The model uses the spatial distribution of the Laser Induced Fluorescence (LIF) intensity of SiH (X2Π) radicals and the total SiH4 consumption measured by mass spectrometry, in order to calculate the spatial distribution of the SiH4 electron impact dissociation rate. The distribution of the ground state SiH radicals over the interelectrode space, as measured using LIF, for different values of the applied voltage and in conditions of high dilution of SiH₄ in H₂ is presented in fig. 1.

Figure 1: Spatial LIF intensity distribution of SiH(X2Π) at four different peak to peak voltages (Vpp) in highly diluted SiH4 in H2 (1/150).

The shape of the LIF profiles reflects the existence of the powered sheath as well as the less pronounced grounded sheath electron heating mechanisms, indicating an analogy of the spatial distribution of the LIF intensity to the distribution of the SiH₄ electron induced dissociation rate. The shape of the LIF intensity distribution is determined by the mass balance of the SiH radical in the discharge and can be written

 (1)

where is the rate constant of the SiH+SiH₄ reaction, the SiH₄ number density, d the interelectrode distance and D_SiH the SiH diffusion coefficient. This ratio has values greater than seven over the entire range of the conditions studied here, ensuring that SiH reaction dominates by far SiH diffusional transport.

Thus, the effective electron density for the SiH₄ dissociation can be considered analogous to the LIF intensity and the rate of production of radicals at every point of the discharge space can be expressed as

  (2)

where k_e is the SiH₄ dissociation rate constant, n_e(x) and I(x) are the electron density and the LIF intensity at point x in the discharge and I_TOT  the total LIF intensity.

As  where n_e(x) the average electron density in the discharge, the previous expression can be written as

   (3)

where k_d is the product of the SiH₄ dissociation rate constant and the average electron density.

The term attributed to SiH₄ consumption by electron impact dissociation  together with the term concerning SiH₄ consumption through secondary gas phase reactions W_sec are then introduced to the steady-state mass balance differential equation for the neutral species

  (4)

where D_n is the diffusion coefficient of species n in the gas mixture calculated according to the Chapman-Enscog theory, u is the gas flow velocity and c_n the concentration of species n.

In the term W_sec are included the twenty-seven most important radical-molecule, ion-molecule and radical-radical reactions as well as the electron impact dissociation of Si₂H₆, Si₃H₈ and H₂. The dissociation rates of these molecules have been calculated in accordance with SiH₄ dissociation rate k_d, by compensating the differences in the cross sections and the energy thresholds between these processes. The relations k_d(H₂)=0.08k_d(SiH₄), k_d(Si₂H₆)=2.5k_d(SiH₄), k_d(Si₃H₈)=3.5k_d(SiH₄), k_ion(H₂)=0.02 k_d(SiH₄) and k_ion(SiH₄)=0.22 k_d(SiH₄) have been used to correlate these processes with SiH₄ dissociation. Branching ratios have been adopted to express the different paths of SiH₄ and Si₂H₆ electron induced dissociation, whereas for Si₃H₈ only one dissociation channel has been used. In the case of SiH₄ the branching ratio of SiH₄ ionization has been used (α=46%, β=36%, γ=11% and δ=7%,) , whereas the values (α₁=0.91, β₁=0.09) have been used for the Si₂H₆ dissociation channels.

Ion density balance and ion chemistry are introduced in the simulation by applying the drift-diffusion approximation for the charged particles flux leading to the following modification of eq. (4):

  (5)

where D_i, μ_i are the diffusion and the mobility transport coefficient for ion i, E the effective electric field and W_ion the term accounting for the production and consumption of ion i in the discharge. For simplicity, only two kinds of ions have been included in the model (H₂⁺ and SiH₃⁺). The high and low field mobilities for these ions in the SiH₄/H₂ gas mixture are calculated according to the Langevin formulas. The diffusion coefficient D_i are calculated using the Einstein relation D_i= μ_ik_bT_i/e, where T_i is the ion temperature that is assumed to be equal to the gas temperature. The spatial distribution of the electric field in the discharge, required for the solution of eq. (5), is also an input to the model and is always pre-calculated according to a method that is based on the electrical measurements as described in detail elsewhere.

The boundary conditions required for the solution of eqs. (4) and (5) for the  powered and grounded electrode are different for each specie, taking into account its probability to interact with the surfaces.

Thus, for the neutral radicals (Si_xH_y, H) the boundary condition proposed by Gallagher has been used

  (6) 

where c_n is the radical concentration in a distance of one mean free path from the surface and u_n the radical thermal velocity. The loss probabilities β_n of radicals on the surfaces, required in eq. (6), have been taken from literature. Thus, a value of β=0.5 has been used for Si₂H_2x+1 radicals and a value of β=0.7 for H atoms. Concerning SiH₂ and H₃SiSiH radicals we have followed the assumption of β=0.8 used by Perrin et al.,³⁴ as no measurements of loss probabilities on a-Si:H and μc-Si:H surfaces are available. Disilene (H₂SiSiH₂) has been treated separately by assuming a lower value of β=0.4.

Concerning positive ions, the ion flux towards the electrodes has been assumed to be governed by the ion drift motion in the high field powered and grounded sheaths. Thus, H₂⁺ and SiH₃⁺ ions in both powered and grounded electrode have to satisfy the conditions:

   (7)

Namely, the density gradient of ions close to the electrodes has been set equal to zero, while the sum of the drift fluxes of both ions have to equalize the ion conduction current J_i that is calculated according to the Child-Langmuir law for collisional sheaths.

Concerning the stable molecules (SiH₄, H₂, Si₂H₆ and Si₃H₈), which do not interact with the surfaces, and considering the production of these molecules through surface reactions, the boundary condition can be written as

                (8)

where γ_j is the probability of radical j to recombine at the surface and produce molecule n, and β_j is the total surface loss probability of radical j.

The set of equations concerning the seventeen most important species was solved numerically for the entire interelectrode space using a fixed step centered finite difference scheme, whereas for the first and the last point (corresponding to the powered and the grounded electrode respectively) the boundary conditions have been solved by a forward and a backward finite difference scheme respectively.

The procedure that is followed is that the value of the SiH₄ dissociation rate k_d is adjusted to the specific total SiH₄ consumption as measured by mass spectrometry. Namely, the normalized LIF profiles are used to provide the correct shape of SiH₄ dissociation in the discharge and the total SiH₄ consumption to transform the normalized values into real SiH₄ dissociation rate values.

In addition an initial guess of the values of radical recombination probabilities γ_j is required in eq. (8). These recombination probabilities are not known and the most common assumption is that H, SiH₃ and Si₂H₅ recombine only as H₂, SiH₄ and Si₂H₆ respectively. However, in the case of high deposition rates that require high radical fluxes, this assumption can lead to significant errors especially in the case of the SiH₃ radical for which the relative probability to recombine as Si₂H₆ is enhanced, for fluxes above 10¹⁵ cm^-2 s^-1. Thus, in order to have a better estimation of the recombination probabilities γ_j required in eq. (6), surface kinetics have been treated more precisely by applying an analytical simulation of the film growth, similar to that reported by Guizot et al. Briefly, the evolution of the different surface sites due to radical-surface reactions and surface reconstruction have been taken into account and the chemical composition of the surface was evaluated by solving the balance equations for each surface site-type present:

 (9)

Nine different surface sites have been considered. Namely:

1. dangling bond sites ºSi^o (θ_o), =SiH^o(θ₁) and –SiH₂^o(θ₂)

2. H covered sites ºSiH(θ₃), =SiH₂(θ₄) and -SiH₃(θ₅) and

3. SiH₃ physisorpted sites ºSiHSiH₃ (θ₆), =SiH₂SiH₃(θ₇) and –SiH₃SiH₃ (θ₇)

The surface and gas-surface reactions that have been taken into account in the balance equations together with the frequency, the activation energies and the probabilities of the processes are given in the database page. For the reactions between radicals and dangling bonds a unity probability has been considered, assuming no energy barrier for these processes. The same assumption was applied for the reactions of radicals with SiH₃ in the physisorpted state, considering that surface acts as a third body that stabilizes the volatile products of these reactions. Surface reconstruction through Langmuir-Hinshelwood abstractions has been assumed to lead to silicon network formation rather than to the formation of dangling bond sites. The frequency and the activation energy of reconstruction between sites having a different number of bonded hydrogen atoms have been estimated from the average values of the reconstruction of similar sites.

Having calculated the fraction of the different sites present on the surface, the sticking coefficient of SiH₃ and the relative probability of H atoms, SiH₃ and Si₂H₅ radicals to recombine as H₂, SiH₄, Si₂H₆, and Si₃H₈ can be evaluated.

The system of the nine non-linear site balance equations is solved according to a modified Powel hybrid algorithm, using as input the radicals flux that result from the gas phase chemistry module with the initial guess of γ_j. The calculated new set of the values of γ_j is then returned to eq. (8) and the procedure is iterated until a convergence between the gas phase and the surface module is achieved.

The main results of the combined gas-phase and surface models are the spatial concentration of each of the species in the discharge, the average value of SiH₄ dissociation rate and the radical fluxes towards the powered and the deposition electrode. The deposition rate can be then calculated by summing the contribution of each of the radicals to the film growth, including silicon etching by H atoms

(10)

where p=2.26 gr/mol the mass density of μc-Si:H, m the molar mass assuming a hydrogen content of 5%, α_n the number of silicon atoms in radical n, s_n the sticking coefficient of radical n and γ_e/γ_Η the ratio of the etching probability γ_e of a silicon atom by H atoms to the recombination probability γ_Η to molecular H₂.

The rate constants of the gas phase reactions, the sticking probabilities and coefficients of radicals and the frequency of the surface properties that have been used in the model are summarized in the database page.

 

The model has been applied for the investigation of the effect of the main SiH4/H2 discharge parameters (frequency, power, pressure, silane fraction in the mixture) on the microcrystalline silicon deposition rate. The most striking features are summarized in the results page.