A theory of the energy dependent STM image of a charge density wave

In a previous paper [1] we reported an intriguing bias-dependent contrast in the STM images of 2H-NbSe2 taken at low-temperature (4.2 K). This layered material is in a charge density wave (CDW) state below 35 K, with a period nearly commensurate to three times the atomic lattice. The CDW energy gap is 2 Delta approximate to 70 meV, but still the material remains metallic, even superconducting below 7.2 K. The bias-dependent phase shift in the STM image, in particular the comparison between occupied and unoccupied states, was shown to be attributable to the CDW, as no phase shift was observed in the atomic pattern. This phase shift does not result in a contrast reversal, which could be expected by analogy to some semi-conductor band gaps. The electron versus hole distribution has not been solved for a CDW gap, which is quite complex due to the many-band situation near the Fermi level. In the present work we write a general expression for the local density of states (LDOS) due to a commensurate CDW. Its amplitude and phase can be related in a simple way to the hand structure, if one assumes an approximate form for the surface Bloch functions. We apply the method to the example of NbSe2, where its electronic structure is taken to be two-dimensional and the CDW is exactly commensurate to 3a. The new band structure and Fermi surface in the CDW state is calculated in perturbation theory, with a suitable pseudopotential. The energy-dependent contrast is due to new states on the order of E-F +/- Delta, having characteristic phases. The amplitude of the CDW is largest at the particular energies (or voltages) where tunneling occurs to the high symmetry points of the Brillouin zone. At these energies, the phase of the LDOS varies considerably, which gives a number of possible motifs in the STM image. For a shift in voltage corresponding to the energy gap edges, E = E-F +/- Delta, We find that the maxima of the corrugation are shifted along the diagonal of the conventional unit cell.