Transient Negative Mobility and Negative Differential Conductivity Effects in Xenon

This model simulates two interesting effects that can be explained by solving the Boltzmann equation in the two-term approximation: (i) the Negative Differential Conductivity (NDC) in gases that occurs when the electron drift velocity decreases with the increase of the electric field; and (ii) the Transient Negative Mobility (TNM) that occurs when the temporal variation of the Electron Energy Distribution Function (EEDF) is much faster than the variation of the electron number density and the electron mobility becomes negative during the relaxation of the EEDF. The model results here presented agree well with the results from Ref. 1.

Model Definition

The theory and formulation used in this tutorial is based on Ref. 2. In Ref. 2 only a approximated version of the stationary Boltzmann equation is solved. Here, to study the TNM effect the time derivative of the electron energy distribution function is introduced in the same way as in Ref. 1.

The time dependent Boltzmann equation in the two-term approximation can be written as

where F0 is the normalized EEDF (eV−3/2) and

(1)

and

(2)

For definitions of the quantities in the equations Equation 1 and Equation 2, see the chapter The Boltzmann Equation, Two-Term Approximation Interface in the Plasma Module User’s Guide.

At zero energy, the condition that energy flux is zero must hold:

and as ε → ∞, F0 → 0.

The EEDF is defined by how electrons gain energy from the electric field and lose (or gain) their energy in collisions with the background gas. The electron collisions are characterized by cross sections that need to be provided by the user. In this model, the background gas is xenon and the electron impact collisions cross-sections are obtained from Ref. 3.

Results and Discussion

To study the NDC it is used the stationary Boltzmann equation in the two-term approximation. The simulations are done at 1 atm and cover a range of reduced electric fields between 0.1 and 10 Td, and ionization degrees between 0 and 10-5. The results are presented in Figure 1to Figure 4. The electron drift velocity is presented in Figure 4 as a function of the reduced electric field. Without electron-electron collisions the drift velocity is a monotonic increasing function of the electric field. However, introducing electron-electron collisions a negative slope on the drift velocity appears indicating the presence of NDC.

To study the TNM it is used the time dependent Boltzmann equation in the two-term approximation. The simulations are done at 10 atm and solve the temporal relaxation of the EEDF to 0.01 Td starting from a stationary solution at 2.2 Td. Figure 5 to Figure 7 show simulation results related to the TNM. Figure 7 shows the electron drift velocity as a function of time. After switching the electric field to 0.01 Td there is a fast drop (~3 ns) in the drift velocity followed by long relaxation period of the order of 100 ns. For the case with electron-electron collisions (ionization degree of 10-7) the drift velocity is allays positive and reaches a steady just after 100 ns. Without electron-electron collisions the drift velocity passes by a region of negative values clearly showing the TNM effect and a steady state is still not reached at 200 ns.

Figure 1: EEDFs for the stationary case for β=0, 1 atm, and for several reduced electric fields between 0.1 and 10 Td. Compare with figure 2 of Ref. 1.

Figure 2: EEDFs for the stationary case for β=10-6, 1 atm, and for several reduced electric fields between 0.1 and 10 Td. Compare with figure 2 of Ref. 1.

Figure 3: Mean electron energy as a function of the reduced electric field for several ionization degrees at 1 atm. Compare with figure 2 of Ref. 1.

Figure 4: Electron drift velocity as a function of the reduced electric field for several ionization degrees at 1 atm. Compare with figure 4 of Ref. 1.

Figure 5: EEDFs for several time instants during the relaxation from 2.2 to 0.01 Td at 10 atm and β=0. Compare with figure 8 of Ref. 1.

Figure 6: EEDFs for several time instants during the relaxation from 2.2 to 0.01 Td at 10 atm and β=10-7. Compare with figure 8 of Ref. 1.

Figure 7: Temporal evolution of the electron drift velocity at 10 atm for β=0 and 10-7. Compare with figure 7 of Ref. 1.

References

1. Z. Donko and N. Dyatko, “First-Principles Particle Simulation and Boltzmann Equation of Negative Differential Conductivity and Transient Negative Effects in Xenon”, Eur.Phys. J. D vol. 70, pp. 135–146, 2016.

2. G.J.M. Hagelaar and L.C. Pitchford, “Solving the Boltzmann Equation to Obtain Electron Transport Coefficients and Rate Coefficients for Fluid Models”, Plasma Sources Science and Technology, vol. 14, pp. 722–733, 2005.

3. M. Hayashi (2003) database, www.lxcat.net, retrieved 2021.

Plasma_Module/Two-Term_Boltzmann_Equation/transient_negative_mobility

Modeling Instructions

Select a 0D space dimension to have access to the Boltzmann Equation, Two-Term Approximation interface.

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Boltzmann Equation, Two-Term Approximation (be)

Select to include electron-electron collision, add more elements, refine the region at 0 eV, and set the maximum energy to 30 eV.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Select the check box.

In the N text field, type 300.

In the R text field, type 30.

In the εmax text field, type 30.

Import electron impact cross sections for Xenon.

Cross Section Import 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file Xe_xsecs.txt.

Click

Set some parameters to be used in the model. Some of the parameters are only relevant for the time dependent study.

Add a rectangle function that will work as a switch to be used in the time dependent model.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Tgas	300[K]	300 K	Gas temperature
P0	1[atm]	1.0133E5 Pa	Gas pressure
Ngas	P0/(k_B_const*Tgas)	2.4463E25 1/m³	Gas density
Beta	1e-7	1E-7	Ionization degree
ne	Beta*Ngas+eps	2.4463E18 1/m³	Electron density