Magnetic borophenes from evolutionary search

TL;DR

This paper introduces a computational approach combining ab initio evolutionary algorithms and spin-polarized density functional theory to predict novel 2D magnetic materials, revealing a stable, magnetic borophene with unique properties.

Contribution

It develops a new methodology for predicting 2D magnetic materials and discovers a stable, magnetic borophene with auxetic and magnetic properties simultaneously.

Findings

Stable borophene is an antiferromagnetic semiconductor.

Finite electron doping turns borophene into a half metal.

Borophene exhibits out-of-plane negative Poisson's ratio.

Abstract

A computation methodology based on ab initio evolutionary algorithms and the spin-polarized density functional theory was developed to predict two-dimensional (2D) magnetic materials. Its application to a model system borophene reveals an unexpected rich magnetism and polymorphism. A stable borophene with nonzero thickness was an antiferromagnetic (AFM) semiconductor from first-principles calculations, which can be further turned into a half metal by finite electron doping. In this borophene, the buckling and coupling among three atomic layers are not only responsible for the magnetism, but also result in an out-of-plane negative Poissons ratios under uniaxial tension, making it the first elemental material possessing auxetic and magnetic properties simultaneously.

Figures

a, our calculations demonstrated that the most stable structure has ferromagnetic order and $P31m$ symmetry, and its lattice constants and magnetization (3 $\mu_{B}$ per Cr atom) are in good agreement with the experimental results. [R02 ](#bib.bib2) Compared with high-throughput computational screening, [R04 ](#bib.bib4); [R13 ](#bib.bib13); [R14 ](#bib.bib14) this search uncovered many new metastable magnetic structures which do not have analogous known parent bulk materials in the database, suggesting that our search is not biased by the database and thus offers a more complete sampling of t

c), non-collinear $120^{\circ}$ AFM order (ncl-AFM, see Figs. 2a and 2d), FM, and NM states are -6.162, -6.162, -6.161, and -6.152 eV/atom, which are higher in energies than previously synthesized 2-Pmmn (-6.19 eV/atom), [R32 ](#bib.bib32) $\chi_{3}$ (-6.24 eV/atom), [R34 ](#bib.bib34) $\beta_{12}$ (-6.23 eV/atom), [R34 ](#bib.bib34) and $\alpha$-sheet structures (-6.28 eV/atom), [R20 ](#bib.bib20) but are lower in energy than graphene-like borophene (-5.42 eV/atom), [R35 ](#bib.bib35) indicating that $19-P6/mmm$ borophene is a metastable structure with the ground-state AFM order. There is onl

d, the lowest conduction band with minority spin is very close to the Fermi level along $\Gamma$–X and $\Gamma$–Y directions, while the conduction band with majority spin is relatively far from the Fermi level. Obviously, it may be tuned from the FM semiconductor to a FM half metal by shifting the Fermi level upward a little, which can be realized by electron doping. Note that a charge doping of $10^{15}$ $cm^{-2}$ had already been experimentally achieved in some 2D materials by a gate-voltage. [R46 ](#bib.bib46) For the electron doping concentration of 0.02 ($1.2\times 10^{14}$ $cm^{-2}$) and

a), which are much softer than previously reported structures due to its large empty spaces among three layers. [R32 ](#bib.bib32); [R50 ](#bib.bib50) In sharp contrast to the positive in-plane Poisson’s ratio ($\nu_{xy}$ = $\nu_{yx}$ = 0.63), $19-P6/mmm$ borophene, regardless of its magnetic order, possesses an unexpected out-of-plane negative Poisson’s ratios (NPR) when tensile strain was applied in the $x$ (parallel to the $a$ axis) and $y$ (parallel to $b$ axis) directions. As shown in Figs. 7b and 7c, the out-of-plane strains $\epsilon_{z}$ of striped-AFM $19-P6/mmm$ borophene have the li

Full text

I INTRODUCTION

Two-dimensional (2D) magnetic materials have attracted huge interest owing to their potential applications in spintronics and data storage. R01 ; R02 ; R03 ; R04 ; R05 ; R06 ; R07 ; R08 ; R09 ; R10 ; R11 ; R12 The ultimate thinness changes the physical properties dramatically compared to the corresponding bulk materials. Although a number of 2D magnetic materials have been proposed theoretically, R09 ; R10 ; R11 ; R12 ; R13 ; R14 few of them were synthesized experimentally. Recent experimental studies found that 2D forms of CrI3 and Cr2Ge2Te6 can inherit the ferromagnetic (FM) orders at low temperatures from their bulk forms. R01 ; R02 Strikingly, even for paramagnetic (PM) bulk material, i.e., VSe2, its monolayer form shows an unexpected strong room-temperature ferromagnetism on van der Waals substrates. R03 Stimulated by these exiting experimental reports, 56 new magnetically ordered monolayer structures were predicted from high-throughput computation to be exfoliable from known magnetic bulk materials. R04 However, discovering new 2D magnetic materials beyond exfoliation from the parent bulk compounds is still governed by trial and error approaches. Among the light element-based 2D materials, graphene is not magnetic, R15 but may be a promising material for spintronics by optimizing defects, substrate, or adsorbing hydrogen atoms. R16 ; R17 ; R18 ; R19 Boron, with only three valence electrons, is electron deficient, resulting in the formation of multicenter B-B bonds and rich polymorphism with great chemical complexity in borophenes. R20 ; R21 ; R22 ; R23 ; R24 ; R25 ; R26 ; R27 ; R28 ; R29 ; R30 ; R31 Various borophenes have been successfully synthesized on Ag, Cu, or Al substrates under ultrahigh-vacuum conditions. R32 ; R33 ; R34 ; R35 ; R36 The atomically thin borophenes display remarkable properties, such as the emergence of superconductivity, massless Dirac fermions, superior transport and mechanical properties. R20 ; R21 ; R22 ; R23 ; R24 ; R25 ; R26 ; R27 ; R28 ; R29 ; R30 ; R31 ; R32 ; R33 ; R34 ; R35 ; R36 However, the phase diagram of borophene remains largely unexplored due to lack of efficient structure prediction method. With the goal of uncovering more 2D mateials with superior magnetic properties besides magnetic doping that have been experimentally confirmed as one of simple and efficient methods, R29 ; R30 ; R31 we developed a new strategy based on evolutionary structure prediction and explored the magnetic borophenes in a systematic manner.

II METHOD

To search for the stable 2D magnetic structures, we further developed a new computational scheme based on the ab initio evolutionary algorithm uspex R37 ; R38 ; R39 ; R40 combined with the spin-polarized density functional theory (DFT). The initial structures are produced with a randomly assigned layer group symmetry and a user-defined initial thickness of 2 Å, while the initial magnetic moment was generated by random numbers, which are classified as 0 (0,0.1], 1 (0.1, 0.3], 4 (0.3, 0.5], $\pm 1$ (0.5, 0.75], and $\pm 4$ (0.75, 1.0], and then the structures are treated as nonmagnetic (NM) order, FM with low spin (FM-LS), FM with high spin (FM-HS), AFM with low spin (AFM-LS), and AFM with high spin (AFM-HS), R41 respectively. The structures with odd number of atoms are supposed to have FM order. The ratio of structures with different magnetic orders (NM, FM-LS, FM-HS, AFM-LS, and AFM-HS) can be either uniform or user-defined. The relaxed energies associated with the magnetic moment were used as a criterion for parent structures selection to generate new structures by various evolutionary operators, such as random structure generator, heredity and mutations (including lattice, coordination, and spin heredity/mutations). For each relaxed structure, we sum over the (absolute) values of the magnetic moments for all atoms ($\sum M_{i}$ and $\sum|M_{i}|$). If $\sum|M_{i}|$ is close to zero ($<0.03\times N_{atoms}$, where $N_{atoms}$ is the number of magnetic atoms), we assign it to be NM; otherwise, the structure would be either AFM ($|\sum M_{i}|<0.25\times N_{atoms}$) or FM ($|\sum M_{i}|\geq 0.25\times N_{atoms}$). We also assign the structure to be LS {if $\max(|M_{i}|)<1.5$} or HS {$\min(|M_{i}|)>1.5$}. Otherwise, the structure would be labelled as a hybrid HSLS. We limited our structural search within an unit cell of 5-20 atoms with a vacuum of 20 Å. Structure relaxations and total energy calculations were performed with the projector-augmented wave R42 (PAW) method as implemented in the VASP package, R43 and the exchange-correlation energy was computed within the generalized gradient approximation (GGA) with the functional of Perdew, Burke, and Ernzerhof (PBE). R44 The plane-wave cutoff energy of 600 eV and uniform $\Gamma$-centered k-points grids with the resolution of $2\pi\times 0.025$  Å*-1* were used. The convergence for terminating the electronic self-consistency cycle and the force criterion for structure relaxation were set at 10*-6* eV and 10*-2* eV/Å. Phonon dispersion curves were calculated by the supercell method (the $2\times 2\times 1$ supercell for NM, FM and AFM states) using the phonopy package R45 with the energy convergence of 10*-7* eV.

III RESULTS AND DISCUSSION

To test the reliability and accuracy of this new method, we first investigated 2D CrI3 system. As shown in Fig. 1a, our calculations demonstrated that the most stable structure has ferromagnetic order and $P31m$ symmetry, and its lattice constants and magnetization (3 $\mu_{B}$ per Cr atom) are in good agreement with the experimental results. R02 Compared with high-throughput computational screening, R04 ; R13 ; R14 this search uncovered many new metastable magnetic structures which do not have analogous known parent bulk materials in the database, suggesting that our search is not biased by the database and thus offers a more complete sampling of the configurational space. We then applied this method to borophene. Both magnetic and nonmagnetic borophenes are unveiled in Fig. 1b. Fewer than 300 magnetic borophenes were predicted among 3500 structures in total, after the removal of duplicate structures which were found more than once in the search. We also note that magnetic calculations may be numerically sensitive to the choice of computational parameters, and magnetism of some planar monolayers may disappear in more converged calculations because high density of states (DOS) at the Fermi level can be lowered by structural (Peierls) distortion rather than spin polarization. Since elemental boron prefers NM state, all magnetic borophenes are metastable phases with respect to its ground-state structure. However, rigorous convergence of calculations for all structures would be prohibitive because it is beyond our computational resource. Hence, we focus on these low-energy structures in which ferromagnetism persists, as they are more likely to be synthesized on a suitably chosen substrate. The most stable magnetic structure contains 19 atoms per unit cell (Fig. 2) and is designated as $19-P6/mmm$ borophenes according to its symmetry. For $19-P6/mmm$ borophene with NM order (NM $19-P6/mmm$ borophene), the lattice parameters are a = b = 6.033  Å, and c = 19.97  Å. Three inequivalent atomic positions are B1 (0.0,0.0,0.5), B2 (0.362,0.181,0.5), and B3 (0.576,0.153,0.434), which form triatomic 2.64  Å thick layers (see Figs. 2a and 2b). The middle layer, through which a mirror plane passes, consists of isolated hexagonal B7 clusters (meanwhile, leading to large empty spaces, see Fig. 2a), while the top and bottom layers, are composed of triangular units, and they connect together to form dodecagonal vacancies on top of the B7 clusters, which forms particular symmetric voided structure.

The calculated total energies for the $19-P6/mmm$ borophene with striped-AFM (Fig. 2c), non-collinear $120^{\circ}$ AFM order (ncl-AFM, see Figs. 2a and 2d), FM, and NM states are -6.162, -6.162, -6.161, and -6.152 eV/atom, which are higher in energies than previously synthesized 2-Pmmn (-6.19 eV/atom), R32 $\chi_{3}$ (-6.24 eV/atom), R34 $\beta_{12}$ (-6.23 eV/atom), R34 and $\alpha$-sheet structures (-6.28 eV/atom), R20 but are lower in energy than graphene-like borophene (-5.42 eV/atom), R35 indicating that $19-P6/mmm$ borophene is a metastable structure with the ground-state AFM order. There is only one magnetic atom per unit cell of FM $19-P6/mmm$ borophene (Fig. 2a), which is located at the center of B7 clusters (B1 atoms). The local magnetic moment of each B1 atom is about 1.0 $\mu_{B}$. The geometry of B1 atoms forms a magnetic triangular lattice and thus many configurations include FM, striped-AFM and ncl-AFM orders are considered (Fig. 2). The calculations show that ncl-AFM phase, which are considerably, by 2.2 meV per 19 atoms, and 24.5 meV per 19 atoms, lower in energy than those of striped-AFM and FM orders. Therefore, ncl-AFM borophene is the ground-state structure, which is an indirect semiconductor with bandgap of 0.54 eV (Fig. 3a). Moreover, by mapping the DFT energy difference to the classical spin model, the nearest-neighbor exchange parameter $J_{1}$ is estimated to be -5.45 meV and the next nearest-neighbor exchange parameter $J_{2}$ is $\sim$ -0.14 meV. $J_{1}$ is much bigger than $J_{2}$, indicating that $19-P6/mmm$ borophene has a ground-state ncl-AFM structure, which is dominated by the AFM $J_{1}$ in the triangular lattice. However, under minor electron doping, ncl-AFM phase is less stable than striped-AFM and FM phases, and there is even an AFM-FM transition when doping concentration is higher than $\sim$ 0.006 electrons per atom ($3.6\times 10^{13}$ $cm^{-2}$). Hence the ncl-AFM structure will not be discussed further in subsequent discussions for the doped cases.

Electronic band structure calculations show that NM $19-P6/mmm$ borophene is metallic (Fig. 3b). In contrast, striped-AFM and FM $19-P6/mmm$ borophenes are semiconductors with an indirect bandgap of 0.41 eV and a direct bandgap of 0.18 eV, respectively (Figs. 3c and 3d). The electronic stability of different states (NM metal $<$ FM or AFM semiconductor) is consistent with the energetic stability. To explore the physical origin of magnetism in striped-AFM borophene, we analyzed the atomic orbital-resolved band structures as shown in Figs. 3e and 3f. In Fig. 3e, two flat bands around the Fermi level are primarily contributed by the $p_{z}$ orbitals of magnetic B1 atoms. Between them, one flat band with majority spin is colored in pink, while the other with minority spin is colored in dark blue. Interestingly, there also exist two more flat bands in Fig. 3f, which mainly originate from the other B1 atoms, but their spin characters are opposite. That is, two highest valence bands or two lowest conduction bands (flat bands) with opposite spin electrons are offseted each other (Figs. 3e and 3f) and thus, the whole structure exhibits AFM order. As mentioned above, these flat bands are dominated by the unpaired electrons ($p_{z}$ orbitals), which can be confirmed by the spin charge density distribution (Figs. 4a and 4b). Large bubble-like spin density maxima are localized on top of B1 atoms (center of B7 clusters). These bubbles represent the majority-spin (colored in red) and minority-spin electrons (colored in blue) and have the same size and shape due to mirror symmetry. The special arrangement of spin densities among B1 atoms in the plane are responsible for striped-AFM order. In addition, the electron localization function (ELF) of FM $19-P6/mmm$ borophene in Figs. 4c and 4d shows that each B1 atom has an unpaired electron, and consequently has a local magnetic moment of 1 $\mu_{B}$.

FM $19-P6/mmm$ borophene is a direct narrow-gap semiconductor. As see in Fig. 3d, the lowest conduction band with minority spin is very close to the Fermi level along $\Gamma$–X and $\Gamma$–Y directions, while the conduction band with majority spin is relatively far from the Fermi level. Obviously, it may be tuned from the FM semiconductor to a FM half metal by shifting the Fermi level upward a little, which can be realized by electron doping. Note that a charge doping of $10^{15}$ $cm^{-2}$ had already been experimentally achieved in some 2D materials by a gate-voltage. R46 For the electron doping concentration of 0.02 ($1.2\times 10^{14}$ $cm^{-2}$) and 0.008 ($4.8\times 10^{13}$ $cm^{-2}$) electrons per atom (Figs. 5a and 5b), the bands in the majority and minority spin channels are separated, suggesting intrinsic ferromagnetism. Most importantly, minority spin channels are completely polarized, indicating that $19-P6/mmm$ borophene is a half-metal whose majority spin electrons behave like semiconductor, while minority spin electrons display metallic conduction. All of these suggest that this structure may have significant potential applications in spintronics. Figure 5c shows the variation of relative energy difference between FM and striped-AFM phases via the electron doping concentration from 0 to $1.2\times 10^{14}$ $cm^{-2}$. Actually, the $19-P6/mmm$ borophene is a ground-state FM half metal above the concentration of 0.006 electrons per atom. We also calculated the phonon dispersion curves with finite electron doping. The absence of imaginary frequencies in the phonon spectrum (Fig. 5d) is suggesting that the structure remains dynamically stable upon doping. Note that the rectangular $19-P6/mmm$ borophene has a large unit cell and the excess electrons are dominantly distributed on the magnetic B1 atoms in both FM and striped-AFM states (Fig. 6), the weak spurious electrostatic interaction can be cancelled out, so the energy differences between the charged FM and striped-AFM states are correct. R47

Because of the strong covalent B-B networks, borophenes are usually expected to be hard 2D materials, e.g., the in-plane Young s modulus of $2-Pmmn$ borophene (398 GPa$\cdot$nm along the a axis), potentially rivals graphene (the hardest 2D material) at 340 GPa$\cdot$nm. R32 ; R48 It is natural to study the mechanical properties of $19-P6/mmm$ borophene with different magnetic states. With its reconstructed rectangular lattice, using the standard Voigt notation, the elastic strain energy per unit area can be expressed asR49 Uϵ = (1/2)C11$\epsilon^{2}_{xx}$ + (1/2)C22$\epsilon^{2}_{yy}$ + C12\epsilon_{xx}$$\epsilon_{yy} + 2C66$\epsilon^{2}_{xy}$, where C11, C22, C12, and C66 are the elastic constants, corresponding to second partial derivatives of the energy with respect to strain. The in-plane Young’s modulus and Poisson’s ratio can be derived from the elastic constants as: Ex = (C11C22–C12C21)/C22, Ey = (C11C22–C12C21)/C11, $\nu_{xy}$ = C21/C22, and $\nu_{yx}$ = C12/C11. The calculated C11, C22, C12, and C66 for striped-AFM $19-P6/mmm$ borophene are 227, 227, 143, and 42 GPa$\cdot$nm, so the Young s modulus is equal to $E_{x}$ = $E_{y}$ = 137 GPa$\cdot$nm, and the corresponding Poisson’s ratio is equal to 0.63 (Fig. 7a), which are much softer than previously reported structures due to its large empty spaces among three layers. R32 ; R50 In sharp contrast to the positive in-plane Poisson’s ratio ($\nu_{xy}$ = $\nu_{yx}$ = 0.63), $19-P6/mmm$ borophene, regardless of its magnetic order, possesses an unexpected out-of-plane negative Poisson’s ratios (NPR) when tensile strain was applied in the $x$ (parallel to the $a$ axis) and $y$ (parallel to $b$ axis) directions. As shown in Figs. 7b and 7c, the out-of-plane strains $\epsilon_{z}$ of striped-AFM $19-P6/mmm$ borophene have the linear relationship with $\epsilon_{x}$ and $\epsilon_{y}$, respectively. When it was compressed, the data were fitted to the function of $y$ = –0.119(–0.269)$x$ in the $x$ ($y$) direction. Since the out-of-plane Poisson’s ratio R51 is defined as $\nu_{zx}$ = –$\partial$ $\epsilon_{z}$/$\partial$ $\epsilon_{x}$ or $\nu_{zy}$ = –$\partial$ $\epsilon_{z}$/$\partial$ $\epsilon_{y}$, these results represent positive Poisson’s ratios in both directions. While for tension, the data were fitted to the function of $y$ = 0.381(0.162)$x$, so they are $\nu_{zx}$ = –0.381 and $\nu_{zy}$ = –0.162, showing NPR effect on the contrary, which are different from those of phosphorene or borophane (NPR for both tension and compression only either in the $y$ or in the $x$ direction). R52 ; R53 The length of specific B2-B3 bonds (named as $x$-bonds, colored in red, see Fig. 7d) increases under tensile strain along the $x$ direction, so does for the other specific B2-B3 bonds (named as $y$-bonds, colored in blue, see Fig. 7d) along the $y$ direction. The slope of $x$-bonds is more than that of $y$-bonds, implying that $x$-bonds elongate faster than $y$-bonds upon stretching and resulting in|$$\nu_{zx}$$| $>$ |$$\nu_{zy}$$|. Owing to very similar lattice constants among AFM, FM, and NM states, the $\nu_{zx}$ and $\nu_{zy}$ of NM (FM) $19-P6/mmm$ borophene are –0.416 (–0.379) and –0.194 (–0.167) by using the same methods. Therefore, such exotic buckling bonding configurations are responsible for the NPR effect (see the inset of Fig. 7d). At last, the energy difference of striped-AFM, ncl-AFM refer to the FM phases via the biaxial tensile (negative strain) or compressive strain (positive strain) was plotted in Fig. 8. It showed that the striped-AFM structure is the most stable phase among different states between the range of 1% and 3%, while the ncl-AFM structure is the most stable one under biaxial tensile strain.

IV CONCLUSION

In conclusion, we developed and performed a systematic evolutionary search for stable magnetic borophenes and identified that $19-P6/mmm$ borophene is a stable striped-AFM semiconductor, which not only can be further tuned into a half metal, but also has intrinsic out-of-plane negative Poisson’s ratios. The $19-P6/mmm$ borophene may be grown on Au (111) substrate because of the small lattice mismatch ($<$ 5%), or be made on suitable substrates of transition-metal borides. R54 Our calculations show that the intrinsic magnetism is destroyed by the interaction between $19-P6/mmm$ borophene and the substrate, i.e, Au (111). However, if the freestanding form can be realized from experiments, the outstanding properties make it a promising candidate in spintronics and nanoelectromechanical devices simultaneously.

ACKNOWLEDGMENTS This work was supported by the National Science Foundation of China (Grants 11674176 and 11874224), the Tianjin Science Foundation for Distinguished Young Scholars (Grant No. 17JCJQJC44400). A.R.O. thanks Russian Science Foundation (Grant No. 16-13-10459), and the Foreign Talents Introduction and Academic Exchange Program (Grant No. B08040). Q. Z. is grateful for support from the National Nuclear Security Administration under the Stewardship Science Academic Alliances program through DOE Cooperative Agreement DE-NA0001982. X. D. and X. F. Z thanks the computing resources of Tianhe II and the support of Chinese National Supercomputer Center in Guangzhou.