Pentagons in the Si$(331)-(12\times1)$ surface reconstruction

TL;DR

This study combines microscopy and ab initio calculations to propose a structural model of the Si(331)-(12×1) surface, highlighting the role of pentagons in reducing surface energy and predicting multiple buckled configurations.

Contribution

The paper introduces a novel structural model of the Si(331) surface featuring pentagons, advancing understanding of high-index silicon surface reconstructions.

Findings

Pentagons lower the surface energy of Si(331)

Multiple buckled configurations with similar energies are predicted

Structural model aligns with experimental microscopy data

Abstract

The microscopic structure of the high-index Si$(331)-(12\times1)$ surface is investigated combining scanning tunneling microscopy with ab initio calculations. We present a structural model of the Si(331) surface, employing a reconstruction element composed of six pentagons integrated to the structure of the adjacent pentamer with an interstitial atom. We demonstrate that appropriately arranged additional pentagons significantly lower the surface energy of the high-index surface. The model predicts the existence of multiple Si(331) buckled configurations with similar energies.

Equations

$${\mathrm{\Gamma }}_{(331)}{\mathrm{S}}_{(331)}={\gamma }_{(111)}{\mathrm{S}}_{(111)}+{\gamma }_{(110)}{\mathrm{S}}_{(110)},$$

Full text

High-index surfaces of Si are interesting for both fundamental research and technological applications. The technological interest is based on the demonstrated improved heteroepitaxial growth on such surfaces and the use of them as templates for nanostructures growth.Zhachuk et al. (2004) Such surfaces, however, often demonstrate complex surface reconstructions. The problem of finding the atomic structure of surface reconstructions is still a formidable challenge. The main difficulty is the existence of a large number of atomic configurations for surface cells even with a moderate number of atoms. Scanning tunneling microscopy (STM) and density functional theory (DFT) calculations are two complementary methods often used in conjunction for surface structure determination. Although DFT calculations offer accurate total energies, the surface structure prediction of materials with large surface cells is very hard nowadays due to a high computational cost of such calculations. The experimental STM data help a lot to narrow down the search for possible atomic configurations by showing the actual structure of a surface at the atomic scale of a real sample. However, the interpreting of high resolution STM images can be very tricky, since STM does not actually show positions of atomic nuclei. In the most simplified view, the STM images represent a mixture of surface topography and a map of local density of electronic states of a sample surface.Hofer (2003); Tersoff and Hamann (1985) Consequently, the interpreting of such images, in its part, may require the knowledge of surface atomic structure and ab initio calculations.

Si$(331)$ is a flat silicon surface exhibiting a complex reconstruction. The surface structure is often designated as $(12\times 1)$ or $(6\times 2)$, although the correct notation can only be given by matrix.Battaglia et al. (2009a); Olshanetsky et al. (1998) The study of $(12\times 1)$ surface reconstruction has long history. Three structural models were proposed.Battaglia et al. (2009a); Olshanetsky et al. (1998); Gai et al. (2001) It was recognized from the very beginning that the rectangular surface unit cell contains two identical structural units (Fig. 1(a)).Hibino et al. (1993) The first structural unit is located at the surface cell corner. The second unit is shifted by $a/2$ from the center to $[\bar{1}10]$ or $[1\bar{1}0]$, where $a$ is a basic translational unit of the unreconstructed $(331)$ plane in that direction. The surface has a glide plane symmetry along the $[\bar{1}\bar{1}6]$ direction running through the center of the zigzag chain of structural units (dashed line in Fig. 1(a)).

There were several attempts to construct the observed structural units from the elementary building blocks known from the previous studies of silicon surfaces.Olshanetsky et al. (1998); Gai et al. (2001); Battaglia et al. (2009a) It was proposed that the structural units consist of adatomsOlshanetsky et al. (1998) or adatoms and dimers.Gai et al. (2001) In the most recent structural model proposed by Battaglia et al.,Battaglia et al. (2009a) those units were constructed from the pentamer with an interstitial atom (hereafter pentamer) and two adatoms. Originally, the pentamers were suggested as a structural building block on the silicon $(113)$ surfaceDąbrowski et al. (1995) and were used to explain the structure of Si$(110)$ later.Stekolnikov et al. (2004) The model by Battaglia *et al.*Battaglia et al. (2009a) basically represents an adaptation of the adatom-tetramer-interstitial (ATI) model of the Si$(110)-(16\times 2)$ surface reconstruction by Stekolnikov *et al.*Stekolnikov et al. (2004) for the Si$(331)-(12\times 1)$ surface. We, therefore, refer to the structural model proposed in Ref. Battaglia et al., 2009a as the ATI model. It was demonstrated that the pentamers indeed adequately describe the groups of five bright spots observed in the experimental STM images of the Si$(331)$ surface.Battaglia et al. (2009a) Nevertheless, the ATI model of Si$(331)-(12\times 1)$ is questionable as it shows a poor agreement with STM images of the areas between the pentamers and it leads to the high surface energy, as demonstrated below.

The aim of our work is to develop a relistic Si$(331)-(12\times 1)$ surface reconstruction model by a combined experimental and theoretical study. We propose a microscopic model of the $(12\times 1)$ reconstruction which shows a remarkably low surface energy and explains the experimental STM data.

The STM images were recorded at room temperature in the constant-current mode using an electrochemically etched tungsten tip. The measurements were performed in an ultrahigh vacuum chamber ($7\times 10^{-11}\,\mathrm{Torr}$) on a system equipped with an Omicron STM. A clean Si$(331)$ surface was prepared by sample flash annealing at $1250\,\mathrm{{{}^{\circ}}C}$ for one minute followed by stepwise cooling with $2\,\mathrm{{{}^{\circ}}C}$ per minute steps within temperature range $400\text{\textminus 850}\,\mathrm{{{}^{\circ}}C}$. More details on the experimental procedure can be found in Ref. Zhachuk et al., 2013. The WSXM software was used to process the experimental and calculated STM images.Horcas et al. (2007)

The calculations were carried out using the pseudopotentialTroullier and Martins (1991) DFT siesta codeSoler et al. (2002) within the local density approximation to the exchange and correlation interactions between electrons.Perdew and Wang (1992) The valence states were expressed as linear combinations of numerical atomic orbitals of the Sankey-Niklewski type.Soler et al. (2002) In the present calculations, the polarized double-$\mathrm{\zeta}$ functions were assigned for all species. This means two sets of* *$s$ and $p$ orbitals plus one set of $d$ orbitals on Si atoms, and two sets of $s$ orbitals plus a set of $p$ orbitals on H. The electron density and potential terms were calculated on a real space grid with the spacing equivalent to a plane-wave cut-off of $200\,\mathrm{Ry}$.

The surface energy (per unit area) of the reconstructed Si$(331)$ surface ($\gamma_{rec}$) was calculated as $\gamma_{rec}=\gamma_{unrec}+\triangle\gamma_{rec}$, following the procedure described in Refs. Stekolnikov et al., 2002; Zhachuk et al., 2013. Here $\gamma_{unrec}$ is the energy of the unreconstructed and unrelaxed Si$(331)$ surface, and $\triangle\gamma_{rec}$ is the energy gain due to surface reconstruction and relaxation. $\mathrm{\gamma_{unrec}}$ was calculated using a symmetric slab, 20 Si bilayers thick. $\mathrm{\triangle\gamma_{rec}}$ were calculated using 10 bilayers thick slabs terminated by hydrogen from one side. A $10\,\mathrm{\mathring{A}}$ thick vacuum layer was used. The rectangular surface unit cell, as outlined in Fig. 1(a), was employed. The Brillouin zone was sampled using a $4\times 4\times 1$ $\mathbf{k}$-point grid.Monkhorst and Pack (1976) The geometry was optimized until all atomic forces became less than $1\,\mathrm{meV/\mathring{A}}$. The constant-current STM images were produced within the Tersoff-Hamann approachTersoff and Hamann (1985) using eigenvalues and eigenfunctions of the Kohn-Sham equationKohn and Sham (1965) for a relaxed atomic structure.

The tests were carried out to monitor the convergence of simulated STM images and surface energies with respect to basis set, Brillouin zone integration, slab thickness and separation between slabs. We estimate an error less than $1\,\mathrm{meV/\mathring{A}^{2}}$ for the calculated surface energy differences between relaxed structures. The absolute values of surface energies are overestimated by about $3\text{\textminus 7}\,\mathrm{\mathrm{meV/\mathring{A}^{2}}}$.

The ATI structural model by Battaglia *et al.*Battaglia et al. (2009a) has two main flaws. First, the calculated surface energy, according to that model, is too high. The upper limit for the Si$(331)$ surface energy can be estimated by requiring surface stability to faceting to Si$(111)$ and Si$(110)$. All three planes are schematically shown in Fig. 1(c). Therefore,

[TABLE]

where $\mathrm{\Gamma_{(331)}}$ is the upper limit for the Si$(331)$ surface energy, $\mathrm{\gamma_{(111)}}$ and $\mathrm{\gamma_{(110)}}$ are surface energies for Si$(111)$ and Si$(110)$, respectively. $\mathrm{S_{(331)}}$, $\mathrm{S_{(111)}}$, $\mathrm{S_{(110)}}$ are the surface areas of $(331)$, $(111)$ and $(110)$, which are mutually dependent due to geometrical constraints (Fig. 1(c)): $\mathrm{S_{(110)}\approx 0.649\cdot S_{(331)}}$, $\mathrm{S_{(111)}\approx 0.397\cdot S_{(331)}}$. The surface energy of Si$(111)-(7\times 7)$, according to the dimer-adatom stacking fault model by Takayanagi et al.,Takayanagi et al. (1985) is $84.9\,\mathrm{meV/\mathring{A}^{2}}$,Stekolnikov et al. (2002) while the surface energy of Si$(110)-(16\times 2)$ is $103.7\,\mathrm{meV/\mathring{A}^{2}}$ according to the structural model by Stekolnikov *et al.*Stekolnikov et al. (2004) Thus, the estimated upper limit for the Si$(331)-(12\times 1)$ surface energy according to the Eq. 1 is $101.0\,\mathrm{meV/\mathring{A}^{2}}$, which is $\approx 7\,\mathrm{meV/\mathring{A}^{2}}$ less than the value given in Ref. Battaglia et al., 2009b. This means that, according to the ATI model of the Si$(331)$ surface, it should be decomposed into Si$(111)$ and Si$(110)$ facet surfaces in the obvious contradiction with experiments.

Second, our ab initio investigation demonstrates that the relaxed ATI model by Battaglia et al. cannot account for the important surface features observed in experiments. The calculated constant-current STM images of Si$(331)-(12\times 1)$, based on the ATI structural model, are shown in Figs. 2(a) and 2(b). The pentamers, indeed, reproduce the brightest STM image features in Figs. 1(a) and 1(b). On the other hand the vertical dark stripes in the $[\bar{1}\bar{1}6]$ direction clearly visible in experimental STM images, are not reproduced. The dark stripes, representing surface depressions or trenches, have been observed almost in every STM study of the Si$(331)$ surface and, therefore, the correct structural model should account for this surface feature.Battaglia et al. (2009a); Gai et al. (2001); Hibino et al. (1993) All these problems - incorrect STM images and too high surface energy

• taken together imply that the ATI model of the $(12\times 1)$ by Battaglia et al. is not a good model for Si$(331)$.

The new structural building block, proposed in this work, is shown in Figs. 3(a) and 3(b). It contains a 6-pentagon unit (6PU) and the pentamer with an interstitial atom. The 6PU structure can be represented as two mirror-symmetrical groups with three pentagons in each of them (3-pentagon unit, 3PU). The pentagons in 3PU are folded into a trefoil with one of its lobes being side of the pentamer structure. This makes 6PU to be closely integrated into the structure of the adjacent pentamer. The silicon interconnections in 3PU are similar to that in $\mathrm{C{}_{20}}$ - the smallest fullerene.Jarrold (2000) The 3PU surface is concaved, like the $\mathrm{C{}_{20}}$ surface, if viewed from the inside of a fullerene. The 6PU, as shown in Figs. 3(a) and 3(b), has only four dangling bonds (four under-coordinated Si atoms). The pentamer with an interstitial atom introduces two additional pentagons: one at the top of the pentamer and the other - on the side away from 6PU. Therefore, we refer to the complete structure, composed of a pentamer and 6PU, as an 8-pentagon unit (8PU).

The atomic model of the Si$(331)-(12\times 1)$ surface, composed of 8PUs and presented in Fig. 3(c), is named 8P. The 8P model has two less unsaturated bonds (under-coordinated Si atoms) per unit cell, as compared to the ATI structural model proposed in Ref. Battaglia et al., 2009a. According to the 8P structural model, only six additional Si atoms per 8PU are required to form the $(12\times 1)$ reconstruction on the initially unreconstructed surface (these atoms are marked by black circles in Fig. 3(c)).

The ideal unrelaxed 8PU has a mirror symmetry in the $(\bar{1}10)$ plane (Fig. 3(a)). This symmetric atomic configuration is, however, unstable against buckling. When relaxing the structure, the under-coordinated Si atoms are displaced either away (raised) or toward the bulk (lowered), as marked by red/blue balls in Fig. 3(c). Similar structural transformations are well known for dimers on Si$(100)-(2\times 1)$Ramstad et al. (1995) and also have been observed for more complex structures on the triple step edges of the Si$(7\,7\,10)$ surface.Zhachuk et al. (2014); Teys et al. (2006) Thus, the mirror symmetry of relaxed 8PU breaks due to buckling of surface atoms, although the glide plane symmetry of the $(12\times 1)$ reconstruction along the $[\bar{1}\bar{1}6]$ direction retains.

The three bonds of raised atoms become strongly $p$-like, and a fully occupied dangling bond state, mostly $s$-like, is formed. Conversely, the lowered atoms become approximately $sp^{2}$-coordinated. They produce high-energy $p$-like dangling bond states, whose electrons are donated to the $s$-type radicals on raised atoms. The raised/lowered silicon atoms interact with each other due to a charge transfer between them and the locally induced tensile/compressive strain.

Due to the buckling of the surface atoms in 8PU, multiple configurations of the $(12\times 1)$ reconstruction are possible. There are 8 symmetry nonequivalent atoms with dangling bonds per $(12\times 1)$ unit cell (Fig. 3(c)). In the absence of the interaction between them, their buckling would be uncorrelated and we could expect $2^{8}=256$ configurations with the glide plane symmetry. We have found, however, only 8 atomic configurations which are, at least, metastable out of 98 (most probable) relaxed structures with a glide plane symmetry. These configurations are shown in Supplementary Fig. 1. The surface energies of most of them cluster in the $2\,\mathrm{meV/\mathring{A}^{2}}$ energy window. The mixed configurations ij, composed of symmetric configurations i and j are also metastable (see Supplementary Fig. 2, for an example of such structure). These configurations have no glide plane symmetry. The Si$(331)$ surface, in principle, should adopt the configuration with the lowest energy. However, the influence of the STM tip (electric field, injected charge) cannot be excluded since the calculated structures are quasi-degenerate. The Si$(331)-(12\times 1)$ surface configuration, which demonstrates the best agreement with the experimental STM images, is shown in Fig. 3(c) and discussed below.

Local and reversible modification of the buckled Ge$(100)$ atomic structure by STM tip has been reported.Takagi et al. (2004) The results have been discussed in the context of realizing a rewritable nanometer-scale memory.Cho and Joannopoulos (1996) The existance of multiple buckled configurations of the Si$(331)$ surface with similar energies imply that these effects can be observed on Si$(331)$ as well. This idea deserves furher research.

The formation energy of the unreconstructed and unrelaxed Si$(331)$ surface is $129.7\,\mathrm{meV/\mathring{A}^{2}}$ according to our calculation. The energy gain due to the $(12\times 1)$ surface reconstruction and relaxation, according to the ATI model proposed by Battaglia et al.,Battaglia et al. (2009a) is $15.8\,\mathrm{meV/\mathring{A}^{2}}$ (our data). Thus, the surface energy according to that model is $113.9\,\mathrm{meV/\mathring{A}^{2}}$. These values are in a reasonable agreement with the data reported in Ref. Battaglia et al., 2009b. The energy gain due to the surface reconstruction, according to the 8P model, shown in Fig. 3(c), is $31.2\,\mathrm{meV/\mathring{A}^{2}}$. Therefore, according to the 8P model, the Si$(331)-(12\times 1)$ surface energy is $15.4\,\mathrm{meV/\mathring{A}^{2}}$ lower than in the ATI model proposed in Ref. Battaglia et al., 2009a. Such huge energy difference is far beyond the possible error in computed relative surface energies. The surface energy of Si(331)-(12×1), according to the 8P model, is $98.5\,\mathrm{meV/\mathring{A}^{2}}$, which is below its estimated upper limit, calculated using the Eq. 1. Moreover, the calculated surface energy is close to that of the Si$(111)-(7\times 7)$, which is $92.1\,\mathrm{meV/\mathring{A}^{2}}$ according to our results obtained using a similar calculation procedure.Zhachuk et al. (2013) The Si(111)-(7×7) surface is, in turn, known to be the most stable silicon surface with the lowest energy.Stekolnikov et al. (2002); Stekolnikov and Bechstedt (2005)

There are several reasons for the low surface energy of Si(331)-(12×1) in the 8P model. First, the number of dangling bonds in the 8P model is less than in the ATI model. Second, the bond lengths in 8P are nearly the bulk bond length and they are less stretched than in the ATI model. Third, the bond angles are only slightly distorted with respect to the tetrahedral structure. Fourth, the surface energy is additionally decreased due to the buckling of surface atoms.Bechstedt (2003)

The structure of 6PU is difficult to visualize in STM because most of its bonds are saturated and its surface is concaved. The same difficulty exists for the dimers in the Si$(111)-(7\times 7)$ reconstruction, which, to our knowledge, have never been observed in STM. The high-resolution STM image of the Si$(331)$ surface exhibiting the $(12\times 1)$ reconstruction is presented in Fig. 1(b). The image agrees with the study of Battaglia *et al.,*Battaglia et al. (2009a) but it reveals more details between pentamers (Fig. 1(b)). There are a few surface defects visible in the presented STM image, but the repeating structural units are easily recognized. Besides the pentamer structure, clearly resolved in Fig. 1(b), three symmetry nonequivalent atoms can be distinguished in the experimental STM image. These atoms are numbered $1-3$ in the experimental STM image in Fig. 1(b), in the calculated STM image in Fig. 2(d) and in the atomic model in Fig. 3(c). Atom 3 is also visible in the STM images by Battaglia *et al.*Battaglia et al. (2009a) and it was attributed to the adatom in the ATI atomic model. According to the 8P model, however, atoms 2 and 3 correspond to the under-coordinated buckled Si atoms in the 6PU structure (atom 2 is lowered, atom 3 is raised), while atom 1 is a rest-atom of the Si$(331)$ surface.

The 8P model correctly reproduces the trenches in the $[\bar{1}\bar{1}6]$ direction as one can see in the large scale calculated STM image in Fig. 2(c). The trench area is located between the zig-zag rows of 8PUs. Due to the 3D structure of 8PUs, the atoms in the trench appear relatively lower (darker) in STM images. One may suggest that the trenches serve for the strain relaxation introduced by 8PUs similar to the dislocations formed in the strained systems during growth.

In summary, we have presented a novel model of the Si(331) surface. The new model consistently describes the experimental STM data and demonstrates the remarkably low surface formation energy. The model predicts that many surface configurations are possible depending on the buckling states of Si(331) reconstruction elements. This can potentially be used for information storage and requires further research.