= Current-distance test of vacuum tunnelling
<Tunneling through a controllable vacuum gap by Gerd Binnig and others> made barrier width experimentally adjustable. Their apparatus used a pointed <electrical conductor> opposite a conducting plate, with fine positioning by <piezoelectricity>. Rubber-tube support and magnetic levitation above a cooled bowl exhibiting <superconductivity> suppressed vibrations; the junction itself remained near room temperature.<Tunneling through a controllable vacuum gap by Gerd Binnig and others>[p. 178, Fig. 1: tungsten tip, platinum plate, piezodrive and helium-cooled superconducting lead bowl]{ref}
The readout was <electrical resistance> versus displacement at a fixed <electric potential difference> of 60 mV of either polarity. They observed an <exponential function> dependence spanning four orders of magnitude. Cleaning and better <vacuum> increased the inferred barrier from approximately 0.6-0.7 eV to 3.2 eV. Reproducible curves and the cleaning response argued against deforming contamination layers; an appropriate slope also helped exclude an ordinary conducting contact. The paper reports residual contamination, electronics-limited range and a jump attributed to release of thermal stress; its curve origins generally do not determine absolute gap widths.<Tunneling through a controllable vacuum gap by Gerd Binnig and others>[p. 179, equation (1), Fig. 2 and accompanying control/limitation discussion]{ref}
The ideal <tunnel junction> model explains the slope. At fixed small <electric potential difference>, fixed tip geometry and approximately unchanged <density of states>, take an effective barrier <energy> $\Phi$ above the relevant <Electron> states and approximate the current magnitude by
$$
I(s)\simeq C\exp(-2\kappa s),\qquad
\kappa=\frac{\sqrt{2m_e\Phi}}{\hbar},\qquad
-\frac{d\ln I}{ds}\simeq2\kappa.
$$
The coefficient $C$ includes the applied <electric potential difference>, available occupied and empty <quantum states>, and tip geometry. A single <transmission probability> is not an <electric current>: a junction combines contributions from many <Electron> states in both directions, with the <electric potential difference> producing a net imbalance. Its tilted, material-dependent barrier is only approximated by the equal-exterior <rectangular potential barrier>; the exact single-particle formula above is not a complete junction model.
For the chosen model value $\Phi=4$ <electronvolts>, $\kappa\simeq10.25\,\mathrm{nm}^{-1}$. Increasing separation by $0.1\,\mathrm{nm}$ multiplies $I$ by about $0.129$, a reduction by about $7.8$. The following diagram chooses $I(0.4\,\mathrm{nm})=1\,\mathrm{nA}$ solely for illustration; it contains no measured data and is not a fit to the 1982 curves.
\Image[/vacuum-tunnelling-current-distance.png]
{title=Vacuum-junction controls and the exponential tunnelling-current distance law}
{description=Original <tunnel junction> schematic and illustrative <electric current> versus distance model at effective barrier <energy> four <electronvolts>. The <electric current> scale is chosen, the <logarithm> axis slope is calculated, and the apparatus is not drawn to scale. Distance sweeps and constant-current imaging are distinct operating procedures.}
Distance calibration is essential. In <Scanning Tunneling Microscopy: From Birth to Adolescence by Gerd Binnig and others>, the authors describe an initial inferred barrier near 1 eV. Recalibration found only half the quoted piezo displacement sensitivity and raised the inferred barrier above 4 eV. Thus an exponential trace alone was insufficient evidence for tunnelling through <vacuum>. The model makes the sensitivity explicit: the inferred $\Phi=\hbar^2(d\ln I/ds)^2/(8m_e)$ depends quadratically on the distance scale.<Scanning Tunneling Microscopy: From Birth to Adolescence by Gerd Binnig and others>[p. 397, piezo-sensitivity calibration discussion]{ref}
For <scanning tunnelling microscopy>, <feedback control> adjusts the vertical position to hold <electric current> fixed during a lateral scan; the recorded height follows separation only when the surface is electronically homogeneous. Changes in the local <density of states> can also change apparent height. This limitation is already explicit in the authors' account of constant-current imaging.<Scanning Tunneling Microscopy: From Birth to Adolescence by Gerd Binnig and others>[p. 391, Fig. 2, joint local density of states; p. 396, Fig. 5(a), electronically homogeneous surface condition]{ref}
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