Ion Energy Distribution Function in a Capacitively Coupled Plasma Reactor

Plasma processing techniques are widely used in the industry to modify the chemical and physical properties of surfaces. Some processes require energetic ion bombardment and a high degree of ion velocity anisotropy. Therefore, it is of great value to know the ion energy distribution function (IEDF) and the velocity dispersion at the surface.

In this example, the IEDF at the electrode surface is computed for a commercial capacitively coupled plasma reactor. The computed IEDF is compared with measurements from Ref. 1 and a reasonable agreement is found.

This application requires the Plasma Module and the Particle Tracing Module.

Model Definition

The reactor that was used in Ref. 1 to measure the IEDF is simulated using the COMSOL Multiphysics Plasma Module. The reactor is the Plasmalab System 100 parallel plate, capacitively coupled, RIE plasma tool. It is an asymmetric capacitively coupled reactor with a 200 mm diameter powered electrode, and a gap between electrodes of 4.5 cm. The other dimensions of the reactor where inferred from diagrams and photos. Since the plasma volume and the area ratio between electrodes is not exactly known a direct comparison between simulations and measurements for absorbed power, applied voltage, and DC self–bias is only approximate.

Two physics interfaces are used in two consecutive steps. First, the interface is used to solve for the space and time-periodic evolution of the plasma. After, the interface is used to obtained the IEDF collected at the powered electrode.

Plasma Simulation

The simulations are for an Argon plasma sustained at a pressure of 20 mTorr with a periodic electric excitation of 13.56 MHz. The model is 2-dimensional and describes the space and time–periodic evolution of several macroscopic properties of the discharge.

The electron mobility and other electron transport properties are automatically computed from the electron impact reactions. For the ions, the mobility is given as a function of the reduced electric field using a lookup table. The ion diffusivity is obtained from the Einstein relation where the ion temperature comes from a local field approximation.

IEDF simulation

To simulate the IEDF at the powered electrode particles representing ions are released from a point in the plasma and the ion trajectories are computed over several RF cycles until all the released particles reach the wall. The ions should be released from a point outside the plasma sheath in order to experience the full range of the sheath electric field. This point can be estimated by looking at the time averaged electric potential and identifying a region where the potential starts to have smaller variations in the axial direction. Also important is to release particles along the excitation period in order to sample the sheath motion.

The ions are released with a random Maxwellian distribution at 300 K. After the release, the ions are accelerated by a time varying electric field toward the surface. Eventually some ions experience collisions with the background gas. In this simulation two different types of collision events are used: elastic and resonant charge exchange collisions. The collisions are specified by energy dependent cross sections. For the charge exchange collision a scattering angle of 5 degrees is arbitrarily defined.

Electric excitation

The driven electrode has a fixed power and computes the self DC bias. This corresponds to the following expression and set of constraints on the electric potential:

(1)

(2)

(3)

The constraint in Equation 2 is used to compute the self DC bias, Vdc,b. The constraint in Equation 3 is used to compute the RF potential, Va such that a fixed amount of power is deposited into the plasma.

plasma chemistry

Argon plasmas have one of the simplest reactions schemes. We use a simplified plasma chemistry that comprises 7 volume reactions involving electrons, atomic ion, and a lumped level representing the argon 4s states (electron impact cross-sections are obtained Ref. 3).

Table 1: table of collisions and reactions modeled.
Reaction	Formula	Type
1	e+Ar=>e+Ar	Elastic	0
2	e+Ar=>e+Ars	Excitation	11.5
3	e+Ars=>e+Ar	Superelastic	-11.5
4	e+Ar=>2e+Ar+	Ionization	15.8
5	e+Ars=>2e+Ar+	Ionization	4.24
6	Ars+Ars =>e+Ar+Ar+	Penning ionization	-
7	Ars+Ar => Ar+Ar	Metastable quenching	-

In addition to volumetric reactions, the following surface reactions are implemented:

Table 2: table of surface reactions.
Reaction	Formula	Sticking coefficient	Secondary emission coefficient	Mean energy of secondary electrons (V)
1	Ars=>Ar	1	0.07	5.8
2	Ar+=>Ar	0	0.07	5.8

When an excited states make contact with the wall, they revert to the ground state argon with some probability. At the metal electrodes, the ions and the excited state use their internal energy to extract one electron from the wall with a probability of 0.07 and a mean energy of 5.8 V. For the ions, the sticking coefficient is zero meaning that losses to the wall are assumed to be due to migration only.

Results and Discussion

Figure 1, Figure 2, and Figure 3 show the time averaged electron density, electron temperature and electric potential for a plasma sustained at 20 mTorr and 30 W. The period averaged electron density is largest in magnitude in the reactor center, decreasing considerable along the radial and axial directions as expected. The period averaged electron temperature is highest near the electrodes where intense electric fields exist. Since it is an asymmetric reactor the smaller electrode (the powered electrode in this model) has a sheath with more intense electric fields. This is reflected in higher electron temperature at the power electrode. In Figure 3 it is possible to observe the DC self–bias potential developed in order to ensure that there is no period average conduction current through the electrode.

Figure 4 and Figure 5 presents the IEDF and the angular dispersion of the ion velocity at the power electrode. A double peak structure is observed at high energies meaning that the most energetic ions cross the sheath in a time comparable with the RF cycle. The midpoint energy between the two energetic peaks corresponds to the energy that the ions would gain while accelerating through the time–averaged electric potential. From Figure 3 it is possible to estimate that the midpoint energy should be around 350 V (potential difference from −200 V at the wall to 150 V in the discharge bulk), which is in agreement with the results of Figure 4.

The width between the two high energy peaks ΔE can be estimated from analytical models such as the model from Charles and others (from Ref. 1)

(4)

and the model from Sobolewski and others (from Ref. 1)

(5)

where Vpp is the peak-to-peak sheath voltage, Vs is the time-averaged sheath potential, τi is the ion transit time through the sheath, τrf is the RF period, and Te is the electron temperature. The ΔE computed form Equation 4 and Equation 5 gives 42 V and 71 V, respectively, which is in good agreement with the 70 V from the simulated IEDF. The values used to compute ΔE are obtain from the simulations and are presented in Table 3.

Table 3: Values obtained from simulations and used to compute ΔE.
Te	4 V
Vs	340 V
Vpp	1070
τi	900 ns

The ions that cross the sheath without colliding (or with a collision in the beginning of the trajectory) contribute to the high-energy part of the IEDF. The low-energy part of the IEDF also has several well-defined peaks. These structures are created by ions that undergo resonant charge exchange collisions along their motion in direction to the wall. After such collision events, a slow ion is created within the sheath, and consequently it experiences only part of the electric field range.

In Figure 5 it is possible to observe that the high energy ions have a −5 degree tilt relatively to their axial velocity. This is a consequence of the existence of important radial electric fields in this region. Also from Figure 5 you can see that there is a dispersion angle at the low-energy range. This is because it is used an arbitrary 5 degree scattering angle for the resonant charge exchange collision. To obtain realistic results for the ion dispersion velocity, a physical model for the scattering angle is needed.