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Predicting Novel 2D MB (M=Ti, Hf, V, Nb, Ta) Monolayers with Ultrafast Dirac Transport Channel and Electron-Orbital Controlled Negative Poisson’s Ratio Chunmei Zhang, Tianwei He, Sri Kasi Matta, Ting Liao, Liangzhi Kou, Zhongfang Chen, and Aijun Du J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00762 • Publication Date (Web): 03 May 2019 Downloaded from http://pubs.acs.org on May 3, 2019

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The Journal of Physical Chemistry Letters

Predicting Novel 2D MB2 (M=Ti, Hf, V, Nb, Ta) Monolayers with Ultrafast Dirac Transport Channel and Electronorbital Controlled Negative Poisson’s Ratio Chunmei Zhang1, Tianwei He1, Sri Kasi Matta1, Ting Liao1, Liangzhi Kou1, Zhongfang Chen2, Aijun Du1,* 1School

of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Gardens Point Campus, Brisbane, QLD 4001, Australia

2Department

of Chemistry, University of Puerto Rico, Rio Piedras Campus, San Juan, Puerto Rico 00931, USA

Abstract Three-dimensional diborides MB2, featuring in stacking the M layer above the middle of the honeycomb boron layer, has been extensively studied. However, little information of the twodimensional counterparts of MB2 is available. Here, by means of evolutionary algorithm and firstprinciples calculations, we extensively studied the monolayer MB2 crystal with M elements ranging from group IIA to IVA covering 34 candidates. Our computations screened out eight stable monolayers MB2 (M=Be, Mg, Fe, Ti, Hf, V, Nb, Ta), and they exhibit Dirac-like band structures. Dramatically, among them, groups IVB-VB transition-metal diboride MB2 (M=Ti, Hf, V, Nb, Ta) are predicted to be a new class of auxetic materials. They harbour in-plane negative Poisson’s ratio (NPR) arising mainly from the orbital hybridization between M d and Boron p orbitals, which is distinct from previously reported auxetic materials. The unusual NPR and the Dirac transport channel of these materials are applicable to nanoelectronics and nanomechanics.

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Materials with a negative Poisson’s ratio (NPR) known as auxetic, exhibit unusual mechanical behavior contradicting common sense: they expand under stretching, while contract upon compression. These materials possess some novel properties, such as enhanced toughness and superior sound or vibration absorption1. NPR is theoretically allowable, and mainly observed in engineered three-dimensional (3D) bulk structures2. In comparison, two-dimensional (2D) auxetic materials are rather rare. Generally, the known 2D auxetic materials can be divided into two categories (i) 2D auxetic material with in-plane NPR: pentagonal monolayers B2N4, B4N23, Pentagraphene4, Silicon Dioxide5, Be5C26, δ-phosphorene7, and graphene-related structure8-10; and (ii) 2D auxetic material with out-of-plane NPR: SnSe11, BP512, borophene13, borophane14, TiN15, ABP2X6 (A=Ag, Cu; B=Bi, In; X=S, Se)16, and black phosphrous17. These materials possess re-entrant or hinged geometric structures, which is believed to be the main reason of the auxetic behavior. However, depending on only such geometric features to design 2D auxetic materials may exclude some promising candidates, as Yu et al. recently demonstrated that some auxetic behaviour in certain compounds is determined by their distinct electronic structures18. Thus, it is highly demanded to take electronic properties into consideration when designing auxetic materials. The 3D transition-metal diborides, MB2 (M: transition metal atom), with the hexagonal AlB2 structure (space group: P6/mmm), is a representative of electron interaction systems with strong hybridization of the M d and B p states19-22. The physical properties of 3D MB2, such as high melting temperature, high stiffness, high electrical, thermal conductivity23, and positive Poisson’s ratio24, have been well studied. In contrast, for the 2D counterpart of these diborides, little information is currently available, let alone the mechanical properties. The elastic behaviour in 2D MB2 can be totally different from the bulk. Note that the relatively strong interlayer interaction in bulk MB2 25 could constrict the stretch along the z axis, indicating that the Poisson’s ratio might be sensitive to the number of layers. Thus, novel mechanical properties might be hidden in the 2D transition-metal diborides MB2. In addition, compared with the 3D metallic bulk, 2D MB2 systems may have novel electronic properties due to the quantum confinement effect. For example, theoretically it has been predicted that the monolayer TiB2 26, ZrB2 (but with negative frequencies in the phonon spectrum)27, 28, and FeB2 29 are stable Dirac materials with ultrahigh Fermi velocity. To our best knowledge, these are the only works on monolayer transition-metal diborides, thus monolayer MB2 has yet to be explored.


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Modern computational sciences are not satisfied with only simulating a single material. As exemplified by the materials project30, 31 and the Material Genome Initiative (MGI)32-35, scientists are exploring collective properties from all the elements in the periodic table to find potential functional materials for applications such as electronic and mechanical devices. Along this line, it is desirable to simulate 2D MB2 systems with other elements (including non-transition metal) in the periodic table since they may prove to be a versatile, and useful class of materials. In this Letter, we examined 34 MB2 monolayers with M elements ranging from group IIA to IVA to screen out stable materials for promising applications by means of particle swarm optimization (PSO) and density functional theory (DFT) computations. Our screening process identified eight stable MB2 monolayers (M=Be, Mg, Ti, Hf, V, Nb, Ta, Fe), among them four being reported previously (M=Be, Ti, Fe, Mg) and their electronic band structures are featured by Dirac cone and p-, n-type Dirac points, which endow these 2D materials with ultrafast transport channels comparable to that of graphene. Remarkably, out of the eight stable 2D monolayers, five MB2 compounds with M belonging to IVBVB groups (M= Ti, Hf, V, Nb, Ta) exhibit in-plane auxetic behavior distinct from other 2D auxetic materials, which could not be explained only by the geometric structures, but is determined by the strong coupling between M d orbital and B pz orbital. The unique auxetic behavior and Dirac transport channel in these new 2D MB2 monolayers would lead to novel multi-functionalities, such as nanoelectronics, nanoscale auxetic electrodes. Structural search for 2D MB2 were carried out by using the particle-swarm method as implemented in the CALYPSO code36,

37.

The structural relaxation and electronic structure calculations were

performed by DFT methods using the Vienna Ab Initio Simulation (VASP) package38-40. The generalized gradient approximation (GGA) in the Perdew−Burke−Ernzerhof (PBE)41 form for the exchange and correlation potential, together with the projector-augmented wave (PAW) method, were

adopted.

A

hybrid

functional

based

on

the

Heyd−Scuseria−Ernzerhof

(HSE)

exchange−correlation functional42 was adopted for accurately calculating band structure. A dispersion correction of total energy (DFT-D3 method)43 was used to incorporate the long-range vdW interaction. To study 2D systems under the periodic boundary conditions, a vacuum layer with a thickness of 15 Å was set to minimize artificial interactions between neighbouring layers. The plane wave energy cut-off was set to 500 eV. The structures were fully relaxed until energy and force were converged to 10-6 eV and 0.001eV/Å, respectively. The Brillouin zone integration was sampled by an 11 × 11 × 1 k-grid mesh for a honeycomb unit cell and 11 × 7 × 1 k-grid mesh for a rectangular unitcell. For mechanical property computations, the structure model is based on rectangular unit-cell 3 ACS Paragon Plus Environment


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which is periodic in the x−y plane. When the lattice is strained under a specific loading condition, the lattice constants in other directions are fully relaxed. The strain is defined as ϵ = (a − a0)/a0, while the strain along out-of-the-plane direction is defined as ϵz = (t − t0)/t0, where a0 and t0 are the lattice constant and buckling height of the freestanding film, respectively, and a, t are the corresponding values at the strained states. Ab initio molecular dynamics (AIMD) simulations with canonical ensemble were performed to evaluate the thermodynamic stability. The single layer MB2 consists of a layer of boron in a honeycomb lattice alternated by a layer of M atoms sitting above the centres of the honeycomb tiles (Figure 1a). We first narrowed our candidate pool by applying a stability criterion (without imaginary frequencies in phonon band structure) to thirty-four different types of MB2 monolayers with M being Be, Mg, Ca, Sr, Ba, Sc, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Mn, Tc, Re, Fe, Ru, Os, Co, Rh, Ir, Ni, Pd, Cu, Ag, Cu, Al, Ga, In, Ge, Sn, retaining the materials for which the phonon spectrum is stable with only real frequencies. From this set, eight stable monolayers MB2 (M=Be, Mg, Ti, Hf, V, Nb, Ta, Fe) Figure 1b-h are retained, and the lattice parameters were listed in Table S1. The particle swarm search method with 3 atoms in a hexagonal unit cell are used to search for the lowest-energy structures and further confirmed the stability of these materials (more details please see support information in Figure S1 and Table S2). These remove materials that maybe challenging to synthesize (Table S3). The vertical distance between the M and B layer is smaller than that in the bulk MB2, indicating a much stronger interaction between the M atoms and B layer within the single layer. Moreover, the AIMD simulations44 were further carried out for 10ps at 700K as shown in Figure S2. These show no evidence of structural destruction, suggesting a robust thermal stability for monolayer MB2 (M=Be, Mg, Ti, Hf, V, Nb, Ta, Fe). Among them, several stable monolayer diborides TiB226, MgB245, BeB246, and FeB229 have been proposed before.


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Figure 1. (a) Top and side view of MB2 lattice structure with the blue and green balls representing M and boron atoms, respectively. The honeycomb lattice unit-cell MB2 is marked by black dots. The rectangular grey shade displays the unit-cell adopted for our mechanical property calculation, which contains two MB2 formula units. The zigzag (a) and armchair (b) directions are chosen as the x and y axis, respectively. (b)-(h) the phonon band structure for stable monolayer MB2 (M= Ti, Hf, V, Nb, Ta, Be, Mg).

Having confirmed the stability of MB2 (M=Ti, Hf, V, Nb, Ta, Mg, Be, Fe) monolayer materials, we next study their electrical property. We observed Dirac-like bands along high symmetry path in the first Brillouin zone by adopting the honeycomb lattice unit cell (Figure 2). As we can see the band structure in Figure 2a-d, Dirac cones (red square) are visualized at the Fermi surface for TiB2, HfB2, BeB2, and FeB2 monolayers and Dirac points are observed above (green circle) or below (blue circle) the Fermi surface for MB2 (M=Ti, Hf, V, Nb, Ta, Mg, Be, Fe) monolayers behaving like n- and p-type Dirac fermions, respectively (Figure 2e-h). The Dirac cone or Dirac points of the 2D MB2 monolayers were further checked by HSE method47 (Figure S3). Around these Dirac points the bands (valence or conduction) exhibit a linear dispersion, indicating that charge carriers (electrons or holes) in these bands behave as small mass of Dirac fermions. Since the different kind of Dirac points are at different energies, it is possible to control the types of Dirac fermions in a certain material by tuning the doping level, probing each kind separately. Remarkably, the Fermi velocity for the Dirac cone and the Dirac points are quite high comparable to that of graphene (11x105m/s)48 (Table 1). Thus, monolayer MB2 (M=Ti, Hf, V, Nb, Ta, Mg, Be, Fe) are Dirac materials with ultrahigh electron mobility, which could be ideal nanoelectronics49.



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Compared with graphene, the boron layer is not stable because the boron hexagonal ring lacks two electrons50. For MB2 (M=Ti, Hf, Be, Fe) (Figure 2a-d) monolayers, the Dirac state arises from the M atoms donating two electrons to the boron layer. This creates the isoelectronic analogue of graphene. While for MB2 (M= V, Nb, Ta) monolayers, there are bands crossing the Fermi level due to the extra electrons. For MgB2 monolayer (Figure 2h), it can be a Dirac material if the Fermi surface could be shifted upward (~2eV) by doping holes.

Figure 2. Band structure of MB2 calculated by adopting honeycomb lattice unit-cell, where high-symmetry kpoints are shown in the inset of (a). Among them TiB2, HfB2, BeB2, and FeB2 have Dirac cone (red square) at the Fermi surface. MB2 (M=Ti, Hf, V, Nb, Ta, Be, Mg, Fe) have Dirac points above (green circle) or below (blue circle) the Fermi surface. The Fermi level is set at the energy zero point.

Table 1. The largest Fermi velocity for Dirac cone and the p- and n-type Dirac fermions of MB2 (M=Ti, Hf, V, Nb, Ta, Be, Mg, Fe) within the energy of -3eV to 3eV. The corresponding references are cited. Structures TiB2

HfB2 BeB2 FeB2

VB2

Carrier types p Dirac cone n p Dirac cone n Dirac cone p Dirac cone n p 6

Fermi velocity (x 105 m/s) 6.24 5.48；5.7 [ref 26] 5.58 3.64 86.2 7.60 13.0 3.74 6.53；6.54 [ref 29] 9.03 4.33

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n p n p n p n

NbB2 TaB2 MgB2

8.99 4.56 7.42 6.48 9.89 6.48 13.17

To probe the mechanical properties of stable MB2 (M=Ti, Hf, V, Nb, Ta, Mg, Be, Fe) monolayer materials, the elastic constants are calculated (Table S4) using rectangular unit-cell. We note that C12 is negative for this MB2, (M atoms belongs to group IVB and VB) nanosheet, leading to an NPR. 𝐶12

Take HfB2 for example, 𝑣12 = 𝐶22 = - 0.195 = 𝑣21. For a validation, we applied a uniaxial strain from 5% to +5% in x (Figure 3) and y (Figure S4) directions of MB2 monolayer, respectively. The buckling height and strain relationships along x and y directions are basically identical and the buckling height decreases with the increasing strain, while the in-plane direction presents NPR. Generally, the values of the Poisson’s ratio (ν) for covalent materials (ν ∼ 0.1) are smaller than the metallic materials (ν ∼0.33)51. In this case, the values of ν for group VB MB2 (M=V, Nb, Ta) is larger than group IVB MB2 (M=Ti, Hf) as a whole, indicating an increase of metal–metal bonding for the diborides when the dorbital metal goes from group IVB to group VB. This reflects the well-known situation52-54 that the metallicity of MB2 phases increases with the filling of d orbitals. Figure 3 further demonstrates that for group IVB MB2 (M=Ti, Hf) Figure 3a-b , the in-plane Poisson’s ratio is negative for the entire strain range (-5% to +5%), while for group VB MB2 (M=V, Nb, Ta) Figure 3c-e, the Poisson’s ratio is negative at partial strain range (green enclosure area). Remarkably, the NPR values for MB2 monolayer in both x- and y-directions are quite high compared with Penta graphene (ν = −0.068)4, borophene (νx = −0.04, νy = −0.02)13, and Be5C2 (νx = −0.04, νy = −0.16)6. Most interestingly, we find that MB2 (M=Fe, Be, Mg) with the same structure as that of group IVB and VB MB2 monolayers are ill-suited for this NPR behaviour (Figure 3f-h and Figure S4f-h).



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Figure 3. (a)-(h) Poisson’s ratios for MB2 as a function of strain applied along the a axis (-5%-5%). The axis x, y, and z correspond to the in-plane lateral and thickness directions, respectively.

Considering the fact that both auxetic and non-auxetic materials are found in the same crystal structured materials, it can be inferred that the geometric property is not the only reason responsible for the NPR. Thus, we further analysed the density of states (DOS) (Figure 4) to obtain deeper insight into the electronic properties of the MB2 monolayers (M=Ti, Hf, V, Nb, Ta, Be, Mg, Fe). The strong hybridization between B 2p and M d states should be taken into consideration. Due to the C6ν symmetry of MB2 monolayer and D6h symmetry of triangular M lattice, the M d orbitals split into e1 (𝑑𝑥𝑦,𝑑𝑥2 ― 𝑦2), e1* ( 𝑑𝑥𝑧, 𝑑𝑦𝑧 ), and a1 (𝑑𝑧2) in the triangular lattice. The absence of dstates in Mg and Be atoms makes the electronic structure of MgB2 and BeB2 (Figure 4a-b) fundamentally different from the transition metal diborides. Thus, we propose a hypothesis that the lack of hybridized bonds between the d-orbital of M atom and p-orbital of B atom account for the non-auxetic behaviour in BeB2 and MgB2 monolayers (Figure 4a-b). Comparing the projected density of state (PDOS) for MB2 (M=Ti, Hf, V, Nb, Ta) (Figure 4 c-g) and FeB2 (Figure 4h), it is seen that there is negligible overlap between a1 and 𝑝𝑧 orbital in FeB2, while for M-a1 (M belongs to group IVB and VB), the overlap is obvious and strong, as shown by their similar DOS peak shapes and positions in energy. This results in a fundamental difference in the bonding between the metal and boron layers as a1-𝑝𝑧 coupling force attracts atoms M and B towards each other. Thus, the d8 ACS Paragon Plus Environment

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electron relying on the dependence of a1–𝑝𝑧 interaction direction plays a main role in determining the structure deformation presented below. To check the robustness of our results, we also explored the Poisson’s ratio and the electronic properties for other group single layer transitionmetal diborides MB2, and found positive Poisson’s ratio (Figure S5) in combination with the negligible overlap between al-pz orbital (Figure S6) in CrB2, TcB2, NiB2. The auxetic behaviour is generally believed to originate at the structural level, so the theoretical search such as machine learning for prediction of new auxetic materials are mainly focused on reentrant or hinged geometric structures2. This would exclude some materials from consideration as materials with strong electron coupling can harbour NPR. Thus, we propose that the orbital hybridization might be regarded as an important descriptor in the promising machine learning55 process to guide new auxetic material design.

Figure 4. DOS of MB2. The M_d-B_p orbital coupling manifests itself in the overlap of their DOS. The DOSs shown in the figure are 𝑒1 = 𝑑𝑥𝑦 + 𝑑𝑥2 ― 𝑦2, 𝑒1 ∗ = 𝑑𝑥𝑧 + 𝑑𝑦𝑧 and 𝑎1 = 𝑑𝑧2. The Fermi level is set to 0.



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Investigations show that the bulk transition metal diborides MB2 present positive Poisson’s ratio24. To examine the possible quantum size effect, we also calculated the bi-layer and tri-layer Poisson’s ratio for MB2 with M in group IVB and VB, and found positive Poisson’s ratio for bi- and tri-layers of TiB2 and NbB2 (Figure S7). We attribute this to the spatial constrains in z axis in 3D material, that is to say, the buckling height of monolayer MB2 can enlarge or decrease at free along the buckling direction, while there is a restriction in the multi-layer or non-single layer MB2 due to the atoms potential repulsion. Moreover, a further negligible overlap between al-pz orbital (Figure S8) in bi- ,tri- layer TiB2 and NbB2 demonstrate that orbital coupling might still play an important role for Poisson’s ratio in bi-, tri-layer, and even bulk MB2. Therefore, we believe that the distinction between these two situations stems from the interlayer coupling of MB2 as well as the orbital coupling between M and B atoms. After analysing the different mechanical behaviour in 3D, multi-layer, and single-layer MB2 (M in group IVB and VB), we further analyse the geometry to know the reason for the NPR in monolayer MB2. Take TiB2 for an example, when compressive strain is applied along b direction (-5%), the buckling height increases significantly (1.32 Å) compared with strain-free buckling height (1.22 Å) (Figure 5a). This trend effectively wrinkles the pucker of single-layer MB2 sheets, which significantly enlarges the required strain energy. To accommodate the elongation in this direction, the strong orbital coupling between a1-𝑝𝑧 enables C atoms move towards D atom along a direction (Figure 5b). And the interaction of d orbital of M atom can attract them to each other, thus A atom can also move towards D atom along a axis (from 3.13 Å to 3.11 Å). This renders a compressive strain along a direction (corresponding to the in-plane NPR).


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Figure 5. The explanation of Poisson’s ratio for MB2 structures with the blue and green balls representing M and B atoms, respectively. Each M atom is surrounded by six boron atoms. When the compressive strain is applied to b (y) direction, the thickness t (z) of the MB2 enlarges (a) and a (x) axis becomes short (b).

Our first principles calculations combined with the particle swarm search approaches successfully screened out eight stable monolayer MB2 (M =Be, Mg, Fe, Ti, Hf, V, Nb, Ta) Dirac materials from thirty-four candidates. They exhibit Dirac cone with ultrafast transport channel. In addition, five MB2 (M =Ti, Hf, V, Nb, Ta) monolayers harbour orbital hybridization dominated auxetic behaviour. And we propose that the orbital hybridization could be regarded as an important descriptor in the promising machine learning process to guide new auxetic material design. The auxetic effect in the puckered structure exists in both x and y directions, and is sensitive to the number of layers due to the strong van der Waals interaction. These superior electronic properties, along with the NPR, make monolayer MB2 promising materials for the design of nano-electromechanical devices. In experiments, the preferred growth method of stable single-layer MB2 is epitaxial growth on a substrate. By employing hexagonal MoS2 as a representative substrate, we take BeB2, TiB2, VB2, and FeB2 for examples to verify the experimental feasibility and robust Dirac cone of MB2 monolayers (Figure S9). We hope our study would stimulate further experimental effort on this subject.

Corresponding Author *Email: [email protected]

ORCID Aijun Du: 0000-0002-3369-3283 Zhongfang Chen: 0000-0002-1445-9184 Ting Liao: 0000-0001-7488-6244 Liangzhi Kou: 0000-0002-3978-117X Sri Kasi Matta: 0000-0003-4465-640X

Notes The authors declare no competing financial interest.

Acknowledgements



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A.D acknowledges the financial support by Australian Research Council under Discovery Project (DP170103598) and computer resources provided by high-performance computer time from computing facility at the Queensland University of Technology, NCI National Facility, and The Pawsey Supercomputing Centre through the National Computational Merit Allocation Scheme supported by the Australian Government and the Government of Western Australia.

Supporting information Available: The optimized lattice constants and the buckling height for stable MB2 monolayers; Comparison between three different phases searched by CALYPSO code; The optimized lattice constants and buckling height for explored unstable MB2 monolayers; Ab initio MD simulations of the evolution of energy of MB2 monolayers under 700K; The elastic constants for MB2 monolayers; Poisson’s ratios for MB2 (M=Ti, Hf, V, Nb, Ta, Be, Mg, Fe) as a function of strain applied along the b axis; Poisson’s ratios for MB2 (M=Cr, Tc, Ni) as a function of strain applied along a axis and b axis; DOS of MB2; Poisson’s ratios and DOS for bi- and tri-layer MB2 (M=Ti, Nb) as a function of strain applied along a axis and b axis; Structures and band structures of MB2 monolayer grow on MoS2 substrate.

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