The electrons a field pulls from cold metal

The wall that is not quite a wall found that a particle meeting a barrier higher than its energy is not stopped but attenuated: its wave decays inside the barrier, and a fraction gets through that falls exponentially with the barrier’s width and with the square root of its excess height. A wall that a factor of two makes impassable applied that to alpha particles escaping nuclei, and listed among the barriers still to be treated “a barrier that a field tilts”, which is field emission, and the reason a sharp point emits electrons at a voltage a flat plate would ignore.

It was one of the first two problems quantum tunnelling ever solved. Gamow, Gurney and Condon explained alpha decay in 1928; in the same year Ralph Fowler and Lothar Nordheim explained why cold metal points give off electrons in strong fields, a puzzle that had sat unexplained since Robert Wood saw it in 1897. Their answer is still the working law of every field emitter, and the curve it gives is one of the steepest in physics.

A step tilted into a triangle #

The conduction electrons in a metal fill states up to the Fermi level, and the pressure that is not a temperature found that even at absolute zero the most energetic of them move at about a million metres a second. They do not leave because the metal’s surface is a step: an electron outside, at rest, has more energy than one at the Fermi level inside by the work function, about four and a half electronvolts for tungsten. Light can lift an electron over it — the photoelectric effect — and so can heat, at temperatures of two or three thousand kelvin.

An electric field outside the surface changes the shape of the step. Outside the metal an electron’s potential energy now falls with distance, by $eE$ per metre, so the flat step becomes a triangle: still four and a half electronvolts high at the surface, but coming down to the Fermi level a distance $\varphi/eE$ out. At two gigavolts per metre that is 2.25 nanometres. At five, it is less than a nanometre, three or four atoms’ width.

An electron at the Fermi level arriving at the surface now faces a barrier of finite width, and it tunnels through with a probability set, in the WKB approximation, by the integral of the decaying wave’s exponent across the triangle:

$$D \approx \exp\!\left(-\frac{4\sqrt{2m}\,\varphi^{3/2}}{3\hbar e E}\right).$$

The field sits in the denominator of the exponent. That is the whole character of field emission: every doubling of the field squares the transmission’s dependence on everything else, and the current is negligible until the field is a few gigavolts per metre and then enormous.

The three-halves power has a simple origin. The exponent is the integral, across the barrier, of how fast the wave decays, and the decay rate at each point goes as the square root of the barrier’s height there. Across a triangle the height falls linearly from $\varphi$ to zero over a width $\varphi/eE$, so the integral is the average of a square root, two-thirds of $\sqrt{\varphi}$, times the width: $\tfrac{2}{3}\sqrt{2m\varphi}\cdot\varphi/eE$. One power of $\varphi$ comes from the width and a half-power from the height, and the field appears only once, in the width. A rectangular barrier of the same height and width would decay faster, by the factor three-halves, because a triangle is lower on average than its peak.

That one appearance of the field is why the law is so steep. Halve the barrier’s width by doubling the field and the logarithm of the transmission halves — the transmission is square-rooted, which for a number like $10^{-60}$ is an increase by thirty orders of magnitude. The alpha particle escaping a nucleus faces a different shape, the Coulomb barrier falling as the inverse of distance, and its exponent goes as the square root of the barrier’s height over the energy; both are the same integral of a decaying wave’s rate over a forbidden region, evaluated for two different walls.

Two features of the picture deserve a moment. The field stops at the surface: the inside of a conductor holds no field, because the metal’s electrons rearrange within a fraction of an atomic spacing to cancel it, so the tilt acts only outside and the electrons inside are untouched until they arrive at the wall. And the electrons arrive constantly. A cubic metre of tungsten’s conduction electrons, moving at Fermi speeds, strikes each square metre of surface some $10^{34}$ times a second, a current of about $10^{15}$ amperes per square metre if every one got out. At four gigavolts per metre half a million amperes per square metre do, so each electron’s chance of tunnelling on a given arrival is about one in three thousand million. Field emission is not a rare process made common; it is an extremely rare process applied to an enormous number of attempts, and the exponential sets which fraction of the attempts succeeds.

Thirty decades in a factor of ten #

Summing the transmission over all the electrons in the Fermi sea that arrive at the surface, weighted by how often they arrive, Fowler and Nordheim found the current density:

$$J = \frac{aE^2}{\varphi}\exp\!\left(-\frac{b\,\varphi^{3/2}}{E}\right),$$

with $a$ and $b$ made only of the electron’s charge and mass and Planck’s constant.

For tungsten the current density is about $10^{-17}$ amperes per square metre at one gigavolt per metre, a hundredth of an ampere at two, half a million at four, and six thousand million at eight — close to thirty decades over a factor of ten in field. A current density of a million amperes per square metre, the working value of a practical emitter, needs about 4.2 gigavolts per metre.

The work function enters as its own three-halves power inside the exponent, so the surface’s chemistry matters almost as much as the field. Lanthanum hexaboride, with a work function of 2.7 electronvolts, gives two hundred thousand times as much current as tungsten at the same field, and a single layer of adsorbed atoms that changes the work function by a few tenths of an electronvolt changes the current by orders of magnitude. That sensitivity is a nuisance in a device, since gas molecules landing on an emitter make its current flicker, and a gift to anyone who wants to study surfaces.

Three ways out of a metal #

Field emission is one of three ways electrons leave a metal, and the comparison shows what each is good for. In thermionic emission the metal is heated until the fastest electrons in the thermal tail of the Fermi sea have enough energy to climb over the step. Richardson’s law gives the current density as $AT^2e^{-\varphi/kT}$, and tungsten at 2,500 kelvin delivers some thousands of amperes per square metre — the source in every vacuum tube and in the cathode-ray tubes of old televisions, and still in many electron microscopes. In photoemission each electron is lifted over the step by a single photon whose energy exceeds the work function, which is how light arrives in lumps was first established. In field emission nothing lifts the electron at all; the step is made thin enough to walk through.

The three can be combined. A Schottky emitter is a tungsten tip coated with zirconium oxide to lower its work function, heated to about 1,800 kelvin and held in a moderate field, which both lowers the top of the barrier and thins it; most of its electrons go over a barrier that the field has pulled down, rather than through one. It is less bright and less monochromatic than a cold field emitter, and far more stable, because a hot surface does not hold the adsorbed gas that makes a cold one flicker. Commercial electron microscopes mostly use it, and keep cold field emitters for the instruments that need the last factor of two in resolution.

A straight line that gives the step’s height #

Divide the current by the square of the field and take the logarithm, and the law becomes a straight line against the reciprocal of the field, with slope $-b\varphi^{3/2}$. Two things follow. The slope gives the work function without any knowledge of how much area is emitting or of the current’s absolute size, both of which are usually unknown — and the work functions of individual crystal faces of tungsten, which differ by more than an electronvolt between faces, were first measured this way. And a straight Fowler–Nordheim plot is the standard test that a current is tunnelling at all: electrons boiled over the barrier by heat, or leaking through defects, bend the line.

The plot hides a correction. A real surface’s barrier is not a sharp triangle: an electron just outside the metal is attracted back by the charge it induces in the surface, its image, which lowers and rounds the top of the barrier. The elementary law drawn here leaves that out; including it, as Nordheim and later Murphy and Good did, multiplies the current by a factor of tens to hundreds at practical fields and makes the line very slightly curved, and modern analyses fit the corrected form.

The voltage a sharp point multiplies #

Gigavolts per metre are not available between flat plates: a gap a millimetre wide would need millions of volts, and long before that it would have broken down. The way to the field is through shape.

The field at the surface of a charged conductor is greatest where the surface curves most sharply — how much charge a shape will hold found the charge crowding onto points — and at the apex of a needle of radius $r$ held at voltage $V$ it is roughly $V/5r$. A tungsten wire etched electrochemically to a point fifty nanometres in radius and held at a thousand volts sees four gigavolts per metre at its tip and emits a few nanoamperes; the same wire with a tip four times blunter sees a quarter of the field and, through the exponential, nothing measurable. The current depends on the tip’s radius far more than on the voltage, and a field emitter’s whole art is making and keeping the tip sharp and clean.

The same physics works against engineers elsewhere. The spark that needs room to start found that gas breakdown between electrodes a few micrometres apart departs from Paschen’s law because electrons are pulled straight out of the metal, and a microscopic whisker or a speck of dust on an electrode in vacuum can reach field-emission fields at voltages the gap as a whole would hold easily, starting the breakdown that limits vacuum insulation, particle accelerators’ cavities and X-ray tubes.

Particle accelerators meet the same problem on a larger scale. The radio-frequency cavities that push particles to high energy run with surface fields of tens of megavolts per metre, far below what field emission needs from a flat surface. But a polished niobium or copper surface carries microscopic protrusions and particles that multiply the local field by factors of a hundred or more, and from these points a dark current of field-emitted electrons flows each cycle, wasting power, heating the cavity and eventually triggering an arc. The fitted Fowler–Nordheim plots of such currents give enhancement factors far larger than the visible bumps would suggest, and the gradient a cavity can sustain — and so how long a collider must be — is set more by the cleanliness of its surface than by any property of the metal. Cleaning cavities with high-pressure jets of ultrapure water, to remove particles a micrometre across, raised the attainable gradients more than any change of material.

A source with one energy #

Tunnelling favours electrons near the top of the barrier so strongly that almost all the emitted current comes from a narrow band just below the Fermi level. Electrons a few tenths of an electronvolt deeper face a wider, taller barrier and are exponentially suppressed; electrons above the Fermi level hardly exist in a cold metal. The emitted electrons therefore arrive with an energy spread of a couple of hundred millielectronvolts, where a white-hot filament gives one to three electronvolts.

That matters because an electron microscope’s magnetic lenses, like glass lenses with colour, focus different energies at different places, and a spread of energies blurs the image. A cold field emitter is both the brightest electron source there is — a few nanoamperes from an area a few nanometres across, so that the beam can be focused to a spot an atom wide — and the most monochromatic. Albert Crewe’s group at Chicago used such a source in 1970 to make the first images of single heavy atoms with a scanning electron microscope, and the best transmission and scanning microscopes still use them.

Seeing the tip itself #

The electrons leaving a tip travel almost straight out along the field lines, which diverge radially from the apex, so a phosphor screen a few centimetres away shows a map of the tip’s emission magnified a million times. Erwin Müller built this field-emission microscope in 1936, and it showed the crystal faces of a tungsten tip as bright and dark regions — bright where the work function was low, dark where it was high — the Fowler–Nordheim exponent made visible across a single crystal.

Müller then reversed the polarity, made the tip positive and filled the tube with a little helium. Helium atoms drifting to the tip were ionised by the field — an electron tunnelling from the atom into the metal — preferentially above the most protruding surface atoms, and the ions flew out to the screen. In 1955 his field-ion microscope showed the individual atoms of a tungsten tip, the first images in which atoms could be seen one by one. The tunnelling microscope that the last atom does all the seeing described thirty years later is the same physics moved sideways: a field across a gap so narrow that the barrier is a few ångströms, and a current exquisitely sensitive to its width.

What the elementary law leaves out #

The figures use the elementary Fowler–Nordheim law: a triangular barrier with no image-charge rounding, a free-electron metal at zero temperature, a uniform field, and emission from a flat patch. Each of these is corrected in practice. The image charge raises the current by one to two orders of magnitude at working fields. Real metals’ electrons are not free, and their band structure changes which electrons arrive at the surface and at what angles. At high temperatures the emitted electrons come from above the Fermi level too, and field and thermal emission merge into a combined regime, used in the Schottky emitters common in commercial microscopes. The tip-field estimate $V/5r$ is a rule of thumb whose factor depends on the tip’s shape and the distance to the counter-electrode. And at high current densities the emitted electrons’ own space charge reduces the field at the surface, while the current heats the tip, so that a real emitter’s current eventually stops following the law and the tip blunts or melts.

The domain of the argument is a clean metal surface in a field of one to ten gigavolts per metre, at temperatures where few electrons are above the Fermi level, emitting currents small enough that their own charge does not matter. Within it the current is set by a tunnelling exponent in which the field and the work function appear together, and the law, with its image-charge correction, holds across many decades. Outside it — at fields above about fourteen gigavolts per metre for tungsten, where the image charge pulls the barrier’s top down to the Fermi level and electrons simply flow over it, or at currents where the tip heats — the law fails, and what replaces it is no longer tunnelling.

Still open: emission from things that are not metals #

Field emission from carbon nanotubes, graphene edges and nanostructured semiconductors behaves differently from emission from a metal tip: the emitters have few electrons, their own band structure and sometimes quantised levels, and their Fowler–Nordheim plots are often curved in ways that the metal law does not predict. Large arrays of such emitters have been proposed for flat displays, compact X-ray sources and vacuum electronics that work at high frequencies and temperatures where semiconductor devices fail, and whether their emission can be made uniform and stable enough across thousands of tips — when each tip’s current depends exponentially on its own sharpness and cleanliness — is the question that has kept most of them in the laboratory.

The law itself is the simplest tunnelling there is. A field E outside a metal tilts its work-function step φ into a triangle φ/eE wide, and electrons at the Fermi level tunnel through it at a rate falling as exp(−bφ^(3/2)/E), so tungsten’s current rises thirty decades between 1 and 10 GV/m, a sharp tip reaches those fields at a kilovolt, and the electrons leave cold with a spread of energies a tenth of a hot filament’s. Electrons leave a metal not only by climbing out but by walking through the wall, once a field has thinned it to a few atoms.