Spherical tip model in the theory of the scanning tunneling microscope

Abstract

The tunneling current contributed by angular momentum states, l >0, is obtained from a spherical tip model of the scanning tunneling microscope. It is found that for this model, the energy of the bound state depends upon the quantum number n, l , and the tip radius R . Here n is approximately the number of zeros of the l th spherical Bessel function j l inside the well. Since degeneracy is not, in general, present in the spherical square-well model, the dominant contribution to the tunneling current is due to a single state at the Fermi level. However, it is found that accidental degeneracy, e.g., between the s- and d-states for n ≥3 has to be considered. In this case, the d-state contribution is approximately 10% of the s-state contribution to the current. For this model, the p-state energy levels are usually well separated from the s-state level, though they might be accidentally (nearly) degenerate with f-levels. However, for R 20 A, the energy separation, ∼ ( h 2 /2m ) ( π/2R ) 2 , between the s- and p-states is comparable to k B T at room temperature. Therefore, when T ≈300 K, both the s- and p-state contributions must be included. Finally, analysis of the lateral resolution suggests that the smallest resolvable wavelength of the corrugation of the sample surface is [4 ln 2( R+d )/ K )] 1/2 , which is larger by the factor of √¯2 In 2 than the value quoted by Tersoff and Hamann.