The Stoll Formula

In these equations, k is the decay constant, defined by Eq. (1.8), whose typical value is 1 A .  

Stoll s formula, Eq. (1.25), was found to be in excellent agreement with the results of numerical calculations for large a and large d, as well as experimental results on Au(110)-2X1 and Au(110)-3X1, For small d, deviations are found, see Fig. 1.24. 

Some interesting features of the Stoll formula are worth noting. First, only the sum R + d) appears in the formula. Therefore, only the distance of the center of curvature of the tip to the sample surface matters. The radius R becomes irrelevant. Second, the STM corrugation decays exponentially with tip-sample separation. By extrapolating the formula to (/ +1/) = 0, the corrugation coincides with that of the metal surface. These features are also found to be consistent with experimental results and with the results of the Tersoff-Hamann theory, as described below. 

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Fig. 1.24. Comparison of Stoll formula with numerical calculations. Solid curves arc from the Stoll formula. Dots are results of numerical calculations. Periodicities (a) a- 8.1.5 A, (b) a = 12.23 A, corresponding to the case of Au(110)-2X1 and Au(T10)-3Xl, re.spectively. (Reproduced from Stoll, 1984, with permission.)...

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. 

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