Ion Energy Distribution Function

One of the most useful quantities of interest after solving a self-consistent plasma model is the ion energy distribution function (IEDF). The magnitude and shape of the IEDF depends on many of the discharge parameters; pressure, plasma potential, sheath width and so forth. At very low pressures the plasma sheath is said to be collisionless, meaning that the ion energy is not retarded by collisions with the background gas. At higher pressures the ions collide with the background gas molecules in the sheath and their energy at the moment of impact with a surface is reduced.

This application requires the Plasma Module, Particle Tracing Module, and AC/DC Module.

Model Definition

The equations of motion for ions in an electric field and background gas are

where m is the ion mass (SI unit: kg), v is the particle velocity (SI unit: m/s), q is unit charge (SI unit: C), and Z is the ion charge number (SI unit: dimensionless).

When an ion undergoes a collision with the background gas, its velocity vector changes. The probability of a collision event occurring depends in the ion-neutral collision frequency, ν (SI unit: 1/s), which is defined as:

where Nd is the background number density (SI unit: 1/m3), σ is the ion-neutral charge exchange collision cross section (SI unit: m2), vp is the particle velocity, and vg is the velocity of the background gas atoms or molecules. In this example the collision cross section is assumed to be constant, 6x10-19m2, as given in Ref. 2.

The collision probability defined as:

If P is greater than a random number between 0 and 1 then the particle velocity is reinitialized to the following expression:

where vp is the precollision particle velocity, mg is the mass of the background gas atoms or molecules, R is a uniformly distributed random unit vector, and vg is the velocity of the background gas atoms or molecules, which is sampled from a Maxwellian distribution function:

where T is the temperature of the background gas.

The particles are released 5 mm away from the wafer surface and 20 mm radially inward from the center of the reactor. There are 22,500 ions modeled. They all start at the same point in space and have an initial Maxwellian distribution function, which is for each velocity direction

where T is the initial temperature of the ions, in this case 400 K. This results in a distribution function of

When the ions strike the wafer their velocity is frozen for all subsequent time steps. This allows the velocity and energy distribution function to be recovered once all the ions have made contact with the wall.

The angle at which the ions strike the wafer is also of interest. This can be recovered by plotting a histogram of the inverse tangent of the radial and axial particle velocities.

Results and Discussion

The electric potential for an argon plasma at an operating pressure of 20 mTorr is plotted in Figure 1. The initial starting position for the ions is also shown:

Figure 1: Plot of the plasma potential used to compute the IEDF.

The IEDF is plotted in Figure 2. Most of the ions have kinetic energy between around 20 and 22 eV. At the initial starting coordinate, the plasma potential is 20.98 V, so it is expected that the kinetic energy of the ions at the wall is of similar order. As the pressure decreases the collision frequency between ions and neutrals is reduced, so the IEDF should shift to the right, toward the maximum value given by the plasma potential. As the pressure increases the sheath becomes more collisional which inhibits the ions from reaching higher energies. Of course, changing the pressure results in a change in the discharge characteristics which may alter the shape and magnitude of the IEDF in a nonlinear way.

Figure 2: Plot of the ion energy distribution function (IEDF) in an inductively coupled plasma.

The angle at which the ions are striking the wafer surface are plotted in Figure 3. Due to the fact that the ions are released relatively close to the wafer surface, the range of angles is small, typically between -20 and 20 degrees. The plot is not quite symmetric due to the presence of a small outward component of the ambipolar electric field.

Figure 3: Plot of the angle at which the ions strike the surface of the wafer.

The ion angular energy distribution function is plotted in Figure 4. The angle is not quite symmetric about zero due to the profile of the electric field at the release point.

Figure 4: Plot of the ion angular energy distribution function.

Notes About the COMSOL Implementation

This model is most conveniently solved by opening an existing model in the Plasma Module Application Library, then computing the ion trajectories.

References

1. O.V. Vozniy, G.Y. Yeom, and A. Yu. Kropotov, “Plasma Potential Influence on Ion Energy Distribution Function in ICP Source”, PSE, vol 5, no 1-2, pp. 28–33, http://www.pse.scpt.org.ua/en/jornal/1-2_07/3.pdf.

2. M. Surendra, “Radiofrequency Discharge Benchmark Model Comparison”, Plasma Sources Sci. Technol, vol. 4, pp 56–73, 1995.

3. A. V. Phelps, “The application of scattering cross sections to ion flux models in discharge sheaths”, J. Appl. Phys. vol. 76, pp. 747–753, 1994.

Plasma_Module/Inductively_Coupled_Plasmas/ion_energy_distribution_function

Modeling Instructions

Start by opening the model of the inductively coupled GEC reference cell from the Application Library.

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file argon_gec_icp.mph.

Component 1 (comp1)

Now add a interface to compute the ion energy distribution function.

From the menu, choose .

Add Physics

Go to the window.

In the tree, select .

Click in the window toolbar.

From the menu, choose .

Root

From the menu, choose .

Add Study

Go to the window.

Find the subsection. In the table, clear the check boxes for and .

Find the subsection. In the tree, select .

Click in the window toolbar.

From the menu, choose .

Charged Particle Tracing (cpt)

In the window, expand the node, then click .

In the window for , locate the section.

Click

Select Domain 3 only.

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
Mw	0.04[kg/mol]	0.04 kg/mol	Ion molecular weight
mi	Mw/N_A_const	6.6422E-26 kg	Ion mass

Definitions

Analytic 1 (an1)

In the toolbar, click

and choose .

Enter the analytic approximation for momentum cross section for elastic scattering between Ar+ ions and neutral Ar atoms, which depends on the kinetic energy of the particles.

In the window for , type Qm in the text field.

Locate the section. In the text field, type 1.15e-18*x^(-0.1)*(1+0.015/x)^0.6.

Locate the section. In the text field, type eV.

In the text field, type m^2.

Analytic 2 (an2)

In the toolbar, click

and choose .

Enter the analytic approximation for isotropic elastic collision between Ar+ ions and neutral Ar atoms from, which depends on the kinetic energy of the particles.

In the window for , type Qi in the text field.

Locate the section. In the text field, type 2e-19/(x^(0.5)*(1+x))+3e-19*x/(1+x/3)^(2.3).

Locate the section. In the text field, type eV.

In the text field, type m^2.

Charged Particle Tracing (cpt)

Particle Properties 1

In the window, under click .

In the window for , locate the section.

In the mp text field, type mi.

Locate the section. In the Z text field, type 1.

Now add an feature. The electric potential comes from the solved plasma model. Specify that piecewise polynomial recovery should be used when computing the electric force. This results in a more accurate reconstruction of the electric field.

Electric Force 1

In the toolbar, click