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Child’s Law Benchmark
Introduction
Space charge limited emission is a phenomenon that restricts the number of electrons or other charged particles that can be released from a surface. As the current of electrons released by a cathode increases, so does the charge density in the immediate vicinity of the cathode. This distribution of charge density exerts an electric force on the emitted electrons, directed toward the cathode. The space charge limited current is the maximum current that can be released such that the emitted particles are not repelled back toward the cathode.
In this example, the space charge limited current in a plane-parallel vacuum diode is computed using the dedicated Space Charge Limited Emission particle release feature. A bidirectional coupling is set up in which the electrons produce a distribution of space charge density that contributes to the electric potential being solved for. The resulting electric potential distribution and current are compared with the analytical solution given by Child’s law.
The Application Library example Thermionic Emission in a Planar Diode expands upon this example by employing the more detailed Langmuir-Fry model, in which the initial thermal distribution of electron velocity at the cathode is also taken into account.
Model Definition
The model geometry is two-dimensional but can be thought of as quasi-1D because the current and electric potential are uniform in the y direction. Electrons are emitted from a grounded cathode located at position x = 0 and propagate toward an extraction grid at a fixed potential V0 (SI unit: V) located at x = d, where d > 0. Space charge limited emission occurs when any further increase in current would repel electrons back toward the cathode; this is equivalent to enforcing the boundary condition
on the cathode surface, in addition to the usual boundary condition
for a grounded surface.
Ensuring Numerical Convergence
The space charge limited current is computed using an iterative process in which the particle trajectories and electric potential are computed in alternating steps. The values of variables not solved for in one step are always taken from the most recent iteration of the other. In this way, successively better approximations of the current can be computed. However, if the space charge limited current is significantly overestimated during any iteration, then all of the electrons would immediately be repelled back toward the cathode and the charge density would become saturated in the mesh elements adjacent to the cathode, causing the model to fail. To avoid this overestimation of the space charge limited current, only a small fraction of the space charge density is used to calculate the electric potential during early iterations. This fraction of the space charge density is gradually increased until it reaches unity in the Number of iterations specified in the Settings window for the Electric Particle Field Interaction node.
To further improve the stability and robustness of the model, the electrons are not released precisely at the cathode, but rather a short distance away from it where the gradient of the electric potential is slightly greater than zero. To do so, the Space Charge Limited Emission feature is applied to the boundary at x = 0. Instead of representing the cathode, the left side of the geometry represents an emission surface a short distance dbuf away from the cathode. The electric potential is computed in the narrow region between the cathode and the emission surface, allowing the initial particle velocity at the emission surface to be defined. The electric potential to the right of the emission surface can then be computed using the space charge density computed by the Electric Particle Field Interaction node.
The Bidirectionally Coupled Particle Tracing study step sets up a solver loop in which the particle trajectories are computed using a time-dependent solver, while the electric potential is computed using a stationary solver. The solver sequence alternates between these two solvers for a specified number of iterations, always using the previous solution for the variables not solved for. In this way, a self-consistent solution, in which the particle trajectories and the electric potential affect each other, is obtained.
Results and Discussion
For space charge limited emission (Ref. 1), the electric potential varies with position as
(1)
where d is the distance from the emission surface to the anode and x is the distance from the emission surface to a given point. The space charge limited current is given by Child’s law,
(2)
Where
•
J (SI unit: A/m2) is the norm of the current density,
•
ε0 = 8.854187817 × 10−12 F/m is the vacuum permittivity,
•
e = 1.602176634 × 10−19 C is the elementary charge,
•
me = 9.10938291 × 10−31 kg is the electron mass,
•
V (SI unit: V) is the potential at the anode, and
•
d (SI unit: m) is the distance between the cathode and the anode.
The comparison between the analytical potential distribution given by Equation 1 and the computed result is shown in Figure 1. The relative error is shown in Figure 2.
The model also includes Global Evaluation nodes to compare the total emitted current to the analytical solution given by Equation 2. The relative error in the space charge limited current is about 2%.
Figure 1: Comparison of the computed electric potential and the analytic solution given by Child’s law.
Figure 2: Relative error between the computed electric potential and the analytic solution given by Child’s law.
Reference
1. S. Humphries, Charged Particle Beams, Dover Publications, New York, 2013.
Application Library path: Particle_Tracing_Module/Charged_Particle_Tracing/childs_law_benchmark
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select AC/DC > Particle Tracing > Particle–Field Interaction, Nonrelativistic.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select Preset Studies for Selected Physics Interfaces > Charged Particle Tracing > Bidirectionally Coupled Particle Tracing.
6
Click  Done.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
V0
1000[V]
1000 V
Potential difference across gap
d
1[cm]
0.01 m
Length of the modeling domain
dbuf
0.01[cm]
1E-4 m
Length of the buffer region
H
2[cm]
0.02 m
Height of the modeling domain
Depth
1[m]
1 m
Depth into the modeling domain
Jan
(4*epsilon0_const/9)*sqrt(2*V0*e_const/me_const)*V0/d^2
738.06 A/m²
Analytic current density
Ian
Jan*H*Depth
14.761 A
Analytic total current
Geometry 1
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type d.
4
In the Height text field, type H.
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Van
V0*((x+dbuf)/(d+dbuf))^(4/3)
V
Analytic potential distribution
rel_err
(V-Van)/Van
 
Error in potential distribution
Charged Particle Tracing (cpt)
Particle Properties 1
1
In the Model Builder window, under Component 1 (comp1) > Charged Particle Tracing (cpt) click Particle Properties 1.
2
In the Settings window for Particle Properties, locate the Particle Species section.
3
From the Particle species list, choose Electron.
Electric Force 1
1
In the Model Builder window, click Electric Force 1.
2
In the Settings window for Electric Force, locate the Electric Force section.
3
From the E list, choose Electric field (es/fsp1).
Electrostatics (es)
In the Model Builder window, under Component 1 (comp1) click Electrostatics (es).
Electric Potential 1
1
In the Physics toolbar, click  Boundaries and choose Electric Potential.
2
Select Boundary 4 only.
3
In the Settings window for Electric Potential, locate the Electric Potential section.
4
In the V0 text field, type V0.
Multiphysics
Set up space charge limited emission of electrons at the left boundary.
Space Charge Limited Emission 1 (scle1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Boundary > Space Charge Limited Emission.
2
Select Boundary 1 only.
3
In the Settings window for Space Charge Limited Emission, locate the Space Charge Limited Emission section.
4
In the os text field, type dbuf.
The iterative scheme employed when solving can be made more stable by ramping up the computed space charge density over a number of iterations. The rate at which the space charge density is ramped up is controlled by the Number of iterations checkbox in the settings window for the Electric Particle Field Interaction node. Further stability can be achieved by selecting the Use cumulative space charge density checkbox, which treats the charge density as the average over successive iterations. The number of iterations should typically be around 15, depending on the size of the buffer zone specified in the Space Charge Limited Emission feature. The smaller the buffer zone, the larger the number of iterations should be. In addition, the number of iterations over which to ramp up to full space charge density should be 10-15 less than the number specified in the Bidirectionally Coupled Particle Tracing study settings.
Electric Particle–Field Interaction 1 (epfi1)
1
In the Model Builder window, click Electric Particle–Field Interaction 1 (epfi1).
2
In the Settings window for Electric Particle–Field Interaction, locate the Continuation Settings section.
3
Select the Use cumulative space charge density checkbox.
4
In the β text field, type 15.
Mesh 1
Use a boundary layer mesh immediately in front of the emission surface for higher accuracy.
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size
1
In the Model Builder window, click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Finer.
Boundary Layers 1
1
In the Mesh toolbar, click  Boundary Layers.
2
In the Settings window for Boundary Layers, click to expand the Transition section.
3
Clear the Smooth transition to interior mesh checkbox.
Boundary Layer Properties
1
In the Model Builder window, click Boundary Layer Properties.
2
Select Boundary 1 only.
3
In the Settings window for Boundary Layer Properties, locate the Layers section.
4
In the Number of layers text field, type 4.
5
Click  Build All.
Study 1
Step 1: Bidirectionally Coupled Particle Tracing
1
In the Model Builder window, under Study 1 click Step 1: Bidirectionally Coupled Particle Tracing.
2
In the Settings window for Bidirectionally Coupled Particle Tracing, locate the Study Settings section.
3
From the Time unit list, choose ns.
4
In the Output times text field, type range(0,0.1,3).
5
Locate the Iterations section. From the Termination method list, choose Convergence of global variable.
6
In the Global variable text field, type scle1.rc.
7
In the Maximum number of iterations text field, type 35.
8
In the Study toolbar, click  Compute.
Results
Use the Cut Line 2D dataset to visualize the relative error in the electric potential versus the prediction of Child’s Law.
Cut Line 2D 1
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets and choose Cut Line 2D.
3
In the Settings window for Cut Line 2D, locate the Line Data section.
4
In row Point 1, set y to 1[cm].
5
In row Point 2, set x to 1[cm] and y to 1[cm].
Comparison With Child’s Law
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Comparison With Child's Law in the Label text field.
3
Locate the Data section. From the Dataset list, choose Cut Line 2D 1.
4
From the Time selection list, choose Last.
5
Locate the Legend section. From the Position list, choose Upper left.
Line Graph 1
1
Right-click Comparison With Child’s Law and choose Line Graph.
2
In the Settings window for Line Graph, click to expand the Legends section.
3
Select the Show legends checkbox.
4
From the Legends list, choose Manual.
5
In the table, enter the following settings:
Legends
Computed potential
Line Graph 2
1
In the Model Builder window, right-click Comparison With Child’s Law and choose Line Graph.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type Van.
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose None.
5
Find the Line markers subsection. From the Marker list, choose Point.
6
From the Positioning list, choose Interpolated.
7
In the Number text field, type 20.
8
Locate the Legends section. Select the Show legends checkbox.
9
From the Legends list, choose Manual.
10
In the table, enter the following settings:
Legends
Analytical Solution
11
In the Comparison With Child’s Law toolbar, click  Plot. The plot should look like Figure 1.
Relative Error
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Relative Error in the Label text field.
3
Locate the Data section. From the Dataset list, choose Cut Line 2D 1.
4
From the Time selection list, choose Last.
Line Graph 1
1
Right-click Relative Error and choose Line Graph.
2
In the Settings window for Line Graph, locate the y-Axis Data section.
3
In the Expression text field, type rel_err.
4
In the Relative Error toolbar, click  Plot. The plot should look like Figure 2.
Finally, check that the space charge limited current matches the analytical solution.
Global Evaluation 1
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Time selection list, choose Last.
4
Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Global definitions > Parameters > Ian - Analytic total current - A.
5
Click  Evaluate.
Global Evaluation 2
1
In the Results toolbar, click  Global Evaluation.
2
In the Settings window for Global Evaluation, locate the Data section.
3
From the Time selection list, choose Last.
4
Click Replace Expression in the upper-right corner of the Expressions section. From the menu, choose Component 1 (comp1) > Currents and charge > scle1.rc - Release current magnitude - A.
5
Click the arrow next to the Evaluate button and click Table 1 - Global Evaluation 1 (Ian).