J. Phys. Soc. Jpn. 93, 074601 (2024) [6 Pages]

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A New Superstructure Model of the Si(111)-\(\sqrt{7} \times \sqrt{3} \)-In Surface

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, Jun-Ichi Iwata²^,³, and Yu-ichiro Matsushita²^,³^,⁴

+ Affiliations

¹Department of Physics, Kyoto Sangyo University, Kyoto 603-8555, Japan²Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan³Quemix Inc., Chuo, Tokyo 103-0027, Japan⁴Quantum Materials and Applications Research Center, National Institutes for Quantum Science and Technology (QST), Meguro, Tokyo 152-8550, Japan

Received January 9, 2024; Accepted May 20, 2024; Published June 18, 2024

We propose a new atomistic-scale model (2.429-ML model) for experimentally-observed superstructures of the Si(111)-\(\sqrt{7} \times \sqrt{3} \)-In surface, using first-principles calculations based on the density-functional theory. It successfully explains the long-range images observed in scanning-tunneling microscopy (STM) experiments, as well as the band structure obtained from angle-resolved photoemission spectroscopy (ARPES) experiments. Our new model is more plausible than the previous 2.4-ML model that only reproduces the ARPES experiment, providing a basis for application studies of this surface. We also show that the arrangement of the bright spots in the STM images can be seen as a beat pattern, i.e., one-dimensional Moiré.

https://doi.org/10.7566/JPSJ.93.074601

1. Introduction

Superstructures or superlattices are very important research topics, as they can exhibit physical properties that are quite different from those of their constituents.¹^,²⁾ Occasionally, superstructures can be nested and higher-order superstructures can emerge.³⁾ Crystal surfaces with deposited adatoms are typical systems in which such complex superstructures can appear depending on their coverage.

In this work, we focus on the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface.⁴⁾ This surface has attracted much attention because of its uniqueness in the electronic structure: The topmost In atomic layer exhibits metallic features down to low temperatures,⁵⁾ while the Si substrate shows semiconducting characters; This ultrathin metallic layer is known to become superconducting⁶^,⁷⁾ below ∼3 K, and was first discovered as a two-dimensional superconductor a few atoms thick. Recent experiments⁸⁾ have opened up new prospects for spintronics on this surface, and have gathered further attention in terms of applications. In order to investigate these electronic properties in more detail, it is necessary to fully understand their atomic structure. So far, the surface period is believed to be \(\sqrt{7}\times \sqrt{3}\) based on low-energy electron diffraction (LEED) experiments.⁴^,⁵^,⁷⁾ An atomic structure model with the period of \(\sqrt{7}\times \sqrt{3}\), named 2.4-monolayer (ML) model, has been proposed⁹^,¹⁰⁾ and widely accepted¹¹⁾ due to its excellent reproducibility of the angle-resolved photoemission spectroscopy (ARPES) data.¹²⁾

However, we note here that the 2.4-ML model is still open for discussion. This is because the STM images reported in many experiments⁵^–⁷⁾ are often accompanied by features that cannot be explained by the 2.4-ML model. In Fig. 1, we show a schematic illustration of a typical image in scanning-tunneling microscopy (STM) experiments.⁷⁾ The parallelogram in the figure represents a \(\sqrt{7}\times \sqrt{3}\) unit cell. Bright dimers are observed at some locations [Fig. 1(a)], which can be reproduced by calculations using the 2.4-ML model.⁹^,¹³^,¹⁴⁾ However, bright trimers are also present in the experimentally observed STM images [Fig. 1(b)], which cannot be explained by the 2.4-ML model. Looking at a large area [Fig. 1(c)], we find that bright dimer regions and bright trimer regions appear alternately with a long-range period \({\sim}60{\text{–}}90\) Å along the \((1\bar{1}0)\)-direction.¹³⁾ These observations infer that the surface identity is not the naive 2.4-ML model with the primitive \(\sqrt{7}\times \sqrt{3}\) unit cell, but a superstructure due to some long-range modulation imposed on it. A similar idea has been suggested in an earlier paper,¹³⁾ but specific models of its atomic structure have not yet been presented. It should be noted that, in general, experimental data on superstructures obtained by LEED and/or ARPES may be reported for “primitive” unit cells without recognizing weak long-range orders.¹⁵⁾ Therefore, there is no need to stick to the \(\sqrt{7}\times \sqrt{3}\) period.

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Figure 1. (Color online) Schematic illustrations of a typical image in STM experiments⁷⁾ for the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface. At one location (a) and another (b), bright dimers and bright trimers are seen in a \(\sqrt{7}\times \sqrt{3}\) cell, respectively. In an overview (c), we find a superstructure in which bright-dimer regions and bright-trimer regions alternate with a long-range period along the \((1\bar{1}0)\)-direction. The period depends on the number of dimers and trimers involved in the superstructure, and is experimentally reported¹³⁾ to be \({\sim}60{\text{–}}90\) Å.

In this paper, we revisit the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface from the viewpoint of its superstructure. The purpose of this paper is to propose a promising superstructure model of this surface from first principles. We start from the 2.4-ML model represented by a rectangular supercell of appropriate size. By slightly changing the surface coverage from 2.4-ML to 2.429-ML, we obtain a new superstructure model that reproduces both the STM⁵^–⁷⁾ and ARPES¹²⁾ experiments. Our new atomic structure model replaces the conventional 2.4-ML model⁹^,¹⁰⁾ that reproduces only the ARPES experiment,¹²⁾ and is expected to serve as a basis for previous and future studies on the electronic properties of the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface. We also find that the arrangement of the bright spots in the STM images can be seen as a beat pattern, i.e., one-dimensional Moiré.

2. Method

We have performed first-principles calculations based on the density functional theory (DFT)¹⁶^,¹⁷⁾ within the limit of the generalized gradient approximation (GGA).¹⁸⁾ The electron–ion interactions were described by norm-conserving pseudo-potentials.¹⁹⁾ We used the real-space scheme,²⁰⁾ in which wave functions, charge density and local potentials were calculated at grid points in the real space. The intergrid distance d was chosen to be 0.22 Å in optimizing the atomic structures, which corresponds to the cutoff energy of 52 Ry for a plain-wave basis set. For atomic structures optimized in that way, \(d=0.35\) Å was enough to obtain converged band structures. We used repeated slab models with periodic boundary conditions. The width of the vacuum region separating the slabs is larger than ∼7 Å, which is sufficient to avoid artificial slab–slab interactions. Structural optimizations were performed until the forces on the atoms became less than \(5\times 10^{-4}\) hartree/bohr. All calculations were performed with the Quloud-RSDFT program package, which has been developed by Iwata et al.²⁰^,²¹⁾

3. Results and Discussions

Since the conventional 2.4-ML model reproduces the ARPES experiment¹²⁾ for the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface but cannot reproduce the periodic alternation of the dimer and trimer images in the STM experiments,⁵^–⁷⁾ we construct a new superstructure model that explains both ARPES and STM experiments through the following steps: (i) Start with a supercell of the 2.4-ML model in the size that is approximately the same as that of the superstructure of the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface observed in STM experiments; (ii) Slightly modify the 2.4-ML supercell model in (i) so that the dimer and trimer regions appear alternately in the calculated STM image; (iii) Verify that the modification in (ii) does not compromise the reproducibility of the ARPES data inherent to the original 2.4-ML model.

First, a supercell of the 2.4-ML model of the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface⁹^,¹⁰⁾ is shown in Fig. 2(a) as a benchmark for constructing a superstructure model. We use a slab model in which two In layers are deposited on a substrate consisting of four Si layers, with the dangling bonds on the opposite side of the slab terminated by H atoms. The parallelogram depicted by the solid line in the top view is a primitive unit cell with a period of \(\sqrt{7}\times \sqrt{3}\) in the direction parallel to the surface. The cell contains 12 In atoms, half of them in the upper In layer and the other half in the lower In layer. They are arranged on 5 buckled hexagons of Si atoms located in the top two layers of the substrate. The coverage of the surface, 2.4-ML (\(=12/5\)), is calculated as the ratio of them. The In atom marked with \(^{*}\) in Fig. 2(a) is placed at \(x=0\), and then the entire system is structurally optimized to be mirror symmetric with respect to \(x=0\).

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Figure 2. (Color online) (a) The side and top views of the 2.4-ML model.⁹^,¹⁰⁾ The primitive unit cell shown by the parallelogram is \(\sqrt{7}\times \sqrt{3}\) in size. The \(15\times \sqrt{3}\) supercell is shown by the large box, with 72 In atoms deposited on 30 buckled Si hexagons in the substrate. (b) The side and top views of the 2.429-ML model proposed by us. The \(14\times \sqrt{3}\) primitive cell is shown by the large box, with 68 In atoms deposited on 28 buckled Si hexagons in the substrate. The local arrangement of the atoms in (b) is similar to that in (a) at the left and right ends of the cell, and shifted from that in (a) at the center.

Experimentally, the period of the superstructure is reported¹³⁾ to be \({\sim}60{\text{–}}90\) Å. Taking account of this scale, a \(15\times \sqrt{3}\) supercell of the 2.4-ML model is shown by a large box in Fig. 2(a), which is 58.1 Å in the x-direction with 72 In atoms deposited on 30 buckled Si hexagons. Though the shape is rectangular and the size is six times larger than the primitive \(\sqrt{7}\times \sqrt{3}\) cell, the arrangement of atoms and the coverage are identical to the original 2.4-ML model. We show the band structure of the 2.4-ML model calculated for the \(\sqrt{7}\times \sqrt{3}\) cell in Fig. 3(a). It is excellently consistent with the ARPES experiments¹²⁾ as has already been reported.⁹^,¹⁰⁾ We also show the STM image calculated for the 2.4-ML model in the top panel of Fig. 4(a). We find a bright dimer (two adjacent bright spots) in each of the \(\sqrt{7}\times \sqrt{3}\) cells. The bright spots, labeled A and B, are on the upper-layer In atoms located very close to the hollow sites of the Si hexagons in the underlying substrate. As found in the bottom panel, they are at the lowest positions in the corrugated structure of the upper In layer. The bright dimers found in the 2.4-ML model explain a local part of the experimental STM image [Fig. 1(a)] but not the entire STM image [Fig. 1(c)]. Thus, the superstructure model to be constructed in the next step must improve the STM image with the reproducibility of the ARPES data maintained.

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Figure 3. (Color online) The band structure of the 2.4-ML model (a) and the 2.429-ML model (b), calculated for the Brillouin zone (BZ) of the \(\sqrt{7}\times \sqrt{3}\) cell with the Fermi level \(E_{\text{F}}\) set to be 0. Band unfolding calculations were done to obtain (b).

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Figure 4. (Color online) For the (a) 2.4-ML and (b) 2.429-ML models, we show the STM image calculated as the local density of states integrated over the energy range from the Fermi level \(E_{\text{F}}\) to \(E_{\text{F}} + 500\) meV, on a horizontal plain 1 Å above the topmost In atom (top). We also show the z-coordinates of the In atoms in the upper In layer measured from the top layer of the Si substrate (bottom).

As the second step, in Fig. 2(b), we present a new superstructure model of the Si(111)-\(\sqrt{7}\times\sqrt{3}\)-In surface. Though it looks similar to the previous 2.4-ML model [Fig. 2(a)] with the \(15\times \sqrt{3}\) cell, the cell of our new superstructure model is \(14\times \sqrt{3}\) in size. The cell consists of 68 In atoms and 28 buckled Si hexagons, and the surface coverage is calculated to be 2.429-ML (\(=\frac{68}{28}\)). Thus, the essential difference between the previous model and our new model is the very slight modification of the surface coverage. The In atom denoted by \(^{**}\) in Fig. 2(b) is placed at \(x=0\), and structural optimization is performed assuming mirror symmetry with respect to \(x=0\).

Around the left and right edges of the cell, the local arrangement of the atoms in the 2.429-ML superstructure model [Fig. 2(b)] is similar to the 2.4-ML model represented by the \(15\times \sqrt{3}\) supercell [Fig. 2(a)]. In places closer to the center of the cell, on the other hand, we find larger shift of the In atoms with respect to the Si substrate reflecting the difference in the coverage between the models: For example, the In atoms located at the exact center of the cell [indicated by # and ## in Figs. 2(a) and 2(b)] are in the lower and upper In layers of the 2.4-ML and 2.429-ML models, respectively. In the top panel of Fig. 4(b), we show the STM image calculated for the 2.429-ML superstructure model. Desirably, bright dimer and bright trimer regions are observed in stripes, which successfully illustrate the overall STM image of the experiment⁷⁾ [Fig. 1(c)]. The bright dimer regions are around the left and right edges of the supercell, while the bright trimer region is at the center. They correspond to the regions where the local arrangement of atoms is similar to and shifted from the original 2.4-ML model, respectively. We thus find that the appearance of the trimer around the cell center is caused by the relative shifts of the In atoms with respect to the Si substrate. The transition from the dimer to the trimer region is continuous, reflecting the gradual change in the local arrangement of the atoms in between. The top panel of Fig. 4(b) also shows that the bright spots are on the upper-layer In atoms located very close to the hollow sites of the Si hexagons in the underlying substrate, just like those in the original 2.4-ML model [Fig. 4(a)]. The bottom panel of Fig. 4(b) shows that they are at the lowest positions in the corrugated structure of the upper In layer.

The close correlation between the bright In atoms and the hollow sites of the Si hexagons as discussed above infers that the arrangement of the bright spots can be essentially described by an interference of two simple waves: \begin{equation} f_{\text{In}}(x) + f_{\text{Si}}(x) = -{\cos}\left(2\pi \frac{n}{L}x\right) + \cos\left(2\pi \frac{m}{L}x\right). \end{equation} (1) The first and second terms model the periodic distribution of the n In atoms in the top layer and that of the m Si hexagons in the substrate, respectively. The \(f_{\text{In}}(x)+f_{\text{Si}}(x)\) has a period L, and is larger at positions where these two distributions with close periods (\(\frac{L}{n}\) and \(\frac{L}{m}\)) interfere and reinforce each other. The results of the DFT calculations [Figs. 4(a) and 4(b)] suggest that such reinforced positions correspond to bright spots. The calculated \(f_{\text{In}}(x)+f_{\text{Si}}(x)\) and its envelope function for the 2.4-ML model [\((n,m)=(18,15)\) and \(L=58.1\) Å] and those for the 2.429-ML model [\((n,m)=(17,14)\) and \(L=54.3\) Å] are shown in Figs. 5(a) and 5(b), respectively. We find that the antinodes in the envelope functions correspond to the bright regions. Also, local maximum points of \(f_{\text{In}}(x)+f_{\text{Si}}(x)\) which are denoted by A, B, i, ii, iii, iv, and v well reproduces the dimer and trimer arrangements in the STM images [of the second row in Figs. 4(a) and 4(b)]. In this sense, the array of the bright spots can be regarded as a beat pattern, i.e., one-dimensional Moiré.

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Figure 5. (Color online) The interference model to explain the arrangement of the bright spots in the (a) 2.4-ML model and the (b) 2.429-ML model. We focus on the second row in the STM image [row2s in Figs. 4(a) and 4(b)]. We use cosine waves to model the distribution of the In atoms in the top layer, \(f_{\text{In}}(x)\), and that of the Si hexagons in the substrate, \(f_{\text{Si}}(x)\). The sum of them \(f_{\text{In}}(x)+f_{\text{Si}}(x)\) shows a beat pattern with antinodes corresponding to bright regions. Local maximum points indicated by A, B, i, ii, iii, iv, and v correspond to the bright spots in Figs. 4(a) and 4(b).

The third step is to calculate the band structure for the 2.429-ML superstructure model. A naive calculation leads to a band structure for the small BZ corresponding to the \(14\times \sqrt{3}\) cell. For a direct comparison with the experimental ARPES data reported for the large BZ of the \(\sqrt{7}\times \sqrt{3}\) cell, we have subsequently performed a band-unfolding calculation according to the procedure formulated by Popescu²²^,²³⁾ (see also Appendix of the present paper). The result is shown in Fig. 3(b), which is very similar to that in Fig. 3(a) and excellently consistent with the ARPES data.¹²^,²⁴⁾

As has been discussed, the In atoms are shifted relative to the Si substrate in the center of the supercell, so that a trimer is generated at the center of the supercell in the STM image. However, Figs. 3(a) and 3(b) show that this shift of the In atoms has minimal impact on the band structure. The key to understanding the difference in the impact on the STM image and the band structure lies in the similarity of the atomic arrangements within the dotted lines in Figs. 2(a) and 2(b). As seen from the figures, the shift of the In atoms occurs at the center of the supercell, but almost the same structure reappears only a short distance from the center of the supercell as before the shift of the In atoms.²⁵⁾ Therefore, the contribution to the unfolded band structure from near the center of the supercell is almost independent of the shift of the In atoms. This is why the band structure of the modulated model (2.429-ML) is almost identical to that of the original model (2.4-ML), even though the STM images are very different.

In this way, we find that the 2.429-ML superstructure model successfully explains both the ARPES¹²⁾ and STM⁵^–⁷⁾ experiments performed for the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface. Our superstructure model is superior to the conventional 2.4-ML model⁹^,¹⁰⁾ that only reproduces the ARPES experiment.

Similar calculations were performed for several other superstructures which are also obtained by slightly modulating the 2.4-ML model represented by the \(15\times \sqrt{3}\) supercell. The results are summarized in Table I, showing that the 2.375-ML model also reproduces the ARPES¹²⁾ and STM⁵^–⁷⁾ experiments for the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface [Figs. 6(c) and 7(c)]. Thus, the 2.375-ML model is another possible candidate for the identity of the surface (though the 2.429-ML model is energetically more favored as will be discussed in the paragraph after the next). For the other models in Table I, both the ARPES and the STM images deviate from the experimental ones.

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Figure 6. (Color online) The band structures of the 2.250-ML (a), 2.267-ML (b), 2.375-ML (c), 2.533-ML (d), and 2.571-ML (e) models, unfolded to the BZ of the \(\sqrt{7}\times \sqrt{3}\) cell with the Fermi level set to be 0.

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Figure 7. (Color online) The STM images calculated for the 2.250-ML (a), 2.267-ML (b), 2.375-ML (c), 2.533-ML (d), and 2.571-ML (e) models.

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Table I. Summary on the superstructure models which are constructed by slightly modulating the 2.4-ML model represented by the \(15\times \sqrt{3}\) supercell. The columns correspond to surface coverage, cell size, change in the number of In atoms/Si hexagons with respect to the 2.4-ML model represented by the \(15\times \sqrt{3}\) cell, period in the x-direction, reproducibility of the STM experiments, and that of the ARPES experiments.

Figure 6 also shows that the models with similar coverages have similar band structures (ARPES data): the results for the 2.250-ML and 2.267-ML models are very similar [Figs. 6(a) and 6(b)]; the results for the 2.533-ML and 2.571-ML models are similar, too [Figs. 6(d) and 6(e)]. This indicates that the close similarity in coverage between the 2.375-ML model (as well as the 2.429-ML model) and the 2.4-ML model is also essential to the result that their band structures are nearly identical to each other.

We finally discuss the energetics of the 2.429-ML and 2.375-ML superstructure models. We define the surface energy \(E^{\text{surf}}\) of the superstructure by \begin{equation*} E^{\text{surf}} = E^{\text{total}}-\mu_{\text{In}}N_{\text{In}}-\mu_{\text{Si}}N_{\text{Si}}-\mu_{\text{H}}N_{\text{H}}-F_{\text{0ML}}, \end{equation*} where \(E^{\text{total}}\), \(N_{j}\), and \(\mu_{j}\) are the total energy of the cell obtained by the DFT calculations, the number of atoms and the chemical potential of the atomic species j included in the cell, respectively. The final term, defined by \(F_{\text{0ML}} = E^{\text{total}}_{\text{0ML}} -\mu_{\text{Si}}N_{\text{Si}} -\mu_{\text{H}}N_{\text{H}}\), is the formation energy of the 0-ML surface represented by the cell of the same size as the superstructure. Considering the experimental situation where In atoms deposited on Si substrate is annealed, the In chemical potential \(\mu_{\text{In}}\) is set somewhere between that of an In crystal and that of an isolated In atom. The calculated surface energy \(E^{\text{surf}}\) per \(1\times 1\) cell (denoted by \(\epsilon^{\text{surf}}\)) measured from that of the 2.4-ML model \(\epsilon^{\text{surf}}_{\text{2.4ML}}\) is shown in Fig. 8, as a function of the In chemical potential \(\mu_{\text{In}}\). The results show that the 2.429-ML model is the most promising in the sense that it can be energetically more stable than both of the 2.4-ML and 2.375-ML models, which supports our arguments that the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface should be identified as the superstructure with the long-range modulation.

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Figure 8. (Color online) The surface energy of the systems per \(1\times 1\) cell, measured from that of the 2.4-ML model. The In chemical potential \(\mu_{\text{In}}\) is set somewhere between that of an In crystal and that of an isolated In atom \(\mu_{\text{In}}^{\text{atom}}\).

The three models shown in Fig. 8 differ in the number of In atoms per unit area, the stretching of the In layer along the surface, and the arrangement of dimers and trimers. The relative stability of the models may reflect these differences. In particular, the relative stability of the 2.429-ML and 2.4-ML surfaces is reversed when the In chemical potential \(\mu_{\text{In}}\) is varied, indicating that the difference in the amount of In atoms per unit area is the most essential factor in this case. For a more quantitative discussion, it may be necessary to consider spin degrees of freedom²⁶⁾ or to increase the number of Si layers in the substrate,²⁷⁾ though we believe that the conclusions of this paper remain essentially the same even in such cases.

4. Summary

In summary, we have proposed the 2.429-ML model as the identity for the superstructure of the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface observed in experiments, using first-principles calculations based on the density-functional theory. Our superstructure model reproduces both the ARPES¹²⁾ and STM⁵^–⁷⁾ experiments, and is therefore superior to the conventional 2.4-ML⁹^,¹⁰⁾ model that only reproduces the ARPES data. Our superstructure model was constructed by introducing sufficiently small deviations of the surface coverage from 2.4-ML. Though the 2.375-ML model is also found to reproduce the both experiments, the 2.429-ML model is energetically more favored. We thus believe that the 2.429-ML model is the true picture of the Si(111)-\(\sqrt{7}\times \sqrt{3}\)-In surface. We have also found that the arrangement of the bright spots in the STM images can be seen as a one-dimensional Moiré. The importance of our discovery is that it again lays the foundation for previous and future research on this surface, especially with regard to its unique electronic properties.

Acknowledgements

This work was partly supported by the Grants-in-Aid for scientific research under the contract number JP 24K08251 conducted by MEXT, Japan. The computation in this work has been done using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo. We used the VESTA tool²⁸⁾ to draw figures used in this manuscript. We thank Professor T. Uchihashi, Professor K. Sakamoto, and Professor T. Abukawa.

Appendix

In the original band-unfolding formalism proposed by Popescu,²²⁾ the cell volume must be an integer multiple of the primitive unit cell. When we calculate the band structure of the \(14\times \sqrt{3}\) cell unfolded to the BZ of the primitive \(\sqrt{7}\times \sqrt{3}\) cell, however, the ratio of the cell volumes (the ratio of the numbers of the Si hexagons) is not an integer, \(\frac{28}{5}=5.6\). Thus, only when we calculate the band structure, we used a \(70\times \sqrt{3}\) cell which is constructed by five \(14\times \sqrt{3}\) cells arranged along the x-direction. This results in an integer ratio of \(\frac{28\times 5}{5} = 28\).

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(25) This is because all In atoms (Si atoms) in each In layer (each Si layer) are equivalent if we focus only on the In atoms (Si atoms). At the very center of the supercell, the x-coordinate of an In atom leaves that of a Si atom in its vicinity. At the same time, however, the x-coordinate of another In atom a short distance away from the center of the supercell approaches that of another Si atom in its vicinity. As a result, the shift of the In atoms with respect to the Si substrate, which influences the band structure, is effectively reduced to be about \(\frac{1}{5}\) of its apparent value. Google Scholar
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A New Superstructure Model of the Si(111)-\(\sqrt{7} \times \sqrt{3} \)-In Surface

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