Voltage-dependent STM image of a charge density wave

Abstract

In the present work we write a general expression for the local density of states (LDOS) due to a commensurate charge density wave (CDW). The main goal is to investigate the voltage dependence of the contrast in the scanning tunneling microscope (STM) images of materials showing CDW's. For layered materials, having nearly two-dimensional electronic structures, the problem is complicated by the many-band situation near the Fermi level, and by the incomplete band gapping. Nevertheless, a simple perturbation method allows one to relate the amplitude and phase of the CDW to features of the band structure. We emphasize the role of particular characteristic energies, at which the CDW has a large contribution from special k points of the surface Brillouin zone, leading to different modulations in the STM image. In a second part of the paper, we consider the voltage-dependent contrast of ${\mathrm{NbSe}}_{2}.$ For this material, we find that the amplitude and the phase of the CDW change significantly as a function of energy (or voltage), resulting in a number of different possible motifs. For example, the direct comparison between occupied and empty states reveals that new states on the order of ${E}_{F}\ifmmode\pm\else\textpm\fi{}\ensuremath{\Delta},$ giving a dominant contribution to the LDOS, have different phases. As a result, in the corresponding STM images, the maxima of the corrugation have shifted positions along the diagonal of the conventional unit cell.