In this section, we have carried out the structural search for the most stable configuration with different coverage of Sb atoms on pure Ag (111) substrate. In Figure 1A, we depicted the variation of formation energies with C _Sb on Ag (111) substrate. There are formation energy valleys at the corresponding coverages of C _Sb = 2/3, 2/3, 4/9 for type A , B , and C substrates, respectively. The configuration with the lowest formation energy is at C _Sb = 2/3 on B -type Ag (111) substrate, in which the Sb atoms are arranged in a zigzag shape on the substrate, which is consistent with the half layer of α-Sb, as shown in Figure 4 (4/6).

The upper four subfigures in Figure 2 show the structures with the lowest energy at different coverage of antimony on an A -type of pure Ag (111) substrate. Overall, as C _Sb increased from 1/3 to 4/3, there were two kinds of antimony sheets present, one with the flat honeycomb lattice at C _Sb = 2/3, and the other with the buckled honeycomb lattice at C _Sb = 3/3 and 4/3. At C _Sb = 1/3, the Sb atoms sit on the fcc-hollow sites, which is the most favorable adsorbed site. According to our tests, the fcc- and hcp-hollow sites have very similar adsorbed energies, about a difference of 0.04 eV/atom. Therefore, both the fcc- and hcp-hollow sites could be stable during actual production. At C _Sb = 2/3, the Sb atoms of the lowest energy configuration sit on the hcp-hollow sites, and formed a flat honeycomb lattice with a 23.1% stretched lattice constant of a = 5.067 Å compared to the freestanding β-Sb(4.117 Å). This configuration is the most stable on A -type Ag (111) substrate and well agreed with the results of the work by Shao et al. [27]. It is interesting to note that the C _Sb = 3/3 structure is split into two layers, with the upper layer being a buckled honeycomb lattice and the discrete Sb atoms in the lower layer sitting almost on the bridge sites. When the deposition of antimony atoms increased to more than a single layer, the structure did not grow from the bottom to the up, layer by layer, like building blocks. The flat honeycomb lattice was no longer available and has been replaced by the buckled honeycomb ones. When C _Sb increased to 4/3, a two-layer buckling honeycomb lattice with AB stacks of different buckling heights was formed. The buckling height of the upper layer is 0.968 Å and that of the lower layer is 0.712 Å. The electron localization function of C _Sb from 2/3 to 4/3 in Figure 3 showed that the electrons were localized around the Sb atoms. The interaction between Ag–Sb was weaker than Sb–Sb. The stronger Sb–Sb interaction from the z-direction leads to the fact that the antimonene no longer maintains the flat structure when the number of layers is more than one.

As it is shown in Figure 4, from C_Sb = 1/6 to 4/6, there existed a clear pathway to form a half layer of α-Sb on B -type Ag (111) substrate. From C_Sb = 5/6 to 8/6, all the configurations were based on C _Sb = 4/6, and as the Sb atoms continued to be deposited, the atoms were regularly arranged on this half-layer structure, and when C _Sb = 8/6, a full layer of α-Sb was formed. The lattice constants of α-Sb on Ag (111) substrate are a = 4.388 Å and b = 5.067 Å, which have a stretch of 0.9 and 6.6% along each of the a (Zigzag) and b (Armchair) directions compared to the freestanding α-Sb. The thickness of α-Sb was 2.95 Å and its sublayers were almost flat.

On C -type Ag (111) substrates, we did the calculations for Sb coverage from 1/9 to 9/9, a range that contains a monolayer of antimony sheet, as shown in Figure 5. Same as A - and B -type Ag (111) substrates, the single Sb sits on the hollow sites. From C _Sb = 3/9 to 6/9, the most stable structures of each coverage are triangular, quadrilateral, pentagonal, and hexagonal lattices, respectively. This is quite different from the results of the other two types of substrate. Small flat polygonal Sb clusters seem to dominate in the early stages of growth, and the formation energy of the quadrilateral structure is the lowest. With the increase of C _Sb from 4/9, the formation energy shows an overall increasing trend, and the Sb atoms did not spread over the entire surface until C _Sb = 7/9. A valley point appears at C _Sb = 8/9, which indicates that the structure is more stable at that coverage. However, the lowest energy structure of C _Sb = 8/9 we obtained from the structural search is a bilayer structure shown in Figure 5. It was an irregular mesh of seven atoms in the bottom layer and an additional atom located at the top site of the center of a square of four atoms in the bottom layer. We also obtained the structure of β-Sb at C _Sb = 8/9 with the lattice constant of about 4.390 Å and thickness of 1.552 Å (see Supplementary Figure S3, its formation energy is 0.035 eV/atom higher than that of the lowest one.) These results indicated that C _Sb = 8/9 maybe a magic coverage.

To further examine the structure of antimony clusters grown on the surface of Ag (111), we calculated the various structures of 1-3 Sb₄ clusters falling on the Ag (111) surface by using ab initio molecular dynamics (AIMD) embedded in the package of CP2K [46]. The substrate of Ag (111) was set to three layers with the bottom layer fixed, and the orthogonal lattice parameter was set to a = 26.327 Å, b = 25.333 Å, and c = 32.314 Å to avoid the interaction between the periodic clusters. The AIMD results showed the process of how small Sb₄ clusters spread over the Ag 111) substrate into the flat ones after 5 ps at 300 K. Figure 6A plots the total energy of the systems as a function of time for three Sb₄ clusters on a pure Ag (111) surface. Two of these clusters disintegrate and rapidly tile to the substrate after 0.7 ps of simulations. About 4.2 ps, the atoms in all the clusters spread out. We found that the flat tetragonal structures, which have the same structure as the C _Sb = 4/9 on pure Ag (111) surface, exhibit unusual stability during the simulation. To analyze the interaction between the Sb and Ag atoms, we have drawn the curve of radial distribution function (RDF), as shown in Figure 7. The curve shows that the average distance between Sb–Sb stays within 2.935 Å, and the Sb–Ag bond increases slightly with the increase of the number of Sb atoms, from about 2.815 to 2.885 Å. As the Sb _N cluster gets larger, the distance between Sb _N and the substrate gets farther, indicating that the interaction between Sb and Sb is much stronger than that between Sb and Ag. Thus, the growth of small clusters is much easier than that of Ag–Sb alloys. The results of the other two configurations are plotted in Supplementary Figure S1 and Supplementary Figure S2. We have also examined the integration curve of RDF. The average coordination number of these three systems increases from 1 to 2 with the increasing number of Sb₄ clusters. It can be seen from its structural variations, where dimer, chain, and polygonal structures occupy the majority of conformations. For more visual observation of the structural changes, we exported the animations of the AIMD trajectory, as shown in the MP4 files in SI. The flat square Sb₄ clusters are very stable in this simulation. This is consistent with the fact that the most stable configuration on C -type Ag (111) substrate in Figure 1 is Sb₄ at C _Sb = 4/9.

Theoretically, the larger the substrate, the closer the results are to reality. Combining the results of these three substrates, it can be seen that at lower coverages, flat 2D clusters occupy the major part, and when the antimony atoms increase enough to spread over the substrate or even more, it is quite possible to grow the so-called β- and α-Sb structures. However, in terms of energy comparison, the most stable configuration is the flat half layer of α-Sb.

Before calculating the deposition of Sb atoms on the Ag₂Sb/Ag (111) substrate, the climbing image nudged elastic band (cNEB) method [47] was applied to search the transition state to analyze the process of penetration and replacement of Sb to Ag on the Ag (111) surface. We started with modeling the Sb/Ag (111) interface by adding a single Sb atom onto 3 × 3 supercell Ag (111) surface (see Figure 8 (a), the structure diagram on the left). The adsorption energy was calculated by the formula: E ads Sb = E Ag ( 111 ) + E atom Sb − E tot Sb/Ag ( 111 ) , where, E atom Sb is the total energy of per Sb atom in bulk. The calculated E ads Sb value is 0.037 eV on the 3 × 3 Ag (111) surface. The minimum energy path (MEP) of the penetration of one Sb atom into the Ag (111) surface plotted in Figure 8A, it shows a forward energy barrier of E a ⃗ = 0.779 eV, and a reverse energy barrier of E a ⃖ = 0.672 eV. The E a ⃗ is much higher than E ads Sb . Therefore, the replacement process should not occur unless the system is in a heat-absorbing state. The environment provides enough energy to make it cross this energy barrier.

Next, we added the second Sb atom to the substrate while removing the Ag atom that was expelled. Since there was already a penetrated Sb atom after step (A), so we chose to calculate the MEPs for the penetration at the neighbor and subneighbor site separately, see Figure 8B. From the results, it is clear that path B (II) is more favorable than path B (I), with a lower energy of the final state and a relatively shorter reaction path. Based on this, we continued to add a 3rd Sb atom to the final state of path B (II) and made it penetrate at the subneighbor site, and the MEP of this process was plotted in Figure 8C. All the energies of MEPs are listed in Table 1, the ΔE _a are all positive, which proves that Sb penetration is a heat absorption process. Such subneighbor atomic arrangement characteristics predetermine the impossibility of continuous replacement of Ag by Sb, and with Sb penetration occurring all over the surface, and an Ag₂Sb alloy monolayer is formed. Till now, we have confirmed that if a surface alloy could be formed, it would be Ag₂Sb.

Then, we continued to study the adsorption of Sb atoms on the Ag₂Sb/Ag (111) substrate. Same as pure Ag (111) substrates, three types of substrates were constructed, and the curve of the formation energies vs C _Sb were plotted in Figure 1B. Overall, the formation energies on 3 × 3 substrates were generally lower than on the other two types of substrates. Unlike pure Ag (111) substrates, Sb exhibits different growth characteristics on the Ag₂Sb monolayer, the small antimony clusters are no longer more stable, and the formation energy continues to decline until at coverage of 8/9. At C _Sb = 8/9, the Sb layer shows a 2 × 2 reconstructed β-phase structure with a minor deformation on C -type Ag₂Sb alloy monolayer. One of the Sb atoms in the lower sublayer is affected by the Sb atom of the substrate and shifts upward, causing the original double sublayers structure to split into three sublayers from the side view. The lattice constant is a = 4.388 Å, with about 6.58% stretching compared to the free state of antimonene (4.117 Å). The layer height is about 4.377 Å. This structure is in good agreement with the results of Sun et al.‘s work [28].

For A-type of Ag₂Sb/Ag (111) substrates, the results of single Sb on Ag₂Sb/Ag (111) shows that the Ag–Ag bridge site was the most favorable, but a monolayer of β-Sb with a layer height of 0.736 Å formed at C _Sb = 2/3 and Sb atoms occupy the top sites of both the Ag and Sb atoms in the substrate, see Figure 2A 2/3). At C _Sb = 3/3, the bilayer structure consisted of a layer of β-Sb and intercalated atoms, while the intercalated atoms were located at the Ag–Ag bridge-sites of the substrate. At C _Sb = 4/3, the bilayer structure consisted of two layers of β-Sb with AB stacking, the bottom layer was almost flat, and the top layer was a buckled honeycomb lattice with buckling higher the 1.022 Å. On such substrates, the flat honeycomb antimonene cannot be stabilized due to the surface alloy, but is replaced by a honeycomb structure with buckling. The interaction between the Ag and Sb atoms affects the structures of antimonene layers. It is worth mentioning that the formation energies of A -type is the highest one of the three types.

The lowest formation energy on the B -type Ag₂Sb/Ag (111) substrates was at C _Sb = 8/6, and a full-layer of α-Sb formed, see Figure 9. It was almost the same structural evolution path compared to the pure Ag (111) substrates, but, the formation energy of the full-layer of α-Sb is much lower than that of the half-layer, which means that the full-layer α-Sb is energetically stable on such substrate. The adsorbed Sb atoms first formed a half layer of α-Sb, which then grow to a full layer. Unlike on the pure Ag (111) substrate, the bottom layer of α-Sb was no longer flat. One of the Sb atoms in the unit cell sits on the top site of the Sb atom in the surface alloy, while the other Sb atoms sit on the bridge sites of the Ag–Ag pair. The difference in the strength of the interactions of Ag–Sb and Sb–Sb makes the interatomic spacing different.

On C-type Ag₂Sb/Ag (111) substrates, the lowest formation energy was at C _Sb = 8/9 of a reconstructed 2 × 2 supercell of β-Sb, see Figure 10. The reconstructed supercell contains eight Sb atoms, divided into two sublayers, with three of the four Sb atoms in the bottom layer at the bridge site of the Ag–Ag pair and the other Sb atom at the top site of the Sb atom in the surface alloy. This structure is highly consistent with that in Sun’s work [28]. It can be seen from the green dashed line in Figure 1 that the formation energy decreases with the increase of C _Sb. The existence of the alloy surface changes the potential energy surface of the Ag (111) substrate. Under its impact, the flat polygonal clusters, that appear on the pure silver substrate, are no longer the more stable configurations.

From the calculation results of the abovementioned three substrates, we can see that the presence of the surface alloy changes the adsorption site. The Sb–Sb interaction is somewhat stronger than that of Sb–Ag, so the presence of Sb atoms in the surface alloy can affect the growth of antimonene, especially in the earlier stage. The energetically stable adsorption sites change from the hollow sites to the Ag–Ag bridge sites. Moreover, unlike the pure silver substrate, the Sb top site is also a more stable adsorption site.

Our calculation shows that the formation energies of β-Sb (-0.014 eV/atom) and α-Sb (-0.026 eV/atom) on the Ag₂Sb/Ag (111) substrates are very close. Since both α- and β-Sb monolayers are stable in ambient conditions, we need to compare their structural stability and find their structural growth paths to determine which phrase is easier to grow on and which one is relatively stable. Therefore, we carefully examined all the structures, including the substable ones, from the half-layer to the full layer and found two evolutionary paths for each phrase on the Ag₂Sb/Ag (111) substrate. In Figure 11A, the formation energies increased and then decreased with the increasing coverage, where the two relatively stable configurations were half- and full-layer α-Sb at C _Sb = 4/6 and 8/6, respectively. The structures from C _Sb = 5/6 to 8/6 were all grown based on the stable structure at C _Sb = 4/6. It is clear that α-Sb is grown half-layer by half-layer on this substrate. Although, there exists a small energy barrier of 0.069 eV/atom along this path. For 3 × 3 Ag₂Sb/Ag (111) substrate in Figure 11B, at C _Sb = 4/9 and 5/9, there were quadrilateral and pentagonal rings with pucker, in which the Sb₅ is a very special one, two of the five atoms are sitting on the top site of the Sb atoms in the substrate, and the other three are on the top site of the Ag atoms. This pucker structure is exactly a part of the β-Sb at C_Sb = 8/9. The continuously deposited Sb atoms placed in their gap positions can form the structures of C_Sb = 6/9 and 7/9, and the structure of C_Sb = 7/9 is exactly β-Sb with a Sb vacancy. The formation energy decreases with the increasing concentration of the adsorbed Sb atoms, which indicates that the deposition process is exothermic and can react autonomously. The growth of β-Sb is more advantageous under lower-temperature, while at a high-temperature, there exists a competition between the two phases. Therefore, we believe that it should be possible to observe the growth of both the structures simultaneously during a given experiment.

Considering the results of all the types of substrates together, both the size of reconstructed supercells and the surface alloy have a significant effect on growth. No matter with or without the surface alloy, the structure of antimonene on the A -type of substrates always show a large stretch whether forming the flat honeycomb antimonene or low-buckled honeycomb antimonene. On the B -type of substrates, the α-Sb occupy the main structures, and there exists a very clear evolutionary path of growing half-layer by half-layer. The half layer α-Sb has high stability, regardless of the deformation by the substrates. On the largest substrates in this work, the β-Sb exhibits excellent stability on the substrate with surface alloy and much less stability on the pure Ag (111) substrates compared to the flat honeycomb and half-layer α-phase antimonene. It can be seen that the phase of antimonene is deeply influenced by the size of the reconstructed supercells. Especially, all the configurations are at lower energy levels at the coverage of 2/3, which is included in all types of substrates, and the structures of antimonene are significantly modulated by the size of the reconstructed supercells. When adsorbed on the smaller substrates ( A and B ), the Sb atoms could fully cover the substrates, but not for the larger ones ( C ) . The direct factor, which determines the different phases, is the size of the reconstructed supercells. If we can control the size of the reconstructed supercells, the phase of antimonene will be determined.

The existence of the surface alloy of Ag₂Sb has also played an important role in the growth of antimonene. At lower coverages (C_Sb < 2/3), clusters of Sb _N (N = 3, 4, and 5), although present, are no longer the most stable structures. On the other hand, at higher coverages (C_Sb > 2/3), the formation energies of α- and β-Sb shift from the higher to the lowest levels with the presence of the surface alloy. This has a positive effect on the formation of a stable monolayer of α-antimonene.

Note that the minimum value of the required coverage to form a monolayer are 2/3, 2/3, and 7/9 for the substrates of A , B , and C types, respectively. Our simulations are based on the complete relaxation of the Sb atoms on the substrate, and if the speed of the process of atomic relaxation can be controlled, then the phase of the antimonene should also be controllable. In the molecular beam epitaxy experiments, the control of relaxation is usually multifactorial, such as temperature, pressure, and source evaporation rate. Which one is the main factor affecting the relaxation speed remains to be finely verified.