The surface optical properties, i.e. the behaviour of the electromagnetic field in the vicinity of the surface is of considerable importance in surface science. There are two main reasons: the first is the fundamental importance of the electromagnetic field for many processes, e.g. for photoemission. The other reason is the practical usefulness of light as a surface probe. Remember that most of the surface science techniques we have looked at so far are based on electron spectroscopy. This has the fundamental disadvantage that it only works in UHV. With light we have the possibility to study surface properties even in “real” environments present in heterogeneous catalysis or in semiconductor growth.
As we have seen in one of the first lectures, the main problem associated with light is the lack of surface sensitivity. The penetration depth is large such that the surface signal is relatively small. The situation is especially bad for the spectral regions where the material under investigation is “transparent”, i.e. below the fundamental adsorption edge of semiconductors and above the plasma energy of metals. Two ways have been established to do surface-sensitive spectroscopy despite this problem. The first is a difference technique. One measures the optical properties of a clean and of an adsorbate-covered surface and takes the difference between these two in order to learn something about the adsorbate. We will see an example of this in the second part of the lecture (IRAS). The other approach is to make an optical measurement which results in a 0 signal from the bulk because of symmetry. If this symmetry is broken at the surface, the measured signal stems from the surface region. We give two examples for this approach below.
The classical reflection and refraction of a plane wave at a surface is described by Snell’s law
| sinΘiñi=sinΘtñt, (9.1) |
where Θi and Θt are the angles of incidence and refraction, respectively, and ñi and ñt are the complex refractive indices. In order to calculate the actual fields one has to consider the Maxwell equations together with the boundary condition that the tangential components of E and H are continuous over the interface. This leads to the famous Fresnel equations for the (amplitude) reflection coefficients
| rp= |
| (9.2) |
| rs=− |
| (9.3) |
| tp= |
| (9.4) |
| ts= |
| (9.5) |
These equations permit the calculation of all the desired properties:
Fig. 9.1 shows the reflection coefficients for light
polarized parallel and perpendicular to the plane of incidence,
respectively. The complex index of refraction is chosen to be
n=3, k=30, representing a metal surface. Fig. 9.1 shows a highly
reflecting surface with a Brewster angle near 90∘.
Figure 9.1: (Intensity) reflection coefficients for light polarized parallel (p) and perpendicular (s) to the plane of incidence. The complex index of refraction is n=3,k=30.
It is also interesting to calculate the electric field components
at the surface for the same material. The result is shown in Fig.
9.2. It can be seen that the most important field at
the surface is polarized in the plane of incidence and it is
perpendicular to the surface. This field is strongly enhanced
(over
the initial amplitude of 1) at a grazing angle of incidence. An
experiment in which we want to look at the interaction between
molecules on a surface and the light should be build
in such a geometry.
Figure 9.2: Electric field components at the surface of the material with n=3,k=30. The plane of incidence is the x/z plane. The z-axis is perpendicular to the surface.
For the purpose of surface science, however, the Fresnel-description of the fields is insufficient. This is easily seen by looking at the D field which has to be continuos at the surface. In a classical theory this is achieved by a delta charge-sheet right at the surface. This is of course physically meaningless in a microscopic theory. In such a theory a non-local relation between D and E has to be assumed and the fields have to be calculated in a self-consistent way. It turns out that such a complicated approach is essential if things like photoemission intensity from surface states are to be described.
In our discussion of the surface plasmon in section 4.4.5 we have found the surface plasmon frequency to be ωsp=ωP / √2 and we did not consider the possibility of a surface plasmon dispersion ωsp(q) which can in fact be found and measured with EELS. Whereas this dispersion is usually not very big, something rather dramatic happens when the wave vector q becomes very small, in the order of the light wave vector ω/c. For these very long wavelengths we have to consider the fact that a surface plasmon carries an electric field and it has to be treated on equal footing with an ordinary electric field, i.e. it has to fulfil the Maxwell equations. This leads to a dramatic change in the dispersion which is shown inf Fig. 9.3. The surface plasmon energy is no longer constant, as it would be on this q scale, but it shows an avoided-crossing behaviour with the light dispersion line ω=cqz. The lower branch is called the surface plasmon polariton.
Figure 9.3: Dispersion of a surface plasmon polariton.
As a consequence of this, it is impossible to excite surface plasmons with light because energy and momentum can not be conserved simultaneously. This is easily seen from Fig. 9.3. The light dispersion line can be changed by changing the angle of incidence. For normal incidence q∥=0 and the dispersion is a vertical line in the figure. For grazing incidence qz=0 and the dispersion is ω=cq∥. For any angle in between ω=c(q∥2+qz2)1/2. This means that the light dispersion and the surface plasmon polariton dispersion never cross and hence there can not be any excitation.
In short, there are essentially two ways of circumventing this. The first is to introduce a long periodic structure, e.g. a grating, on the surface such that the surface plasmon polariton gets back-folded and this back-folded mode crosses the light dispersion lines. The same effect can be achieved by a rough surface which can be viewed as a superposition of many gratings with different periodicities. The excitation of surface plasmons via surface roughness is thought to play a role in surface-enhanced Raman scattering. The other way to achieve the coupling is to use ‘slow’ photons with an qz which is imaginary. Then the dispersion line can be moved down, as indicated in the figure. Such light can be produced by a total reflection inside a prism mounted in a short distance over the surface. In this case, an evanescent electric field penetrates the gap between prism and surface. The field decays exponentially, i.e. it possesses an imaginary q in the z direction.
The surface plasmon is not the only mode which leads to the formation of polaritons at long wavelength. In fact, all modes of the crystal which carry an electromagnetic field have to be considered, e.g. optical phonons for materials with several atoms in the unit cell.
The technique of Reflection Anisotropy Spectroscopy (RAS) is based
on a symmetry-trick in order to make the optical spectroscopy
surface-sensitive. The optical response of a solid is dictated by its complex
dielectric tensor є or by the complex refraction
tensor ñ. In the case of a centrosymmetric material such as
a cubic crystal the tensor is reduced to a complex scalar.
Consequently the normal-incidence reflectivity of a cubic crystal
should not depend on the azimuthal orientation of the polarization
vector. This is only true, however, for the dielectric response of
the bulk crystal; at the surface the inversion symmetry is broken.
Any azimuthal anisotropy in the normal-incidence reflectivity of
cubic crystals must therefore have its origin in the surface region.
In a RAS experiment (see Fig. 9.4) one probes the difference in
the normal-incidence reflectivity along two mutually perpendicular
orientations of the polarization vector. Usually one or both
of these directions coincide with the principal crystallographic
directions in the surface.
Figure 9.4: Setup for a RAS experiment. The complex difference in reflectance along two mutually perpendicular directions is measured. After Ref. [73].
This technique is of course rather restricted: the only possible measuring geometry is normal incidence, the bulk crystal has to have inversion symmetry and the surface must be chosen such that it has two mutually perpendicular directions which are not symmetry-equivalent (i.e. fcc(110) works but fcc(100) does not).
One important example for the usefulness of RAS is the study of
surface states on metal surfaces. Fig. 9.5 shows
the electronic structure in the vicinity of the Ȳ point of
the Surface Brillouin Zone on Ag(110). Two surface states are found
in the projected band gap. One state above and one state below the
Fermi level.
Figure 9.5: Surface electronic structure of Ag(110) in the vicinity of the Ȳ point of the SBZ. One occupied and one unoccupied surface state can be found.
Fig. 9.6 shows photoemission and RAS spectra for the
clean and oxygen-covered Ag(110) surface
Figure 9.6: Photoemission and RAS spectra for the clean and oxygen-covered Ag(110) surface. The photoemission spectrum taken at the Ȳ point shows the surface state right below the Fermi level. In the RAS spectrum both surface states give rise to a peak corresponding to an interband transition between them. After [74].
RAS does thus allow the direct observation of surface states. This could be used to monitor chemical reactions time-resolved with a simple optical technique.
The non-linear technique of Second Harmonic Generation (SHG) is based on a similar idea as RAS, i.e. using optical properties which vanishes in the bulk of a centro-symmetric material. We only touch this technique very briefly here. Consider the current driven by an AC field. It can be written as
| j=σE+χ : EE… (9.6) |
where χ is the third-rank tensor of second harmonic generation. In the bulk of a centrosymmetric material this tensor must vanish but at the surface it does not. Equ. 9.6 practically means that by shining light with a frequency ω on a crystal, scattered light with a frequency 2ω can be generated. This second-harmonic signal can be safely ignored for all conventional light sources, including synchrotron radiation. If one uses a laser as light source it does, however, lead to measurable intensities.
We illustrate this technique by using the same example as for RAS,
the surface state on Ag(110) in Fig. 9.7. The
interband transition can only be induced for one polarization
direction. This leads to a different absorption of light by the
surface and this, in turn, is seen in the SHG data.
Figure 9.7: SHG spectrum from clean Ag(110) and two different polarization directions. For one polarization direction the surface state transition contributes to the SHG signal. After Ref. [75].
SHG does have a similar disadvantages as RAS: the bulk material has to be centrosymmetric. The experimental geometry is, however, much less restricted. On the other hand, it is very difficult to tune the laser light and SHG cross section is small. This means that it takes many weeks of work to take a spectrum as in Fig. 9.7 while the corresponding RAS spectrum in Fig. 9.6 can be taken in a couple of minutes.