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Breakdown of the electron delocalization in hexagonal borophene toward tunable energy gap
Journal Pre-proofs Full Length Article Breakdown of the Electron Delocalization in Hexagonal Borophene Toward Tunable Energy Gap Jia Liu, Xianfei Chen, Yi Huang, Wentao Zhang, Pan Xiang, Beibei Xiao PII: DOI: Reference:
S0169-4332(19)33757-2 https://doi.org/10.1016/j.apsusc.2019.144940 APSUSC 144940
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Applied Surface Science
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Please cite this article as: J. Liu, X. Chen, Y. Huang, W. Zhang, P. Xiang, B. Xiao, Breakdown of the Electron Delocalization in Hexagonal Borophene Toward Tunable Energy Gap, Applied Surface Science (2019), doi: https://doi.org/10.1016/j.apsusc.2019.144940
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Breakdown of the Electron Delocalization in Hexagonal Borophene Toward Tunable Energy Gap Jia Liua, Xianfei Chena*, Yi, Huangb, c*, Wentao Zhanga, Pan Xianga, Beibei Xiaod a
College of Materials and Chemistry & Chemical Engineering, Chengdu University of Technology, Chengdu 610059, China b College of Environment and Ecology, Chengdu University of Technology, Chengdu 610059, China c State Environmental Protection Key Laboratory of Synergetic Control and Joint Remediation for Soil & Water Pollution, Chengdu University of Technology, Chengdu 610059, China dSchool of Energy and Power Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China Abstract: The absence of energy gap (Eg) in borophene polymorphs, where multi-center bonds favor electron delocalization and form metallic phases, restricts their potential applications in microelectronic and optoelectronic industry. Herein, we propose to incorporate chalcogenide atoms to breakdown the electron delocalization in hexagonal borophene and introduce an Eg. Based on first principle calculation, we report a group of semiconducting boron chalcogenide monolayers, denoted as B2X (X = O, Se and Te), with an indirect Eg of 2.84, 0.97 and 1.04 eV under HSE06 levels, respectively. Moreover, the introduced Eg can be modulated by external strains (± 6%) affordable by substrates, with electronic effective mass being comparable or even lower than those of MoS2 along certain directions. Furthermore, B2X monolayers exhibit desirable light absorption properties under visible light, which can be improved by stacking bilayers. The current study presents an alternative route to introduce the Eg in borophene, endows them with new vitality and highlights their promise in flexible electronics, optoelectronics and photovoltaic devices.
*
Corresponding author. Email: [email protected], [email protected]
1
Keywords: 2D materials, first principles, hexagonal borophene, bandgap modulation.
Introduction Two-dimensional (2D) boron monolayers, known as borophenes, have emerged as a new 2D platform with unique physical and chemical properties [1-4]. Boron (IIIA) sits next to carbon in the periodic table and possesses one less electron. Consequently, multi-center bonds, rather than the traditional two-center-two-electrons interactions (e.g., in graphene), dominate the interactions in borophene due to its electron-deficiency in octet rule. This endows more flexibility and complexity for B atoms to form 2D boron structure [5, 6]. In the past decade, several borophene polymorphs in atomic thickness, such as α-, β-, δ-sheets [5, 7, 8] and planer B36 cluster [9], have been predicted, which can be distinguished by diverse geometrical profiles and vacant concentration. In these allotropes, pristine honeycomb centers are always partially occupied by extra boron atoms to offset the insufficient electrons for structural stability. Based on this, different metal atoms can also be incorporated into the hexagon centers to compensate the electronic deficiency. For instance, several 2D metalized borophene monolayers have been predicated with outstanding properties, including FeB2 [10], TiB2 [11], BeB2 [12] and MgB2 [13, 14]. Moreover, net charges have been successfully applied to control the phase structure of borophene polymorphs [15, 16]. However, the electrons in borophene and its derivatives are highly delocalized due to the emergence of multi-center bonds, where the itinerant electrons favor metallic phase according to the band theory. Therefore, as was expected, most of the theoretically predicted and experimentally synthesized borophene polymorphs behave like metals, which limits their potential utilization in the semiconductor industry such as logic devices and semiconductor-based optical devices. Given that, focused research efforts are being made to introduce appropriate energy gap (Eg) in borophene either by quantum confinement, such as rolling borophene into small nanotubes [17], cutting it into nanoribbons with armchair edges [18], or by applying strain engineering [19, 20]. For instance, Kistanov et al. proposed to open the Eg of borophene through chemical functionalization and defect engineering [21]. Unfortunately, the metallicity in borophene is quite robust against H, O or F surface decoration and even the incorporation of various vacancies, which has been proved to be effective in graphene, also failed to endow it with an Eg [22-27]. Recently, a group of semiconducting boron monolayers has been determined by Bhattacharyya et al. [19], which contain particular triangular and heptagonal regions in addition to the hexagonal vacancies between them. Tunable band gap was achieved by modifying the quantity of β unit cells (defined as boron sheet with coordination number CN = 4, 5, and 6) in heptagonal region or blending different heptagonal regions in boron sheet [28]. Nevertheless, the complex decoration methods are experimentally less desirable and the fabrication of finely tuned boron monolayer becomes a challenging task due to the presence of irregular regions. Herein, we aimed to introduce the Eg in hexagonal borophene, which was synthesized recently on Al (111) substrate through molecular beam epitaxy in ultrahigh vacuum [3], by chalcogen incorporation. Based on the particle swarm optimization (PSO) algorithm in CALYPSO package [29], Zhao et.al predicated a new 2D anisotropic Dirac cone material B2S, which showed a graphene-like hexagonal lattice [30, 31]. According to their results, we considered the siuation of replacing the S atoms with O, Se and Te. In this work, we identified three types of 2D semiconducting boron chalcogenide monolayers (B2X, X=O, Se and Te) but with broken electron 2
delocalization of B-atom network. The introduction of chalcogen compensated the electron-deficiency of B by forming atomic rings isoelectronic to carbon. The resulting compounds possess graphene-like hexagonal lattice with corrugations, whose stabilities have been confirmed by phonon dispersions and molecule dynamic simulations at elevated temperature. In contrast with pristine borophene, weak indirect Eg and anisotropic electron-hole mobility have been observed with desirable visible light absorptions. Furthermore, Eg can be tuned by applying uniaxial and biaxial strains and exhibit indirect to direct transition at a certain strain. Therefore, the tunable Eg and unique visible light response endow B2X with great potential in electronic and optical applications.
Computational methods The geometry optimization and electronic properties have been calculated by using the CASTEP package, which is based on density functional theory (DFT) [32]. The exchange-correlation interactions are described by the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) as function [33]. The interactions between ionic core and valence electrons are characterized by norm conserving pseudopotentials [34]. Meanwhile, Brillouin zone integrations in the reciprocal space are performed by Monkhorst-Pack scheme with 7×7×1 k-point mesh. The plane wave cutoff energy of 750 eV was employed to guarantee the accuracy of the results. The structures were fully relaxed until the energy, displacement and force converged to 1.0 × 10−5 eV/atom, 0.03 eV/Å and 0.001 Å, respectively. To weaken the interactions between periodic images, we have adopted a vacuum layer larger than 14 Å. The highest occupied molecular orbitals (HOMO) and lowest unoccupied molecular orbitals (LUMO) of B2X at Γ points have been calculated by DMol3 module [35]. The molecule dynamic simulations with NVT ensemble and massive GGM themostat were carried out by using 2×2 supercells to further examine the stability of B2X. The k points of 7×7×1 were used and T was set to be 300 ~ 1200 K. The exchange-correlation interactions were described by the GGA and PBE function in the dynamic simulations. The electron population in B2X were calculated using the Bader charge analysis code [36]. Results and discussion The unit cells of B2O, B2Se and B2Te are presented in Figure 1a-c, exhibiting a hexagonal crystal with lattice parameters of a = b = 4.51, 5.49 and 5.77 Å, respectively. Note that the configurations of B2X are not completely flat, where X atoms are alternatively protruded from the 2D plane. The wrinkle height (d in Figure 1a-c) increased from 0.27 to 1.34 Å with increasing radius of chalcogen atom. Different from BxO structures predicated by zhang et al. [37], where O prefers to unzip a B-B bond or form OB3 tetrahedron structure motifs, X atoms in B2X incline to adopt an interposition distribution. The cohesive energy (Ec) of B2X can be given as: Ec = [Etotal – (mEB + nEX)] / (m + n) (1) where Etotal, EB and EX refer to the total energy of B2X, single boron and X atom, and m and n represent the number of boron and X atoms. The calculated Ec for B2O, B2Se and B2Te is –5.99, – 4.83 and –4.51 eV, respectively, which is much larger than the experimentally synthesized InSe and InTe (–2.57 and –2.77 eV, respectively) [38], indicating that these compounds could be experimentally synthesized. Moreover, the Ec decreased with increasing radius of X-atom, which can be rationalized by the weakened bonding from B-O to B-Te and increased lattice strain in the 3
hex-atomic ring due to the incorporation of a larger X atom. Furthermore, the formation of B-X bond has been analyzed by electron location function (ELF), as shown in Figure S1. Electrons have been partially accumulated between B and O atoms and formed covalent σ-like sp2 bonding (Figure S1a). Moreover, out-of-plane π-hybridization, similar to graphene, has been confirmed by ELF slice (Figure S1d). Besides, there are also some ionic components with reduced electron accumulation between B-O bonds due to their electronegativity difference, where 1.62e were transferred from B to O atom according to the Bader charge analysis. Similar electronic distributions have also been found in B2Se and B2Te (Figure S1b-c and e-f). As for B2Se, about 0.86e were transferred from B to Se. However, 0.03e were migrated from Te to B due to the similar electronegativities between B (2.04) and Te (2.10) atoms. Figure 1e-g presents the phonon dispersion of B2X along their high symmetry sites in the Brillouin zone. The absence of any imaginary modes in phonon dispersions indicate that the considered B2X configurations are indeed the local minimums. In addition, the highest frequency of B2O (42 THz) is comparable to the hexagonal boron nitride monolayer (~ 48 THz) [39], MoS2 (~ 14 THz) [40] and black phosphorus (~ 14 THz) [41], which indicates the robustness of B-O bond in B2O monolayer. However, the highest frequency value of B2Se (35 THz) and B2Te (33 THz) is much lower than B2O, which is consistent with the calculated Ec. The molecule dynamic simulations were carried out at elevated temperature (300 ~ 1200 K) by using a 2×2 supercell to further examine the thermal stability of B2O, B2Se and B2Te. As shown in Figure S2, despite some atomic fluctuations around their equilibrium position, the overall structure of B2X remained stable at a high temperature of 1200 K.
Figure 1. Top and side views of (a) B2O, (b) B2Se and (c) B2Te unit cells, where d refers to the corresponding corrugation height and the pink, red, yellow and brown balls represent B, O, Se and Te atoms, respectively. (d) Schematic illustration of the Brillouin zone path, used for phonon dispersion calculations, and the as-obtained phonon dispersions of (e) B2O, (f) B2Se and (g) B2Te. The mechanical stability of B2X has been confirmed by calculating the in-plane stiffness, which can be given as: 4
Y = 1 / S0 × (∂2Estr / ∂ε2) ε=0 (2) where the strain (ε) can be defined as ε = (l – l0) / l0. Moreover, l and l0 represent the lattice parameters of the strained and unstrained B2X, S0 denotes the area of the equilibrium structure, and Estr refers to the strain energy, i.e., the energy difference between the strained and equilibrium structure. The stress-strain curves of B2X under different strains, ranging from −6% to 6%, are presented in Figure S3. Herein, the orthorhombic B2X structure has been constructed to calculate the elastic constant (Y) under uniaxial and biaxial strains, as shown in the inset of Figure S3. Unlike the strong anisotropy in T-borophene (Yborophene = 398 and 170 N/m along zigzag and armchair directions) [1], B2X exhibits isotropic stiffness along armchair and zigzag directions. Table 1 shows that the B2O, B2Se and B2Te exhibited Yzigzag values of 204.48, 122.78 and 108.34 N/m and Yarmchair values of 213.56, 131.98 and 105.56 N/m, respectively. These values are smaller than the T-borophene, indicating the reduced mechanical strength of B2X. However, these values are much larger than phosphorene (27.73 along armchair and 90.53 N/m along zigzag directions) [42], which indicates that the as-proposed B2X structures render superior mechanical properties than phosphorene. In addition, molecule dynamic simulations of 2×2 B2X subjected to 6% external strain were performed at 300 K to further confirm their stability. As shown in Figure S4, the overall structures of B2X remained stable even under strain up to 6%. Table 1. The calculated elastic constant (Y) of B2X under zigzag, armchair and biaxial strains. The previously reported elastic constant values of other 2D materials are also included for comparison. Structures B 2O B2Se B2Te T-borophene [1] Phosphorene [42]
zigzag
Y (N/m) armchair
biaxial
204.48 122.78 108.34 398 90.53
213.56 131.98 105.56 170 27.73
499.83 303.87 250.02 -
Figure 2. The band structures under PBE and HSE06 level and the partial density of states (PDOS) per PBE of (a) B2O, (b) B2Se and (c) B2Te. The valence band maximum was set at zero in all band structures and the horizontal grey dashed lines indicate the Fermi levels. In contrast to the metallic band of pristine borophene, the studied B2O, B2S and B2Te compounds have exhibited an indirect band gap of 1.96, 0.52 and 0.58 eV, respectively, under PBE level (Figure 2). The indirect gap stemmed from the mismatch of valence band maximum (VBM) and conduction band minimum (CBM), where VBM was located at Γ points and CBM favored the M points. However, in the case of B2O and B2Se, the energy difference between Γ and M points in 5
the conduction band (ΔEc) is quite small (< 0.09 eV) due to their nearly flat band feature along the Γ-M direction (Figure 2a and 2b). In contrast, the band of B2Te is remarkably dispersive along Γ-M, which resulted in a much larger ΔEc of 0.63 eV. The detailed analysis of the PDOS revealed that the valence and conduction bands of B2X sheets had been significantly contributed by B-2p orbitals with some Xp component. Figure 3 presents the distribution of HOMO and LUMO of B2X in real space. One can readily observe that the HOMOs of B2X are mainly accumulated on the B-B bonds (triangular B motifs), showing π-bond characteristic with few contributions from the localized X-2p orbitals. Moreover, the delocalized electronic states in hexagonal borophene were broken by the lone-pair electrons of X atoms, leading to the emergence of separated boron-island, where the electrons remained highly delocalized. Similar observations have also been found in 2D PC6 system [43]. In the case of LUMO, the electrons are mainly contributed by the 2pz orbital of B, which is connected by two X atoms, forming slightly hybridized anti-bonding orbitals with adjacent Se-2p, Te-2p and O-2p orbitals. Similarly, these electron states are high localized around the B atoms. Therefore, the breakdown of electron-delocalization in B-atoms network due to the presence of chalcogen atom is believed to be the origin of the induced Eg in B2X. As the PBE function underestimates the band gap, the screened hybrid density function has been utilized to determine the band gap, which rendered better consistency with experimental results. The band structures of B2X, calculated by HSE06 function, are marked by the black-colored solid lines in Figure 2a-c. HSE06 function delivers much higher gaps of 2.84, 0.97 and 1.04 eV for B2O, B2Se and B2Te, respectively. These gaps are also larger than the Eg of recently reported semi-conductive boron sheets β1s (0.74 eV) [28] and H-functionalized borophene nanoribbons (around 0.93 eV) [21].
Figure 3. HOMO and LUMO orbitals of B2X at Γ points. The equipotential values are set to be 0.03 e/Å3. The blue (yellow) volumes in upper images and red (blue) areas in lower images represent different phases of wave functions. Further, the modulation of the Eg in B2X can be achieved by restoring the external strains, as shown in Figure 4. Herein, we have only considered the small biaxial strain (−6% ~ 6%), which can be afforded by lattice mismatch of the substrate or stretched by external load without damaging the structural integrity. The results revealed that the Eg of B2O is sensitive to the compressive biaxial strains (0 to −6%) and increased from 2.84 to 4.45 eV with increasing compressive strain (inset in Figure 4a). On the other hand, the tensile strain exhibited a negligible influence on Eg and the lattice dilatation of 6% resulted in an Eg modulation of only 0.24 eV. One should note that the asymmetric modulation of Eg-B2O originated from the synergetic movement of valence (−5.09 ~ −4.16 eV) and conduction bands (−0.60 ~ −1.56 eV) as shown in Figure 4d. 6
Specifically, the CBM of B2O continuously moved downwards under the strain of −6% to 6%, whereas the VBM initially moved upwards under the compressive strain of −6% to 0%, leading to a rapid decrease in Eg, and, then, moved downwards under the tensile strain of 0% to 6%. The downward movement of VBM compensated the decrease of CBM and led to a negligible change in Eg under tensile stretching. However, the Eg-B2Se and Eg-B2Te exhibited obvious critical points for the band gap modulation, as shown in the inset of Figure 4b and 4c. It can be readily observed that the Eg of B2Se initially decreased under the uniaxial strain of −6% to 2%, followed by a gradual increase. On the other hand, the Eg of B2Te initially increased under compressing strains from −6% to −2%, and then decreased in the strain range of −2% to 6%. The corresponding shift in CBM and VBM of B2Se and B2Te is illustrated in Figure 4e-f. Besides, we have observed an indirect-to-direct band gap transition in B2Se and B2Te with a strain larger than 2% and 6%, respectively. On the other hand, the indirect band structure of B2O remained stable under the application of external strain as the energy at Γ point always remained a little higher than the energy at K point (Figure 4d).
Figure 4. Eg of (a) B2O, (b) B2Se and (c) B2Te under biaxial strain, ranging from −6% to 6%. The inset images represent the stretching methods of B2X. Movements of valence bands and conduction bands of (d) B2O, (e) B2Se and (f) B2Te under biaxial strains. The circles on valance and conduction bands represent the VBM and CBM under different strains. Table 2 The calculated effective mass of holes (mh*) and electrons (me*) along different directions, where m0 refers to the mass of a free electron. Structure B2O B2Se B2Te
m h* / m 0 Γ-K Γ-M
m e* / m 0 M-Γ M-K
0.84 0.39 0.56
6.57 12.54 1.85
0.83 0.39 0.55
0.64 0.40 0.56
We have also investigated the influence of strain on the effective mass of holes and electrons in B2X structures, which determines the carrier mobility. The effective mass (m*) can be given as: m* = ħ2 ∙ (∂2E(k) / ∂k2)−1 (3) where ħ represents the reduced Plank constant, k denotes the wave vector and E(k) refers to the band dispersion relation. The effective mass of holes (mh*) and electrons (me*) along different directions 7
is summarized in Table 2. In the case of B2O, mh* along Γ-K and Γ-M remained isotropic and found to be 0.84 and 0.83 m0, respectively, whereas the me* along M-Γ and M-K directions exhibited anisotropic behavior and found to be 6.57 and 0.64 m0, respectively. Similar observations have been made in the case of B2Se and B2Te. Importantly, the mh-ΓK*, mh-ΓM* and me-MK* of the B2Se are much lighter than B2O and B2Te structures. Moreover, the mh-ΓK*, mh-ΓM* and me-MK* of the B2Se are comparable or even lower than the MoS2 (mh* = 0.57 and 0.60 m0, me* = 0.46 and 0.48 m0 along two different directions) [44]. Hence, B2Se is expected to exhibit higher mobility than MoS2 along certain direction, which makes it more appealing in applications as high-speed electronics. Furthermore, the influence of external biaxial strain on mh* and me* of B2X has been studied and the results are shown Figure S5. Both mh-ΓK* and mh-ΓM* in B2O increased under increasing compressive strain, whereas remained unchanged under tensile strains (Figure S5a). A similar trend has also been observed in B2Se and B2Te, where critical point emerged at the compressive strain of −4% and −2%, respectively, and further compression resulted in lower effective mass (Figure S5b-c). It is worth noting that, the mh-ΓK* remained almost equal to mh-ΓM* under different strains because of the similar conduction band dispersion (Figure 2). On the other hand, me* exhibited significant anisotropy along M-Γ and M-K directions (Figure S5d-f).
Figure 5. The light absorption of (a) B2O, (b) B2Se and (c) B2Te monolayers and bilayers by using HSE06 approach. Based on the semiconducting electronic properties of B2X, the optical performance of the as-proposed B2X structures have investigated for potential application in optoelectronics. Figure 5 presents the light absorption coefficients (I) of B2X under HSE06 level. It can be readily observed that the B2X nanosheets can absorb the lights from ultraviolet to visible light regions with I in the order of 104 cm-1. Nevertheless, only a small part of visible light (close to violet) could be absorbed by B2O. In contrast, B2Se and B2Te have demonstrated considerable absorption in the overall visible light region, which can even assimilate infrared light due to their relatively narrow Eg. Recently, it has been demonstrated that van der Waals stackings [45] and the transformations between physical and chemical absorption states [46] can significantly modulate the electronic properties (such as Schottky barriers and rectifying behaviors) of 2D materials. Here, we investigated the influence of bilayer stacking on the electronic and optical performance of B2X. Different stacking between bilayer B2X has been considered to found the most energetic one. The binding energy (Eb) between two layers were defined as: Eb = Ebilayer – 2Emonolayer (4) where Ebilayer and Emonolayer represents the energy of B2X bilayer and monolayer, respectively. The atomic structures of B2X bilayers with various stacking are depicted in Figure S6-8, respectively. Eb are also given below the structures. The ground state structures of B2X are shown in Figure 6 with Eb = −3.54, −5.19 and −2.12 eV, respectively. The lattice constants a (b) of the energy 8
favorable B2O, B2Se and B2Te bilayers are 4.55 (4.52), 5.45 (5.40) and 5.76 (5.87) Å, respectively. Note that the upper and lower layers possess covalent bonding in B2O and B2Se bilayers, whereas the two layers of B2Te possess van der Waals interactions. As shown in Figure S9, the B2O and B2Se bilayers could from covalent combination, with negligible barriers, when two pre-positioned counterparts move close to each other. However, in the case of B2Te, high energy-barrier (~ 4.04 eV) must be conquered before forming a covalently bonded bilayer structure. Therefore, van der Waals interactions would dominate the interlayer bonding in B2Te.
Figure 6. The most stable configurations of 2×2 (a) B2O, (b) B2Se and (c) B2Te bilayers. dh indicates the interlayer distance.
Figure 7. The calculated band structure of B2O, B2Se and B2Te bilayers under (a-c) PBE and (d-f) HSE06 level. The band structure of B2X bilayers was presented in Figure 7. B2O, B2Se and B2Te bilayers have exhibited semiconducting behavior with an indirect band gap of 1.41, 0.99 and 0.53 eV, respectively, under the PBE level. Meanwhile, Eg was further increased to 2.55, 1.66 and 0.73 eV under HSE06 level. The Eg-B2O and Eg-B2Te are lower than the Eg of its monolayer counterparts (2.84 and 1.04 eV for B2O and B2Te, respectively), whereas Eg-B2Se is larger than Eg of monolayer B2Se monolayer (0.97 eV). In the case of monolayer and bilayer B2O, the highest peak increased from 9
6.39×104 (90 nm) to 8.71×104 cm−1 (122 nm), whereas the peak at 289 nm declined from 4.31×104 to 1.65×104 cm−1 (at 336 nm) (Figure 5). As for B2Se and B2Te, the peaks around 100 ~ 300 nm merged into one peak and the light adsorption coefficient of B2Se and B2Te increased to 1.46×105 and 1.39×105 cm−1, respectively. Moreover, a red shift has been observed in adsorption peaks of B2O and B2Te due to the reduced Eg (2.84 to 2.55 eV and 1.04 to 0.73 eV), whereas a blue shift has been observed in B2Se, which has exhibited an increase in Eg (0.97 to 1.66 eV). Owing to the excellent visible light response, the proposed B2X structures exhibit promise in solar-energy harvesting applications. Another borophene analogue with nonzero thickness, Pmmn8 sheet, is predicted by Zhou, et al. [47] in 2014. It was found that this sheet is more stable than the α-sheet and other analogues [47]. Hence, we also considered the possibility of chalcogenide atom doping in Pmmn8 sheet for energy gap. The atomic configuration and band structure under PBE level of Pmmn8 sheet is depicted in Figure S10, respectively. However, incorporation of chalcogenide atom would destroy the symmetry of Pmmn8 borophene and could not create localize electronic states to open energy gap.
Conclusions In summary, we have reported three 2D B2X monolayer structures with versatile electronic and optical properties by using PSO and first-principle calculations. The results demonstrate that the breakdown of the electron delocalization in hexagonal borophene introduced an indirect Eg of 2.84, 0.97 and 1.04 eV in B2O, B2Se and B2Te, respectively. Moreover, the effective mass of electrons in B2O, B2Se and B2Te is comparable to MoS2 along certain directions. Furthermore, the introduced Eg and carrier mass can be modulated by external strains, which remarkably influence the electronic properties. It is worth mentioning that the proposed B2X structures exhibited visible light response, which makes them promising for light-harvesting-based applications. Our results propose an alternative way to open an energy gap in hexagonal borophene, which endows it with great potential applications in microelectronic and optoelectronic domain. ■ ASSOCIATED CONTENT Supporting information Electron location function of B2X; molecular dynamics simulations; strain energy-strain curves; carrier effective masses under external strain; and estimated energy barrier for spontaneous combination of B2X bilayer ■ AUTHOR INFORMATION Corresponding Authors Xianfei Chen *E-mail: [email protected] Yi Huang *E-mail: [email protected] ORCID Xianfei Chen: 0000-0002-5078-3950 Notes The authors declare no competing financial interest. 10
■ ACKNOWLEDGMENTS This work was supported by the China Postdoctoral Science Foundation (2017M623306XB), the Science and Technology Department of Sichuan Province (2018SZDZX0022, 2017SZ0185), and the National Natural Scientific Foundation of China (41673109). ■ REFERENCES [1] A.J. Mannix, X.-F. Zhou, B. Kiraly, J.D. Wood, D. Alducin, B.D. Myers, X. Liu, B.L. Fisher, U. Santiago, J.R. Guest, M.J. Yacaman, A. Ponce, A.R. Oganov, M.C. Hersam, N.P. Guisinger, Synthesis of borophenes: Anisotropic, two-dimensional boron polymorphs, Science, 350 (2015) 1513. [2] B. Feng, J. Zhang, Q. Zhong, W. Li, S. Li, H. Li, P. Cheng, S. Meng, L. Chen, K. Wu, Experimental realization of two-dimensional boron sheets, Nature Chemistry, 8 (2016) 563. [3] W.B. Li, L.J. Kong, C.Y. Chen, J. Gou, S.X. Sheng, W.F. Zhang, H. Li, L. Chen, P. Cheng, K.H. Wu, Experimental realization of honeycomb borophene, Science Bulletin, 63 (2018) 282-286. [4] R. Wu, I.K. Drozdov, S. Eltinge, P. Zahl, S. Ismail-Beigi, I. Božović, A. Gozar, Large-area single-crystal sheets of borophene on Cu(111) surfaces, Nature Nanotechnology, 14 (2019) 44-49. [5] X. Wu, J. Dai, Y. Zhao, Z. Zhuo, J. Yang, X.C. Zeng, Two-Dimensional Boron Monolayer Sheets, ACS Nano, 6 (2012) 7443-7453. [6] N. Karmodak, E.D. Jemmis, The Role of Holes in Borophenes: An Ab Initio Study of Their Structure and Stability with and without Metal Templates, Angew Chem Int Edit, 56 (2017) 10093-10097. [7] Z. Zhang, E.S. Penev, B.I. Yakobson, Polyphony in B flat, Nature Chemistry, 8 (2016) 525. [8] E.S. Penev, S. Bhowmick, A. Sadrzadeh, B.I. Yakobson, Polymorphism of Two-Dimensional Boron, Nano Letters, 12 (2012) 2441-2445. [9] Z.A. Piazza, H.-S. Hu, W.-L. Li, Y.-F. Zhao, J. Li, L.-S. Wang, Planar hexagonal B36 as a potential basis for extended single-atom layer boron sheets, Nature Communications, 5 (2014) 3113. [10] H. Zhang, Y. Li, J. Hou, A. Du, Z. Chen, Dirac State in the FeB2 Monolayer with Graphene-Like Boron Sheet, Nano Letters, 16 (2016) 6124-6129. [11] L.Z. Zhang, Z.F. Wang, S.X. Du, H.J. Gao, F. Liu, Prediction of a Dirac state in monolayer TiB2, Physical Review B, 90 (2014). [12] P. Zhang, V.H. Crespi, Theory of B2O and BeB2 Nanotubes: New Semiconductors and Metals in One Dimension, Physical Review Letters, 89 (2002) 056403. [13] J. Bekaert, A. Aperis, B. Partoens, P.M. Oppeneer, M.V. Milošević, Evolution of multigap superconductivity in the atomically thin limit: Strain-enhanced three-gap superconductivity in monolayer MgB2, Physical Review B, 96 (2017) 094510. [14] H. Tang, S. Ismail-Beigi, Self-doping in boron sheets from first principles: A route to structural design of metal boride nanostructures, Physical Review B, 80 (2009) 134113. [15] Z. Zhang, S.N. Shirodkar, Y. Yang, B.I. Yakobson, Gate-Voltage Control of Borophene Structure Formation, Angewandte Chemie-International Edition, 56 (2017) 15421-15426. [16] D. Liu, D. Tománek, Effect of Net Charge on the Relative Stability of 2D Boron Allotropes, Nano Lett., 19 (2019) 1359-1365. [17] X. Yang, Y. Ding, J. Ni, Ab initio prediction of stable boron sheets and boron nanotubes: Structure, stability, and electronic properties, Physical Review B, 77 (2008) 041402. 11
[18] S. Saxena, T.A. Tyson, Insights on the Atomic and Electronic Structure of Boron Nanoribbons, Physical Review Letters, 104 (2010) 245502. [19] B. Gargee, M. Arup, C. Indrani, P. Biswarup, Semiconducting phase in borophene: role of defect and strain, Journal of Physics D: Applied Physics, 50 (2017) 405103. [20] S. Ipek, M.E. Kilic, A. Mogulkoc, S. Cahangirov, E. Durgun, Semiconducting defect-free polymorph of borophene: Peierls distortion in two dimensions, Physical Review B, 98 (2018). [21] A.A. Kistanov, Y.Q. Cai, K. Zhou, N. Srikanth, S.V. Dmitriev, Y.W. Zhang, Exploring the charge localization and band gap opening of borophene: a first-principles study, Nanoscale, 10 (2018) 1403-1410. [22] R. Martinazzo, S. Casolo, G.F. Tantardini, Symmetry-induced band-gap opening in graphene superlattices, Physical Review B, 81 (2010) 245420. [23] X. Fan, D. Ma, B. Fu, C.-C. Liu, Y. Yao, Cat's-cradle-like Dirac semimetals in layer groups with multiple screw axes: Application to two-dimensional borophene and borophane, Physical Review B, 98 (2018) 195437. [24] L.-C. Xu, A. Du, L. Kou, Hydrogenated borophene as a stable two-dimensional Dirac material with an ultrahigh Fermi velocity, Physical Chemistry Chemical Physics, 18 (2016) 27284-27289. [25] J.E. Padilha, R.H. Miwa, A. Fazzio, Directional dependence of the electronic and transport properties of 2D borophene and borophane, Physical Chemistry Chemical Physics, 18 (2016) 25491-25496. [26] J. Khanifaev, R. Peköz, M. Konuk, E. Durgun, The interaction of halogen atoms and molecules with borophene, Physical Chemistry Chemical Physics, 19 (2017) 28963-28969. [27] R. Peköz, M. Konuk, M.E. Kilic, E. Durgun, Two-Dimensional Fluorinated Boron Sheets: Mechanical, Electronic, and Thermal Properties, ACS Omega, 3 (2018) 1815-1822. [28] S.-G. Xu, X.-T. Li, Y.-J. Zhao, J.-H. Liao, W.-P. Xu, X.-B. Yang, H. Xu, Two-Dimensional Semiconducting Boron Monolayers, Journal of the American Chemical Society, 139 (2017) 17233-17236. [29] Y. Wang, J. Lv, L. Zhu, Y. Ma, Crystal structure prediction via particle-swarm optimization, Physical Review B, 82 (2010) 094116. [30] Y. Zhao, X. Li, J. Liu, C. Zhang, Q. Wang, A New Anisotropic Dirac Cone Material: A B 2S Honeycomb Monolayer, J Phys Chem Lett, 9 (2018) 1815-1820. [31] P. Li, Z.Y. Li, J.L. Yang, Rational Design of Two-dimensional Anode Materials: B2S as a Strained Graphene, Journal of Physical Chemistry Letters, 9 (2018) 4852-4856. [32] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne, First principles methods using CASTEP, Z. Kristall., 220 (2005) 567-570. [33] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Physical Review Letters, 77 (1996) 3865-3868. [34] J.P. Perdew, K. Burke, Y. Wang, Generalized gradient approximation for the exchange-correlation hole of a many-electron system, Physical Review B, 54 (1996) 16533-16539. [35] B. Delley, An allǦ electron numerical method for solving the local density functional for polyatomic molecules, The Journal of Chemical Physics, 92 (1990) 508-517. [36] W. Tang, E. Sanville, G. Henkelman, A grid-based Bader analysis algorithm without lattice bias, J. Phys.: Condens. Matter, 21 (2009) 084204. 12
[37] R. Zhang, Z. Li, J. Yang, Two-Dimensional Stoichiometric Boron Oxides as a Versatile Platform for Electronic Structure Engineering, The Journal of Physical Chemistry Letters, 8 (2017) 4347-4353. [38] S. Demirci, N. Avazlı, E. Durgun, S. Cahangirov, Structural and electronic properties of monolayer group III monochalcogenides, Physical Review B, 95 (2017) 115409. [39] M. Topsakal, E. Aktürk, S. Ciraci, First-principles study of two- and one-dimensional honeycomb structures of boron nitride, Physical Review B, 79 (2009) 115442. [40] A. Molina-Sánchez, L. Wirtz, Phonons in single-layer and few-layer MoS2 and WS2, Physical Review B, 84 (2011) 155413. [41] T. Lin, X. Cong, M.-L. Lin, X.-L. Liu, P.-H. Tan, The phonon confinement effect in two-dimensional nanocrystals of black phosphorus with anisotropic phonon dispersions, Nanoscale, 10 (2018) 8704-8711. [42] A. Priydarshi, Y.S. Chauhan, S. Bhowmick, A. Agarwal, Strain-tunable charge carrier mobility of atomically thin phosphorus allotropes, Physical Review B, 97 (2018) 115434. [43] T. Yu, Z. Zhao, Y. Sun, A. Bergara, J. Lin, S. Zhang, H. Xu, L. Zhang, G. Yang, Y. Liu, Two-Dimensional PC6 with Direct Band Gap and Anisotropic Carrier Mobility, Journal of the American Chemical Society, 141 (2019) 1599-1605. [44] Y. Cai, G. Zhang, Y.-W. Zhang, Polarity-Reversed Robust Carrier Mobility in Monolayer MoS2 Nanoribbons, Journal of the American Chemical Society, 136 (2014) 6269-6275. [45] T. Shen, J.-C. Ren, X. Liu, S. Li, W. Liu, van der Waals Stacking Induced Transition from Schottky to Ohmic Contacts: 2D Metals on Multilayer InSe, JACS, (2019). [46] G. Su, S. Yang, S. Li, C.J. Butch, S.N. Filimonov, J.-C. Ren, W. Liu, Switchable Schottky Contacts: Simultaneously Enhanced Output Current and Reduced Leakage Current, JACS, 141 (2019) 1628-1635. [47] X.-F. Zhou, X. Dong, A.R. Oganov, Q. Zhu, Y. Tian, H.-T. Wang, Semimetallic Two-Dimensional Boron Allotrope with Massless Dirac Fermions, Physical Review Letters, 112 (2014) 085502.
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Graphical abstract
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Highlights 1. Incorporation of O, Se and Te atoms into 2D hexagonal borophene (B2X) leads to an indirect energy gap. 2. High thermodynamic, dynamic, and mechanical stability were found in B 2X. 3. B2X exhibited desirable light absorptions in visible light regions with strain tunable energy gap. 4. The electronic effective masses of B2X are comparable to MoS2
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Declaration of interests ܈The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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S0169-4332(19)33757-2 https://doi.org/10.1016/j.apsusc.2019.144940 APSUSC 144940
To appear in:
Applied Surface Science
Received Date: Revised Date: Accepted Date:
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Please cite this article as: J. Liu, X. Chen, Y. Huang, W. Zhang, P. Xiang, B. Xiao, Breakdown of the Electron Delocalization in Hexagonal Borophene Toward Tunable Energy Gap, Applied Surface Science (2019), doi: https://doi.org/10.1016/j.apsusc.2019.144940
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Breakdown of the Electron Delocalization in Hexagonal Borophene Toward Tunable Energy Gap Jia Liua, Xianfei Chena*, Yi, Huangb, c*, Wentao Zhanga, Pan Xianga, Beibei Xiaod a
College of Materials and Chemistry & Chemical Engineering, Chengdu University of Technology, Chengdu 610059, China b College of Environment and Ecology, Chengdu University of Technology, Chengdu 610059, China c State Environmental Protection Key Laboratory of Synergetic Control and Joint Remediation for Soil & Water Pollution, Chengdu University of Technology, Chengdu 610059, China dSchool of Energy and Power Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China Abstract: The absence of energy gap (Eg) in borophene polymorphs, where multi-center bonds favor electron delocalization and form metallic phases, restricts their potential applications in microelectronic and optoelectronic industry. Herein, we propose to incorporate chalcogenide atoms to breakdown the electron delocalization in hexagonal borophene and introduce an Eg. Based on first principle calculation, we report a group of semiconducting boron chalcogenide monolayers, denoted as B2X (X = O, Se and Te), with an indirect Eg of 2.84, 0.97 and 1.04 eV under HSE06 levels, respectively. Moreover, the introduced Eg can be modulated by external strains (± 6%) affordable by substrates, with electronic effective mass being comparable or even lower than those of MoS2 along certain directions. Furthermore, B2X monolayers exhibit desirable light absorption properties under visible light, which can be improved by stacking bilayers. The current study presents an alternative route to introduce the Eg in borophene, endows them with new vitality and highlights their promise in flexible electronics, optoelectronics and photovoltaic devices.
*
Corresponding author. Email: [email protected], [email protected]
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Keywords: 2D materials, first principles, hexagonal borophene, bandgap modulation.
Introduction Two-dimensional (2D) boron monolayers, known as borophenes, have emerged as a new 2D platform with unique physical and chemical properties [1-4]. Boron (IIIA) sits next to carbon in the periodic table and possesses one less electron. Consequently, multi-center bonds, rather than the traditional two-center-two-electrons interactions (e.g., in graphene), dominate the interactions in borophene due to its electron-deficiency in octet rule. This endows more flexibility and complexity for B atoms to form 2D boron structure [5, 6]. In the past decade, several borophene polymorphs in atomic thickness, such as α-, β-, δ-sheets [5, 7, 8] and planer B36 cluster [9], have been predicted, which can be distinguished by diverse geometrical profiles and vacant concentration. In these allotropes, pristine honeycomb centers are always partially occupied by extra boron atoms to offset the insufficient electrons for structural stability. Based on this, different metal atoms can also be incorporated into the hexagon centers to compensate the electronic deficiency. For instance, several 2D metalized borophene monolayers have been predicated with outstanding properties, including FeB2 [10], TiB2 [11], BeB2 [12] and MgB2 [13, 14]. Moreover, net charges have been successfully applied to control the phase structure of borophene polymorphs [15, 16]. However, the electrons in borophene and its derivatives are highly delocalized due to the emergence of multi-center bonds, where the itinerant electrons favor metallic phase according to the band theory. Therefore, as was expected, most of the theoretically predicted and experimentally synthesized borophene polymorphs behave like metals, which limits their potential utilization in the semiconductor industry such as logic devices and semiconductor-based optical devices. Given that, focused research efforts are being made to introduce appropriate energy gap (Eg) in borophene either by quantum confinement, such as rolling borophene into small nanotubes [17], cutting it into nanoribbons with armchair edges [18], or by applying strain engineering [19, 20]. For instance, Kistanov et al. proposed to open the Eg of borophene through chemical functionalization and defect engineering [21]. Unfortunately, the metallicity in borophene is quite robust against H, O or F surface decoration and even the incorporation of various vacancies, which has been proved to be effective in graphene, also failed to endow it with an Eg [22-27]. Recently, a group of semiconducting boron monolayers has been determined by Bhattacharyya et al. [19], which contain particular triangular and heptagonal regions in addition to the hexagonal vacancies between them. Tunable band gap was achieved by modifying the quantity of β unit cells (defined as boron sheet with coordination number CN = 4, 5, and 6) in heptagonal region or blending different heptagonal regions in boron sheet [28]. Nevertheless, the complex decoration methods are experimentally less desirable and the fabrication of finely tuned boron monolayer becomes a challenging task due to the presence of irregular regions. Herein, we aimed to introduce the Eg in hexagonal borophene, which was synthesized recently on Al (111) substrate through molecular beam epitaxy in ultrahigh vacuum [3], by chalcogen incorporation. Based on the particle swarm optimization (PSO) algorithm in CALYPSO package [29], Zhao et.al predicated a new 2D anisotropic Dirac cone material B2S, which showed a graphene-like hexagonal lattice [30, 31]. According to their results, we considered the siuation of replacing the S atoms with O, Se and Te. In this work, we identified three types of 2D semiconducting boron chalcogenide monolayers (B2X, X=O, Se and Te) but with broken electron 2
delocalization of B-atom network. The introduction of chalcogen compensated the electron-deficiency of B by forming atomic rings isoelectronic to carbon. The resulting compounds possess graphene-like hexagonal lattice with corrugations, whose stabilities have been confirmed by phonon dispersions and molecule dynamic simulations at elevated temperature. In contrast with pristine borophene, weak indirect Eg and anisotropic electron-hole mobility have been observed with desirable visible light absorptions. Furthermore, Eg can be tuned by applying uniaxial and biaxial strains and exhibit indirect to direct transition at a certain strain. Therefore, the tunable Eg and unique visible light response endow B2X with great potential in electronic and optical applications.
Computational methods The geometry optimization and electronic properties have been calculated by using the CASTEP package, which is based on density functional theory (DFT) [32]. The exchange-correlation interactions are described by the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) as function [33]. The interactions between ionic core and valence electrons are characterized by norm conserving pseudopotentials [34]. Meanwhile, Brillouin zone integrations in the reciprocal space are performed by Monkhorst-Pack scheme with 7×7×1 k-point mesh. The plane wave cutoff energy of 750 eV was employed to guarantee the accuracy of the results. The structures were fully relaxed until the energy, displacement and force converged to 1.0 × 10−5 eV/atom, 0.03 eV/Å and 0.001 Å, respectively. To weaken the interactions between periodic images, we have adopted a vacuum layer larger than 14 Å. The highest occupied molecular orbitals (HOMO) and lowest unoccupied molecular orbitals (LUMO) of B2X at Γ points have been calculated by DMol3 module [35]. The molecule dynamic simulations with NVT ensemble and massive GGM themostat were carried out by using 2×2 supercells to further examine the stability of B2X. The k points of 7×7×1 were used and T was set to be 300 ~ 1200 K. The exchange-correlation interactions were described by the GGA and PBE function in the dynamic simulations. The electron population in B2X were calculated using the Bader charge analysis code [36]. Results and discussion The unit cells of B2O, B2Se and B2Te are presented in Figure 1a-c, exhibiting a hexagonal crystal with lattice parameters of a = b = 4.51, 5.49 and 5.77 Å, respectively. Note that the configurations of B2X are not completely flat, where X atoms are alternatively protruded from the 2D plane. The wrinkle height (d in Figure 1a-c) increased from 0.27 to 1.34 Å with increasing radius of chalcogen atom. Different from BxO structures predicated by zhang et al. [37], where O prefers to unzip a B-B bond or form OB3 tetrahedron structure motifs, X atoms in B2X incline to adopt an interposition distribution. The cohesive energy (Ec) of B2X can be given as: Ec = [Etotal – (mEB + nEX)] / (m + n) (1) where Etotal, EB and EX refer to the total energy of B2X, single boron and X atom, and m and n represent the number of boron and X atoms. The calculated Ec for B2O, B2Se and B2Te is –5.99, – 4.83 and –4.51 eV, respectively, which is much larger than the experimentally synthesized InSe and InTe (–2.57 and –2.77 eV, respectively) [38], indicating that these compounds could be experimentally synthesized. Moreover, the Ec decreased with increasing radius of X-atom, which can be rationalized by the weakened bonding from B-O to B-Te and increased lattice strain in the 3
hex-atomic ring due to the incorporation of a larger X atom. Furthermore, the formation of B-X bond has been analyzed by electron location function (ELF), as shown in Figure S1. Electrons have been partially accumulated between B and O atoms and formed covalent σ-like sp2 bonding (Figure S1a). Moreover, out-of-plane π-hybridization, similar to graphene, has been confirmed by ELF slice (Figure S1d). Besides, there are also some ionic components with reduced electron accumulation between B-O bonds due to their electronegativity difference, where 1.62e were transferred from B to O atom according to the Bader charge analysis. Similar electronic distributions have also been found in B2Se and B2Te (Figure S1b-c and e-f). As for B2Se, about 0.86e were transferred from B to Se. However, 0.03e were migrated from Te to B due to the similar electronegativities between B (2.04) and Te (2.10) atoms. Figure 1e-g presents the phonon dispersion of B2X along their high symmetry sites in the Brillouin zone. The absence of any imaginary modes in phonon dispersions indicate that the considered B2X configurations are indeed the local minimums. In addition, the highest frequency of B2O (42 THz) is comparable to the hexagonal boron nitride monolayer (~ 48 THz) [39], MoS2 (~ 14 THz) [40] and black phosphorus (~ 14 THz) [41], which indicates the robustness of B-O bond in B2O monolayer. However, the highest frequency value of B2Se (35 THz) and B2Te (33 THz) is much lower than B2O, which is consistent with the calculated Ec. The molecule dynamic simulations were carried out at elevated temperature (300 ~ 1200 K) by using a 2×2 supercell to further examine the thermal stability of B2O, B2Se and B2Te. As shown in Figure S2, despite some atomic fluctuations around their equilibrium position, the overall structure of B2X remained stable at a high temperature of 1200 K.
Figure 1. Top and side views of (a) B2O, (b) B2Se and (c) B2Te unit cells, where d refers to the corresponding corrugation height and the pink, red, yellow and brown balls represent B, O, Se and Te atoms, respectively. (d) Schematic illustration of the Brillouin zone path, used for phonon dispersion calculations, and the as-obtained phonon dispersions of (e) B2O, (f) B2Se and (g) B2Te. The mechanical stability of B2X has been confirmed by calculating the in-plane stiffness, which can be given as: 4
Y = 1 / S0 × (∂2Estr / ∂ε2) ε=0 (2) where the strain (ε) can be defined as ε = (l – l0) / l0. Moreover, l and l0 represent the lattice parameters of the strained and unstrained B2X, S0 denotes the area of the equilibrium structure, and Estr refers to the strain energy, i.e., the energy difference between the strained and equilibrium structure. The stress-strain curves of B2X under different strains, ranging from −6% to 6%, are presented in Figure S3. Herein, the orthorhombic B2X structure has been constructed to calculate the elastic constant (Y) under uniaxial and biaxial strains, as shown in the inset of Figure S3. Unlike the strong anisotropy in T-borophene (Yborophene = 398 and 170 N/m along zigzag and armchair directions) [1], B2X exhibits isotropic stiffness along armchair and zigzag directions. Table 1 shows that the B2O, B2Se and B2Te exhibited Yzigzag values of 204.48, 122.78 and 108.34 N/m and Yarmchair values of 213.56, 131.98 and 105.56 N/m, respectively. These values are smaller than the T-borophene, indicating the reduced mechanical strength of B2X. However, these values are much larger than phosphorene (27.73 along armchair and 90.53 N/m along zigzag directions) [42], which indicates that the as-proposed B2X structures render superior mechanical properties than phosphorene. In addition, molecule dynamic simulations of 2×2 B2X subjected to 6% external strain were performed at 300 K to further confirm their stability. As shown in Figure S4, the overall structures of B2X remained stable even under strain up to 6%. Table 1. The calculated elastic constant (Y) of B2X under zigzag, armchair and biaxial strains. The previously reported elastic constant values of other 2D materials are also included for comparison. Structures B 2O B2Se B2Te T-borophene [1] Phosphorene [42]
zigzag
Y (N/m) armchair
biaxial
204.48 122.78 108.34 398 90.53
213.56 131.98 105.56 170 27.73
499.83 303.87 250.02 -
Figure 2. The band structures under PBE and HSE06 level and the partial density of states (PDOS) per PBE of (a) B2O, (b) B2Se and (c) B2Te. The valence band maximum was set at zero in all band structures and the horizontal grey dashed lines indicate the Fermi levels. In contrast to the metallic band of pristine borophene, the studied B2O, B2S and B2Te compounds have exhibited an indirect band gap of 1.96, 0.52 and 0.58 eV, respectively, under PBE level (Figure 2). The indirect gap stemmed from the mismatch of valence band maximum (VBM) and conduction band minimum (CBM), where VBM was located at Γ points and CBM favored the M points. However, in the case of B2O and B2Se, the energy difference between Γ and M points in 5
the conduction band (ΔEc) is quite small (< 0.09 eV) due to their nearly flat band feature along the Γ-M direction (Figure 2a and 2b). In contrast, the band of B2Te is remarkably dispersive along Γ-M, which resulted in a much larger ΔEc of 0.63 eV. The detailed analysis of the PDOS revealed that the valence and conduction bands of B2X sheets had been significantly contributed by B-2p orbitals with some Xp component. Figure 3 presents the distribution of HOMO and LUMO of B2X in real space. One can readily observe that the HOMOs of B2X are mainly accumulated on the B-B bonds (triangular B motifs), showing π-bond characteristic with few contributions from the localized X-2p orbitals. Moreover, the delocalized electronic states in hexagonal borophene were broken by the lone-pair electrons of X atoms, leading to the emergence of separated boron-island, where the electrons remained highly delocalized. Similar observations have also been found in 2D PC6 system [43]. In the case of LUMO, the electrons are mainly contributed by the 2pz orbital of B, which is connected by two X atoms, forming slightly hybridized anti-bonding orbitals with adjacent Se-2p, Te-2p and O-2p orbitals. Similarly, these electron states are high localized around the B atoms. Therefore, the breakdown of electron-delocalization in B-atoms network due to the presence of chalcogen atom is believed to be the origin of the induced Eg in B2X. As the PBE function underestimates the band gap, the screened hybrid density function has been utilized to determine the band gap, which rendered better consistency with experimental results. The band structures of B2X, calculated by HSE06 function, are marked by the black-colored solid lines in Figure 2a-c. HSE06 function delivers much higher gaps of 2.84, 0.97 and 1.04 eV for B2O, B2Se and B2Te, respectively. These gaps are also larger than the Eg of recently reported semi-conductive boron sheets β1s (0.74 eV) [28] and H-functionalized borophene nanoribbons (around 0.93 eV) [21].
Figure 3. HOMO and LUMO orbitals of B2X at Γ points. The equipotential values are set to be 0.03 e/Å3. The blue (yellow) volumes in upper images and red (blue) areas in lower images represent different phases of wave functions. Further, the modulation of the Eg in B2X can be achieved by restoring the external strains, as shown in Figure 4. Herein, we have only considered the small biaxial strain (−6% ~ 6%), which can be afforded by lattice mismatch of the substrate or stretched by external load without damaging the structural integrity. The results revealed that the Eg of B2O is sensitive to the compressive biaxial strains (0 to −6%) and increased from 2.84 to 4.45 eV with increasing compressive strain (inset in Figure 4a). On the other hand, the tensile strain exhibited a negligible influence on Eg and the lattice dilatation of 6% resulted in an Eg modulation of only 0.24 eV. One should note that the asymmetric modulation of Eg-B2O originated from the synergetic movement of valence (−5.09 ~ −4.16 eV) and conduction bands (−0.60 ~ −1.56 eV) as shown in Figure 4d. 6
Specifically, the CBM of B2O continuously moved downwards under the strain of −6% to 6%, whereas the VBM initially moved upwards under the compressive strain of −6% to 0%, leading to a rapid decrease in Eg, and, then, moved downwards under the tensile strain of 0% to 6%. The downward movement of VBM compensated the decrease of CBM and led to a negligible change in Eg under tensile stretching. However, the Eg-B2Se and Eg-B2Te exhibited obvious critical points for the band gap modulation, as shown in the inset of Figure 4b and 4c. It can be readily observed that the Eg of B2Se initially decreased under the uniaxial strain of −6% to 2%, followed by a gradual increase. On the other hand, the Eg of B2Te initially increased under compressing strains from −6% to −2%, and then decreased in the strain range of −2% to 6%. The corresponding shift in CBM and VBM of B2Se and B2Te is illustrated in Figure 4e-f. Besides, we have observed an indirect-to-direct band gap transition in B2Se and B2Te with a strain larger than 2% and 6%, respectively. On the other hand, the indirect band structure of B2O remained stable under the application of external strain as the energy at Γ point always remained a little higher than the energy at K point (Figure 4d).
Figure 4. Eg of (a) B2O, (b) B2Se and (c) B2Te under biaxial strain, ranging from −6% to 6%. The inset images represent the stretching methods of B2X. Movements of valence bands and conduction bands of (d) B2O, (e) B2Se and (f) B2Te under biaxial strains. The circles on valance and conduction bands represent the VBM and CBM under different strains. Table 2 The calculated effective mass of holes (mh*) and electrons (me*) along different directions, where m0 refers to the mass of a free electron. Structure B2O B2Se B2Te
m h* / m 0 Γ-K Γ-M
m e* / m 0 M-Γ M-K
0.84 0.39 0.56
6.57 12.54 1.85
0.83 0.39 0.55
0.64 0.40 0.56
We have also investigated the influence of strain on the effective mass of holes and electrons in B2X structures, which determines the carrier mobility. The effective mass (m*) can be given as: m* = ħ2 ∙ (∂2E(k) / ∂k2)−1 (3) where ħ represents the reduced Plank constant, k denotes the wave vector and E(k) refers to the band dispersion relation. The effective mass of holes (mh*) and electrons (me*) along different directions 7
is summarized in Table 2. In the case of B2O, mh* along Γ-K and Γ-M remained isotropic and found to be 0.84 and 0.83 m0, respectively, whereas the me* along M-Γ and M-K directions exhibited anisotropic behavior and found to be 6.57 and 0.64 m0, respectively. Similar observations have been made in the case of B2Se and B2Te. Importantly, the mh-ΓK*, mh-ΓM* and me-MK* of the B2Se are much lighter than B2O and B2Te structures. Moreover, the mh-ΓK*, mh-ΓM* and me-MK* of the B2Se are comparable or even lower than the MoS2 (mh* = 0.57 and 0.60 m0, me* = 0.46 and 0.48 m0 along two different directions) [44]. Hence, B2Se is expected to exhibit higher mobility than MoS2 along certain direction, which makes it more appealing in applications as high-speed electronics. Furthermore, the influence of external biaxial strain on mh* and me* of B2X has been studied and the results are shown Figure S5. Both mh-ΓK* and mh-ΓM* in B2O increased under increasing compressive strain, whereas remained unchanged under tensile strains (Figure S5a). A similar trend has also been observed in B2Se and B2Te, where critical point emerged at the compressive strain of −4% and −2%, respectively, and further compression resulted in lower effective mass (Figure S5b-c). It is worth noting that, the mh-ΓK* remained almost equal to mh-ΓM* under different strains because of the similar conduction band dispersion (Figure 2). On the other hand, me* exhibited significant anisotropy along M-Γ and M-K directions (Figure S5d-f).
Figure 5. The light absorption of (a) B2O, (b) B2Se and (c) B2Te monolayers and bilayers by using HSE06 approach. Based on the semiconducting electronic properties of B2X, the optical performance of the as-proposed B2X structures have investigated for potential application in optoelectronics. Figure 5 presents the light absorption coefficients (I) of B2X under HSE06 level. It can be readily observed that the B2X nanosheets can absorb the lights from ultraviolet to visible light regions with I in the order of 104 cm-1. Nevertheless, only a small part of visible light (close to violet) could be absorbed by B2O. In contrast, B2Se and B2Te have demonstrated considerable absorption in the overall visible light region, which can even assimilate infrared light due to their relatively narrow Eg. Recently, it has been demonstrated that van der Waals stackings [45] and the transformations between physical and chemical absorption states [46] can significantly modulate the electronic properties (such as Schottky barriers and rectifying behaviors) of 2D materials. Here, we investigated the influence of bilayer stacking on the electronic and optical performance of B2X. Different stacking between bilayer B2X has been considered to found the most energetic one. The binding energy (Eb) between two layers were defined as: Eb = Ebilayer – 2Emonolayer (4) where Ebilayer and Emonolayer represents the energy of B2X bilayer and monolayer, respectively. The atomic structures of B2X bilayers with various stacking are depicted in Figure S6-8, respectively. Eb are also given below the structures. The ground state structures of B2X are shown in Figure 6 with Eb = −3.54, −5.19 and −2.12 eV, respectively. The lattice constants a (b) of the energy 8
favorable B2O, B2Se and B2Te bilayers are 4.55 (4.52), 5.45 (5.40) and 5.76 (5.87) Å, respectively. Note that the upper and lower layers possess covalent bonding in B2O and B2Se bilayers, whereas the two layers of B2Te possess van der Waals interactions. As shown in Figure S9, the B2O and B2Se bilayers could from covalent combination, with negligible barriers, when two pre-positioned counterparts move close to each other. However, in the case of B2Te, high energy-barrier (~ 4.04 eV) must be conquered before forming a covalently bonded bilayer structure. Therefore, van der Waals interactions would dominate the interlayer bonding in B2Te.
Figure 6. The most stable configurations of 2×2 (a) B2O, (b) B2Se and (c) B2Te bilayers. dh indicates the interlayer distance.
Figure 7. The calculated band structure of B2O, B2Se and B2Te bilayers under (a-c) PBE and (d-f) HSE06 level. The band structure of B2X bilayers was presented in Figure 7. B2O, B2Se and B2Te bilayers have exhibited semiconducting behavior with an indirect band gap of 1.41, 0.99 and 0.53 eV, respectively, under the PBE level. Meanwhile, Eg was further increased to 2.55, 1.66 and 0.73 eV under HSE06 level. The Eg-B2O and Eg-B2Te are lower than the Eg of its monolayer counterparts (2.84 and 1.04 eV for B2O and B2Te, respectively), whereas Eg-B2Se is larger than Eg of monolayer B2Se monolayer (0.97 eV). In the case of monolayer and bilayer B2O, the highest peak increased from 9
6.39×104 (90 nm) to 8.71×104 cm−1 (122 nm), whereas the peak at 289 nm declined from 4.31×104 to 1.65×104 cm−1 (at 336 nm) (Figure 5). As for B2Se and B2Te, the peaks around 100 ~ 300 nm merged into one peak and the light adsorption coefficient of B2Se and B2Te increased to 1.46×105 and 1.39×105 cm−1, respectively. Moreover, a red shift has been observed in adsorption peaks of B2O and B2Te due to the reduced Eg (2.84 to 2.55 eV and 1.04 to 0.73 eV), whereas a blue shift has been observed in B2Se, which has exhibited an increase in Eg (0.97 to 1.66 eV). Owing to the excellent visible light response, the proposed B2X structures exhibit promise in solar-energy harvesting applications. Another borophene analogue with nonzero thickness, Pmmn8 sheet, is predicted by Zhou, et al. [47] in 2014. It was found that this sheet is more stable than the α-sheet and other analogues [47]. Hence, we also considered the possibility of chalcogenide atom doping in Pmmn8 sheet for energy gap. The atomic configuration and band structure under PBE level of Pmmn8 sheet is depicted in Figure S10, respectively. However, incorporation of chalcogenide atom would destroy the symmetry of Pmmn8 borophene and could not create localize electronic states to open energy gap.
Conclusions In summary, we have reported three 2D B2X monolayer structures with versatile electronic and optical properties by using PSO and first-principle calculations. The results demonstrate that the breakdown of the electron delocalization in hexagonal borophene introduced an indirect Eg of 2.84, 0.97 and 1.04 eV in B2O, B2Se and B2Te, respectively. Moreover, the effective mass of electrons in B2O, B2Se and B2Te is comparable to MoS2 along certain directions. Furthermore, the introduced Eg and carrier mass can be modulated by external strains, which remarkably influence the electronic properties. It is worth mentioning that the proposed B2X structures exhibited visible light response, which makes them promising for light-harvesting-based applications. Our results propose an alternative way to open an energy gap in hexagonal borophene, which endows it with great potential applications in microelectronic and optoelectronic domain. ■ ASSOCIATED CONTENT Supporting information Electron location function of B2X; molecular dynamics simulations; strain energy-strain curves; carrier effective masses under external strain; and estimated energy barrier for spontaneous combination of B2X bilayer ■ AUTHOR INFORMATION Corresponding Authors Xianfei Chen *E-mail: [email protected] Yi Huang *E-mail: [email protected] ORCID Xianfei Chen: 0000-0002-5078-3950 Notes The authors declare no competing financial interest. 10
■ ACKNOWLEDGMENTS This work was supported by the China Postdoctoral Science Foundation (2017M623306XB), the Science and Technology Department of Sichuan Province (2018SZDZX0022, 2017SZ0185), and the National Natural Scientific Foundation of China (41673109). ■ REFERENCES [1] A.J. Mannix, X.-F. Zhou, B. Kiraly, J.D. Wood, D. Alducin, B.D. Myers, X. Liu, B.L. Fisher, U. Santiago, J.R. Guest, M.J. Yacaman, A. Ponce, A.R. Oganov, M.C. Hersam, N.P. Guisinger, Synthesis of borophenes: Anisotropic, two-dimensional boron polymorphs, Science, 350 (2015) 1513. [2] B. Feng, J. Zhang, Q. Zhong, W. Li, S. Li, H. Li, P. Cheng, S. Meng, L. Chen, K. Wu, Experimental realization of two-dimensional boron sheets, Nature Chemistry, 8 (2016) 563. [3] W.B. Li, L.J. Kong, C.Y. Chen, J. Gou, S.X. Sheng, W.F. Zhang, H. Li, L. Chen, P. Cheng, K.H. Wu, Experimental realization of honeycomb borophene, Science Bulletin, 63 (2018) 282-286. [4] R. Wu, I.K. Drozdov, S. Eltinge, P. Zahl, S. Ismail-Beigi, I. Božović, A. Gozar, Large-area single-crystal sheets of borophene on Cu(111) surfaces, Nature Nanotechnology, 14 (2019) 44-49. [5] X. Wu, J. Dai, Y. Zhao, Z. Zhuo, J. Yang, X.C. Zeng, Two-Dimensional Boron Monolayer Sheets, ACS Nano, 6 (2012) 7443-7453. [6] N. Karmodak, E.D. Jemmis, The Role of Holes in Borophenes: An Ab Initio Study of Their Structure and Stability with and without Metal Templates, Angew Chem Int Edit, 56 (2017) 10093-10097. [7] Z. Zhang, E.S. Penev, B.I. Yakobson, Polyphony in B flat, Nature Chemistry, 8 (2016) 525. [8] E.S. Penev, S. Bhowmick, A. Sadrzadeh, B.I. Yakobson, Polymorphism of Two-Dimensional Boron, Nano Letters, 12 (2012) 2441-2445. [9] Z.A. Piazza, H.-S. Hu, W.-L. Li, Y.-F. Zhao, J. Li, L.-S. Wang, Planar hexagonal B36 as a potential basis for extended single-atom layer boron sheets, Nature Communications, 5 (2014) 3113. [10] H. Zhang, Y. Li, J. Hou, A. Du, Z. Chen, Dirac State in the FeB2 Monolayer with Graphene-Like Boron Sheet, Nano Letters, 16 (2016) 6124-6129. [11] L.Z. Zhang, Z.F. Wang, S.X. Du, H.J. Gao, F. Liu, Prediction of a Dirac state in monolayer TiB2, Physical Review B, 90 (2014). [12] P. Zhang, V.H. Crespi, Theory of B2O and BeB2 Nanotubes: New Semiconductors and Metals in One Dimension, Physical Review Letters, 89 (2002) 056403. [13] J. Bekaert, A. Aperis, B. Partoens, P.M. Oppeneer, M.V. Milošević, Evolution of multigap superconductivity in the atomically thin limit: Strain-enhanced three-gap superconductivity in monolayer MgB2, Physical Review B, 96 (2017) 094510. [14] H. Tang, S. Ismail-Beigi, Self-doping in boron sheets from first principles: A route to structural design of metal boride nanostructures, Physical Review B, 80 (2009) 134113. [15] Z. Zhang, S.N. Shirodkar, Y. Yang, B.I. Yakobson, Gate-Voltage Control of Borophene Structure Formation, Angewandte Chemie-International Edition, 56 (2017) 15421-15426. [16] D. Liu, D. Tománek, Effect of Net Charge on the Relative Stability of 2D Boron Allotropes, Nano Lett., 19 (2019) 1359-1365. [17] X. Yang, Y. Ding, J. Ni, Ab initio prediction of stable boron sheets and boron nanotubes: Structure, stability, and electronic properties, Physical Review B, 77 (2008) 041402. 11
[18] S. Saxena, T.A. Tyson, Insights on the Atomic and Electronic Structure of Boron Nanoribbons, Physical Review Letters, 104 (2010) 245502. [19] B. Gargee, M. Arup, C. Indrani, P. Biswarup, Semiconducting phase in borophene: role of defect and strain, Journal of Physics D: Applied Physics, 50 (2017) 405103. [20] S. Ipek, M.E. Kilic, A. Mogulkoc, S. Cahangirov, E. Durgun, Semiconducting defect-free polymorph of borophene: Peierls distortion in two dimensions, Physical Review B, 98 (2018). [21] A.A. Kistanov, Y.Q. Cai, K. Zhou, N. Srikanth, S.V. Dmitriev, Y.W. Zhang, Exploring the charge localization and band gap opening of borophene: a first-principles study, Nanoscale, 10 (2018) 1403-1410. [22] R. Martinazzo, S. Casolo, G.F. Tantardini, Symmetry-induced band-gap opening in graphene superlattices, Physical Review B, 81 (2010) 245420. [23] X. Fan, D. Ma, B. Fu, C.-C. Liu, Y. Yao, Cat's-cradle-like Dirac semimetals in layer groups with multiple screw axes: Application to two-dimensional borophene and borophane, Physical Review B, 98 (2018) 195437. [24] L.-C. Xu, A. Du, L. Kou, Hydrogenated borophene as a stable two-dimensional Dirac material with an ultrahigh Fermi velocity, Physical Chemistry Chemical Physics, 18 (2016) 27284-27289. [25] J.E. Padilha, R.H. Miwa, A. Fazzio, Directional dependence of the electronic and transport properties of 2D borophene and borophane, Physical Chemistry Chemical Physics, 18 (2016) 25491-25496. [26] J. Khanifaev, R. Peköz, M. Konuk, E. Durgun, The interaction of halogen atoms and molecules with borophene, Physical Chemistry Chemical Physics, 19 (2017) 28963-28969. [27] R. Peköz, M. Konuk, M.E. Kilic, E. Durgun, Two-Dimensional Fluorinated Boron Sheets: Mechanical, Electronic, and Thermal Properties, ACS Omega, 3 (2018) 1815-1822. [28] S.-G. Xu, X.-T. Li, Y.-J. Zhao, J.-H. Liao, W.-P. Xu, X.-B. Yang, H. Xu, Two-Dimensional Semiconducting Boron Monolayers, Journal of the American Chemical Society, 139 (2017) 17233-17236. [29] Y. Wang, J. Lv, L. Zhu, Y. Ma, Crystal structure prediction via particle-swarm optimization, Physical Review B, 82 (2010) 094116. [30] Y. Zhao, X. Li, J. Liu, C. Zhang, Q. Wang, A New Anisotropic Dirac Cone Material: A B 2S Honeycomb Monolayer, J Phys Chem Lett, 9 (2018) 1815-1820. [31] P. Li, Z.Y. Li, J.L. Yang, Rational Design of Two-dimensional Anode Materials: B2S as a Strained Graphene, Journal of Physical Chemistry Letters, 9 (2018) 4852-4856. [32] S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.J. Probert, K. Refson, M.C. Payne, First principles methods using CASTEP, Z. Kristall., 220 (2005) 567-570. [33] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation Made Simple, Physical Review Letters, 77 (1996) 3865-3868. [34] J.P. Perdew, K. Burke, Y. Wang, Generalized gradient approximation for the exchange-correlation hole of a many-electron system, Physical Review B, 54 (1996) 16533-16539. [35] B. Delley, An allǦ electron numerical method for solving the local density functional for polyatomic molecules, The Journal of Chemical Physics, 92 (1990) 508-517. [36] W. Tang, E. Sanville, G. Henkelman, A grid-based Bader analysis algorithm without lattice bias, J. Phys.: Condens. Matter, 21 (2009) 084204. 12
[37] R. Zhang, Z. Li, J. Yang, Two-Dimensional Stoichiometric Boron Oxides as a Versatile Platform for Electronic Structure Engineering, The Journal of Physical Chemistry Letters, 8 (2017) 4347-4353. [38] S. Demirci, N. Avazlı, E. Durgun, S. Cahangirov, Structural and electronic properties of monolayer group III monochalcogenides, Physical Review B, 95 (2017) 115409. [39] M. Topsakal, E. Aktürk, S. Ciraci, First-principles study of two- and one-dimensional honeycomb structures of boron nitride, Physical Review B, 79 (2009) 115442. [40] A. Molina-Sánchez, L. Wirtz, Phonons in single-layer and few-layer MoS2 and WS2, Physical Review B, 84 (2011) 155413. [41] T. Lin, X. Cong, M.-L. Lin, X.-L. Liu, P.-H. Tan, The phonon confinement effect in two-dimensional nanocrystals of black phosphorus with anisotropic phonon dispersions, Nanoscale, 10 (2018) 8704-8711. [42] A. Priydarshi, Y.S. Chauhan, S. Bhowmick, A. Agarwal, Strain-tunable charge carrier mobility of atomically thin phosphorus allotropes, Physical Review B, 97 (2018) 115434. [43] T. Yu, Z. Zhao, Y. Sun, A. Bergara, J. Lin, S. Zhang, H. Xu, L. Zhang, G. Yang, Y. Liu, Two-Dimensional PC6 with Direct Band Gap and Anisotropic Carrier Mobility, Journal of the American Chemical Society, 141 (2019) 1599-1605. [44] Y. Cai, G. Zhang, Y.-W. Zhang, Polarity-Reversed Robust Carrier Mobility in Monolayer MoS2 Nanoribbons, Journal of the American Chemical Society, 136 (2014) 6269-6275. [45] T. Shen, J.-C. Ren, X. Liu, S. Li, W. Liu, van der Waals Stacking Induced Transition from Schottky to Ohmic Contacts: 2D Metals on Multilayer InSe, JACS, (2019). [46] G. Su, S. Yang, S. Li, C.J. Butch, S.N. Filimonov, J.-C. Ren, W. Liu, Switchable Schottky Contacts: Simultaneously Enhanced Output Current and Reduced Leakage Current, JACS, 141 (2019) 1628-1635. [47] X.-F. Zhou, X. Dong, A.R. Oganov, Q. Zhu, Y. Tian, H.-T. Wang, Semimetallic Two-Dimensional Boron Allotrope with Massless Dirac Fermions, Physical Review Letters, 112 (2014) 085502.
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Graphical abstract
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Highlights 1. Incorporation of O, Se and Te atoms into 2D hexagonal borophene (B2X) leads to an indirect energy gap. 2. High thermodynamic, dynamic, and mechanical stability were found in B 2X. 3. B2X exhibited desirable light absorptions in visible light regions with strain tunable energy gap. 4. The electronic effective masses of B2X are comparable to MoS2
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Declaration of interests ܈The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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