Tunneling Matrix Elements

This chapter presents systematic methods to evaluate the tunneling matrix elements in the Bardeen tunneling theory. A key problem in applying the Bardeen tunneling theory to STM is the evaluation of the tunneling matrix elements, which is a surface integral of the wavefunctions of the tip and the sample, roughly in the middle of the tunneling gap. By expanding the tip wavefunction in terms of spherical harmonics and spherical modified Bessel functions, very simple analytic expressions for the tunneling matrix elements are derived: the tunneling matrix elements are proportional to the amplitudes or the corresponding x-, y-, or z-derivatives of the sample wavefunction at the center of the tip. Two proofs are presented. The first proof is based on the Green’s function of the Schrödinger’s equation in vacuum. The second proof is based on a power-series expansion of the tip wavefunctions.