Science, Technology and Engineering

Centre for Materials and Surface Science

Toroidal Spectrometer Project - Fermi Surface Mapping

Azimuthal type Fermi surface scans

The analyser can be set to observe emission from the vicinity of the Fermi edge of a conducting sample. Using 10 eV pass energy, the energy window will show emission covering a range of approximately 0.8 eV.  It is therefore not essential to  select a precise value for the Fermi energy before data collection since the most appropriate angular value can be selected subsequently thereby allowing for any variation in the apparent energy position of the actual Fermi level. Indeed, by observing the form of the energy distribution across the energy window at a particular angle , the Fermi function for that angle may be observed and the point of inflection determined exactly.

A Fermi azimuthal scan with this spectrometer involves a rotation of the sample (azimuthal scan mode). 180 degrees of sample rotation in azimuth, when combined with the standard ability of the analyser to acquire 180 degrees of polar angle emission is sufficient to ensure that the data contains intensity information covering a complete hemisphere.

The data is consequently acquired in the form of a grey-scale plot in terms of theta (polar) and phi (azimuthal) rotation settings as shown in Figure 1.  The straight vertical lines originate from the (111) surface state at Gamma, the other structure from the bulk Fermi surface topology via either direct or indirect transitions (see below). The folowing image was obtained with the sample oriented at 35 degrees relative to the incoming photons: hence the difference in intensity for positive and negative polar emission angles.

If a free electron like final state is assumed,  angular data such as that above can be transformed so as to show the data in terms of a 2-dimensional electron wave vector cut through the Fermi surface topology. The combination of 180 degres of polar data with 360 degrees of azimuthal data overspecifies the intensity distribution for the emission hemisphere and we have consequently a number of choices for displaying the results. In this case, data from the relatively intense 90 degree segment of the polar angle range was combined with 360 degrees of azimuthal rotation to produce the symmetrical diagram below. Comparable images at various photon energies may be found in recent publications. (see Toroidal Index Page for details)

By combining this type of information with CIS data (see below) we have been able to show that the relatively sharp Fermi surface "outline" visible above is due to indirect transitions associated with enhanced Kperp broadening from certain regions of the Fermi surface whereas the remaining diffuse structures result from direct (k-conserving) transitions.

Angle resolved CIS type Fermi scans

The full hemisphere, azimuthal rotation method of determining the topology of a Fermi surface as above has the advantage of illustrating the symmetry of the surface in a rather direct manner.  The assumption of a free electron final state and direct transitions, however, implies that the pattern observed is a projection on a k_x-k_y surface of a spherical cut through the bulk Fermi surface.

If direct transitions dominate, a linear slice through the Fermi surface may be examined by performing an angle resolved constant initial state (ARCIS) scan at a selected azimuth angle. In this mode, the detection kinetic energy is scanned synchronously with increasing photon energy so as to maintain the initial energy at the Fermi energy. Structure seen in such an ARCIS plot is expected to correspond to the intersection of an expanding free electron circle (centered on a neighbouring Brillouin zone) as indicated above, with the "original" Fermi surface. Transitions with a final Kparallel value which is positive (towards the surface) and lage enough to imply energy sufficient to overcome the inner potential are indicated above the dashed line. Using the toroidal analyser, such transitions produce an intensity map of polar angle against photon (or kinetic) energy which can be readily tranformed into K space as shown below on the basis of a free electron approximation for the final states.

This data shows CIS intensity maps (with enhanced contrast to make features at high photon energies visible despite the decrease in photoionization cross-section) for two azimuthal directions from a Cu(111) surface. The signature of the surface state in these diagrams is the two parallel vertical lines centered on zero polar angle. Diffuse structures may be seen following the dotted outline of the Fermi surface as known from de Haas van Alfen studies. The ovals highlight the extent of Kperp broadening from those regions of the Fermi surface for which there is a large density of 1-dimensional states. Significant Kperp broadening is evident in all features due to the short lietime of the final state. At the Fermi energy, the initial state lifetime may be taken as negligible.

By combining this information with azi scans (see above) taken at selected photon energies, we have been able to distinguish between those features resulting from direct and indirect transitions. Unexpectedly different results have been obtained from Cu(100) surfaces from which direct transitions appear to play the dominant role unlike the indirect transitions which appear prominently from Cu(111) emission.

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