Size effects in surface-reconstructed $⟨100⟩$ and $⟨110⟩$ silicon nanowires

The geometrical and electronic structure properties of $⟨100⟩$ and $⟨110⟩$ silicon nanowires in the absence of surface passivation are studied by means of density-functional calculations. As we have shown in a recent publication [R. Rurali and N. Lorente, Phys. Rev. Lett. 94, 026805 (2005)] the reconstruction of facets can give rise to surface metallic states. In this work, we analyze the dependence of geometric and electronic structure features on the size of the wire and on the growth direction.

Figure 1

(Color online) Section views of the unrelaxed [(a)–(c)] and relaxed [(d)–(f)] $⟨100⟩$ SiNWs with a diameter of $\sim 8$, $\sim 11$, and $\sim 15\mathrm{\AA }$. The facet geometry is determined by the number of rows of dimers (indicated by the brackets) that can form: (a,d) one-single row of dimers can form; (b,e) the formation of a second row of dimers is frustrated and the presence of an unpaired row of atoms results in an asymmetric relaxation of the facet; and (c,f) two rows of dimers form, yielding a trough in the middle of the {100} facet.

Figure 2

(Color online) $⟨100⟩$ SiNWs of $\sim 8\mathrm{\AA }$ diameter: side view of the reconstructed (pseudo) {100} facet.

Figure 3

(Color online) $⟨100⟩$ SiNWs of $\sim 11\mathrm{\AA }$ diameter: side view of the reconstructed (pseudo) {100} facet.

Figure 4

(Color online) The mechanism that rules the facet relaxation when not all the dangling bonds can form dimers. The gray atom (indicated with an arrow) is unpaired and cannot form a dimer; to reduce the number of dangling bonds it breaks a bond of the underlying layer (red atoms, indicated with a circle) and enters it. The golden atoms (indicated with a cross) form a surface dimer according to the conventional mechanism of Si(100)-like surfaces.

Figure 5

Top view of the Si(100) $c(4\times 2)$. White and gray balls represent the (asymmetric) top dimers, dark balls are the second layer atoms. A strip equivalent to the {100} facets of the $\sim 15\mathrm{\AA }$ $⟨100⟩$ SiNW is drawn. Each of the atoms represented by white balls have only one second layer atom to rest onto (the missing ones are indicated with an arrow). This induces a tilt of the dimer axis when it relaxes, thus achieving a different packing density than the infinite Si(100) surface.

Figure 6

(Color online) Band structure diagram of (a) the $15\mathrm{\AA }$ SiNW of Ref. 29 and (b) the $27\mathrm{\AA }$ SiNW. The comparison between the two panels shows the trend of the band-gap opening as the diameter is increased.

Figure 7

(Color online) Possible geometries considered for $⟨110⟩$ SiNWs. The section that features a central hexagon [shown in panel (b)] turned out to be the most stable.

Figure 8

(Color online) Section views of the unrelaxed [(a)–(c)] and relaxed [(d)–(f)] $⟨110⟩$ SiNWs with a diameter of $\sim 12$, $\sim 20$, and $\sim 26\mathrm{\AA }$.

Figure 9

(Color online) {111} facets of the two stable reconstructions that we have found for the $12\mathrm{\AA }$ $⟨110⟩$ SiNW. The different unit cells of the facet are indicated. The reconstructions have extension of (a) $c$ and (b) $3c$, where $c$ is the axial lattice parameter of the SiNW. In (a) the reconstruction features one row of aligned atoms, while in (b) only one atom of each three (indicated with an arrow) protrudes outward.

Figure 10

(Color online) {111} facets of the two stable reconstructions of the $20\mathrm{\AA }$ $⟨110⟩$ SiNW. The different unit cells of the facet are indicated. The reconstructions have extension of (a) $c$ and (b) $2c$, where $c$ is the axial lattice parameter of the SiNW. In (a) the reconstruction features one row of aligned atoms, while in (b) only one atom of each two (indicated with an arrow) protrudes outward.

Figure 11

(Color online) Energy gained with the relaxation, Eq. (1) (a) and surface energy, as defined in Eq. (2) (b). Black diamonds, $⟨100⟩$ SiNWs; and red triangles, (110) SiNWs.