Deconvolution of the electronic density of states of tip and sample from scanning tunneling spectroscopy data: Proof of principle

It is demonstrated that deconvolution of the density of states (DOS) of tip and sample from scanning tunneling spectroscopy data is possible within the framework of a one-dimensional Wentzel-Kramers-Brillouin approximation if additional information such as data sets taken at two sufficiently different tip-sample separations is provided. The basic concept is to convert the underlying integral equation for the tunneling current by differentiation with respect to the sample bias (first set) and, in addition, with respect to the tip-sample separation (second set) into two sets of Volterra integral equations of the second kind with two equations for the tip and another two for the sample DOS. Though these integro-differential equations can in principle be solved numerically employing the Neumann approximation scheme, it turns out in practice that suitable iteration schemes have to be found to guarantee stable solutions. Employing tunneling data taken at two sufficiently different tip-sample separations, it is demonstrated that iterating suitably through the system of equations results in a recovery and deconvolution of the tip and sample DOS. The underlying formalism is derived, examples are given and limitations discussed. Finally, we apply an adapted procedure to experimental data obtained on Nb (110) and compare the deconvolved sample DOS with density-functional theory data.