Edge-dependent electronic and magnetic characteristics of freestanding \beta_12-Borophene Nanoribbons

Nanoribbons produced from cutting {\beta}_12-Borophene sheet is investigated by density functional theory. The electronic and magnetic properties of Borophene nanoribbons are studied and found that all considered ribbons are metal which is in good agreement with recent experimental results. {\beta}_12-Borophene nanoribbons have a lot of diversity due to existence of 5 Boron atoms in a unit cell of the sheet. The magnetic properties of ribbons are strongly dependent on the cutting direction and the edge profile. It is interesting that a ribbon with a specific width can be a normal or ferromagnetic metal with magnetization in just one edge or two edges. The spin anisotropy is observed in some ribbons so that magnetic moment is not the same in both edges in antiferromagnetic configuration. The effect comes from the edge asymmetry of the ribbons and results in the breaking of spin degeneracy in the bandstructure. Our findings show that {\beta}_12 nanoribbons are potential candidates for next-generation spintronic devices.

Cutting two-dimensional (2D) structures along one-direction makes them as one-dimensional nanoribbons. Nanoribbons have electronic, magnetic, and optical properties which are different from 2D structures. It is well known that the zigzag-edge graphene nanoribbons are metal while the armchair-edge nanoribbons are metal or semiconductor with respect to the ribbons width [34,35]. Garcia-Fuente et al. [8] studied borophene nanoribbons produced from 2Pmmn and 8Pmmn borophene sheets. They found that 8Pmmn borophene nanoribbons are more stable and have more interesting properties. Nanoribbons can be non-magnetic or magnetic dependent on the cutting directions. In addition, the 8Pmmn nanoribbon can be a metal or a semiconductor with respect to the cutting direction and its width. Meng and co-workers [36] investigated 2Pmmn borophene nanoribbons and reported that the ribbons produced by cutting the sheet along x-direction are metal, whereas, the ribbons produced from y-direction can be magnetic. They also found that upon hydrogenation all nanoribbons become non-magnetic.
Zhong et al. [37] has reported successful synthesis of borophene nanoribbons on Ag(110) surface very recently. They observed several phase of the borophene ribbons like 3 , and 8 . Motivated by these experimental works on the borophene sheets and ribbons, we study the electronic and magnetic properties of 12 borophene nanoribbons using density functional theory for the first time. Results show that all considered borophene nanoribbons are metal and the edge magnetization is dependent on the cutting direction. In addition, we observe that some ribbons are magnetic in just one edge. The spin anisotropy of the edge states is also observed that is attributed to the reconstruction of the edge. The electron density analysis reveals that the charge accumulation occurs in some edges which is consistent with recent experimental results [38].
The next section is devoted to simulation method. The simulation results are presented in section 3. We analyze binding energy, electron density, transmission channel, electron localized function, bandstructure, and magnetization of the considered ribbons in detail. Some sentences are given as a summary in the end of the article.

2-Simulation Details
All calculations were performed using density functional theory (DFT) implemented in SIESTA package [39]. The interaction between valance and core electrons was described by norm-conserved Troullier-Martins pseudopotentials [40] and Perdew-Burke-Ernzerhof (PBE) [41] generalized gradient approximation (GGA) was employed as exchange-correlation functional. Cut-off energy was 200Ry and 100 k-points centered at Γ-point was used in direction where the ribbon is periodic. We considered 30 Å vacuum to eliminate interlayer interactions. All ribbons were fully relaxed until force was converged to 0.001eV/ Å. 13 orbitals were employed for each Boron atom consisting of 2 sets of orbitals of s type, 2 sets of p type and 1 set of d type with cut-off radius of 2.8 Å, 3.35 Å and 3.35 Å, respectively. Ref. [37,38] showed that the electrons are accumulated in the boundary of the 12 sheet. In addition, it has been recently reported that the edge of -Borophene ribbons hosts more electrons than the body of the ribbon. The ELF also supports the above results so that the electrons are completely localized in the edge of the ribbons. Our findings show that the edge of Borophene ribbons is able to absorb atoms and molecules.

3-Results and Discussion
Mannix at al. reported that their synthesized Borophene was partially hydrogenated [1] which can be attributed to the edge absorption with respect to our results.
NBBXBNRs are more stable than NAAXBNRs. The bonding length of the edge Boron atoms is 3 percent shorter than the sheet. In addition, bonding of the edge Boron atoms with Boron atoms next to the edge is stronger in the ribbon due to the shorter bond length. The electron density analysis, Figure  In the following, the ribbons obtained by cutting the Borophene sheet along y-direction are investigated.
As it is shown in Figure 1, five Boron atoms of a 12 -Borophene unit cell have different x positions, numbered by 1 to 5 in Figure 1 so that there is a lot of variety for the ribbons. We name each ribbon as NYuvBNR where u, v = 1. . .5 and u and v shows the number of the Boron atom which is in bottom or top edge, respectively. N stands for the number of the Boron atoms in a unit cell of the ribbon and describes the width of the ribbon. We investigate the ribbons with N = 20 to N = 25 so that the maximum width of the ribbon studied here is 24 Å. We found that for each N there are three different edge configurations, therefore, 18 ribbons are studied in details. It is interesting to note that there are just two distinct configurations for ribbons created from 2Pmmn and 8Pmmn Borophene sheets [14] but here, we are faced with more diversity.
The diversity leads to the more complexity of the 12 -Borophene nanoribbons along y-direction.
The total energy per atoms of NYBNRs is plotted in Figure 7. The total energy analysis shows that the NYBNRs can be magnetic in some widths, unlike NXBNRs. First, we study allotropes of 20YBNR and 25YBNR which have the same configurations. The increase of the ribbon width strengthens the stability of the ribbon which is clear with more energy of 25YBNRs than 20YBNRs. Although, some allotropes are magnetic, the ground state, 20Y32BNR and 25Y32BNR, is nearly nonmagnetic. The optimized structures of 20YBNRs are plotted in the supplementary information, Fig. S1. We find that the edge configuration, the bonding length, and the existence of fully occupied hexagonal lattices or hexagonal hole lattices are key factors in electronic and magnetic properties of ribbons. The electron localization function of 20YBNR allotropes is plotted in Figure 8(a). It is clear that the edge significantly effects on the localization of electron. As it is seen in Figure 8 20Y32 is a few meV more than nonmagnetic one. Highest electron localization is observed in 20Y21 and therefore, the energy difference between its magnetic and nonmagnetic states is more. We expect that if the ribbons growth on the substrate, the interaction between Boron atoms and the substrate becomes strong in some ribbons like 20Y21 and electron transfer between the ribbon and the substrate reduces the magnetization. For the magnetic state, we considered both ferromagnetic, two edges with same majority spin orientation, and anti-ferromagnetic, two edges with opposite majority spin orientation, configurations and found that they are degenerate. The spin density of allotropes of 20YBNR is depicted in Figure 8 Figure 9(a) and 9(b) shows ELF and spin density in ferromagnetic configuration, respectively. ELF shows well that why these ribbons are magnetic. The electron localization is observed in both edges. The energy difference between magnetic and nonmagnetic states of 22Y51BNR is more than others because both edges are composed of fully occupied hexagonal lattices. It is clear that the electron localization is weaker in hexagonal hole lattices like 23Y13BNR. The edge atoms are coupled to their neighbor's atoms anti-ferromagnetically in the y-direction and ferromagnetically in the x-direction.
22Y15 exhibits the strongest magnetization so the magnetic moment is 0.75 in both edges. The most spin anisotropy is observed in 24Y31 so that the magnetic moment difference between two edges is equal to 0.54 , and upper edge has the higher magnetic moment. The spin anisotropy gives rise to spin splitting of the bandstructure in antiferromagnetic configuration, see Figure 10. Although one expects that the bandstructure becomes spin degenerate in anti-ferromagnetic configuration, the structural anisotropy of the edges breaks degeneracy and the bands become spin-dependent.
Results show that cutting 12 sheet cannot induce a band gap in the structure, so all ribbons are metal which is in good agreement with recent experimental results [37].  Figure 10. We assumed that the spin orientation of mentioned atom is spin-down in 23Y13, and therefore, the bands are formed from spin-down electrons. About 24Y31, the spin orientation is set to be spin-up for atom with the highest magnetic moment in antiferromagnetic configuration. Note that these bands are a direct consequence of spin anisotropy that were not reported in previous studies about Borophene nanoribbons [14].
We did not consider the role of substrate on the electronic and magnetic properties of nanoribbons. We expect that the substrate reduces the anisotropy of the structure and moderate the conductance of the structure. However, it was shown that 12 Borophene is metal and exhibits Dirac cone in the presence of the supported Ag [6]. On the other hand, recent experimental results show that the synthesized ribbons are metal and they are flat, therefore, we expect that the spin anisotropy reported in the article is a robust feature of some widths of the ribbon and it will be alive in the presence of the substrate. The spin anisotropy makes the 12 nanoribbons a potential candidate for future spintronic and spin filtering devices. Unlike previous nanoribbons like Graphene nanoribbons, Silicene nanoribbons or Germanene nanoribbons, 12 -Borophene nanoribbons have a lot of diversity with individual characteristics which makes them an interesting candidate for next-generation electronic devices.

4-Summary
We have analyzed freestanding 12 -Borophene nanoribbons using density functional theory. The structural, electrical, and magnetical properties of ribbons are studied in details and found that the magnetization of ribbon is strongly dependent on its structural properties. Results show that all considered ribbons are metal and some of them can be magnetic. Magnetization is solely observed in ribbons prepared from cutting the Borophene sheet along y-direction YBNR and in some widths. Generally, YBNRs are more interesting so that some ribbons can be magnetic in one or two edges in specific widths. In addition, spin anisotropy is observed in YBNRs, so, one edge is more magnetic than the other. The spin anisotropy comes from asymmetry in edge of the ribbon. The spin anisotropy makes 12 -Borophene nanoribbons a potential candidate for spintronic applications.