Illustrated Physics
physics, drawn
Concept

Work function — where it appears

The energy needed to remove an electron from a metal into the vacuum outside it, a few electronvolts for every clean surface. It sets the barrier height in a tunnelling junction and therefore the rate at which the current falls with the gap — about a decade for every ångström at four and a half electronvolts.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Photocurrent against applied voltage, at two intensities. Photocurrent against retarding voltage for caesium lit at 12×10¹⁴ hertz, at relative intensities of 1 and 2. Both curves reach zero at the same stopping voltage of 2.82 volts, because the brighter light delivers more photons and not more energetic ones. The saturation currents are in the ratio of the intensities. The slope between the stopping voltage and zero assumes the emitted electrons' energies are spread uniformly below the maximum, which they are not; the crossing point does not depend on that assumption.

Light arrives in lumps, and brightness only changes how many

Shine dim blue light on a metal and electrons come out. Shine intense red light and none do, however long the wait. The frequency decides whether anything happens; the intensity decides only how much.

quantum · Photon A current that falls by a decade for every ångström. Tunnelling current against the width of a vacuum gap, on a logarithmic scale, for three work functions covering the range of clean metal surfaces. The curves are straight because the transmission is an exponential in the gap, and their slopes are 0.833, 0.944, 1.044 decades per ångström — fitted to the drawn curves and agreeing with 2κ/ln 10 to a part in a million. At 4.5 electronvolts a change of ten picometres, a tenth of an atomic radius, changes the current by 24 per cent. That is the sensitivity a scanning tunnelling microscope lives on, and it is why the instrument measures height by holding the current fixed and recording what the piezo had to do: the current is far too steep a function of height to be read as one.

The last atom does all the seeing

A tunnelling rate falls by a factor of eight for every tenth of a nanometre of extra barrier, which is normally quoted as the reason nothing ever tunnels anywhere. Read the other way it is a microscope: the second-nearest atom of a blunt metal tip carries a two-hundredth of the current the nearest one does, so a tip nobody sharpened resolves a single atom, and the resolution comes from an exponential rather than from any piece of engineering.

quantum · Tunnelling A barrier the metal was supposed to choose. The barrier height at a metal contact on n-type silicon against the metal's work function, from Bardeen's model with interface-state densities of none, 10¹² per cm² per eV, 10¹³ per cm² per eV, 10¹⁴ per cm² per eV, a 0.5 nm gap and a neutrality level a third of the gap above the valence band. With no interface states the barrier is Schottky and Mott's φₘ − χ, rising one for one with the work function: from aluminium to platinum, 0.23 to 1.60 eV. With 10¹² per cm² per eV the slope is 0.92 and the same metals give 0.27 to 1.53 eV; with 10¹³ per cm² per eV the slope is 0.53 and the same metals give 0.48 to 1.20 eV; with 10¹⁴ per cm² per eV the slope is 0.10 and the same metals give 0.71 to 0.84 eV. Dense enough interface states pin every barrier near 0.76 eV, and the metal hardly matters.

The barrier the metal cannot choose

Put a metal on a semiconductor and a barrier forms at the interface. The obvious theory says its height is the difference between the metal's work function and the semiconductor's electron affinity. Platinum and aluminium differ by nearly an electronvolt and a half in work function, so they should make wildly different contacts. On silicon and gallium arsenide they do not: nearly every metal gives nearly the same barrier. Bardeen explained why in 1947. A thin layer of electronic states at the interface screens the metal, and it takes only a few such states per hundred surface atoms to pin the barrier wherever they put it.

quantum · Bands A current that grows thirty decades with the field. The current density of electrons tunnelling out of a cold metal against the field at its surface, on a logarithmic scale, from the Fowler–Nordheim law, for work functions of 2.7 eV (lanthanum hexaboride), 4.5 eV (tungsten) and 5.6 eV (a surface barely willing to let electrons go). For tungsten the current is 0.009 A/m² at 2 GV/m, 5·10⁵ at 4 GV/m and 6·10⁹ at 8 GV/m — rising by more than twenty orders of magnitude between 1 and 4 GV/m. The work function sits in the exponent as a three-halves power, so lowering it from 4.5 to 2.7 eV raises the current at 3 GV/m by a factor of 2·10⁵. A current density of a million amperes per square metre, an ordinary working value for a field emitter, needs about 4.2 GV/m on tungsten.

The electrons a field pulls from cold metal

A metal holds its electrons behind a step a few electronvolts high, and to get them out the usual way is to heat the metal white-hot until some climb over it. Put a strong enough field outside instead and the step tilts into a triangle a couple of nanometres wide, and electrons leave a cold metal by tunnelling through it. The current grows thirty decades as the field rises tenfold, a sharp needle at a thousand volts reaches the field that a flat plate would need a million volts for, and the electrons that come out all have nearly the same energy — which is why the brightest, sharpest electron microscopes are fed from a single etched tungsten point.

quantum · Tunnelling

Named alongside it

The objects these essays reach for when they reach for this one.

Fermi levelTunnellingBand bendingBarrierBlackbodyDecay lengthDensity of statesDepletion regionElectric fieldElectron microscopeEnergy conservationEvanescent wave

All concepts