Debye sheath
The Debye sheath (also electrostatic sheath) is a layer in a plasma which has a greater density of positive ions, and hence an overall excess positive charge, that balances an opposite negative charge on the surface of a material with which it is in contact.
The thickness of such a layer is several Debye lengths thick, a value whose size depends on various characteristics of plasma (e.g. temperature, density, etc.).
A Debye sheath arises in a plasma because the electrons usually have a temperature on the order of magnitude or greater than that of the ions and are much lighter.
At the interface to a material surface, therefore, the electrons will fly out of the plasma, charging the surface negative relative to the bulk plasma.
An equilibrium is finally reached when the potential difference is a few times the electron temperature.
The Debye sheath is the transition from a plasma to a solid surface.
Similar physics is involved between two plasma regions that have different characteristics; the transition between these regions is known as a double layer, and features one positive, and one negative layer.
Sheaths were first described by American physicist Irving Langmuir.
In 1923 he wrote: Langmuir and co-author Albert W. Hull further described a sheath formed in a thermionic valve: The quantitative physics of the Debye sheath is determined by four phenomena: Energy conservation of the ions: If we assume for simplicity cold ions of mass
, having charge opposite to the electron, conservation of energy in the sheath potential requires where
Ion continuity: In the steady state, the ions do not build up anywhere, so the flux is everywhere the same: Boltzmann relation for the electrons: Since most of the electrons are reflected, their density is given by Poisson's equation: The curvature of the electrostatic potential is related to the net charge density as follows: Combining these equations and writing them in terms of the dimensionless potential, position, and ion speed, we arrive at the sheath equation: The sheath equation can be integrated once by multiplying by
Nevertheless, an important piece of information can be derived analytically.
Since the left-hand-side is a square, the right-hand-side must also be non-negative for every value of
Ultimately a so-called pre-sheath will develop with a potential drop on the order of
and a scale determined by the physics of the ion source (often the same as the dimensions of the plasma).
Normally the Bohm criterion will hold with equality, but there are some situations where the ions enter the sheath with supersonic speed.
Although the sheath equation must generally be integrated numerically, we can find an approximate solution analytically by neglecting the
For a "floating" surface, i.e. one that draws no net current from the plasma, this is a useful if rough approximation.
For a surface biased strongly negative so that it draws the ion saturation current, the approximation is very good.
It is customary, although not strictly necessary, to further simplify the equation by assuming that
Then the sheath equation takes on the simple form As before, we multiply by
, we have This equation is known as Child's law, after Clement D. Child (1868–1933), who first published it in 1911, or as the Child–Langmuir law, honoring as well Irving Langmuir, who discovered it independently and published in 1913.
It was first used to give the space-charge-limited current in a vacuum diode with electrode spacing d. It can also be inverted to give the thickness of the Debye sheath as a function of the voltage drop by setting
: In recent years, the Child-Langmuir (CL) law have been revised as reported in two review papers.
