Electrical Modelling

Electrical Model Based on Experimental Measurements

Basic assumptions:The model  is one-dimensional, based on the following assumptions:

- Ions respond only to the time-averaged electric field and the ion motion is collisional with a constant mean free path. This assumption limits the model application to total gas pressures higher than 200 mTorr.

- The time-averaged values of the electric field and the electron density in the RF sheath have been calculated assuming that the sheath-plasma boundary oscillates as d_s(t)= d_p(1+sinωt). Results for the sheath of the grounded electrode have been extracted by shifting the phase of solution of RF sheath by ω_t=π.

- An ambipolar transport in the bulk has been assumed and although the electron distribution function is not expected to be Maxwellian, a use of electron temperature has been made for an initial estimation of the ambipolar dc field and the determination of ambipolar diffusion coefficient.

Model Inputs

The formulation of the model is not self-consistent in the sense that it requires the self-bias voltage, the discharge current, the ohmic part of the discharge impedance, and the spatial distribution of the ionization rate.

Model outputs

The output parameters of the model are the spatial distribution of electrons and ions, the distribution of the electric field, the power dissipation to ions and electrons, and the mean electron energy.

Model Formulation

- Ion transport in the sheaths

The transport of ions in the sheath is usually described as a collisionless or collisional, space charge limited motion. The charge conservation condition and the assumption of no ionization or recombination in the sheath impose the requirement of a constant ion flux. In reality, ion density gradually decreases while ion drift velocity increases, from the plasma sheath boundary towards the RF electrode:

      (1)

where, n_i(x) the ion density and u₊(x) the ion drift velocity.

The mobility solution of the Child-Langmuir law for a collisional sheath has been used to calculate the ion flux thus,  the ion mean free path λ₊ have to be  much smaller than the sheath thickness. The values of the sheath electric fields are expected to be high enough in the conditions studied here, allowing thus to express the ion drift velocity in the sheath using the high-field mobility factor k

          (2)

By combining eq. 2 to Poisson’s equation (dE/dx=en_i(x)/ε_ο), the expression of ion current density (eq.1) is transformed to a relation of the form of Child’s law

          (3)

where E is the time averaged value of the sheath field at the position x, and E_o is the value of the electric field at the plasma-sheath boundary (x=0). The use of nonzero value of the field at the plasma-sheath edge is necessary to eliminate the singularity for the ion density in the collisional case while the value of E_o is determined by the relation E_o=T_e/λ_D  describing the plasma sheath transition. According to eq. (3) for constant ion flux, the distribution of the time-averaged field in the sheath can be written as

          (4)

where d is the mean sheath thickness and E_d the field value at the electrode.

The distribution of the electric field can then be combined to Poisson’s equation to calculate the ion density distribution

          (5)

and the voltage distribution in the sheath

          (6)

Furthermore, the electrons in the sheath are assumed to be in Boltzmann equilibrium and have a Maxwellian distribution. Therefore, the electron density distribution is of the form:

          (7)

where n_e(0)  and V(0) is the electron density and the voltage at plasma-sheath boundary.

The time-averaged values of the electric field and the sheath thickness, required for the solution of the set of equations (3-7), are calculated as follows: The mean values of the powered and grounded sheath lengths are found assuming a pure capacitive nature of the two sheaths. The power and grounded sheath reactive impedances can be distinguished, and the sheath lengths can be then calculated using:

          (8)

where I_d is the total discharge current and f a factor related to the fraction of the total current conducted by the grounded electrode, with respect to the applied voltage for each frequency.

The values of the sheath fields E_d at x=d are calculated with the assumption that the sheath current is a displacement current

          (9)

For the values of the field at x=0 (sheath-bulk interface) the Godyak-Sternberg model for the pre-sheath is used and E_o is calculated by equalizing the velocity of ions injected from the plasma into the sheath with the velocity of ions in the sheath, in Eq. (3)

          (10)

The necessary electron temperature estimation results in this case from equation

          (11)

with V_p the plasma potential, and Io the zero-order modified Bessel function of the first kind.

Model Formulation

- Ion transport in the bulk

In order to relate the time-averaged values of the field and the charged particle densities in the sheaths to their values in the bulk, ambipolar diffusion dominated ion transport is considered:

          (12)

where S(x) is the time averaged ionization rate in space and Dais the ambipolar diffusion coefficient. The ambipolar diffusion coefficient D_a, is determined by the electron temperature and the ion mobility:    

         (13)

Since, the values of the electric field in the bulk are expected to be rather weak, a low field mobility  is used instead of the high field mobility k used to describe ion drift motion in the sheaths. Thus, the ambipolar diffusion equation (12) is solved for d_ps < x < d_gs , with the boundary conditions:

          (14)

where, d_pd, d_gd are the mean powered and grounded sheath lengths, J_ip, J_ig the ion conduction current densities, and N_pd, N_gd the charge densities at bulk-powered and grounded sheath boundaries respectively. Furthermore, the distribution of the ambipolar electric field Eab is calculated using the equation

           (15)

Model Formulation

- Power dissipation

The calculation of the ion flux density in both sheaths permits the estimation of the power consumed for ion acceleration

           (16)

The power consumed for electron heating can then be distinguished as the difference between the total power consumed in the discharge and the power consumed for ion acceleration

            (17)

In order to relate the different paths of power dissipation with the ohmic part of the discharge impedance, an in series equivalent circuit is assumed for the discharge.

The resistance related with ion acceleration in the powered and grounded sheath is calculated using the formulas

          (18)

The resistive component Rν related with electron-molecule collisions is then calculated using the expression

          (19)

where, R the total resistive component, is strictly related to the frequency at which the electrons relax their momentum

          (20)

where n_Av , is the average plasma density and d_b the bulk length.

Model Formulation

- Electron heating and mean electron energy

The power consumed for electron heating Pe that has been calculated above represents the total power transferred to electrons. However, significant variations are expected in the spatial distribution of electron heating.  In order to follow the changes in the spatial variation of electron heating, the time-averaged power transferred per electron θ can be used. This parameter can be expressed as a function of the time averaged electric field and the electron-molecule collision frequency

          (21)

The changes in electron heating will be reflected in the value of the mean electron energy, resulting from the balance between the energy gained from the field and the energy lost per collision:

          (22)

where, the value of the effective (ohmic) part of the RF bulk electric field has been used:

          (23)

u_e is the electron drift velocity and k is the mean energy loss factor.

Assuming a sinusoidal RF field and substituting electron drift velocity as

          (24)

where, φ=tan^-1(ω/ν_m) is the angle by which the electron drift motion lags the field eq. (24) results in a differential equation of the form

          (25)

that has the solution

           (26)

where νe is the mean collision frequency for energy transfer, resulting from the product of k times νm. The term    can be attributed to the mean electron energy that would be observed in a dc electric field of strength Eeff.

Model Results

The complete model formulation has been presented in the following work:

        E. Amanatides and D. Mataras
        J. Appl. Phys. 89, 1556 (2001) ©

    while the extension of the model to include reactive gases and mixtures was published in:

        E. Amanatides, D. Mataras, D. E. Rapakoulias
        J. Appl. Phys. 90, 5799 (2001) ©

Application of the model to H₂ and SiH₄/H₂ discharges and the main results are shown below:

Electron Density: Spatial distribution of electron density for four different frequencies - The conditions are: pure H₂ 0.5 Torr discharges and constant power dissipation of 40 mW/cm². The model predicts an increase of electron density and a shift of the peak of electron density towards the center of the discharge with frequency