The wall a plasma builds against itself

A plasma screens out any field imposed on it over a distance of a Debye length, which is why the long-range Coulomb force does not reach and why a plasma can be treated as neutral at every scale larger than that.

Every use of a plasma involves a boundary. It is in a vessel, or it is being probed, or it is depositing ions on a wafer, or it is the solar wind meeting a spacecraft. And a boundary is exactly where the screening argument stops working, because there is nothing on the other side to screen with.

Why the wall goes negative #

At the same temperature, electrons move faster than ions by the square root of the mass ratio — a factor of $43$ for hydrogen and $271$ for xenon.

A surface introduced into a plasma is therefore struck by far more electrons than ions, and it charges negative — the surface charge sitting on it like any other. As it does, its potential begins to repel electrons: the flux arriving is cut by the Boltzmann factor of the potential, while the ions, being attracted, arrive at whatever rate the plasma delivers them. The charging stops when the two fluxes are equal.

Setting the two equal and solving gives the floating potential:

$$\frac{e|\phi_f|}{kT_e} = \frac{1}{2}\ln\!\left(\frac{M}{2\pi m}\right),$$

which is $2.84$ for hydrogen, $3.7$ for helium, $4.68$ for argon and $5.28$ for xenon.

Three features of that result are worth noticing. It depends on nothing but the mass ratio — not on the density, not on the plasma’s size, not on what the wall is made of. It is logarithmic in the mass ratio, which is why four decades of ion mass move it by less than a factor of two. And it is measured in electron temperatures, so a $3$ eV plasma floats its walls about $9$ volts negative and a fusion-edge plasma at $100$ eV floats them nearly $300$ volts negative, with the same arithmetic.

The generator finds the number by bisecting on the flux balance, writing each flux from its own definition, and checks it against the closed form — two routes to one potential.

The layer that holds it #

A potential drop requires a space charge, and here the space charge is a genuine departure from neutrality.

Inside the sheath the electron density falls as the Boltzmann factor of the potential, which is steep. The ion density falls too, but only as the ions speed up — flux conservation means a faster stream is a thinner one — and that is a much weaker dependence. The two therefore separate, and by the wall the ion density exceeds the electron density by a large fraction of itself.

Solving for the profile means solving Poisson’s equation, which is the one place in plasma physics where the usual substitution of $n_e = n_i$ is not allowed. In the standard normalisation, with $\chi$ the potential measured downward in electron temperatures and $\xi$ the distance in Debye lengths,

$$\frac{\mathrm{d}^2\chi}{\mathrm{d}\xi^2} = \left(1 + \frac{2\chi}{M^2}\right)^{-1/2} - e^{-\chi},$$

where $M$ is the ions’ entry speed in units of the ion sound speed. Integrating it gives the profile drawn, and the layer comes out about fifteen Debye lengths thick — an exponential decay is what a screened field does, and here the decay has run out of room — which for a laboratory discharge is a millimetre or two and for the solar wind at a spacecraft is metres.

Why the departure from neutrality is allowed here #

It is worth being explicit about why quasineutrality is a good approximation nearly everywhere and a bad one in the sheath, because the reason is quantitative and not a matter of taste.

Poisson’s equation says that a charge imbalance $\Delta n$ produces a potential varying over a length $L$ with $\Delta n/n \sim (\lambda_D/L)^2$ times the potential in units of the electron temperature. Over a length much larger than the Debye length that ratio is tiny, so a potential of a few $kT_e/e$ requires an imbalance too small to notice — which is what quasineutrality means and why it is safe.

Over a length comparable with the Debye length the ratio is of order one, and holding a potential of a few $kT_e/e$ requires an imbalance of the same order as the density itself. The sheath is exactly that case, and the imbalance it needs is not a correction.

So the two statements — “a plasma is neutral” and “the sheath is not” — are the same equation evaluated at two length scales. There is no conflict and no separate physics; there is one Poisson equation and a ratio of lengths that decides which term dominates.

The ions have to arrive already moving #

The equation has a condition attached that is not obvious and is the subject’s one genuinely surprising result.

Expand both densities for small $\chi$. The ion term gives $1 - \chi/M^2$ and the electron term gives $1 - \chi$, so the curvature near the sheath edge is proportional to $(1/M^2 - 1)\chi$. For $M > 1$ that coefficient is negative in the right way and the solution grows monotonically; for $M < 1$ it changes sign and the solution oscillates back through zero.

An oscillating potential is not a sheath. So a steady sheath exists only if the ions enter it at or above the ion sound speed $\sqrt{kT_e/M}$ — the Bohm criterion, from 1949 — and the threshold is exactly Mach one, with no adjustable numbers in it.

That is a strange requirement, because the ions in the bulk plasma are cold and slow. Something must accelerate them before they reach the sheath, and what does is a much wider and much gentler region called the presheath, in which quasineutrality still holds approximately and a weak field does the work. The potential drop across it is about half an electron temperature — just enough to reach the sound speed — which is why the plasma density at the sheath edge is $e^{-1/2} = 0.61$ of the bulk value rather than equal to it.

Every quantitative statement about a plasma-wall interaction carries that factor of $0.61$, and it comes from a stability condition on a differential equation.

What the criterion costs the plasma #

The presheath is a small effect with a long reach, and it is worth following.

Accelerating the ions to the sound speed requires a potential drop of about $kT_e/2e$, which by the Boltzmann relation reduces the electron density — and hence the plasma density — at the sheath edge to $e^{-1/2}$ of its bulk value. That is $0.61$, and it appears in every flux to every surface.

The presheath is not thin. It has to be quasineutral, so its length is set by whatever produces the gentle field within it — ionisation, collisions, or the geometry of the plasma itself — and it is typically comparable with the size of the whole discharge. A sheath is millimetres and a presheath is centimetres, and the interesting consequence is that the wall’s influence extends much further into the plasma than the layer where neutrality fails.

The bookkeeping is also where a beginner’s error lives. Computing the ion flux to a wall from the bulk density and the Bohm speed overestimates it by $1/0.61$, which is a $64$ per cent error in a quantity that decides an etch rate or a heat load. The factor is not decoration.

One curve, three measurements #

The theory above is a theory of the commonest diagnostic in plasma physics.

Put a small wire into a plasma, bias it, and measure the current. Three regions and three numbers.

Far negative, every electron is turned back and the probe collects only ions. The current saturates at $0.61\,n e u_B A$ with $u_B$ the Bohm speed, which gives the plasma density once the electron temperature is known.

Through the middle, the electron current is the Boltzmann factor of the probe potential, so the logarithm of the electron current against voltage is a straight line whose slope is $e/kT_e$. That is how an electron temperature is measured — by a line on a semi-logarithmic plot, with no thermometer and nothing in thermal contact with anything — the Boltzmann factor read backwards.

And the zero crossing is the floating potential, which is what an insulated surface in this plasma would sit at.

Langmuir built this in 1923, along with most of the vocabulary — he coined “plasma” for the ionised gas and “sheath” for the layer — and the analysis has not changed. It remains the standard diagnostic because it is cheap, local and quantitative, and because a plasma’s own oscillation reports only an average along a line of sight, and because the alternative to a wire in the plasma is usually an optical measurement that averages along a line of sight.

The electron side of the balance is a random flux to a surface — a quarter of the density times the mean speed — and that is the same integral that gives an effusion rate through a hole. It is computed from a thermal distribution of speeds and it is enormous compared with the ion flux, because electrons at the same temperature move about forty times faster than protons and rather more than that in a heavier gas. The whole sheath exists to bring those two fluxes into equality.

Driving the wall, which is an industry #

A floating wall sits a few volts negative and its sheath is a few Debye lengths thick. Driving it hard changes both dramatically.

Hold the wall hundreds of volts negative and the electrons are excluded entirely; the layer is then a pure ion space charge, the current across it is space-charge limited, and the thickness follows Child’s law — proportional to the three-quarter power of the voltage.

That is what a plasma etcher does. A wafer sits on a driven electrode, the sheath above it holds a few hundred volts, and ions crossing it are accelerated to that energy in a direction perpendicular to the surface, because the sheath’s field is perpendicular. They arrive with a well-defined energy and a well-collimated direction, and they etch a trench with vertical walls.

Every feature of that process is a statement about the sheath. The etch is anisotropic because the sheath’s field has a direction; the ion energy is set by the sheath potential; the collimation is spoiled if the sheath is thick enough for the ions to collide while crossing it. The whole of semiconductor patterning below a micrometre depends on the layer in the opening figure.

The same layer, three sizes #

The arithmetic is scale-free in the sense that everything is measured in Debye lengths and electron temperatures, so the same picture describes systems that share nothing else.

A laboratory discharge at $10^{16}$ per cubic metre and $3$ eV has a Debye length of $129$ micrometres and a sheath of a couple of millimetres, holding about nine volts. It is visible: the dark space next to an electrode in a glow discharge is the sheath, dark because the electrons there have been repelled and there is nothing to excite the gas.

A spacecraft in the solar wind sits in a plasma of about $5\times10^6$ per cubic metre at $10$ eV, giving a Debye length of some ten metres. Its sheath is therefore tens of metres across — larger than the spacecraft — and the vehicle floats a few tens of volts negative in shadow. Differential charging between shadowed and sunlit surfaces, where photoemission pushes the sunlit side positive, is a leading cause of spacecraft anomalies, and it is this essay’s balance with one more current in it.

A fusion divertor runs at $10^{20}$ per cubic metre and $10$–$100$ eV, giving a Debye length of micrometres and a sheath thinner than a hair, holding a few hundred volts. The heat it delivers to the target is the ion flux times the sheath potential plus the thermal energy, and designing a divertor is largely a matter of making that product survivable.

Nine orders of magnitude in density and four in temperature, and the same solved profile with the axes relabelled.

Where it stops #

The sheath is assumed collisionless. Ions must cross it without hitting anything, which requires the layer to be thinner than an ion mean free path. In a low-pressure discharge that holds comfortably; raise the pressure and ions arrive with a spread of energies and directions, and the anisotropy of an etch degrades — which is why plasma processing runs at millitorr rather than at atmospheric pressure.

Secondary electron emission can change the sign of everything. An energetic ion or electron striking a surface can knock electrons out of it, and those leave the surface and enter the plasma — a current in the opposite direction to the one they are usually counted in. If the yield exceeds one, the surface can charge positive, and the whole balance above inverts. Dust grains in space and the sunlit side of a spacecraft routinely float positive for exactly this reason, photoemission being another current in the same balance.

A magnetic field changes the geometry. Where the field meets the wall at an angle, electrons are tied to field lines and cannot cross to the wall freely, so the flux balance is modified and a magnetic presheath appears in front of the electrostatic one. That is the standard situation in a tokamak divertor and it makes the simple picture here a first approximation.

And the electrons are assumed Maxwellian. The Boltzmann factor for the electron flux presumes a thermal distribution, and in a low-pressure discharge the electrons are often not thermal — they are heated in one place and lost in another, with a depleted tail. Since it is the tail that reaches a repelling wall, a probe measurement of “the electron temperature” is a measurement of the tail rather than of the bulk, and a plasma can honestly be said to have two.

A metal screens a field in the same spirit and by a quite different mechanism. Its conduction electrons are already there and merely rearrange, so the screening length is a fraction of an ångström and the interior is field-free almost exactly. A plasma’s electrons have to be pushed out of the way, which costs energy and leaves a region depleted rather than rearranged — so the screening length is the Debye length, micrometres to millimetres, and the screening is never complete.

What the profile does not settle #

The solved layer is a one-dimensional, steady, collisionless, unmagnetised, Maxwellian idealisation, and it is worth saying which of those absences bite hardest.

The one-dimensionality is usually harmless: a sheath is thin compared with the surface it sits on, so a flat-plate treatment applies almost everywhere except at edges and around small objects. A probe is a small object, which is why probe theory beyond the simplest case is a subject in itself and why probe radii are chosen to be large compared with the Debye length when they can be.

The steadiness is not harmless at all. Most industrial plasmas are driven at radio frequency, and the sheath in front of an electrode expands and collapses each cycle. Ions, being heavy, cannot follow — they respond to the time-averaged field — while electrons follow easily, so the sheath rectifies: an electrode driven with a symmetric waveform develops a large negative self-bias, sometimes hundreds of volts, with no direct current supplied anywhere. That effect has no counterpart in the steady analysis above and is the basis of how a wafer is biased in practice.

And the ion energy distribution arriving at the wall is not a single value even in the collisionless steady case, because ions entering the sheath have a spread of speeds from the presheath. The width of that distribution is what decides how vertical an etched wall really is, and computing it requires the kinetic problem rather than the fluid one solved here.

Two words Langmuir chose #

The vocabulary is worth a paragraph because it was invented for this problem and has since been applied to a quarter of the visible universe.

“Plasma” arrived in 1928, borrowed from the blood plasma that carries corpuscles about, for the ionised gas in a discharge tube — Langmuir’s point being that the ionised medium carried electrons and ions in the way a fluid carries what is suspended in it. The name stuck, and it is now applied to the solar corona, the interstellar medium, a fusion device and a fluorescent tube, none of which Langmuir had in mind.

“Sheath” arrived earlier, in 1923, and was chosen because the layer wraps a surface the way a sheath wraps a blade. That name is the more accurate of the two: it says that the layer belongs to the object rather than to the plasma, which is exactly right — every object in a plasma has one, its thickness is set by the plasma and its potential by the object, and the plasma proper begins where it ends.

The ladder from here #

Later rungs on this anchor: the Bohm criterion derived properly, with the presheath solved rather than asserted and the $0.61$ obtained from a matched asymptotic expansion; the radio-frequency sheath, which rectifies the applied waveform and produces a large self-bias that no direct current supplies; the dust-grain charging problem, where secondary emission and photoemission compete with the collection currents and grains levitate in the sheath’s field; and the magnetised sheath at oblique incidence, which decides the heat load on a fusion divertor.

The neighbouring ladders are the long-range force that does not reach, which is the screening this layer is the failure of; how far a field gets into metal, which is the same question asked of a conductor; and the inside of a conductor, where the field is excluded completely rather than over a few Debye lengths.