The introduction of a surface gives the possibility of new vibrations
which are localized at the surface. It is the same as we have
already
discussed for the electronic surface states but it is somewhat easier
to
imagine. The new states have to be again in "band-gaps" of the
projected
bulk phonon dispersion. This means that for the same
q∥
there is no bulk
mode with the same vibrational frequency. Otherwise the surface and
bulk
mode would couple and the surface mode would no longer be localized
at the
surface. Fig. 10.1 shows
the projected bulk-phonon structure for Be(0001) together with the
surface phonon modes. The
dashed lines are calculated surface modes based on the same
force-constants which work fine for describing the phonon dispersion
in bulk Be and the dots
are
the result of measurements. Clearly the agreement is not good. The
force-constants at the surface must be different from the bulk values.
Figure 10.1: Calculated surface phonon dispersion for Be(0001) (line, from the bulk force constants) together with a measurement (markers). The continuum is the projected bulk phonon structure. After Ref. [76].
Here the force constants are not as strong as in the bulk leading to surface vibrations with a lower energy than the bulk continuum. There can of course also be surface vibrations above the bulk continuum if the force constants in the surface are stiffer.
Apart from these "intrinsic" vibrational surface states, adsorbates on the surface
lead
to "extrinsic" surface vibrations. In many cases these modes will lie
above the bulk continuum such that they do not couple to bulk
vibrations.
If they lie in the bulk continuum they will couple and act as a source
of
bulk vibrations. The "extrinsic" vibrational modes will be the modes
of
the adsorbate shifted in frequency due to the different chemical
environment. But there are also new modes: the translational and
rotational degrees of freedom of the free molecule will be turned
into new
vibrations, the so-called frustrated translations and rotation.
Fig. 10.2
gives an example for the vibrational modes of a diatomic molecule on
a
surface. Remember that such a molecule has only one vibrational mode
when
it is free.
Figure 10.2: Vibrational modes and corresponding energies (in meV) for CO on a two-fold bridge site. The free molecule has only one vibrational mode, the others are frustrated rotations and translations. After Ref. [77]. The energies are only intended as an order of magnitude and are taken from Ref. [78].
A measurement of the surface phonons requires a probe which is coupling to the vibration and which can be used with sufficiently high resolution. The substrate phonon energies lie in most cases below 50 meV. This means that the spectral resolution should be clearly better than 10 meV. Three common probes are used at present: light, electrons and atoms. As we have already discussed in the section about optical properties, light has one outstanding advantage with respect to the other probes: it can also be used under "real" working conditions of a catalyst, i.e. not in ultra-high vacuum and at elevated temperatures.
We know the design of an EELS or HR(high-resolution)EELS spectrometer
from
the lecture about electron-surface interactions. It is shown in
Fig. 10.3.
Figure 10.3: An EELS spectrometer.
The instrument consists of an electron source, a double electron monochromator, lenses which focus the beam on the sample, lenses which image the beam into a second double analyser and an electron multiplier. The monochromator side of the instrument can be turned around the sample, such that different combinations of incidence / emission angles are possible. With an instrument of this design a spectral resolution of about 1 meV can be achieved.
The most important physical aspect of the scattering is of course the
conservation of energy and momentum. The impinging electrons can
excite or (at not too low temperatures) destroy a phonon (see Fig.
10.4).
Figure 10.4: EELS spectrum from Mg(0001). There are three peaks visible: the elastic peak, a loss peak from a creating a phonon and a gain peak from destroying a phonon [79].
The energy of the phonon can be directly read from the experiment. The momentum transfer for the elastically scattered electrons is simply given by the beam energy and the scattering geometry. The momentum transfer for typical phonon losses is nearly identical to that of the elastic beam because the energy loss is so small with respect to the beam energy.
There are two important mechanisms for the interaction between the electrons and the surface (1) dipole scattering and (2) impact scattering. We want to discuss both.
Consider an electron on its way towards the surface of a metal. The
electric field of the electron is screened in a classical image
charge picture like illustrated in Fig. 10.5.
Figure 10.5: Electric field created by an electron approaching a metal surface (and its image charge).
This means that the electron on its way above the crystal surface creates a changing electrical field perpendicular to the surface, which can couple to the surface vibrations. This mechanism has two important restrictions: only vibrations with a dipole moment perpendicular to the surface can couple to the electric field and can be excited. The interaction is of long range both perpendicular and parallel to the surface. That means that parallel to the surface all the atoms are excited in phase over a long range, hence the k-vector parallel to the surface is small or zero and the electrons which have lost or gained a phonon are scattered again into the specular beam (the beam for which Θin=Θout). The cross section for a loss is given by
| S ∝ |
| (10.1) |
where p is the perpendicular dipole moment and E0 is the impact energy. In other words: dipole scattering can be observed in the specular direction, for low impact energies and of course only for vibrations with a dipole moment perpendicular to the surface.
The other and somewhat complementary scattering mechanism is called impact scattering. Here the electron interacts with the electron shell of the ions on an extremely short length scale. Therefore this type of scattering creates a broad angular distribution. The selection rules for dipole scattering (only dipoles perpendicular to the surface and only phonons near the zone centre) do not apply such that impact scattering can be used to map out phonon dispersions. The cross section for impact scattering is, however, much smaller than for dipole scattering.
The two scattering mechanisms are illustrated in Fig. 10.6.
In the Θs=0∘, i.e. in the specular geometry, only the very strong
loss due
to the symmetric H-metal vibration can be seen. This is the only one,
which is dipole active. When going away from the specular direction
the other two modes can also be seen.
Figure 10.6: EELS spectra for H on W for different scattering geometries. After Ref. [80].
What can vibrational spectroscopy be used for?
A first example is the determination of chemical identity and to some
extend also adsorption geometry. Fig.
10.7
shows an EELS spectrum
for CO
adsorbed on Pt(111) taken in the specular geometry. All the strong
losses which are observed must be dipole-active
. From such a spectrum a lot
can be learned: Two losses can be found in the frequency range typical for the C-O
stretch vibration. This means that there must be two different CO species on
the surface. A comparison with metal-carbonyl complexes suggests that the
higher frequency mode is the one where the CO molecule is adsorbed on
top of a substrate atom and the lower frequency mode is due to a bridge
site. Naive intuition is also helpful: one would argue that if the C atom
has to make two bonds to the surface, then this would weaken the internal
bond to the O atom. Hence the force constant would be softer and (due to
ω=√K/M) the mode lower in energy. One has to be careful,
though, these naive pictures and even the comparison to a metal-carbonyl do
sometimes lead to a wrong adsorption site.
Figure 10.7: EELS spectrum of CO adsorbed on Pt(111). After Ref. [81]
One loss with an unresolved shoulder is observed in the CO-M stretch region. Again the softer vibration is associated with the bridge site for similar arguments as above.
Some losses are not observed: There should be frustrated translations and frustrated rotations of the molecule. These are not seen because they are not dipole active but also because they lie so low in energy (7 meV and 26 meV, respectively) that they could not be resolved by this particular spectrometer.
Whenever dealing with vibrations a central concept in experimental solid state physics is the isotope effect. By using different isotopes one can change the mass of the vibrating atoms without changing the force constants. The most famous example is probably that the observation of a strong isotope effect on the superconducting transition temperature has given a hint that phonons are important in BCS superconductivity. Here the isotope effect can be used to understand vibrational spectra:
Fig. 10.8 shows EELS data for acetylene and deuterated acetylene on
Ni(111). The C-C stretch frequency is not really affected very much
by the isotopic substitution. It shifts from 149 meV to 148 meV. But
the C-H stretch frequency severely changes when going to CD: it changes
from 361 to 272 meV. Interestingly the ratio of the two frequencies is
1.32, not too far away from √2 which would be expected from a naive
mass-difference argument.
Figure 10.8: EELS spectra for C2H2 and C2D2 showing the isotope effect for acetylene on Ni(111). After Ref. [82].
Another application of EELS is the measurement of surface phonon
dispersion curves. The problem is that for many materials the
energies of
the surface phonons are so low that this has only taken off in the
last few years because of the technical progress in building high
resolution instruments (for very low energies one can take He atom scattering, see
below). We have already seen the surface phonon dispersion for
Be(0001) above. Such data is measured by changing the scattering geometry. For
Θin=Θout
one probes the centre of the surface Brillouin zone. If one of
the angles is changed the momentum parallel to the surface is different for
the incoming and scattered electrons. In this way dispersions in the
whole surface Brillouin zone can be mapped. Fig. 10.9 shows experimental
data for Be(0001). This data has been used to draw the measured
phonon dispersion in Fig. 10.1.
Figure 10.9: EELS spectrum showing a phonon losses from Be(0001) as a function of scattering geometry (k-vector). After Ref. [83].
So far we have been concerned with the description of the surface in the harmonic approximation neglecting anharmonic effects. We know that anharmonicity is very important in the real world: it is needed for e.g. thermal expansion and a finite value of the thermal conductivity (it would be infinite for the harmonic solid).
Obviously anharmonicity is already important in a three dimensional solid. There is good reason to assume that it is even more important on a surface. As we have seen for Be(0001) the surface vibrations are often softer than the bulk vibrations. This is intuitively clear since the surface atoms have lost a part of their neighbours. The softer vibrations mean that the mean square amplitude of the vibrations is larger and the anharmonicity of the potential becomes more important.
Fig. 10.10 shows EELS data from Cu(110) taken in the specular
geometry for
temperatures between 21 K and 766 K. At low temperatures a clear
phonon loss is observed just above 20 meV. As the temperature is raised
the loss broadens out and is eventually indistinguishable from the
background. This is exactly what one would expect in an anharmonic solid. The
phonons are no good descriptions of the anharmonic solid, remember that we
have derived them in the harmonic approximation. Nevertheless, one keeps
the phonon concept but one assigns a finite lifetime to the phonons. A
phonon will eventually decay (this simple concept also explains the finite
thermal conductivity). This is exactly what is observed in the data:
for higher temperatures the lifetime gets shorter and the width of the
line gets bigger.
Figure 10.10: EELS data from Cu(110) in the specular geometry as a function of temperature. After Ref. [84].
Another way of probing surface vibrations is the use of infrared radiation. The two main advantages over EELS are that light can work in a gaseous environment and that the spectral resolution is much higher (at least by a factor of 10). Disadvantages are that the spectral region is restricted, that the momentum transfer is always 0 (due to the small momentum of light) and that IR spectroscopy is not as sensitive as EELS. We restrict ourselves to an experimental configuration where the IR light is reflected from the surface and absorption due to the vibrations are observed, Infrared Reflection Absorption Spectroscopy (IRAS).
The selection rules for vibrations in IRAS are essentially the same as for dipole scattering in EELS. Consider the reflection of an infrared beam from a metal surface. The electrical field in the vicinity of the surface can be calculated using the Fresnel equations as we have seen in the first part of the lecture. For the light polarized perpendicular to the plane of incidence (s-polarized) the reflectivity is high for all angles of incidence (see Fig. 9.1) and the phase is reversed upon reflection. The interference of incoming and reflecting wave creates a very small field at the surface (see Fig. 9.2). For the p-polarized light the reflectivity is strongly dependent on the angle. At near grazing incidence a strong enhancement of the perpendicular component can be achieved while the tangential component is always small (see Fig. 9.2).
This results in the same scenario as in EELS: only the component of the electric field perpendicular to the surface can couple to the vibrations and therefore only vibrations with a dipolar moment perpendicular to the surface can be excited.
Infrared spectroscopy has really taken off since the development of
Fourier transform spectrometers. They are also used for surfaces.
Fig. 10.11 shows such a spectrometer: it is based on a Michelson
interferometer. The
light from the source is hitting a beamsplitter. One part of the
light
is reflected from a fixed mirror, the other from a mirror which moves
periodically in x. The detector measures the resulting intensity as
a
function of mirror displacement x. The
signal I(x) is averaged over many moving cycles. Then the spectral
distribution I(λ) can be calculated from I(x) by a simple Fourier
transform.
Figure 10.11: FT-IRAS spectrometer.
All one has to do is to measure the spectrum without a sample (to obtain the instrumental function) and a spectrum with sample. The spectrum with sample is then divided by the empty spectrum.
The resolution of such a set-up is given by the maximal length x. The movement can, however, not be made too long because the coherence of the light source is limited.
In reality a surface science set-up is more complicated because it involves UHV around the sample and high vacuum around the spectrometer in order to get rid of water absorption bands from the air. Such a setup is shown in Fig. 10.12.
Figure 10.12: FT-IRAS spectrometer in UHV. After Ref. [86].
The extremely high resolution of IRAS can be used to study small
effects,
which are completely hidden in the broad peaks of EELS measurements,
for
example small energy shifts of the vibrational frequencies induced by
the
adsorbate-adsorbate interaction. Fig. 10.13 gives
an example from a study of
CO
adsorption on a stepped Pt surface, Pt(533). In the beginning the
CO molecules adsorb at the steps of the surface giving a characteristic
vibrational C-O stretch frequency. For higher coverages, sites on the
terraces are occupied as well, giving rise to a higher frequency
vibration. Eventually the low frequency vibration vanishes: the coupling between
the adsorbed CO molecules is so strong that the molecules at the step can
not keep up their characteristic vibrational frequency.
Figure 10.13: The C-O stretch frequency of CO on a stepped Pt surface as a function of coverage. After Ref. [87].
He atoms can also be used as a probe for detecting surface
vibrations. This technique is only touched very briefly here. The
use of He atoms has the advantage of very high surface
sensitivity,
very
high sensitivity towards contamination and much higher resolution
than
EELS. The disadvantages are that the maximum loss energy is rather
low (about
50 meV due to the energy of the He beam) and that the experiment is
very
complicated (see Fig. 10.14).
Figure 10.14: He-atom scattering apparatus. After Ref. [88].
In He atom scattering the only interaction mechanism for the creation or destruction of phonons is impact scattering. Like in EELS, energy and momentum conservation have to be satisfied in the scattering event, but in the case of He scattering the elastic peak and the losses observed in one scattering geometry correspond to different values of the momentum transfer since the size of the loss is comparable to the energy of the beam.
In vibrational spectroscopy Helium atom scattering is somewhat
complementary to EELS but the applications can be similar. Here we
give
just one short example. Fig. 10.15 shows
the losses for the frustrated
translations
and vibrations of CO
on Pt(111). We saw the higher energy losses
already
in the paragraph about EELS.
Figure 10.15: Low-energy vibrational modes for CO on Pt(111) investigated by He-scattering. After Ref. [89].
The spectroscopic mode of the STM can be used to observe vibrational
excitations in just the same way like the electronic excitations
discussed in section 8.4.
Vibrational spectra of molecules in buried metal-oxide interfaces
where already investigated in the 1960s by tunnelling spectroscopy,
long before the invention of the STM. One observes sharp increases in
the conductance dI/dV when sweeping the tunnelling voltage
through the energy of a vibrational mode because above this energy a new inelastic
tunnelling channel opens. The first vibrational spectra for
single-molecules on a surface were measured in 1998 with the STM.
Fig. 10.16 shows such spectra in the C-H stretch region of
acetylene adsorbed on Cu(100). The C-H stretch peak is clearly
visible in the d2I/dV2(V) spectrum! Fig. 10.16
also illustrate the isotope shift between acetylene and deuterated
acetylene.
Figure 10.16: Vibrational spectra of acetylene and deuterated acetylene taken by STS from a single molecule on Cu(100). After Ref. [90].
This experiment demonstrates a major progress in STS and STM. Using vibrational spectroscopy, one can hope to gain chemical sensitivity in STM. One will also be able to study the influence of nearby adsorbates on the vibrational modes of molecules.
A good introduction about surface optical properties can be found in [3] but for all the basics like the Fresnel equations you should consult a book on optics. A good book on EELS is [77] and a good review article on IRAS is [91]. Detailed information about He scattering can be found in [2].