Theory of the scanning tunneling microscope

We present a theory for tunneling between a real surface and a model probe tip, applicable to the recently developed ``scanning tunneling microscope.'' The tunneling current is found to be proportional to the local density of states of the surface, at the position of the tip. The effective lateral resolution is related to the tip radius R and the vacuum gap distance d approximately as [(2 \AA{})(R+d${)]}^{1/2}$. The theory is applied to the 2\ifmmode\times\else\texttimes\fi{}1 and 3\ifmmode\times\else\texttimes\fi{}1 reconstructions of Au(110); results for the respective corrugation amplitudes and for the gap distance are all in excellent agreement with experimental results of Binnig et al. if a 9-\AA{} tip radius is assumed. In addition, a convenient approximate calculational method based on atom superposition is tested; it gives reasonable agreement with the self-consistent calculation and with experiment for Au(110). This method is used to test the structure sensitivity of the microscope. We conclude that for the Au(110) measurements the experimental ``image'' is relatively insensitive to the positions of atoms beyond the first atomic layer. Finally, tunneling to semiconductor surfaces is considered. Calculations for GaAs(110) illustrate interesting qualitative differences from tunneling to metal surfaces.