Debye sheath

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 ${\displaystyle m_{\mathrm{i}}}$ entering the sheath with a velocity ${\displaystyle u_0}$, having charge opposite to the electron, conservation of energy in the sheath potential requires

${\displaystyle \frac{1}{2}{m}_{\mathrm{i}}u(x{)}^2=\frac{1}{2}{m}_{\mathrm{i}}{u}_0^2-e\phi (x)}$,

where ${\displaystyle e}$ is the charge of the electron taken positively, i.e. ${\displaystyle e=1.602}$ x ${\displaystyle {10}^{-19}}$ ${\displaystyle \mathrm{C}}$.

Ion continuity: In the steady state, the ions do not build up anywhere, so the flux is everywhere the same:

${\displaystyle n_0{u}_0=n_{\mathrm{i}}(x)u(x)}$.

Boltzmann relation for the electrons: Since most of the electrons are reflected, their density is given by

${\displaystyle n_{\mathrm{e}}(x)=n_0\exp (\frac{e\phi (x)}{k_{\mathrm{B}}{T}_{\mathrm{e}}})}$.

Poisson's equation: The curvature of the electrostatic potential is related to the net charge density as follows:

${\displaystyle \frac{d^2\phi (x)}{d{x}^2}=\frac{e(n_{\mathrm{e}}(x)-n_{\mathrm{i}}(x))}{{ϵ}_0}}$.

Combining these equations and writing them in terms of the dimensionless potential, position, and ion speed,

${\displaystyle \chi (\xi )=-\frac{e\phi (\xi )}{k_{\mathrm{B}}{T}_{\mathrm{e}}}}$

${\displaystyle \xi =\frac{x}{{\lambda }_{\mathrm{D}}}}$

${\displaystyle \mathfrak{M}=\frac{u_{\mathrm{o}}}{(k_{\mathrm{B}}{T}_{\mathrm{e}}/m_{\mathrm{i}}{)}^{1/2}}}$

we arrive at the sheath equation:

${\displaystyle {\chi }^{″}={\left(1+\frac{2\chi }{{\mathfrak{M}}^2}\right)}^{-1/2}-e^{-\chi }}$.

Bohm sheath criterion

The sheath equation can be integrated once by multiplying by ${\displaystyle {\chi }^{\prime }}$:

${\displaystyle {\int }_0^{\xi }{\chi }^{\prime }{\chi }^{″}d{\xi }_1={\int }_0^{\xi }{\left(1+\frac{2\chi }{{\mathfrak{M}}^2}\right)}^{-1/2}{\chi }^{\prime }d{\xi }_1-{\int }_0^{\xi }{e}^{-\chi }{\chi }^{\prime }d{\xi }_1}$

At the sheath edge (${\displaystyle \xi =0}$), we can define the potential to be zero (${\displaystyle \chi =0}$) and assume that the electric field is also zero (${\displaystyle {\chi }^{\prime }=0}$). With these boundary conditions, the integrations yield

${\displaystyle \frac{1}{2}{\chi }^{\prime 2}={\mathfrak{M}}^2\left[{\left(1+\frac{2\chi }{{\mathfrak{M}}^2}\right)}^{1/2}-1\right]+e^{-\chi }-1}$

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 ${\displaystyle \chi }$, in particular for small values. Looking at the Taylor expansion around ${\displaystyle \chi =0}$, we see that the first term that does not vanish is the quadratic one, so that we can require

${\displaystyle \frac{1}{2}{\chi }^2\left(-\frac{1}{{\mathfrak{M}}^2}+1\right)\ge 0}$,

or

${\displaystyle {\mathfrak{M}}^2\ge 1}$,

or

${\displaystyle u_0\ge (k_{\mathrm{B}}{T}_{\mathrm{e}}/m_{\mathrm{i}}{)}^{1/2}}$.

This inequality is known as the Bohm sheath criterion after its discoverer, David Bohm who discussed it in 1949. 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 ${\displaystyle (k_{\mathrm{B}}{T}_{\mathrm{e}}/2e)}$ 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.

Child–Langmuir law

Although the sheath equation must generally be integrated numerically, we can find an approximate solution analytically by neglecting the ${\displaystyle e^{-\chi }}$ 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 ${\displaystyle 2\chi /{\mathfrak{M}}^2}$ is much larger than unity. Then the sheath equation takes on the simple form

${\displaystyle {\chi }^{″}=\frac{\mathfrak{M}}{(2\chi {)}^{1/2}}}$.

As before, we multiply by ${\displaystyle {\chi }^{\prime }}$ and integrate to obtain

${\displaystyle \frac{1}{2}{\chi }^{\prime 2}=\mathfrak{M}(2\chi {)}^{1/2}}$,

or

${\displaystyle {\chi }^{-1/4}{\chi }^{\prime }=2^{3/4}{\mathfrak{M}}^{1/2}}$.

This is easily integrated over ξ to yield

${\displaystyle \frac{4}{3}{\chi }_{\mathrm{w}}^{3/4}=2^{3/4}{\mathfrak{M}}^{1/2}d}$,

where ${\displaystyle {\chi }_{\mathrm{w}}}$ 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 ${\displaystyle u_0}$ and ${\displaystyle \phi }$ and noting that the ion current into the wall is ${\displaystyle J=e{n}_0{u}_0}$, we have

${\displaystyle J=\frac{4}{9}{\left(\frac{2e}{m_i}\right)}^{1/2}\frac{|{\phi }_w{|}^{3/2}}{4\pi d^2}}$.

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 ${\displaystyle J=j_{\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{s}\mathrm{a}\mathrm{t}}}$:

${\displaystyle d=\frac{2}{3}{\left(\frac{2e}{m_{\mathrm{i}}}\right)}^{1/4}\frac{|{\phi }_{\mathrm{w}}{|}^{3/4}}{2\sqrt{\pi j_{\mathrm{i}\mathrm{o}\mathrm{n}}^{\mathrm{s}\mathrm{a}\mathrm{t}}}}}$.

In recent years, the Child-Langmuir (CL) law have been revised as reported in two review papers.