Le Bel Array AlB6 with High Stability

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Two-dimensional anti-van’t Hoff/Le Bel array AlB6 with high stability, unique motif, triple Dirac cones and superconductivity Bingyi Song, Yuan Zhou, Hui-Min Yang, Ji-Hai Liao, Li-Ming Yang, Xiao-Bao Yang, and Eric Ganz J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.8b13075 • Publication Date (Web): 29 Jan 2019 Downloaded from http://pubs.acs.org on January 29, 2019

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Journal of the American Chemical Society

Two-dimensional anti-van’t Hoff/Le Bel array AlB6 with high stability, unique motif, triple Dirac cones and superconductivity Bingyi Song,1 Yuan Zhou,§1 Hui-Min Yang,§1 Ji-Hai Liao,2 Li-Ming Yang,*1 Xiao-Bao Yang,2 and Eric Ganz3

1School

of Chemistry and Chemical Engineering, Huazhong University of Science and Technology, Wuhan 430074,

China. 2Department of Physics, South China University of Technology, Guangzhou 510640, China. 3School of Physics and Astronomy, University of Minnesota, 116 Church St. SE, Minneapolis, Minnesota 55455, USA. (email: [email protected], [email protected]) §these authors contribute equally to the work and are co-second authors

Abstract: We report the discovery of a rule breaking two-dimensional aluminum boride (AlB6−ptAl−array) nanosheet with a planar tetracoordinate aluminum (ptAl) array in a tetragonal lattice by comprehensive crystal structure search, first principle calculations and molecular dynamics simulations. It is a brand new 2D material with a unique motif, high stability and exotic properties. These anti-van’t Hoff/Le Bel ptAl−arrays are arranged in a highly ordered way and connected by two sheets of boron rhomboidal strips above and below the array. The regular alignment and strong bonding between the constituents of this material lead to very strong mechanical strength (in plane Young modulus Yx=379, Yy=437 N/m, larger than that of graphene Y=340 N/m) and high thermal stability (the framework survived simulated annealing at 2080K for 10 ps). Additionally, electronic structure calculations indicate that it is a rare new topological phase material with triple Dirac cones, Dirac-like Fermions, and node-loop features. Remarkably, this material is predicted to be a 2D phonon-mediated superconductor with Tc=4.7K, higher than the boiling point of liquid helium (4.2K). Surprisingly, the Tc can be greatly enhanced up to 30K by applying tensile strain at 12%. This is much higher than the temperature of liquid hydrogen (20.3K). These outstanding properties may pave the way for potential applications of AlB6−ptAl−array in nanoelectronics and nanomechanics. This work opens up a new branch of two dimensional aluminum boride materials for exploration. The present study also opens a field of two-dimensional arrays of anti-van’t Hoff/Le Bel motifs for study.

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Introduction The design of novel two dimensional (2D) materials with unique atomistic configurations and exotic properties is always highly desirable for material science innovation and potential technology application.1 Due to the universal polymorphism, structural diversity and fascinating properties, the 2D forms of boron (i.e., borophene)2-16 are an ideal platform for the design of new functional materials. As a counterpart of graphene,17 honeycomb borophene is a fascinating target for chemists to pursue. However, unlike graphene, free standing honeycomb borophene is unstable due to electronic deficiency. Thus, it has been very difficult to experimentally fabricate honeycomb borophene. Recently, however, honeycomb borophene has been experimentally fabricated on an Al(111) substrate after substantial efforts.18 The substantial charge transfer and strong interaction between borophene and Al(111) stabilize the honeycomb configuration. This indicates that the aluminum is playing an important role in stabilizing the structure and modulating the properties of the borophene. The successful fabrication of honeycomb borophene on an aluminum substrate reveals that aluminum is a unique partner for boron in the formation of novel 2D materials. This suggested to us that these materials together might form interesting 2D aluminum borides with unique topology and exotic properties. Inspired by this, we carried out an in-depth study of 2D AlB6 alloys using first principles calculations together with a comprehensive crystal structure search. 2D aluminum boron alloys are extremely rare in the literature. In this work, on the basis of a comprehensive crystal structure search, first principles calculations, and molecular dynamics simulations, we predict a new and unusually stable 2D aluminum boride nanosheet (AlB6−ptAl−array). This material has a unique highly ordered antivan’t Hoff/Le Bel19 planar tetracoordinate aluminum (ptAl) 2D array. The structure consists of one layer of aluminum atoms which are spaced far apart and stabilized together with two slightly buckled parallel boron sheets. The boron sheets consist of chains of hexagonal holes and strips of boron rhombuses. The resultant architecture is highly stable since the component atoms are strongly bound. The AlB6−ptAl−array is the global minimum in 2D space. The framework of the material survived 10 ps of simulated annealing up to 2080K in brief molecular dynamics simulations. Unexpectedly, electronic structure calculations indicate that this layer is a novel Dirac material with triple Dirac cones and node-loop features, which is extremely rare. ACS Paragon Plus Environment

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Surprisingly, the AlB6−ptAl−array is predicted to be a phonon-mediated superconductor with Tc=4.7K. Remarkably, the transition temperature can be increased to 30K by applying a tensile strain of 12%. This strain could potentially be produced experimentally by growing the array on a suitable substrate. The strong cohesive energy and stability of this material are promising for experimental fabrication.

2. Computational methodologies A number of different computational codes were used for the calculations. The comprehensive crystal structure search for the two dimensional AlB6 nanosheets was carried out using the evolutionary algorithm USPEX code20 (with the local minimization schemes within unit cells of 7, 14, 21, and 28 atoms). In these calculations, initial structures are randomly produced using planar group symmetry. All newly produced structures are relaxed and relaxed energies area used for selecting structures as parents for the new generation of structures (produced by carefully designed variation operators, such as heredity and soft mutation). To identify the ground state, the population size was set to be 60 and the maximum number of generations was maintained at 60. With 60% of the lowest-enthalpy structures allowed to produce the next generation through heredity (60%), lattice mutation (30%), and atomic permutation (10%). In addition, two lowest-enthalpy structures were allowed to survive into the next generation. Local structure relaxations were performed within the framework of density functional theory (DFT) using the Vienna Ab initio simulation package (VASP).21 The projector−augmented−wave (PAW)22 method was used to describe the ion–electron interactions and the Perdew−Burke−Ernzerhof (PBE), and generalized gradient approximation (GGA)23 were chosen for the exchange correlation functional to treat the interactions between electrons. A plane wave cutoff of 500 eV was used for the kinetic energy. A uniform Г−centered Monkhorst−Pack24 k−points grid with a resolution of 2π × 0.025 Å–1 was used for the Brillouin zone integrations. The van der Waals (vdW) interactions were corrected via Grimme’s semiempirical corrections (DFT−D3).25 The structural model for the 2D AlB6 film is periodic in the xy plane, and separated by at least 15Å in the z direction to avoid interactions between adjacent layers. The energy convergence threshold of 10−6 eV was chosen for the structural relaxation and geometry optimization calculations. All of the atoms in the unit cell were fully relaxed without



any symmetry constraints until the force on each atom was less than 0.01 eV/Å. The phonon dispersion analysis was carried out for the low-lying allotropes using the Phonopy code26 with the finite displacement method, interfaced with density functional perturbation theory implemented in VASP. To obtain reliable results of phonon dispersion, more accurate DFT calculations were performed for further geometry optimization accuracy. The more stringent energy convergence criterion was set to 10−8 eV for the total energy and 10−4 eV/Å for the force convergence during the phonon calculations. Ab initio molecular dynamics (AIMD) simulations with the canonical ensemble were performed to evaluate the stability of the low-lying structures at a series of different temperatures. To test the thermal stability of these materials, they were subjected to simulated annealing at a series of temperatures up to 2200K. A 4 × 4 × 1 supercell was adopted during the molecular dynamics simulations. The AIMD simulations in the constant number, volume, temperature (NVT) ensemble lasted for 10 ps with a time step of 1 fs. The temperature was controlled using the Nosé−Hoover thermostat method.27 Using the optimized structures of the stable 2D AlB6 nanosheets from the above calculations (structural search, phonon analysis, AIMD), the band structure and density of states were calculated to characterize the electronic properties of these materials. Both the global minimum structure, as well as several low-lying allotropes were evaluated. These calculations used a uniform Г−centered Monkhorst−Pack k−points grid with a resolution of 2π × 0.015 Å–1. Furthermore, to increase the reliability of the electronic structure calculations, the screened hybrid Heyd−Scuseria−Ernzerhof functional (HSE06)28 is used. The Quantum Espresso (QE) package29 was used to compute the phonon-mediated superconducting properties of these materials and to estimate the superconducting critical temperature Tc. The VASP optimized structure was re-optimized within QE. For this calculation, we adopted the LDA functional and ultrasoft pseudopotentials to model the electron-ion interactions. After full convergence, the kinetic energy cut-off was chosen to be 90 Ry, and the charge density cut-off of the plane wave basis was chosen to be 900 Ry. The Methfessel-Paxton smearing technique of width 0.02 Ry was used. The self-consistent electron density was evaluated on a 16 × 16 × 1 k−point. The dynamical matrices and EPC matrix elements were calculated on a 4 × 4 × 1 q−point, resulting in a well-converged superconducting transition temperature. In the calculations of Tc based on the McMillan-Allen-Dynes formula, and


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estimated retarded Coulomb pseudopotential μ* of 0.05 was used to estimate Tc. The elastic constants were calculated to evaluate the mechanical properties of the AlB6 nanosheets. Furthermore, the in-plane Young’s modulus and the Poisson’s ratio were calculated based on the elastic modulus tensors. The evolution of the in-plane Young’s modulus and Poisson’s ratio in different directions were also calculated. Electron localization function (ELF)30,31 analyses were performed to get insight into the bonding interactions. Bader charge32 analysis was performed using the code developed by Henkelman’s group. The bond overlap population (BOP) values were calculated with on the fly pseudopotentials as implemented in the CASTEP code33 as a part of Materials Studio 2016.

3. Results and discussion 3.1 Structure information of AlB6 The crystal structures of two dimensional AlB6 nanosheets are very complicated, and many diverse motifs have been identified during the process of the crystal structure search. To simplify the discussion, we label each structure as AlB6-i (i=1, 2, 3, ...), AlB6-1 is the global minimum structure, based on its structural features, it also called the AlB6−ptAl−array. AlB6-2 is the second low-lying allotrope, and so on. To achieve a fundamental understanding of the structures of 2D AlB6 nanostructures, we will discuss them in two parts. The low-lying allotropes of 2D AlB6 will be discussed in the SI.

3.1.1 Crystal structure search A comprehensive global minimum structural search was performed using the USPEX code to look for various 2D configurations of AlB6 thin films. USPEX generated 15,000 initial random structures. Then VASP was used to perform geometry optimizations. After several iterations of optimization and generation of new structures using the evolutionary algorithm, final structures were generated. These structures were sorted by relative energy. The distribution of energies are shown in Fig.1a. We can see the number of different structures are essentially distributed as a Gaussian normal distribution with a peak at −5.7 eV/atom (Fig. 1b).



(a)

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(b)

Fig.1. The distribution (ratio and number) of different structures of two dimensional AlB6 nanosheets. (a) Pie diagram of different structures, the left side of the legend is the relative energy, the right side is the percentage of total structures, and (b) when the energy is increasing with 0.1 eV/atom interval, the evolution of the number of different configurations.

3.1.2 Structure evolution Different from the previous studies, which only show the global minimum or a few lowlying isomers, here, we display the evolution of different crystal structures of 2D AlB6 nanosheets. This is quite crucial for us to understand how the motifs changes, and further get insight into the stability of different configuration when the atomic positions and the shape of unit cell change. In Fig. 2, we show the range of possible structures within 1 eV energy. In Fig.2, we can see that the AlB6−ptAl−array is the most stable configuration from a thermodynamic viewpoint. The least stable isomer is the atomically thick monolayer where all boron atoms have pentacoordinate bonding and all aluminum atoms have hexacoordinate configurations. The buckled sheets with some holes are intermediate in energy. In general, the 2D structures of the AlB6 nanosheets become more stable with increasing thickness since they are approaching the bulk state and get additional stabilization energies. The global minimum is an array of planar tetracoordinate Aluminum (ptAl) structures embedded in two boron sheets, which is extremely rare and different from all previous reported 2D crystals.


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Fig. 2. The evolution of different structures of two dimensional AlB6 nanosheets within 1 eV energy of the ground state.

3.2 Stability of different isomers of AlB6 sheets After thoroughly exploring the 2D AlB6 configuration space, several well-ordered and stable AlB6 nanosheets were identified. Detailed analyses regarding the electronic structures, lattice dynamics properties, thermal stabilities, mechanical properties, and chemical bonding of these predicted 2D AlB6 nanosheets were performed. On the basis of the above discussion and analysis, we now focus on the global minimum structure and a few low-lying allotropes for further analysis. 3.2.1 Thermodynamic stability On the basis of extensive structural searches and analysis of the relative energies, we discovered that the global minimum AlB6−ptAl−array is a brand new structure. This structure is completely different from any known 2D structure with the same formula and stoichiometry, and has never been reported before. The character of the ground state AlB6−ptAl−array consists of three parts: the borophene composed of strips of Boron rhombuses, hexagonal hole arrays, and highly ordered arrays of planar tetracoordinate Aluminum (ptAl) motifs. This non-classical structural motif (planar tetracoordinate Carbon) was first proposed in carbon chemistry by Hoffmann et al.,34 and has triggered great interest in the community.35 Two borophene layers are connected by the ptAl layer and form a unique architecture. Each ptAl layer holds two borophene layers and keeps them from fusing together. Meanwhile, the ptAl array is anchored and stabilized



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by the two borophene sheets on either side. This strong interaction leads to high stability for the complete structure. The aluminum and boron atoms have strong bonds which form interlocking networks, this greatly stabilizes this material. The AlB6−ptAl−array is a highly symmetric structure with a tetragonal lattice with space group Pmmm (No.47). The lattice parameters of AlB6−ptAl−array are a = 2.91, b = 3.41Å. The unit cell contains one Al and six B atoms. The coordination environment is unusual. Planar tetracoordinate Aluminum (ptAl), quasiplanar tetracoordinate boron (ptB), and umbrella hexacoordinate boron (uhB) coexist in the AlB6−ptAl−array. Three symmetrically distinct atom types exist: one for Al, and two for B (labeled B1 and B2). The key Al−B1 distance is 2.262 Å, Al−B2 is 2.409 Å, and B1−B2 is 1.708 Å. There are two types of B2−B2 bonds: the bond with the side shared by hexagonal holes is 1.688 Å, the bond along the diagonal of the rhombus is 1.723 Å. The Al−Al distances are 3.411 and 2.911 Å for intralayer and interlayer of the ptAl array, respectively. This is too long to form chemical bonds between the Al atoms. Considering the additive covalent radii data (RXY = rX + rY),36 the single-bond radii for Al and B atoms are 1.26 Å and 0.85Å, respectively. Therefore, the estimated reference distance for Al−Al is 2.52 Å, for Al−B is 2.11 Å, and for B−B is 1.70 Å. Thus, each Al atom is isolated, but connected through B atoms in the borophene to form an extended 2D network. To further evaluate the stability of these layers, we calculated the cohesive energy Ecoh and formation energy Ef using the following formulas: Ecoh 

EAl  6 EB  EAlB6 7

,

Ef 

EAlB6  Al  6 B 7

where EAlB6 is the total energy of global minimum AlB6−ptAl−array, EAl is the energy of an isolated Al atom, and EB is the energy of an isolated B. The cohesive energy of AlB6−ptAl−array is 6.101 eV/atom. For the formation energy, μAl is the chemical potential of bulk Al, μB is the chemical potential of borophene. Due to the polymorphism, several types of borophene have been chosen as references including three that have been experimentally fabricated (triangular, β12, χ3) and one theoretical (α-boron). The formation energy of AlB6−ptAl−array is –0.20, –0.16, –0.15, and –0.12 eV/atom using triangular, β12, χ3, and α-boron respectively. This means that all four processes are exothermic, independent of the borophene reference selected. These formation energies predict very high stability and experimental


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accessibility of AlB6−ptAl−array. Usually, the absolute magnitude of the cohesive energy is much larger than the formation energy. Compared with the previously computationally predicted37 and later experimentally fabricated38 binary 2D crystal Cu2Si (Ecoh = 3.46, Ef = – 0.679 eV/atom), MoS2 (Ecoh = 5.322, Ef = –0.805 eV/atom), Ni2Si (Ecoh=4.80 eV/atom),39 and Cu2Ge (Ecoh=3.17 eV/atom),40 we find that the cohesive energy is much higher than for the Cu2Si, Ni2Si, Cu2Ge, and MoS2 monolayers. This demonstrates the high stability of the proposed AlB6−ptAl−array. The formation energy is lower (between –0.12 and –0.20 eV/atom), but should still be large enough to enable fabrication. The atomic distribution of the AlB6−ptAl−array is the most stable configuration. This has the maximum interaction strength between the components, and therefore the largest stabilization energy.

Fig. 3. Top view, side view, and tilt view of the global minimum structure AlB6−ptAl−array (AlB6-1) are displayed in the left and right panels, respectively. Al pink, B beige. The space group is Pmmm, and the 7 atoms in the unit cell are indicated. In the top view, the red dotted line indicates the unit cell, the green dotted line indicates the B rhombus stripe. Symmetric nonequivalent atoms are labeled Al, B1 and B2 in blue. The planar tetracoordinate Al atom configuration is highlighted with a blue circle. The representative distances between two atoms are shown with black dotted lines and two-way arrows. The ptAl array can also be seen in the side view-I and tilt view-I.



Experimental fabrication of hypothetical materials predicted from computation is not just limited to the global minimum. Some low-lying or even relatively higher energetic allotropes may also be possible under suitable conditions. Prototypical examples are the borophene allotropes. Several allotropes (e.g., β12, χ3, triangular, and honeycomb) with relatively higher energy have been fabricated experimentally.10,11,18 This inspires us to explore and briefly discuss some low-lying allotropes (Table S1) of the AlB6 nanosheets that have relatively high energies. These results are listed in the SI. In general, the thicker allotropes of the AlB6 sheets are more stable than the thinner ones. The stabilities of our predicted AlB6 nanosheets can be explained from the nature of their chemical bonding, which will be analyzed in detail below. In the following sections, we mainly discuss the most stable structure AlB6−ptAl−array, but also consider other low-lying structures.

3.2.2 Lattice dynamic stability The phonon dispersion spectrum along the high-symmetry lines in the first Brillouin zone was evaluated and used to characterize the lattice dynamic stability of AlB6−ptAl−array. From Fig. 4, we can see that there are no imaginary modes in the entire Brillouin zone. This confirms the lattice dynamic stability of the proposed ptAl−array. Furthermore, the highest phonon frequency is 1150 cm−1, which is much higher than that of MoS2 (473 cm−1),41 but smaller than that of graphene (ca. 1600 cm−1).42 This high frequency indicates strong bonding interactions between the component atoms. This strong bonding contributes to the high stability of the AlB6−ptAl−array. Regarding the contribution of each type of atom to the Phonon density of state (PhDOS), we display the PhDOS in Fig. 4. From Fig. 4 we can see that the contribution from aluminum atoms is mainly below 500 cm−1, there is almost no contribution above 500 cm−1. From the PhDOS, we can see that the major contribution from boron atoms is under 500 cm−1, the PhDOS above 500 cm−1 is almost exclusively contributed by the boron atoms. This is understandable because the atomic mass of boron is lighter than that of aluminum. The lower frequency region is dominated by the heavier atoms. The higher frequency region is dominated by the lighter atoms.


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Fig. 4. Phonon dispersion spectrum (left) and phonon density of states (right) of the global minimum AlB6−ptAl−array. Γ (0, 0, 0), Y (0.0, 0.5, 0.0), S (0.5, 0.5, 0.0), and X (0.5, 0.0, 0.0) refer to high symmetry points in the first Brillouin zone in reciprocal space.

3.2.3 Thermal stability For practical application of 2D AlB6 crystals in nanoelectronic devices or electrode films, it is very important to know the thermal stability at elevated temperatures, such as room temperature or even higher. In order to evaluate the thermal stability of ultrathin sheets of AlB6, we performed 10 ps long AIMD simulations at a series of different temperatures (500, 1000, 1500, 2000, 2050, 2080, 2100, 2200 K). To minimize the effects of periodic boundary conditions and to explore possible structural reconstruction, a 4 × 4 × 1 supercell was used for the simulations without any symmetry constraints. The snapshots after 10 ps of simulated annealing at each temperature are shown in Fig. 5. We find that the structures are able to maintain the planar tetracoordinate motif, the ptAl−array, the framework and structural integrity up to a rather high temperature of 2080 K for 10 ps. At 2100 K, the ptAl motif breaks down, the framework of the array is melting, holes appear, and the framework is distorted to a large extent. At 2200 K, the distortion is larger and the holes are larger. The high thermal stability may pave the way for



fabrication and application of AlB6 as a component of devices in nanoelectronics and electrode materials under ambient condition and higher temperature situation.

Fig. 5. Snapshots of the final frame of each molecular dynamics simulation from 500 to 2200 K (top and side views) after 10 ps of simulated annealing. Bonds to atoms outside this 4 × 4 section exist but are not shown.

3.3 Electronic structure, Triple Dirac cones and Superconductivity To evaluate the electronic performance of the AlB6 nanosheets, spin polarized DFT calculations were used to investigate the band structure (Fermi surface and high symmetry lines in the first Brillouin zone are shown in Fig.6d) and the corresponding total and partial density of states (TDOS and PDOS). From Fig. 6, we find that these materials are nonmagnetic and metallic with substantial density of states at the Fermi level. Electronic structure calculations indicate that other low-lying allotropes of AlB6 nanosheets are also nonmagnetic and metallic (see Table S1). From Fig. 6b, we can see that the contribution to the DOS from B atoms is much larger than that from Al atoms. The p-states of the B atoms dominate the TDOS (with B2 contribution larger than B1). From Fig. 6c, we can see that pz > py ≈ px for B2, and pz ≈ py > px for B1. The dominant contribution to the electronic states at the Fermi level comes from pz states from B1 and B2, with negligible contribution from the Al atoms. From the PDOS, we can see that the electronic states


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at the Fermi level are dominated by contributions from the pz bands of B, followed by some contributions from px and py bands of B.

Fig. 6. Electronic structure of the AlB6−ptAl−array. (a) band structure, (b) total density of states (TDOS) and projected s and p density of states (PDOS), (c) projected px, py, pz density of states (PDOS), (d) Fermi surface and high symmetry lines in the first Brillouin zone, (e) three dimensional band structure showing Dirac node loop. Dirac cones DC1, DC2, and DC3 are indicated in red in Fig. 6a. The Fermi level is set at 0 eV.

Some bands cross over the Fermi level. Interestingly, there are three Dirac cones which can be identified in the 2D band structure (Fig. 6a). The first Dirac cone (DC1) is formed by two linear bands crossing at the Fermi level located between Г and S, with the crossing point exactly



at the Fermi level, close to the high symmetry point S in the first Brillouin zone. The second Dirac cone (DC2) is located at the highly symmetric line S–X, close to the S point. DC2 is slightly above the Fermi level. The third Dirac cone (DC3) is located at the highly symmetric line Y–S, close to the S point. DC3 is slightly above the Fermi level. DC2 and DC3 are formed by four linear bands crossing just slightly above the Fermi level. In general, three Dirac cones are close to the S point. For further understanding the band dispersion and for a clear illustration of the Dirac cones, we plot the 3D band structure in Fig. 6e. We can see that this structure is not merely a Dirac cone material, but in fact there is a ring of Dirac nodes in the Brillouin zone. Three Dirac cones are connected together to form node-loop features in 3D. Three Dirac cones and node-loop features within the same material is extremely rare. To the best of our knowledge, this is the first example of a triple Dirac cone material featuring a node-loop with two dimensional ptAl array formed by all main-group elements. One of the most effective approaches to modulate the electronic properties is strain engineering (straintronics).43 In order to systematically tune the electronic behavior of the proposed materials, we applied both tensile and compressive biaxial strains ranging from –20% to +20% (but we found the stable interval for phonon dispersion is 0% ≤ ε ≤ 12%, see below for details). For each strain, we optimized the structure, and then performed electronic structure calculations. The band structures under different strains are plotted in Fig. S2 and S3. From there we can see that the three Dirac cones are maintained with just a slight deviation from the Fermi level. In order to cross check the PBE functional results and also to get accurate band structures of these materials, a more accurate hybrid functional HSE06, which includes screened exact exchange interaction, was employed to verify the metallicity of the AlB6 nanosheets. The results are plotted in Fig. S4. Excitingly, three Dirac cones maintain very well in HSE06 band structure even though there are some slight shifts above the Fermi level compared with that of PBE. The Dirac cones of the AlB6-ptAl-array at the HSE06 level is located just about 0.4 eV above the Fermi level. The partially occupied band-crossing features at the Fermi level remain, confirming the metallicity of the AlB6 nanosheet. After performing the HSE06 calculations for the AlB6 systems, we found that the metallic features of the AlB6 allotropes are also retained at the hybrid functional level (except that the HSE06 predicted energy peaks are slightly different from the


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PBE results). Therefore, metallic electronic states are intrinsic ground states for the AlB6 nanosheets. The Dirac cone-like electronic states generally produce excellent electronic transport properties. Near the Dirac points, the bands exhibit linear dispersion. We calculate the Fermi velocity (ʋF) of these materials using the expression F 

E E , where the is the slope of hk k

the linear valence band (VB) or conduction band (CB), and the ħ is the reduced Planck’s constant. The calculated ʋF values along the Г → S (DC1), X → S (DC2), and Y → S (DC3) directions were calculated using the DFT/PBE method. At the PBE level, the slopes of the bands in the kx direction are –5.0 and 6.5 eV Å. The corresponding Fermi velocities are 7.6 × 105 and 9.8 × 105 m/s along the Г → S pathway close to DC1. The slope of the bands in the ky direction are –5.0 and 6.2 eV Å. The corresponding Fermi velocities are 7.6 × 105 and 9.4 × 105 m/s along the X → S direction close to DC2. The slope of the bands in the kx direction are 5.7 and –7.4 eV Å. The corresponding Fermi velocities are 8.7 × 105 and 1.1 × 106 m/s along the Y → S direction close to DC3. All of these results are similar to or even higher than those for graphene. To confirm the reliability of the present calculation methods, they were applied to graphene as a test. The calculated Fermi velocity of graphene was 9.0 × 105 m/s along the Г → K direction, which is in good agreement with the previous result (8.22 × 105 m/s).44 According to the 1

 d2 E  k   definition of the effective mass of the charge carriers m  h   , the linear dispersion 2  dk  *

of the energy bands suggests close to zero effective mass for the carriers near the Fermi level. Therefore, the AlB6−ptAl−array is expected to have ultrahigh carrier mobility at the Dirac points. The high Fermi velocity and the massless carrier character observed in the AlB6 nanosheet could benefit future electronics. The recently discovered superconductor MgB2 with high Tc of 39K45 inspired us to explore the electron-phonon coupling (EPC) and potential superconductivity of 2D AlB6 nanosheets. The component elements Al and B are light elements, the metallic nature with substantial electronic states at the Fermi level, and the existing strong covalent bonding of AlB6 are similar to MgB2 despite completely different geometric structures.46 The Quantum Espresso (QE) package was used to calculate the phonon dispersion, and the electron-phonon coupling. The results are shown in Fig.7. The QE calculated phonon dispersion is used for two purposes. One is to cross check ACS Paragon Plus Environment


the phonon dispersion from QE in comparison with the previous VASP results discussed above. The other purpose is to calculate the EPC and potential superconductivity. From the comparison between the VASP phonon results in Fig.4 and QE phonon results in Fig.7a, we can see that the results from the two computational programs are essentially identical. This bolsters the reliability of our results. In the following, we discuss the EPC and superconductivity based on the phonon dispersion from the QE calculations. The Allan-Dynes modified McMillan’s approximation47 of the Eliashberg equation was used to estimate the superconducting temperature:

Tc 

log

 1.04(1   )  exp    * 1.2     (1  0.62 ) 

Where μ* is the Coulomb pseudopotential, λ is the overall electron-phonon coupling strength computed from the frequency-dependent Eliashberg spectral function α2F(ω), and ωlog is the logarithmic average phonon frequency. Because there is no exact method to determine the parameter μ* yet, we opted to compare the calculated Tc with the experimentally known MgB2 results to estimate a value for μ*. To estimate μ*, we calculated the Tc for MgB2. If μ*= 0.05, the Tc of 36.9 K for MgB2 with a λ of 0.71 is obtained, which is consistent with the experimental Tc of 40 K and λexp ~ 0.75.48 Therefore, we use μ*= 0.05 in our estimates of Tc for the 2D AlB6 nanosheets. The calculated total EPC parameter λ is 0.36, and the superconducting critical temperature Tc is estimated to be potentially as large as 4.7 K. This is higher than the boiling point of liquid helium (4.2 K) and comparable to Tc for intercalated graphite compounds. This suggests that the AlB6 nanosheets could be intrinsic BCS-type superconductors.


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Fig. 7. (a) Phonon dispersion with phonon line width, (b) phonon density of states (PhDOS), (c) Eliashberg spectral function α2F(ω) and the overall electron-phonon coupling strength λ of the AlB6 layer from Quantum Espresso calculations, (d) The evolution of the density of states at the Fermi level N(EF), the logarithmic average frequency ωlog versus the applied strain, and (e) Evolution of EPC and Tc versus the applied strain. ACS Paragon Plus Environment


The above discussion corresponds to a freestanding AlB6 layer. In reality, these layers might need to be fabricated on a substrate. Due to the lattice mismatch between the freestanding AlB6 layer and the substrate, this could impose compressive or tensile strain on the AlB6 layer. In fact, straintronics is widely used to modulate the electronic properties of 2D systems.43 Applying the strain could also potentially enhance the superconducting properties of these materials. Computationally, we imposed a series of strains on the AlB6 layers. We then optimized the structures and calculate the phonon dispersion at each point. We find that the AlB6−ptAl−array is stable under strains between 0% ≤ ε ≤ 12%. Either compressive or tensile strains beyond 12% will not stable. The phonon spectra of the AlB6−ptAl−array from −1% to +13% strain can be found in Fig. S1. From there, we can see that it displays negative frequencies at −1% and +13%. Between 0% and 12%, all positive frequencies were observed. We calculated the EPC and Eliashberg spectral function at 0 ~ 12%, the results are listed in Fig.7d and 7e. From Fig.7d and 7e, we can see that the density of states is increasing with the applied strain, the Eliashberg spectral function is decreasing with the applied strain, and the EPC and Tc are increasing with applied strain. Remarkably, the predicted superconducting Tc is predicted to be greatly enhanced to 30K. Thus, these structures may be potential future platforms for superconducting applications. The superconductivity predicted in the 2D AlB6 nanosheets could potentially have important applications in nanoscale superconducting devices, such as superconducting quantum interference devices and superconducting transistors.

3.4 Chemical bonding analysis We have found that the global minimum is the unusual AlB6−ptAl−array. It would be interesting to understand the bonding interactions between the components in detail. Why is this the most stable allotrope? What kinds of bonding interactions stabilize such configuration? From the PDOS of AlB6−ptAl−array in Fig.6b and 6c, we can see that both s− and p−states of Al and B contribute to both valence band (VB) and conduction band (CB). These electronic states are distributed energetically in the same range and they are spatially adjacent to each other (see Fig.3), thus the valence band states can effectively overlap and form strong covalent bonds between B atoms. Due to substantial charge transfer between Al and B atoms, dominant ionic


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bonds formed between Al and B atoms. After the charge transfer from Al to B, there is almost no charge located on the Al atoms. Furthermore, the large spatial separation leads to essentially no overlap of the electronic states between the Al atoms. Thus, there is almost no bonding interactions between the Al atoms. To get more information about the bonding, we performed Bader charge analysis on AlB6−ptAl−array. The Bader charge values for Al, B1 and B2 are +2.1, –0.91 and –0.07|e|, respectively. This indicated significant charge transfer occurs from Al to B in AlB6−ptAl−array. The values of bond overlap population (BOP) provide further information on the nature of the chemical bonding and the relative amount of covalency or ionicity. The larger BOP value, the larger covalency and the less ionicity. The B–B bonds (B1–B2 and B2–B2) have large BOP values (1.0 ~ 1.33) indicate a strong covalent bonding interaction. The longer distance, the smaller overlap and BOP value, the less covalency. The BOP values of Al–B1 (0.22) and Al–B2 (–0.14) are quite small, indicating a dominant ionic interaction with a very small part of covalent interaction. To gain a deep insight into the unique bonding nature and stabilization mechanism in AlB6−ptAl−array, we plot the electron localization function (ELF) from different perspectives in Fig.8. The ELF ranges from 0 to 1. Regions close to 1 have strong covalent bonding electrons or lone-pair electrons, the region close to 0 is typical for very low electron density, and the regions with 0.5 are typical for homogenous electron gas. From Fig.8a’, we can see the electron delocalization across the boron sheet, indicating the delocalized multicenter bonding interaction within boron sheet. From Fig.8 we can see that the large ELF values between B atoms, smaller values between Al and B, and negligible values between Al atoms. This indicate the dominant covalent bonds between B atoms, a small part of covalency together with dominant ionicity between Al and B, and almost no interactions between Al atoms.

Fig. 8. The calculated ELF of AlB6−ptAl−array from different perspectives is plotted, (a) – (e) ACS Paragon Plus Environment


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show the plane along which the ELF is being displayed, (a’) – (e’) show the ELF results on each plane.

3.5 Mechanical properties The mechanical properties are the fundamental aspects for the 2D materials towards practical applications. For novel 2D materials, the mechanical properties can often be quite striking. The elastic constants are the basis for the evaluation of the mechanical properties. According to the Born criteria,49 a mechanically stable 2D structure should satisfy C11C22 – C122 > 0 and C66 > 0 (in the Voigt notation, note that in some literature, this is labeled as C44). For small external strains ε near the equilibrium position, the elastic energy Uε can be expressed as: 1 1 U   E  E0  C11ε x 2  C22 ε y 2  C12 ε x ε y  2C66 ε xy 2 2 2

Where Eε and E0 are the total energies of the strained and equilibrium structures, respectively. By fitting the energy curves associated with strains, the elastic constants of our new material AlB6−ptAl−array were derived to be C11=380, C22=438, C12=C21=23, and C66=160 N/m. These values satisfy the Born criteria and so should be mechanically stable. On the basis of the calculated elastic constants, the in-plane Young modulus (Y) of this material along the x and y directions can be calculated by Yx =

 C11C22 - C12C21  =379N/m C22

and Yy =

 C11C22 - C12C21  C11

=437N/m. These values are larger than that of graphene (340 N/m),50 suggesting that the AlB6−ptAl−array has remarkable mechanical properties. The marked difference of Young’s modulus in different directions indicates that this material is mechanically anisotropic. The Poisson’s ratio can be calculated by  x 

C21 C =0.053 and  y  12 =0.061. These numbers are C22 C11

much smaller than those for graphene (ʋ=0.173),51 showing outstanding mechanical properties. If this material is stretched in the x direction, it would only shrink slightly in the y direction (and vice versa). The polar diagrams of Y(θ) and ʋ(θ) along different directions can be found in Fig. S9, which shows the evolution of in-plane Young modulus and the Poisson’s ratio versus the directions.

Conclusions ACS Paragon Plus Environment

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By comprehensive crystal structure search, first-principle calculations and molecular dynamics simulations, we have predicted a series of previously unknown 2D AlB6 crystals. The global minimum AlB6−ptAl−array is a highly stable and exciting new structure featuring a rare planar tetracoordinate aluminum (ptAl) array. It survived molecular dynamics simulated annealing up to 2080K for 10 ps. The in-plane Young modulus of AlB6−ptAl−array is larger than that of graphene, indicating remarkable mechanical properties. This material is completely different from any known metal boride structure. Remarkably, the electronic structure displays triple Dirac cones and node-loop features, which is extremely rare. Furthermore, the triple Dirac cones are robust to applied strain from 0% to +12%. Surprisingly, this new material is predicted to be superconducting with a Tc of 4.7K. The Tc can be greatly enhanced up to 30K by applying tensile strain at 12%. These outstanding properties may pave the way for the AlB6−ptAl−array to be used in nanoelectronics. In addition to the global minimum structure, we also found some low-lying 2D AlB6 allotropes with diverse structural motifs (some featuring quasiplanar octacoordinate motif). The phonon dispersion analysis and molecular dynamics simulations indicate high stability. The electronic structure calculations demonstrate the metallicity or even superconductivity in these low-lying allotropes. The ground state AlB6−ptAl−array together with these low-lying allotropes have high feasibility to be synthesized experimentally. The present study could open a new branch of 2D aluminum boride layers with exotic structure and fascinating properties. We hope that these results will inspire the experimenters to synthesize these materials. Supporting Information The relative energies, atomistic structures (include space group, the number of atoms in the unit cell), phonon dispersion, molecular dynamics simulation results, electronic band structures, density of states and Electron Localization Function (ELF) plots of low-lying allotropes of 2D AlB6 nanosheets. Phonon dispersion spectra for AlB6−ptAl−array under elastic strain from −1% to +13%. Electronic band structures of AlB6−ptAl−array under tensile strain from −20% to +20%. Band structures of AlB6−ptAl−array calculated from HSE06. several characterization tools to reveal the bonding nature of AlB6−ptAl−array. phonon vibration modes of AlB6−ptAl−array at high symmetry points, Elastic constants (Cij, N/m), Layer modulus (γ, N/m), Young’s modules (Y, N/m), Poisson’s ratio (ʋ) of low-lying allotropes of 2D AlB6 nanosheets, Polar diagrams of



in-plane Young’s modules Y(θ) and Poisson’s ratio ʋ(θ) along an arbitrary in-plane direction θ (θ is the angle relative to the x direction) for low-lying allotropes of 2D AlB6 nanosheets.

Acknowledgements B.–Y. S, Y. Z, H.–M. Y, and L.–M. Y. gratefully acknowledge support from the National Natural Science Foundation of China (21673087, 21873032), startup fund (2006013118 and 3004013105) and independent innovation research fund (0118013090) from Huazhong University of Science and Technology. J. H. L. gratefully acknowledge support from the Guangdong Natural Science Funds for Doctoral Program (Grant No. 2017A030310086). The authors thank the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for supercomputing resources. This work is dedicated to Prof. Roald Hoffmann on the occasion of his 80th birthday.


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Table of Contents Graphic and Synopsis

We report the discovery of a rule breaking and brand new two-dimensional anti-van’t Hoff/Le Bel AlB6−ptAl−array with high stability, unique motif, triple Dirac cones and superconductivity


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Le Bel Array AlB6 with High Stability

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