Tersoff Hamann model

The purpose of doing STM is to learn about surface structures, and the tip as such is regarded as an uninteresting probe. In this sense, it is a problem that the electronic structure of the tip is contained in the formula for the tunnel current in the original work by Bardeen 58). Tersoff and Hamann 59,60), however, extended Bardeen s formalism and showed by simple, yet relevant approximations that the impact of the unwanted electronic structure of the tip is in many cases less pronounced for typical tunneling parameters. Fortunately, the Tersoff-Hamann model provides a simple conceptual framework for interpreting STM images, and therefore it is still the most widely used model. 

Fig. 2. STM image (78 A x 76 A) of nitrogen atom adsorbates on an Fe(l 00) surface. Because the nitrogen adsorbates deplete the LDOS at the Fermi level, the nitrogen atoms are imaged as depressions in accord with the Tersoff-Hamann model. From the STM image it is concluded that nitrogen atoms adsorb in fourfold hollow sites on Fe(l 0 0). This is just one of many examples illustrating how the STM contrast may depend on the details of the LDOS around an adsorbate and produce a somewhat counterintuitive picture. Adapted from Reference (dd).

Figure 3 The important quantities in the Tersoff-Hamann model of STM. The matrix element is evaluated on the surface S the conductance is proportional to the sample density of states at the tip centre of curvature Tq.

The most commonly used model in interpreting STM data is the Tersoff-Hamann model, in which the analysis is carried a step further. It is assumed that the tip wavefunction is an s-wave, and decays into the vacuum like... 

It is possible to extend perturbation theory beyond the Tersoff-Hamann model, for example by including tunnelling to or from states of non-zero angular momentum on the tip, or by using states explicitly calculated from a particular atomistic model to find the matrix element in Equation [2]. However, both these approaches require additional information about the geometry of the tip and the electronic states it supports. This is generally not available from experiment, as a tip will be modified by the forces acting in the course of the experiment (as discussed in more detail below) even if the tip is well-characterized before use (for example, by electron microscopy or field-ion microscopy), this information will become out-of-date once the experiment starts. 

See Surface states Tersoff-Hamann approximation See s-wave-tip model Tetragonal symmetry 128 Tip annealing 286, 288 Tip preparation 281—285 cutting 282... 

The difficulty of evaluating the effect on the tunneling current of the tip electronic structure was approached by Tersoff-Hamann by assuming a simple, s-wavc tip model with wave functions centered at a point Fq in the tip. In the limit of low-bias voltages, the total tunnel current can then be expressed as follows ... 

Therefore, the interpretation of such structures should be performed carefully and preferably in combination with other experimental or theoretical techniques. Nevertheless, the large database of STM results, which in combination with other experimental and theoretical surface science investigations have solved many problems correctly, often rely on the Tersoff Hamann interpretation (62-65). In this sense, the Tersoff-Hamann theory has proved itself very successful, but when applying it to STM characterizations of more complicated samples, one has to be aware of the limitations of the model. The sample wave functions may be distorted by the close proximity of the tip to the surface, and the forces between the tip and sample may lead to geometric relaxations of the atoms in the surface layer beneath the tip (66). Moreover, the tip is in this model represented by a simple x-wave, so that the chemical composition of the tip is neglected. In reality the nature of the tip may, however, differ significantly from this situation because of adsorbates or other contaminants present on the tip apex (53). 

To obtain a more quantitative analysis of STM data, three-dimensional wave functions for the tip and the sample are calculated by expHcitly solving the Schrodinger equation for the combined system. In a standard model, commonly referred to as the Tersoff-Hamann modd [16], the tip wave function is approximated by an s-wave function. One can show that, within this model, for small bias voltages, the STM image reflects the SDOS at the Fermi energy at the position of the tip center. 

In the s-wave-tip model (Tersoff and Hamann, 1983, 1985), the tip was also modeled as a protruded piece of Sommerfeld metal, with a radius of curvature R, see Fig. 1.25. The solutions of the Schrodinger equation for a spherical potential well of radius R were taken as tip wavefunctions. Among the numerous solutions of this macroscopic quantum-mechanical problem, Tersoff and Hamann assumed that only the s-wave solution was important. Under such assumptions, the tunneling current has an extremely simple form. At low bias, the tunneling current is proportional to the Fermi-level LDOS at the center of curvature of the tip Pq. 

Fig. 1.25. The s-wave-tip model. The tip was modeled as a spherical potential well of radius R. The distance of nearest approach is d. The center of curvature of tip is To, at a distance (R + d) from the sample surface. Only the 5-wave solution of the spherical-potential-well problem is taken as the tip wavefunction. In the interpretation of the images of the reconstructions on Au(llO), the parameters used are R = 9 A, d = 6 A. The center of curvature of the tip is 15 A from the Au surface. (After Tersoff and Hamann, 1983.)...

In principle, STM does not image the atomic structure of the surface directly, but instead it presents an image of the electronic structure of the surface. In the simple model of Tersoff and Hamann, the tunneling current is proportional to the local density of states (LDOS), which roughly corresponds to the square... 

To evaluate it is necessary to know explicitly the wave functions of the tip and the sample. Unfortunately, the geometry and the structure of the tip at the atomic level are generally poorly defined so that accurate calculations of the tip wave functions are not feasible. This problem can be solved through a different approach proposed by Tersoff and Hamann [14]. This model considers an ideal tip with the smallest dimension compatible with the highest resolution and it intends to measure the properties of the sample surface instead of the properties of tip-sample system. Thus, the tip on the limit i —0, is replaced by a point located at the distance fq perpendicular to the surface sample. For small Vt and r, jt is given by... 

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