Coupling of morphology to surface transport in ion-beam irradiated surfaces: Oblique incidence

We propose and study a continuum model for the dynamics of amorphizable surfaces undergoing ion-beam sputtering (IBS) at intermediate energies and oblique incidence. After considering the current limitations of more standard descriptions in which a single evolution equation is posed for the surface height, we overcome (some of) them by explicitly formulating the dynamics of the species that transport along the surface and by coupling it to that of the surface height proper. In this, we follow recent proposals inspired by “hydrodynamic” descriptions of pattern formation in aeolian sand dunes and ion-sputtered systems. From this enlarged model and by exploiting the time-scale separation among various dynamical processes in the system, we derive a single height equation in which coefficients can be related to experimental parameters. This equation generalizes those obtained by previous continuum models and is able to account for many experimental features of pattern formation by IBS at oblique incidence, such as the evolution of the irradiation-induced thin amorphous layer, transverse ripple motion with nonuniform velocity, ripple coarsening, onset of kinetic roughening, and others. Additionally, the dynamics of the full two-field model is compared with that of the effective interface equation.

Figure 1

Height profiles $h$ (dashed line) and thickness of the mobile material over $h$ (solid line) at different times given by the 1D two-field coupled model with $\overline{\varphi }=0.99$, ${\alpha }_0=0.03$, ${\alpha }_1=-1$, ${\alpha }_2=30$, ${\alpha }_3={\alpha }_4=1$, ${\alpha }_5=-1$, ${\alpha }_6=-3$, $R_{\text{eq}}={\gamma }_0={\gamma }_2=1$, and $D=10$. All units are arbitrary.

Figure 2

(Color online) Height profiles at different times given by the 1D two-field model (black line) for parameters as in Fig. 1 and by the effective equation [Eq. (44)] (green line) with parameters as given by relation (45), namely, $v_0=-3\times {10}^{-4}$, ${\gamma }_x=3\times {10}^{-4}$, ${\nu }_x=-9\times {10}^{-3}$, ${\lambda }_x^{\left(1\right)}=9\times {10}^{-4}$, ${\Omega }_x=-0.297$, ${\xi }_x=3\times {10}^{-4}$, ${\mathcal{K}}_{xx}=1.0993$, and ${\lambda }_{xx}^{\left(2\right)}=0.8901$. All units are arbitrary.

Figure 3

(Color online) Temporal evolution of the global roughness $W\left(t\right)$ given by the two-field model (blue triangles) and by the effective interface equation (green circles) for the same coefficients as in Fig. 2. Error bars are smaller than the symbol sizes. All units are arbitrary.

Figure 4

(Color online) Temporal evolution of the lateral pattern wavelength, $l\left(t\right)$, given by the two-field model (blue triangles) and by the effective interface equation (green circles) for the same coefficients as in Fig. 2. A few representative error bars are given that represent statistical dispersion. The dashed line corresponds to $l\left(t\right)\sim t^{0.12}$. All units are arbitrary.

Figure 5

Height profiles between $t=1.5\times {10}^6$ (top) and $t=1.8\times {10}^6$ (bottom) evaluated at equally spaced intervals of ${10}^4$ time units for the effective Eq. (44) and parameters as in Fig. 2. All units are arbitrary.

Figure 6

Time evolution of relatively disordered ripples with mild wavelength coarsening (see also supplementary video) (Ref. 61). Snapshots at increasing times: (a) $t=10$, (b) $t=106$, and (c) $t=953$ for Eq. (42) with $v_0=0$, ${\gamma }_x=-0.1$, ${\nu }_x=-1$, ${\nu }_y=-0.1$, ${\Omega }_x=1$, ${\Omega }_y=0.5$, ${\xi }_i=0.1$, ${\lambda }_x^{\left(1\right)}=1$, ${\lambda }_y^{\left(1\right)}=5$, ${\lambda }_{i,j}^{\left(2\right)}=5$, and ${\mathcal{K}}_{i,j}=1$. Top views (left column) and corresponding transverse cuts at $y=L/2$ (right column). Inset in (a) is its corresponding height autocorrelation. All units are arbitrary.

Figure 7

Time evolution of relatively ordered ripples with sizeable wavelength coarsening [see also supplementary video (Ref. 61)]. Snapshots at increasing times: (a) $t=10$, (b) $t=106$, and (c) $t=953$ for Eq. (42) with the same parameters as in Fig. 6, except for ${\lambda }_x^{\left(1\right)}=0.1$. Top views (left column) and corresponding transverse cuts at $y=L/2$ (right column). Inset in (a) is its corresponding height autocorrelation. All units are arbitrary.

Figure 8

Time evolution of relatively disordered ripples with sizable wavelength coarsening [see also supplementary video (Ref. 61)]. Ripple orientation as for typical large incidence angle conditions. Snapshots at increasing times: (a) $t=10$, (b) $t=106$, and (c) $t=953$ for Eq. (42) with $v_0=0$, ${\gamma }_x=0.1$, ${\nu }_x=1$, ${\nu }_y=-0.95$, ${\Omega }_i=-0.5$, ${\xi }_i=0.1$, ${\lambda }_x^{\left(1\right)}=0.1$, ${\lambda }_y^{\left(1\right)}=1.0$, ${\lambda }_{i,x}^{\left(2\right)}=0.5$, ${\lambda }_{i,y}^{\left(2\right)}=5.0$, and ${\mathcal{K}}_{i,j}=1$. Top views (left column) and corresponding transverse cuts at $x=L/2$ (right column). Inset in (a) is its corresponding height autocorrelation. All units are arbitrary.

Figure 9

In-plane nonuniform ripple motion as seen from the evolution of transverse cuts of the surface at $y=L/2$ for equally spaced times between $t=0$ and $t=1500$. Results from the numerical integration of Eq. (42) with $v_0=0$, ${\nu }_x=-1$, ${\nu }_y=-0.1$, ${\gamma }_x={\xi }_i=0$, ${\Omega }_i=-2$, ${\lambda }_x^{\left(1\right)}=1$, ${\lambda }_y^{\left(1\right)}=5$, ${\lambda }_{i,x}^{\left(2\right)}=50$, ${\lambda }_{i,y}^{\left(2\right)}=5.0$, and ${\mathcal{K}}_{i,j}=1$. All units are arbitrary.

Figure 10

Change in the direction of in-plane ripple motion as seen from the evolution of transverse cuts of the surface at $y=L/2$ for equally spaced times between $t=0$ and $t=5000$. Results from the numerical integration of Eq. (42) for parameters as in Fig. 9 except for ${\xi }_i=4.5$. All units are arbitrary.

Figure 11

(Color online) Temporal evolution of the lateral ripple velocity, $V\left(t\right)$, given by the numerical integration of Eq. (42) in the nonlinear regime for the same coefficients as in Fig. 9. A few representative error bars are given. All units are arbitrary.

Figure 12

(Color online) Temporal evolution of the global roughness, $W\left(t\right)$, given by the effective equation Eq. (42) with $v_0=0$, ${\nu }_x=-1$, ${\nu }_y=-0.1$, ${\gamma }_x={\xi }_i={\Omega }_i=0$, ${\mathcal{K}}_{i,j}=1$, ${\lambda }_i^{\left(1\right)}=0.1$, and ${\lambda }_{i,j}^{\left(2\right)}=0.1r$ for different values of $r$. The solid and dashed lines show the fit to power laws for ${\lambda }_i^{\left(1\right)}=0$ and $r=100$ where $W\sim t^{1.00}$ and $W\sim t^{0.71}$, respectively. All units are arbitrary.

Figure 13

(Color online) Temporal evolution of the lateral wavelength of the pattern, $l\left(t\right)$, given by the effective equation [Eq. (42)] for the same coefficients as Fig. 12. The dashed line shows the fit to a power law for $r=100$ where $l\sim t^{0.38}$. The fit in the same region for ${\lambda }_i^{\left(1\right)}=0$ yields $l\sim t^{0.48}$ (not shown). All units are arbitrary.

Figure 14

Long time development of oblique ripple patterns at $45°$ to the $x$ axis due to cancellation modes (see text). Top views of morphologies obtained by numerical integration of Eq. (42) with $v_0=0$, ${\nu }_x=-1$, ${\nu }_y=1$, ${\gamma }_x={\xi }_i={\Omega }_i=0$, ${\mathcal{K}}_{i,j}=1$, ${\lambda }_x^{\left(1\right)}=0.1$, ${\lambda }_y^{\left(1\right)}=-0.1$, ${\lambda }_{i,x}^{\left(2\right)}=0.5$, and ${\lambda }_{i,y}^{\left(2\right)}=-0.5$ at increasing times: (a) $t=25$, (b) $t=50$, and (c) $t=185$. All units are arbitrary.