High-order Gundlach resonances at exceptional large voltages: Consequences for determining work functions

We report a combined experimental and theoretical study of Gundlach resonances U-n in scanning tunneling spectroscopy at constant current over an exceptional range of energy and number, typically tens of an eV and over thirty in order n. By performing (1) three-dimensional electrostatic calculations, (2) WKB quantum calculations of the current, and (3) one-dimensional solutions of the Schrodinger equation along the perpendicular line from the surface to the tip apex, we provide a theoretical understanding and prediction of the experimental U(n) curve. Unlike commonly assumed, the triangular potential well is not found to be a good approximation for the high-n states. We show that although the spectroscopy mode assures a constant electric field at the tip apex, this leads only for the intermediate resonance states (approximately 2 < n < 6) to reside in a linear potential between the tip and the surface. Whereas the low lying (n < 6) states all lie approximately in the same quantum well, at higher tip-sample distances d and bias voltages V(d), the quantum well is no longer triangular but attains a curvature, which is d dependent. Each high-n state resides in its own well that can be well-approximated by a polynomial of second order. Hence, the range of U-n to be analyzed in terms of spectroscopic positions needs to be chosen with great care when deducing surface work functions.