Fourier-transform scanning tunnelling spectroscopy: the possibility to obtain constant-energy maps and band dispersion using a local measurement

We present here an overview of the Fourier-transform scanning tunnelling spectroscopy technique (FT-STS). This technique allows one to probe the electronic properties of a two-dimensional system by analysing the standing waves formed in the vicinity of defects. We review both the experimental and theoretical aspects of this approach, basing our analysis on some of our previous results, as well as on other results described in the literature. We explain how the topology of the constant-energy maps can be deduced from the FT of dI/dV map images which exhibit standing waves patterns. We show that not only the position of the features observed in the FT maps but also their shape can be explained using different theoretical models of different levels of approximation. Thus, starting with the classical and well known expression of the Lindhard susceptibility which describes the screening of electron in a free electron gas, we show that from the momentum dependence of the susceptibility we can deduce the topology of the constant-energy maps in a joint-density-of-states approximation (JDOS). We describe how some of the specific features predicted by the JDOS are (or are not) observed experimentally in the FT maps. The role of the phase factors which are neglected in the rough JDOS approximation is described using the stationary-phase conditions. We present also the technique of the T-matrix approximation, which accurately takes into account these phase factors. This technique has been successfully applied to normal metals, as well as to systems with more complicated constant-energy contours. We present results recently obtained on graphene systems which demonstrate the power of this technique, and the usefulness of local measurements for determining the band structure, the map of the Fermi energy and the constant-energy maps.