# Small atoms fall into bulk from non-close-packed surfaces?

Shota Ono<EMAIL_ADDRESS>Honoka Satomi  Department of Electrical,
Electronic and Computer Engineering, Gifu University, Gifu 501-1193, Japan
Junji Yuhara  Graduate School of Engineering, Nagoya University, Nagoya
464-8603, Japan

###### Abstract

Surface rippling has been observed when atoms of $X$ and $A$ are mixed on the
$A$ substrate surface. The rippling amplitude has been estimated using hard
sphere models. We present a gedanken experiment predicting a penetration of
small atoms into bulk through the (100) surface. To understand how the
electronic effects alter this picture, we investigate the surface rippling of
$X/A(100)$ from first-principles, assuming $X=$ H to Bi except for noble gases
and $A=$ Cu, Ag, and Au. We show that the small atoms (such as H, C, N, O and
F) attract electrons from the substrate due to the large electronegativity,
which prevent them from passing through the void in the (100) surface. The
behaviors of small atoms are further explored by studying lateral
displacements of the top layer in the $A$ substrate and a formation of the $X$
dimer above, below, and across the top layer. The present work provides an
example to understand when atoms are not hard spheres.

##  I Introduction

In solid state physics, atoms are often treated as hard spheres. For example,
a close packing of equally sized spheres explains the crystal structure of
simple metals; an assembly of hard spheres shows a crystallization that is
known as the Alder transition alder ; and a hard sphere model provides
rationale for the phase transition between B1 (NaCl-type) and B2 (CsCl-type)
structures in ionic crystals AM . More recently, hard sphere models have been
used to understand stable structures in ceramic and refractory materials
smirnov , superconductors zhang_sc , binary and ternary systems ozaki1 ;
ozaki2 , and two-dimensional ionic crystals ono_ion . It has also played an
important role in classification of materials saad . The advantage of the hard
sphere model is its simplicity. On the contrary, it needs to be clarified when
the hard sphere model breaks down.

Overbury and Ku observed the surface rippling of Sn atoms on Cu(111), Ni(111),
and Pt(111) surfaces, where they form the
$(\sqrt{3}\times\sqrt{3})R30^{\circ}$ structure and the Sn atoms buckle
outward from the first layer overbury . The amplitude of the surface rippling
decreases with the lattice constant of substrate metals, which has been
attributed to the atomic size mismatch between the Sn and substrate atoms.
Since the seminal work of them, the surface alloy has served as a playground
for investigating the validity of hard sphere models overbury ; wuttig ;
woodruff ; harrison ; kimura ; xun . The library of surface alloys has been
expanding Bi_Au111 ; osiecki ; lee ; sadhukhan ; shah ; Ge_Ag111 , and the
surface alloy has played an important role in creating or identifying two-
dimensional materials yuhara ; pzhang .

The description of surface alloys within the hard sphere approximation is
based on several factors. First, it depends on the definition of atomic radii
such as Clementi’s (isolated atom), metallic, and covalent radii clementi ;
wells . The hard sphere model using metallic radii has been known to
overestimate the magnitude of rippling amplitude woodruff , which has been
attributed to a lack of interatomic interaction between the foreign atom and
the substrate atom well below the surface xun . Second, it depends on the
combination of atomic species for the foreign and substrate atoms. Recently,
the rippling amplitude of 15 atoms, which are taken from group III, IV, and V
in the periodic table, on the Ag surface has been investigated from first-
principles ishii . This search is, however, restricted to the Ag(111) surface.
Third, it depends on the surface orientation of the substrate. It has been
suggested that the rippling amplitude is not sensitive to the orientation of
the substrate xun ; li . However, a few combination of atomic species, such as
Sn on Pt(111) and Pt(100) li , was investigated. In this paper, we point out,
through a gedanken experiment, a fundamental problem regarding the
applicability of the hard sphere model to the (100) surface, present density-
functional theory (DFT) calculations highlighting the effects beyond the hard
sphere approximation, and shed light on a peculiarity of small radius elements
including H, B, C, and N for surface alloys.

##  II Gedanken experiment with hard sphere model

Let us consider a surface alloy, which consists of $X$ foreign atoms on $A$
substrate in the face-centered cubic (fcc) structure, forming a c$(2\times 2)$
structure on the $A(100)$ surface, i.e., two interpenetrating square lattices
of $X$ and $A$, displaced along the diagonal of the square cell by a half
length of the diagonal (see Fig. 1(a)). With a hard sphere model, we denote
the atomic radii of the $X$ and $A$ atoms by $R_{X}$ and $R_{A}$,
respectively. When $R_{A}=R_{X}$, the atoms in the surface layer have no
corrugation. When $R_{X}>R_{A}$, the $X$ atom is pushed up from the surface,
as shown in Fig. 1(b). The magnitude of the displacement $\delta z$ from the
flat surface is

$\displaystyle\delta z=\sqrt{(R_{A}+R_{X})^{2}-(2R_{A})^{2}}.$ (1)

When $R_{X}<R_{A}$, the $X$ atom is embedded into the first and second layers,
yielding a negative value of $\delta z$ expressed by

$\displaystyle\delta z=-\sqrt{2}R_{A}+\sqrt{(R_{A}+R_{X})^{2}-2{R}_{A}^{2}}.$
(2)

It should be noted that Eq. (2) breaks down when the atomic radius is small
enough to satisfy the following inequality

$\displaystyle R_{X}<(\sqrt{2}-1)R_{A}.$ (3)

In this case, the $X$ atom passes through the void created in the square
lattice of the second layer (see Fig. 1(c)), collides with the face-centered
atom in the third layer, and eventually falls into the bulk by passing through
the void in deep layers. If the $X$ atom is trapped to the third layer (i.e.,
assuming no displacements parallel to the surface), the displacement is
expressed by

$\displaystyle\delta z=-a+R_{A}+R_{X},$ (4)

where $a$ is the lattice constant of $A$ in the fcc structure and thus $z=-a$
is the $z$-component of the third layer of the $A(100)$ surface.

Table 1 lists the Clementi’s atomic radii $R_{\rm C}$ clementi  for $X=$ H and
Li to F and $A=$ Cu, Ag, and Au. These values are derived from solving the
Hartree-Fock equation for an isolated atom. We extract the values of $R_{\rm
C}$ by using pymatgen pymatgen . The condition of Eq. (3) is satisfied when
$X=$ H, C, N, O, and F (except for $X=$ C and $A=$ Cu). This clearly shows
that the atoms with small atomic radii fall into the bulk through the (100)
surface of noble metals.

The magnitude of the atomic radii depends on the definition. We derive the
metallic radii $R_{\rm M}$ from the magnitude of $a$ of $X$ in the fcc
structure (i.e., 12-coordination): $R_{\rm M}=\sqrt{2}a/4$. The optimization
of $a$ is done by using Quantum ESPRESSO (QE) qe , where the computational
details are the same as those used in the slab model calculations described
below, but a 24$\times$24$\times$24 $k$ grid is used in the self-consistent
field (scf) calculations MK . When $R_{\rm M}$ is used (see also Table 1), the
condition of Eq. (3) is never satisfied even for $X=$ H. We should keep in
mind the following tendency: When $X$ is changed from B to F, $R_{\rm M}$
deviates around $1.1$ but $R_{\rm C}$ decreases from 0.87 to 0.42. This
implies that for $X=$ B to F the coordination has a strong impact on the
electronic distribution of valence electrons and enlarges the atomic radius,
which may prevent the $X$ atom from passing through the void in the square
lattice.

It is useful to investigate to what extent the electron charge is distributed
between different atomic species. We use the Pauling’s electronegativity
$\chi_{\rm P}$ for atoms of $X$ and $A$ to study which atoms attract electrons
strongly, and in Table 1 list the values of $\chi_{\rm P}$ that are extracted
from pymatgen pymatgen . The $\chi_{\rm P}$ increases monotonically as one
goes from Li to F, so that the $\chi_{\rm P}$ of B to F becomes larger than
that of $A=$ Cu, Ag, and Au (except for $X=$ B and $A=$ Au). The effective
radius will be larger than $R_{\rm C}$, implying that the atoms of $X$ are
trapped near the $A(100)$ surface. This needs to be studied in detail within
DFT.

Figure 1: (a) Schematic structure of $X/A(100)$ surface alloy with the first six layers, where the $A$ substrate atoms are alloyed with the $X$ foreign atoms to form the c$(2\times 2)$ structure. This is illustrated using VESTA vesta . (b) Top and side views of the surface alloy within the hard sphere model. The unit cell is represented by the dashed line. When the atomic radii satisfy the relation $R_{X}>R_{A}$, $R_{X}=R_{A}$, and $R_{X}<R_{A}$, the atom displacement $\delta z$ is positive, zero, and negative, respectively. When $R_{X}$ is small enough to satisfy the relation of $R_{X}<(\sqrt{2}-1)R_{A}$, the $X$ atom can pass through any voids in the square lattice. (c) The void in the second layer. The dashed circle indicates the atomic sphere in the third layer.  Table 1: The Clementi’s and metallic atomic radii, $R_{\rm C}$ and $R_{\rm M}$, in units of Å and the Pauling’s electronegativity $\chi_{\rm P}$. The figures in a parenthesis is $(\sqrt{2}-1)R$ with $R=R_{\rm C}$ and $R_{\rm M}$ for Cu, Ag, and Au.  | H | Li | Be | B | C | N | O | F
---|---|---|---|---|---|---|---|---
$R_{\rm C}$ | 0.53 | 1.67 | 1.12 | 0.87 | 0.67 | 0.56 | 0.48 | 0.42
$R_{\rm M}$ | 0.80 | 1.53 | 1.09 | 1.01 | 1.04 | 1.10 | 1.11 | 1.08
$\chi_{\rm P}$ | 2.20 | 0.98 | 1.57 | 2.04 | 2.55 | 3.04 | 3.44 | 3.98
| Cu |  | Ag |  | Au |  |  |
$R_{\rm C}$ | 1.45 | (0.60) | 1.65 | (0.68) | 1.74 | (0.72) |  |
$R_{\rm M}$ | 1.28 | (0.53) | 1.47 | (0.61) | 1.47 | (0.61) |  |
$\chi_{\rm P}$ | 1.90 |  | 1.93 |  | 2.54 |  |  |
Figure 2: Comparison between the atomic displacements for $X/A(100)$ within
DFT ($\delta z^{\rm DFT}$) and those within the hard sphere model ($\delta
z^{\rm model}$) using Clementi’s atomic radii (left) and metallic radii
(right). When Eq. (3) is satisfied, Eq. (4) is used. The dashed line indicates
$\delta z^{\rm DFT}=\delta z^{\rm model}$. Figure 3: The atomic number $Z_{X}$
dependence of $\delta z$ for the $A(100)$ surface within the DFT and hard
sphere model using the metallic radii of $R_{\rm M}$ with $A$ being Cu, Ag,
and Au. The vertical lines (dashed) indicate $Z_{X}=29$ (Cu), $47$ (Ag), and
$79$ (Au). Within the hard sphere model, the curves for the Ag and Au almost
overlap due to the similar values of $R_{\rm M}$.

##  III DFT calculations

We use an electronic structure calculation program of QE qe  to perform DFT
calculations for thin films. The metallic substrate $A$ with the (100) surface
has 11 atomic layers including 22 atoms. We use the Perdew-Burke-Ernzerhof
(PBE) pbe  functionals of the generalized gradient approximation for the
exchange-correlation energy and use the ultrasoft pseudopotentials provided in
pslibrary.1.0.0 dalcorso . The cutoff energies for the wavefunction and the
charge density are 55 Ry and 550 Ry, respectively. A 20$\times$20$\times$1 $k$
grid is used in the scf calculations MK . The smearing parameter of 0.02 Ry
within Methfessel and Paxton approach is used smearing_MP . The vacuum layer
is taken to be 15 Å. The interlayer spacing along the [100] direction is fixed
to $a/2=1.82,2.06$, and $2.08$ Å for Cu, Ag, and Au, respectively. The
$z$-component of atoms in the first and last three layers (i.e., 1-3 and
9-11th layers) are relaxed in the geometry optimization. For the first and
eleventh layers, we next replace the $A$ atom with the $X$ atom to form the
c$(2\times 2)$ structure, still having an inversion symmetry in the model. We
again optimize the $z$-component of the $X$ and $A$ atoms within the first and
last three layers, i.e., an initial guess of $\delta z=0$ Å. In the present
work, we study $A=$ Cu, Ag, and Au, allowing us to understand a trend in the
group 11 elements. The atom of $X$ ranges from H to Bi, where noble gases are
excluded. The magnetic effect on $\delta z$ will be important especially for
$X=$ Mn wuttig ; harrison , but is not investigated in the present work. The
parameters of $R_{\rm C}$, $R_{\rm M}$, $a$, and $\chi_{\rm P}$ for $X$, and
the calculated values of $\delta z$ for $X/A(100)$ are provided in
Supplemental Material SM .

###  III.1 Breakdown of hard sphere model

Figure 2 shows a relationship between $\delta z$s calculated within DFT and
those within hard sphere models using $R_{\rm C}$ (left) and $R_{\rm M}$
(right). When $R_{\rm C}$ is used, the agreement is not good because the
calculated data are distributed around the vertical line of $\delta z^{\rm
DFT}=0$. On the other hand, when $R_{\rm M}$ is used, $\delta z^{\rm model}$
is well correlated with $\delta z^{\rm DFT}$. This is consistent with
experiments overbury , where a positive value of $\delta z$ is caused by the
lattice constant mismatch between $X$ and $A$ (i.e., $a=4R_{\rm M}/\sqrt{2}$).
This result implies that the atomic environment near the surface is similar to
that in the fcc structure. We also find that $\delta z^{\rm model}$ is larger
than $\delta z^{\rm DFT}$. Such an overestimation has also been reported in
Refs. woodruff ; xun , while it has been remedied by including long-range
forces between hard spheres xun . Below the line of $\delta z^{\rm DFT}=\delta
z^{\rm model}$, we observe some anomalies corresponding to F/$A(100)$ and
Cl/$A(100)$ except for Cl/Cu(100). This is due to the large electronegativity
of F and Cl, which will be discussed below.

Figure 3 shows $\delta z$ as a function of the atomic number $Z_{X}$ of the
$X$ atom on noble metal substrates. The trend for $\delta z$ as a function of
$Z_{X}$ is similar between the model and DFT calculations: (i) The magnitude
of $\delta z$ is maximum for the alkali atoms of $X$, which is due to the
small $Z_{X}$ in alkali metals (group I), yielding an electron delocalization
around the nucleus and an enhancement of atomic radius; (ii) $\delta z$ for
$A=$ Cu is larger than that for $A=$ Ag and Au, which may be due to the large
atomic radii of Ag and Au compared with Cu atom; and (iii) a peak of $\delta
z$ is observed at $Z_{X}=63$, europium (Eu). The Eu crystal has the body-
centered cubic structure as the ground state, which is similar to alkali
metals, while the other lanthanides have closed-packed structures. In this
way, the hard sphere model can be used to interpret DFT results.

Let us move on to the cases of $X=$ H and Li to F. When $X=$ H, C and N, a
negatively large value of $\delta z$ is observed. This indicates that the $X$
atoms are trapped to the second layer from the surface (i.e., $\delta
z\simeq-a/2$). We also observe anomalous decrease in $\delta z$ for B/Au(100).
This is not described by the hard sphere model using $R_{\rm M}$. When $X=$ O
and F, a small negative or positive value of $\delta z$ is obtained. This is
not described by the hard sphere model using $R_{\rm C}$. In this way, no
consistent picture can be obtained within hard sphere models, and thus the
electronic property beyond the hard sphere approximation is investigated below
by studying the case of $A=$ Cu.

Figure 4: Electronic band structure (left) and projected density-of-states
(right) for B/Cu(100), C/Cu(100), and N/Cu(100). The electron energy
$\varepsilon$ is measured from the Fermi level.

Note that we have calculated the potential energy as a function of $\delta z$
by assuming $X=$ H and Li to F on Cu(100) surface. We observed parabola for
each $X$, confirming no dependence of the initial condition for $\delta z$.
This is provided in Supplemental Material SM . We have also studied the van
der Waals effect on $\delta z$ and performed DFT calculations including a
Grimme’s van der Waals correction (DFT-D3) grimme , assuming $X=$ H and Li to
F and $A=$ Cu, Ag, and Au. Such calculations also predict a trap to the second
layer of the surface, while the value of $\delta z$ increases by less than 0.1
Å. In addition, we have studied the energetic stability of $X/A(100)$ by
calculating the formation energy $\Delta E$ and demonstrated that the
$X/A(100)$ is stable for $X=$ B, C, and N, confirming a correlation between
$\Delta E$ and $\delta z$ (except for $X=$ H). This is also provided in
Supplemental Material SM .

We emphasize that no atomic trap to the second layer is observed for the
$A(111)$ surface in the $(\sqrt{3}\times\sqrt{3})R30^{\circ}$ structure.
Within the hard sphere model, the rippling is expressed by $\delta
z=-\sqrt{8/3}R_{A}+\sqrt{(R_{A}+R_{X})^{2}-4{R}_{A}^{2}/3}$ when $R_{X}<R_{A}$
ishii . This expression breaks down when $R_{X}<(2/\sqrt{3}-1)R_{A}\simeq
0.15R_{A}$, whereas the factor of $0.15$ is very small. Although the B, C, and
N atoms take negative value of $\delta z$ within the DFT, they are located in
between the first and second layers rather than trapped to the latter. Note
that the H atom is close to the second layer because $\delta z=-1.6$ and
$-1.5$ Å for $A=$ Ag and Au, respectively. The $Z_{X}$-dependence of $\delta
z$ for $A(111)$ surface is compared with that for $A(100)$ surface in
Supplemental Material SM .

Figure 5: Total electron density (Å-3) subtracted by the superposition of
electron density of atoms. The $x$-$y$ (top) and $x$-$z$ (bottom) planes
including the $X$ atom are shown for $X$/Cu(100) with $X=$ H and Li to F. The
positive (red) and negative (blue) values indicate an increase and decrease in
the electron density compared to that of the isolated atom, respectively.

###  III.2 Electronic effect

Figure 4 shows the electron band structure and projected density-of-states
(PDOS) for $X=$ B, C, and N on Cu(100). As can be seen in a Cu bulk, we
observe parabolic bands with the minimum electron energy at the $\Gamma$ point
and non-dispersive bands from $\varepsilon=-5$ to $-1.5$ eV, which originate
from the Cu $4s$ and $3d$ electrons, respectively. In addition, a cosine-type
band is located well below the Fermi level. As the $Z_{X}$ is increased, the
location of the band becomes deep. From the PDOS, we identify that such a
cosine band originates from the $X$ $2s$ and Cu $4p$ electrons. We also find
that the $X$ $2p$ electron contributes to the PDOS for $\varepsilon\geq-9$ eV,
implying that $X$ $2p$ orbital hybridizes with Cu orbitals.

Figure 5 shows a difference between the total electron density and the
superposition of atomic electron density, assuming the Cu(100) surface. The
electron transfer occurs from Cu to $X$ atoms except for $X=$ Li and Be, which
is consistent with the electronegativity analysis using Table 1. For $X=$ C
and N, anisotropic distribution in the electron density is observed around the
second layer of the surface: The electron density around the $X$ atom
increases within the $x$-$y$ plane, which gives rise to electron
redistribution of Cu atoms in the second ($z=1.82\times 4$ Å) as well as the
third ($z=1.82\times 3$ Å) layers. The electron redistribution for $X=$ H is
spherically symmetric, compared to that for $X=$ C and N, which is intrinsic
to the $1s$ orbital. For $X=$ O and F, the effective atomic radius is large
due to the large $\chi_{\rm P}$, explaining no penetration into a bulk.

Figure 6: (Top) The $\delta z$ for $A(100)$ surface with $X=$ Li to F and $A=$
Cu, Ag, and Au. $1\times 1$ (dashed) and $2\times 2$ (solid) supercells are
assumed, where the former results are the same as those obtained for
c$(2\times 2)$ calculations. (Bottom) The atomic displacements obtained by the
$2\times 2$ supercell calculations. The $A$ atoms (blue) approach to and go
away from the $X$ atom (orange) for type 1 and type 2, respectively.

###  III.3 Atomic density dependence

We study to what extent the lateral force within the surface (i.e., the
periodic boundary condition along the $x$\- and $y$-directions) is important
for yielding negatively large values of $\delta z$. We have performed $2\times
2$ supercell calculations, where the surface of the slab consists of seven $A$
atoms and one $X$ atom in the unit cell and 88 atoms are present in the slab.
The $x$\- and $y$\- as well as $z$-components of atoms are relaxed in the
first and last three layers. A 5$\times$5$\times$1 $k$ grid is used in the scf
calculations MK . We still observe a trap around the second layer when $X=$ H,
B and C, except for B/Cu(100), as shown in Fig. 6(top). It is interesting that
the atomic distribution is classed to two types illustrated in Fig. 6(bottom):
the $A$ atoms are displaced to approach the $X$ atom (type 1) and to go away
from the $X$ atom (type 2). The type 1 includes Be/Cu, B/Cu, F/Cu, Be/Ag, and
F/Ag, and type 2 otherwise. Only type 2 systems can show negatively large
$\delta z$. However, for $X=$ N taking the type 2, the value of $\delta z$
increases by about 1 Å, so that the N atoms are located in between the first
($\delta z=0$ Å) and second ($\delta z\simeq 2$ Å) layers.

We next explore surface properties specific to the H, B, C, and N atoms. The
negatively large $\delta z$ indicates the presence of a deep hole on the (100)
surface in the c$(2\times 2)$ structure, which may accommodate more atoms. We
have added one $X$ atom to form a $X$ dimer whose symmetry axis is parallel to
the $z$ axis and optimized the $z$-components of $X$ and $A$ atoms. We have
obtained three patterns for alignment of two $X$ atoms: a $X$ dimer formation
above ($\delta z_{1},\delta z_{2}>0$), below ($\delta z_{1},\delta z_{2}<0$),
and across ($\delta z_{1}<0,\delta z_{2}>0$) the first layer, as listed in
Table 2. The Ag(100) and Au(100) surfaces push the N dimer away from the
surface and an isolated N dimer is formed, indicating that a N dimer is
immiscible with Ag and Au surfaces. On the other hand, the Cu(100), Ag(100),
and Au(100) surfaces accommodate the N dimer, the H dimer, and the H and B
dimers, respectively. The interatomic distance $d$ of the N and B dimers below
the first layer is 1.25 and 1.68 Å, respectively, which is larger than that of
the isolated N and B dimers ($d=$ 1.11 and 1.64 Å within DFT-PBE). The $d$ of
the H dimer is much larger than that of the isolated $H$ dimer ($d=0.75$ Å).
This implies that two H atoms are trapped to the first and second $A$ layers
rather than that a H dimer is formed below the first layer, while for $A=$ Au
the H atoms tend to be trapped to the second and third layers. For the other
cases, the $X$ dimer is formed across the first layer of $A(100)$ surface.
These results suggest that tuning the interatomic distance of $X$ dimer can be
possible by changing the substrate.

Table 2: The $\delta z$ for two $X$ atoms relative to the first layer of the $A(100)$ surface. $d=\delta z_{2}-\delta z_{1}$ is the interatomic distance of $X_{2}$. The $d$ for an isolated $X$ dimer is also listed. $X_{2}/A$ | $\delta z_{1}$ | $\delta z_{2}$ | $d$
---|---|---|---
H2/Cu | $-1.99$ | 0.08 | 2.07
H2/Ag | $-2.39$ | $-0.07$ | 2.32
H2/Au | $-3.18$ | $-1.52$ | 1.66
H2 | - | - | 0.75
B2/Cu | $-1.22$ | 0.37 | 1.59
B2/Ag | $-1.44$ | 0.15 | 1.58
B2/Au | $-3.11$ | $-1.42$ | 1.68
B2 | - | - | 1.64
C2/Cu | $-1.02$ | 0.29 | 1.31
C2/Ag | $-1.07$ | 0.22 | 1.29
C2/Au | $-1.13$ | 0.19 | 1.32
C2 | - | - | 1.28
N2/Cu | $-1.75$ | $-0.50$ | 1.25
N2/Ag | 3.59 | 4.70 | 1.11
N2/Au | 3.57 | 4.68 | 1.11
N2 | - | - | 1.11

##  IV Summary

We have presented a thought experiment that the small atoms, $X=$ H and C to
F, fall into a bulk through the (100) surface, which is based on a hard sphere
model using the Clementi’s atomic radii. To check whether this is true, we
have performed DFT calculations on the surface rippling for $X/A(100)$ in the
c$(2\times 2)$ structure with $A=$ Cu, Ag, and Au. The small atoms stay near
the surface rather than penetrate into a bulk: the H, C, and N atoms are
trapped to the second layer of the (100) surface, while the O and F atoms stay
around the first layer. The effects beyond the hard sphere approximation have
been revealed by studying electronic properties of $X/A(100)$ and performing
$2\times 2$ supercell calculations. These calculations have suggested an
importance of the electronegativity difference between atomic species and also
the atomic displacement along the parallel to the surface. We have also
studied a formation of $X$ dimer around the $A(100)$ surface and shown that
the N atoms tend to form a N dimer, while the H atoms trap to different
layers.

We have exhaustively investigated the surface rippling of $X=$ H to Bi except
for noble gases, providing a benchmark for the surface rippling in the
c$(2\times 2)$ structure. We demonstrate that (i) a hard sphere model
parametrized by metallic radii can capture an overall trend of the rippling
amplitude, suggesting that a consideration of atomic environment into atomic
sphere is important, and (ii) such a model fails to predict the structure of
non-close-packed surfaces including a small atom. The electronic effect
overcomes the geometric effect on the structural property. On the contrary, we
can expect that surfaces including small radius elements such as H, B, C, and
N would have interesting structures beyond the hard sphere description, which
potentially show useful physical and chemical properties. In particular, the
behavior of H atoms will be of importance in high-temperature
superconductivity zhang_sc  and hydrogen storage ren .

###### Acknowledgements.

This work was supported by JSPS KAKENHI (Grant No. 21K04628). The computation
was carried out using the facilities of the Supercomputer Center, the
Institute for Solid State Physics, the University of Tokyo, and also using the
supercomputer “Flow” at Information Technology center, Nagoya University.

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