Corrugation amplitude

Fig. 1.16. Atom-resolved image observed on Cu(lll). The atomic distance of Cu(lll) is 2.8 A. A skew dislocation appears on the surface. Near the dislocation, every single Cu atom on the surface is observed. The corrugation amplitude of the flat...

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) ... 

Fig. 1.26. General features of Fermi-level LDOS corrugation amplitude. The...

An early systematic experimental study on the imaging mechanism was conducted on Al(lll) (Wintterlin et al., 1989). The observed corrugation amplitude was more than one order of magnitude larger than the Fermi-level LDOS corrugation. Aluminum is a textbook example of simple metals. The electronic states on the AI(lll) surface have been studied thoroughly. 

The surface charge density of Al(lll) has been well characterized by first-principles calculations as well as helium scattering experiments. The asymptote of the corrugation amplitude Az of equal-LDOS surface contours follows an exponential law, as obtained from a first-principles calculation of the electronic structure of the Al(l 11) surface (Mednick and Kleinman, 1980) ... 

Fig. 1.27. Quantitative results from STM images of Al(lll). An exponential relation between the corrugation amplitude and the tip-sample distance is observed. The best corrugation observed is more than 20 times greater than the maximum corrugation amplitude expected from the Fermi-level LDOS contour. (Reproduced from Wintterlin et al., 1989, with permission.)...

An exponential dependence of corrugation amplitude with distance is clearly observed. 

The detailed data from He-scattering experiments provide information about the electron density distribution on crystalline solid surfaces. Especially, it provides direct information on the corrugation amplitude of the surface charge density at the classical turning point of the incident He atom, as shown in Fig. 4.13. As a classical particle, an incident He atom can reach a point at the solid surface where its vertical kinetic energy equals the repulsive energy at that point. The corrugation amplitude of the surface electron density on that plane determines the intensity of the diffracted atomic beam. 

The corrugation amplitude of the Fermi-level LDOS for a metal surface with one-dimensional corrugation can be obtained using Equations (5.7) and (5.18),... 

Fig. 5.2. Corrugation enhancement arising from different tip states. Solid curves, enhancement of tunneling matrix elements arising from different tip states. The tunneling current is proportional to the square of the tunneling matrix element. Therefore, the enhancement factor for the corrugation amplitude is the square of the enhancement factor for the tunneling matrix element, dotted curves. (Reproduced from Chen, 1990b, with permission.)...

Therefore, the corrugation amplitude arising from a p, tip state gains a factor of [l over that of the charge-density contour see Fig. 5.2. This is... 

The enhancement for the tunneling matrix element is shown in Fig. 5.2. The enhancement factor for the corrugation amplitude, Cl + (3iyV2K-)T, could be substantial. For example, on most closc-packcd metal surfaces, a= 2.5 A,... 

A straightforward calculation using the tunneling matrix elements listed in Table 3.2 shows that the state results in a large but inverted corrugation amplitude on metal surface, because the tunneling matrix element for the sample wavefunction at the F point vanishes. The role of this state and the state in the inverted corrugation will be discussed in Section 5.5. 

Fig. 5.4. The corrugation amplitude of the STM images for Cu(OOl). Calculated for different m = 0 lip states and different tip-sample distances. The corrugation amplitude of charge density contours is obtained from Gay ct al. (1977).

Figure 5.4 shows the calculated corrugation amplitudes of STM images of Cu(OOl) with different tip states. The, r-wave tip state does not provide atomic resolution, as expected. The large corrugation amplitude observed on Cu(OOl) (Samsavar et al., 1990) is probably due to an 4- Op state. 

Similarly, 3 = 7 - 2k. The ratio (Ci/Cq) can be determined by comparing Eq. (5.49) with the corrugation amplitudes of the charge-density contours obtained from first-principles calculations. For example, from Fig. 5.7, averaged from five contours ranging from three contours of thinnest densities, we find (C /Co) 5.7 1.0. Following the procedure for the one-dimensional ca,se, the STM image for the p- tip state is... 

Fig. 5.8. Interpretation of the STM corrugation observed on Al(lll). The predicted corrugation amplitude with a 4- tip state (solid curve) agrees well with the experimental data from Wintterlin et al. (1989) (circles with error bars). The parameters of the theoretical curve are taken from Fig. 5.7. The corrugation from an i-wave tip state (dashed curve), that is, the corrugation of Fermi-level LDOS contour, is included for comparison, (Reproduced from Chen, 1990, with permission.)...

The enhancement factor E, that is, the quantity in the parenthesis in this equation, is displayed in Fig. 5.10. Because the corrugation amplitude depends only on the relative intensities of different components, we normalize it through... 

Fig. 5.10. Enhancement factor for different tip states. The shaded area near E=0 is the area where the corrugation amplitude is within the limit of the Fermi-level LDOS contours. In the hatched area near the bottom, the theoretical amplitude of the negative corrugation shows a spurious divergence. (Reproduced from Chen, 1992c, with permission.)...

Using these equations and the conductance distribution functions listed in Table 6.1, the corrugation amplitudes for a tetragonal close-packed surface with different tip states and sample states can be obtained. For example, for a Is state, using Eq. (6.32), we have... 

Table 6.2. Independent-state model corrugation amplitudes for surfaces with tetragonal symmetry ...

The definition of (3 is identical to that of Hams and Liebsch (1982, 1982a), as is given by Eq. (5.20) = y - 2k. For p and d states, similar results can be obtained using Eq. (6.33). A list of corrugation amplitudes for =1 states is shown in Table 6.2. Clearly, with p. and d.j states on the tip as well as on the sample, substantial corrugation enhancements should be observed. Using similar method, the corrugations for n> and m>0 cases as well as mixed states, such as sp states, can be obtained. 

The dependence of corrugation amplitudes on tip-sample distances are calculated using the independent-state model. The corrugation in the [211] direction is much easier and much less dependent on tip electronic states than the corrugations in the [011] direction. 

The corrugation amplitudes in the [oTl] direetion and in the [211] direction are displayed in Fig. 6.7. Some general features are worth noting First, the corrugation in the [211] direction is much easier to observe and much less dependent on tip states than the corrugation in the [oTl] direction. Second, the decay constant for the corrugations in the two directions are quite different. Third, those decay constants arc independent of the tip stale. 

By comparing this with the results for surfaces with tetragonal symmetry, it is clear that the only difference is the factor of 8 in Eq. (6.29) is replaced by 9/2. With the same lattice constant, the corrugation amplitude of a surface with hexagonal symmetry is smaller than that for a surface with tetragonal symmetry by a factor of 9/16=0.5625. The decay constant of the corrugation is... 

Figure 6.9 is a comparison of the results discussed previously with the first-principles calculation of the AI(lll) surface as well as the experimental results of STM images on Al(lll) by Wintterlin et al. (1989). A very simple model of the Al(lll) surface is used On each surface Al atom, there is an independent Is state near the Fermi level. The charge density contour (i.e., the image with an. r-wave tip state) agrees with the extrapolated corrugation amplitudes of the first-principles calculation (Mednick and Kleinman, 1980 ... 

Fig. 6.9. Corrugation amplitudes of a hexagonal close-packed surface. Solid curve, theoretical corrugation amplitude for an s and a d,- tip state, on a close-packed metal surface with a=2.88 A and 4>=3.5 eV. The orbitals on each metal atom on the sample is assumed to be 1 i-type. Measured STM corrugation amplitudes are from the data of Wintterlin et al. (1989). The first-principle calculation of Al(lll) is taken from Mednick and Kleinman (1980). The corrugation amplitude for a 4-wave tip state is more than one order of magnitude smaller then the experimental corrugation. (Reproduced from Chen, 1991, with permission.)...

On surfaces of some d band metals, the 4= states dominated the surface Fermi-level LDOS. Therefore, the corrugation of charge density near the Fermi level is much higher than that of free-electron metals. This fact has been verified by helium-beam diffraction experiments and theoretical calculations (Drakova, Doyen, and Trentini, 1985). If the tip state is also a d state, the corrugation amplitude can be two orders of magnitude greater than the predictions of the 4-wave tip theory, Eq. (1.27) (Tersoff and Hamann, 1985). The maximum enhancement factor, when both the surface and the tip have d- states, can be calculated from the last row of Table 6.2. For Pt(lll), the lattice constant is 2.79 A, and b = 2.60 A . The value of the work function is c() w 4 cV, and k 1.02 A . From Eq. (6.54), y 3.31 A . The enhancement factor is... 

The modification of an x-wave sample state due to the existence of the tip is similar to the case of the hydrogen molecule ion. For nearly free-electron metals, the surface electron density can be considered as the superposition of the x-wave electron densities of individual atoms. In the presence of an exotic atom, the tip, the electron density of each atom is multiplied by a numerical constant, 4/e 1.472. Therefore, the total density of the valence electron of the metal surface in the gap is multiplied by the same constant, 1.472. Consequently, the corrugation amplitude remains unchanged. 

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