Vacuum Arc Equipment for Mass Production of ta-C Coatings

Abstract

The outstanding tribological properties of ta-C and partially also of hard a-C films (see Vol. II, Part VII) suggest their broad application as wear protecting and friction reducing coatings. The realization of the corresponding mass production depends on the basic problem of any thin film technology, the extra costs in comparison to the gained performance.

Explanations

1.1 E1 Deposition Rate on the Central Axis (for 13.10, 13.13)

The angular variation of the plasma beam may be described by a cos^kϑ distribution. The exponent k is (for k ≥ 1) approximately related to the FWHM value Δϑ, i.e. cos^k (Δϑ/2) = 0.5, by k ≈ (132°/Δϑ)². Narrower distributions correspond to larger k values (Fig. 9.35).

The plasma sources may be rotationally symmetric distributed over the cathode surface. The substrate is parallel to the cathode in the distance d_c−s. The local deposition rate w = dh/dt is related to the particle flux density j = dJ/dA and the particle density n_f in the film by w = j/n_f. The particle flux dJ from an areal element dA_c on the cathode to an areal element dA on the parallel substrate in the direction ϑ is given by

$$ dJ = \frac{k + 1}{2\pi }\frac{{J_{0} }}{{A_{c} }}{ \cos }^{k} \vartheta \,d\Omega \,dA_{c} $$

(13.18)

where J₀ denotes the total particle flux from the cathode. dΩ represents the solid angle, the element dA in the distance r = d/cosϑ is seen from dA_c: dΩ = cosϑ dA/r² = cos³ϑ dA/d ²_c−s .

The angle ϑ is related to the axial distance R by tgϑ = R/d_c−s, cosϑ = 1/(1 + (R/d_c−s)²)^1/2 (Fig. 13.32).

Fig. 13.32

Geometric relations between the direction of the particle beam and the deposition on a parallel substrate

Uniform source distribution on a circular area

The plasma sources may be uniformly distributed over an circle with radius R₀. The total particle flux from all elements on the active cathode area is given by

$$ \begin{aligned} J & = \frac{k + 1}{2\pi }\frac{{J_{0} }}{{\pi R_{0}^{2} }}\frac{1}{{d_{c - s}^{2} }}\int {dA_{c} \,{ \cos }^{k + 3} \vartheta } = \frac{k + 1}{2\pi }\frac{{J_{0} }}{{\pi R_{0}^{2} }}\frac{1}{{d_{c - s}^{2} }}2\pi \int\limits_{0}^{{R_{0} }} {dR\frac{R}{{\left( {1 + \left( {R/d_{c - s} } \right)^{2} } \right)^{(k + 3)/2} }}} \\ & = \frac{k + 1}{2\pi }\frac{{J_{0} }}{{\pi R_{0}^{2} }}\frac{1}{{d_{c - s}^{2} }}2\pi \frac{{d_{s - c}^{2} }}{k + 1}\left. {\frac{ - 1}{{\left( {1 + \left( {R/d_{c - s} } \right)^{2} } \right)^{(k + 1)/2} }}} \right|_{0}^{{R_{0} }} \\ \end{aligned} $$

(13.19)

For the deposition rate w follows

$$ w = \frac{{J_{0} }}{{n_{f} \,\pi R_{0}^{2} }}\left( {1 - \frac{1}{{\left( {1 + \left( {R_{0} /d_{c - s} } \right)^{2} } \right)^{(k + 1)/2} }}} \right) $$

(13.20)

At small distances ${\text{d}}_{{\text{c}}\text{-}{\text{s}}} \ll {\text{R}}_{0}$, the central deposition rate does not depend on the substrate distance and is given by the emission flux density:

$$ {\text{w}} \approx {\text{J}}_{ 0} \,/\,(\uppi{\text{R}}_{0}^{2} )\;1\,/\,{\text{n}}_{\text{f}} = {\text{j}}_{ 0} \,/\,{\text{n}}_{\text{f}} \quad \left({\text{d}}_{{\text{c}} {-}{\text{s}}} \ll {\text{R}}_{ 0} \right) $$

(13.21)

At large distances ${\text{d}}_{{\text{c}}\text{-}{\text{s}}} \gg {\text{R}}_{0}$, the rate corresponds to a point source:

$$ w = \frac{k + 1}{2\pi }\frac{1}{{n_{f} }}\frac{{J_{0} }}{{n_{f} \,d_{c - s}^{2} }} $$

(13.22)

Uniform source distribution on a ring

The plasma sources may be uniformly distributed over a ring with radius R_0. The total particle flux from all elements on the active cathode areas on the radius R₀ with width ΔR is given by

$$ J = \frac{k + 1}{2\pi }\frac{{J_{0} }}{{2\pi R_{0} \,\Delta R}}\frac{1}{{d_{c - s}^{2} }}\int {dA_{c} \,{ \cos }^{k + 3} \vartheta } = \frac{k + 1}{2\pi }\frac{{J_{0} }}{{2\pi R_{0} \,\Delta R}}\frac{1}{{d_{c - s}^{2} }}2\pi \,\Delta R\frac{{R_{0} }}{{\left( {1 + \left( {R_{0} /d_{s - c} } \right)^{2} } \right)^{(k + 3)/2} }} $$

(13.23)

For the deposition rate w follows

$$ w = \frac{{\left( {k + 1} \right)\,J_{0} }}{{n_{f} \,2\pi R_{0}^{2} }}\frac{{\left( {R_{0} /d_{c - s} } \right)^{2} }}{{\left( {1 + \left( {R_{0} /d_{c - s} } \right)^{2} } \right)^{(k + 3)/2} }} $$

(13.24)

The development for small distances ${\text{d}}_{{\text{c}}\text{-}{\text{s}}} \ll {\text{R}}_{0}$ yield a central deposition rate

$$ w \approx \frac{{\left( {k + 1} \right)J_{0} }}{{n_{f} \,2\pi R_{0}^{2} }}\left( {\frac{{d_{c - s} }}{{R_{0} }}} \right)^{k + 1} $$

(13.25)

increasing with the substrate distance ~d ^k+1_c−s . For large distances ${\text{d}}_{{\text{c}}\text{-}{\text{s}}} \gg {\text{R}}_{0}$ the rate is described by the dependence ~1/d ²_c−s for a point source (13.22), independent on the specific source distribution. The deposition rate is maximum at

$$ {\text{d}}_{{\text{c}} - {\text{s}}} \,/\,{\text{R}}_{ 0} = (({\text{k}} + 1)\,/\,2)^{1/2} $$

(13.26)

1.2 E2 Deposition Rate with Rotating Carriers (for 13.16, 13.17)

Single rotation

The substrates are attached on a uniformly rotating cylindrical carrier (radius R). The axis is parallel to the line source of length L in the distance R + d_c−s from the source. Here d_c−s denotes the shortest distance between the cathodic line source and the substrates. The particle flux dJ into the angular segment dϑ is described by

$$ {\text{dJ}} =\upgamma\,{ \cos }^{\text{k}}\upvartheta\,{\text{d}}\upvartheta $$

(13.27)

Only the fraction between −ϑ₀ and +ϑ₀ hit the cylindrical surface and is during the rotation distributed over the surface with the area 2π R L. (Edge effects are neglected.) The limiting angle ϑ₀ is given by the tangent from the source to the cylinder:

$$ { \sin }\,\upvartheta_{0} = {\text{R}}\,/\,\left( {{\text{R}} + {\text{d}}_{{\text{s}} {-}{\text{c}}}} \right) $$

(13.28)

The mean particle flux density on the cylindrical carrier amounts to

$$ j = \frac{\gamma }{2\pi \,R\,L}\int\limits_{{ - \vartheta_{0} }}^{{\vartheta_{0} }} {d\vartheta \,{ \cos }^{k} \vartheta } $$

(13.29)

The mean deposition rate w = dh/dt is given by

$$ {\text{w}} = {\text{j}}\,/\,{\text{n}}_{\text{f}} $$

(13.30)

where n_f denotes the particle density within the film.

In the case of immovable substrate, fixed in the distance d_c−s, the particle flux is given by

$$ {\text{dJ}}_{ 0} =\upgamma\,{\text{d}}\upvartheta =\upgamma\,{\text{ds}}\,/\,{\text{d}}_{{\text{c}}\text{-}{\text{s}}} =\upgamma\,{\text{L}} \times {\text{ds}}\,/\,({\text{L}} \times {\text{d}}_{{\text{c}} {-}{\text{s}}}) =\upgamma\,{\text{dA}}\,/\,({\text{L}} \times {\text{d}}_{{\text{c}} {-}{\text{s}}}) = {\text{j}}_{ 0} \,{\text{dA}} $$

(13.31)

Hence, the ratio of the deposition rate w of rotating samples to fixed ones amounts to

$$ \frac{w}{{w_{0} }} = \frac{{d_{c - s} }}{2\pi \,R}\int\limits_{{ - \vartheta_{0} }}^{{\vartheta_{0} }} {d\vartheta \,{ \cos }^{k} \vartheta } < \frac{{d_{c - s} }}{2R} $$

(13.32)

For large distances ${\text{d}}_{{\text{c}}\text{-}{\text{s}}} \gg {\text{R}}$, the limiting angle can be approximated by ϑ₀ ≈ R/d_c−s, leading to w/w₀ ≈ 1/π. It corresponds to the ratio of the projected area 2 R · L to the cylinder surface 2π R · L.

The integral in (13.24) depends only weakly on the exponent k. The rotation blurs the specifics of the angular distribution. Hence, the expression can be approximated by the value for k = 1:

$$ \frac{w}{{w_{0} }} \approx \frac{{d_{c - s} }}{2\pi R}\int\limits_{{ - \vartheta_{0} }}^{{\vartheta_{0} }} {d\vartheta \,{ \cos }^{1} \vartheta } = \frac{{d_{c - s} }}{\pi R}{ \sin }\,\vartheta_{0} = \frac{{d_{c - s} }}{{\pi \,\left( {R + d_{s - c} } \right)}} $$

(13.33)

Twofold and threefold rotations

In the case of the additional rotation of cylindrical samples around their axes, more samples with a correspondingly increased surface can be arranged within the coater. By the rotations, the particle flux is in the mean uniformly distributed over the surfaces, notwithstanding temporarily shadowing. The circumference of the cylindrical carrier (radius R) can be completely filled by 2πR/2r cylindrical samples (radius r). The total surface amounts to A₂ = 2πR/2r 2πr · L = 2π² R · L in comparison to the surface A₁ = 2πR · L of the carrier cylinder. The particle flux is now distributed over a larger area. Due to w = j/n = J/A 1/n, the ratio of the growth rates w₂/w₁ is given by the inverse ratio of the coated areas:

$$ {\text{w}}_{ 2} /{\text{w}}_{ 1} = {\text{A}}_{ 1} /{\text{A}}_{ 2} = 1/\uppi $$

(13.34)

The same argumentation holds also for the relation between the growth rates with threefold and twofold rotations: w₃/w₂ = A₂/A₃ = 1/π. The total reduction of the growth rate in a planetary with three axes in comparison to a single rotation is given by 1/π² = 0.101.