Scanning tunneling microscopy comes in several flavors; vanilla (traditional STM) is constant current where the tip height is varied to maintain a constant current at a set constant bias1 and the signal controlling the tip height is reported, and neapolitan is Scanning tunneling spectroscopy or dI/dV which can be implemented in various ways but usually uses a high frequency AC component to the bias above the frequency response of the feedback loop to "dither" and sense the differential conductivity at some average bias.
This answer to Quantifying electronic overlap? briefly mentions in passing quantum ESPRESSO and CP2K in the context of STM, and this answer to About the surface Greens function method for calculating the surface state briefly mentions in passing the open-source package WannierTools in the context of STM as well, but as far as I can tell this issue hasn't really been addressed in this site yet.
- How can scanning tunneling microscopy images (constant current and dI/dV conductivity) be simulated? I see that the linked Wikipedia articles contain some integrals, but in practice this is a complicate problem involving DOS of both the tip material and sample material, tunneling geometry and matrix elements for coupling into/out of (or simply between) those states. Has this ever actually been worked out in detail? (Assume DOS has previously been calculated, I'm only asking about simulation of the tunneling data.)
- What's under the hood in software packages that do this (well)? How far do they go down the rabbit hole of modeling this complex process?
1Tip bias can be positive to sample (tunnel electrons from) filled states, or negative to sample (tunnel electrons into) empty states.