Tersoff-Hamann approach

In parallel, the related activity was in the field of single-electron shuttles and quantum shuttles [143-153]. Finally, based on the Bardeen s tunneling Hamiltonian method [154-158] and Tersoff-Hamann approach [159,160], the theory of inelastic electron tunneling spectroscopy (IETS) was developed [113-116,161-163],... 

The additional approximation in the Tersoff-Hamann approach concerns only the shape of the tip orbital, in particular the substitution of the matrix... 

Recently, the question was taken up by another group, which used a slight modification of the Tersoff-Hamann approach to calculate the STM contours, obtaining quite similar images to the ones presented here [60]. The authors interpreted features II and III of the experimental scans in a quite different manner feature II supposedly arises from an end-bridge configuration (see (C) 2 of Fig. 11), while feature III is thought to be due to two... 

The combination of state-of-the-art first-principles calculations of the electronic structure with the Tersoff-Hamann method [38] to simulate STM images provides a successful approach to interpret the STM images from oxide surfaces at the atomic scale. Typically, the local energy-resolved density of states (DOS) is evaluated and isosurfaces of constant charge density are determined. The comparison between simulated and measured high-resolution STM images at different tunneling... 

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 ... 

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before use (for example, by electron microscopy or field-ion microscopy), this information will become out-of-date once the experiment starts. 

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.)...

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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