Nanoscale structures formed in silicon cleavage studied with large-scale electronic structure calculations: Surface reconstruction, steps, and bending

The $10\text{−}\mathrm{nm}$-scale structure formed in silicon cleavage is studied by the quantum mechanical calculations of large-scale electronic structure. The cleavage process was simulated and the results show not only the elementary process of the (experimentally observed) (111)-$(2\times 1)$ surface reconstruction but also several step-formation processes. These processes are studied by analyzing electronic freedom and compared with scanning tunneling microscopy experiments. The stability mechanism of the (111)-$(2\times 1)$ cleavage mode is presented beyond the traditional approach with surface energy. In other results, the cleavage path was bent into the experimentally observed planes, owing to the relative stability among cleavage modes. Several common aspects between cleavage and other phenomena are discussed from the viewpoints of nonequilibrium process and $10\text{−}\mathrm{nm}$-scale structure.

Figure 1

(a)–(c) Toy model of cleavage, in which surface reconstruction is illustrated as dimerization. (d) Possible cleavage paths on Si(111) plane.

Figure 2

Cleavage process of silicon with (111)-$(2\times 1)$ cleaved surface. (a)–(e) Successive snapshots with a time interval of approximately $2\mathrm{ps}$. (f) A part of the lower cleavage surface shown in the snapshot (e). A step is found and is classified into the $\left[\overline{2}11\right]$-type or via-$\left(1\overline{1}\overline{1}\right)$-plane type that can be decomposed into the successive bending of cleavage planes as $\left(111\right)\to \left(1\overline{1}\overline{1}\right)\to \left(111\right)$.

Figure 3

(a)–(d) Initiation of cleavage while forming the (111)-$(2\times 1)$ structure. (e), and (f) Schematic figures of the transition from (e) the buckled $(2\times 1)$ structure to (f) the (tilted) $\pi$-bonded $(2\times 1)$ structure. The former structure appears on the lower cleaved surface in (b) and the latter appears in (d). The oval in (e) indicates the presence of a lone pair state on the $D$ atom site. The oval in (f) indicates the $\pi$-bonding zigzag chain in the direction perpendicular to the page. (g)–(i) Quantum mechanical analysis of a process in which the bonding wave function $\left({\varphi }_i\right)$ between $A$ and $B$ sites changes into that between $A$ and $C$ sites: (g) the one-electron energy ${\epsilon }_i\equiv ⟨{\varphi }_i\mid H\mid {\varphi }_i⟩$ and the weight of $s$ orbitals $f_s^{\left(i\right)}$ for the wave function ${\varphi }_i$; (h) the atomic distance between $A$ and $B$ sites $\left(d_{AB}\right)$ and that between $A$ and $C$ sites $\left(d_{AC}\right)$; (i) the spatial weight distribution on the $A$, $B$, and $C$ atom sites $(n_{\mathrm{A}},n_{\mathrm{B}},n_{\mathrm{C}})$ for the wave function ${\varphi }_i$. The rest of the weight $(1-n_{\mathrm{A}}-n_{\mathrm{B}}-n_{\mathrm{C}})$ is also plotted.

Figure 4

(a) Si(111) surface with coexistence of the buckled and $\pi$-bonded $(2\times 1)$ structures. The induced strain is shown as an arrow at the boundary between the buckled and $\pi$-bonded structures. The dashed line indicates the bond site that will appear after the next induced BP transition. (b) and (c) Two snapshots of cleavage process. The solid arrow corresponds to the direction of the possible surface strain force shown in (a). The dashed arrow denotes the cleavage propagation direction.

Figure 5

Step formation process on (111) cleavage plane. The step is classified into the $\left[\overline{2}11\right]$-type or via-$\left(1\overline{1}\overline{1}\right)$-plane-type step [see Fig. 2f]. Two snapshots with a time interval of approximately $0.6\mathrm{ps}$ are shown. The dashed lines indicate the initial (crystalline) bonds. The + and − symbols on atoms indicate excess and deficit electron populations, respectively.

Figure 6

(a) (111) cleaved surface with $\left[2\overline{1}\overline{1}\right]$-type or via-(100)-plane-type step. The arrow indicates the cleavage propagation direction. (b), (c): Two “slices” of (a) (see text for details).

Figure 7

Defective area on cleaved Si(111) surface. Three slices are picked out. The six-membered rings marked (I) and (II) are geometrically equivalent among slices (a)–(c).

Figure 8

Cleavage process of silicon with bending of cleavage planes. (a)–(e) Snapshots with a time interval of approximately $1\mathrm{ps}$. Atoms are drawn as balls from the viewpoint of the $\left[\overline{1}10\right]$ direction. (f) The broken bond sites in snapshot (e) are plotted as bold lines in the ideal (crystalline) geometry. (g) Indices of geometry in (f).

Figure 9

Appearance of well-defined reconstructed surfaces after bending in cleavage path. The figures are drawn by projection from the $\left[\overline{1}10\right]$ direction. (a) Bending from the (111) plane to (110) plane. After the bending, the buckled (110) surface structures appears, as indicated by b. (b) and (c) Two snapshots of the bending from (001) plane to (111) plane. The resultant (111) plane contains the $\pi$-bonded $(2\times 1)$ structure with seven-membered and five-membered rings, indicated as 7 and 5 in (c).

Figure 10

(Left) Computational time for molecular dynamics simulation with up to 11 315 021 Si atoms. Our calculations are compared with the conventional calculation for the eigenstates. See text for explanations. (Right) Dimer geometry on (001) surface of group IV elements (C, Si, and Ge). The angle $\theta$ is the tilt angle and $\varphi$ is the angle between the surface dimer and the plane that contains the two back bonds of the upper atom.