## Abstract

The Fermi surface that characterizes the electronic band structure of crystalline solids can be difficult to image experimentally in a way that reveals local variations. We show that Fermi surfaces can be imaged in real space with a low-temperature scanning tunneling microscope when subsurface point scatterers are present: in this case, cobalt impurities under a copper surface. Even the very simple Fermi surface of copper causes strongly anisotropic propagation characteristics of bulk electrons that are confined in beamlike paths on the nanoscale. The induced charge density oscillations on the nearby surface can be used for mapping buried defects and interfaces and some of their properties.

The coherent propagation of electrons in solids is central for a variety of phenomena that are at the core of modern physics. Scanning tunneling microscopy (STM) has been used to manipulate atoms and create structures that allow standing electron wave patterns to be visualized (*1*). C. R. Moon *et al*. (*2*) extended this line of investigations to the retrieval of quantum-phase information in nanostructures with the scanning tunneling microscope. Another facet of electron propagation has been revealed by measurements (*3*) of exchange interaction between adatoms and wires mediated through the Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism (*4*) on a platinum surface. All of these effects depend on a fundamental property of the electron sea: It rearranges itself to minimize the disturbance caused by foreign atoms. These Friedel oscillations (*5*) may cause technologically important effects such as the formation of diluted magnetic semiconductors, spin-glasses, or the interlayer exchange coupling between magnetic layers (*6*) exploited in read heads of magnetic hard discs.

In contrast to surface impurities, research on subsurface defects has been less intense (*7*–*10*) because of the inherent experimental and theoretical difficulties involved in the investigation process. We show that a single buried impurity can be used to visualize an unexpected directional propagation of electrons, even in a rather simple metal such as copper. Combining an STM experiment and density functional calculations, we demonstrate the existence of unusually strong anisotropies in the resulting charge densities. This peculiar behavior is caused by the shape of the Fermi surface, which focuses electrons in very narrow directions. These findings are general and should be observable in other crystalline solids.

The presence of the Fermi surface in metals is the ultimate signature that electrons of a system can be treated as quasi-particles within the Fermi liquid theory. The determination of the Fermi surface of a material is crucial to understanding its thermal, electrical, magnetic, and optical properties. Experimentally, the Fermi surface is usually measured with the De Haas–van Alphen effect, after the pioneering work of D. Shoenberg (*11*), or with angle-resolved photoemission spectroscopy. The importance of the Fermi surface on macroscopic electron transport through single crystals has been demonstrated by Y. V. Sharvin *et al*. (*12*–*14*). Their explanation uses the electrons group velocities that are normal to the Fermi surface of the particular system (here, *ħ* is Planck's constant *h* divided by 2π, ∇_{k} is the gradient with respect to the wave vector **k**, and *E*(** k**) is the band structure). An accumulation of these vectors in a certain direction results in enhanced electron flux, which is also called electron focusing. Our investigations suggest that an unforeseen tool, the scanning tunneling microscope, can be used to provide a real space visualization of Fermi surfaces.

Furthermore, we propose different possible applications of this electron focusing effect. We prepared isolated subsurface Cobalt impurities in single crystals of Cu(100) and Cu(111) and investigated their influence on the local density of states (LDOS) with the use of low-temperature STM. The impurities were embedded in different depths below the surface. We added 0.1% Co to the topmost 15 monolayers by simultaneous deposition of a host metal and the impurity compound from two electron-beam evaporators.

A 9-by-9–nm STM-topography (Fig. 1A) illustrates the influence of different Co atom depths (6, 7, 9, and 10 monolayers) below the Cu(111) surface on the electron's spectral density at the surface. Apart from a long-wavelength standing-wave pattern caused by surface state electrons scattered at a monoatomic step-edge in the upper left corner, four ringlike structures can be observed. These correspond to short-wavelength oscillations with an amplitude of ∼2 pm, varying in diameter but being constant in their radial envelope width (∼1.5 oscillations). As the diameter expands with increasing depth of the impurity below the surface, the LDOS within a certain angle does not seem to be perturbed by the impurity.

The experiments performed on the (100) surface elucidate the propagation of electrons in Cu from another perspective (Fig. 1B). Here we observe the LDOS being influenced only in a squarelike region above the impurity and within four beamlike regions extending diagonally outside the four vertices. It is also observed that the wavelength of the standing wave pattern is very close to the lattice constant. This similarity occurs because the corresponding Fermi wave vectors are near the boundary of the Brillouin zone.

From these results, we can demonstrate that the surface LDOS—especially at greater distances from the impurity atoms—is influenced only in narrowly confined directions. These regions are not perfectly circular-symmetric, but they reflect the symmetry of the underlying band structure [i.e., threefold symmetry for (111) and fourfold symmetry for (100) surfaces].

All of these observed effects are related to the propagation of electrons and can be theoretically described with one fundamental concept: the quantum-mechanical propagator [i.e., the one-electron retarded Green function *G*(** x**,

**′,ϵ) of the given system]. It describes how electrons of energy ϵ propagate from a point source at**

*x***′ to other positions**

*x***in the crystal. The ripple extending from a stone impact in still water and the Huygens wavelet in optics are comparable examples of propagators in other areas of physics. Huygens principle states that the behavior of extended systems (the diffraction pattern of a grating, for example) can be expressed as a superposition of multiple Huygens-waves—that is, by the integration over all possible propagation paths. The single electron propagator in vacuum is a spherical wave, in full analogy to the Huygens wavelet in isotropic optical media. If an electron propagates within a crystal potential, the propagation becomes anisotropic. With the use of spectral representation, the single-electron propagator of a given system is connected to its electronic structure [the wave functions Ψ**

*x*_{k}(

**) and the band structure**

*x**E*(

**)] via: (1) In the above equation, η is an infinitesimal positive quantity and**

*k**i*

^{2}= –1. The imaginary part of Eq. 1 can be transformed into a superposition of all wave functions having energy ϵ. Thus, the propagator is directly associated with the k-space geometry of the corresponding constant energy-surface. In the case of free electrons, the Fermi surface is a sphere (Fig. 2A) with radius of the Fermi wave vector

*k*

_{F}, and the corresponding Green function is a spherical wave decaying with |

**–**

*x***′|**

*x*^{–1}in amplitude (Fig. 2B). Interesting effects arise if the Fermi surface deviates from a spherical shape, which is the case for the vast majority of crystals. In the most extreme case of a Fermi surface being composed of flat areas (Fig. 2C), the wave functions interfere constructively in beamlike regions perpendicular to these facets and destructively elsewhere in space (Fig. 2D). In strong contrast to the isotropic case, the amplitude within the beams does not decay with increasing distance, as is the case for the Green function in one dimension. Thus, whenever electrons are emitted from pointlike-sources or scattered at pointlike defects, they do not propagate in a spherical wave as in free space, but instead are, in general, focused in preferential directions and are detectable at much larger distances from the source than in the isotropic case.

For copper, the Fermi surface is rather spherical but shows areas having strongly reduced curvature (Fig. 3A). The propagator for such a metal can be calculated quickly by considering plane waves as wave functions and using *E*(** k**) obtained by a simple linear combination of atomic orbitals (LCAO) technique with band structure parameters taken from (

*15*). The calculated Green function (Fig. 3B) demonstrates the premises of a substantial electron focusing in copper. In contrast to the spherical wave of a free electron, a preferred propagation can be observed along eight slightly distorted hollow cones around the [111] directions. The explicit connection with the shape of the Fermi surface is settled when evaluating the LDOS change induced by a subsurface impurity (2) with

**,**

*x*

*x*_{i}, and

*x*_{j}being arbitrary positions in the system, whereas Im stands for the imaginary part. For our problem,

**corresponds to a vacuum site probed by STM. Here,**

*x**t*is the t-matrix describing the scattering of electrons by the impurity caused by the potential change Δ

*V*=

*V*

_{new}–

*V*

_{host}.

At large distances *R* between the impurity and the vacuum site, we can apply the stationary phase approximation and end up with a result , similar to that given by the well-known theory of interlayer exchange coupling (*16*). Here, *c* is the Gaussian curvature (*17*) of the isoenergy surface at energy ϵ. Thus, the observed charge density oscillations (CDO) depend strongly on the shape of the Fermi surface: A small value of the curvature means that the Fermi surface has a flat region leading to big values of the LDOS and to strong focusing of intensity in this space region.

This statement is reinforced after comparing the results of Eq. 2 using the simple LCAO model (shown in the lower right insets in Fig. 1) with the density functional theory (DFT) calculations (depicted in the left insets of Fig. 1) [spin-polarized full-potential Korringa-Kohn-Rostoker Green function method (*18*)] that are in total accordance with our experimental measurements. The lower left inset in Fig. 1A shows the case of a substitutional Co impurity sitting at the 6th layer below the surface. The induced CDO has been calculated around the Fermi energy *E*_{F} in the vacuum region at ∼6.1 Å above the center of the surface layer after removing the ideal surface background. The same period of oscillations was also noticed in the experiment. Figure 3C shows the DFT Fermi surface of Cu, on which the Gaussian curvature is represented with color. The low curvatures depicted in red correspond to less curved regions of the Fermi surface and are in accordance with the high-intensity points calculated for the charge variation in the vacuum. Diagonally opposite of the high intensities, we get only small oscillations resulting from the stronger curved regions shown in Fig. 3C. This result explains the triangular shape with a three fold symmetry instead of the six fold symmetry of the face-centered cubic-(111) surface.

Knowing the cone angle β of the focusing beams (Fig. 3A; β∼60°) and the depth *Z*_{imp} of the impurity allows us to evaluate the diameter *D* of the LDOS pattern and vice versa. This result allows us to experimentally determine the position of impurities below any noble metal surface. As a proof, a circle having the theoretical diameter *D* ∼ 21 Å [obtained from *D* = 2*Z*_{imp}tan(β/2)] is shown in the left inset of Fig. 1A. In addition, we performed the identical procedure to determine the depths of the impurities observed in the STM measurements.

The strong directionality of electron propagation is not unique to copper and has many consequences for other materials. Once the shape of the propagator is known, either by STM investigation, calculations based on the band-structure, or quick estimates from a knowledge of the Fermi surface, many physical effects can be easily predicted. We have demonstrated and explained the very anisotropic CDO caused by subsurface point defects. Of course, similar behavior will be found for both spin channels if the impurity is magnetic, in the arrangement of the Kondo-screening cloud if the system is below the Kondo-temperature, and for the indirect exchange (RKKY)-interaction between two magnetic atoms (Fig. 4C).

By scrutinizing the real-space properties of the single-electron propagator and settling the explicit relation between the Fermi surface shape and the anisotropic charge oscillations, we demonstrate a new application of STM: visualization of Fermi surfaces in real space. The LDOS pattern as function of energy is determined by (i) the dispersion of host-metal band structure, (ii) the energy-dependent coherence length of the quasi-particles, and (iii) the energy-dependent scattering phase shift of the impurity. With the use of spectroscopic STM techniques, our approach can provide access to constant-energy surfaces apart from the Fermi surface. For energies close to *E*_{F}, effects due to decoherence can be neglected, and even for |ϵ – *E*_{F}| = 1 eV, the mean free path is much larger than the distances between impurity and surface (*19*, *20*). Thus, for magnetic impurities, for instance, the information in the focusing pattern is dominated by the scattering behavior of the impurity, and the focusing effect can be applied to investigate the electronic properties of subsurface impurities spectroscopically.

The beamlike propagation paths could, for example, be used to construct a “nano-sonar” (Fig. 4A). Such a device could determine not only the depth and reflectivity of buried interfaces but also the position and magnetism of the interface. Here, in addition to the LDOS-oscillations of Fig. 1 caused by processes of direct propagation between impurity and surface, larger concentric rings should appear that are produced by electrons propagating from the impurity toward the interface and being reflected toward the surface. In fact, we extracted CDO and spin-dependent CDO for an interface of Co with Cu(111) using DFT calculations. These results allowed us to determine the position as well as the magnetism of the interface (supporting online material).

In ferromagnets, the Fermi surfaces and, consequently, the electron propagators are obviously different for the two spin channels. This should allow the design of effective spin filters (Fig. 4B). A mixture of both spin species enters the ferromagnet at a pointlike source contact (S↑↓) from which spin up– and spin down–electrons propagate in different directions and are collected at different drain contacts D_{↑} and D_{↓}. According to recent theoretical calculations (*21*), body-centered cubic europium may be a good candidate for such an application.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/323/5918/1190/DC1

Materials and Methods

SOM Text

Fig. S1

References