Growth of ta-C Films

Abstract

The understanding of the growth of tetrahedrally bonded amorphous carbon is based on the experimentally selected conditions for the preparation of ta-C films: sufficient high kinetic energy of the impinging carbon particles of at least some ten electron-volt and sufficient low temperatures, usually below 150 °C (see Chap. 7).

Explanations

1.1 E1 Stopping Power S with Power Law (for (8.5), (8.32))

For the impact of carbon ions on carbon targets, the stopping power S(ε) is according to (8.4) approximately given by

$$ \text{S}(\upvarepsilon) = - \text{d}\,\upvarepsilon/\text{ds}_{\text{i}} \approx \text{A}\,\upvarepsilon^{{1 - \text{B}}} $$

(8.80)

with B ≈ 0.6. For an ion with the initial energy ε₀, the relation between path length s_i and residual energy ε is given by

$$ s_{i} \left( \varepsilon \right) = - \int\limits_{{\varepsilon_{0} }}^{\varepsilon } {d\varepsilon^{\prime } } \frac{1}{{S\left( {\varepsilon^{\prime } } \right)}} = \frac{1}{A}\int\limits_{{\varepsilon_{0} }}^{\varepsilon } {d\varepsilon^{\prime } } \varepsilon^{{\prime^{B - 1} }} = \frac{1}{AB}\left( {\varepsilon_{0}^{B} - \varepsilon^{B} } \right) $$

(8.81a)

and

$$ \upvarepsilon\left( \text{s}_{\text{i}} \right) =\upvarepsilon_{0} \left( {1{-}\text{s}_{\text{i}} /\text{s}_{0} } \right)^{{1/\text{B}}} $$

(8.81b)

where s₀ = ε ^B₀ /AB denotes the total path length up to the ion stop. The fraction ε/ε₀ of the ion energy is lost on the first part s_i/s₀ = 1 − (ε/ε₀)^B. For ε/ε₀ = 0.5 and B = 0.6 follows s_i/s₀ = 0.34.

With (8.81a, b), the variation of the stopping power along the ion track follows to

$$ \text{S}\left( {\text{s}_{\text{i}} } \right) = \text{A}\left( {\text{AB}} \right)^{{\left( {1 - \text{B}} \right)/\text{B}}} \left( {\text{s}_{0} {-}\text{s}_{\text{i}} } \right)^{{\left( {1 - \text{B}} \right)/\text{B}}} = \left( {\text{AB}} \right)^{{1/\text{B}}} /\text{B}\left( {\text{s}_{0} {-}\text{s}_{\text{i}} } \right)^{{1/\text{B}{-}1}} $$

(8.82)

The stopping power decreases along the path.

From S = −dε/dt 1/v follows for the time t_imp of a collision cascade:

$$ t_{imp} \left( {\varepsilon_{0} } \right) = \sqrt {\frac{m}{2}} \int\limits_{0}^{{\varepsilon_{0} }} {d\varepsilon } \frac{1}{\sqrt \varepsilon \,S\left( \varepsilon \right)} = \sqrt {\frac{m}{2}} \frac{1}{A}\int\limits_{0}^{{\varepsilon_{0} }} {d\varepsilon \,\varepsilon^{B - 1/2} } = \sqrt {\frac{m}{2}} \frac{1}{{A\,\left( {B - 1/2} \right)}}\varepsilon_{0}^{B - 1/2} $$

1.2 E2 Binary Collision Approximation and TRIM Program (for Sect. 8.2)

TRIM (=Transport of Ions in Matter) is a Monte Carlo program for the simulation of the impact of energetic ions into amorphous carbons, developed by the groups of J.P. Biersack and J.F. Ziegler [23, 95] and continuously updated [24]. It is based on the binary collision approximation, i.e. the influence of neighbored atoms is neglected.

An ion (mass m_i, velocity v₀, energy ε₀ = m_i/2 v ²₀ ) is colliding with a resting target atom (mass m_t). The collision process is best treated within the center of mass system (CMS), which is moved with the velocity v_C = m_i/(m_i + m_t) v₀ against the laboratory system. If the interaction of the both particles depends only on their mutual distance r, their motion can be reduced to the movement of one particle with the reduced mass

$$ \upmu_{\text{C}} = \text{m}_{\text{i}} \text{m}_{\text{t}} /\left( {\text{m}_{\text{i}} + \text{m}_{\text{t}} } \right) $$

(8.83)

in the interaction potential V(r). Due to the vanishing total momentum, particles move always in the opposite direction. In CMS, the system momentum is zero before and after the collision: m_i v_i = m_t v_t, corresponding to a constant velocity ratio v_i/v_t = m_t/m_i. From the energy conservation follows 2 ε = m_i v ²_i  + m_t v ²_t  = v ²_t (m_i (v_i/v_t)² + m_t) = v ²_t (m ²_t /m_i + m_t) = const. That means, that the absolute values of the velocities remain unchanged. For balancing the momentums, the particles must fly into opposite directions. Hence, the scattering angles Φ of the target atom and Θ of the ion are related by Φ + Θ = π in the CMS. They can be transformed into the corresponding angles φ and ϑ within the laboratory system

$$ \begin{array}{*{20}l} {{\varphi } =\Phi /2} \hfill & {\tan \,\upvartheta = \left( {\text{m}_{\text{t}} /\text{m}_{\text{i}} } \right)\,\sin \,\Theta /(1 + \left( {\text{m}_{\text{t}} /\text{m}_{\text{i}} } \right)\cos \,\Theta )} \hfill \\ \end{array} $$

(8.84)

The kinetic energy T transferred by the collision from the ion to the target atom is given by

$$ \text{T} = 4\,\upvarepsilon_{0} \text{m}_{\text{i}} \,\text{m}_{\text{t}} /\left( {\text{m}_{\text{i}} + \text{m}_{\text{t}} } \right)^{2} \,\text{sin}^{2}\Theta /2 $$

(8.85)

The angle Θ can be calculated from the “general orbit equation”

$$ \Theta = \pi - 2\int\limits_{{r_{0} }}^{\infty } {\frac{p\,dr}{{r^{2} \sqrt {1 - \frac{{V(r)(1 + m_{i} /m_{t} )}}{{\varepsilon_{0} }} - \frac{{p^{2} }}{{r^{2} }}} }}} $$

(8.86)

where r₀ denotes the distance of closest approach of both particles. The impact parameter p constitutes the perpendicular distance of the ion direction from the resting atom, i.e. the smallest distance, the ion would pass the target atom, if it would not be deflected.

In the case of coincident kinds of projectile and target atoms (m_i = m_t) holds

$$ \upvartheta =\Theta /2,\quad {\varphi } =\uppi/2 -\upvartheta $$

(8.87)

$$ \text{T} =\upvarepsilon_{0} \text{sin}^{2}\upvartheta $$

(8.88)

In the TRIM program, the interatomic potential V(r) is described by the electrostatic repulsion of the both nuclei, partially screened by their merging electron clouds:

$$ V(r) = \frac{1}{{4\pi \,\varepsilon_{0} }}\frac{{Z_{i} \,Z_{t} \,e^{2} }}{r}\,\Phi _{i,t} (r) $$

(8.89)

The effect of the electrons is considered by the screening function Φ_i,t(r). It is determined by the interaction of fixed electron distributions without any reconfiguration during the collision. For the ions and the target atoms as well electron distributions of their solid states are used. They are spherically symmetrized by averaging over all spatial orientations. Besides the electrostatic interactions also quantum-mechanical effects are included in a simplified manner. The individual functions Φ_I,t can be approximated by a universal function Φ(r/a), where the atomic distance r is scaled by the “screening length” a = 0.8854 a₀/ (Z ^0.23_i  + Z ^0.23_t ) and a₀ = 0.0529 nm (Bohr radius):

$$ \Phi (x) = 0.1818\,e^{ - 3.2x} + 0.5099\,e^{ - 0.9423x} + 0.2802\,e^{ - 0.4028x} + 0.2817\,e^{ - .2016x} $$

(8.90)

The individual collision is determined by the energy of the impacting ion ε₀ and the impact parameter p. According to the random atomic arrangement, the probability W(p) for an impact parameter below p is given by the corresponding area fraction

$$ \begin{array}{*{20}l} {\text{W}\left( \text{p} \right) =\uppi\,\text{p}/\text{L}^{2} } \hfill & {\text{for}\,\text{p} < \text{L}/\surd\uppi} \hfill \\ \end{array} $$

(8.91)

For larger p values the collision is assigned to the neighbored atom. In the Monte Carlo simulation the impact parameter is selected from

$$ \text{p} = \text{L}(\text{R}/\uppi)^{1/2} $$

(8.92)

where R denotes a random number, evenly distributed between 0 and 1. For the determination of Θ, TRIM uses the expression

$$ \cos \,\Theta /2 = \frac{\rho + p + \delta }{{\rho + r_{0} }} $$

(8.93)

instead of the general orbit equation for saving computer capacity. Herein the distance of closest approach r₀ is given by the condition

$$ 1 - \frac{{V(r_{0} )(1 + m_{i} /m_{t} )}}{{E_{0} }} - \frac{{p^{2} }}{{r_{0}^{2} }} = 0 $$

(8.94)

The quantity ρ = ρ_i + ρ_t, the sum of the radii of curvature of the trajectories at closest approach, can be obtained from

$$ \rho = \frac{{2\left( {\frac{{E_{0} }}{{1 + m_{i} m_{t} }} - V(r_{0} )} \right)}}{{ - V^{\prime } (r_{0} )}} $$

(8.95)

The only approximation consists in the treatment of the usually small correction term δ. According to a fitting procedure it is derived from an analytical expression, depending on E₀, p, m_i/m_t, Z_i and Z_t.

After the collision, it is assumed, that the particles fly straightforward in the asymptotic directions up to next collision after a path length λ. The maximum flight distance is estimated by λ_max ≈ L = n^−1/3 (n = atomic density of the target). For high energy ions, the free flight path approximation of TRIM saves CPU time by using larger free path length according to the mean distances, where a relevant collision with scattering by some degrees could be expected. The statistical arrangement in the amorphous material is considered by a random factor R between 0 and 1 (λ = R · λ_max).

In this way the path of an impinging ion is followed from collision to collision until the ion energy is reduced to a fixed value. The same procedure is made with the collided atoms, if the transferred energy ε_t exceeds the displacement energy ε_d. Then it will be the starting point of a new collision cascade with the initial energy ε_t − ε_b, where ε_b denotes the binding energy.

Atoms, which cross the surface plane leave the target, if their kinetic energy in normal direction exceeds the surface binding energy ε_sb. It is usually identified with the sublimation energy, notwithstanding possible modifications by roughness and ion bombardment.

1.3 E3 Film Growth by Subplantation (for (8.28))

Under the continuous impact of ions the surface region of a ta-C film grows by incorporation of the incident ions beneath the surface (subplantation). The incident ion flux density j(x, t) decreases with the depth due to stopping in upper layers. The flux density of the ions, stopping in Δx, is given by ∂j/∂x Δx, where x is directed inwards opposite to the growth direction. The particle number (per area) ΔN in a thin slab of thickness Δx in the depth x amounts to ΔN = n(x, t) Δx. It increases in time according to

$$ - \partial \text{j}/\partial \text{x}\,\Delta \text{x} = \text{d}\Delta \text{N}/\text{dt} = \Delta \text{x}\,\partial \text{n}/\partial \text{t} + \text{n}\partial \Delta \text{x}/\partial \text{t} $$

(8.96)

It is assumed, that by elastic expansion the width Δx will increase in proportion to the relative change of the density:

$$ \frac{\partial \Delta x/\partial t}{\Delta x} = c\frac{\partial n/\partial t}{n} $$

(8.97)

Then (8.96) reduces to

$$ - \partial \text{j}/\partial \text{x} = \left( {1 + \text{c}} \right)\partial \text{n}/\partial \text{t} $$

(8.98)

In the stationary state the density distribution shifts outwards with the linear growth rate w = dh/dt:

$$ \text{n}\left( {\text{x} - \text{wt},\text{t}} \right) = \text{n}\left( {\text{x},0} \right) = \text{const}. $$

(8.99)

This allows to transform the time dependence into a depth dependence:

$$ \text{dn}/\text{dt} = - \text{w}\,{\partial }\,\text{n}/{\partial }\,\text{x} + {\partial }\,\text{n}/{\partial }\,\text{t} = 0 \Rightarrow {\partial }\,\text{n}/{\partial }\,\text{t} = \text{w}\,{\partial }\,\text{n}/{\partial }\,\text{x} $$

(8.100)

The incident particle flux density j_s = j(x_s) leads in δt on the area A to an increase of the film volume δV = A δh = v j_s A δt, where v = 1/n₀ denotes the atomic volume in the bulk film with the density n₀:

$$ \text{w} = \text{dh}/\text{dt} = \text{j}_{\text{s}} /\text{n}_{0} $$

(8.101)

With these equations, (8.98) can be transformed into

$$ \partial \text{j}/\partial \text{x} = - \left( {1 + \text{c}} \right)\text{j}_{\text{s}} /\text{n}_{0} \,\partial \text{n}/\partial \text{x} $$

(8.102)

The stationary density distribution n(x) is then given by

$$ n(x) = n(x,0) = - \frac{{n_{0} }}{{(1 + c)j_{s} }}\int\limits_{0}^{x} {dx^{\prime } } \frac{{\partial j(x^{\prime } ,0)}}{\partial x} + b = \frac{{n_{0} }}{{(1 + c)j_{s} }}\left( {j(0,0) - j(x,0)} \right) + b $$

(8.103)

The both unknown constants b and c are determined from the density n(0) = n_s at the surface, where j(0, 0) = j_s and from the bulk density n₀ in the depth beyond the ion range, where j = 0.

The resulting density distribution within the surface region

$$ \text{n}\left( \text{x} \right) = \text{n}_{0} - \left( {\text{n}_{0} {-}\text{n}_{\text{s}} } \right)\text{j}\left( \text{x} \right)/\text{j}_{\text{s}} $$

(8.104)

changes accordingly to the local ion flux density j(x).

1.4 E4 Kinchin-Pease Model (for Sect. 8.2)

The Kinchin-Pease model determines the number of displaced atoms by using the following approximations for the formation of the collision cascade [32, 42]:

• separation into a series of two-body collisions,

• elastic collisions (only exchange of kinetic energy, no electronic excitations, no lattice excitation),

• atoms as hard spheres (interaction only in direct contact, changed direction according to regular reflection)

• random arrangement of the atoms (no orientation effects such as channeling)

• displacement of an atom from a stable site to a metastable position if and only if the energy of the moving atom exceeds the displacement energy ε_d, If the remaining energy of the colliding atom has been dropped below ε_d, it rests at the site of the displaced atom (replacement collision).

When two identical hard and smooth spheres (radius R) collide, there is only some transfer of momentum perpendicular to their contact plane. Hence, in the center-of-mass system the deflection of the spheres corresponds to regular reflection with the same angle ϑ for entrance and exit. With cosϑ = b/2R it is determined by the collision parameter b (Fig. 8.47). According to (8.84) the energy T transferred by a collision with the kinetic energy ε₀ is given by the asymptotic scattering angle Θ = 2ϑ:

Fig. 8.47

Scheme of the collision of two identical hard spheres in their center-of-mass system

$$ \text{T} =\upvarepsilon_{0} \text{sin}^{2}\Theta /2 =\upvarepsilon_{0} \left( {1{-}\left( {\text{b}/2\text{R}} \right)^{2} } \right) $$

(8.105)

The probability dP for a specific energy transfer T or for the corresponding b value is given by the area ratio of the b range dA = 2π b db to the total cross section A = π (2R)²:

$$ \text{dP} = \text{dA}/\text{A} = \text{b}/2\text{R}^{2} \text{db} = \left| {\text{dT}} \right|/\upvarepsilon_{0} $$

(8.106)

The energy transfer is equally distributed between 0 and ε₀. In the mean the energy is symmetrically split between the moving particle and the particle in rest. The mean transferred energy <T> = ε₀/2 corresponds to scattering angles Θ = 90° in the CMS and ϑ = 45° in the laboratory system.

When by successive collisions the energy of all atoms in the cascade lies in the range between 0 and 2ε_d, the formation of the cascade has been completed and the maximum number N_d of displaced atoms has been achieved. Atoms with a transferred energy below ε_d must remain on their original positions. Atoms with an energy between ε_d and 2ε_d leave their sites. They may (1) displace another atom by transferring T > ε_d, but have now not enough energy to leave the freed site or (2) they transfer less energy <ε_d and will itself further run as an interstitial. In both cases the number of displaced atoms is not changed. Thus the atoms with final energies in the range between ε_d and 2ε_d (or their successors) are the sought N_d displaced atoms. Due to the equipartition of the transferred energy, the number of atoms in these both ranges of the same width ε_d coincides. The total number N of atoms, which all carry now some of the impact energy is given by number 2 N_d of bounced atoms plus the incident ion:

$$ N = 2\,N_{d} + 1 = \int\limits_{0}^{{2\varepsilon_{d} }} {d\varepsilon \,g\left( \varepsilon \right)} = \int\limits_{0}^{{2\varepsilon_{d} }} {d\varepsilon \,g = 2\varepsilon_{d} \,g} $$

(8.107)

where g(ε) = g = const. describes the final equipartitioned energy distribution. The total energy is given by

$$ \varepsilon_{0} = \int\limits_{0}^{{2\varepsilon_{d} }} {d\varepsilon \,g\left( \varepsilon \right)\,} \varepsilon = \frac{{\left( {2\varepsilon_{d} } \right)^{2} }}{2}\,g = \varepsilon_{d} \left( {2N_{d} + 1} \right) $$

(8.108)

This yields the number of displaced atoms in the collision cascade

$$ \text{N}_{\text{d}} = (\upvarepsilon_{0} -\upvarepsilon_{\text{d}} )/2\upvarepsilon_{\text{d}} $$

(8.109)

Equation (8.109) represents a modification of the well-known Kinchin-Pease formula [32, 42]

$$ \text{N}_{\text{d}} =\upvarepsilon_{0} /2\upvarepsilon_{\text{d}} $$

(8.110)

for lower energies, comparable with the displacement energy and correspondingly few displacements.

1.5 E5 Thermal Spike Model (for Sect. 8.3)

The thermal spike model treats the thermal induced processes after an ion impact in the approximation of a continuum with classical heat conduction.

The transient temperature field due to radial heat conduction from a concentrated point-like energy pulse ε (corresponding to the Green’s function of the three-dimensional heat conduction equation) is given by

$$ T\left( {r,t} \right) = T_{0} + \frac{\varepsilon }{{8\pi^{3/2} \rho c}}\frac{1}{{\left( {a\,t} \right)^{3/2} }}e^{{ - \frac{{r^{2} }}{4at}}} $$

(8.111)

where ρ, c and a denote the density, the mass specific heat capacity and the thermal diffusivity, respectively. The representative heat diffusion length amounts to l_th = √<r²> = (6 a t)^1/2.

The frequency f of atomic displacements is described by a thermally activated process with the activation energy q:

$$ f = f_{0} \,e^{{ - \frac{q}{kT}}} $$

(8.112)

The total number of rearrangements is then given by

$$ N_{d} = n_{0} \int {dt} \,\int {dV} \,f_{0} \,e^{{ - \frac{q}{{kT\left( {r,t} \right)}}}} $$

(8.113)

For the thermally induced processes during the ion impact with temperatures of some thousand Kelvin within the spike, the contribution of the basis temperature T₀ can be neglected. The exponentials are approximated by unit step functions: exp(−x) ≈ 1 for x < 1 and exp(−x) ≈ 0 for x > 1.

In the case of a point-like source, the thermal activated processes are concentrated in a central sphere with radius r(t) = l_th(t) = (6a · t)^1/2 and volume V(t) = 4π/3 r³(t). Inside the sphere, a homogeneous temperature T(t) = T(r(t), t) is assumed. The transition probability is one, as long kT(t) ≥ q. The number N_d of displacements is then given by:

$$ N_{d} \approx n_{0} \,f_{0} \int\limits_{0}^{{t_{d} }} {dt\,V_{d} \left( t \right) = n_{0} \,f_{0} \frac{64\pi }{15}a^{3/2} t_{d}^{5/2} } $$

(8.114)

The upper bound t_d is given by the condition kT(t_d) = q:

$$ t_{d} = \frac{1}{4\pi \,a}\left( {\frac{k\,\varepsilon }{q\,\rho c}} \right)^{2/3} $$

(8.115)

leading to [17, 73]

$$ N_{d} \approx 0.024\frac{{n_{0} \,f_{0} }}{a}\left( {\frac{k\,\varepsilon }{q\,\rho c}} \right)^{5/3} $$

(8.116)

With the approximations n₀ ≈ 1/d³, a ≈ f₀ d²/3 and ρc ≈ 3 nk = 3 k/d³, (8.116) simplifies to

$$ N_{d} \approx 0.012\left( {\frac{\varepsilon }{q}} \right)^{5/3} $$

(8.117)

According to the assumed relations between n₀, a and ρc, sometimes slightly deviation prefactors in (8.117) are used.

For a concentrated line source of length L, the pulse energy ε is distributed by the two-dimensional radial heat conduction according to

$$ T(r,t) = T_{0} + \frac{\varepsilon /L}{4\pi \,\rho c}\frac{1}{a\,t}e^{{ - \frac{{r^{2} }}{4at}}} $$

(8.118)

with l_th = √<r²> = 2 (a t)^1/2.

The thermal activated processes are now concentrated in a cylindrical environment with radius r(t) = l_th(t) = 2 (a · t)^1/2 and volume V(t) = π r²(t)·L. Inside the cylinder, a homogeneous temperature T(t) = T(r(t), t) is assumed. The number N_d of displacements is then given by [17, 74]:

$$ N_{d} \approx n_{0} \,f_{0} \int\limits_{0}^{{t_{d} }} {dt\,V_{d} \left( t \right)} = n_{0} \,f_{0} 2\pi \,L\,a\left( {t_{d} } \right)^{2} $$

(8.119)

The upper bound t_d is given by the condition kT(t_d) = q:

$$ t_{d} = \frac{1}{4\pi \,a}\frac{k\,\varepsilon }{q\,\rho c\,L} $$

(8.120)

leading to

$$ N_{d} \approx 0.040\frac{d}{L}\left( {\frac{\varepsilon }{q}} \right)^{2} $$

(8.121)

With the approximations n₀ ≈ 1/d³, a ≈ f₀ d²/3 and ρc ≈ 3 nk = 3 k/d³, (8.121) transforms into

$$ N_{d} \approx 0.013\frac{d}{L}\left( {\frac{\varepsilon }{q}} \right)^{2} $$

(8.122)

1.6 E6 Surface Force on Point Defects (for Sect. 8.4)

According to the continuum theory of lattice defects, an isotropic interstitial with an extra volume ΔV may be described by a spherical displacement dipole, characterized by the tensor ΔV/3 · Ι (Ι = unit tensor). Apart from the different tensorial nature, there is a strong correspondence to the electric dipole in electrostatics with the equivalence of ΔV/3 with the dipole moment p. The force F_x on an electric dipole (in x direction) is determined by the field gradient: F_x = p · dE_x/dx. The modification of the electric field of a dipole by a free surface in the distance d is given by the field of a “mirror dipole” beyond the surface. Its field is proportional to p/x³, leading to

$$ \text{F}_{\text{x}} \sim \text{p}^{2} /\text{d}^{4} $$

(8.123)

The analog in continuum mechanics amounts to

$$ F_{x} = \frac{G}{12\pi }\frac{{(1 + v)^{2} }}{1 - v}\frac{{(\Delta V)^{2} }}{{d^{4} }} $$

(8.124)

With G = shear modulus and ν = Poisson modulus [96].