Theory for the Space Charge Limited Emission Node

Use the Space Charge Limited Emission node to model space charge limited emission of electrons from the selected boundary. To use this feature effectively, the following requirements must be met:

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In the settings for the Charged Particle Tracing interface, must be selected from the list.

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The node does not consider the thermal distribution of released particle velocities. Instead, particles are assumed to begin at rest and accelerate due to the electric force close to the cathode. Therefore the speed of the electrons throughout most of the modeling domain should be significantly greater than the thermal velocity. If this criterion cannot be met, consider instead using the Thermionic Emission node in The Charged Particle Tracing Interface.

One example of space charge limited emission for which an analytic solution is readily available, is a plane parallel vacuum diode. Electrons are released from a flat cathode and propagate toward a parallel flat anode. If the potential difference across the diode is V0 (SI unit: V), The magnitude of the space charge limited current density J (SI unit: A/m2) is given by Child’s Law (Ref. 1) as outlined below.

This problem is essentially one-dimensional. The equation for the electrostatic potential is

where ε0 = 8.854187817 × 10-12 F/m is the vacuum permittivity. The charge density ρ (SI unit: C/m3) is due to the released electrons.

The next step is to establish a relationship between the current density J and the charge density ρ. Consider a surface element dΩ (SI unit: m2) parallel to the yz-plane at some distance x from the cathode. In a time interval dt, the number of electrons to cross this surface element is

and these electrons will occupy a volume of

where v (SI unit: m/s) is the electron velocity.

The space charge density in this volume is

The velocity at position x is determined from the principle of conservation of energy, noting that the initial particle velocity is assumed to be zero (no thermal contribution),

So the space charge density at any position can be expressed in terms of the electric potential and the current density,

Then the equation for the electric potential becomes

with boundary conditions

The first two boundary conditions are self-explanatory but the third requires some explanation. Ordinarily, a 1D second-order differential equation would be overconstrained with two Dirichlet boundary conditions and an additional Neumann condition. However, the current density J is not yet known, and the additional boundary condition is needed to determine its value.

The physical justification for the additional Neumann condition is illustrated in Figure 6-1. If the emitted current is too low (2), it would be possible to release even more electrons, and the electrons would not encounter any potential barrier close to the cathode. However, if the emitted current is too high (3), then they would hit a potential barrier and get sent backward before they could accelerate.

Solution (1) shows the potential distribution at the space charge limit because it is the largest perturbation of the potential from a linear solution that can be obtained without creating a potential barrier (V < 0 somewhere).

The solution to Equation 6-5 is

Substituting Equation 6-7 back into Equation 6-5 yields

Solving for J yields Child’s law,

Note that Ref. 1 uses Gaussian units whereas SI units are used here. This explains the additional factor of 4πε0. Also note the difference in sign convention: here J is the inward current density, hence it is negative because the released particles are electrons.

Although the special case of a plane parallel vacuum diode has an analytic solution, for more general shapes such a solution might not exist. Then an approximate numerical solution can be obtained by iterating between particle trajectory calculations and electric potential calculations. As described in the Space Charge Density Calculation section, when electrons are released from a boundary at a constant rate, the space charge density of electrons in each mesh element can be estimated, and this can be used as an additional source term when computing the electric potential (using, for example, the Electrostatics interface).

From a numerical standpoint, a key challenge with the above formulation is that it neglects the thermal distribution of emitted particle velocity. Therefore, if electrons were released at the cathode surface itself, they would have zero initial velocity. This causes the numerical calculation of the space charge density to become numerically unstable because any potential barrier adjacent to the cathode would repel all electrons back to the cathode, leaving them unable to propagate at all.

The solution used by the feature is to treat the selected boundaries as an emission surface a short distance away from the actual cathode. The space between the emission surface and the cathode should be significantly shorter than the geometric length scale.

Because the space charge density of the emitted electrons can change rapidly as the electrons begin to accelerate, a fine boundary layer mesh should be used. The element thickness in the wall normal direction should be smaller than the gap between the cathode and the emission surface, for at least the first few rows of elements.

The node then defines appropriate boundary conditions on the electric potential at the emission surface, by treating the gap between the cathode and the emission surface as a plane parallel vacuum diode. Thus, if Ve is the potential at the emission surface, the cathode potential is 0, and the gap thickness is L, then Equation 6-7 becomes

Differentiating both sides with respect to x yields

Substituting x = L then yields

Thus, the node applies the following boundary condition on the electric potential in the simulation domain:

A negative sign is prepended to the left-hand side of Equation 6-9 because, by the usual COMSOL convention, n is the outward normal. The model particles are released at the emission surface with initial velocity

For the purpose of applying bidirecionally coupled particle field interactions, the current density of the emitted model particles is obtained by substituting the electric potential at the emission surface into Child’s Law (Equation 6-8).

For brevity, the above equations have all been written for a cathode potential of zero. However, the anode potential here can be understood as the potential relative to that of the cathode. The conclusions of the above analysis could also be applied when a nonzero cathode potential is specified.

When modeling particle-field interaction using the Electric Particle Field Interaction node and the study step, it is recommended to gradually ramp up the space charge density over the first few iterations. To learn more about ramping up the space charge density term over multiple iterations, see the Continuation Settings section of the settings window for the Electric Particle Field Interaction node. The corresponding theory is outlined in the Stabilization of the Space Charge Density Calculation section.