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).

Figure 1. A scanning probe over an atomically corrugated surface (top), the interaction law (Lennard-Jones force for AFM, or the exponential tunnelling decay for STM, middle), and the scan signal versus tip position (bottom). Method: closed-form Lennard-Jones potential/force and the STM tunnelling law I ~ V exp(-2 kappa d) (gate-tested sim.js), Canvas2D; the live readouts are the gap, kappa, the per-angstrom current factor, and the instantaneous signal.

mode

gap / setpoint5.00

work fn phi (eV)5.00

WHAT TO TRY

In the default constant-height mode, drag the gap: the whole current map brightens as the tip drops and goes dark as it lifts, because I falls off exponentially with the gap.
Raise the work function phi: kappa grows, the per-angstrom current factor climbs toward ten, and the atomic spots sharpen. This is why STM resolves single atoms.
Switch to constant-current to see the topograph that faithfully reproduces the surface, or to AFM to read the Lennard-Jones force curve with the operating point marked.