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Stability and Strength of Atomically Thin Borophene from First Principles Calculations

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Stability and Strength of Atomically Thin Borophene from First Principles Calculations Materials Research Letters ISSN: (Print) 2166-3831 (Online) Journal homepage: http://www.tandfonline.com/loi/tmrl20 Stability and strength of atomically thin borophene from first principles calculations Bo Peng, Hao Zhang, Hezhu Shao, Zeyu Ning, Yuanfeng Xu, Gang Ni, Hongliang Lu, David Wei Zhang & Heyuan Zhu To cite this article: Bo Peng, Hao Zhang, Hezhu Shao, Zeyu Ning, Yuanfeng Xu, Gang Ni, Hongliang Lu, David Wei Zhang & Heyuan Zhu (2017): Stability and strength of atomically thin borophene from first principles calculations, Materials Research Letters, DOI: 10.1080/21663831.2017.1298539 To link to this article: http://dx.doi.org/10.1080/21663831.2017.1298539 © 2017 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. Published online: 19 Mar 2017. Submit your article to this journal View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tmrl20 Download by: [Iowa State University] Date: 20 March 2017, At: 08:24 MATER. RES. LETT., 2017 http://dx.doi.org/10.1080/21663831.2017.1298539 ORIGINAL REPORT Stability and strength of atomically thin borophene from first principles calculations Bo Penga, Hao Zhang a, Hezhu Shaob, Zeyu Ninga, Yuanfeng Xua,GangNia, Hongliang Luc, David Wei Zhangc and Heyuan Zhua aShanghai Ultra-precision Optical Manufacturing Engineering Research Center, Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai, People’s Republic of China; bNingbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, People’s Republic of China; cState Key Laboratory of ASIC and System, Institute of Advanced Nanodevices, School of Microelectronics, Fudan University, Shanghai, People’s Republic of China ABSTRACT ARTICLE HISTORY A new 2D material, borophene, has been grown successfully recently on single crystal Ag substrates. Received 4 September 2016 β χ Three main structures have been proposed ( 12, 3 and striped borophene). However, the stability KEYWORDS of three structures is still in debate. Using first principles calculations, we examine the dynamical, β χ β Two-dimensional materials; thermodynamical and mechanical stability of 12, 3 and striped borophene. Free-standing 12 and borophene; stability; χ3 borophene is dynamically, thermodynamically and mechanically stable, while striped borophene strength is dynamically and thermodynamically unstable due to high stiffness along a direction. The origin of high stiffness and high instability in striped borophene along a direction can both be attributed to strong directional bonding. Our work shows a deep connection between stability and strength, and helps researchers to estimate accurately the mechanical performance of 2D materials. IMPACT STATEMENT • A benchmark for examining the relative stability of different structures of borophene is provided. • Strong directional bonding in striped borophene leads to high stiffness and high brittleness. 1. Introduction β12 and χ3 borophene has planar structure with periodic Recent years have witnessed many breakthroughs in holes [17], while striped borophene has buckled struc- research on 2D materials due to their potential applica- ture with anisotropic corrugation [16]. The following tions in next-generation electronic and energy conver- first principles calculations have predicted that striped sion devices [1–14]. Recently, a new type of 2D material, borophene possesses remarkable mechanical proper- borophene (2D boron sheet) [15], has been successfully ties [35,36], which may rival graphene [16]. However, grown on single crystal Ag(111) substrates by two par- phonon instability in striped borophene is observed [37], allel experiments [16,17]. Although various proposals of which may challenge previous results demonstrating stable 2D boron sheets and quasiplanar boron clusters that borophene is stiffer than graphene along a direc- have been made [18–34], three main structures (β12, χ3 tion [16,35,36]. and striped borophene) have been observed by scan- Theoretical investigation of the formation of boron ning tunnelling microscopy in these two experiments: sheet on Ag(111) surface has demonstrated that stable CONTACT Hao Zhang [email protected] Shanghai Ultra-precision Optical Manufacturing Engineering Research Center, Key Laboratory of Micro and Nano Photonic Structures (Ministry of Education), Department of Optical Science and Engineering, Fudan University, Shanghai, 200433, People’s Republic of China © 2017 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 B. PENG ET AL. 1 2. Computational details boron sheet should contain 6 vacancies in a striped pat- tern [38]. This is consistent with previous theoretical All the calculations are performed using the Vienna studies indicating that planar boron sheets with vacancies ab initio simulation package (VASP) based on density aremorestable[21,23–25,29–31,39,40]. The ground state functional theory (DFT) [58,59] under the generalized of2Dboronisindebate[16,17,21,31,35,36,38–45]. These gradient approximation expressed by the Perdew–Burke– debates raise several questions: (i) Are these three struc- Ernzerhof functional [60]. A plane-wave basis set is β χ tures stable? (ii) What is the relative stability of 12, 3 employed with kinetic energy cutoff of 700 eV with and striped borophene? (iii) Could these structures pos- 15×25×1, 29×29×1and27×15×1 k-mesh for β12, χ3 sess high hardness in thermodynamic aspect? In fact, due and striped borophene respectively during structural to the structural complexity of boron, even the relative relaxation, until the energy differences are converged α β − stability of -and -rhombohedral boron has been dis- within 10 8 eV,withaHellman–Feynmanforcecon- − cussed for over 30 years [46–52]. Thus a systematic inves- vergence threshold of 10 6 eV/Å. The vacuum space is tigation is needed to examine the stability and strength of around 15 Å. The harmonic interatomic force constants these three structures. (IFCs) are obtained using density functional perturba- When discussing the stability of crystal structures, it tion theory (DFPT) within a supercell approach [61]. is important to distinguish dynamical, thermodynamical A5×7×1, 5×5×1and7×5×1supercellwith7×7×1, and mechanical stability [53,54].Amaterialisdynam- 5×5×1and7×5×1 k-meshisusedforβ12, χ3 and ically stable when no imaginary phonon frequencies striped borophene, respectively. The phonon dispersion exist. Thermodynamical stability can be described by the and thermodynamical properties are calculated from the Helmholtz free energy [55], which demonstrates how harmonic IFCs using the PHONOPY code [62,63]. We phonons determine the relative stability at finite temper- calculate the elastic tensor coefficients including ionic atures. Regarding mechanical stability, the Born–Huang relaxations using the finite differences method [64,65]. criteria for elastic constants must be fulfilled56 [ ,57]. The Born–Huang mechanical stability criteria provide a nec- essary condition for the dynamical stability, but not a 3. Results and discussion sufficient one [54]. Therefore, although the mechanical 3.1. Crystal structure and dynamical stability properties of striped borophene have been studied inten- sively [16,35,36], the questions of stability still remain Three structures of borophene are shown in Figure 1.For unanswered. In this work, the dynamical, thermodynam- β12 (χ3) borophene, five (four) B atoms in the unit cell ical and mechanical stabilities of β12, χ3 and striped are arranged in the same plane. For striped borophene, borophene are evaluated using first principles calcula- there is no corrugation along a direction, while a ver- tions.Wefurtherstudyforthefirsttimethemechanical tical buckling along the b direction is observed. The properties of β12 and χ3 borophene, and compare the major difference between striped borophene and the stiffness of all these structures. The bonding character- othertwostructuresistheabsenceofvacancies.The istics are also examined to understand the stability and introduction of vacancies leads to lower cohesive energy, strength of striped borophene. as shown in Table 1, which is in good agreement with Figure 1. Top view and side view of (a) β12,(b)χ3 and (c) striped borophene. MATER. RES. LETT. 3 Table 1. Calculated lattice constant a and b,bucklingheighth, cohesive energy Ec and ZPE of three structures of borophene. abhEc ZPE Space group (Å) (Å) (Å) (eV/atom) (eV/atom) β12 borophene Pmm2 5.07 2.93 0 −6.147 0.116 –5.0[17]2.9[17]– – χ3 borophene Cmmm 4.45 4.45 0 −6.159 0.114 –4.3[17]4.3[17]– – Striped borophene Pmmn 1.613 2.864 0.911 −6.099 0.109 -5.1±0.2 [16]2.9±0.2 [16]– – Note: Previous experimental data are also listed for comparison. Figure 2. Calculated band structures of (a) β12 borophene and (b) striped borophene along different symmetry lines. previous results [21,39]. The optimized geometries of For newly proposed 2D materials, stability is an three structures are listed in Table 1.Thepredicteda important aspect for experimental realization and large- 1 of striped borophene corresponds to 3 the a observed
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