Stability and electronic properties of (Ge)x(BN)y monolayers

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Stability and electronic properties of Ge

x

ðBNÞ

_y

monolayers

A. Freitas

a,*

_{, L.D. Machado}

a

_{, R.M. Tromer}

a

_{, C.G. Bezerra}

a

_{, S. Azevedo}

b a_{Universidade Federal do Rio Grande do Norte, Departamento de Física, Caixa Postal 1641, 59072-970, Natal, RN, Brazil} b_{Universidade Federal da Paraíba, Departamento de Física, CCEN, Caixa Postal 5008, 58051-970, Jo~ao Pessoa, PB, Brazil}

a r t i c l e i n f o

Article history: Received 27 May 2017 Accepted 11 August 2017 Available online 19 August 2017

a b s t r a c t

In this work, we employ ab initio simulations to propose a new class of monolayers with stoichiometry GexðBNÞy. These monolayers belong to a family of 2D materials combining B,

N and group IV atoms, such as BxCyNzand SixByNz. We calculated the formation energy for

ten atomic arrangements, and found that it increases when the number of BeGe and NeGe bonds increases, and decreases when the number of BeN and GeeGe bonds increases. We found that the lowest energy monolayer presented a Ge2BN stoichiometry, and maximized

the number of BeN and GeeGe bonds. This structure also presented mixed sp2_{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 GexðBNÞymonolayers 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.

1. Introduction

Two-dimensional materials with a single layer of hybridized sp2_{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 be-tween 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], BC2N[11,12]and BC4N[13]. In general,

theoretical and experimental studies reveal that, unlike graphene and h-BN, BxCyNz 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, andfinite temperature molecular dynamics predict that silicene and germanene present a puckered hexagonal lattice with sp3 _{bonds, instead of a} _{flat configuration of sp}2 _bonds _[16_e21]_{. Nevertheless, these structures have already been}

* Corresponding author.

E-mail address:alilianeﬁsica@yahoo.com.br(A. Freitas).

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Superlattices and Microstructures

j o u r n a l h o me p a g e : w w w . e l s e v i e r . c o m/ l o ca t e / s u p e r l a t t i c e s

http://dx.doi.org/10.1016/j.spmi.2017.08.032

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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 106_m_,s1_[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 sp2 _{bonds and}

stoichiometry Si2BN, without out-of-plane buckling. Since Si atoms do not tend to form a 2Dﬂat hexagonal structure, the

prediction that Si2BN can exist as a stableﬂat 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 stoichiometryðBNÞxSi1x, 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 (seeFig. 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].

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2. 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 GexðBNÞymonolayers. We use a double-

z

basis set, composed of numerical atomic orbitals ofﬁnite

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 approxi-mation (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 104. Three-dimensional periodic boundary conditions were applied, resulting in a monolayer composed of repeating cells in the x-z plane (seeFig. 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 GexðBNÞy monolayers investigated in this work, with different stoichiometries ðx; yÞ. These

structures have unit cells containing n atoms (n¼ 24 or 32 atoms) and different numbers of BeN, GeeGe, BeGe, and NeGe bonds. As mentioned previously, the chosen atomic arrangements for the GexðBNÞymonolayers 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 Ge2BN, all arranged in a graphene-like hexagonal lattice, where all bonds are sp2 and all

angles are around 120+(seeFig. 1). Meanwhile,Fig. 1(g) shows the non-optimized and optimized structures for another monolayer with stoichiometry Ge2BN. The non-optimized structure consists of zigzag BN chains connected to zigzag GeeGe

chains which form a hexagonal lattice with sp2 _{bonds. However, our DFT optimization resulted in a structure with}

out-of-plane 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 sp2_{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 Ge2BN 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 Si2BN monolayers.

Additionally, we also investigated hexagonal structures with stoichiometries GeBN, GeB, and GeN (seeFig. 1(h)-(j)). The GeN monolayer exhibits buckling, mixing sp2_{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]. 3. Results and discussion

In order to analyze the stability of the GexðBNÞymonolayers, we have calculated the energy required to form a given

structure (EForm), 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

EForm¼ ðETot nB

m

B nN

m

N nGe

m

GeÞ=nT; (1)

where ETotis the calculated total energy provided by the SIESTA code, niis the number of atoms for each element (i¼ B, N, Ge),

m

iis the corresponding chemical potential for each element, and nTis the total number of atoms in the structure. The chemical

potentials should depend on the atomic reservoirs used in the synthesis reaction. Using gaseous N2for the N-rich reservoir,

we obtained

m

_N¼ 270.51 eV. For the B-rich reservoir, we used

a

b

boron, and found

m

_B¼ 77.22 eV[38]. The chemical potentials

m

B,

m

N and

m

Geshould fulﬁll the constraints of thermodynamic equilibrium:

m

BN¼

m

Bþ

m

N and

m

GeGe¼

m

Geþ

m

Ge; (2)

where the constants

m

_BN and

m

_GeGeare 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

BN¼ 350.75 eV and

m

GeGe¼ 213.84 eV. Finally,

for the Ge2BN and GeBN structures, with equal number of BN and GeGe pairs, the formation energies have been calculated

using the following expressions: EGe2BN

Form ¼ ðETot 8ð

m

BNþ

m

GeGeÞÞ=nT; (3)

E_FormGeBN¼ ðETot 4ð2

m

BNþ

m

GeGeÞÞ=nT; (4)

and for the structures with stoichiometries GeB and GeN, we have:

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Table 1

Formation energy (EForm), number of bondsni(i¼ BeN, GeeGe, BeGe and NeGe) and energy band gap (Eg), for allGexðBNÞymonolayers. The underlined number indicates the most stable structure.

Monolayer (seeFig. 1) EfromðeV=nTÞ nBN nGeGe nBGe nNGe EgðeVÞ

Ge2BN-(a) 0.93 8 4 14 14 0.00 Ge2BN-(b) 0.88 8 4 14 14 0.00 Ge2BN-(c) 0.96 8 4 14 14 0.00 Ge2BN-(d) 1.13 0 0 18 22 0.00 Ge2BN-(e) 1.20 0 0 22 18 0.25 Ge2BN-(f) 1.10 0 0 22 18 0.74 Ge2BN-(g) 0.36 14 18 4 8 0.00 GeBN 0.92 11 0 8 10 0.00 GeB 0.69 0 0 40 0 0.00 GeN 0.44 0 0 0 40 0.72

Fig. 2. Snapshots of the GexðBNÞymonolayers after thermalization at T¼ 300 K. We employed ab initio molecular dynamics to evolve each layer for 20 ps. The providedDE values represent the energy difference between thermalized and optimized structures.

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E_FormGeN ¼ ðETot 8ð2

m

Nþ

m

GeGeÞÞ=nT: (6)

The values for the formation energy and the number of BeN, GeeGe, BeGe and NeGe bonds of GexðBNÞymonolayers, are

reported inTable 1.

From this data one can see that the Ge2BN-(g) monolayer, with the maximum number of GeeGe and BeN bonds, presents

the lowest value of EForm, so that it is the most stable structure. This result is in agreement with those found for stable

monolayers with stoichiometry BxCyNzreported 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/nT). The reason for the

stability of the Ge2BN-(g) and GeN structures is the mixture of sp2and sp3bonds. It is well known that sp3bonds exhibit a

stronger character, when compared to sp2_{bonds, which provides more stability to the lattice. The most unstable structures}

are the Ge2BN from (a) to (f), GeBN, and GeB monolayers, which have higher values of EForm (at least 0.33 eV/nT above

Ge2BN-(g)). The common characteristic of these structures is that they exhibit a hexagonal lattice formed only by sp2bonds,

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 Mo-lecular 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 Nose 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. InFig. 2, we present snapshots of the structures proposed in this work after the completion of the thermalization process. The provided

D

E values represent the energy difference between thermalized and optimized structures. Inspection of the snapshots reveals that bond breakage and formation was only observed for Ge2BN-(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, Ge2BN-(g). Our AIMD simulations

indicate that this conﬁguration is indeed very stable.

With the exception of the Ge2BN-(g) monolayer, the length of the BeN and GeeGe bonds is around 1.44 and 2.40 Å,

respectively. For the Ge2BN-(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 inter-mediate 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 Ge2BN-(b) and Ge2BN-(c). Their detailed structural information is presented inFig. 3.

For the x-direction, the same single condition needs to be satisﬁed for both structures:

Fig. 3. Detailed structural information for Ge2BN-(b) and Ge2BN-(c). Notice Ge2BN-(b) is rather assymetric. Periodicity imposes more constraints on structure Ge2BN-(c), leading to higher symmetry but also to higher formation energy.

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a_{¼ 2d}BNþ 2dGeGeþ 2dBGecosð

b

=2Þ þ 2dNGecosð

a

=2Þ. For the z-direction, however, periodicity imposes: (i) one

constraint to monolayer Ge2BN-(b); (ii) two constraints to monolayer Ge2BN-(c). Namely,

b¼ 4dNGesinð

a

=2Þ þ 4dBGesinð

b

=2Þ for Ge2BN-(b), and b¼ 8dBGesinð

b

=2Þ ¼ 8dNGesinð

a

=2Þ for Ge2BN-(c). Because of this,

the lengths of dNGeand dBGeare uncorrelated in monolayer Ge2BN-(b), and bond distances can be rather different. On the

other hand, the lengths of dNGeand dBGeare correlated for monolayer Ge2BN-(c): dBGesinð

b

=2Þ ¼ dNGesinð

a

=2Þ. Although

this results in a more symmetric appearance for structure (c), it also leads to a higher formation energy.

Fig. 4shows the band structures calculated for the ten GexðBNÞymonolayers investigated in this work. The values of the

energy band gap Egare shown in the last column ofTable 1- we found Egvalues ranging from 0.0 to 0.74 eV. The Ge2BN (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 Ge2BN

(e)-(f) and GeN structures behave as semiconductors. For such cases, the presence of mixed sp2 _{and sp}3 _{orbitals contribute}

signiﬁcantly to the increase in the electronic band gap.

By means of projected density of states (PDOS), shown inFig. 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 GexðBNÞymonolayer. Complementary calculations of the localized density of states (LDOS) (seeFig. 6), reveal that the bottom

of the conduction band and the top of the valence band are associated with the pzorbitals 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 GexðBNÞy monolayers investigated, which

means that the valence band is completelyﬁlled 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 do-mains are very stable structures with peculiar electronic properties. Since the most stable GexðBNÞymonolayer 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

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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 GexðBNÞy monolayers studied in this work. We intend to study these

structures in the future. In the present study, DFT calculations for GexðBNÞymonolayers 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 GexðBNÞymonolayer (the Ge2BN-(g) structure inFig. 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.

4. Conclusions

Usingﬁrst principles calculations, we have investigated the stability and electronic properties of a new class of monolayers with stoichiometry Gex(BN)y. We have found that the most stable Gex(BN)ymonolayer exhibits stoichiometry Ge2BN, while

Fig. 5. Calculated projected density of states (PDOS) of selected GexðBNÞymonolayers. The Fermi energy Ef is indicated by the solid vertical line.

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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 BxNyCz, where their most stable structures also maximize the number of BeN

bonds. Another interesting feature of the most stable Gex(BN)ymonolayer is the mixture of sp2and sp3bonds in an exotic

lattice of squares and heptagons, which presents out-of-plane buckling. Thus, future efforts to synthesize the Gex(BN)y

monolayers must take into account their non-hexagonal morphologies, in contrast to the hexagonal BxNyCz and Si2BN

monolayers. Molecular dynamics simulations, performed at room temperature, revealed structural rearrangement only for the Ge2BN-(c) monolayer. No bond breakage was observed after thermalization for the other structures. In particular, the

monolayer with lowest formation energy (Ge2BN-(g)) underwent minimal deformation, a result which indicates high

sta-bility. 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 (Ge2BN-(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.

Acknowledgments

We would like to thank the Brazilian Research Agencies CNPq, PNPD/CAPES and INCT of Space Studies forﬁnancial support. We are also grateful to the CLIMA cluster - UFRN and the Laboratory of Computational Physics - UFPB, where the numerical calculations were performed.

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