Atomistic simulation study of the structure and dynamics of a faceted crystal-melt interface

A detailed analysis of the structure and dynamics of the crystal-melt interface region in silicon, modeled with the Stillinger-Weber potential, is performed via molecular dynamics simulations. The focus is on the faceted (111) crystal-melt interface, but properties of the rough (100) interface are also determined. We find an intrinsic 10-90 interface width of $0.681\pm 0.001\mathrm{nm}$ for the coarse-grained density profile at the (111) interface and a $0.570\pm 0.005\mathrm{nm}$ width at the (100) interface. Coarse-grained profiles of a suitably defined local order parameter are found to show a smaller width anisotropy between (111) and (100) interfaces while the order profiles exhibit a $0.20–0.25\mathrm{nm}$ shift in position toward the crystal phase relative to the corresponding density profiles. The structural analysis of the layer of melt adjacent to the (111) facet of the crystal finds ordered clusters with average lifetimes of $16\mathrm{ps}$, as determined from autocorrelations of time-dependent layer structure factors, and cluster radii of gyration from $0.2\mathrm{nm}$ for the smallest cells to as large as $1.5\mathrm{nm}$.

Figure 1

(Color online) Snapshots of (111) (top) and (100) (bottom) crystal-melt interface regions in silicon. The atom coordinates are projected onto the plane of the page. The viewpoints, as defined by the directions perpendicular to the page, are $\left(10\overline{1}\right)$ and (010), respectively. The points representing the atoms are colored in proportion to a local order parameter [Eq. (3)]: from blue (dark gray) for an order parameter value of 1 (order) to green (light gray) for a value of 0 (no order).

Figure 2

Fine-scale density profile $n\left(z\right)$ in the (111) direction ($z$ coordinate) of the melt-crystal-melt Si system at coexistence. The bin size is $3\mathrm{pm}$.

Figure 3

Fine-scale density profile $n\left(z\right)$ in the (100) direction ($z$ coordinate) of the melt-crystal-melt Si system at coexistence. The bin size is $0.01\mathrm{nm}$.

Figure 4

Illustration of the computation of the coarse-scale density profile $n\left(z\right)$ using nonuniform bins that match the crystalline layers and the density oscillations near the interface. (a) shows a portion of the fine-scale density profile at the (111) crystal-melt interface with the coarse-scale bins marked by dashed vertical lines. The coarse-grained profile for the same region is displayed in (b).

Figure 5

Logarithm of the smoothness function $S$ computed for filtered profiles obtained from the fine-scale profile of Fig. 2. The six lines correspond to six values of the filter parameter $N$ in the range 150–400. Each line shows a clear minimum in the dependence on the second filter parameter $ϵ$.

Figure 6

Coarse-grained density profiles for the melt-crystal-melt system with (111) interfaces: FIR method (solid line) and nonuniform bins (dashed line).

Figure 7

Closeup of the fine-scale density profile $n\left(z\right)$ in the second interface region (crystal-melt) from Fig. 2 with five layers marked by dashed line boundaries. The envelope of the density peaks is sketched with a dashed line (long dashes).

Figure 8

Probability distribution of the interface position for the (111) (filled symbols) and (100) (open symbols) interfaces. The histogram bin is $0.01\mathrm{nm}$.

Figure 9

Coarse-grained density profiles for the melt-crystal-melt system with (100) interfaces: FIR method (solid line) and nonuniform bins (dashed line).

Figure 10

Fine-scale (a) and smoothed profiles (b) of the density of the local tetrahedral order parameter (solid line) and number density of atoms (dashed line). The pair of vertical grid lines marks the location of layer $C$, as labeled in Fig. 7.

Figure 11

Two-dimensional pair distribution functions in the $xy$ plane for the layers $C$ and $E$ from Fig. 7. The dotted line represents the difference between the two distribution functions.

Figure 12

Two-dimensional pair distribution functions in the $xy$ plane for the layers $C$ and $B$ from Fig. 7.

Figure 13

(Color online) Projected snapshots of the layers $A$, $B$, $C$, and $D$ (left column) with the two-dimensional density type plots of time-averaged structure factors (right column) for the corresponding layers. The layers are labeled in Fig. 7. The range of $\mathbf{k}$ vectors covered in the $x$ and $y$ directions is $-150\times 2\pi ∕L_x$ to $150\times 2\pi ∕L_x$ and $-150\times 2\pi ∕L_y$ to $150\times 2\pi ∕L_y$, respectively. For clarity, the linear gray scale of the density plots for the structure factors is restricted to a range of structure factors between 0.5 and 1.5. Values outside this interval are represented by black if smaller than 0.5 and white if larger than 1.5. The disks representing the atoms in the layer snapshots are colored in proportion to the local order parameter of Eq. (3): from blue (dark gray) for an order parameter value of 1 to green (light gray) for a value of 0.

Figure 14

(Color online) Plan views of layer $C$ in three different snapshots of the (111) crystal-melt system. The three configurations are presented in the order of increasing layer structure factor at the first peak, $S_{2\mathrm{D}}(\mid {\mathbf{G}}_1\mid ,t)$: 7.1 for configuration (a), 9.7 for (b), and 18.8 for (c). The coloring convention is the same as the one used for the real space snapshots of Fig. 13.

Figure 15

Normalized autocorrelation function of the two-dimensional structure factor for the projections of layers $A$, $B$, $C$, and $D$ (as labeled in Fig. 7) onto the $xy$ plane. The structure factors are the average of three values computed at three reciprocal space locations with $\mid \mathbf{k}\mid =\mid {\mathbf{G}}_1\mid$, the magnitude of the smallest vectors in the reciprocal lattice of the projected crystal layers.