Electron Transport Theory
The equation for the electron density is given by
(4-5)
Here (the subscript e refers to an electron:
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ne denotes the electron density (SI unit: 1/m3).
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Re is the electron rate expression (SI unit: 1/(m3·s)).
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μe is the electron mobility which is either a scalar or tensor (SI unit: m2/(V·s)).
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E is the electric field (SI unit: V/m).
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De is the electron diffusivity which is either a scalar or a tensor (SI unit: m2/s).
The electron energy density is given by the following equation
(4-6)
where (the subscript ε refers to electron energy):
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nε is the electron energy density (SI unit: V/m3).
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Sen is the energy loss/gain due to inelastic collisions (SI unit: V/(m3·s)).
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Q is an external heat source (SI unit: W/m3).
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Qgen is a generalized heat source (SI unit: W/m3).
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με is the electron energy mobility which is either a scalar or a tensor (SI unit: m2/(V·s)).
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E is the electric field (SI unit: V/m).
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Dε is the electron energy diffusivity (SI unit: m2/s).
The mean electron energy, (SI unit: V) is computed through the expression
(4-7)
Because of the high degree of nonlinearity inherent in the drift diffusion equation, the electron number density can span 10 orders of magnitude over a very small distance. In this region (the plasma sheath), the difference in the mobility and diffusivity between the ions and electrons creates a separation of space charge. This in turn produces a large electric field which can lead to a substantial increase in the mean electron energy. The best way of handling with this from a numerical point of view is to solve for the log of the electron number and energy density. This also prevents a divide by zero from occurring when equation Equation 4-7 is evaluated. So, letting Ne = ln ne, equation Equation 4-5 becomes
or
similarly, for the electron energy density:
The resulting equation system is inherently more stable than the original equation system. A useful quantity for results and analysis is the electron “temperature” (SI unit: V) which is defined as: