Theory of real space imaging of Fermi surface parts

A scanning tunneling microscope can be used to visualize in real space effects provided by Fermi surfaces with buried impurities far below substrates acting as local probes [Weismann et al. Science 323, 1190 (2009)]. After scattering at buried impurities, anisotropic electronic wave oscillations are observed on the surface as hot spots: The experiments exhibit strongly enhanced intensities in certain directions and much weaker intensities in other directions. A theory describing these features is developed based on the stationary phase approximation for the Friedel oscillations and taking into account the band structure of the host material. It is demonstrated how the Fermi surface of a material, for instance, through Fermi contours’ critical points, acts as a mirror focusing electrons that scatter at hidden impurities which allow the projection of parts of the Fermi surface, a quantity defined in reciprocal space, onto real space.

Figure 1

Fermi surface of copper represented along three directions: (111) is shown in (a), (001) in (b), and (110) in (c). The inverse mass tensor corresponding to the square root of the denominator of Eq. (20) is represented by the color in units of the inverse electron mass. Small values represented in red lead to high intensities of the charge variation.

Figure 2

[(a) and (b)] Impurity-induced charge density around $E_F$ at a height of $\approx$6.1 Å above the Cu(111) surface with an Co impurity sitting in the sixth layer (a) and in the third layer (b) below the surface. (c) The case of an impurity buried at eight layers below a Cu(001) surface is shown. Red/blue color means enhancement/reduction of the local density of states at $E_F$.

Figure 3

Examples of Fermi contours for illustrative purposes. (a) An ellipsoidal Fermi surface, which for a given $\mathrm{R⃗}$ has one critical point with a group velocity $\mathrm{v⃗}$ parallel to $\mathrm{R⃗}$. The length of the thick line defines the oscillation period. The Fermi surface sections [(b), (c), and (d)] are similar to the “dog’s bone” of the copper Fermi surface. For an oblique orientation of $\mathrm{R⃗}$ with respect to the contour’s long axis represented in (b), several critical points contribute to the oscillatory behavior. In (c), however, $\mathrm{R⃗}$ is parallel to the short axis, which reduces the number of critical points. In (d), thick red points show possible inflection points corresponding to higher-order critical points discussed in the text. One possible direction of $\mathrm{R⃗}$ probing this region is shown.

Figure 4

Fermi surface of copper showing the diagonal elements of the inverse mass tensor in the $X$ and $y$ direction in (a) and (b), respectively. (c) The Fermi surface of copper is colored following the curvature of the Cu-Fermi surface, which determines the amplitude and the phase of the oscillations in Eqs. (18) and (20). Blue indicates an inward curvature like the curvature of an ellipsoid, and green means an outward curvature. At the boundary line between the green and blue areas the Gaussian curvature vanishes, indicating higher-order critical behavior as in Eq. (24).