# Honeycomb germanene flakes on Ag nano-islands

R Bernard, K Zhang, M-C Hanf, Y Borensztein, H Cruguel, P Sonnet, R

Stephan, G Prévot

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R Bernard, K Zhang, M-C Hanf, Y Borensztein, H Cruguel, et al.. Honeycomb germanene flakes on Ag nano-islands. Journal of Physics: Condensed Matter, 2025, 37 (41), 10.1088/1361-648x/ae0bdd hal-05376816

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# Honeycomb germanene flakes on Ag nano-islands

To cite this article: R Bernard et al 2025 *J. Phys.: Condens. Matter* **37** 415001

View the [article online](#) for updates and enhancements.

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# Honeycomb germanene flakes on Ag nano-islands

R Bernard<fnref n="1" />, K Zhang<fnref n="1" />, M-C Hanf<fnref n="2" />,<fnref n="3" /> ID, Y Borensztein<fnref n="1" />, H Cruguel<fnref n="1" />, P Sonnet<fnref n="2" />,<fnref n="3" /> ID,
R Stephan<fnref n="2" />,<fnref n="3" /> ID and G Prévot<fnref n="1" />,* ID

E-mail: prevot@insp.jussieu.fr

Received 23 May 2025, revised 1 September 2025

Accepted for publication 25 September 2025

Published 13 October 2025

<figure><img src="image_3.png" /></figure>

CrossMark

## Abstract

Since germanene cannot be exfoliated from a bulk material, and since Ge-based surface alloys are more stable than pure Ge monolayers (MLs) on metal surfaces, the synthesis of germanene flakes remains challenging. Here we report on the synthesis of germanene nanoflakes on Ag ML islands grown on Ag(111). The maximum achievable size is of the order of 30 nm, and germanene growth occurs due to the heavily oversaturated chemical potential for the Ge atoms confined on top of the Ag islands during growth. The nucleation of three-dimensional clusters, however, prevents the formation of larger germanene flakes.

Supplementary material for this article is available [online](#)

Keywords: germanene, nanoflakes, silver, scanning tunneling microscopy, density functional theory

## 1. Introduction

Like graphene, several mono-elemental two-dimensional (2D) materials have been discovered in recent years, now referred to as the Xenes family [1]. These materials have the potential to host exotic quasiparticles such as massless Dirac fermions, Majorana fermions or Weyl fermions. However, only a few possess a lamellar three-dimensional (3D) parent crystal and can therefore be obtained, like graphene or black phosphorus, by exfoliation from the bulk. The others must instead be synthesized by epitaxial growth on suitable substrates. This is for example the case of silicene, germanene, stanene and plumbene that belong to the same column of the periodic table as graphene. Important research has been devoted to the synthesis of such group-IV Xenes [2–5]. First reports on experimental realization of silicene [6], germanene [7], stanene [8]

and plumbene [9] have been made over the past twelve years. While some of these systems, such as silicene/Ag(111) are now well established as honeycomb 2D layers through multiple independent studies [3], others remain the subject of debate. This is especially the case of germanene. Compared to silicene and graphene, free-standing germanene exhibits a larger spin-orbit coupling-induced gap of about 24 meV [10], making it a more promising candidate for a topological insulator operable at room temperature.

Synthesis of epitaxial germanene has been first reported on Pt(111) [7], whereas previous results have interpreted Ge/Pt growth within the framework of alloy formation [11]. Similarly, Ge growth on Au(111) and Al(111) has been described either as germanene [12–16] or as alloy formation [17–19]. Germanene growth has also been reported on lamellar substrates such as MoS₂ [20] or HOPG [21] although the HOPG results were later refuted [22]. To date, most studies of epitaxial germanene have focused on Ag(111) [23–27], motivated by the analogy with the well-known silicene

<footnote n="1">¹ Sorbonne Université, Centre National de la Recherche Scientifique, Institut des NanoSciences de Paris, INSP, F-75005 Paris, France</footnote>

<footnote n="2">² Université de Haute Alsace, CNRS, IS2M UMR7361, F-68100 Mulhouse, France</footnote>

<footnote n="3">³ Université de Strasbourg, F-67081 Strasbourg, France</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
formation on Ag(111). Since Si and Ge share similar struc-
tural and electronic properties, and both exhibit low solubil-
ity in silver at room temperature, Ag(111) appears as the best
substrate for germanene growth. However, honeycomb 2D Ge
formation is usually hindered by alloying, as pointed several
times [28–30]. We have recently identified the structure of
two ordered phases corresponding to different Ge coverages:
the so-called striped phase, corresponding to a Ag₂Ge sur-
face alloy with an atomic density 6.45% higher than that of
Ag(111) [31], and what has been identified as the ‘ground state
of epitaxial germanene on Ag(111)’ [32], corresponding to
a mixed pentagonal/hexagonal/heptagonal organization of Ge
incorporating a few Ag atoms. This layer, with composition
Ge<sub>98</sub>Ag<sub>6</sub>, and displaying a (√109 × √109) R ± 24.5° recon-
struction with respect to the Ag surface, is shown to coex-
ist with Ge islands at the growth temperature, and thus cor-
responds to the highest achievable Ge surface concentration
under thermodynamic equilibrium.

Here, we show that a Ge supersaturation leads to the form-
ation of nanodomains of pure honeycomb germanene. Such
growth is realized at 425 K on Ag islands of size 20–30 nm
where the transition towards 3D Ge islands is hindered by the
impossibility to reach the critical size for 3D nucleation. Such
germanene islands had never been observed before due to the
very narrow conditions of synthesis.

## 2. Methods

Sample preparation

The preparation of the Ag(111) single crystal consisted in
several cycles of Ar ion sputtering (P = 7 × 10⁻⁵ mbar,
600 eV) and annealing at a temperature of 870 K). A Ge flux
of ~0.12 ML h⁻¹, from a graphite crucible, was obtained by a
Focus GmbH e-beam evaporator located in front of the scan-
ning tunneling microscope (STM). The monolayer (ML) is
related to a Ge layer. The striped Ge layer on Ag(111) [31] was
used as a reference for calibration. The associated Ge atomic
density is 4.91 10¹⁴ at cm⁻², or 0.335 ML [31, 32].

STM measurements

After growth, a Omicron variable-temperature STM apparatus
was used to perform STM measurements at room temperature.
Corrections of STM images were achieved in order to align
the Ag[111] axis along the z direction. The effects of abrupt
tip changes were corrected using a home-made procedure.

Density functional theory (DFT) calculations

The DFT calculations were carried out using the VASP
(Vienna ab initio simulation package) code [33–36]. The
projector augmented plane wave (PAW) [37, 38] method
was employed within the generalized gradient approximation,
and the electron–electron exchange-correlation interactions
were described with the Perdew–Burke–Ernzerhof functional
[39, 40]. Van der Waals were included through a dispersion
correction term computed using Grimme’s DFT-D3 method
with a zero-damping function [41–43]. The calculations were

performed until the forces were below 0.2 eV·nm⁻¹, with a
cutoff energy of 450 eV. The Brillouin zone was sampled by
a Γ- centered (2 × 2 × 1) *k*-points mesh.

For both reconstructions, the Ag lattice constant and the √37 hexagonal mesh parameter were 0.4075 nm and 1.7527 nm, respectively. The slab consisted of 10 Ag(111) layers of 37 atoms and a pure Ge layer of 38 atoms. Only the bottom Ag plane was kept fixed during relaxation. The local density of states (LDOS) images were obtained using the Tersoff–Hamann approximation [44]. Data postprocessing was performed with the P4vasp software, and atomic structures were visualized with the PyMOL software.

3. Results

Figure 1(a) shows a large scale scanning tunneling microscopy (STM) image of the surface after deposition of 1 ML of germanium on Ag(111) at 425 K. Small 2D islands are visible on the Ag terraces. As can be seen in the profile drawn in figure 1(b), their height is equal to 0.24 nm, which also corresponds to a monoatomic Ag step. These islands have the same surface termination as the terraces, above an additional Ag plane. As previously observed, these additional islands are formed during Ge/Ag(111) growth since surface alloying leads to the release of Ag atoms that either diffuse to a step edge where they are incorporated, or condense into small single-layer Ag islands that are instantaneously covered with Ge [45]. Note that due to the low miscibility of Ge in bulk Ag [46], the incorporation of Ge atoms below the surface layer is energetically unfavorable [32]. Higher islands, with a height of 1 to a few nm, and adopting an irregular shape, can be attributed to 3D germanium clusters (such a cluster is marked ‘c’ in figure 1(a)).

Figure 1(c) shows a detailed view of three 2D islands in another part of the sample. Between the islands, the surface layer displays a carpet of protrusions, mostly arranged in groups of six, forming a flower motif. Some regions are locally ordered. They are indicated by yellow circles in figure 1(c). In these regions, the surface layer adopts a (√109 × √109) R ± 24.5° reconstruction, previously identified as a Ge<sub>98</sub>Ag<sub>6</sub> surface alloy and recognized as the thermodynamical stable configuration of the surface for a Ge chemical potential equal to that of bulk Ge [32]. Indeed, when growth is performed at a slightly higher temperature, namely 455 K, this reconstruction entirely covers the Ag(111) surface [32].

Apart from these ordered regions, other areas between the
islands show a higher density of protrusions. The same over-
all structure is observed on the two 2D islands ('i2' and 'i3')
in figure 1(c) on the right, with a variable density of protru-
sions. In contrast, the smaller island to the left of figure 1(c)
('i1') shows a much denser array of ordered protrusions, and
a long-range periodicity, indicated by a black lozenge, differ-
ent from the previous one. A detailed view of a similar 2D
island grown at 428 K during another experiment is shown
in figure 1(d). These islands exhibit a higher apparent height
relative to the average surface, about 0.28 nm, as determined
<figure><img src="image_4.png" /><figcaption>**Figure 1.** STM images (1.2 V, 50 pA) of 1 ML Ge/Ag(111) deposited at 425 K. (a) Large scale view (340 × 250 nm²) showing 2D islands and a large germanium cluster (marked ‘c’). (b) height profile corresponding to the line drawn in white in (a). (c) Detailed view (86 × 86 nm²) on another position. 2D islands are marked ‘i1’, ‘i2’, ‘i3’, blue arrows indicate the position of small Ge clusters, yellow circles indicate ordered region with the (√109 × √109) R ± 24.5° reconstruction drawn as yellow lozenges. The black lozenge corresponds to the unit cell of the (√37 × √37) R + 25.3° reconstruction. (d) Detailed view of a nano-island grown at 428 K (17 × 15 nm²), with the (√37 × √37) R + 25.3° reconstruction drawn as a black lozenge. (e)–(f) height profiles corresponding to the lines drawn in white in (c) and (d) respectively.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 2.** (a) STM image of the (√109 × √109) R = 24.5° reconstruction (6 × 6 nm², -0.2 V, 50 pA). (b) STM image of the (√37 × √37) R = 25.3° reconstruction (6 × 6 nm², 0.7 V, 20 pA). Growth temperature: 425 K. (c) and (d): top view of the corresponding atomic models relaxed by DFT. The yellow balls correspond to Ge atoms located on top of Ag atoms and the red ones to the other Ge atoms, except orange atoms in an intermediate position. Grey balls represent Ag atoms. Black circles highlight the flower motifs. (e) and (f) are corresponding side views. In the lower parts of figures (a) and (b), balls superimposed on the STM images represent the Ge atoms of the atomic model.</figcaption></figure>

from the profiles shown in figures 1(e) and (f). Their atomic lattices exhibit a hexagonal symmetry. Their unit cells, shown by black lozenges, have a size of approximately 1.8 nm and are rotated by ±25° with respect to the [110] axis of the Ag(111) surface. Relative to the Ag surface unit cell, these lattices correspond to a (√37 × √37) R ±25.3° reconstruction (of theoretical size equal to 1.757 nm).

The comparison between the different structures observed
on the surface allows us to propose an atomic model for
this novel reconstruction. Figure 2(a) shows a detailed view
of an ordered region of the surface locally displaying a

(√109 × √109) R ± 24.5° reconstruction. As a reminder, a
schematic of the corresponding atomic structure, as determ-
ined from surface x-ray diffraction and DFT [32] is shown
in figures 2(c) and (e). In the unit cell, 98 Ge atoms form
pentagons, heptagons and hexagons. They are adsorbed on
three-fold or bridge positions (red balls), except for 6 of them
(yellow balls), located on top of Ag atoms, which form the
flower motif clearly identified on the STM images (black
circles in figures 2(a) and (c)). In addition to these 98 Ge
atoms, 6 Ag atoms are incorporated in the germanene layer
(light grey balls).
As can be seen in the profile drawn in figure 1(e), the protrusions on island i1 and i3 are measured at the same apparent height as those on the terraces, corresponding as well to the apparent height of the flower motifs. We attribute thus all these protrusions to Ge atoms on top of Ag atoms. The formation of a higher protrusion density, as exhibited by island ‘i1’ suggests that when the remaining Ag atoms are replaced by Ge atoms in the surface layer, the probability that a Ge atom sits on top of a Ag atom increases.

A detailed view of the $(\sqrt{37} \times \sqrt{37}) R - 25.3^{\circ}$ reconstruction is shown in figure 2(b). Six protrusions per unit cell are clearly visible. Assuming that these protrusions correspond to Ge atoms on top of Ag atoms, it is straightforward to derive an atomic model for this layer, associated with honeycomb germanene on top of a Ag plane, where a $(\sqrt{19} \times \sqrt{19})$ germanene unit cell fits the $(\sqrt{37} \times \sqrt{37})$ Ag unit cell. This results in a germanene lattice constant of 0.403 nm, within the calculated range (0.397–0.406 nm) for free-standing germanene [2]. This germanene model was further relaxed by DFT. The atomic positions after relaxation for the $-25.3^{\circ}$ orientation are shown in figures 2(d) and (f). Our calculations show that the germanene structure is stable, but adopts a slightly distorted configuration. Ge-Ge distances range between 0.247 and 0.255 nm, slightly larger than in free-standing germanene (0.244 nm using the same DFT parameters, corresponding to a germanene lattice constant of 0.406 nm). 6 Ge atoms (yellow balls) lie 0.116 nm above the average Ge plane. The Ge-Ge-Ge bond angles range between 105° and 124°, with a mean value of 112°, except for the highest Ge atoms (yellow balls) which present a mean Ge-Ge$_{top}$-Ge angle of 89°. Inside the ‘flower’ motifs (black circles in figures 2(b) and (d)), three other atoms are faintly visible in the STM images, and correspond to the atoms indicated by orange balls in figure 2(d). They also lie slightly above (0.057 nm) the average Ge plane. The variation in apparent height of the six atoms inside the flower motifs in STM images may be related to a vertical mobility of these atoms.

The comparison between experimental and simulated images for empty states of the $(\sqrt{37} \times \sqrt{37}) R - 25.3^{\circ}$ reconstruction in figures 3(a) and (c) shows a nice agreement. The 6 brighter protrusions correspond perfectly to the 6 Ge atoms located on top of Ag atoms. In particular, the tilt angle of the hexagon formed by these atoms with respect to the unit cell orientation, illustrated by the black parallelograms, is exactly the same. The three weaker protrusions visible inside the hexagons in the simulated image (figure 3(b)) also reproduce the faint ones seen in the experimental image (figure 3(a)). A comparison over a wide range of experimental biases is provided in the supplementary information (figure S1), showing no noticeable changes in the STM images as a function of the bias. The same comparison for the $(\sqrt{37} \times \sqrt{37}) R + 25.3^{\circ}$ reconstruction is shown in figures 3(b) and (d). In the experimental image (figure 3(b)), due to a lower tip resolution, the weaker protrusions inside the hexagons cannot be distinguished. The two lattices are symmetrical through a mirror line along the $[1\bar{1}0]$ axis, which is not a mirror line of the Ag(111) surface. Indeed, such a mirror symmetry operation is

equivalent to changing the abc stacking of fcc planes into an acb stacking. This could explain the small differences between the appearance of these two reconstructions.

From a thermodynamic point of view, our DFT calculations indicate that both $(\sqrt{37} \times \sqrt{37})$ reconstructions have the same energy. The computed adsorption energy per Ge atom in the honeycomb $(\sqrt{37} \times \sqrt{37})$ structure is $E_{\sqrt{37}}^{\text{ads}} = -0.12 \text{ eV at}^{-1}$, referenced to the bulk Ge chemical potential.

## 4. Discussion

In this part, we first discuss the possible origin of the formation of honeycomb germanene on Ag(111).

The adsorption energy per Ge atom in the honeycomb $(\sqrt{37} \times \sqrt{37})$ structure, $E_{\sqrt{37}}^{\text{ads}} = -0.12 \text{ eV at}^{-1}$, is markedly weaker than that found for the alloyed $(\sqrt{109} \times \sqrt{109})$ structure, with $E_{\sqrt{109}}^{\text{ads}} = -0.26 \text{ eV at}^{-1}$. This confirms the lower stability of a pure Ge honeycomb structure compared with a Ge$_{98}$Ag$_6$ surface alloy. One may wonder if substituting Ge atoms with Ag atoms in the $(\sqrt{37} \times \sqrt{37})$ layer improves its stability. We have computed various structures where 1, 2 or 3 Ge atoms are replaced by Ag atoms. The results are shown in the supplementary information (table S1). Taking as reference the bulk chemical potential for Ag and Ge, we always find that the system energy increases upon replacement. Namely, replacing one Ge atom with a Ag one costs around 0.5 eV per unit cell, while replacing two Ge atoms costs around 1.1 eV and replacing three Ge atoms increases the energy by around 1.7 eV per unit cell. The corresponding LDOS images shows small variations as compared to the one of the pure Ge structure (see figure S2).

Figure 4 shows the difference $\Delta\gamma$ of formation energy between the different alloyed honeycomb structures computed and the pure Ge honeycomb phase. The formation energy is given by:

$$ \gamma = (E_{\text{Ge-Ag}} - N_{\text{Ge}}\mu_{\text{Ge}} - N_{\text{Ag}}E_{\text{Ag bulk}})/A - \gamma_{\text{Ag}} $$

where $E_{\text{Ge-Ag}}$ is the total energy of the considered model, $N_{\text{Ge}}$ and $N_{\text{Ag}}$ are the number of Ge and Ag atoms in the slab, $E_{\text{Ag bulk}}$ is the energy of a Ag atom in the bulk, while A is the surface area of the slab. $\gamma_{\text{Ag}}$ is the surface energy of a bare Ag(111) surface and corresponds to the energy of the backside of the slab considered.

As can be seen, when the Ge chemical potential increases, the pure Ge honeycomb model is always thermodynamically more stable than the alloyed honeycomb models. The value of $\Delta\gamma$ for the alloyed $(\sqrt{109} \times \sqrt{109})$ structure (Ge$_{98}$Ag$_6$) is also shown in figure 4 as a grey dotted line. As expected from its stronger adsorption energy, the $(\sqrt{109} \times \sqrt{109})$ structure is more stable than the Ge$_{38}$ $(\sqrt{37} \times \sqrt{37})$ structure for a chemical potential of Ge atoms equal to the one in the bulk, i.e. for $\Delta\mu_{\text{Ge}} = 0$. However, as the $(\sqrt{37} \times \sqrt{37})$ structure contains a higher surface density d of Ge atoms, it becomes more stable under supersaturated conditions. The difference of
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Comparison between experimental STM images for empty states (a: *V* = 0.7 V; b: *V* = 1.7 V) and LDOS images taken between *E*<sub>Fermi</sub> = 0 eV and *E* = 0.7 eV (c) or *E* = 1.7 V (d). (a) and (c) images correspond to the (√37 × √37) *R* − 25.3° reconstruction. (b) and (d) images correspond to the (√37 × √37) *R* + 25.3° reconstruction. The white lozenges indicate the unit cell of the reconstructions. The black parallelograms evidence the different tilt angles of two adjacent hexagons. Images (a) and (b) have been obtained on the islands shown in figures 2(b) and 1(c) respectively. The calculated images (c) and (d) were obtained in the same conditions of brightness contrast and isodensity. Image (d) is lighter than image (c) as the range of LDOS lies on a larger range.</figcaption></figure>

formation energy (per unit area) between the two superstruc-
tures is $d_{\sqrt{37}} (E_{\sqrt{37}}^{\text{ads}} - \Delta\mu) - d_{\sqrt{109}} (E_{\sqrt{109}}^{\text{ads}} - \Delta\mu)$, where $\Delta\mu$
is the chemical potential of Ge atoms, with respect to bulk
Ge. For $\Delta\mu > \frac{d_{\sqrt{37}} E_{\sqrt{37}}^{\text{ads}} - d_{\sqrt{109}} E_{\sqrt{109}}^{\text{ads}}}{d_{\sqrt{37}} - d_{\sqrt{109}}} = 0.86$ eV, this difference is
negative, favoring the formation of the $(\sqrt{37} \times \sqrt{37})$ struc-
ture. Note that this value of $\Delta\mu$ should be taken with caution,
as it relies on the difference of adsorption energies determined
by DFT for two systems of different size.

$\Delta\mu$ is related to the evolution of the system during Ge
growth: after formation of the Ge<sub>98</sub>Ag<sub>6</sub> layer, and before form-
ation of 3D Ge islands, the chemical potential increases and
is given by the adatom concentration: $\mu_{\text{Ge}} = E_{\text{ad}} + kT\ln(c)$
where $E_{\text{ad}}$ is the creation energy of an isolated Ge adatom and
$c$ is the adatom concentration. The adatom creation energy
on top of various sites of the (4 × 4) silicene reconstruc-
tion on Ag(111) has been computed, and found to be in
the 0.62 eV–1.52 eV energy range, with respect to bulk Si
[47]. Similar formation energy should be obtained for Ge
adatoms on the Ge<sub>98</sub>Ag<sub>6</sub> layer. Our STM observations for

small islands show that the initial (√109 × √109) reconstruc-
tion evolves towards a (√37 × √37) reconstruction, demon-
strating that the germanene layer is more stable than a Ge<sub>98</sub>Ag<sub>6</sub>
surface covered with Ge adatoms. On the terraces, and on
top of large islands, our observations show that the forma-
tion of Ge clusters on a Ge<sub>98</sub>Ag<sub>6</sub> surface is preferred com-
pared with the formation of germanene. However, an energy
barrier must be overcome to initiate the growth of these Ge
clusters. Indeed, in the classical nucleation theory [48], there
is a critical radius for the growth of a 3D cluster, which is
given by $r_c = \frac{2\gamma\Omega}{\Delta\mu}$, where $\gamma$ is the Ge surface energy, $\Delta\mu$
is the difference of chemical potential for Ge atoms in the
honeycomb germanene layer and in the bulk, and $\Omega$ is the
atomic volume. As the (√109 × √109) is in equilibrium with
large Ge islands [32], the chemical potential for Ge atoms
in the bulk and in the surface layer is the same. Thus $\Delta\mu \approx$
$E_{\sqrt{37}}^{\text{ads}} - E_{\sqrt{109}}^{\text{ads}}$. Using a value of 1.06 Jm⁻² for the Ge surface
energy [49], we obtain a critical radius of 2.1 nm. For a hemi-
spherical cluster, this represents around $N_c = 900$ Ge atoms.
This number should correspond to the number of Ge atoms
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Difference of formation energy $\Delta\gamma$ between the different structures computed and the pure honeycomb phase (Ge$_{38}$), corresponding to the continuous black line. Blue lines: Ge$_{37}$Ag$_{1}$ models (solid line: model b, dashed line: model c, dotted line: model d, dash-dotted line: model e). Green lines: Ge$_{36}$Ag$_{2}$ models (continuous line: model f, dotted line: model g). Red lines: Ge$_{35}$Ag$_{3}$ models (continuous line: model h, dotted line: model i). Models are shown in the supplementary information. Grey dotted line: Ge$_{98}$Ag$_{6}$.</figcaption></figure>

released by the transformation of a ($\sqrt{37} \times \sqrt{37}$) domain area into a ($\sqrt{109} \times \sqrt{109}$) one. Indeed, there is a difference of atomic density between the Ge$_{98}$Ag$_{6}$ surface alloy and the germane layer given by $\Delta d = d_{\sqrt{37}} - d_{\sqrt{109}} = 1.77$ at nm$^{-2}$. For a source of matter limited to the size of a Ge island, the critical radius can thus only be reached if the island area is larger than $\frac{N_c}{\Delta d} = 530$ nm$^2$. This corresponds to the area of a circular island of diameter 26 nm, which is in excellent agreement with the experimental observations. Of course, these values should be interpreted with caution as they depend on the precise shape of the clusters. Moreover we cannot exclude the fact that the presence of defects may increase the probability of cluster nucleation.

In figure 1(c), the two larger islands marked ‘i2’ and ‘i3’ have individual areas of 900 nm², whereas the smallest island marked ‘i1’ has an area equal to 650 nm². As can be seen, small Ge clusters (indicated by blue arrows) have nucleated on islands ‘i2’ and ‘i3’, leading to a decrease of the Ge concentration in the flat regions of these 2D islands. On the contrary, during the time of preparation, no Ge cluster nucleated on island ‘i1’, ensuring the metastability of the honeycomb germanene layer. The difference of Ge concentration on top of the island and next to them is likely related to a high Ehrlich–Schwoebel barrier to the atomic diffusion between the terraces and the 2D islands [50, 51]. We have obtained an estimate for the diffusion barrier needed to keep a higher concentration on top of the islands than in their vicinity. For a circular island of size r, having an excess Ge concentration $\Delta c$, the integrated

flux of Ge atoms at the periphery is $F_{\text{out}} = 2\pi r a v_d e^{-E_d/kT} \Delta c$, where $v_d$ is an attempt frequency, $E_d$ is the diffusion barrier, which is the sum of the diffusion barrier for Ge atoms on the flat terraces and the extra Ehrlich–Schwoebel barrier, and a, the characteristic length of the barrier. The integrated incoming flux is $F_{\text{in}} = \pi r^2 R$ where R is the incoming deposition rate. These two quantities equilibrate for $E_d = kT \ln(a v_d \Delta c / R)$. For an island of radius $r = 15$ nm, using the experimental value $R = 6.0 \ 10^{-4}$ at nm$^{-2}$, an attempt frequency $v_d = 10^{13}$ Hz, and a equal to the Ag–Ag lattice spacing (0.29 nm), $\Delta c$ being the difference in Ge concentration between the two phases, we obtain $E_d = 1.2$ eV. It is interesting to remark that, since $F_{\text{in}}$ scales as the island area, and $F_{\text{out}}$ as the island perimeter, for a given Ehrlich–Schwoebel barrier, the excess concentration in an island increases linearly with island size. There is thus also a critical island size below which the excess concentration in the islands is not high enough to induce the transition from a Ge$_{98}$Ag$_{6}$ to a pure Ge layer.

These stringent conditions account for the narrow temperature window in which germanene nanoflakes were observed. Under a comparable Ge flux, no such structures have been observed for growth at temperatures of 455 K and above, or at 385 K and below.

It is also noteworthy that the determined germanene lattice constant of 0.403 nm matches the value reported for free-standing germanene [2]. In contrast, with the exception of Pt(111) [7], significantly larger values have been proposed for germanene layers on other metal surfaces. For instance, lattice constants of 0.43 nm, 0.44 nm, and 0.44 nm have been reported for Ge/Al(111), Ge/Au(111), and Ge/Cu(111), respectively [12, 13, 52]. For all these systems, the formation of alloyed phases consistent with the experimental observations has also been proposed or demonstrated [11, 17, 18, 53, 54]. In the present case, we have demonstrated that the nanoflakes observed were pure germanene flakes. Our study is thus the first evidence of the formation of epitaxial germanene nanoflakes.

## 5. Conclusion

In this work, we demonstrate unambiguously the synthesis of nanoflakes of pure honeycomb germanene supported on Ag nano-islands. We underline that despite the absence of a bulk ordered alloyed Ag–Ge phase, of the low miscibility of Ge in bulk Ag and the negligible solubility of Ag in bulk Ge, a surface alloy is thermodynamically more stable than a pure Ge monolayer, up to an oversaturated Ge chemical potential difference of $\Delta\mu = 0.86$ eV. We propose the following hypothesis: such a chemical potential could be achieved for a germanene layer on a small Ag nano-island, where the critical size for the formation of a stable 3D Ge cluster cannot be reached and where the Ehrlich–Schwoebel barrier prevents Ge diffusion out of the Ag nano-island towards large Ge clusters on another part of the surface. These very narrow conditions for obtaining germanene nanoflakes on Ag(111) call into question the reported formation of large germanene flakes on other metal surfaces such as Au(111), Cu(111), and
Pt(111) [7, 12, 55]. Such a system appears to be a promising platform for the study of edge states in topological insulators.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Author contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally.

## Acknowledgments

This study is financially supported the French National Research Agency (Germanene project ANR-17-CE09-0021-02). K Z was supported by the Chinese Scholarship Council (CSC contract 201808070070). The authors would like to acknowledge the High Performance Computing Center of the University of Strasbourg for supporting this work by providing scientific support and access to computing resources. Part of the computing resources were funded by the Equipex Equip@Meso project (Programme Investissements d'Avenir) and the CPER Alsac calcul/Big Data.

## ORCID iDs

M-C Hanf ID 0000-0003-1820-504X

P Sonnet ID 0000-0001-5891-5500

R Stephan ID 0000-0003-0345-1387

G Prévot ID 0000-0001-7960-5587

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