# Driving Force of Phase Transition in Indium Nanowires on Si(111)

Hyun-Jung Kim and Jun-Hyung Cho∗ Department of Physics and Research Institute
for Natural Sciences, Hanyang University, 17 Haengdang-Dong, Seongdong-Ku,
Seoul 133-791, Korea

###### Abstract

The precise driving force of the phase transition in indium nanowires on
Si(111) has been controversial whether it is driven by a Peierls instability
or by a simple energy lowering due to a periodic lattice distortion. The
present van der Waals (vdW) corrected hybrid density functional calculation
predicts that the low-temperature 8${\times}$2 structure whose building blocks
are indium hexagons is energetically favored over the room-temperature
4${\times}$1 structure. We show that the correction of self-interaction error
and the inclusion of vdW interactions play crucial roles in describing the
covalent bonding, band-gap opening, and energetics of hexagon structures. The
results manifest that the formation of hexagons occurs by a simple energy
lowering due to the lattice distortion, not by a charge density wave formation
arising from Fermi surface nesting.

###### pacs:

73.20.At, 68.35.Md, 71.30.+h

One-dimensional (1D) electronic systems have attracted much attention because
of the richness of exotic physical phenomena such as charge density wave
formation due to the Peierls instability peierls , non-Fermi liquid behavior
stewart , or Jahn-Teller distortion jahn . A prototypical example of quasi-1D
systems is self-organized indium nanowires on the Si(111) surface snijders ;
bunk ; yeom1 . Each nanowire is composed of two zigzag chains of In atoms, and
the nanowires are separated by a zigzag chain of Si atoms (see Fig. 1) bunk .
Below ${\sim}$120 K, this quasi-1D system undergoes a reversible phase
transition initially from a 4${\times}$1 structure to a 4${\times}$2
structure, then to an 8${\times}$2 structure yeom1 ; kump , showing a period
doubling both parallel and perpendicular to the In wires. This
(4${\times}$1)${\leftrightarrow}$(8${\times}$2) phase transition is
accompanied by a metal-insulator transition yeom1 ; ahn ; yeom2 . These
intriguing results have stimulated many experimental park ; ahn ; gon ;
chandola  and theoretical studies cho1 ; cho2 ; tsay ; lopez ; gon1 ; gon2 ;
cho3 ; rii ; steko . However, the precise driving force of the phase
transition has been elusive for a long time. It has been suggested that the
phase transition is driven by a Peierls instability yeom1 ; ahn ; yeom2 ; park
or by a simple energy lowering due to a periodic lattice distortion cho1 ;
cho2 ; tsay ; lopez ; gon1 ; gon2 ; rii ; steko . The former mechanism
involves the strong coupling between lattice vibrations and electrons near the
Fermi level caused by Fermi surface nesting. Consequently, the charge density
wave formation together with the lattice distortion occurs because of a larger
electronic energy gain compared to an elastic energy cost. On the other hand,
the latter mechanism involves either the trimer formation cho1 ; cho2 ; tsay ;
lopez  in In chains with an elastic energy gain arising from the lattice
distortion or the hexagon formation cho3 ; gon1 ; gon2 ; rii ; steko  with an
elastic energy gain from the lattice distortion as well as an electronic
energy gain from the band-gap opening.

Figure 1: Top view of the optimized (a) 4${\times}$1 and (b) 8${\times}$2
structures of the In/Si(111) surface system. The dark and gray circles
represent In and Si atoms, respectively. For distinction, Si atoms in the
subsurface are drawn with small circles. Each unit cell is indicated by the
solid line.

Despite the above-mentioned controversial issue on the origin of the phase
transition in the In/Si(111) system, the so-called hexagon model describes
well several observed features of the low-temperature phase such as an
insulating character yeom1 ; ahn ; yeom2 , scanning tunneling microscopy
images gon , and anisotropic optical interband transitions chandola .
Initially, Gonz$\acute{\rm a}$lez, Ortega, and Flores proposed that a shear
distortion, whereby neighboring In chains are displaced in opposite
directions, allows for the formation of hexagon in the 4${\times}$2 unit cell
gon1 ; gon2 . Since this shear phonon mode gon  is different from a phonon
mode with the observed yeom1  Fermi surface nesting vector 2$k_{F}$ =
${\pi}$/$a_{\rm x}$ ($a_{\rm x}$ is the lattice constant along the In chains),
the Peierls mechanism is unlikely to be the driving force of the phase
transition in the In/Si(111) system. Moreover, the stabilization of the
8${\times}$2 structure by doubling the unit cell perpendicular to the In wires
is irrelevant with an electron-phonon coupling due to the observed Fermi
surface nesting along the direction parallel to the In wires. The density-
functional theory (DFT) calculations of Gonz$\acute{\rm a}$lez, Ortega, and
Flores showed that the 4${\times}$2 or 8${\times}$2 hexagon structure is
energetically favored over the 4${\times}$1 structure gon1 ; flores , but
subsequent more accurate DFT calculations steko ; cho3  within the local
density ca  as well as generalized gradient approximation pw  (LDA/GGA) did
not support the energetic preference of the hexagon structures (see Table I).
According to an LDA calculation with keeping the In 4$d$ electrons frozen, the
8${\times}$2 hexagon structure was predicted to be energetically favored over
the 4${\times}$1 structure steko . However, this result is an artifact of the
relatively less accurate scheme. Because of the proper prediction for the
energy stability between the 4${\times}$1 and 8${\times}$2 structures, the LDA
scheme with a frozen core of In 4$d$ electrons has been employed to calculate
the entropy difference wippermann  or the energy barrier between the two
structures wall . We note, however, that the LDA and GGA calculations with the
treatment of the In 4$d$ states as valence electrons predicted that the
4${\times}$2 and 8${\times}$2 hexagon structures are less stable than the
4${\times}$1 structure steko .

In this Letter, we present a new theoretical study which extends the previous
work by considering a hybrid exchange-correlation functional hse  and by
taking van der Waals (vdW) vdw  interactions into account. We will show that
the correction of self-interaction error (SIE) cures over-delocalization of
surface-state electrons inherent in the DFT and therefore describes adequately
the covalent bonding, band-gap opening, and energetics of hexagon structures.
Furthermore, we find that the vdW interactions between In atoms play an
important role in further stabilizing the 4${\times}$2 and 8${\times}$2
hexagon structures. Since the formation of hexagons and the more stabilization
of the 8${\times}$2 structure are not associated with an electron-phonon
coupling due to Fermi surface nesting, we can say that the phase transition in
the In/Si(111) system is driven by a simple energy lowering due to the hexagon
formation rather than by a Peierls-like mechanism.

The present vdW corrected hybrid DFT calculations were performed using the
FHI-aims aims  code for an accurate, all-electron description based on numeric
atom-centered orbitals, with “tight” computational settings. For the exchange-
correlation energy, we employed the hybrid functional of HSE hse  as well as
the GGA functional of PBE per . The ${\bf k}$-space integrations in various
unit-cell calculations were done equivalently with 64 ${\bf k}$ points in the
surface Brillouin zone of the 4${\times}$1 unit cell. The Si(111) substrate
(with the Si lattice constant $a_{0}$ = 5.482 Å) was modeled by a 6-layer slab
(not including the Si surface chain bonded to the In chains) with ${\sim}$15 Å
of vacuum in between the slabs. Each Si atom in the bottom layer was
passivated by one H atom. All atoms except the bottom layer were allowed to
relax along the calculated forces until all the residual force components were
less than 0.02 eV/Å.

Table 1: Calculated total energies (in meV per 4${\times}$1 unit cell) of the 4${\times}$2 and 8${\times}$2 structures relative to the 4${\times}$1 structure, together with the band gaps (in eV). For comparison, the previous LDA ca  and GGA pw  results are also given. “Valence $d$ (core $d$)” represents the treatment of the In 4d states as valence (core) electrons. | 4${\times}$2 | 8${\times}$2
---|---|---
| ${\Delta}E$ | $E_{\rm g}$ | ${\Delta}E$ | $E_{\rm g}$
PBE | 33 | no | 26 | 0.08
PBE+vdW | 22 | 0.05 | 13 | 0.08
HSE | 3 | 0.10 |  $-$15 | 0.19
HSE+vdW |  $-$23 | 0.21 |  $-$40 | 0.21
LDA (Ref. gon1 ; gon2 ) |  $-$80 |  |  $-$100 |
GGA$-$valence $d$ (Ref. steko ) | 48 |  | 27 | 0.05
GGA$-$core $d$ (Ref. steko ) | 36 |  | 25 |
LDA$-$valence $d$ (Ref. steko ) | 15 |  | 2 |
LDA$-$core $d$ (Ref. steko ) | 5 |  |  $-$12 |

We begin to optimize the 4${\times}$1, 4${\times}$2, and 8${\times}$2
structures using the PBE functional. The optimized 4${\times}$1 and
8${\times}$2 structures are displayed in Fig. 1(a) and 1(b), respectively. We
find that the 4${\times}$2 and 8${\times}$2 structures show the formation of
hexagons. Unlike previous pseudopotential calculations cho1 ; cho2 ; steko ,
the present all-electron calculations were not able to find the stabilization
of trimers in the 4${\times}$2 and 8${\times}$2 structures, which were
converged to the 4${\times}$1 structure. The calculated total energies (in meV
per 4${\times}$1 unit cell) of the 4${\times}$2 and 8${\times}$2 structures
relative to the 4${\times}$1 structure are given in Table I. We find that the
4${\times}$2 and 8${\times}$2 structures are less stable than the 4${\times}$1
structure with ${\Delta}E_{\rm 4{\times}2-4{\times}1}$ = 33 meV and
${\Delta}E_{\rm 8{\times}2-4{\times}1}$ = 26 meV, respectively, consistent
with those (48 and 27 meV in Table I) obtained by a previous GGA calculation
steko  with the Perdew-Wang exchange-correlation functional pw . The
calculated interatomic distances of In atoms are given in Table II. In the
4${\times}$1 structure, the In-In distance $d_{{\rm In}_{1}-{\rm In}_{3}}$
($d_{{\rm In}_{2}-{\rm In}_{4}}$) within an In chain is 3.045 (3.047) Å, while
$d_{{\rm In}_{3}-{\rm In}_{4}}$ between the two In chains is 3.115 Å. However,
in the 4${\times}$2 (8${\times}$2) structure, the In-In distances between the
two In chains are shortened as $d_{{\rm In}_{3}-{\rm In}_{4}}$ = 3.054 (3.030)
and $d_{{\rm In}_{7}-{\rm In}_{8}}$ = 3.027 (3.011) Å, leading to the
formation of hexagon [see Fig. 1(b)]. We note that each In-In distance in the
8${\times}$2 structure slightly changes compared to the corresponding one in
the 4${\times}$2 structure because of the formation of hexagons in two
opposite orientations (see Table II).

Table 2: Calculated interatomic distances (in Å) of In atoms in the 4${\times}$1, 4${\times}$2, and 8${\times}$2 structures using PBE. The results obtained using PBE+vdW are also given in parentheses. The labeling of In atoms are shown in Fig. 1. |  | 4${\times}$1 |  | 4${\times}$2 |  | 8${\times}$2
---|---|---|---|---|---|---
In1$-$In3 |  | 3.045 (3.040) |  | 2.949 (2.904) |  | 2.941 (2.912)
In2$-$In4 |  | 3.047 (3.053) |  | 2.999 (3.006) |  | 2.996 (2.994)
In3$-$In4 |  | 3.115 (3.092) |  | 3.054 (3.041) |  | 3.030 (3.017)
In3$-$In5 |  |  |  | 3.018 (2.993) |  | 2.999 (2.975)
In2$-$In8 |  |  |  | 3.024 (3.005) |  | 2.995 (2.983)
In5$-$In7 |  |  |  | 3.010 (3.021) |  | 3.028 (3.026)
In8$-$In6 |  |  |  | 2.952 (2.913) |  | 2.952 (2.928)
In7$-$In8 |  |  |  | 3.027 (3.029) |  | 3.011 (3.011)
In1$-$In${}_{7^{\prime}}$ |  |  |  | 3.171 (3.201) |  | 3.128 (3.153)
In4$-$In${}_{7^{\prime}}$ |  |  |  | 3.294 (3.306) |  | 3.347 (3.348)
In4$-$In${}_{6^{\prime}}$ |  |  |  | 3.130 (3.178) |  | 3.119 (3.160)

To examine the influence of vdW interactions on the geometry and energetics,
we use the PBE+vdW scheme developed by Tkatchenko and Scheffler vdw , where
the vdW coefficients and radii are determined using the self-consistent
electron density guo . As shown in Table II, several In-In distances obtained
using PBE and PBE+vdW exhibit some differences by less than 0.05 Å. The
calculated PBE+vdW total energies of the 4${\times}$2 and 8${\times}$2
structures relative to the 4${\times}$1 structure are also listed in Table I.
We find that the 4${\times}$2 and 8${\times}$2 structures are still less
stable than the 4${\times}$1 structure with ${\Delta}E_{\rm
4{\times}2-4{\times}1}$ = 22 meV and ${\Delta}E_{\rm 8{\times}2-4{\times}1}$ =
13 meV, respectively. Thus, the inclusion of vdW interactions within the
PBE+vdW scheme does not reverse the stability of the 4${\times}$2 (or
8${\times}$2) and 4${\times}$1 structures.

The calculated surface band structures of the 4${\times}$2 structure obtained
using PBE and PBE+vdW are displayed in Fig. 2(a) and 2(b), respectively. It is
seen that PBE gives a metallic feature while PBE+vdW gives an insulating
feature with a band gap opening ($E_{\rm g}$) of 0.05 eV. The PBE and PBE+vdW
calculations for the 8${\times}$2 structure give almost the same value of
$E_{\rm g}$ = 0.08 eV [see Fig. 1(a) and 1(b) of the Supplemental Material
(SI)]. For the 4${\times}$1 structure, both PBE and PBE+vdW predict well the
observed metallic feature ahn , where three surface bands cross the Fermi
level [see Fig. 2(a) and 2(b) of the SI]. Thus, we can say that PBE cannot
predict the observed yeom1 ; ahn ; yeom2  insulating feature for the low-
temperature phase, consistent with a previous pseudopotential calculation cho3
.

Figure 2: Surface band structure of the 4${\times}$2 structure obtained using
(a) PBE, (b) PBE+vdW, (c) HSE, and (d) HSE+vdW. The inset in (a) shows the
surface Brillouin zone for the 4${\times}$1 and 4${\times}$2 unit cells within
that for the 1${\times}$1 unit cell. The energy zero represents the Fermi
level.

The local and semi-local DFT have challenged for a reasonable description of
the energetics of different structures involved in the phase transition or the
kinetics of chemical reactions because of their intrinsic SIE sie1 ; sie2 .
Especially, the GGA tends to stabilize artificially delocalized states due to
the SIE, since delocalization reduces the self-repulsion. It is thus likely
that the present PBE functional would give a lower energy for the metallic
4${\times}$1 structure, compared to the 4${\times}$2 and 8${\times}$2
structures. In order to correct the SIE, we use the hybrid HSE functional hse
to calculate the total energies of the 4${\times}$1, 4${\times}$2, and
8${\times}$2 structures with the PBE geometries relax . We find that the
correction of SIE enhances the stability of the 4${\times}$2 and 8${\times}$2
structures relative to the 4${\times}$1 structure, giving rise to
${\Delta}E_{\rm 4{\times}2-4{\times}1}$ = 3 meV and ${\Delta}E_{\rm
8{\times}2-4{\times}1}$ = $-$15 meV, respectively. This enhanced HSE stability
of the 4${\times}$2 (8${\times}$2) structure is caused by the electronic
energy gain arising from an increased band gap of $E_{\rm g}$ = 0.10 (0.19)
eV, as shown in Fig. 2(c) [Fig. 1(c) of the SI]. Thus, HSE predicts well the
observed yeom1 ; ahn ; yeom2  insulating feature for the 4${\times}$2 and
8${\times}$2 structures.

To see the effects of the SIE on the charge density distribution, we plot the
charge density difference defined as

${\Delta}{\rho}={\rho}_{\rm HSE}-{\rho}_{\rm PBE},$ (1)

where ${\rho}_{\rm HSE}$ (${\rho}_{\rm PBE}$) is the charge density obtained
using the HSE (PBE) functional. The results for the 4${\times}$1 and
4${\times}$2 structures are displayed in Fig. 3(a) and 3(b), respectively. We
find a conspicuous difference between the 4${\times}$1 and 4${\times}$2
structures for ${\Delta}{\rho}$. It is seen that the insulating 4${\times}$2
structure has a larger ${\Delta}{\rho}$ between In atoms compared with the
metallic 4${\times}$1 structure, indicating that the relatively localized
surface states in the former are more affected by the SIE than the delocalized
surface states in the latter. This fact also reflects that in the 4${\times}$2
structure, the correction of the SIE by HSE recovers the charge localization
in the covalently bonding between In atoms. As a consequence, we obtain an
increase of band gap with $E_{\rm g}$ = 0.10 eV, leading to a decrease of
${\Delta}E_{\rm 4{\times}2-4{\times}1}$ = 3 meV compared to the PBE result
(${\Delta}E_{\rm 4{\times}2-4{\times}1}$ = 33 meV). For the 8${\times}$2
structure, ${\Delta}{\rho}$ shows a similar pattern with the 4${\times}$2 case
(see Fig. 3 of the SI), yielding $E_{\rm g}$ = 0.19 eV and ${\Delta}E_{\rm
8{\times}2-4{\times}1}$ = $-$15 meV. We note that there is a general trend
that the 8${\times}$2 structure is more stable than the 4${\times}$2 structure
(see Table I). This indicates some energy gain caused by the correlation
between two In nanowires in the 8${\times}$2 structure, as pointed out by a
previous theoretical study steko .

Figure 3: (Color on line) Charge density difference between ${\rho}_{\rm HSE}$
and ${\rho}_{\rm PBE}$ for the (a) 4${\times}$1 and (b) 4${\times}$2
structures. The dark (gray) color represents the isosurface of 0.02 ($-$0.02)
e/Å3.

Using the HSE+vdW schme, we calculate the total energies of the 4${\times}$1,
4${\times}$2, and 8${\times}$2 structures with the PBE+vdW geometries relax .
We find that the 4${\times}$2 and 8${\times}$2 structures are more stable than
the 4${\times}$1 structure with ${\Delta}E_{\rm 4{\times}2-4{\times}1}$ =
$-$23 meV and ${\Delta}E_{\rm 8{\times}2-4{\times}1}$ = $-$40 meV,
respectively. Since the total energy is composed of the HSE energy ($E_{\rm
HSE}$) and the vdW energy ($E_{\rm vdW}$) which is given by a sum of pairwise
interatomic C6$R^{-6}$ terms, the total energy difference between the
4${\times}$2 (or 8${\times}$2) and 4${\times}$1 structures is determined by

${\Delta}E={\Delta}E_{\rm HSE}+{\Delta}E_{\rm vdW}.$ (2)

Figure 4 shows ${\Delta}E_{\rm vdW}$ in ${\Delta}E_{\rm
4{\times}2-4{\times}1}$ and ${\Delta}E_{\rm 8{\times}2-4{\times}1}$, together
with its components originating from In$-$In, In$-$Si, and Si$-$Si atoms. We
find that ${\Delta}E_{\rm HSE}$ in ${\Delta}E_{\rm 4{\times}2-4{\times}1}$
(${\Delta}E_{\rm 8{\times}2-4{\times}1}$) is $-$4 ($-$18) meV, while
${\Delta}E_{\rm vdW}$ in ${\Delta}E_{\rm 4{\times}2-4{\times}1}$
(${\Delta}E_{\rm 8{\times}2-4{\times}1}$) is $-$19 ($-$22) meV. Therefore, the
inclusion of vdW interactions largely enhances the stabilization of the
4${\times}$2 and 8${\times}$2 structures. We note that the ${\Delta}E_{\rm
HSE}$ values ($-$4 and $-$18 meV) in ${\Delta}E_{\rm 4{\times}2-4{\times}1}$
and ${\Delta}E_{\rm 8{\times}2-4{\times}1}$ are somewhat different from those
(3 and $-$15 meV) obtained from the HSE calculation due to the use of two
different PBE+vdW and PBE geometries. As shown in Fig. 4, the component of
${\Delta}E_{\rm vdW}$ originating from In$-$In atoms in the 4${\times}$2 and
8${\times}$2 structures is significantly larger in magnitude than those
originating from In$-$Si and Si$-$Si atoms. Thus, we can say that the vdW
interactions between In atoms play an important role in stabilizing the
formation of hexagons.

Figure 4: (Color on line) Contribution of vdW energy to the total-energy
difference ${\Delta}E_{\rm 4{\times}2-4{\times}1}$ (${\Delta}E_{\rm
8{\times}2-4{\times}1}$) obtained using HSE+vdW. The components originating
from In$-$In, In$-$Si, and Si$-$Si atoms are also given.

As shown in Fig. 2(d) [Fig. 1(d) in the SI], the HSE+vdW band structure of the
4${\times}$2 (8${\times}$2) structure gives $E_{\rm g}$ = 0.21 (0.21) eV, in
good agreement with a recent scanning tunneling spectroscopy measurement of
$E_{\rm g}$ = 0.2 eV zhang . We note that the HSE+vdW magnitude of
${\Delta}E_{\rm 8{\times}2-4{\times}1}$ is 40 meV per 4${\times}$1 unit cell,
which is equal to 10 meV per In atom. Although the precise
(4${\times}$1)${\leftrightarrow}$(8${\times}$2) phase transition temperature
can be estimated by comparing the vibrational free energies of the
4${\times}$1 and 8${\times}$2 structures wippermann , the HSE+vdW magnitude of
${\Delta}E_{\rm 8{\times}2-4{\times}1}$ is well comparable with the thermal
energy at the observed phase transition temperature of ${\sim}$120 K yeom1 ;
kump . Therefore, the HSE+vdW scheme is likely to give good band gap and
energetics of the low-temperature phase in the In/Si(111) system.

In summary, using the HSE and HSE+vdW schemes, we investigated the energy
stability of the low-temperature and room-temperature structures in the
In/Si(111) system, which has not been adequately described by previous DFT
calculations cho3 ; steko . We found that the correction of SIE cures the
delocalization error not only to give the insulating feature for the
4${\times}$2 and 8${\times}$2 structures but also to reverse the stability of
the 4${\times}$1 and 8${\times}$2 structures. We also found that the vdW
interactions between In atoms enhance the stability of hexagon structures. Our
results demonstrate that the formation of hexagons in the In/Si(111) system
occurs by a simple energy lowering due to the lattice distortion rather than
by a Peierls instability. We notice that the Sn/Si(111) and Sn/Ge(111)
surfaces have been the object of a large number of studies for determining the
exact crystallographic arrangement, the electronic structure, and the
mechanism of the phase transition teje . We anticipate that the correction of
self-interaction error and the inclusion of vdW interactions would give more
accurate description for the structural and electronic properties of such
prototype two-dimensional electron systems.

This work was supported by National Research Foundation of Korea (NRF) grant
funded by the Korean Government (NRF-2011-0015754). The calculations were
performed by KISTI supercomputing center through the strategic support program
(KSC-2012-C3-11) for the supercomputing application research.

∗ Corresponding author<EMAIL_ADDRESS>
## References

  * (1) R. E. Peierls, ”Quantum Theory of Solids”, Clarendon: Oxford, (1964).
  * (2) G. R. Stewart, Rev. Mod. Phys. 73, 797 (2001).
  * (3) H. A. Jahn and E. Teller, Proc. R. Soc London, Ser. A 161, 220 (1937).
  * (4) P. Snijders and H. Weitering, Phys. Mod. Phys. 82, 307 (2010), and references therein.
  * (5) O. Bunk, G. Falkenberg, J. H. Zeysing, L. Lottermoser, R. L. Johnson, M. Nielsen, F. Berg-Rasmussen, J. Baker, and R. Feidenhans’l, Phys. Rev. B 59, 12228 (1999).
  * (6) H. W. Yeom, S. Takeda, E. Rotenberg, I. Matsuda, K. Horikoshi, J. Schaefer, C. M. Lee, S. D. Kevan, T. Ohta, T. Nagao, and S. Hasegawa, Phys. Rev. Lett. 82, 4898 (1999).
  * (7) C. Kumpf, O. Bunk, J. H. Zeysing, Y. Su, M. Nielsen, R. L. Johnson, R. Feidenhans’l, and K. Bechgaard, Phys. Rev. Lett. 85, 4916 (2000).
  * (8) H. W. Yeom, K. Horikoshi, H. M. Zhang, K. Ono, and R. I. G. Uhrberg, Phys. Rev. B 65, 241307(R) (2002).
  * (9) J. R. Ahn, J. H. Byun, H. Koh, E. Rotenberg, S. D. Kevan, and H. W. Yeom, Phys. Rev. Lett. 93, 106401 (2004).
  * (10) S. J. Park, H. W. Yeom, S. H. Min, D. H. Park, and I.-W. Lyo, Phys. Rev. Lett. 93, 106402 (2004).
  * (11) C. González, J. Guo, J. Ortega, F. Flores, and H. H. Weitering, Phys. Rev. Lett. 102, 115501 (2009).
  * (12) S. Chandola, K. Hinrichs, M. Gensch, N. Esser, S. Wippermann, W. G. Schmidt, F. Bechstedt, K. Fleischer, and J. F. McGilp, Phys. Rev. Lett. 102, 226805 (2009).
  * (13) J.-H. Cho, D. H. Oh, K. S. Kim, and L. Kleinman, Phys. Rev B 64, 235302 (2001).
  * (14) J.-H. Cho, J.-Y. Lee, and L. Kleinman, Phys. Rev B 71, 081310(R) (2005).
  * (15) S.-F. Tsay, Phys. Rev. B 71, 035207 (2005).
  * (16) X. López-Lozano, A. Krivosheeva, A. A. Stekolnikov, L. Meza-Montes, C. Noguez, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 73, 035430 (2006).
  * (17) C. González, J. Ortega, and F. Flores, New J. Phys. 7, 100 (2005).
  * (18) C. González, F. Flores, and J. Ortega, Phys. Rev. Lett. 96, 136101 (2006).
  * (19) S. Riikonen, A. Ayuela, and D. Sánchez-Portal, Surf. Sci. 600, 3821 (2006)
  * (20) A. A. Stekolnikov, K. Seino, F. Bechstedt, S. Wippermann, W. G. Schmidt, A. Calzolari, and M. Buongiorno Nardelli , Phys. Rev. Lett. 98, 026105 (2007).
  * (21) J.-H. Cho and J.-Y. Lee, Phys. Rev B 76, 033405 (2007).
  * (22) González, J. Ortega, and F. Flores performed a first-principles tight-binding molecular dynamics calculation with a basis set of optimized atomic-like orbitals and the LDA using the FIREBALL code. Unlike the traditional use of the LDA which generally underestimates the band gap for insulators, their employed local orbital basis set with the LDA tends to overestimate band gaps [see Jelínek $et$ $al$., Phys. Rev. B 71, 235101 (2005), and references therein], thereby leading to the stabilization of the insulating 4${\times}$2 and 8${\times}$2 structures over the metallic 4${\times}$1 structure.
  * (23) D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
  * (24) J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).
  * (25) S. Wippermann and W. G. Schmidt, Phys. Rev. Lett. 105, 126102 (2010).
  * (26) S. Wall, B. Krenzer, S. Wippermann, S. Sanna, F. Klasing, A. Hanisch-Blicharski, M. Kammler, W. G. Schmidt, and M. Horn-vonHoegen, Phys. Rev. Lett. 109, 186101 (2012).
  * (27) A. V. Krukau, O. A. Vydrov, A. F. Izmaylov, and G. E. Scuseria, J. Chem. Phys. 125, 224106 (2006).
  * (28) A. Tkatchenko and M. Scheffler, Phys. Rev. Lett. 102, 073005 (2009).
  * (29) V. Blum, R. Gehrke, F. Hanke, P. Havu, V. Havu, X. Ren, K. Reuter, and M. Scheffler, Comput. Phys. Commun. 180, 2175 (2009).
  * (30) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); 78 1396(E) (1997).
  * (31) G.-X. Zhang, A. Tkatchenko, J. Paier, H. Appel, and M. Scheffler, Phys. Rev. Lett. 107, 245501 (2011).
  * (32) J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
  * (33) S. Kümmel and L. Kronik, Rev. Mod. Phys. 80, 3 (2008).
  * (34) The PBE (PBE+vdW) geometry was kept fixed for the HSE (HSE+vdW) calculations. It was found that the fully relaxation of the geometries using HSE and HSE+vdW changes the energy difference between the 4${\times}$1 and 4${\times}$2 (or 8${\times}$2) structures by less than 3 meV per 4x1 unit cell.
  * (35) H. Zhang, J.-H. Choi, Y. Xu, X. Wang, X. Zhai, B. Wang, C. Zeng, J.-H. Cho, Z. Zhang, and J. G. Hou, Phys. Rev. Lett. 106, 026801 (2011).
  * (36) A. Tejeda, Y. Fagot-Révurat, R. Cortés, D. Malterre, E. G. Miche, and A. Mascaraque, Phys. Status Solidi A 209, 614 (2012), and references therein.