The s-wave-tip model

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. 

For simple metal surfaces with fundamental periodicity a, the corrugation amplitude of the Fermi-level LDOS as a function of tip-sample distance can be estimated with reasonable accuracy (Tersoff and Hamann, 1985)  

This expression coincides with the Stoll formula, Eq. (1.25), within a constant factor. Therefore, both theories provide an adequate explanation of the topographic STM images of the superstructures on Au(llO) surfaces. 

The superstructures of the Au(l 10) surface, with periodicities 8.15 and 12.23 A, respectively, are much larger than the resolution of STM observed 

The original 5-wave-tip model described the tip as a macroscopic spherical potential well, for example, with r 9 A. It describes the protruded end of a free-electron-metal tip. Another incarnation of the 5-wave-tip model is the Na-atom-tip model. It assumes that the tip is an alkali metal atom, for example, a Na atom, weakly adsorbed on a metal surface (Lang, 1986 see Section 6.3). Similar to the original 5-wave model, the Na-atom-tip model predicts a very low intrinsic lateral resolution. 

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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.)...

Similar to the. y-wave model, the Na-atom-tip model predicts a poor resolution. The agreement of the Na-atom-tip model with the y-wave-tip model does not mean that the s-wave-tip model describes the actual experimental condition in STM. According to the analysis of Tersoff and Lang (1990), real tips are neither Na or Ca, but rather transition metals, probably contaminated with atoms from the surface (for example. Si and C are common sample materials). For a Si-atom tip, the p state dominates the Fermi-level LDOS of the tip. For a Mo-atom tip, while the p contribution is reduced, this is more than compensated by the large contribution from states of d like symmetry. The STM images from a Si, C, or Mo tip, as predicted by Tersoff and... 

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 ... 

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... 

The survey of spiral-wave behavior in the Oregonator model by Jahnke and Winfree [41] revealed a number of trajectories with high rotation symmetry in a region of the s, f) plane marked with the name meander by the authors. In this region the path of the spiral tip forms closed floral patterns for specific choices of the parameters (e, / ). The center of these regular patterns have... 

When the Gaussian curvature of the surface is not constant (for example, for a prolate spheroid) the wave s tip experiences varying curvature F as it moves over the surface. This results in the systematic drift of spiral waves on the nonuniformly curved surfaces [51 ]. To check this prediction, the numerical simulation of the spiral wave on the surface of a prolate spheroid has been performed in [28] using full reaction-diffusion equations of the model (60)-(62). Figure 13 shows the computed trajectory of motion of the tip of a spiral wave on the coordinate plane (0, ) where 6 and are the spherical angles. We see that the spiral wave drifts approximately along the equator of the spheroid. 

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. 

Numerical simulation was conducted based on the proposed model and the obtained parameters. The beam-shaped gel was discretized every 6l= [mm] along the longitudinal direction for numerical integration. We applied the same current density to the gel for 400[s]. The time step for numerical integration was St=l [s]. The waving motion of the tip was observed and plotted in Fig. 7.20. The speed was faster than the experimental results. The times of extremum were ti=43 [s], <2 = 133[s], tz = 271[s]. The angles of the tip were < )i=2.13[rad], < )2=1-31 [rad], 3=1.68[rad], respectively. The shapes of the gel, which were obtained numerically, are shown in Fig. 7.21. Therefore, the numerical simulations qualitatively confirm the wave-shape pattern formation observed in the experiments. 

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