The recent development of the “scanning tunneling microscope” (STM) by Binnig et al. [8.1–5] has made possible the direct real-space imaging of surface topography. In this technique, a metal tip is scanned along the surface while ad justing its height to maintain constant vacuum tunneling current. The result is essentially a contour map of the surface. This contribution reviews the the ory [8.6–8] of STM, with illustrative examples. Because the microscopic structure of the tip is unknown, the tip wave functions are modeled as s-wave functions in the present approach [8.6, 7]. This approximation works best for small effective tip size. The tunneling current is found to be proportional to the surface local density of states (at the Fermi level), evaluated at the position of the tip. The effective resolution is roughly [2A(R+d)]1/2, where R is the effective tip radius and d is the gap distance. When applied to the 2x1 and 3x1 reconstructions of the Au(l10) surface, the theory gives excellent agreement with experiment [8.4] if a 9 A tip radius is assumed. For dealing with more complex or aperiodic surfaces, a crude but convenient calculational technique based on atom charge superposition is introduced; it reproduces the Au(l10) results reasonably well. This method is used to test the structure-sensitivity of STM. The Au(l10) image is found to be rather insensitive to the position of atoms beyond the first atomic layer.

Theory of the scanning tunneling microscope

Scanning tunneling microscopy

Localized scatterers can be expected to give rise to spatial variations in the electric field and in the current distribution. The transport equation allowing for spatial variations is solved by first considering the homogeneous transport equation which omits electric fields. The homogeneous solution gives the purely diffusive motion of current carriers and involves large space charges. The electric field is then found, and approximate space charge neutrality is restored, by adding a particular solution of the transport equation in which the electric field is associated only with space charge but not with a current. The presence of point scatterers leads to a dipole field about each scatterer. The spatial average of a number of these dipole fields is the same as that obtained by the usual approach which does not explicitly consider the spatial variation. Infinite plane obstacles with a reflection coefficient r are also considered. These produce a resistance proportional to r/(1-r).

Spatial variation of currents and fields due to localized scatterers in metallic conduction

Tunneling spectroscopy has played a central role in the experimental verification of the microscopic theory of superconductivity in classical superconductors. Initial attempts to apply the same approach to high-temperature superconductors were hampered by various problems related to the complexity of these materials. The use of scanning tunneling microscopy and spectroscopy (STM and STS) on these compounds allowed the main difficulties to be overcome. This success motivated a rapidly growing scientific community to apply this technique to high-temperature superconductors. This paper reviews the experimental highlights obtained over the last decade. The crucial efforts to gain control over the technique and to obtain reproducible results are first recalled. Then a discussion on how the STM and STS techniques have contributed to the study of some of the most unusual and remarkable properties of high-temperature superconductors is presented: the unusually large gap values and the absence of scaling with the critical temperature, the pseudogap and its relation to superconductivity, the unprecedented small size of the vortex cores and its influence on vortex matter, the unexpected electronic properties of the vortex cores, and the combination of atomic resolution and spectroscopy leading to the observation of periodic local density of states modulations in the superconducting and pseudogap states and in the vortex cores.

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Scanning tunneling spectroscopy of high-temperature superconductors

The effects of material, thermal, and geometrical asymmetries on the nonlinearity of the I-V characteristics of a scanning tunneling microscope (STM) tunneling junction are investigated theoretically and experimentally. As a consequence of the nonlinearity, an STM junction can rectify an incident laser beam to produce both DC and AC tunneling current components. Using the dependence of the DC rectified current on the tip-sample distance s and the laser frequency v, an operational definition of a tunneling time is proposed. An experiment using a laser-irradiated STM junction is also performed to measure the rectified current as a function of s, for a fixed v. From the preliminary results of this experiment, a tunneling time on the order of 1.8 fs is determined. This time is in reasonable agreement with those obtained in other experimental and theoretical studies of tunneling junctions of comparable dimensions and energies. It is also shown that the AC current contains components at the laser frequency and its higher harmonics. This implies that the STM junction can be used as a nonlinear device to detect, generate, and measure laser frequency harmonics. >

Mechanisms of current rectification in an STM tunnel junction and the measurement of an operational tunneling time

Derivation of the image interaction for non-planar pointed emitter geometries: application to field emission I–V characteristics

The WKB method is commonly used in semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of a one-dimensional potential. In a multidimensional space, the method can be adapted to construct wave functions in an ``allowed'' region from classical trajectories or wave normals. However, in the ``forbidden'' region the WKB wave function is in general specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces or equivalently by two sets of paths defined to be normal to these surfaces, respectively. We present a Huygens-type construction for obtaining these wave fronts and paths, which reveals that for non-normal incidence the paths are coupled to each other. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. A special and important case occurs when the incident wave is normal to the turning surface. We show that for normal incidence the path equations are decoupled and are equivalent to Newton's equations of motion for the inverted potential and energy.

T.E. Feuchtwang

Papers

Mechanisms of current rectification in an STM tunnel junction and the measurement of an operational tunneling time

Derivation of the image interaction for non-planar pointed emitter geometries: application to field emission I–V characteristics

Wentzel-Kramers-Brillouin method in multidimensional tunneling

Scattering-theoretic approach to elastic one-electron tunneling through localized barriers: Application to scanning tunneling microscopy.

A theory of vacuum tunneling microscopy