain
model is larger than for the buckling model. This is consistent with
intuition because the close neighbourhood of the atoms responsible for
the bands should lead to a larger bandwidth.
Figure 8.21: Two possible models for the (2x1) reconstruction of Si(111) and their electronic surface state dispersion. Note that the energy zero is the valence band maximum and not the Fermi energy. Both surfaces are semiconducting. After Ref. [64].
The dispute over the right model was eventually solved by ARPES.
Fig. 8.22 shows the convincing agreement between experiment and theory for
the π-bonded chain model.
Figure 8.22: Comparison between experiment and theory for the π-bonded chain model of Si(111)-(2x1). After Ref. [65].
Another interesting example of a semiconductor reconstruction
is the
Si(100)-(2x1)
structure. Let us look at it again first with simple
electron counting arguments. Si(100) has two dangling bonds per
surface unit cell which are located on the same atom. In the (2x1)
unit cell one would find four dangling bonds, i.e. four electrons on
two atoms.
These could be distributed for example in two filled electronic bands.
A first result from photoemission was that a reconstruction in which
two surface atoms are paired to form dimers is in better agreement with the
experimental data (shown in Fig. 8.23) than other models.
The disputed question is whether the dimers
are symmetric or asymmetric.
In a dimer-type reconstruction, two of the four electrons are used to
form the dimer bond. They are in a stable configuration and give rise
to the low-lying surface state band S3 in Fig. 8.23. The question of a
symmetric or asymmetric dimer mainly influences the electronic states
of the remaining two electrons localized on each surface dimer.
Fig. 8.24 shows that the symmetric and the asymmetric dimers
lead to very different theoretical predictions. In the asymmetric
dimer model, the is a considerable charge-transfer from the “down”
atom to the “up” atom. The model gives rise to
two well separated bands. Since we have two electrons left
this corresponds to a non-metallic surface, consistent with all
experimental data. The symmetric dimer also gives two bands but the
bands overlap meaning that the surface has to be metallic. A detailed
comparison of the measured dispersion of the S2 state with more
modern calculations shows a good agreement favouring the asymmetric
dimer. However, there are still many open questions in this
complicated problem for example concerning the nature of the
S1 surface state which should
not appear in an asymmetric dimer model.
Figure 8.23: Experimental surface band structure from Si(100)-(2x1) in the Γ−J′ direction. After Ref. [66].
Figure 8.24: Calculated dispersion for the upper surface states of Si(100)-(2x1) for a symmetric and an asymmetric dimer configuration. After Ref. [68].
In the case of a doped semiconductor, the existence of surface or
interface states in the gap can have a dramatic effect on the
electronic structure rather far away from the surface.
Consider the situation shown in Fig. 8.25 (left). We have an n-doped
semiconductor which supports two surface state bands, one empty and one
occupied, with the Fermi level in between. The bulk Fermi level is set
such that it is close to the conduction band.
The position of the surface states in the gap is such that the surface Fermi
level lies at a much lower energy than the bulk Fermi level. This is
an unstable situation because it brings bulk donors above the surface
Fermi level. Electrons from the donors will flow into the previously
unoccupied surface state and partially fill it. At the same time,
they will leave a layer of positively charged bulk donors behind. This
will go on until the energy gain by filling up the lower-lying states
is equal to the energy cost for setting up an electric field in the
surface region. In the end, the Fermi level is constant over the whole
crystal and surface like in Fig. 8.25 (right). One can also think about
this effect as a band bending in the surface region of the crystal.
Bulk donor levels are ionized because the electrostatic field lifts
them over the Fermi level. The ionized bulk donors represent a
positive space charge. This positive space layer can extend very
deeply into the surface because the density of bulk dopants parallel
to the surface (between 108 and 1012cm−2) is much lower
than the density of surface states (about 1015cm−2). The
layer is called a depletion layer
Figure 8.25: Left: Surface state bands in the bulk gap and bulk electronic structure of an n-doped semiconductor. The situation is unstable because of the non-constant Fermi level. Right: electrons from the bulk donors flow into the previously unoccupied surface states and leave a positive space charge layer behind.
This effect has some interesting consequences. The first is, that one can use a heavily doped semiconductor to study states with ARPES which would be unoccupied if the semiconductor where intrinsic. Another, less pleasant, aspect is that in lightly doped semiconductors without intrinsic surface states in the gap, very small quantities of adsorbates might induce sufficient states in the gap to induce a band bending. This is a real problem because it shifts the whole spectrum with respect to the Fermi level (the reference level for the spectrometer).
A related consequence is the so-called Fermi level pinning
by surface
states in the gap shown in Fig. 8.26. In the bulk the Fermi
level can be changed from just below the conduction band to just above
the valence band by changing the doping. One would expect that the
workfunction, i.e. the distance between the Fermi level and the Vacuum
level would follow this change. But for certain surfaces with states
in the gap this is not the case. Upon n-doping, a depletion layer is
build up and the surface states are charged negatively. This dipole
layer slows down the electrons from the bulk and this almost exactly
compensates the gain of kinetic energy due to the higher Fermi level.
In the end, the measured Fermi level is almost independent of the
doping level. This effect is called Fermi level pinning.
Figure 8.26: Fermi level pinning for Si(111). The measured workfunction is almost independent of the doping level. The straight line in the middle illustrates the expected behaviour due to the doping. After Ref. [69].
It is clear that the forgoing discussion is only relevant for semiconductors. Space charge regions do not exist for metals because of the good screening in a metal. The conduction electrons will screen away the surface such that everything looks like bulk already after a few Å(see Fig. 3.5).
We have seen that there is a big difference between the surfaces of metals and those of semiconductors. In most metals, covalent bonding is unimportant and most surfaces do not show reconstructions. On semiconductors the opposite is found. Creating the surface requires the breaking of covalent bonds and leaves so-called dangling bonds which could give rise to half-filled and therefore metallic bands. However, it turns out that on most semiconductor surfaces the atoms reconstruct such that the dangling bonds are removed and the surface is again a semiconductor, not a metal. Semimetals lie in between these two cases. On one hand, a semimetal is close to being a semiconductor since directional bonding is important and the valence and conduction bands are almost separated by a gap. On the other hand, a very small overlap between both bands is found at some point of the Brillouin zone such that the material is formally a metal. This delicate balance between being a metal and a semiconductor depends crucially on the structural details and is disturbed severely at the surface. A semimetal surface can be expected to turn either into a better metal or into a semiconductor. The former case is more interesting because a good metallic surface on a semimetal (or, indeed, on a semiconductor) can be taken as a model for a nearly two-dimensional metal.
The (110) surface of Bi is such a case. The truncated bulk structure is such that one dangling bond can be found in a surface unit cell containing two atoms. This relatively low density of dangling bonds is not sufficient to drive a reconstruction. Therefore the surface is a good metal, in contrast to bulk Bi. Photoemission data from Bi(110) is shown in Fig. 8.27 together with a bulk band structure projection. The data is presented in a way which is different from what we have seen so far. Very many EDCs have been taken for k∥ points along all the high symmetry lines of the surface Brillouin zone. The plot shows the logarithm of the photoemission intensity as a function of k∥ and binding energy. This presentation has the advantage that the dispersion is directly seen. Among other things, several surface state Fermi level crossings are observed.
Figure 8.27: Projected band structure of Bi(110) and experimental data along several high symmetry lines of the surface Brilloin zone. The data are the logarithm of the photoemission intensity. The grey scale is defined such that black means low intensity and white high intensity. The features A,B,C,D and E are the electronic surface states. After Ref. [67].
In the structure lecture we have only discussed the ability of the STM to provide topographic information. It is evident, however, that one can also use the STM in a spectroscopic mode in order to learn something about the density of states of the sample. This approach is called scanning tunnelling spectroscopy (STS). STS experiments are usually carried out by parking the tip at a fixed distance over the surface and measuring the tunnelling current as a function of applied voltage, an I-V spectrum. From the I-V spectrum one tries to obtain information about the sample density of states. One can also measure dI/dV(V) or d2I/dV2(V) using a small modulation voltage and a lock-in amplifier. The STS approach faces two formidable problems. The first one is of technical nature and the second lies in the data analysis.
It is difficult to measure reproducible STS data. The tip has to be at a constant height above the surface while the spectrum is taken. A small change in height will change the current much more than any spectral feature! Therefore the tip has to be held at constant height within of 0.01 Å or so. If one simultaneously wants lateral resolution one also has to keep the tip parked above the same spot on the sample although it is enough to do this within 0.1 Å. Achieving this against the thermal drift and vibrations from outside is hard. STS spectra have to be taken many times to obtain an acceptable signal to noise ration. It is desirable, that the tip does not change during these many spectra. On the other hand, it is also a good idea to take spectra for differently prepared tips in order to learn which spectral features are due to the tip and which are due to the sample.
So let’s look at the black magic involved in analysing the spectra. We restrict ourselves to a very simplified discussion for relatively small tunnelling voltages, i.e. smaller than the workfunctions of tip and sample. We write down a simple expression for the tunnelling current.
| I= | ∫ |
| ρs(r,E)ρt(r,−eV+E)T(E,eV,r)dE, (8.19) |
where ρs and ρt are the density of states of sample and tip, r is the location of the tip, E is the energy measured with respect to individual Fermi levels of sample and tip, V is the tunnelling voltage and T(E,eV,r) describes the tunnelling probability. We further simplify this by making the assumption that the tip density of states is constant.
| ρt(E)=const. (8.20) |
A featureless tip DOS might be expected for a blunt and disordered tip. But our ideal tip for obtaining good spacial resolution will be one with just one atom at the end. So the approximation of a featureless tip might not be very good. On the other hand, one might be able to sort out the tip contribution to the I−V curves because the tip DOS does not depend on the position on the sample. If we now also say that
| T(E,eV)=const. (8.21) |
then we can obtain the sample DOS from equ. 8.19 simply by plotting dI/dV. But a constant T is a very bad assumption. In fact, it turns out that the tunnelling probability is always highest for the states at the Fermi level. This means that if we set V such that we tunnel from the tip into the sample, most of the electrons will come from the Fermi level of the tip and we will measure the unoccupied DOS of the sample. If we reverse the sign of V most of the tunnelling electrons come from the Fermi level of the sample. We will unfortunately not learn very much about the occupied DOS of the sample. It turns out that the errors caused by 8.21 can be reduced by taking (dI/dV)/(I/V) instead of simply dI/dV.
In view of all this, it is needless to say that STS results have to be treated with great care.
We illustrate the capabilities of STS by going back to the (2x1)
dimer reconstruction of Si(100).
As we have seen above, photoemission
is not quite conclusive on the question if the dimers are symmetric or
not. LEED seems to favour an asymmetric dimer. STM in the topography
mode has shown both symmetric and asymmetric dimers, depending on the
sample temperature. The STM image shown in Fig. 7.47 clearly
shows the asymmetric dimers. In the same paper, STS has been
performed on the individual atoms of the dimers. The result is shown
in Fig. 8.28. The STS data have been taken simultaneously
with the topography data shown in the inset. In Fig. 8.28(a)
the average spectrum of this surface is shown. In (b) two spectra from
the different atoms in the dimer are displayed. In the unoccupied
states, a significantly higher DOS of atom 1 at about 1 eV is found.
Since atom 1 is associated with the “down” atom this increase in
the unoccupied DOS is consistent with the predicted charge transfer of the “down”
to the “up” atom. The increase of occupied DOS in the “up” atom
is not observed. This is partly due to the difficulty of obtaining
spectroscopic information about occupied sample states.
Figure 8.28: (a) Averaged tunnelling spectrum on the marked area in the inset. (b) corresponding spectra for the two positions marked in the inset. After Ref. [45].
For a discussion of the work function a good reference is the original paper of Lang and Kohn [70]. A good (but old) reference on angle-resolved photoemission which I have used a lot to prepare these notes is [59] but see also [71] and [72]. The issue of work function is theoretically discussed E.W. Plummer and W. Eberhardt, Adv. Chem. Phys. 49, 533 (1982).
Angle-resolved photoemission, Ed. S.D. Kevan, Elsevier 1992