Stability and electronic properties of Ge x ð BN Þ y monolayers

In this work, we employ ab initio simulations to propose a new class of monolayers with stoichiometry Ge x ð BN Þ y . These monolayers belong to a family of 2D materials combining B, N and group IV atoms, such as B x C y N z and Si x B y N z . We calculated the formation energy for ten atomic arrangements, and found that it increases when the number of B e Ge and N e Ge bonds increases, and decreases when the number of B e N and Ge e Ge bonds increases. We found that the lowest energy monolayer presented a Ge 2 BN stoichiometry, and maximized the number of B e N and Ge e Ge bonds. This structure also presented mixed sp 2 and sp 3 bonds and out-of-plane buckling. Moreover, it remained stable in our ab initio molecular dynamics simulations carried out at T ¼ 300 K. The calculated electronic properties revealed that Ge x ð BN Þ y monolayers might present conductor or semiconductor behavior, with band gaps ranging from 0.0 to 0.74 eV, depending on atomic arrangement. Tunable values of band gap can be useful in applications. In optoelectronics, for instance, this property might be employed to control absorbed light wavelengths. Our calculations add a new class of monolayers to the increasing library of 2D materials. © 2017 Elsevier Ltd. All rights reserved.


Introduction
Two-dimensional materials with a single layer of hybridized sp 2 atoms, arranged in a honeycomb lattice, have received increasing interest due to their novel properties and potential applications.Among these materials, the most widely studied are graphene and h-BN, which are allotropic forms of carbon and boron nitride (BN) [1,2].One of the main differences between these structures is that graphene is a zero-band-gap semiconductor, while h-BN is a large-band-gap semiconductor ( > 4 eV).Both structures have been used as media for new fundamental studies aimed at understanding, for example, the influence of defects, substrates, doping, strain, and electric fields on their mechanical and electronic properties, as well as studying their electronic transport properties [3e9].Moreover, graphene and h-BN have motivated the search for new single layered two-dimensional materials, such as hybrid monolayers of B, C, and N: BCN [10], BC 2 N [11,12] and BC 4 N [13].In general, theoretical and experimental studies reveal that, unlike graphene and h-BN, B x C y N z hybrid systems present small energy band gaps ( < 2 eV).For this reason, several proposals exist to use these structures in electronic devices [14,15].
In this context, it is important to point out that silicon (Si) and germanium (Ge) also have four valence electrons, and also present graphene-like allotropes: silicene and germanene.First-principle calculations of structure optimization, phonon modes, and finite temperature molecular dynamics predict that silicene and germanene present a puckered hexagonal lattice with sp 3 bonds, instead of a flat configuration of sp 2 bonds [16e21].Nevertheless, these structures have already been synthesized [22e24].Curiously, the electronic band structures of silicene and germanene are similar to that of graphene, where p and p* bands cross linearly at the Fermi level of the Brillouin zone, forming the so-called Dirac cones [25,26].In these materials, electrons behave as massless Dirac fermions, with Fermi velocities of magnitude 10 6 m,s À1 [27,28].
Very recently, Andriotis et al [29] and Sandoval et al [30] have employed ab initio simulations to propose two-dimensional hybrid monolayers formed by B, N, and Si atoms, with peculiar characteristics.These monolayers are analogous to the BeCeN hybrid structures.Andriotis et al showed that the most stable structure is one with a hexagonal lattice of sp 2 bonds and stoichiometry Si 2 BN, without out-of-plane buckling.Since Si atoms do not tend to form a 2D flat hexagonal structure, the prediction that Si 2 BN can exist as a stable flat monolayer is quite surprising.The electronic band structure revealed metallic behavior.All phonon frequencies were found to be real, indicating a high degree of stability.On the other hand, Sandoval at al found monolayers with stoichiometry ðBNÞ x Si 1Àx , with even lower energies.These works suggest that new hybrid structures might exist, composed by B, N, and other atoms of group IV, such as Ge and Sn.
With the above motivation in mind, in here we apply density functional theory (DFT) to investigate the structural stability and electronic properties of a new class of two-dimensional materials, formed by Ge, B, and N atoms (see Fig. 1).In order to limit the total number of structures, we chose to investigate structures presenting only GeeGe, BeN, GeeN, or GeeB bonds.In this manner, we avoid the energetically adverse BeB and NeN bonds [29].

Computational details
We employed density functional theory, as implemented in the SIESTA code [31,32], to optimize the structure and calculate the electronic properties of Ge x ðBNÞ y monolayers.We use a double-z basis set, composed of numerical atomic orbitals of finite range (with cutoff radius of about 21 Å).The norm-conserving pseudopotential for Ge, B, and N were generated using the Troullier-Martins scheme [33], in the Kleinman-Bylander factorized form [34].We apply the generalized gradient approximation (GGA) for the exchange-correlation potential [35,36].The special k-points were generated by the Monkhorst-Pack scheme [18].During the calculations, all geometries have been relaxed until the total force on each atom reached a value lower than 0.1 eV/Å.It was assumed a convergence criterion where the self-consistency is achieved when the maximum difference between the output and the input of each element of the density matrix, in a self-consistent cycle, is smaller than 10 À4 .Three-dimensional periodic boundary conditions were applied, resulting in a monolayer composed of repeating cells in the x-z plane (see Fig. 1).In the perpendicular direction -along the y axis -there is a distance of 15 Å between each infinite monolayer, forming a sufficiently thick vacuum region, so that neighboring layers do not interact.
Fig. 1 shows the relaxed Ge x ðBNÞ y monolayers investigated in this work, with different stoichiometries ðx; yÞ.These structures have unit cells containing n atoms (n ¼ 24 or 32 atoms) and different numbers of BeN, GeeGe, BeGe, and NeGe bonds.As mentioned previously, the chosen atomic arrangements for the Ge x ðBNÞ y monolayers preclude the presence of BeB and NeN bonds.The differences in length for the BeN, GeeGe, BeGe, and NeGe bonds result in a substantial amount of structural stress and, in some cases, introduce out-of-plane buckling.We considered six different atomic arrangements for monolayers with stoichiometry Ge 2 BN, all arranged in a graphene-like hexagonal lattice, where all bonds are sp 2 and all angles are around 120 + (see Fig. 1).Meanwhile, Fig. 1 (g) shows the non-optimized and optimized structures for another monolayer with stoichiometry Ge 2 BN.The non-optimized structure consists of zigzag BN chains connected to zigzag GeeGe chains which form a hexagonal lattice with sp 2 bonds.However, our DFT optimization resulted in a structure with out-ofplane buckling, so that the zigzag GeeGe chains spontaneously recombined to form square chains connected to zigzag BN chains.The resulting lattice is composed of squares and pentagons with mixed sp 2 and sp 3 bonds.This result is not surprising, since the length of the GeeGe bond is significantly higher than that of the BeN bond.Therefore, a zigzag GeeGe chain cannot be combined with a zigzag BN one.It is important to note that the Ge 2 BN structures (a), (b), (f), and (g) were built on the basis of atomic arrangements proposed by Andriotis et al [29] and Sandoval et al [30] for Si 2 BN monolayers.
Additionally, we also investigated hexagonal structures with stoichiometries GeBN, GeB, and GeN (see Fig. 1 (h)-(j)).The GeN monolayer exhibits buckling, mixing sp 2 and sp 3 bonds, while the GeBN and GeB monolayers present only sp 2 bonds.Moreover, the GeN and GeB monolayers have atomic arrangements similar to that proposed for a GeC monolayer [37].

Results and discussion
In order to analyze the stability of the Ge x ðBNÞ y monolayers, we have calculated the energy required to form a given structure (E Form ), using a thermodynamical approach based on the determination of the chemical potentials of the atoms involved in a synthesis reaction.The details of this approach are better described in Refs.[38,39], and the formation energy is defined by following expression where E Tot is the calculated total energy provided by the SIESTA code, n i is the number of atoms for each element (i ¼ B, N, Ge), m i is the corresponding chemical potential for each element, and n T is the total number of atoms in the structure.The chemical potentials should depend on the atomic reservoirs used in the synthesis reaction.Using gaseous N 2 for the N-rich reservoir, we obtained m N ¼ À270.51 eV.For the B-rich reservoir, we used a À b boron, and found m B ¼ À77.22 eV [38].The chemical potentials m B , m N and m Ge should fulfill the constraints of thermodynamic equilibrium: where the constants m BN and m GeGe are the chemical potentials for BN and GeGe pairs within infinite BN and germanium hexagonal monolayers.In this calculation, an infinite germanium or BN monolayer is taken as reference, where it is assigned a null value for the formation energy.Using this approach we have obtained m BN ¼ À350.75 eV and m GeGe ¼ À213.84 eV.Finally, for the Ge 2 BN and GeBN structures, with equal number of BN and GeGe pairs, the formation energies have been calculated using the following expressions: and for the structures with stoichiometries GeB and GeN, we have: (5) The values for the formation energy and the number of BeN, GeeGe, BeGe and NeGe bonds of Ge x ðBNÞ y monolayers, are reported in Table 1.
From this data one can see that the Ge 2 BN-(g) monolayer, with the maximum number of GeeGe and BeN bonds, presents the lowest value of E Form , so that it is the most stable structure.This result is in agreement with those found for stable monolayers with stoichiometry B x C y N z reported in Refs.[10e12].Following this line of reasoning, the GeN monolayer, which presents only NeGe bonds, is the second most stable structure (by an energy difference of only 0.08 eV/n T ).The reason for the stability of the Ge 2 BN-(g) and GeN structures is the mixture of sp 2 and sp 3 bonds.It is well known that sp 3 bonds exhibit a stronger character, when compared to sp 2 bonds, which provides more stability to the lattice.The most unstable structures are the Ge 2 BN from (a) to (f), GeBN, and GeB monolayers, which have higher values of E Form (at least 0.33 eV/n T above Ge 2 BN-(g)).The common characteristic of these structures is that they exhibit a hexagonal lattice formed only by sp 2 bonds, which, according to our results, is not a favorable trend for monolayers formed by atoms of Ge, B and N.
In order to verify whether the proposed structures are stable at room temperature [40,41], we performed Ab Initio Molecular Dynamics (AIMD) simulations with the SIESTA code.We used a time step of 1 fs to evolve each of the proposed structures for 20 ps, in the NPT ensemble.We employed the Nos e thermostat to control temperature and the Parrinello-Raman barostat to control the in-plane pressure components.We did not allow the system to relax in the direction perpendicular to the plane, in order to keep the monolayers separated by a 20 Å vacuum slab.In Fig. 2, we present snapshots of the structures proposed in this work after the completion of the thermalization process.The provided DE values represent the energy difference between thermalized and optimized structures.Inspection of the snapshots reveals that bond breakage and formation was only observed for Ge 2 BN-(c).For this structure, reconstruction lowered the total energy, suggesting that the initial atomic arrangement is not stable.All other monolayers remained stable in our simulations.In some cases, however, we did observe structural deformation, with atoms moving out of the basal plane.Finally, note that the thermalization process caused little change to the atomic arrangement of the lowest formation energy structure, Ge 2 BN-(g).Our AIMD simulations indicate that this configuration is indeed very stable.
With the exception of the Ge 2 BN-(g) monolayer, the length of the BeN and GeeGe bonds is around 1.44 and 2.40 Å, respectively.For the Ge 2 BN-(g) case, due to the out-of-plane buckling, the BeN bonds are around 1.43 Å, while the GeeGe bonds were elongated and range from 2.50 to 2.72 Å.These values are close to those experimentally measured: 1.45 Å for the BeN bond [42] and 2.44 Å for the GeeGe bond [23,24].For the case of the BeGe and NeGe bonds, their lengths are intermediate between those obtained for BeN and GeeGe.They range from 1.93 to 2.2 Å for the BeGe bond, and from 1.90 to 2.08 Å for the NeGe bond.Because the NeGe bond tends to be shorter than the BeGe bond, an optimized structure might be fairly assymetric.To understand why some structures present higher or lower symmetry, consider the geometrical constraints imposed by periodicity on monolayers Ge 2 BN-(b) and Ge 2 BN-(c).Their detailed structural information is presented in Fig. 3.For the x-direction, the same single condition needs to be satisfied for both structures: Although this results in a more symmetric appearance for structure (c), it also leads to a higher formation energy.
Fig. 4 shows the band structures calculated for the ten Ge x ðBNÞ y monolayers investigated in this work.The values of the energy band gap E g are shown in the last column of Table 1 -we found E g values ranging from 0.0 to 0.74 eV.The Ge 2 BN (a)-(d) and (g), GeBN, and GeB structures are all metallic.This behavior is due to the electronic configuration of the lattice with p bonds, which is similar to the electronic configuration of the graphene honeycomb lattice.On the other hand, the Ge 2 BN (e)-(f) and GeN structures behave as semiconductors.For such cases, the presence of mixed sp 2 and sp 3 orbitals contribute significantly to the increase in the electronic band gap.
By means of projected density of states (PDOS), shown in Fig. 5, it is possible to see that the contribution of the germanium atoms are, in general, more expressive for the electronic states near the Fermi energy E f .On the other hand, the contribution of the boron and nitrogen atoms, for the electronic states near the Fermi level, depends on the atomic arrangement of the Ge x ðBNÞ y monolayer.Complementary calculations of the localized density of states (LDOS) (see Fig. 6), reveal that the bottom of the conduction band and the top of the valence band are associated with the p z orbitals of the germanium, boron and nitrogen atoms.This result is compatible with those from PDOS.
Additionally, spin-polarization calculations reveal a total spin of zero for all Ge x ðBNÞ y monolayers investigated, which means that the valence band is completely filled and that unpaired electrons are not available.
Recent work by Manna et al [43] showed that graphene sheets containing BN domains and BN sheets containing C domains are very stable structures with peculiar electronic properties.Since the most stable Ge x ðBNÞ y monolayer is the one that alternates a BN chain and a GeGe chain, in a configuration of different BN and GeGe regions, it would be interesting to study germanene monolayers with BN nanodomains and the opposite (BN monolayers with GeGe nanodomains).If such structures occur, the number of BeN and GeeGe bonds increase and the number of BeGe and NeGe bonds decrease, so that these structures can exhibit stabilities higher than the Ge x ðBNÞ y monolayers studied in this work.We intend to study these structures in the future.In the present study, DFT calculations for Ge x ðBNÞ y monolayers with BN and GeGe domains were frustrated due to the small size of the unit cell, with 24 or 32 atoms.On the other hand, we propose that the most stable Ge x ðBNÞ y monolayer (the Ge 2 BN-(g) structure in Fig. 1) can be rolled up to form zigzag and armchair nanotubes, in the same way it is done with graphene and BN.More generally, the proposed monolayers can be rolled up into varied nanotubes, with mechanical and electronic properties that are different from those composed by carbon and BN.We are also planning to study these nanotubes in a forthcoming paper.

Conclusions
Using first principles calculations, we have investigated the stability and electronic properties of a new class of monolayers with stoichiometry Ge x (BN) y .We have found that the most stable Ge x (BN) y monolayer exhibits stoichiometry Ge 2 BN, while maximizing the number of BeN and GeeGe bonds and minimizing the number of GeeN and GeeB bonds.This behavior is similar to that found for hybrid monolayers of B x N y C z , where their most stable structures also maximize the number of BeN bonds.Another interesting feature of the most stable Ge x (BN) y monolayer is the mixture of sp 2 and sp 3 bonds in an exotic lattice of squares and heptagons, which presents out-of-plane buckling.Thus, future efforts to synthesize the Ge x (BN) y monolayers must take into account their non-hexagonal morphologies, in contrast to the hexagonal B x N y C z and Si 2 BN monolayers.Molecular dynamics simulations, performed at room temperature, revealed structural rearrangement only for the Ge 2 BN-(c) monolayer.No bond breakage was observed after thermalization for the other structures.In particular, the monolayer with lowest formation energy (Ge 2 BN-(g)) underwent minimal deformation, a result which indicates high stability.Our calculations also demonstrate that the electronic properties of the monolayers are highly sensitive to the specific arrangement of the Ge, B, and N atoms.We have found band gaps ranging from 0.0 to 0.74 eV.Out of the two most stable structures, one exhibits metallic (Ge 2 BN-(g)) and the other semiconductor (GeN) behavior.This diversity in electronic properties suggests these nanostructures might be good candidates for use in electronic devices.Our calculations add ten new monolayers, belonging to the family of 2D materials combining B, N and group IV atoms, to the 2D library.

Fig. 1 .
Fig. 1.Illustration of the optimized GexðBNÞ y monolayers with different atomic arrangements.

Fig. 3 .
Fig. 3. Detailed structural information for Ge 2 BN-(b) and Ge 2 BN-(c).Notice Ge 2 BN-(b) is rather assymetric.Periodicity imposes more constraints on structure Ge 2 BN-(c), leading to higher symmetry but also to higher formation energy.

Fig. 4 .
Fig. 4. Calculated band structures of the monolayers shown in Fig. 1.The dashed line represents the Fermi level.

Fig. 5 .Fig. 6 .
Fig. 5. Calculated projected density of states (PDOS) of selected GexðBNÞ y monolayers.The Fermi energy E f is indicated by the solid vertical line.

Table 1
Formation energy (E Form ), number of bonds n i (i ¼ BeN, GeeGe, BeGe and NeGe) and energy band gap (Eg), for all GexðBNÞ y monolayers.The underlined number indicates the most stable structure.
Fig. 2. Snapshots of the GexðBNÞ y monolayers after thermalization at T ¼ 300 K. We employed ab initio molecular dynamics to evolve each layer for 20 ps.The provided DE values represent the energy difference between thermalized and optimized structures.A. Freitas et al. / Superlattices and Microstructures 110 (2017) 281e288