# Crucial role of vibrational entropy in the Si$(111)$-$7\times 7$ surface
structure stability

R. A. Zhachuk<EMAIL_ADDRESS>Institute of Semiconductor Physics, pr.
Lavrentyeva 13, Novosibirsk 630090, Russia  J. Coutinho  I3N, Department of
Physics, University of Aveiro, Campus Santiago, 3810-193 Aveiro, Portugal

(August 27, 2024)

###### Abstract

We investigate the relative thermodynamic stability of the $3\times 3$,
$5\times 5$, $7\times 7$, $9\times 9$ and infinitely large structures related
to the dimers-adatoms-stacking faults family of Si$(111)$ surface
reconstructions by means of first-principles calculations. Upon accounting for
the vibrational contribution to the surface free energy, we find that the
$5\times 5$ structure is more stable than the $7\times 7$ at low temperatures.
While a phase transition is anticipated to occur at around room temperature,
the $7\times 7\rightarrow 5\times 5$ transformation upon cooling is hindered
by the limited mobility of Si atoms. The results not only flag a crucial role
of vibrational entropy in the formation of the $7\times 7$ structure at
elevated temperatures, but also point for its metastable nature below room
temperature. _[Pre-print published in Physical Review B_ _105_ _, 245306
(2022)]_ DOI:10.1103/PhysRevB.105.245306

##  I Introduction

First-principles electronic structure methods, most notably density functional
theory (DFT), have been widely used as a powerful tool to solve surface
structures and to study their electronic properties. Due to its technological
importance, semiconductor silicon has been a benchmark for studying surface
physics, with several surface structures being extensively investigated [1, 2,
3, 4, 5, 6, 7, 8, 9, 10]. When Si surface structures are composed of different
building blocks as, for example, $2\times 1$, $2\times 2$ and $7\times 7$
reconstructions on Si$(111)$, their surface energy differences are usually
significant (above $1\ \mathrm{meV/\mathring{A}^{2}}$), facilitating a
reliable identification of the lowest energy atomistic configuration [11, 12].
In the above, the $2\times 1$ structure consists of $\pi$-bonded chains [13],
$2\times 2$ consists of Si adatoms on $\mathrm{T_{4}}$-sites [14], while the
lowest energy $7\times 7$ reconstruction is described by the complex dimers
adatoms stacking faults (DAS) model [15]. The DAS model describes a series of
structures belonging to the $\left(2n+1\right)\times\left(2n+1\right)$ family
of reconstructions observed on Si$(111)$ and Ge$(111)$ surfaces ($n$ being a
positive integer): $3\times 3$, $5\times 5$, $7\times 7$, $9\times 9$ and so
on.

However, if the competing surface structures belong to the same family of
reconstructions, _i.e._ when they are made of common building blocks, it
becomes difficult to determine which one has the lowest formation energy, and
ultimately to find the one that should be observed. For instance, the
$c(4\times 2)$ and $p(2\times 2$) reconstructions on Si$(100)$ are both
composed of Si dimers, but because they buckle with different periodicity,
their reported energy difference is as low as $0.7\,\mathrm{meV/dimer}$
($0.02\ \mathrm{meV/\mathring{A}^{2}}$) [16]. We note that in spite of such
low energy difference, the calculated lowest energy structure [$c(4\times 2)$]
is in agreement with experimental scanning tunneling microscopy (STM) data
obtained at low temperature.

The $5\times 5$ and $7\times 7$ DAS reconstructions on Si$(111)$ surface
provide us with another example of two structures with very close surface
formation energies. Two previous attempts to compare the surface formation
energies of DAS reconstructions on Si$(111)$ within DFT are singled out. In
Ref. 17, formation energies of $3\times 3$, $5\times 5$ and $7\times 7$
reconstructed Si$(111)$ surfaces were compared, and the $7\times 7$
reconstruction was claimed as the most favorable structure. This result agrees
with many experiments demonstrating that the $7\times 7$ is the most
frequently observed structure on Si$(111)$ below the $7\times 7\leftrightarrow
1\times 1$ order-disorder transition temperature at around $1100\,\mathrm{K}$
[18]. However, two years later, Needels [19] provided evidence that the above
calculations suffered from poor Brillouin zone (BZ) sampling, that once
adequately corrected, resulted in a different energy ordering, with the
$5\times 5$ surface becoming the ground state (more stable than the $7\times
7$ reconstruction).

The second attempt to calculate the formation energy of Si$(111)$ DAS
reconstructions was carried out by Solares _et al._ [20]. The
Si$(111)\textrm{-}7\times 7$ came out as the one with lowest formation energy,
and once again, was deemed the most stable. However, the calculated energy
difference between $5\times 5$ and $7\times 7$ structures of only $4\
\mathrm{meV/1\times 1\ cell}$ ($0.3\ \mathrm{meV/\mathring{A}^{2}}$), which
combined with a reported accuracy of about $28\ \mathrm{meV/1\times 1\ cell}$
($2.2\ \mathrm{meV/\mathring{A}^{2}}$) [20], make their conclusions
questionable. The large error stems from several calculation settings, most
notably the use of thin 4-bilayer (BL) slabs, insufficient number of
$\mathbf{k}$-points to sample the BZ, and the use of the experimental lattice
parameter ($5.43\,\mathrm{\mathring{A}}$) instead of calculated one
($5.48\,\mathrm{\mathring{A}}$) in the construction of the models [20]. The
later factor introduces an artificial strain field across the slabs which
alter the calculated energy differences of $5\times 5$ and $7\times 7$ surface
structures.

In summary, the available picture regarding the stability ordering of the DAS
structures on Si$(111)$ is rather unclear. In particular, our understanding of
the physics that governs the formation of the most stable Si surface is
covered by error bars. Importantly, both studies described above are based on
calculations of potential (static) energies, _i.e._ for $0\ \mathrm{K}$
temperature only. It is a general practice to neglect thermal excitations as
well as the contribution of entropy to the free energy of formation of
surfaces. However, given the small energy difference between $5\times 5$ and
$7\times 7$ surface structures, it would be important not only to account for
the above referred sources of uncertainty (slab thickness, BZ sampling,
artificial strain neutralization), but also to assess the effect of entropy,
especially the contribution of vibrational degrees of freedom, usually the
dominant finite-temperature contribution to the surface free energy of
formation of semiconductor surfaces.

The aim of the present work is to find well converged energy differences
between several members of the DAS family of structures on Si$(111)$ surface,
size the entropy contribution to the free energy of formation of $5\times 5$
and $7\times 7$ reconstructions, and finally understand the driving
stabilization factors. We find that while the $5\times 5$ reconstruction is
the most favorable structure at $T=0$ K, vibrational freedom stabilizes the
$7\times 7$ reconstruction, with a cross-over of the respective free-energies
of formation taking place close to room temperature. The observation of
metastable Si$(111)$-$7\times 7$ below room temperature can be explained by a
large barrier along the $7\times 7\rightarrow 5\times 5$ transition that
involves the motion of Si atoms during cooling.

##  II Calculation details, Results and Discussion

###  II.1 Relative formation energies of Si(111) surfaces with DAS
reconstructions at $0\,\mathrm{K}$ temperature

The calculations were carried out using the pseudopotential [21] density
functional theory SIESTA code [22, 23]. Both the local density approximation
(LDA) of Perdew and Zunger (PZ) [24], as well as the generalized gradient
approximation (GGA) of Perdew _et al._ (commonly referred as PBEsol) [25],
were employed for the description of the exchange-correlation (XC)
interactions between electrons. Although it was previously shown that DAS
Si$(111)$ surfaces are diamagnetic [20], our calculations were spin-
unrestricted. The valence states were expressed as linear combinations of
Sankey-Niklewski-type numerical atomic orbitals [22]. In the present
calculations, polarized double-$\zeta$ functions were placed on all atomic
coordinates. This means two sets of _s_ \- and _p_ -orbitals plus one set of
_d_ -orbitals on silicon atoms, and two sets of _s_ -orbitals plus one set of
_p_ -orbitals on hydrogen atoms. The electron density and potential terms were
calculated on a real space grid with spacing equivalent to a plane-wave cut-
off of 200 Ry.

Figure 1:  (a) Top view of $3\times 3$, $5\times 5$, $7\times 7$, and $9\times
9$ structures on Si(111) surface according to the DAS model. Dark-shaded large
circles are adatoms. Light-shaded small circles are atoms in the upper bilayer
and in the surface dimes. Black dots are atoms in the bulk unreconstructed
bilayer. Thick rods edging the perimeter of the cells are dimers. For the
$9\times 9$ structure the unit cell is delimited by a yellow rhombus,
triangles outline the $2\times 2$ domains in faulted and unfaulted half unit
cells, and the blue rhombus shows a $2\times 2$ unit cell within a faulted
half. (b) Sectional view in the $(1\bar{1}0)$ plane cutting the long diagonal
of the $9\times 9$ surface unit cell, showing the nearest-neighbor bonding
structure.

For total energy calculations we used Si$(111)$-slabs of 6 BLs separated by a
30 Å thick vacuum layer. Dangling bonds at the _bottom side_ of the slabs were
saturated by hydrogen, while the top layers were modified according to the DAS
$3\times 3$, $5\times 5$, $7\times 7$ and $9\times 9$ atomic models [15], as
well as a $2\times 2$ model, consisting of adatoms on $T_{4}$ sites. A
graphical description of the surface structures is presented in Figures 1(a)
(top-view) and 1(b) (cross-sectional view along the $[1\bar{1}0]$ direction).

The total number of Si atoms in the slabs were the following: 106 for $3\times
3$, 300 for $5\times 5$, 592 for $7\times 7$, 982 for $9\times 9$ and 49 for
$2\times 2$. The positions of all slab atoms (except for the H atoms and Si
atoms in the bottom bilayer) were fully optimized until the atomic forces
became less than 1 meV/Å. BZ integration was discretized as summations over
Monkhorst-Pack (MP) $\mathbf{k}$-point meshes [26]. The following BZ-sampling
schemes were applied: MP-$6\times 6\times 1$ for $3\times 3$, MP-$4\times
4\times 1$ for $5\times 5$, MP-$3\times 3\times 1$ for $7\times 7$,
MP-$2\times 2\times 1$ for $9\times 9$ and MP-$10\times 10\times 1$ for
$2\times 2$ surface slabs. These sampling schemes lead to approximately the
same $\mathbf{k}$-point density in all BZs. We found that the above
calculation settings provide a relative surface energy convergence within
$0.1\,\mathrm{meV/\mathring{A}^{2}}$.

Although experiments are normally performed under constant pressure, any
theoretical description of surface thermodynamics is more conveniently carried
out under constant volume. That avoids dealing with anharmonic effects and the
cumbersome connection between temperature and volume via thermal expansion
coefficients. As demonstrated by Estreicher _et al._ [27, 28], the small
thermal expansion of crystalline Si leads to comparable constant-volume and
constant-pressure free energies up to a few hundred degrees Celsius. This is
enough for the present purpose. However, at higher temperatures, anharmonicity
and electronic excitations become important and the results should be
considered as qualitative.

The Helmholtz free energy (per unit area) of an infinite and periodic
semiconductor surface must account for electronic and vibrational
contributions, $f(T)=f_{\textrm{elec}}(T)+f_{\textrm{vib}}(T)$. Magnetic
degrees of freedom are not relevant since all structures considered are
diamagnetic. Rotational motion and configurational entropy are also excluded
for obvious reasons. The magnitude of the electronic term
($f_{\textrm{elec}}$) is dominant, although it has a weak $T$-dependence. On
the other hand, the vibrational term accounts for a small zero-point energy
(ZPE) at $T=0$ K, but shows a relatively stronger $T$-dependence.

Mostly often, $f_{\textrm{elec}}$ is replaced by the internal electronic
surface energy at absolute zero temperature,
$u_{\textrm{elec}}(T=0\,\textrm{K})=\gamma$, as found from the stationary
solution of the many-body electronic problem within DFT. On the other hand,
$f_{\textrm{vib}}$ is usually neglected, making $f(T)=\gamma$, where for the
sake of convenience the electronic potential energy $\gamma$ is hereafter
referred to as surface energy. This approximation is well justified in many
cases, since $\gamma$ is usually much larger than the other terms. However, as
it will be shown below, this approach has been hiding an important driving
force that determines the structure of the Si(111) surface.

We are interested in finding relative surface free energies ($\Delta f$) of
Si(111) reconstructions. For that we will consider the
Si$(111)\textrm{-}5\times 5$ (DAS model) as reference. The energy of a
Si$(111)\textrm{-}N\times N$ surface relative to that of
Si$(111)\textrm{-}5\times 5$ ($\Delta f_{N\times N}=f_{N\times N}-f_{5\times
5}$) is obtained from

$\Delta f_{N\times N}=\frac{5^{2}F_{N\times N}-N^{2}F_{5\times
5}-\mu\left(5^{2}X_{N\times N}-N^{2}X_{5\times 5}\right)}{N^{2}5^{2}S_{1\times
1}},$ (1)

where to first approximation we start by assuming that $\Delta f_{N\times
N}=\Delta\gamma_{N\times N}$, the slab free energy $F_{N\times N}=E_{N\times
N}$ with $E_{N\times N}$ being the total electronic energy of a particular
$N\times N$ relaxed slab made of $X_{N\times N}$ Si atoms ($N=$3, 5, 7 or 9),
and $S_{1\times 1}$ the area of the Si(111)-$1\times 1$ surface unit cell.
Analogously, the Si chemical potential is approximated to the internal
electronic energy per Si atom in the bulk at $T=0$ K, $\mu=\mu_{0}$,
accounting for any stoichiometric mismatch between $N\times N$ and $5\times 5$
slabs. This quantity was found from a $1\times 1$ bulk-like slab of 3 BLs
thickness with MP-$20\times 20\times 8$ for BZ sampling.

In order to estimate the error bar due to different implementations of DFT,
including basis, pseudopotentials and exchange-correlation functionals, we
also calculated $\gamma$ for Si$(111)\textrm{-}7\times 7$ and
Si$(111)\textrm{-}5\times 5$ using the VASP code [29, 30]. These calculations
were carried out within the GGA to spin-unrestricted density functional
theory, both without and with a correction for dispersion forces (PBE [31] and
PBE-D3 [32, 33], respectively). Both VASP and SIESTA calculations used
identical conditions, except that in the former case, being a planewave code,
the Kohn-Sham states were described with planewaves with kinetic energy up to
250 eV.

Table 1: Relative surface energies ($\Delta\gamma_{N\times N}$) of DAS-reconstructed Si$(111)$ surfaces ($\mathrm{meV/\mathring{A}^{2}}$) calculated according to Eq. 1 from first principles, using local basis and planewave DFT software packages, and several XC functionals. The energy of the Si$(111)\textrm{-}5\times 5$ surface was taken as reference. DFT software | XC functional | $\Delta\gamma_{N\times N}$
---|---|---
$3\times 3$ | $5\times 5$ | $7\times 7$ | $9\times 9$ | infinity
SIESTA [22] | LDA (PZ [24]) | 2.58 | 0 | 0.15 | 0.57 | 3.95
GGA (PBEsol [25]) | 2.32 | 0 | 0.23 | 0.7 | 4.16
VASP [29, 30] | GGA (PBE [31]) | - | 0 | 0.31 | - | -
GGA (PBE-D3 [32, 33]) | - | 0 | 0.36 | - | -

It is possible to calculate the formation energy of an infinitely large DAS
surface. To do this, we note that the upper bilayer of DAS unit cell consists
of two halves: one of them with and the other without stacking faults (faulted
and unfaulted halves in short), as represented in Fig. 1. These halves are
linked together by dimers on the surface [15]. They also contain adatoms on
top, arranged according to a $2\times 2$ periodicity. With increasing the size
of the DAS unit cell, the contribution of dimers to the surface energy scales
with $N$, while that from faulted and unfaulted areas scales with $N^{2}$.
Hence, for $N\sim\infty$, the contribution of dimers can be neglected and the
surface energy is simply the average between energies of two
$(111)\textrm{-}2\times 2$ surfaces composed of adatoms: with and without
stacking faults, respectively.

The surface energy difference between $9\times 9$ and $7\times 7$
reconstructed Si$(111)$ surfaces was estimated from experimental STM data in
Ref. 34 as less than $4\,\mathrm{meV/1\times 1\,cell}$
($0.3\,\mathrm{meV/\mathring{A}^{2}}$). This figure is somewhat lower but
comparable to the calculated energy differences of
$0.42\,\mathrm{meV/\mathring{A}^{2}}$ (LDA) and
$0.47\,\mathrm{meV/\mathring{A}^{2}}$ (GGA) as derived from Tab. 1. According
to Tab. 1, the $3\times 3$ reconstructed Si$(111)$ surface has a significantly
higher surface energy than $5\times 5$, $7\times 7$ and $9\times 9$
reconstructed surfaces. The Si$(111)\textrm{-}5\times 5$ surface has the
lowest surface energy, and that increases monotonically with increasing the
size of the unit cell. The surface energy of the infinitely large DAS
reconstruction is substantially (about $1.5\,\mathrm{meV/\mathring{A}^{2}}$)
higher than that of $3\times 3$.

It is well known from experiments that $7\times 7$ is the most frequently
observed surface reconstruction of Si$(111)$ at temperatures below the
$7\times 7\leftrightarrow 1\times 1$ order-disorder structural transition.
Therefore, the data of Tab. 1 are in seeming disagreement with the
experimental results. However, considering the small energy difference between
$5\times 5$ and $7\times 7$ reconstructed surfaces, it is important to inspect
the relative stability of both surfaces at finite temperatures, accounting for
contributions of vibrational and electronic excitations to the surface free
energy.

###  II.2 Contribution of vibrational and electronic excitations to the free
energies of Si$(111)-5\times 5$ and Si$(111)-7\times 7$ surfaces

The vibrational free energy of $5\times 5$ and $7\times 7$ reconstructed
Si$(111)$ surfaces was obtained within the quasi-harmonic approximation,
according to the usual procedure employed for defects and surfaces in
semiconductors and metals [27, 28, 35, 36]. This approach is applicable to
silicon up to few hundred Kelvin [28, 37], and in particular to Si(111) DAS
surfaces, which form unique and stable minimum energy structures. For surfaces
with structural degrees of freedom, _e.g._ Si$(100)$ [16] and Si$(331)$ [38]
one would have to consider configurational entropy and anharmonicity [36, 39].
For those cases, the potential energy surface contains multiple minima
separated by low energy barriers due to two-fold buckling of small Si
structures (forming double well potentials for the case of buckled dimers on
Si$(100)$).

The relative surface free energy can be calculated using Eq. 1, where $\Delta
f_{N\times N}=\Delta\gamma_{N\times N}+\Delta f_{N\times N,\textrm{vib}}$ now
includes a vibrational contribution (in addition to the electronic potential
difference $\Delta\gamma_{N\times N}$ already reported in the previous
section). The quantity $\Delta f_{N\times N,\textrm{vib}}$ can be calculated
by replacing $F_{N\times N}$ in Eq. 1 by the vibrational contribution to the
Helmholtz free energy of a slab [40],

$F_{N\times
N,\textrm{vib}}=\underset{k}{\sum}\frac{\hbar\omega_{k}}{2}+k_{\textrm{B}}T\underset{k}{\sum}\ln\left[1-\exp\left(-\frac{\hbar\omega_{k}}{k_{\textrm{B}}T}\right)\right],$
(2)

and $\mu=\mu_{\textrm{vib}}$ is now the vibrational part of the Si chemical
potential (see below). The summation of Eq. 2 runs over all vibrational mode
frequencies, $\omega_{k}$, of the slab (excludes 3 translational modes), and
$k_{\textrm{B}}$ is the Boltzmann constant. The first term on the right side
of Eq. 2 gives the ZPE. This is a $T$-independent quantity, which could in
principle be regarded as a potential term, and alternatively be considered as
a small correction to the $\gamma$ values of Tab. 1.

The phonon frequencies were calculated using the frozen phonon scheme as
implemented in the VIBRA utility of the SIESTA software package [23]. Only
zone-center phonons ($\mathbf{q}=\Gamma$) were calculated. This is justified
by the large lateral dimensions of $5\times 5$ and $7\times 7$ reconstructions
($a_{5\times 5}\approx 19\,\mathrm{\mathring{A}}$, $a_{7\times 7}\approx
27\,\mathrm{\mathring{A}}$), corresponding to small BZ surface areas and
rather weak phonon dispersion. For the calculation of the dynamical matrix, we
employed 4 BL-thick slabs ($\approx 10\,\mathrm{\mathring{A}}$ thickness). All
Si atoms of the three top-most BLs were displaced by
$0.02\,\mathrm{\mathring{A}}$ from their relaxed positions along all three
Cartesian coordinates. These correspond to a total of 150 and 298 dynamical
atoms for the $5\times 5$ and $7\times 7$ reconstructions, respectively.
Thicker slabs would require a computational power out of our reach. However,
as we will show below, the calculated vibrational free energy of a 64-atom
supercell (with a (super)lattice spacing $a\approx
11\,\mathrm{\mathring{A}}$), is already converged and the corresponding
vibrational entropy agrees well with the measurements.

The vibrational contribution to the Si chemical potential,
$\mu_{\textrm{vib}}$, was calculated using Eq. 2 for a cubic bulk Si cell. In
order to test the convergence with respect to the cell size, we performed
calculations using $\mathrm{Si_{8}}$ ($a\approx 5\,\mathrm{\mathring{A}}$),
$\mathrm{Si_{64}}$ ($a\approx 11\,\mathrm{\mathring{A}}$) and
$\mathrm{Si_{216}}$ ($a\approx 16\,\mathrm{\mathring{A}}$) bulk cubic cells.
Results within GGA PBEsol are shown in Fig. 2; the LDA PZ results are
comparable. It is clear from this figure that the values for
$\mathrm{Si_{64}}$ and $\mathrm{Si_{216}}$ clusters are very close: their
difference amounts to $\Delta\mu_{\textrm{vib}}=4\,\mathrm{meV/atom}$ at
$T=800\,\mathrm{K}$, which translates into an error bar of only
$0.03\,\mathrm{meV/\mathring{A}^{2}}$ in the calculation of $\Delta f_{7\times
7,\textrm{vib}}$ using Eq. 1. These findings are corroborated by the results
of Estreicher et al. [28] and by convergence tests reported by Gomes _et al._
[37], who found no significant improvement in the calculated specific heat of
Si when moving from a 64-atom cell to a 216-atom cell.

Figure 2:  Calculated vibrational contribution to the Si chemical potential
$\mu_{\textrm{vib}}(T)$ using different supercell sizes (see legend) as a
function of temperature. The XC functional employed was GGA PBEsol.

We also calculated the ZPE and vibrational entropy per Si atom,
$s_{\textrm{vib}}=-\left(\partial\mu_{\textrm{vib}}/\partial T\right)_{V}$,
for bulk Si. The results per molar unit are shown in Tab. 2 and they are close
to both experimental data [41, 42] and previous calculations [27].

Table 2: Calculated (Calc.) and experimental (Exp.) values of ZPE (kJ/mol) and vibrational entropy $s_{\textrm{vib}}\,(\textrm{J/K$\cdot$mol})$ of crystalline Si. The first two rows of calculated data correspond to the present work. | ZPE | $s_{\textrm{vib}}(300\,\textrm{K})$ | $s_{\textrm{vib}}(800\,\textrm{K})$
---|---|---|---
Calc.: LDA PZ, $\mathrm{Si_{216}}$ | 5.980 | 18.896 | 40.847
Calc.: GGA PBEsol, $\mathrm{Si_{216}}$ | 5.871 | 19.327 | 41.350
Calc.: LDA PZ, $\mathrm{Si_{64}}$ [27] | 6.201 | 18.395 | 40.308
Exp. [41, 42] | 6.008 | 18.820 | 41.568

The calculated relative energy $\Delta f_{7\times 7,\textrm{vib}}=f_{7\times
7,\textrm{vib}}-f_{5\times 5,\textrm{vib}}$ as a function of $T$ is shown in
Fig. 3 for LDA PZ [24] and GGA PBEsol [25] XC functionals; A
$\mathrm{Si_{216}}$ supercell was used for the calculation of
$\mu_{\textrm{vib}}$. At a glance, LDA and GGA results are very similar and
indicate a decrease of $\Delta f_{7\times 7,\textrm{vib}}$ (stabilization of
the $7\times 7$ reconstruction) with increasing the temperature. We can only
explain this feature with a faster increase of vibrational entropy with
temperature for the Si(111)-$7\times 7$ surface (compared to that of
Si(111)-$5\times 5$). The identification of specific vibrational modes and
surface bonds at the origin of this effect may not be simple. As shown by
Murali _et al._ [35] for the case of defects in metals, the change in
vibrational entropy is not necessarily accounted for by well-identifiable
localized defect modes, but mostly by resonant modes associated with
vibrations in the strained regions surrounding the defects. The lower their
frequency, the more likely is for such resonances to become thermally
populated at a certain temperature, and therefore the higher their
contribution to the entropy change.

Our resultssuggest a relative softening of the Si(111)-$7\times 7$ surface and
subsurface bonds when compared to Si(111)-$5\times 5$. That would make the
excited vibrational spectrum of the Si(111)-$7\times 7$ more dense and
accessible with temperature raising.

At this point, an obvious question is whether electronic entropy also has a
role to play? It has been shown that gap states deeper than $\sim 0.1$ eV from
the band edges impact no more than few tens of meV to the electronic free
energy of formation, even at several hundred Kelvin [27, 28]. Explicit
calculations of the electronic free energy of TiN semiconductor surface
confirms that electronic excitations are only significant above $T\approx
600\textrm{-}700$ K [39]. Sommerfeld’s approximation to the electronic free
energy of metals [43],

$f_{\textrm{elec}}^{\textrm{SOM}}(T)=-\frac{\pi^{2}}{6}(k_{\textrm{B}}T)^{2}D(E_{\textrm{F}}),$
(3)

provides us with an approximate and simple quantification of electronic
entropy of a metallic or semi-metallic surface. Accordingly,
$f_{\textrm{elec}}^{\textrm{SOM}}$ is proportional to the density of
electronic states at the Fermi level, $D(E_{\textrm{F}})$. Even for a
Si$(111)\textrm{-}7\times 7$ reconstruction, where the gap between highest
occupied and lowest unoccupied surface states can be as narrow as few tens of
meV [44], $D(E_{\textrm{F}})$ is rather small, implying that the difference
$\Delta D(E_{\textrm{F}})$ between $D(E_{\textrm{F}})$ of two close surfaces
(such as $5\times 5$ and $7\times 7$ members of the DAS family), should be
also small. Sommerfeld’s electronic free energy of Si(111)-$7\times 7$ with
respect to the same quantity for Si(111)-$5\times 5$, namely $\Delta
f_{7\times 7,\textrm{elec}}^{\textrm{SOM}}$, was calculated using Eqs. 1 and
3, assuming a vanishing electronic entropy from the bulk Si chemical potential
($\mu=\mu_{0}$). We applied a $0.1\,\mathrm{eV}$-wide Gaussian broadening to
the Kohn-Sham energy levels in the evaluation of $D(E_{\textrm{F}})$. The
results obtained within GGA PBEsol are shown in Fig. 3 as open triangles. Like
$\Delta f_{7\times 7,\textrm{vib}}$, the electronic surface free energy
decreases with increasing the temperature (also working toward a stabilization
of the $7\times 7$ reconstruction). This is explained by approximately 8%
higher density of dangling bonds on $7\times 7$ reconstructed surface than on
$5\times 5$ surface, and therefore by a higher density of states of the
$7\times 7$ reconstruction within the Si gap. However, as anticipated, the
calculated values of $\Delta f_{7\times 7,\textrm{elec}}^{\textrm{SOM}}$ are
at least one order of magnitude lower than $\Delta f_{7\times
7,\textrm{vib}}$, suggesting that electronic entropy can be safely neglected.
Hence, to a good approximation, the free energy difference between $7\times 7$
and $5\times 5$ reconstructed surfaces is simply given by $\Delta f_{7\times
7}=\Delta\gamma_{7\times 7}+\Delta f_{\textrm{vib},7\times 7}$.

We reported already (see Tab. 1) that the electronic surface energy
$\Delta\gamma_{7\times 7}$ is positive and dominates the surface free energy
at low temperatures (favoring the $5\times 5$ structure). On the other hand,
$\Delta f_{\textrm{vib},7\times 7}$ (Fig. 3) is increasingly negative with
temperature (favoring the $7\times 7$ structure), meaning that there is a
phase transition temperature above which the Si(111)-$7\times 7$ becomes more
stable. Considering that $\Delta\gamma_{7\times 7}\approx
0.2\textrm{-}0.3\,\mathrm{meV/\mathring{A}^{2}}$ (Tab. 1) the transition
temperature is found at $T_{\textrm{trans}}\approx
300\textrm{-}400\,\mathrm{K}$, _i.e._ , slightly above room temperature (Fig.
3).

The usual way to prepare a clean Si(111) surface involves sample flashing at
$1250\,\mathrm{\text{\textdegree}C}$ for about $1\,\mathrm{min}$ followed by
cooling to the desired temperature [10]. Above the $7\times 7\leftrightarrow
1\times 1$ order-disorder phase transition temperature at $T\approx
1100\,\mathrm{K}$, the Si$(111)$ surface contains a high density of mobile Si
adatoms [34] and exhibits $1\times 1$ diffraction pattern reflecting the Si
substrate below the moving adatoms. The $7\times 7$ reconstruction forms at
around the transition temperature and persists at lower temperatures. Our
results are in agreement with these well established observations, namely, the
Si(111)-$7\times 7$ reconstruction is the most stable across a wide
temperature range up to $T=1100\,\mathrm{K}$.

However, the $7\times 7$ structure on Si$(111)$ was also observed below room
temperature with no sign of structure change to $5\times 5$ [44]. The most
likely explanation for this experimental fact is the existence of a high
kinetic barrier making the $5\times 5\leftrightarrow 7\times 7$ transformation
unlikely. The hindered diffusion of Si atoms on $5\times 5$ and $7\times 7$
reconstructed surfaces below room temperature provides an example of an
effective slowing factor. Other effects, including a barrier for Si atom
detachment may be invoked as well. Indeed, the $5\times 5$ and $7\times 7$
reconstructions have different surface atomic density [34]. Hence, the
$5\times 5\leftrightarrow 7\times 7$ structural transformation requires the
removal (or addition) of excessive (or deficient) Si surface atoms from
Si$(111)$ terraces, for instance by moving them to the always existing step
edges on the sample surface. Si surface diffusion on Si$(111)$-$7\times 7$ at
room temperature is strongly hindered due to the high energy barrier
($E_{\textrm{b}}=1.14\,\mathrm{eV}$) that separates two halves of the $7\times
7$ unit cell [45]. This experimental figure is in agreement with DFT
calculations showing that there is an energy barrier of about $1\,\mathrm{eV}$
separating faulted and unfaulted $7\times 7$ half-cells for diffusing Si
adatoms [46]. The height of energy barriers for adatom migration depend on the
local atomic arrangement of surface atoms within the DAS structures. This is
essentially identical for $5\times 5$ and $7\times 7$ reconstructions,
suggesting that the energy barriers on $(111)\textrm{-}7\times 7$ and
-$5\times 5$ surfaces should be similarly high. Such a conclusion was also
drawn from experimental STM data obtained on Ge$(111)\textrm{-}7\times 7$ and
-$5\times 5$ reconstructed surfaces [47]. Thus, according to our calculations
the $7\times 7$ reconstruction is metastable below room temperature, being
frozen in upon cooling from high temperatures.

Figure 3:  Calculated $\Delta f_{7\times 7,\textrm{vib}}=f_{7\times
7,\textrm{vib}}-f_{5\times 5,\textrm{vib}}$ and $\Delta f_{7\times
7,\textrm{elec}}^{\textrm{SOM}}=f_{7\times
7,\textrm{elec}}^{\textrm{SOM}}-f_{5\times 5,\textrm{elec}}^{\textrm{SOM}}$ as
a function of temperature obtained within LDA PZ and GGA PBEsol. The dashed
lines highlight the ranges of temperatures and $\Delta f_{7\times
7,\textrm{vib}}$ values where the phase transition $5\times 5\leftrightarrow
7\times 7$ is expected to take place.

##  III Conclusions

The relative thermodynamic stability of the $3\times 3$, $5\times 5$, $7\times
7$, $9\times 9$ and infinitely large DAS reconstructions on Si$(111)$ surface
was investigated by means of first-principles calculations within density
functional theory. It was found that the surface energy ordering (from lowest
to highest) at $T=0\,\mathrm{K}$ is: $5\times 5$, $7\times 7$, $9\times 9$,
$3\times 3$, infinitely large. The vibrational entropy contribution to the
free energy of formation was evaluated for the $(111)\textrm{-}7\times 7$ and
$\textrm{-}5\times 5$ surfaces in the temperature range
$T=0\textrm{-}1100\,\mathrm{K}$. The contribution of electronic entropy was
shown to be negligible. Accordingly, the $5\times 5$ reconstruction is
energetically more favorable than $7\times 7$ at low temperatures, with the
$5\times 5\leftrightarrow 7\times 7$ phase transition anticipated to occur at
around room temperature. The invariable observation of the $7\times 7$ phase
is explained by the low mobility of Si atoms on the surface, effectively
freezing it in a metastable state during cooling. Our study shows that
vibrational entropy plays a crucial role in the stabilization of $7\times 7$
reconstruction at elevated temperatures, and reveals the metastable nature of
this structure below room temperature.

###### Acknowledgements.

R. A. Z. would like to thank the Novosibirsk State University for providing
the computational resources. This work was supported by the Russian Science
Foundation (project no. 19-72-30023). J. C. thanks the support of FCT in
Portugal through Projects LA/P/0037/2020, UIDB/50025/2020, UIDP/50025/2020

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11institutetext: 1 European Space Agency (ESA), ESRIN, $\Phi$-lab, Italy, 2
Trust Stamp, 3 VITO

# Evaluating and Benchmarking Foundation Models for Earth Observation and
Geospatial AI

Nikolaos Dionelis1 Casper Fibaek1 Luke Camilleri1,2 Andreas Luyts1,3 Jente
Bosmans1 Bertrand Le Saux1

###### Abstract

When we are primarily interested in solving several problems jointly with a
given prescribed high performance accuracy for each target application, then
Foundation Models should be used rather than problem-specific models. We focus
on the specific vision application of Foundation Models for Earth Observation
(EO) and geospatial AI. These models can solve important problems we are
tackling, including for example land cover classification, crop type mapping,
flood segmentation, building density estimation, and road regression
segmentation. In this paper, we show that for a limited number of labelled
data, Foundation Models achieve improved performance compared to problem-
specific models. In this work, we also present our proposed evaluation
benchmark for Foundation Models for EO. Benchmarking the generalization
performance of Foundation Models is important as it has become difficult to
standardize a fair comparison across the many different models. We present the
results using our evaluation benchmark for EO Foundation Models and show that
Foundation Models are label efficient in the downstream tasks and help us
solve problems we are tackling in EO.

###### Keywords:

Foundation Models for Earth monitoring Evaluation bench.

##  1 Introduction

An advantage of Foundation Models compared to problem-specific models is that
for a limited number of labelled data, Foundation Models achieve improved
performance. Label efficiency is important in real-world applications as for
many use cases, both labelling and continuous re-labelling are needed. In the
specific case of Earth Observation (EO) and remote sensing, labels change over
time. Also, data from satellites are unlabelled. Annotating such data is
difficult, requires expertise, and is costly in terms of time. An additional
advantage of Foundation Models is that they perform sharing across tasks and
learn a common module, for example segmentation, needed for all the target
applications we are trying to solve jointly with a given prescribed high
performance accuracy for each task.

The target applications of EO Foundation Models are important problems we are
trying to solve, such as land cover classification semantic segmentation, crop
type mapping, and crop yield estimation. Additional target applications are
flood segmentation, building density estimation, road regression segmentation,
estimation of the age of buildings, marine litter detection, methane plume
segmentation, and change detection for wildfires, floods, and anomalies.
Furthermore, there are also important EO problems that we would like to solve
for which we have only unlabelled data, i.e. no labels, for example iceberg
detection.

##  2 Solving $M$ tasks jointly with prescribed high accuracy

Given a prescribed high performance for each task, e.g., accuracy $95\%$, we
deal with $M$ problems jointly. For EO Foundation Models, we address
approximately $M=10$ target applications together. The prescribed high
performance is crucial as we want the model to be useful; otherwise, people
will not use it. For Earth monitoring, we want generalization to a big
geographical area/ large inference set. The performance stringent requirement
drives everything. The two alternatives are the following. For the use cases,
for datasets D$1$, D$2$, …, D$M$ that have labels, the alternative A is to
perform supervised learning on the datasets. We name these tasks P$1$, P$2$,
…, P$M$. The alternative B is to perform self-supervised learning on a common
dataset $D^{\prime}$. We name this task $L$. Then, we perform supervised
learning for the target applications. We name these tasks Q$1$, Q$2$, …, Q$M$.
The dataset $D^{\prime}$ contains relevant data, e.g., similar objects or data
from the same satellite. The alternative A is using problem-specific models,
solving each problem on its own, and assuming the existence of a lot of labels
for each use case. The alternative B is using a common model and solving
groups of tasks that are of interest to us. Big common/ shared models are
Foundation Models. For the alternative A, problem-specific models do not have
label efficiency: for limited labelled data, they yield low performance
accuracy (or F1-score or Intersection over Union (IoU)). There is no sample
efficiency for these models and we have to pay too much and wait too long for
the labels. The performance requirement drives everything as the data size
mainly depends on the prescribed high accuracy. The relationship between the
size of the data and the accuracy is approximately linear. We cannot escape
the large size of the dataset because of the performance stringent
requirement. In EO, the data size is: some TBs. Using common/ shared models is
beneficial: we learn the common representations. There is sharing across
tasks: we learn the commonality, i.e. the common operations (segmentation) for
re-usability and efficiency. For the alternative B, i.e. for common models and
Foundation Models, $N\%$ of the labels are needed that would otherwise be
required. For EO Foundation Models, $N\approx 20$ and even $10$. For the
alternative A (problem-specific models), all the labels are needed.

For the alternative A, the cost C$1$ (which is also directly related to the
data size and how large the architecture needs to be as these three are
similar) is:

$\text{C}1=\text{P}1+\text{P}2+...+\text{P}M\approx My\text{,}$ (1)

where typically $M=10$ tasks and $y$ is the cost or data for one task. Because
of the high accuracy requirement, $y$ is large, e.g., $100000$. This is why
for problem-specific models, the cost, as well as the data size and how large
is the architecture, is times the number of tasks. For $M=10$ use cases, for
the alternative A, we have times $10$, i.e. $\text{C}1=10y$ from (1). Next,
for the alternative B, the cost is:

$\text{C}2=L+\text{Q}1+\text{Q}2+...+\text{Q}M\approx
y+N\%\,y\,M=y\,(1+N\%\,M)\text{.}$ (2)

This scales better than C$1$, i.e. $\text{C}2=3y$. Overall, $\text{C}2\approx
300000$, $\text{C}1\approx 1M$, and $\text{C}2<\text{C}1$. Big common models
achieve label efficiency for both segments and semantics. Segment label
efficiency refers to the segments and their shape. For both segment and
semantic label efficiency, in remote sensing, continuous re-labelling is
needed as we live in a dynamic world: Earth is ever-changing. Human annotators
are needed, as well as expert knowledge. Also, imperfect labels exist in EO,
i.e. noisy labels. $\text{C}1$ grows linearly with $M$, i.e. O($M$), while
$\text{C}2$ grows linearly with $N\%\,M$, i.e. O($N\%\,M$). Because of the
accuracy requirement and the linear relationship between the data size and the
accuracy, for problem-specific models, we train $10$ models that are
approximately as large as $10$ Foundation Models, i.e. it is like training
$10$ Foundation Models. Also, for problem-specific models, a lot of labels are
needed which are expensive in terms of both cost and time.

##  3 Our Proposed Evaluation Benchmark for FMs for EO

Evaluating and benchmarking Foundation Models in terms of their generalization
performance is important as it has become increasingly difficult to
standardize a fair comparison across the many different models. For the
specific vision application of Foundation Models for EO and geospatial AI [1,
2, 3], we present our proposed evaluation benchmark and show that for a
limited number of labelled data, Foundation Models achieve improved results
compared to problem-specific models. Foundation Models are label efficient in
the downstream tasks [4, 5]. For semantic segmentation land cover
classification (lc), the evaluation results are presented in Fig. 1. We
examine both fine-tuning (ft) and linear probing (lp).

Figure 1: Evaluating, benchmarking, and ranking Foundation Models for EO and
geospatial AI on the downstream task of semantic segmentation land cover
classification.

Geo-location classification pre-training is used for the models that we have
developed in-house. These are the geo-aware models in Fig. 1. As a pre-text
task, our Foundation Model Version 1.0 performs longitude and latitude
satellite metadata information learning. For this, we have used a global
unlabelled dataset of satellite Sentinel-2 L2A data and $10$ spectral bands.
As a downstream task, we perform fine-tuning (or linear probing) on the
labelled dataset
WorldCover111http://worldcover2020.esa.int/data/docs/WorldCover$\\_$PUM$\\_$V1.1.pdf.
According to the results in Fig. 1, the percentage improvement of Foundation
Models compared to problem-specific models is approximately $18.52\%$ when
there are limited samples of labelled data, e.g., $100$ images per region
(geo-aware U-Net ft and U-Net fully-supervised). We have examined both a
Transformer-based architecture, i.e. Vision Transformer (ViT), and a U-Net-
based architecture.

For the task of estimating the label at the image level (rather than at the
pixel level) for land cover classification, according to our results, the
percentage improvement of Foundation Models compared to problem-specific
models is approximately $16.36\%$ when limited labels are used, e.g., $100$
samples per region (geo-aware U-Net ft vs. U-Net fully-supervised, $0.64$ and
$0.55$ respectively).

Next, for the task of estimating how dense and close to each other buildings
are, the results are presented in Fig. 2. For this regression downstream task,
the evaluation metric is the Mean Squared Error (MSE). We compare $15$ models
in total. For this specific use case, the percentage improvement of Foundation
Models compared to problem-specific models is $86\%$ when there are limited
labelled data: $100$ samples per region (geo-aware U-Net and U-Net fully-
supervised).

Figure 2: Evaluation of Foundation Models for EO on the target application of
estimating how dense and close to each other buildings are, in the MSE metric
(regression task).

##  4 Conclusion

To solve several problems jointly with a prescribed high accuracy for each
task, we use Foundation Models. For the vision application of Foundation
Models for EO, for limited labelled data, Foundation Models outperform
problem-specific models in our proposed evaluation benchmark for Foundation
Models for EO.

## References

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