Scanning tunnelling microscopy and the LDOS

Ask what a sample state at E_F does in the gap. There the Hamiltonian is p²/2m + V₀ with V₀ − E_F = Φ, so the stationary equation reads ψ″ = κ²ψ with κ = √(2mΦ)/ħ, and the admissible solution decays: ψ ∝ e^(−κz). The current is a probability current built from two such tails, one from each electrode, so it carries the square: I ∝ e^(−2κz). Numerically κ = 0.5123 √(Φ/eV) Å⁻¹, so a 4.0 eV barrier gives κ = 1.02 Å⁻¹ and 2κ = 2.05 Å⁻¹ — the current changes by e^2.05 = 7.8, or 0.89 of a decade, per angstrom. That single number is the entire vertical performance of the microscope. A loop holding the current to 1 % holds the height to δz = 0.01/2κ = 5 × 10⁻³ Å, half a picometre, across a junction whose resistance at 1 nA and 1 V is 1 GΩ. Nothing about lateral resolution follows from it; that is a separate and much weaker argument.

You cannot solve the tip-plus-sample Hamiltonian, and Bardeen's move is not to try. Take Hₜ and Hₛ separately, each with a complete set of eigenstates, and treat the overlap of their tails as a perturbation transferring an electron from |μ⟩ in the tip to |ν⟩ in the sample. Fermi's golden rule gives a rate ∝ |M_μν|² δ(Eμ − Eν), and summing over the window the bias opens gives I = (4πe/ħ) ∫₀^(eV) ρₜ(E_F − eV + ε) ρₛ(E_F + ε)|M|² dε at low temperature. The matrix element is not a potential integral but a surface integral of a current-like operator, M = (ħ²/2m)∫_S(ψμ*∇ψν − ψν∇ψμ*)⋅dS, over any surface lying wholly inside the gap; that the answer does not depend on which surface is the statement that no current is lost in the vacuum. Three assumptions are now fixed and stay fixed: the coupling is weak enough for first order, the transfer is elastic, and each electrode is the one you solved, unperturbed by the other.

Bardeen leaves M depending on the tip you happen to have, which is the one thing an experiment does not know. Tersoff and Hamann close that gap by modelling the apex as a single s-wave orbital of radius R centred at r₀, so ψμ in the vacuum is a spherical decaying wave. Evaluating the surface integral with that choice collapses |M|² onto |ψᵥ(r₀)|², and the current becomes I ∝ V ρₜ(E_F) R² e^(2κR) Σᵥ |ψᵥ(r₀)|² δ(Eᵥ − E_F). Every tip property is now a multiplicative constant, and the only laterally varying factor is the sample's local density of states at the Fermi level evaluated at the tip's centre of curvature, ρₛ(r₀, E_F). Holding I fixed while scanning therefore traces a surface of constant ρₛ(r₀, E_F), at a point R + z out in the vacuum where that quantity is exponentially small and keeps only a smoothed, indirect memory of where the nuclei sit.

Constant current means ρₛ(r₀, E_F) is held fixed, and that is ρ e^(−2κz) with ρ the Fermi-level LDOS referred to the surface, so between two sites the tip moves by Δz = ln(ρ₁/ρ₂)/2κ and by nothing else. At Φ = 4.0 eV that turns a factor of 2 in LDOS into 34 pm of apparent height, a factor of 3 into 54 pm, a factor of 10 into 112 pm. None of it is geometry, and the consequences are routine and counter-intuitive. CO on Cu(111) carries its carbon roughly 1.9 Å above the metal plane and still images as a depression at small bias, because it pushes the metal's Fermi-level LDOS down; oxygen on Ni and on Al does the same. Subsurface impurities several layers down can appear, because they modulate ρₛ(E_F) in the vacuum above them. Change the bias and the picture changes with it, since the integration window changes — which is the basis of tunnelling spectroscopy, where dI/dV at fixed z tracks ρₛ(E_F + eV) up to a transmission factor usually divided out as (dI/dV)/(I/V).

Decompose each Fermi-level sample state into lateral Fourier components. A component with in-plane wavevector G obeys the same vacuum equation, so its amplitude decays not with κ but with κ_G = √(κ² + G²). For a state near k∥ = 0 the LDOS corrugation is the cross term of that component with the uniform one, falling as exp[−(κ + κ_G)z] against e^(−2κz) for the mean, so relative to the mean it is suppressed by exp[−(κ_G − κ)z], which for small G is e^{(−G²z/2κ)}. That is a Gaussian low-pass, and with the LDOS sampled at the tip centre, R + z out, the image is the surface LDOS convolved with a real-space Gaussian of width σ = √((R + z)/κ). With R = 5 Å, z = 5 Å and κ = 1.02 Å⁻¹ that is σ = 3.1 Å, wider than the 2.6–2.9 Å atomic spacing of close-packed Cu, Al or Au — Tersoff–Hamann says you should not resolve atoms on Al(111) at all. They are resolved, and there the s-wave tip is what fails: a p_z or d_z² apex reweights the high-G components (Chen's derivative rule). Graphite fails differently. Its hexagonal lattice, a = 2.46 Å, has G = 4π/(√3 a) = 2.95 Å⁻¹; at Φ = 4.5 eV, κ_G − κ = 2.06 Å⁻¹, so even a point tip 5 Å out sees the corrugation suppressed by 3.4 × 10⁻⁵, and a 60 % surface modulation becomes 0.002 pm. Experiments see 0.5 to 3 Å. The exponential is not what fails; the k∥ = 0 assumption is. Graphite's Fermi-level states sit at the zone corner, where the three leading Fourier components, one at each equivalent corner, have the same |k∥| and so the same decay constant, and the corrugation they make does not decay at all.

Weak coupling: below about 100 kΩ of junction resistance the electrodes hybridise, first order loses its small parameter, and the measured d ln I/dz collapses from 2κ towards zero — the apparent barrier height falls from 4 eV to a fraction of an eV as the gap closes past about 3 Å. Elastic transfer: inelastic channels open at thresholds set by vibrational modes, each adding a step to dI/dV, and the peaks those steps make in d²I/dV² are the signal in inelastic tunnelling spectroscopy. A structureless tip: real apices carry their own resonances, and a tip state at E_F prints itself onto every image. Small bias: at 1 V and above the barrier is no longer rectangular and ρₜ can no longer be pulled outside the integral. No forces: chemical and van der Waals forces between apex and surface reach the nanonewton scale and elastically relax both electrodes, which is what AFM measures and STM ignores. The instrument still works; what changes is what the picture is a picture of.