= Solution
In <scanning tunneling microscopy>, $I\propto e^{-2\kappa z}\int_{E_F}^{E_F+eV}\rho_s(\mathbf r,E)\,dE$ under simplified constant-tip-density assumptions, with $\kappa\simeq\sqrt{2m\phi}/\hbar$. A constant-current height map mixes actual surface height with local electronic density, barrier/work-function changes and tip structure. Changing bias can change contrast without moving atoms. <Scanning tunneling spectroscopy> instead records <energy> dependence, often through $dI/dV$; tip states, matrix elements and thermal or modulation broadening still affect its interpretation.
In <atomic force microscopy>, contrast comes from tip-sample <forces> or their gradients: short-range chemical bonding and repulsion, dispersion <forces>, electrostatics and sometimes <magnetism> or capillary <forces>. Contact deflection, oscillation amplitude and <frequency> shift are different observables. A force-gradient image can therefore differ from a geometrical-height image, and changing the tip termination can strongly change atomic contrast.
STM's exponential distance sensitivity can give excellent vertical sensitivity; usable resolution also depends on current noise, feedback, vibration and drift. Lateral contrast depends on the apex orbitals and electronic spatial extent, not merely the nominal tip radius. In AFM, broad-tip geometrical convolution broadens larger features; a sharp terminal atom and short-range <forces> can produce much finer atomic contrast. <Cantilever> stiffness, amplitude, <frequency> noise, <force> range, <thermal fluctuations> and feedback influence height sensitivity. In both techniques, detecting a small displacement is distinct from resolving two neighboring objects.
An example is the <Cu(111) surface-state standing-wave experiment>. Low-temperature STM and spatially resolved spectroscopy connect atomic steps or defects in a surface map with surrounding energy-dependent electronic <interference> patterns. Their period changes with bias, whereas a fixed structural corrugation would not acquire that electronic dispersion. The result shows why spectroscopic maps reveal surface-electron scattering beyond what a height image alone identifies.<Imaging standing waves in a two-dimensional electron gas by M. F. Crommie and others>[1993, observation of Cu(111) local-density-of-states standing waves]{ref} The apparent fringes are not a second set of equally spaced surface atoms.
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