^{1}, Yu. A. Kolesnichenko

^{1,a)}and J. M. van Ruitenbeek

^{2}

### Abstract

In this review we discuss recent theoretical studies of single subsurface defects by means of a scanning tunneling microscope (STM). These investigations are based on quantum interference effects between the electron partial waves that are directly transmitted through the contact and the partial waves scattered by a defect. In particular, we demonstrate the feasibility of imaging the position of a defect below a metal surface by means of STM. Different types of subsurface defects are discussed: point-like magnetic and nonmagnetic defects, magnetic clusters in a nonmagnetic host metal, and nonmagnetic defects in an -wave superconductor. The effect of Fermi surface anisotropy is analyzed. Studies of the effect of high magnetic fields on the STM conductance of tunnel point contacts in the presence of a single defect are also discussed.

I. INTRODUCTION

II. QUANTUM INTERFERENCE OF SCATTERED ELECTRON WAVES IN THE VICINITY OF A POINT CONTACT

A. Model of STM contacts and the Schrödinger equation for these systems

B. Wave function for an inhomogeneous tunnel barrier

C. Electron scattering by a single defect in the vicinity of a tunnel point contact

III. FRIEDEL-LIKE OSCILLATIONS OF THE TUNNEL POINT-CONTACT CONDUCTANCE

A. Voltage dependence of the STM conductance

B. Determination of the defect positions

IV. SIGNATURE OF THE FERMI SURFACE ANISOTROPY

V. SUBSURFACE MAGNETIC DEFECTS

A. Kondo impurity

B. Magnetic cluster

VI. MAGNETO-QUANTUM OSCILLATIONS

A. Conductance oscillations in a perpendicular magnetic field

B. Effect of quantization of the flux through the trajectories of scattered electrons

C. Effect of longitudinal focusing of electrons onto a defect by a magnetic field

VII. NONMAGNETIC DEFECT IN A SUPERCONDUCTOR

VIII. CONCLUSIONS

### Key Topics

- Scanning tunneling microscopy
- 60.0
- Electron scattering
- 40.0
- Fermi surface
- 38.0
- Wave functions
- 30.0
- Magnetic fields
- 29.0

## Figures

Model of a tunnel point contact as an orifice in an interface that is nontransparent for electrons except for a circular hole, where tunneling is allowed. Trajectories are shown schematically for electrons that are reflected from or transmitted through the contact and then scattered back by a defect.

Model of a tunnel point contact as an orifice in an interface that is nontransparent for electrons except for a circular hole, where tunneling is allowed. Trajectories are shown schematically for electrons that are reflected from or transmitted through the contact and then scattered back by a defect.

Spatial distribution of the square of the modulus of the wave function in the vicinity of a contact in a plane perpendicular to the interface passing through the contact and a defect. Distances are given in units of the inverse wave number.^{47}

Spatial distribution of the square of the modulus of the wave function in the vicinity of a contact in a plane perpendicular to the interface passing through the contact and a defect. Distances are given in units of the inverse wave number.^{47}

Dependence of the normalized oscillatory part of the conductance on the STM tip position for different depths of the defect below the surface; , .

Dependence of the normalized oscillatory part of the conductance on the STM tip position for different depths of the defect below the surface; , .

Dependence of the oscillatory part of the conductance, , as a function of the position of a defect in the plane . The shape of the FS (41) is defined by the mass ratios , , and the long axis of the ellipsoid is rotated by about the x axis, away from the y axis. The coordinates are measured in units of (46) and the defect lies at .^{51}

Dependence of the oscillatory part of the conductance, , as a function of the position of a defect in the plane . The shape of the FS (41) is defined by the mass ratios , , and the long axis of the ellipsoid is rotated by about the x axis, away from the y axis. The coordinates are measured in units of (46) and the defect lies at .^{51}

(a) The Fermi surface given by Eq. (48) relative to the contact axis for three principal lattice orientations. (b) A plot of the tunneling point-contact conductance G as a function of the contact position for a defect at the origin, at a depth of and for a (100) surface plane; the x and y directions each correspond to 100 directions. (c) Same plot for a (111) surface orientation; the x and y directions correspond to and directions, respectively. (d) Same plot for a (110) surface orientation; the x and y directions correspond to [001] and directions, respectively.^{69}

(a) The Fermi surface given by Eq. (48) relative to the contact axis for three principal lattice orientations. (b) A plot of the tunneling point-contact conductance G as a function of the contact position for a defect at the origin, at a depth of and for a (100) surface plane; the x and y directions each correspond to 100 directions. (c) Same plot for a (111) surface orientation; the x and y directions correspond to and directions, respectively. (d) Same plot for a (110) surface orientation; the x and y directions correspond to [001] and directions, respectively.^{69}

The difference between the conductance as a function of voltage for a magnetic and a nonmagnetic impurity. The parameters , , and are used in Eq. (49) and (46).

The difference between the conductance as a function of voltage for a magnetic and a nonmagnetic impurity. The parameters , , and are used in Eq. (49) and (46).

The oscillatory part of the conductance as a function of tip position on a metal surface for subsurface magnetic clusters with different cluster diameters. The -coordinate is measured from the point at which the contact is situated directly above the cluster; ; ; ; ; .^{64}

The oscillatory part of the conductance as a function of tip position on a metal surface for subsurface magnetic clusters with different cluster diameters. The -coordinate is measured from the point at which the contact is situated directly above the cluster; ; ; ; ; .^{64}

Schematic representation of the electron trajectories in the vicinity of a point contact in an external magnetic field oriented along the contact axis.

Schematic representation of the electron trajectories in the vicinity of a point contact in an external magnetic field oriented along the contact axis.

Oscillatory part of the conductance of a tunneling point contact with a single defect placed at , . The solid curve is a plot of Eq. (62), while the dashed curve shows the component in the semiclassical approximation (68). The field scale is given in units of , and .

Oscillatory part of the conductance of a tunneling point contact with a single defect placed at , . The solid curve is a plot of Eq. (62), while the dashed curve shows the component in the semiclassical approximation (68). The field scale is given in units of , and .

The oscillatory part of the STM conductance as a function of tip position for different values of magnetic field; , and .

The oscillatory part of the STM conductance as a function of tip position for different values of magnetic field; , and .

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