Interfacial phase-change and geometry modify nanoscale pattern formation in irradiated thin films

Abstract

In this paper, we consider the linear stability of ion-irradiated thin films where the typical no-penetration boundary condition has been relaxed to a phase-change or mass conservation boundary condition. This results in the modification of the bulk velocity field by the density jump across the amorphous–crystalline interface as new material enters the film and instantaneously changes volume. In other physical systems, phase change at a moving boundary is known to affect linear stability, but such an effect has not yet been considered in the context of continuum models of ion-induced nanopatterning. We also determine simple closed-form expressions for the amorphous–crystalline interface in terms of the free interface, appealing directly to the physics of the collision cascade, which was recently shown to strongly modify the critical angle at which pattern formation is predicted to begin on an irradiated target. We find that phase-change at the amorphous–crystalline boundary imparts a strong ion, target, and energy dependence and, alongside a precise description of the interfacial geometry, may contribute to a unified, predictive, and continuum-type model of ion-induced nanopatterning valid across a wide range of systems. In particular, we consider argon-irradiated silicon, where the presence of phase-change at the amorphous–crystalline interface appears to predict an experimentally observed, strong suppression of pattern formation near 1.5 keV for that system.

Appendices

Appendix A: Details of linear stability analysis

1.1 A.1 Expansion: small perturbative wavenumber k

Here, we show the details of the long-wave expansion. As described in the main text, we expand as

$$\begin{aligned} \sigma = 0 + k \sigma _1 + k^2 \sigma _2 + {\mathcal {O}}(k^3), \end{aligned}$$

(A.1)

and we obtain the following systems at each order in k.

At ${\mathcal {O}}(1)$:

$$\begin{aligned} \begin{aligned}&\rho _{0z}{\tilde{w}}_{10} + \rho _0{\tilde{w}}_{10}' + {\tilde{\rho }}_{10}'w_{0} + {\tilde{\rho }}_{10}w_{0z} = 0, \\&\quad {\tilde{u}}_{10}'' = 0, \\&\quad -{\tilde{p}}_{ 10}' + 2\eta w_{10}'' = 0, \\&\quad {\tilde{\rho }}_{10} = \frac{-\rho _a {\tilde{\Delta }}_{10}}{(1+\Delta _0)^2}, \\&\quad w_0{\tilde{\Delta }}_{10}' + {\tilde{w}}_{10}\Delta _{0z} = 0. \end{aligned} \end{aligned}$$

(A.2)

At $z=0$,

$$\begin{aligned} \begin{aligned} \Delta _{0z}g_1 + {\tilde{\Delta }}_{10}&= 0, \\ {\tilde{u}}_{10} + u_{0z}{\tilde{g}}_1&= 0 ,\\ {\tilde{w}}_{10} + w_{0z}{\tilde{g}}_1&= 0. \end{aligned} \end{aligned}$$

(A.3)

At $z=h_0$,

$$\begin{aligned} \begin{aligned}&{\tilde{w}}_{10} + {\tilde{h}}_1 w_{0z} + \frac{V\rho ^*}{\rho _0^2}(\rho _{0z}{\tilde{h}}_{1} + {\tilde{\rho }}_{10}) = 0 ,\\&\quad \eta {\tilde{u}}_{10}' = 0, \\&\quad -{\tilde{p}}_{10} + 2\eta {\tilde{w}}_{10}' + T_{0z}^{33} = 0. \end{aligned} \end{aligned}$$

(A.4)

At ${\mathcal {O}}(\epsilon k)$:

$$\begin{aligned} \begin{aligned}&\sigma _1 {\tilde{\rho }}_{10} + i\rho _{0}{\tilde{u}}_{10} + \rho _{0z}{\tilde{w}}_{11} + \rho _0{\tilde{w}}_{11}' + {\tilde{\rho }}_{11}'w_0 + {\tilde{\rho }}_{11}w_{0z} = 0, \\&\quad -i{\tilde{p}}_{10} + \eta ( {\tilde{u}}_{11}'' + i{\tilde{w}}_{10}' )= 0 ,\\&\quad -{\tilde{p}}_{11}' + \eta ( 2{\tilde{w}}_{11}'' + i{\tilde{u}}_{10}') = 0 ,\\&\quad {\tilde{\rho }}_{11} = \frac{-\rho _a{\tilde{\Delta }}_{11}}{(1+\Delta _0)^2}, \\&\quad \sigma _1 {\tilde{\Delta }}_{10} + iu_0{\tilde{\Delta }}_{10} + w_0{\tilde{\Delta }}_{11}' + {\tilde{w}}_{11}\Delta _{0z} = 0. \end{aligned} \end{aligned}$$

(A.5)

At $z=0$,

$$\begin{aligned} \begin{aligned}&{\tilde{\Delta }}_{11} = 0, \\&\quad {\tilde{u}}_{11} + i{\tilde{g}}_1(w_0-V) = 0 ,\\&\quad -i{\tilde{g}}_1u_0 + {\tilde{w}}_{11} = \left( \frac{\rho _a-\rho _c}{\rho _a}\right) \sigma _1{\tilde{g}}_1. \end{aligned} \end{aligned}$$

(A.6)

At $z=h_0$,

$$\begin{aligned} \begin{aligned}&\sigma _{1}{\tilde{h}}_{1} = {\tilde{w}}_{11} -u_0i{\tilde{h}}_1 + \frac{V \rho _c}{\rho _0^2}{\tilde{\rho }}_{11}, \\&\quad \eta ({\tilde{u}}_{11}' + i{\tilde{w}}_{10}) - i {\tilde{h}}_{1}T_0^{11} = 0 ,\\&\quad -{\tilde{p}}_{11} + 2\eta {\tilde{w}}_{11}' = 0. \end{aligned} \end{aligned}$$

(A.7)

At ${\mathcal {O}}(\epsilon k^2$):

$$\begin{aligned} \begin{aligned}&\sigma _{1}{\tilde{\rho }}_{11} + \sigma _{2} {\tilde{\rho }}_{10} + i\rho _{0}{\tilde{u}}_{11} + \rho _{0z} {\tilde{w}}_{12} + \rho _0 {\tilde{w}}_{12}' + {\tilde{\rho }}_{12}' w_0 + {\tilde{\rho }}_{12} w_{0z} = 0, \\&\quad -i {\tilde{p}}_{11} + \eta (-2{\tilde{u}}_{10} + {\tilde{u}}_{12}'' + i{\tilde{w}}_{11}') = 0, \\&\quad -{\tilde{p}}_{12}' + \eta (-{\tilde{w}}_{10} + 2{\tilde{w}}_{12}'' + i{\tilde{u}}_{11}') = 0 ,\\&\quad {\tilde{\rho }}_{12} = \frac{-\rho _a{\tilde{\Delta }}_{12}}{(1+\Delta _0)^2}, \\&\quad \sigma _{1}{\tilde{\Delta }}_{11} + \sigma _{2}{\tilde{\Delta }}_{10} + iu_0{\tilde{\Delta }}_{11} + w_{0}{\tilde{\Delta }}_{12}' + {\tilde{w}}_{12}\Delta _{0z} = 0 \end{aligned} \end{aligned}$$

(A.8)

at z = 0,

$$\begin{aligned} \begin{aligned} {\tilde{\Delta }}_{12}&= 0, \\ {\tilde{u}}_{12}&= 0, \\ {\tilde{w}}_{12}&= \left( \frac{\rho _a-\rho _c}{\rho _a} \right) \sigma _2{\tilde{g}}_1, \end{aligned} \end{aligned}$$

(A.9)

and at z = $h_0$,

$$\begin{aligned} \begin{aligned} \sigma _{2} {\tilde{h}}_{1}&= {\tilde{w}}_{12} + \frac{V \rho _c}{\rho _0^2} {\tilde{\rho }}_{12,} \\ \eta ({\tilde{u}}_{12}' + i{\tilde{w}}_{11})&= 0, \\ -{\tilde{p}}_{12} + 2\eta {\tilde{w}}_{12}'&= 0. \end{aligned} \end{aligned}$$

(A.10)

1.2 A.2 Third expansion: small swelling rate $fA_I$

A third expansion in small swelling rate $fA_I$ is motivated by two observations. First, from previous results [64], it is known that the effect of even large swelling rates is highly self-similar at all wave numbers, and uniformly stabilizing for long waves. Second, as was seen in [63, 65], the expansion in small swelling rate is conducive to analytical solution; while it may be possible to solve the long-wave equations for arbitrary swelling rate analytically (as in [64]), the Appendix in [65] suggests that the linearized equations are substantially more complicated even in the long-wave limit. Hence we take

$$\begin{aligned} \begin{aligned} \alpha&= \alpha _{0} + fA_I\alpha _{1} +... ,\\ \rho _{0}&= \rho _{00} + fA_I\rho _{01} +...,\\&... \\ \sigma _{1}&= \sigma _{10} + fA_I\sigma _{11} +..., \\ {\tilde{\rho }}_{10}&= {\tilde{\rho }}_{100} + fA_I{\tilde{\rho }}_{111} +... ,\\&... \end{aligned} \end{aligned}$$

(A.11)

In following with [65], we shall only write out explicitly the equations expanded in $fA_I$, as the equations for the leading order terms are obvious from the above (simply by appending a “0” to the subscript of each term). We then obtain the following.

Steady state at ${\mathcal {O}}(fA_I)$:

$$\begin{aligned} \begin{aligned}&\frac{\partial }{\partial z}(\rho _{00}w_{01} + \rho _{01}w_{00}) = 0 ,\\&\quad \eta u_{01zz} = 0, \\&\quad -p_{01z} + 2\eta w_{01zz} = 0 \\&\quad \rho _{01} = \frac{-\rho _a \Delta _{01}}{(1+\Delta _{00})^2}, \\&\quad w_{00}\Delta _{01z} + w_{01}\Delta _{00z} = 1. \end{aligned} \end{aligned}$$

(A.12)

At z = 0:

$$\begin{aligned} \begin{aligned} \Delta _{01}&= 0, \\ u_{01}&= 0 ,\\ w_{01}&= 0. \end{aligned} \end{aligned}$$

(A.13)

At z = $h_0$,

$$\begin{aligned} \begin{aligned} w_{01} + V\rho _c\frac{\rho _{01}}{\rho _{00}^2}&= 0, \\ u_{01z}&= 0, \\ -p_{01} + 2\eta w_{01z}&= 0. \end{aligned} \end{aligned}$$

(A.14)

At ${\mathcal {O}}(\epsilon fA_I)$:

$$\begin{aligned} \begin{aligned}&\rho _{00z}{\tilde{w}}_{101} + \rho _{01z}{\tilde{w}}_{100} + \rho _{00}{\tilde{w}}_{101z} + \rho _{01}{\tilde{w}}_{100z} + {\tilde{\rho }}_{100z}w_{01} ,\\&\quad + \hspace{.125cm} {\tilde{\rho }}_{101z}w_{00} + {\tilde{\rho }}_{100}w_{01z} + {\tilde{\rho }}_{101}w_{00z} = 0 \\&\quad {\tilde{u}}_{101zz} = 0 ,\\&\quad -{\tilde{p}}_{101z} + 2\eta {\tilde{w}}_{101zz} = 0, \\&\quad w_{00}{\tilde{\Delta }}_{101z} + w_{01}{\tilde{\Delta }}_{100z} + {\tilde{w}}_{100}\Delta _{01z} + {\tilde{w}}_{101}\Delta _{00z} = 0 ,\\&\quad {\tilde{\rho }}_{101} = \frac{-\rho _a(\Delta _{00}{\tilde{\Delta }}_{101} + {\tilde{\Delta }}_{101} - 2\Delta _{01}{\tilde{\Delta }}_{100} )}{(1+\Delta _{00})^3}. \end{aligned} \end{aligned}$$

(A.15)

At z = 0:

$$\begin{aligned} \begin{aligned}&\Delta _{01z}{\tilde{g}}_{1} + {\tilde{\Delta }}_{101} = 0, \\&\quad {\tilde{u}}_{101} + u_{01z}g_{1} = 0, \\&\quad {\tilde{w}}_{101} + w_{01z}g_{1} = 0. \end{aligned} \end{aligned}$$

(A.16)

At z = $h_{0}$:

$$\begin{aligned} \begin{aligned}&{\tilde{w}}_{101} + {\tilde{h}}_{1}w_{01z} + V\rho _c\frac{( \rho _{00}(\rho _{01z}h_{1} + {\tilde{\rho }}_{101}) -2\rho _{01}(\rho _{00z}h_{1} + {\tilde{\rho }}_{100}))}{\rho _{00}^3} = 0,\\&{\tilde{u}}_{101}' = 0 ,\\&\quad -{\tilde{p}}_{101} + 2\eta {\tilde{w}}_{101}' + T_{01z}^{33}= 0. \end{aligned} \end{aligned}$$

(A.17)

At ${\mathcal {O}}(\epsilon kfA_I)$:

$$\begin{aligned} \begin{aligned}&\sigma _{10}{\tilde{\rho }}_{101} + \sigma _{11}{\tilde{\rho }}_{100} + i(\rho _{00}{\tilde{u}}_{101} + \rho _{01}{\tilde{u}}_{100}) + (\rho _{00z}{\tilde{w}}_{111} + \rho _{01z}{\tilde{w}}_{110}) ,\\&\quad + (\rho _{00}{\tilde{w}}_{111}' + \rho _{01}{\tilde{w}}_{110}') + ({\tilde{\rho }}_{110}'w_{01} + {\tilde{\rho }}_{111}'w_{00}) + ({\tilde{\rho }}_{110}w_{01z} + {\tilde{\rho }}_{111}{\tilde{w}}_{00z}) = 0 ,\\&\quad -i{\tilde{p}}_{101} + \eta ({\tilde{u}}_{111}'' + i{\tilde{w}}_{101}') = 0 ,\\&\quad -{\tilde{p}}_{111}' + \eta (2{\tilde{w}}_{111}'' + i{\tilde{u}}_{101}') = 0, \\&\quad \sigma _{10}{\tilde{\Delta }}_{101} + \sigma _{11}{\tilde{\Delta }}_{100} + i(u_{00}{\tilde{\Delta }}_{101} + u_{01}{\tilde{\Delta }}_{100}) + w_{00}{\tilde{\Delta }}_{111}', \\&\quad + w_{01}{\tilde{\Delta }}_{110}' + {\tilde{w}}_{110}\Delta _{01z} + {\tilde{w}}_{111}\Delta _{00z} = 0, \\&\quad {\tilde{\rho }}_{111} = \frac{-\rho _a(\Delta _{00}{\tilde{\Delta }}_{111} + {\tilde{\Delta }}_{111} - 2\Delta _{01}{\tilde{\Delta }}_{110} )}{(1+\Delta _{00})^3}. \end{aligned} \end{aligned}$$

(A.18)

At z = 0:

$$\begin{aligned} \begin{aligned} {\tilde{\Delta }}_{111}&= 0, \\ {\tilde{u}}_{111} + i{\tilde{g}}_1w_{01}&= 0 , \\ -i{\tilde{g}}_1u_{01} + {\tilde{w}}_{111}&= \left( \frac{\rho _a-\rho _c}{\rho _a}\right) \sigma _{11}{\tilde{g}}_1. \end{aligned} \end{aligned}$$

(A.19)

At z = $h_{0}$:

$$\begin{aligned} \begin{aligned}&\sigma _{11}{\tilde{h}}_{1} = {\tilde{w}}_{111} -u_{01}i{\tilde{h}}_1+ \frac{V\rho _c(\rho _{00}{\tilde{\rho }}_{111} - 2\rho _{01}{\tilde{\rho }}_{110})}{\rho _{00}^3} , \\&\quad -i{\tilde{h}}_1T_{01}^{11} + \eta \{{\tilde{u}}_{111}' + i{\tilde{w}}_{101}\} = 0, \\&\quad -{\tilde{p}}_{111} + 2\eta {\tilde{w}}_{111}' = 0. \end{aligned} \end{aligned}$$

(A.20)

At ${\mathcal {O}}(\epsilon k^2fA_I)$:

$$\begin{aligned} \begin{aligned}&\sigma _{10}{\tilde{\rho }}_{111} + \sigma _{11}{\tilde{\rho }}_{110} + (\sigma _{20}{\tilde{\rho }}_{101} + \sigma _{21}{\tilde{\rho }}_{100}) + i(\rho _{00}{\tilde{u}}_{111} + \rho _{01}{\tilde{u}}_{110}) ,\\&\quad + (\rho _{00z}{\tilde{w}}_{121} + \rho _{01z}{\tilde{w}}_{120}) + (\rho _{00}{\tilde{w}}_{121}' + \rho _{01}{\tilde{w}}_{120}'), \\&\quad + ({\tilde{\rho }}_{120}'w_{01} + {\tilde{\rho }}_{121}'w_{00}) + ({\tilde{\rho }}_{120}w_{01z} + {\tilde{\rho }}_{121}w_{00z}) = 0 ,\\&\quad -i{\tilde{p}}_{111} + \eta (-2{\tilde{u}}_{101} + {\tilde{u}}_{121}'' + i{\tilde{w}}_{111}') = 0, \\&\quad -{\tilde{p}}_{121}' + \eta (-{\tilde{w}}_{101} + 2{\tilde{w}}_{121}'' + i{\tilde{u}}_{111}') = 0,\\&\quad \sigma _{10}{\tilde{\Delta }}_{111} + \sigma _{11}{\tilde{\Delta }}_{110} + \sigma _{20}{\tilde{\Delta }}_{101} + \sigma _{21}{\tilde{\Delta }}_{100} + i(u_{00}{\tilde{\Delta }}_{111} + u_{01}{\tilde{\Delta }}_{110}), \\&\quad + w_{00}{\tilde{\Delta }}_{121}' + w_{01}{\tilde{\Delta }}_{120}' + {\tilde{w}}_{120}\Delta _{01z} + {\tilde{w}}_{121}\Delta _{00z} = 0, \\&\quad {\tilde{\rho }}_{121} = \frac{-\rho _a(\Delta _{00}{\tilde{\Delta }}_{121} + {\tilde{\Delta }}_{121} - 2\Delta _{01}{\tilde{\Delta }}_{120} )}{(1+\Delta _{00})^3}. \end{aligned} \end{aligned}$$

(A.21)

At z = 0:

$$\begin{aligned} \begin{aligned} {\tilde{\Delta }}_{121} = 0, \\ {\tilde{u}}_{121} = 0, \\ {\tilde{w}}_{121} = \left( \frac{\rho _a-\rho _c}{\rho _a} \right) \sigma _{21}{\tilde{g}}_1. \end{aligned} \end{aligned}$$

(A.22)

At z=$h_0$:

$$\begin{aligned} \begin{aligned} \sigma _{21}{\tilde{h}}_1&= {\tilde{w}}_{121} + V\rho _c\bigg (\frac{\rho _{00}{\tilde{\rho }}_{121} - 2\rho _{01}{\tilde{\rho }}_{120} }{\rho _{00}^3}\bigg ) ,\\ {\tilde{u}}_{121}' + i{\tilde{w}}_{111}&= 0, \\ -{\tilde{p}}_{121} + 2\eta {\tilde{w}}_{121}'&= 0. \end{aligned} \end{aligned}$$

(A.23)

1.3 A.3 Solution

In the limit of small cross-terms, the same as in [65] and discussed in the main text, we obtain

$$\begin{aligned} \sigma =&0 + \sigma _{10}(kh_0) + fA_I\sigma _{11}(kh_0) + \sigma _{20}(kh_0)^2 + fA_I\sigma _{21}(kh_0)^2 \nonumber \\&+{\mathcal {O}}\big ((kh_0)^3,(fA_I)^2\big ) \end{aligned}$$

(A.24)

where

$$\begin{aligned} \sigma _{10}= & {} \frac{-2fA_DiD_{13}\big [1 + \frac{{\tilde{g}}_1}{{\tilde{h}}_1} \big ] }{\big [ 1 - \big (1 - \frac{\rho _c}{\rho _a} \big )\frac{{\tilde{g}}_1}{{\tilde{h}}_1} \big ]}, \end{aligned}$$

(A.25)

$$\begin{aligned} \sigma _{11}= & {} 0, \end{aligned}$$

(A.26)

$$\begin{aligned} \sigma _{20}= & {} \frac{\big [-2fA_D\big (D_{11}-D_{33}\big ) + \frac{{\tilde{g}}_1}{{\tilde{h}}_1}\frac{V}{h_0}\big (1 - \frac{\rho _c}{\rho _a}\big ) \big ] }{\big [ 1 - \big (1 - \frac{\rho _c}{\rho _a} \big )\frac{{\tilde{g}}_1}{{\tilde{h}}_1} \big ]} \end{aligned}$$

(A.27)

and

$$\begin{aligned} \sigma _{21} = \frac{-\frac{\rho _a}{\rho _c}\big [2\frac{\rho _c}{\rho _a} + \big (\frac{{\tilde{g}}_1}{{\tilde{h}}_1}\big )^2\big (1 - \frac{\rho _c}{\rho _a}\big )\big (1 + 2\frac{\rho _c}{\rho _a}\big ) - \frac{{\tilde{g}}_1}{{\tilde{h}}_1}\big (1 + 2\frac{\rho _c}{\rho _a} - 2\frac{\rho _c}{\rho _a}^2\big ) \big ]}{2\big [ 1 - \big (1 - \frac{\rho _c}{\rho _a} \big )\frac{{\tilde{g}}_1}{{\tilde{h}}_1} \big ]^2}. \end{aligned}$$

(A.28)

Taking

$$\begin{aligned} \frac{{\tilde{g}}_1}{{\tilde{h}}_1} = \exp \big (-ikx_0(\theta )\big ) \end{aligned}$$

(A.29)

as described in the main text and collecting terms at each order of k recovers our main result, Eq. (44).

Appendix B: Mass-conservation boundary condition

We may express conservation of mass as

$$\begin{aligned} \frac{{\text {d}}}{{\text {d}}t}\int _{\Omega }\rho (\vec {x},t) {\text {d}}V{} & {} = \int _{\partial \Omega }\rho (\vec {x},t)\big [v_I - \vec {v}(\vec {x},t)\cdot {\hat{n}}\big ] {\text {d}}A \nonumber \\{} & {} \quad + \int _{\partial \Omega }S_1(\vec {x},t) {\text {d}}A + \int _{\Omega }S_2(\vec {x},t){\text {d}}V, \end{aligned}$$

(B.30)

where $\rho (\vec {x},t)$ is the scalar density field, $\Omega$ is a control volume, $V_I$ is the normal velocity of the interface $\partial \Omega$ whose differential surface element is dA, $\vec {v}$ is the bulk velocity field of the substrate that the interface moves through, and ${\hat{n}}$ is the normal vector to the differential surface element dA. $S_1$ represents a surface source and $S_2$ represents a bulk (volumetric) source. Because we are interested in the conservation of mass at the amorphous–crystalline boundary $z=g$, and we expect no mass-sources either at the surface or in the bulk, we take the sources $S_1,S_2 \rightarrow 0$. Letting the control volume $\text {Vol}(\Omega ) \rightarrow 0$, conservation requires

$$\begin{aligned} \int _{\partial \Omega }\rho (\vec {x},t)\big [v_I - \vec {v}(\vec {x},t)\cdot {\hat{n}}_g\big ]dA = 0, \end{aligned}$$

(B.31)

hence

$$\begin{aligned} \bigg (\rho (\vec {x},t)\big [v_I - \vec {v}(\vec {x},t)\cdot {\hat{n}}_g\big ]\bigg )_{\text {amorphous}} = \bigg (\rho (\vec {x},t)\big [v_I - \vec {v}(\vec {x},t)\cdot {\hat{n}}_g\big ]\bigg )_{\text {crystalline}}, \end{aligned}$$

(B.32)

or

$$\begin{aligned} \bigg (\rho _a\big [v_I - \vec {v}_a\cdot {\hat{n}}_g\big ]\bigg ) = \bigg (\rho _c\big [v_I - \vec {v}_c\cdot {\hat{n}}_g\big ]\bigg ). \end{aligned}$$

(B.33)

Since the underlying crystalline substrate receives vanishingly little energy compared to the amorphous layer, we anticipate that $|\vec {v}_c| \ll |\vec {v}_a|$, such that $\vec {v}_c \approx \vec {0}$ in comparison. Then rearrangement leads to

$$\begin{aligned} \vec {v}_a\cdot {\hat{n}} = \bigg (1 - \frac{\rho _c}{\rho _a}\bigg )v_{I} \end{aligned}$$

(B.34)

at z = g, or, as a jump relation,

$$\begin{aligned}{}[[\rho \vec {v}]\cdot {\hat{n}} = [[\rho ]]v_{I,g}, \end{aligned}$$

(B.35)

as in the main text. It is noteworthy that when $[[\rho ]] = 0$, such that the density of the crystalline and amorphous phases are assumed to be equal, we immediately restore the more typical no-penetration condition. Equivalently, we point out that the common use of the no-penetration condition throughout the literature on hydrodynamic-type approaches to ion-induced pattern formation literature implicitly takes the crystalline and amorphous phases to have the same density. One of the primary results of the present work is that this irradiation-induced change of phase significantly affects the linear stability of the film.

Next, we convert to the traveling frame via

$$\begin{aligned} \begin{aligned} \vec {v} \rightarrow \vec {v} - V{\hat{k}}. \end{aligned} \end{aligned}$$

(B.36)

Then, we have

$$\begin{aligned} v_{I,g} \rightarrow v_{I,g} - V{\hat{k}}\cdot {\hat{n}}_g \end{aligned}$$

(B.37)

due to ongoing erosion. These lead to

$$\begin{aligned} \begin{aligned} \rho _{a}\vec {v}_{a}\cdot {\hat{n}}_g&= (\rho _a-\rho _c)(v_{I,g} - V{\hat{k}}\cdot {\hat{n}}_g) + \rho _{a}V{\hat{k}}\cdot {\hat{n}}_g ,\\ \vec {v}_{a}\cdot {\hat{n}}_g&= \left( \frac{\rho _a-\rho _c}{\rho _{a}}\right) (v_{I,g} - V{\hat{k}}\cdot {\hat{n}}_g) + V{\hat{k}}\cdot {\hat{n}}_g ,\\ \vec {v}_{a}\cdot {\hat{n}}_g&= \left( 1 - \frac{\rho _c}{\rho _a}\right) v_{I,g} - \left( 1 - \frac{\rho _c}{\rho _a}\right) V{\hat{k}}\cdot {\hat{n}}_g + V{\hat{k}}\cdot {\hat{n}}_g ,\\ \vec {v}_a\cdot {\hat{n}}_g&= \left( 1 - \frac{\rho _c}{\rho _a}\right) v_{I,g} + \bigg (\frac{\rho _c}{\rho _a}\bigg )V{\hat{k}}\cdot {\hat{n}}_g. \end{aligned} \end{aligned}$$

(B.38)

We now drop the subscript a as it is clear that the only bulk velocity field under consideration is that of the amorphous layer. In principle, we have made the assumption that the motion of the amorphous bulk is much faster than that of the underlying crystalline substrate. It is also clear that when $\rho _{a} = \rho _{c}$ (i.e., there is no density drop across the interface), and if we assume the typical no-slip condition $u = 0$ at $z=g$, we recover

$$\begin{aligned} \vec {v} = V{\hat{k}}. \end{aligned}$$

(B.39)

This is as was seen in [63]. From

$$\begin{aligned} \vec {v}\cdot {\hat{n}}_g = \left( 1 - \frac{\rho _c}{\rho _a}\right) v_{I,g} + \bigg (\frac{\rho _c}{\rho _a}\bigg )V{\hat{k}}\cdot {\hat{n}}_g, \end{aligned}$$

(B.40)

the steady-state equation is easily obtained, and we find the following expansions in Fourier modes.

At ${\mathcal {O}}(\epsilon k^0)$,

$$\begin{aligned} {\tilde{g}}_1w_{0z} + {\tilde{w}}_{100} = 0. \end{aligned}$$

(B.41)

At ${\mathcal {O}}(\epsilon k^1)$,

$$\begin{aligned} -i{\tilde{g}}_1u_0 + {\tilde{w}}_{110} = \sigma _{10}\bigg (1 - \frac{\rho _c}{\rho _a}\bigg ){\tilde{g}}_1. \end{aligned}$$

(B.42)

At ${\mathcal {O}}(\epsilon k^2)$,

$$\begin{aligned} {\tilde{w}}_{120} = \sigma _{20}\bigg (1 - \frac{\rho _c}{\rho _a} \bigg ){\tilde{g}}_1. \end{aligned}$$

(B.43)

Expansion in $fA_I$ is straightforward. Then we arrive at the boundary conditions in the main text. We note that this condition is more typical of the solidification theory literature, and is featured prominently in Chapter 9 of [66] and elsewhere, while being largely absent from most other resources on continuum mechanics, where phase transitions are seldom of interest.

Appendix C: Coefficients for small-slope expansion of lower interface

From the $|h_x| \ll 1$ expansion in the main text, and, bringing our notation into alignment with Fig. 2, we obtain

$$\begin{aligned} \begin{aligned} z^B(\theta , h_x) = z^T - \left[ a\cos (\theta ) + 2\left( \sqrt{\alpha ^2\cos ^2(\theta ) + \beta ^2\sin ^2(\theta )}\sqrt{\frac{\ln (\frac{E_0}{E_A})}{2}}\right) \right] + {\mathcal {O}}(h_x^2), \end{aligned}\nonumber \\ \end{aligned}$$

(C.44)

and

$$\begin{aligned} \begin{aligned}&x^B(\theta , h_x)\\&\quad = x^T + a\sin (\theta ) + 2\left( \frac{(\alpha ^2-\beta ^2)\sin (\theta )\cos (\theta )}{\sqrt{\alpha ^2\cos ^2(\theta ) + \beta ^2\sin ^2(\theta )}}\sqrt{\frac{\ln (\frac{E_0}{E_A})}{2}} \right) \\&\qquad - h_x\left( a\cos (\theta ) + \frac{(\alpha ^2-\beta ^2)\left( 4(\alpha ^2+\beta ^2)\cos (2\theta ) + (\alpha ^2-\beta ^2)(3+\cos (4\theta )) \right) }{ (\alpha ^2\cos ^2(\theta ) + \beta ^2\sin ^2(\theta ))^{3/2} }\right) \\&\qquad + {\mathcal {O}}(h_x^2). \end{aligned}\nonumber \\ \end{aligned}$$

(C.45)

In the above, it is clear that the ${\mathcal {O}}(h_x)$ term in $z^B(\theta ,h_x)$ is identically zero, and the next term in the expansion is nonlinear in $h_x$; its form is therefore irrelevant to the linear stability analysis of our present interest. The ${\mathcal {O}}(h_x)$ correction to $x^B(\theta ,h_x)$ remains irrelevant as discussed in the main text.