Spin splitting and electric field modulated electron-hole pockets in antimonene nanoribbons

The one-dimensional nanoribbons have great potential applications in novel multifunctional electronic devices. Herewith, we investigate the electronic structure of antimonene nanoribbons using the first-principles calculations. Four ribbon models are considered. All of them are semiconductors with appropriate band gaps. Because of the loss of inversion symmetry, spin-orbit splitting appears in the asymmetric washboard nanoribbons. Significantly, spin splittings of 62 and 44 meV are found at the valence band maximum and conduction band minimum for zigzag asymmetric washboard nanoribbon (aW-zSbNR). Further applying an in-plane electric field introduces a potential difference between the two zigzag edges. The energies for the localized edge states are modified. Eventually spin-orbit-coupling induced band inversion and electron-hole pockets with appropriately the same size could be triggered in a W-zSbNR, which is interesting for exploiting the magnetoresistance effect. Electrically-tunable electron-hole pockets could make antimonene nanoribbons a platform for probing unusual magnetoresistance effects. Some material changes their resistance when a magnetic field is applied—an effect known as magnetoresistance. This phenomenon is harnessed for a range of technological applications but the origin of the recently observed large, non-saturating magnetoresistance in certain materials is somewhat puzzling. Using first-principles calculations, Wenbo Mi and Yan Song from Tianjin University, along with Xiaocha Wang from Tianjin University of Technology, predict that one-dimensional antimonene nanoribbons with a zigzag asymmetric washboard structure could have electron-hole pockets, which are the key features necessary for observing a non-saturating magnetoresistance. As these pockets could be modulated using in-plane electric fields, these nanoribbons could be a tunable platform for probing this exotic phenomenon.

The intriguing behavior of 2D monolayers is that they can be etched or carved into one-dimensional (1D) nanoribbons along a specific direction. With quantum confinement effect, nanoribbons exhibit many extraordinary electronic, [13][14][15] optical 16 and magnetic properties. 17,18 Half-metallicity was predicted in zigzag graphene nanoribbons by applying an in-plane electric field across the zigzag-shaped edges. 19 Dimensional reduction facilitates the manipulation of electronic and magnetic properties by modifying the edge states. Decorating the edges with different chemical groups can make half-metallicity appear in zigzag graphene, 20 boron nitride, 21 and MoS 2 nanoribbons. 22 Very recently, multilayer arsenene and antimonene nanoribbons are successfully synthesized using plasma-assisted process. Controllable thickness and room-temperature orange light emission are realized, respectively. 23,24 Experimental breakthroughs in fabricating nanoribbons convince the bright future of 1D structure in nanoelectronic applications. In this work, we simulate the electronic properties of antimonene nanoribbons with two stable structures predicted in previous studies. 25 Considering that the spin-orbit coupling (SOC) in heavy element Sb is significant, we include the SOC explicitly in our calculations. It is found that the intrinsic SOC in asymmetric washboard nanoribbons removes the spin degeneracy. Applying an in-plane electric field in a specific model can realize the electron-hole-pocket structure, which is attractive for studying the magnetoresistance (MR) effect in these nanoribbons. 26

RESULTS
We construct antimonene nanoribbons within two stable structures: buckled honeycomb and asymmetric washboard structures, using the names adopted in ref. 25. The armchair and zigzag edge geometries are considered. The structures of four nanoribbon models are shown in Fig. 1. Comparing the balance of edge coupling and consumption of computations, we choose nine rows of atoms for asymmetric washboard armchair antimonene nanoribbon (aW-aSbNR), asymmetric washboard zigzag antimonene nanoribbon (aW-zSbNR), buckled honeycomb armchair antimonene nanoribbon (B-aSbNR), and six rows of atoms for buckled honeycomb zigzag antimonene nanoribbon (B-zSbNR) ( Fig. 1) as the ribbon width. The asymmetric washboard antimonene (Fig. 1a, b) contains two parallel atomic planes and presents a rectangular lattice similar to phosphorene. The primitive unit cell contains four Sb atoms. Each Sb is threefold coordinated with in-plane and inter-planar Sb atoms. Different from phosphorene, one of the two Sb atoms located in either plane in the primitive cell slightly displaces outwards to prevent the structure from long-wave instability, 25 as seen in the side view of Fig. 1b.
Another stable structure is the buckled honeycomb structure (Fig. 1c, d) related to the bulk layered Sb with a space group of R3mh. In hexagonal lattice, the sp 2 bonds of the unstable planar honeycomb structure rehybridize to maintain the stability through buckling, 25 as seen in the side view of Fig.1d. Zhang et al. 12 demonstrated that antimonene with these two structures exhibit thermodynamic stability. The energy of the buckled honeycomb structure is 14 meV/atom lower than that of asymmetric washboard one. 12 We construct nanoribbons along two typical directions: armchair and zigzag directions. The width of each model is depicted in Fig. 1. In ref. 27, the spin density wave state was predicted to be the ground state in zigzag phosphorene, arsenene, and antimonene nanoribbons over the charge density wave state. Hence, we adopt the intra-and inter-edge antiferromagnetic (AFM) orders for the zigzag nanoribbons. 27 The initialization of the edge magnetism is depicted in Fig. 1b, d. The structure optimizations are performed using the conjugate gradient algorithm. We characterize the optimized structures by the relevant structural parameters including the bond angels (α, β) and bond lengths (a, b) ( Fig. 1), as listed in Table 1. For zigzag nanoribbons, spin density wave is indeed observed. The spin density differences for aW-zSbNR and B-zSbNR are plotted in Fig. 2a, b, with red (green) labels for the spin up (spin down) density with the isosurface value of 0.001 e/Å. The edge atoms are coupled antiferromagnetically. For aW-zSbNR, the spin distribution of the edge atoms shows the p x and p z characteristic. In B-zSbNR, the spin distribution shows the p x characteristic. The magnetic moments of the atoms marked in Fig. 2a, b are given in Table 1. The magnetic moments of edge atoms in a W-zSbNR are −0.10 and 0.08 μ B . In ref. 27, the edge magnetic moment of zigzag arsenene nanoribbon calculated using Perdew-Burke-Ernzerhof (PBE) method is 0.07 μ B . In B-zSbNR, it is 0.11 μ B . For the two armchair nanoribbons, they are both indirect-band-gap  Table 1. Values of relevant structural parameters a, b, α, and β in Fig. 1 and magnetic moments of Sb1~Sb4 atoms in Fig. 2a Figure 2c, d gives the orbital-resolved density of states (DOS) for Sb1-Sb4 of aW-zSbNR and B-zSbNR, respectively. As shown in Fig. 2c, the spin polarization of Sb1 and Sb4 comes from the p z and p x orbitals. For Sb2 and Sb3, the spin polarization is attributed to the p x and p y orbitals. In Fig. 2d, it is seen that the spin polarization of Sb1 and Sb4 mainly comes from p x orbitals and that of Sb2 and Sb3 mainly comes from p y orbitals. The spin polarization of the p states in the valence band is consistent with the magnetic configurations of spin density difference shown in Fig. 2a, b.
The band structures of the four nanoribbons with and without SOC, respectively are shown in Fig. 3a-d. A comparison reveals several similarities: (1) No matter whether SOC is included or not, the zigzag nanoribbons are direct-band-gap semiconductors, while the armchair nanoribbons only have indirect band gaps; (2) SOC reduces the band gap in all the configurations, and the reductions of 20, 50, 31, and 190 meV are induced in aW-aSbNR, aW-zSbNR, B-aSbNR, and B-zSbNR, respectively; (3) for the two asymmetric washboard nanoribbons, SOC lifts the spin degeneracy of the bands. More noticeable in a W-zSbNR is that the spin splittings at the valence band maximum (VBM) and conduction band minimum (CBM) are 62 and 44 meV, respectively. Hence, SOC cannot be neglected in simulating the electronic structure since it modifies the frontier states, especially in a W-zSbNR. It is worthy to be mentioned that the band structures of the four types of antimonene nanoribbons were investigated in ref. 25. The ribbon width of aW-aSbNR, B-aSbNR, and B-zSbNR used to Especially, B-zSbNR was predicted to be a ferromagnetic (FM) semiconductor in ref. 25, while in our results the bands are spin-degenerate. The discrepancy is attributed to the different model adopted. And more, in our calculations, the initialization of the edge magnetism is chosen to be the AFM order. For comparison, the band structure for B-zSbNR with FM order is also calculated, and the structure is similar to that with the AFM order but the gap is 35 meV smaller. More importantly, the total energy for the FM order is 8.8 meV higher than that for the AFM order. It again demonstrates that the spin density wave is the ground state for the zigzag nanoribbons, consistent with earlier prediction in ref. 27.
An explanation of the SOC-induced band-gap reduction is provided by an investigation of the orbital contributions to the uppermost valance band (UVB) and lowermost conduction band (LCB) along the Г-Y line. For all the nanoribbons, the orbital contributions of all the atoms show no significant difference before and after an inclusion of SOC. The orbital contributions for the four nanoribbons with and without SO Care available in Fig. S1 in the Supplementary information. However, analysis of the partial charge densities at the edges of valance and conduction bands shows that the band edge states are localized at the edges of the nanoribbons. The partial charge densities for aW-zSbNR are shown in Fig. 3e, f and the data for aW-aSbNR, B-aSbNR, and B-zSbNR are given in Fig. S2 in the Supplementary information. The orbital contributions of the edge atoms are plotted in Fig. 4.
For aW-aSbNR, the UVB and LCB are contributed from the p x and p y states. SOC has little modification effect on these orbitals corresponding to the smallest band-gap reduction. SOC alters the contributions of the p x and p z states for aW-zSbNR and the p x and p y states for B-armchair. However, the general trends shown in these plots can still be recognized, in line with the moderate gap reductions. For B-zSbNR with the largest gap reduction, the UVB and LCB are contributed from the p x states. With turn-on of SOC, the contributions are changed significantly. Especially close to the Y point at the UVB, the contribution is enhanced remarkably. Hence, it is suggested that the SOC-induced band-gap reduction is attributed to the modification on the orbital contributions of the localized edge states.
For the symmetry arguments, the absence of inversion symmetry in an asymmetric washboard structure is the reason that causes a spin-splitting of the energy bands. The symmetry of the asymmetric washboard antimonene is Pmn2 1 (C 7 2v ). A proper cutting of the nanoribbons aW-aSbNR and aW-zSbNR allows the symmetry reduction to Pm (C 1 s ). The both cases are noncentrosymmetric. So SOC will break the spin-degeneracy along the Г-Y line (Fig. 3a, b). On the contrary, the buckled antimonene and antimonene nanoribbons are centrosymmetric with the symmetry groups P3m1ðD 3 3d Þ and P2/mðC 1 2h Þ, respectively, which satisfy the inversion symmetry. Accordingly, the energy bands of B-aSbNR and B-zSbNR are spin-degenerate (Fig. 3c, d).
From Fig. 3e, f, it is seen that both the VBM and CBM are constructed from the edge atoms that present an AFM order  . 2a). It indicates that the splitting at the Y point in a W-zSbNR is attributed to the AFM order that further reduces the symmetry at this point. Moreover, there exists an imbalance of charge distribution, whereas other nanoribbons do not have this property (Fig. S2). The VBM has more population at the right edge, while the CBM has more population at the left edge. The spatially separated edge states suggest that it is possible to modulate the energy bands around the Fermi level. 19 Therefore, we apply an in-plane electric field across the edges of aW-zSbNR (along the x direction in Fig. 1). The variations of band structure under electric field, without (a) and with (b) SOC are given in Fig. 5a, b, respectively. In case of no SOC, the band gap first decreases and then increases with the increasing field. When SOC is turned on, the twofold spin degeneracy is lifted. At a field of 2 V/nm, the bands around the VBM and CBM transform from the "Λ shape" to the "M shape". This variation is reminiscent of the band inversion, just like the case in 2D SbAs exposed to biaxial strain. 28 Further increasing the electric field allows the appearance of electron and hole pockets with appropriately the same size at 4 V/nm. A clearer image can be seen in the zoom-in area. The semiconductor-semimetal transition originates from the fact that the electric field induces the energy-level shifts of opposite signs for the spatially separated band edge states. The electrostatic potential is raised on the right side and lowered on the left side as electric field increases. Because of the imbalanced distribution, the effects of electric field on them are opposite. Correspondingly, the energies for the localized edge states on the right side (mainly contributed to the VBM) are shifted upwards and those on the left side (mainly contributed to the CBM) downwards. The schematic DOS diagrams of aW-zSbNR with and without electric field are given in Fig. 5c. Further increasing the electric field can dismiss the electron-hole-pocket structure. It is well known that the PBE functional tends to underestimate the band gap, but the variation trend of bands that the VBM is shifted upwards while the CBM is shifted downwards by the electric field is valid. For example, in ref. 28, the topological transition in monolayer SbAs under applied strain is predicted using PBE method. Subsequently, the band structures at the hybrid functional level are given in the Supplementary Information. The results show that both the variation trend of bands and the topological properties are not influenced by the functional adopted. The gap closure occurs at a tensile strain of 12% at the PBE level (14.8% at the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid functional level). Given the mechanism proposed in Fig. 5c, it is also expected that the semiconductor-semimetal transition could be realized but merely under a strong electric field.

DISCUSSION
The electron-hole pockets are very attractive because of the extraordinary MR effect recently found in WTe 2 (ref. 26). Extremely large-positive magnetoresistance (XMR) without any sign of saturation was reported at magnetic field up to 60 T (ref. 26). The MR is characterized by a near-quadratic field dependence and a field-induced up-turn in the resistivity, followed by a plateau at low temperature. 29 Soon after, these characters were also observed in several semimetals. [30][31][32][33][34] Different mechanisms have been proposed to explain the exotic quadratic XMR behaviors. 26,30,33,35 Using high-resolution angle-resolved photoemission spectroscopy, Pletikosić et al. 36 demonstrated that the tiny electron and hole pockets of equal size are the fundamental ingredients of physics for the unusual transport properties of WTe 2 . Cai et al. 37 reported that the sizes of the electron-hole pockets show increased difference with pressure. The pressure dependent MR supports the perfect balance scenario. A recent study on the dopant effect in WTe 2 also supported the electron-holecompensation theory. 38 Nevertheless, a non-saturated XMR was found in compensated semimetal LaSb. 29 The suppression of MR without observable change of the Fermi surfaces with increasing temperature was observed. 29 This is contradictory to the above results. Moreover, recent studies showed deviations from the perfect electron-hole compensation in gated WTe 2 thin flakes and WTe 2 under magnetic field, whereas the XMR effects still exist. 39,40 Hence, so far the nature of the exotic XMR effect is still controversial.
Here, we report that the electric-field-modulated carrier compensation can be realized in a W-zSbNR. If the unsaturated XMR of WTe 2 is attributed to this special resonance according to ref. 26, the aW-zSbNR may serve as a platform for testing the cause of this extraordinary MR effect. In addition, aW-zSbNR has more similarities with the elemental materials graphite and bismuth, i.e., low carrier density, small effective mass, and an equal number of electrons and holes, which display XMR at low temperature. 41 These features specific to semimetals give rise to the field-induced metal-insulator behavior. 41 Thus, aW-zSbNR is attractive for exploring the positive MR effect. At high-magnetic field, the MR of graphite and bismuth saturates to a fieldindependent value. 26 The unsaturated MR of WTe 2 is attributed to the maintenance of the perfect charge compensation even at high field. 26 The ratio between the p-and n-type carriers becomes more and more severe with increasing magnetic field. A slight deviation will lead to the saturation of MR, which is the case of graphite and bismuth. 26 Hence, it is suggested that the aW-zSbNR is particularly interested for studying the MR effect.
Besides, it is also demonstrated that the balance between the electron and hole states in WTe 2 is established only beyond finite number of layers (three), showing that the behavior of WTe 2 is not strictly two dimensional. 42 Nevertheless, for nanoelectronics, it is desirable to obtain low-dimensional systems that have the performance as good as, or even better than, their bulk counterparts, favoring the potentials of antimonene nanoribbons.
A problem in the theory-oriented works 19,43,44 on the electric field effects is that the involved electric field strength is relatively strong. It is still a great challenge to be realized in experimental research. Nevertheless, the electrically tunable property of antimonene nanoribbons is a good demonstration of the theoretical concept and does have implications for potential applications. This issue also stimulates further study of new principles or paradigms that are more feasible for achieving electric-field-tunable properties. Instead of applying strong external electric fields, using the built-in electric fields in materials or at the interfaces is another direction deserving further attention.
In summary, we have studied the electronic properties of antimonene nanoribbons by means of first-principles calculations. The spin density wave state is observed in the zigzag models. Because of the loss of inversion symmetry, SOC lifts the spin degeneracy for the asymmetric washboard nanoribbons. More remarkable in aW-zSbNR is the spin-splittings of 62 and 44 meV, respectively, at the VBM and CBM. The imbalanced charge distributions enable the electric modulation of bands around the Fermi level in a W-aSbNR. The in-plane electric field across the edges modifies the spatially separated localized states. As a result, the SOC-induced band inversion and electron-hole pockets with appropriately the same size appear. The unusual electronic properties predicted in a W-zSbNR provide another platform to exploit and understand the XMR effect.

METHODS
The calculations are performed using the density functional theory in conjunction with the projector augmented wave potentials 45 and PBE generalized gradient approximation 46 to the electronic exchange and correlation as implemented in the Vienna ab initio simulation package (VASP) code. 47,48 The Monkhorst-Pack k-point sets 1 × 11 × 3 for aW-aSbNR, 1 × 7 × 3 for aW-zSbNR, B-aSbNR, and B-zSbNR are used. The energy cutoff for the plane-wave basis set is 500 eV. In all, 10 −5 eV and 10 −2 eV/Å are the criteria of energy and force convergence. Sufficient vacuum layers of 15 Å perpendicular to the plane and 20 Å along the ribbon width are adopted to avoid the interactions between the periodic images. In order to simulate the influence of in-plane electric field using VASP, an artificial dipole layer in the middle of the vacuum in the x direction ( Fig. 1) is added.