Firstly, we construct the stable configurations of the bulk α-B and β-B. The optimized configurations in the present study keep the same perfect structure as previously proposed
]. Also, according to the structural characteristic of the bulk α-B and β-B, in the following study, six possible representative nanowires are considered. Three were obtained from the unit cell of α-B, growing along three base vectors, respectively. The other three were from the unit cell of β-B, also growing respectively along the base vectors. The corresponding boron nanowires are denoted according to the based bulk boron and their growth direction, named by α-a , α-b , α-c , β-a , β-b , and β-c . For all these constructed boron nanowires, we perform a complete geometry optimization including spin polarization. Their equilibrium configurations are respectively shown in Figure
,b,c,d,e,f, where the left and right are respectively the side and top views for the same configuration. These results thus reveal that the optimized configurations of the six under-considered boron nanowires still keep the same perfect B-B bond structure as those in the bulk boron. To evaluate the stability of these boron nanowires, we calculate their cohesive energies by determining the cohesive energies according to the definition discussed previously. The calculated cohesive energies are listed in the first column of Table
. For comparison, in Table
, we also give the cohesive energies calculated at the same theoretical level of the bulk α-B and β-B. A negative cohesive energy value indicates that the chemical energies processed to form the boron nanowires are exothermic in reaction. The cohesive energies of all the considered boron nanowires are negative and have the absolute value larger than 6.70 eV/atom. This indicates that the dispersive B atoms prefer to bind together and form novel nanostructures, which can be seen from literatures about the multi-shaped one-dimensional nanowires
]. Simultaneously, by comparison, the cohesive energies of the considered boron nanowires are a little smaller than those of the bulk α-B and β-B, which are the two most stable of the various B bulks. Therefore, we conclude that all these under-considered boron nanowires are chemically stable. However, due to the relatively higher cohesive energy, some of the considered boron nanowires may be metastable, and experimental researchers need to seek the path of synthesizing these materials. Nevertheless, the typical one-dimensional structural characteristic and the attractive electronic and magnetic properties, as shown below, may stimulate experimental efforts in searching for a synthesizing path of this material.
Figure 1 Optimized configurations of the considered boron nanowires (red circles). (a) α-a , (b) α-b , (c) α-c , (d) β-a , (e) β-b , and (f) β-c . Herein, for the same configuration, (more ...)
Cohesive energies and total magnetic moments of considered boron nanowires and of bulk α-B and β-B
To lend further understanding of the nature of the boron nanowires considered above, we studied the electronic structures of all configurations using the spin-polarized calculations. The calculated total magnetic moments of the six nanowires are listed in the second column of Table
. It is obvious that for the three boron nanowires obtained from the unit cell of α-B, the nanowires α-a  and α-b  have the total magnetic moments of approximately equal to zero, while the nanowire α-c  has a distinctly different total magnetic moment of 1.98 μB. Moreover, for the three boron nanowires obtained from the unit cell of β-B, the same trend about the total magnetic moments occurs. The nanowires β-a  and α-b  both have the total magnetic moments also approximately equal to zero, and the nanowire β-c  has the total magnetic moments of 2.62 μB. Additionally, in Table
, we also presented the calculated total magnetic moments of bulk α-B and β-B. Thus, these results indicate that both of the two kinds of boron bulks have no magnetism, with the total magnetic moments equal to zero.
For the two magnetic nanowires, α-c  and β-c , we also set the initial spin configurations to the antiferromagnetic (AFM) order. The difference of the total energy between AFM and ferromagnetic (FM) (ΔE
) is 0.031 and 0.100 eV, respectively, corresponding to nanowires α-c  and β-c . This result indicates that both of the two magnetic nanowires are in the FM ground state. To lend further understanding about magnetic properties of the considered boron nanowires, we calculate the projected total electronic density of states for all considered boron nanowires, as plotted in Figure
. Clearly, we can see that for both of the two magnetic nanowires, the majority (spin-up) state and minority (spin-down) state are not compensated, which resulted in the residue of net spin states, as seen in Figure
,f. However, as shown in Figure
,d,e,f, the other boron nanowires are spin-compensated, with the spin-up and spin-down states equally occupied.
Figure 2 PDOS of the considered systems. (a) α-a , (b) α-b , (c) α-c , (d) β-a , (e) β-b , and (f) β-c . Positive and negative values represent the DOSs projected on the spin up and (more ...)
To pursue the physical origin of the magnetic moments of the two magnetic boron nanowires, we plot the isosurface of spin density of the supercells of the two magnetic boron nanowires, respectively, as shown in Figure
,b. The isovalue is set to 0.30 e/Å3
. It thus is obvious that for the boron nanowire α-c , the total magnetic moment of the system is essentially contributed from the atoms near two vertexes of one diagonals of the cross section. The spin density is symmetrically distributed around the two ends of the diagonals. For the boron nanowire β-c , the spin density is mainly distributed near one vertex of the diagonals in the cross section, which is in agreement with the previous report
]. The key to understand why the magnetic boron nanowires have the magnetic moments around the vertexes of one diagonals of the cross section is the atomic structural characteristic and especially the structural deformation of the magnetic boron nanowires tailored from the bulk boron. By analyzing, we find out that the reasons of the induced magnetic moments are mainly from two aspects. One is the unsaturated chemical bonds of the atoms at the vertexes of the diagonal, which make the electron states redistributed and cause the asymmetry of the spin-up and spin-down states. Another aspect is the local magnetic moments around the ends of the diagonal act by the interaction of spin-spin coupling, which enhances the total magnetic moments of the two magnetic boron nanowires and makes them show distinct and much larger total magnetic moments.
To complete the description of the study on boron nanowires, it is important to analyze their electronic properties of all configurations. The electronic energy band structure of the considered boron nanowires are shown in Figure
, in which the Fermi levels are denoted by the dashed line in this figure. Herein, for boron nanowires having no magnetic moments, we recalculated the band structure by performing DFT without spin polarization, as shown in Figure
,b,d,e. While for both of the two magnetic nanowires, we give the band structures calculated using the spin-polarized DFT. The calculated band energy structures are shown in Figure
,f, wherein the left and right respectively represent the bands of spin-up and spin-down electron states. Clearly, we can see that most of the boron nanowires under study are metallic with the electronic energy bands across the EF, as shown in Figure
. However, as seen in Figure
, the band structure of the boron nanowire α-c  is obviously different from that of the other metallic nanowires. In detail, the boron nanowire α-c  is a narrow bandgap semiconductor with a direct energy gap of 0.19 eV at X point. Due to the well-known shortcoming of DFT in describing the excited states, DFT calculations are always used to understand the bandgaps of materials. Therefore, the bandgap value, 0.19 eV, obtained from the present calculations may be underestimated. However, this value clearly indicates that the electronic property of the boron nanowire α-c  is distinct from that of the bulk boron and other under-considered boron nanowires. In addition, the electronic properties of these considered boron nanowires obtained from the unit cell of the bulk α-B are also direction-dependent. Thus, these results of direction dependence of the electronic and magnetic properties of boron nanowires would be reflected on the photoelectronic properties of these materials and bring them to have many promising applications that are novel for the bulk boron.
Figure 4 The band structures near the Fermi level. (a) α-a , (b) α-b , (c) α-c , (d) β-a , (e) β-b , and (f) β-c . For (c) and (f), the left and right respectively represent the bands (more ...)