Plasma Deposition

Argon discharge for plasma-assisted deposition: electron temperature from particle balance, plasma density from power balance, ion flux and ion energy at the substrate, Debye length.

Model: Textbook approximation

Power absorbed by the plasma (W)Argon pressure (mTorr)Chamber radius R (cm)Chamber length l (electrode gap) (cm)Gas temperature (K)

Electron temperature Tₑ

2.55 eV

particle balance K_iz/u_B = 1/(n_g d_eff)

Centre plasma density n₀

2.15e+17 m⁻³

P_abs / (e u_B A_eff E_T)

Ion current density at the electrode

1309.4 µA/cm²

flux h_l n₀ u_B = 8.17e+19 m⁻² s⁻¹

Ion energy at a floating substrate

13.2 eV

V_s + Tₑ/2, V_s = Tₑ ln √(M/2πm)

Energy lost per ion lost E_T

96.9 eV

E_c = 78.6 eV + 2Tₑ + E_i

Debye length

25.6 µm

√(ε₀ Tₑ / (e n₀))

Bohm speed u_B

2.48 km/s

√(e Tₑ / M)

Ion mean free path λᵢ

6.21 mm

n_g = 1.61e+20 m⁻³

h_l, h_R, d_eff

0.153, 0.137

d_eff = 35.2 cm, D_a = 97.0 m²/s

Model

λᵢ = 1/(n_gσᵢ), σᵢ = 10⁻¹⁸ m²; D_a = Dᵢ(1 + Tₑ/Tᵢ), Dᵢ = eTᵢλᵢ/(M v̄ᵢ), Tᵢ = T_gas

h_l = 0.86 [3 + l/2λᵢ + (0.86 l u_B/πD_a)²]^−½, h_R = 0.80 [4 + R/λᵢ + (0.80 R u_B/χ₀₁J₁(χ₀₁)D_a)²]^−½

d_eff = Rl / (2(Rh_l + lh_R)); A_eff = 2πR(Rh_l + lh_R)

E_c = E_iz + E_exK_ex/K_iz + (3m/M)TₑK_el/K_iz with the argon rate constants of Lieberman & Lichtenberg

A volume-averaged argon discharge in a cylinder: particle balance fixes Tₑ (d_eff depends on Tₑ through D_a and u_B, so they are solved together), and the power sets the density. The h-factors join the low-pressure and the diffusion (high-pressure) solutions. The substrate sits on an end wall (an electrode). The wall is floating; an rf-biased or self-biased electrode (capacitive discharge) draws the same ion flux but at the bias voltage, set by the matching and the electrode areas. Deposition gases (SiH₄, O₂, N₂) attach electrons and change Tₑ and the loss channels, so the page models the argon carrier only. The ion current density is what the ion-assisted deposition page takes.

0.1–100 mTorr at the current power and chamber; pressures where Tₑ leaves the 1–7 eV rate-constant fits are left out.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Argon global model in a cylinder: Maxwellian electrons, ground-state ionisation, floating walls, collisionless sheath, h-factors from low pressure to ambipolar diffusion. No deposition chemistry; P is absorbed power.

Lieberman M. A., Lichtenberg A. J. (2005). Principles of Plasma Discharges and Materials Processing, 2nd ed., Table 3.3 (fits for 1 < T_e ≤ 7 V), eqs. 5.1.5 and 5.1.14 (ambipolar diffusion), 5.3.16–5.3.17 (h-factors) and §10.2 (global model, Example 1). Wiley. doi:10.1002/0471724254
Lee C., Lieberman M. A. (1995). Global model of Ar, O2, Cl2, and Ar/O2 high-density plasma discharges. J. Vac. Sci. Technol. A 13, 368 (source of the joined h-factors, as cited by Lieberman & Lichtenberg). doi:10.1116/1.579366

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