AFM and STM: Tip-Surface Interaction

The two workhorses of nanoscale imaging. A sharp tip scans an atomically corrugated surface. In atomic force microscopy the tip feels the Lennard-Jones interaction $V(d)=4\varepsilon[(\sigma/d)^{12}-(\sigma/d)^6]$: strongly repulsive when it touches, weakly attractive further out, zero force at $d=2^{1/6}\sigma$. In scanning tunnelling microscopy the tip draws a current that decays exponentially with the gap, $I\propto V\,e^{-2\kappa d}$ with $\kappa=\sqrt{2m\phi}/\hbar$; for a metallic work function ($\phi\sim5$ eV) that is roughly a factor of ten in current per angstrom of gap, which is why STM resolves single atoms. Pick AFM (force curve and scan), STM constant-height (atomic-contrast current map) or STM constant-current (the topograph that reproduces the surface). The tip sweeps across the lattice and the bottom panel traces the signal. Everything is closed form (gate-tested).