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. Consequently, they are faster than the ions by at least a factor of formula_1. 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. Due to Debye shielding, the scale length of the transition region will be the Debye length formula_2. As the potential increases, more and more electrons are reflected by the sheath potential. 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.
Description.
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:
Mathematical treatment.
The planar sheath equation.
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 formula_3 entering the sheath with a velocity formula_4, having charge opposite to the electron, conservation of energy in the sheath potential requires
where formula_6 is the charge of the electron taken positively, i.e. formula_7 x formula_8 formula_9.
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 Bohm sheath criterion.
The sheath equation can be integrated once by multiplying by formula_17:
At the sheath edge (formula_19), we can define the potential to be zero (formula_20) and assume that the electric field is also zero (formula_21). With these boundary conditions, the integrations yield
This is easily rewritten as an integral in closed form, although one that can only be solved numerically. 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 formula_23, in particular for small values. Looking at the Taylor expansion around formula_20, we see that the first term that does not vanish is the quadratic one, so that we can require
or
or
This inequality is known as the Bohm sheath criterion after its discoverer, David Bohm. If the ions are entering the sheath too slowly, the sheath potential will "eat" its way into the plasma to accelerate them. Ultimately a so-called pre-sheath will develop with a potential drop on the order of formula_28 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.
The Child–Langmuir law.
Although the sheath equation must generally be integrated numerically, we can find an approximate solution analytically by neglecting the formula_29 term. This amounts to neglecting the electron density in the sheath, or only analyzing that part of the sheath where there are no electrons. 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 formula_30 is much larger than unity. Then the sheath equation takes on the simple form
As before, we multiply by formula_17 and integrate to obtain
or
This is easily integrated over ξ to yield
where formula_36 is the (normalized) potential at the wall (relative to the sheath edge), and "d" is the thickness of the sheath. Changing back to the variables formula_4 and formula_38 and noting that the ion current into the wall is formula_39, 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 formula_41: