Evaluation of scanning tunneling spectroscopy data : Approaching a quantitative determination of the electronic density of states

We introduce a method for recovering the electronic density of states (DOS) from scanning tunneling spectroscopy data. For this purpose, starting from one-dimensional WKB approximation, expressions are derived allowing the reconstruction of the DOS from a measured tunneling current ($I\text{\ensuremath{-}}V$ curve) and its derivative with respect to the tunneling voltage, $V$. In a first step, assuming a constant DOS for the tip, $I\text{\ensuremath{-}}V$ curves are calculated for various model DOS of the sample and the derived expressions are applied to recover the sample DOS. It turns out that in this way the original DOS can be recovered to an accuracy of some percent in the energy range $\ifmmode\pm\else\textpm\fi{}2\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. In a second step, we rewrite the differential conductivity of the tunnel junction to form a Volterra integral equation of the second kind and, consequently, exploit the Neumann approximation scheme to optimize the recovered DOS for a wide class of original DOS to an unprecedented accuracy. In a third step, an energy-dependent DOS of the tip is included resulting in two Volterra integral equations, one for the sample and one for the tip DOS, allowing alternately optimizing the DOS of either side. By analyzing the distance dependence of spectroscopic data, i.e., the energy-resolved differential barrier height, we obtain additional information on the DOS which enables a self-consistent solution of the two integral equations and, thus, to deconvolute the sample and tip DOS.