Model for roughening and ripple instability due to ion-induced mass redistribution [Addendum to H. Hofsäss, Appl. Phys. A 114 (2014) 401, “Surface instability and pattern formation by ion-induced erosion and mass redistribution”]

Abstract

Carter and Vishnyakov introduced a model (CV model) to describe roughening and ripple instability due to ion-induced mass redistribution. This model is based on the assumption that the irradiated surface layer on a static solid substrate is described by a viscous incompressible thin film bound to the substrate by a “no slip” and “no transport” kinematic boundary condition, i.e. similar to a thin film of viscous paint. However, this boundary condition is incomplete for a layer under ion irradiation. The boundary condition must allow exchange of atoms between the substrate and the irradiated film, so that the thickness of the film is always determined by the size of the collision cascade, independent of the evolution of the surface height profile. In addition, the film thickness depends on the local ion incidence angle, which leads to a time dependence of the film thickness at a given position. The equation of motion of the surface and interface profiles for these boundary conditions is introduced, and a new curvature-dependent coefficient is found which is absent in the CV model. This curvature coefficient depends on the angular derivative of the layer thickness and the atomic drift velocity at the film surface induced by recoil events. Such a stabilizing curvature coefficient was introduced in Appl. Phys. A 114 (2014) 401 and is most pronounced at intermediate angles.

Appendices

Appendix 1: Calculation of the surface drift velocity

The calculations of the three-dimensional collision cascades were done with the Monte Carlo Code SDTrimSP V5.05 [11, 12]. SDTrimSP provides a large variety of input and output options. The proper selection of input parameters and further details of the simulations are described in Ref. [10].

The simulation results shown in Figs. 1 and 2 are obtained for ion incidence angular steps of 5°, and typically taking into account N _ion = 10⁵ incident ions and mean values of the recoil distributions, surface and average drift velocities, the number N _ko(θ) of knock-on collision events and N _D (θ) of displacements per ion were derived from all recoil events (up to several million events).

SDTrimSP and TRIDYN use a coordinate system (X, Y, Z) with the X axis pointing opposite to the surface normal and the Y axis lying in the surface along the projected beam direction. The coordinate system (u, v, w) commonly used to describe pattern formation, e.g. in the BH model [1], has the w axis parallel to the global surface normal and the u axis along the projected direction of the ion beam. Here I use the system (u, v, w) = (Y, −Z, −X), so we can retain the (X, Y, Z) denotation used in the SDTrimSP output files.

The mean depth of the recoil distribution D(θ) is obtained from the X _end positions of N _R = N _ion·N _D calculated recoils events, where X _end is the X coordinate (normal to the surface) of the end coordinate of recoil atoms.

$$D(\theta ) = \frac{1}{{N_{\text{R}} }}\sum\limits_{i = 1}^{{N_{\text{R}} }} {X_{{i,{\text{end}}}} (\theta )} .$$

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Values for mass transport parallel to the surface of N _R target atoms recoiled with energy larger than E _D and stopped inside the target are directly obtained as mean value of all recoil events. For a number of N _R recoils (typically several million), the average parallel mass redistribution distance per ion δ _u (θ) is given by

$$\delta_{u} (\theta ) = N_{\text{ko}} (\theta )\;\frac{1}{{N_{\text{R}} }}\sum\limits_{i = 1}^{{N_{\text{R}} }} {\left( {Y_{{i , {\text{end}}}} (\theta ) - Y_{{i , {\text{start}}}} (\theta )} \right)} .$$

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This value δ _u (θ) is used in Eq. (1) to calculate the average drift velocity.

For a certain depth interval {w, w + dw}, we obtain a recoil distance δ _u (w, θ) per incident ion. Here we consider recoils generated within this interval but may become stopped outside. The accuracy of the depth interval can be evaluated with the standard deviation for a given range of start positions X _i,start.

$$\sigma \approx \sqrt {\frac{1}{N}\sum\limits_{i = 1}^{N} {\left( {X_{{i,{\text{end}}}} - X_{{i,{\text{start}}}} } \right)^{2} } } .$$

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Typically σ is about 0.1–0.2 nm for low ion energies and about 0.5–0.8 nm for an ion energy of 10 keV.

The fraction f(z, θ) of recoils generated in a specific depth interval {w, w + dw} is given by the ratio ${{f(w,\theta ) = N_{{{\text{R}},w}} } \mathord{\left/ {\vphantom {{f(w,\theta ) = N_{{{\text{R}},w}} } {N_{\text{R}} }}} \right. \kern-0pt} {N_{\text{R}} }}$, where N _R is the total number of simulated recoils and N _R,w the recoils occurring in the depth interval {w,w + dw}. We obtain

$$\delta_{u} (w,\theta ) = N_{ko} (\theta )\;\frac{1}{{N_{R} }}\sum\limits_{i = 1}^{{N_{R, - X} }} {\left( {Y_{i,end} (\theta ) - Y_{i,start} ( - X,\theta )} \right)} .$$

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For the mean mass transport distance per ion at the surface, we sum only over all recoil events in a certain top surface layer between X = 0 and X = d _top, with d _top corresponding to a sufficiently thin surface layer of 2–3 atomic layers or about d _top ≈ 0.6 − 0.9 nm. Some recoils which are generated in the uppermost atomic layer in the SDTrimSP simulation may have the chance to be recoiled along the surface, resulting in artificially increased recoil distances. This can be seen in Fig. 1 for 1 keV Ar, where the drift velocity in the 0.1-nm bin closest to the surface is rather large. It is therefore not recommended to calculate the mean mass transport distance per ion for a single atomic surface layer or a thickness smaller than 0.6 nm. From Fig. 1, it can be seen that a surface drift velocity as average over the first 6 bins (0.6 nm) is a reasonable choice to determine the surface drift velocity.

$$\delta_{u,top} (\theta ) = N_{\text{ko}} (\theta )\;\frac{1}{{N_{\text{R}} }}\sum\limits_{i = 1}^{{N_{\text{R,top}} }} {\left( {Y_{{i , {\text{end}}}} (\theta ) - Y_{{i , {\text{start}}}} (\theta )} \right)} .$$

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For a given ion incidence angle θ, the forward drift velocity profiles v _u (w, θ) can be determined using $\delta_{u} (w,\theta )$ given by Eq. (37). The drift velocity in the layer at depth w with thickness dz is then given by (the θ dependence of the parameters is omitted)

$$v_{u} (w,\theta ) = \frac{J}{n}\cos \theta \;\frac{{\delta_{u} (w)}}{{{\text{d}}w}} = \frac{J}{n}\frac{{\cos \theta \;N_{\text{ko}} }}{{{\text{d}}w\;N_{\text{R}} }}\;\sum\limits_{i = 1}^{{N_{{{\text{R}}, - X}} }} {\left( {Y_{{i , {\text{end}}}} - Y_{{i , {\text{start}}}} ( - X)} \right)} .$$

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For the top surface layer with thickness d _top ≈ 0.6 nm of about two atomic layers, we obtain surface drift velocity as:

$$v_{u} (h,\theta ) = \frac{J}{n}\cos \theta \;\frac{{\delta_{{x , {\text{top}}}} }}{{d_{\text{top}} }} = \frac{J}{n}\frac{{\cos \theta \;N_{\text{ko}} }}{{d_{\text{top}} N_{\text{R}} }}\;\sum\limits_{i = 1}^{{N_{\text{R,top}} }} {\left( {Y_{{i , {\text{end}}}} - Y_{{i , {\text{start}}}} (0 \le - X \le d_{\text{top}} )} \right)} .$$

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Appendix 2: Evaluation of the layer thickness dependence

The layer thickness at point u ₀ with angle θ between surface normal and ion direction is determined by ion impact at point $u^{*} = \, u_{0} - \Delta u$ at an angle $\theta - \gamma^{*}$ with respect to the local surface normal. Here, Δx is given by Δu ≈ d(θ) · tanθ.

The difference in height between u ₀ and ${\text{u}}^{*}$ is $h^{*} = \frac{1}{2}K(\Delta u)^{2}$, where K is the curvature of the surface. The angle $\gamma ^{*}$ is then given by $\gamma ^{*} = - \frac{{{\text{d}}h^{*}}}{{{\text{d}}x}} = - K \cdot \Delta u$. In linear expansion, we find for the layer thickness $d(\theta - \gamma ) = d(\theta ) - \frac{\partial d}{\partial \theta }\gamma$. The layer thickness $d^{*}$ for ion impact at $u^{*}$ with local angle $\theta - \gamma^{*}$ is then $d(\theta - \gamma ^{*}) = d(\theta ) - \frac{\partial d}{\partial \theta }\gamma^{*}$. Substituting $\gamma ^{*}$, we find $d^{*} = d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K \cdot \;\tan \theta } \right)$. At point u ₀, we have a layer thickness d(u ₀) which is calculated from the $d^{*}$ projected to the surface normal at u ₀ and subtracting the height differences between u ₀ and $u^{*}$

$$\begin{aligned} d(u_{0} ) & = & d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K\;\tan \theta } \right)\frac{1}{\cos \gamma ^{*}} - h^{*} - \varDelta u\gamma ^{*} \\ & = & d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K\;\tan \theta } \right)\frac{1}{\cos \gamma ^{*}} - \frac{1}{2}Kd^{2} (\theta )\;\tan^{2} \theta + Kd^{2} (\theta )\;\tan^{2} \theta \\ & = & d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K\;\tan \theta } \right)\frac{1}{\cos \gamma ^{*}} + \frac{1}{2}Kd^{2} (\theta ) \cdot \tan^{2} \theta \\ \end{aligned}$$

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Now we consider an impact point shifted by du away from $u^{*}$. The height different between u ₀ + du and $u^{**} \, = \, u^{*} + {\text{d}}u$ with respect to the local surface normal at u ₀ is $h^{**} = \frac{1}{2}K(\Delta u - {\text{d}}u)^{2} - \frac{1}{2}K({\text{d}}u)^{2} = \frac{1}{2}K(\Delta u)^{2} - K\Delta u{\text{d}}u$, where K is the curvature of the surface. The angle $\gamma^{**}$ is then given by $\gamma^{**} = - \frac{{{\text{d}}h^{**}}}{{{\text{d}}x}} = - K\;\left( {\Delta u - {\text{d}}u} \right)$. We find in analogy to the calculation above $d^{**} = d(\theta )\left( {1 + d(\theta )\;\frac{\partial d}{\partial \theta }K\;\tan \theta } \right) - \frac{\partial d}{\partial \theta }K{\text{d}}u$. At point u ₀ + du, we find a layer thickness

$$\begin{aligned} d(u_{0} + {\text{d}}u) & = & \left[ {d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K\;d(\theta )\;\tan \theta } \right) - \frac{\partial d}{\partial \theta }K\;{\text{d}}u} \right]\frac{1}{\cos \gamma ^{*}} - h^{**} - \varDelta u\gamma^{**} \\ & = & \left[ {d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K\;d(\theta )\;\tan \theta } \right) - \frac{\partial d}{\partial \theta }K\;{\text{d}}u} \right]\frac{1}{\cos \gamma ^{*}} \\ \quad - \frac{1}{2}K\;d^{2} (\theta )\;\tan^{2} \theta + Kd(\theta )\;\tan \theta {\text{d}}u + K \cdot d(\theta )\;\tan \theta \left( {d(\theta )\;\tan \theta - {\text{d}}u} \right) \\ & = & \left[ {d(\theta )\left( {1 + \frac{\partial d}{\partial \theta }K\;d(\theta )\;\tan \theta } \right) - \frac{\partial d}{\partial \theta }K\;{\text{d}}u} \right]\frac{1}{\cos \gamma ^{*}} + \frac{1}{2}K\;d^{2} (\theta )\;\tan^{2} \theta \\ \end{aligned}$$

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Finally, we obtain with ${ \cos } \gamma^{*} \approx 1$

$$\frac{\partial d}{\partial u} = \mathop {\lim }\limits_{dx \to 0} \frac{{d(u_{0} + {\text{d}}u) - d(u_{0} )}}{{{\text{d}}u}} = - K\frac{\partial d}{\partial \theta }\frac{1}{\cos \gamma ^{*}} \approx - \frac{\partial d}{\partial \theta }\frac{{\partial^{2} h}}{{\partial u^{2} }}$$

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