Stability and electronic properties of gallenene

Two-dimensional metals offer intriguing possibilities to explore the metallic and other related properties in systems with reduced dimensionality. Here, following recent experimental reports of synthesis of two-dimensional metallic gallium (gallenene) on insulating substrates, we conduct a computational search of gallenene structures using the Particle Swarm Optimization algorithm, and identify stable low energy structures. Our calculations of the critical temperature for conventional superconductivity yield values of ∼7 K for gallenene. We also emulate the presence of the substrate by introducing the external confining potential and test its effect on the structures with unstable phonons.


Introduction
Since their conception, two-dimensional (2D) materials have been of great fundamental and technological interest. [1][2][3] Early 2D materials were envisaged as easily exfoliable, weakly coupled 2D layers with strong in-plane binding (e.g., graphene, and transition metal dichalcogenides). More recently, however, there has been growing interest in exploring the possibility of two-dimensional forms where layering is not typical in the bulk form. One of the prominent examples of such a material is monoelemental boron, where none of the natural bulk phases are layered, but can exist in the 2D form. 4,5 Further examples include silicene, germanene, stanene, 6 and other non-van der Waals materials. 7 Recently, another monoelemental 2D material composed of galliumgallenenehas been isolated experimentally by solid-melt exfoliation on silica, 8 grown epitaxially on the Si(100) and Si(111) substrates, 9,10 or achieved by intercalation of epitaxial graphene on SiC. 11 Located two rows down from boron in the periodic table, gallium in its 2D form is metallic, similar to borophene 12,13 and other monoelemental 2D metals. 14 Unlike borophene, however, which was originally grown on another metal (silver), gallenene was obtained on an insulating substrate, allowing exploration of the interesting properties of 2D metals with minimal interference from the substrate.

Results and discussion
Motivated by these advances, we conduct a computational exploration of the structural stability and properties of gallenene. Unlike previous studies, 8,15,16 which focused on a limited number of gallenene structures, we aim at carrying out a comprehensive search employing the Particle Swarm Optimization (PSO) algorithm that allows the identication of low energy structures. In the structures with unstable phonon modes, we tested external potential connement to emulate the stabilizing effect of the substrate, rather than applying an inplane strain. We also evaluated the critical temperature for the superconducting transition from rst-principles density functional theory (DFT) calculations in several gallenenes to study the effects of low dimensionality and nanoconnement. 17,18 The CALYPSO code 19 based on the PSO algorithm has been used extensively for predicting two-dimensional materials, [20][21][22] and is adopted here to explore gallenene. Initially, random structures with certain symmetries are constructed with atomic coordinates generated by the crystallographic symmetry operations. To compare formation energies, structural optimization experiments are performed using the Quantum ESPRESSO code. 23 Scalar relativistic ultraso pseudopotentials were used to represent ionic cores as provided by the pslibrary. 24 The Perdew-Zunger local-density approximation (LDA) functional was employed. A plane-wave basis set with a kinetic energy cutoff of 60 Ry was used for the wavefunctions, and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm was used for structural relaxations. Atomic positions were relaxed until forces on all atoms were less than 0.5 meVÅ À1 . In each generation, 70% of the structures are generated from the lowest energy structures in the previous generation using particle swarm optimization, while the rest of the structures are randomly generated. Here, we use a typical setting 19 of 30 generations of gallenene structures, each with a population size of 20, yielding 600 structures in total. As shown in Fig. 1, gray circles correspond to all algorithmically-generated, optimized structures with relatively high energies. These points are shown for completeness and are not notated, since they are never referred to specically but only as a collective set canvassed by CALYPSO. Among them, the structures with lowest energies, shown with red circles in Fig. 1, were investigated further.
The total energy E tot of a layered material as a function of the number of layers n can be well approximated with the following expression: where E 2D is the total energy of a single layer per unit area, E bind is the binding energy between layers, and A is the area of the 2D cell. Here, a similar expression is adopted for bulk-like materials such as Ga. Substituting n À 1 ¼ w/t and nA ¼ N/r 0 , where w is the slab width (dened as the difference between the maximum and minimum of its atomic coordinates normal to layers), t is the thickness of a single layer, N is the total number of atoms in the system, and r 0 is the assumed constant areal atom density in a single layer, and one obtains the following expression for the relative energy per unit area per layer: The rst term on the le side of eqn (2) is essentially (within a constant r 0 ) the energy per atom, or chemical potential of Ga at T ¼ 0 K. For Ga, the layer thickness t was set to 4.4Å, equal to the LDA c lattice constant of bulk a-Ga, giving r 0 ¼ 0.24Å À2 . For a non-layered material such as gallium, E 2D is the energy of a hypothetical atomically-thin structure obtained by extrapolating eqn (2) to n ¼ 1. To estimate the "degree of 2D-ness" of gallenene, as shown in Fig. 1, we plot the le part of eqn (2) for the 2D Ga structures as a function of 1/n h t/(w + t) and compare with both 2D-like and bulk-like prototypical materials such as graphite 25 and silicon. 26 We also plot the line based on the theoretical surface energy 26 of the lowest-energy Ga surface, Ga {100}. The line connecting the lowest-energy gallenene structures and bulk a-Ga is also drawn, with its slope yielding the effective binding energy. Note that the interlayer binding energy E bind , i.e., the energy of cohesion between the two layers, and the conventionally dened surface energy g are related as E bind x 2g. The experimental point is shown for comparison, based on the reported 8 thickness of $4 nm for exfoliated gallenene.
One can see that in gallenene the energy trend with the thickness is quite close to one that would be obtained by cleavage at the lowest-energy surface, and actually shows a higher slope, i.e., atomically thin Ga displays a bulk-like behavior. The effective binding energy obtained from the slope is 89 meVÅ À2 ; this value should be compared with the binding energies of 26 meVÅ À2 for graphite, 25 63 meVÅ À2 for Ga{100}, and 176 meVÅ À2 for the (2 Â 1)-reconstructed Si{100}. 26 The large slope may reect the tendency of freestanding gallenene sheets to assume the bulk form. On the other hand, the low surface energy of bulk Ga may facilitate exfoliation, although the substrate and/or other kinds of connement may play an important role in obtaining atomically thin layers.
Here the 2D structures are named based on the total Ga coverage, i.e., the total number of atoms per area, taking the Ga honeycomb as a reference, which is assigned a value of 10. The energies of the previously considered 8 gallenene structures, honeycomb a 100 and rectangular b 010 , here designated as structures 10 and 14 1 , respectively, shown in Fig. 1, are seen to be rather high. Allowing out of plane displacements in the honeycomb a 100 structure leads to a triangular-like structure with the 6-fold coordination, with an energy similar to that of  thus of less interest. The evaluation of the critical temperature is based on the microscopic theory of Bardeen, Cooper, and Schrieffer (BCS), 28 with the rigorous treatment of electronphonon interactions introduced by Migdal 29 and Eliashberg. 30 Phonon frequencies and electron-phonon coupling coefficients were calculated using the density-functional perturbation theory. The settings used in these calculations are given in the ESI. † The T c values were obtained from the analytical approximation given by the McMillan equation, 27 further modied by Allen and Dynes: 31 k B T c ¼ ħu ln 1:2 exp À 1:04ð1 þ lÞ l À m* À 0:62lm* The prefactor u ln is the logarithmically averaged phonon frequency and the effective electron-electron repulsion is treated as an empirical parameter with the value m* ¼ 0.1. 32,33 The obtained values of the T c are in the 4-8 K range, as shown in Fig. 2, where the corresponding structures are also displayed. Similar values of the critical temperature are a result of similarity of phonon frequencies and electron-phonon coupling coefficients in different gallenene structures. A recent relevant paper 34 brought to our attention aer the submission of the present work has estimated the T c to be 7-10 K for the structures taken from ref. 8 (unstable 10 and metastable 14 1 in our notations, Fig. 1).
The phonon spectra of structures 40, 14 1 , and 14 2 are shown in Fig. 3. The 4-layer structure 40 is stable, whereas structures 14 1 and 14 2 show phonon instabilities. Dynamic instabilities are common in freestanding atomically-thin 2D materials, manifesting as imaginary frequencies in the phonon dispersions obtained with DFT. Using supercells and applying small in-plane strains are some of the useful approaches for stabilizing the phonons in such cases. However, larger cell use and extra optimization are time consuming, while strains on the cell cannot imitate the stabilization effects from the substrate, as these strains are applied in the plane of the 2D materials (xy), but the role of the substrate is much like a connement in the z direction. These connement effects from the substrate can however be mimicked by applying an external potential.
The full interaction potential between the 2D adsorbate and the substrate is an unknown function of all the atomic coordinates. In the harmonic approximation however, one only needs to consider the second order term of the interaction. Therefore, an external potential of the parabolic form can be used to represent this interaction in phonon calculations. With the potential, the nuclear equations of motion are modied as follows.
In the adiabatic (Born-Oppenheimer) approximation and classical limit, the lattice-dynamical properties are determined from the harmonic phonon frequency u which is the solution of the following secular equation: where R I is the coordinate of the Ith nucleus, M I is its mass, R h {R I } is the set of all the nuclear coordinates, and E(R) is the adiabatic potential energy surface. For monoelemental compounds such as gallenene Aer applying the external potential, the total energy E tot becomes where, E(R) is the energy of the system without the external potential and E ext (R) is the energy of the external force eld. A quadratic conning potential of the following form is applied: where, k is the force constant and R I3 is the coordinate in the zdirection of the Ith nucleus. The corresponding force on the Ith nucleus is given by and its force constant k is added to the appropriate elements of the energy Hessian matrix. Based on the above equations, energy, force, and phonon subroutines of the Quantum ESPRESSO code were modied to implement the quadratic external potential, by adding the terms in eqn (6) and (7), and force constant k to the corresponding variables in the code. In this work, a spring constant k ¼ 3.7 kg s À2 was used. This value corresponds to a harmonic oscillation frequency of 30 cm À1 for the gallium atom, which is also the frequency shi of the ZA branch at the G point. The strength of this stabilizing potential should not exceed the interaction strength between gallenene and real substrates. Our DFT calculations show that this condition should hold for most substrates. For example, growth of gallenene on the Si(111) substrate has been reported recently. 9 We carried out calculations for gallenene on the Si(100) substrate and obtained the second order interaction term k $ 56 kg s À2 , which is $15 times larger than the value used to stabilize the gallenenes. The interaction curve between gallenene and the Si(100) substrate obtained from DFT is shown in the ESI. † The obtained phonon spectra of structures 14 1 and 14 2 with the external connement are shown in Fig. 4a and b. In the former structure, aer re-optimization in the external potential, the imaginary frequencies are removed. The latter structure shows instability even in connement, with imaginary frequencies existing between G and J 1 points. The atomic motions in these unstable modes are in fact in-plane, whereas the forces from the external potential act in the z direction and thus can only stabilize the out-of-plane modes. This is the case for structure 14 1 , where the unstable modes along the G-X line are out-of-plane, and are stabilized by the external potential. The visualization of the unstable vibrational modes is provided in the ESI. †

Conclusions
In summary, we explored feasibility of two-dimensional gallium (gallenene) by performing a computational search of various two-dimensional polymorphs using the Particle Swarm Optimization algorithm. The stability trend of the structures found in the search points at the bulk-like behavior in few-layer   Fig. 3b and c, respectively. Both structures flatten slightly after being put into the external potential. During optimization, in-plane lattice constants, angles, and atom positions are all relaxed. Notice that the ZA branches are all shifted to 30 cm À1 . In the second structure, imaginary frequencies for in-plane phonon modes remain.
gallium. An overall important conclusion seems to be that gallenene is unlikely to maintain the 2D-form, stable only at a rather thick "4-layer" form (structure 40) and gravitates toward thin lms of bulk Ga. Only in the presence of the substrate can some thinner polymorphs be stabilized. This suggests that chemical vapor deposition or molecular beam epitaxy could possibly produce stable gallenene on substrates. Calculations of the critical temperature for conventional superconductivity have yielded values of $7 K in gallenene. Finally, external potential imitating the conning effect of the substrate has been applied to the structures that displayed unstable phonon modes. The potential is shown to be instrumental in stabilizing phonon modes with out of plane atomic displacements.

Conflicts of interest
There are no conicts to declare.