Giant Spin-valley Polarization and Multiple Hall Effect in Functionalized Bi Monolayers

Valleytronic materials, characterized by local extrema (valley) in their bands, and topological insulators have separately attracted great interest recently. However, the interplay between valleytronic and topological properties in one single system, likely to enable important unexplored phenomena and applications, has been largely overlooked so far. Here, by combining a tight-binding model with first-principles calculations, we find the large-band-gap quantum spin Hall effects (QSHEs) and valley Hall effects (VHEs) appear simultaneously in the Bi monolayers decorated with halogen elements, denoted as Bi2XY (X, Y = H, F, Cl, Br, or I). A staggered exchange field is introduced into the Bi2XY monolayers by transition metal atom (Cr, Mo, or W) doping or LaFeO3 magnetic substrates, which together with the strong SOC of Bi atoms generates a time-reversal-symmetry-broken QSHE and a huge valley splitting (up to 513 meV) in the system. With gate control, QSHE and anomalous charge, spin, valley Hall effects can be observed in the single system. These predicted multiple and exotic Hall effects, associated with various degrees of freedom of electrons, could enable applications of the functionalized Bi monolayers in electronics, spintronics, and valleytronics.


INTRODUCTION
Tailoring valley degrees of freedom offers fascinating opportunities to realize novel phenomena and emerging applications, often referred to as valleytronics [1][2][3][4] . While valley effects have been studied for decades in materials such as silicon 5 , diamond 6 , AlAs 7 , and graphene [8][9][10][11] , despite the effort to emulate the better known manipulation of spin and spintronic applications 12 , the related success has been modest 2 . Often the progress in harnessing the valley degrees of freedom was limited not by the lack of ideas 13 , but by the material properties, including the small valley polarization and a weak spin-orbit coupling (SOC) inherent to graphene 1,8,11 . The resurgence of interest in valley effects was recently spurred by the discovery of monolayer (ML) transition metal dichalcogenides (TMDs) with broken inversion symmetry and strong SOC [14][15][16][17][18][19][20][21][22] . A hallmark of ML TMDs is their valley-spin coupling which leads to a valley-dependent helicity of optical transitions [23][24][25] as well as important implications for transport, such as the discovery of the valley Hall effect (VHE) 4 . Lifting the degeneracy between the valleys K and K' to generate the valley polarization was identified as the key step in manipulating valley pseudospin degrees of freedom [15][16][17][18][19][20][21][22] . Common approaches were focused either on optical pumping by circularly polarized light [23][24][25] or very large magnetic fields required by a small Zeeman splitting of ~0.1 meV/T 26,27 . Instead of these external methods to realize valley polarization that could be impractical or limited by the carrier lifetime 3 , transition metal (TM) doping [16][17][18] or magnetic proximity effects [19][20][21][22] have been found effective ways to get permanent valley polarizations in TMDs. Since topological properties are very important for materials 1 , the coupling of the topological behaviors and valley polarizations in the valleytronic materials may give rise to new physics and applications.
The combination of topological and valleytronic has been rarely explored up to the present. It would be very interesting to search for such topologically nontrivial valleytronic materials and explore their valley-dependent transport properties. 3 Since the topological properties and transverse velocities of the electrons in the VHEs are closely related to the SOC strength of the system, it is desirable to search for and to explore the valley related phenomenon in large SOC material systems. Very strong SOC can be generally induced in materials containing heavy elements, such as bismuth, and drives the appearance of the topologically nontrivial states in the Bi-related systems 28,29 . In experiments, the buckled ML structure of Bi (111) with a hexagonal lattice has been successfully synthesized and characterized 28 . A nontrivial quantum spin Hall effect (QSHE) band gap of 0.8 eV was observed experimentally in ML Bi on the top of a SiC substrate 28 .

RESULTS AND DISCUSSION
The structure of ML Bi2XY (X, Y = H, F, Cl, Br, or I) is shown in Figs. 1a and b, where the Bi atoms construct a quasi-planar honeycomb lattice, the X atoms bond the upper surface Bi atoms, and the Y atoms bond the lower surface Bi atoms. When X = Y, the Bi2XY structures form the ML Bi2X2 (X = H, F, Cl, Br, or I), which has been predicted to be stable up to 600 K and show QSH effects 31 . The ML Bi2X2 has inversion symmetry with point group of D3d 31 . The Bi pz orbitals are saturated by the later added X atoms. The strong SOC from Bi px and py orbitals near the Fermi level (EF), thus, opens large nontrivial band gaps at the K and K' valleys [30][31][32] . The calculated local band gaps of Bi2X2 (X = H, F, Cl, Br, or I) are found in the range from 1.160 eV to 1.306 eV as shown in Table S1 and Fig. S1 in Supplementary Information. A large SOC in the ML Bi2X2 could be employed for the potential valleytronic applications after the inversion symmetry of the system is broken.
To explore the valley contrasting physics in the functionalized Bi systems, a TB model is built for the ML Bi2XY. We adopt the spherical harmonic functions . t1 and t2 in Eq. (4) represent the hopping amplitudes. For the ML Bi2X2 there are Dirac points existing at the K and K' points without the SOC, as shown in Fig. 1c. When SOC is considered, nontrivial band gaps of 2λso are opened at the K and K' points shown in Fig. 1d, realizing a QSH insulator, in agreement with the ab-initio calculations (see Fig. S1 in Supplementary Information) and previous studies [30][31][32] . The local band gap opened at the Dirac point K (K') is a result of the first-order relativistic effect related to px and py orbitals of Bi elements 31 . Thus, these band gaps are giant and robust (1.160 ~ 1.306 eV, see Table S1 in Supplementary Information).
To break the inversion symmetry of the Bi2X2 system, we induce a staggered potential (HU) between the sublattices A and B, equivalently forming the Bi2XY system. Simultaneously, the Rashba SOC 12 exists in the system due to the different decorations of X and Y. The HU can be written as   for simplicity and to reduce the number of the independent parameters. The Rashba SOC is important in some systems, which can induce very interesting effects. For example, it opens the topologically nontrivial gap of quantum anomalous Hall effects in graphene system 34,35 . However, in the Bi 2 XY system, the Rashba SOC is small compared to intrinsic SOC and staggered potential, which will In the real ML Bi (111), the staggered potential can be acquired with different chemical decorations on the upper and lower ML surfaces, namely forming the Bi2XY (X, Y = H, F, Cl, Br, or I) structure.
The calculated staggered potentials are found in the range from 6 meV to 300 meV, as shown in Table   S2 and Fig. S2 in Supplementary Information. With the SOC, large nontrivial band gaps from 0.891 eV to 1.256 eV (see Table S1 and Fig. S3 in Supplementary  To identify the QSH effect and VHE emerging in the ML Bi2HF, the spin Berry curvature Ω s (k) 14 and Berry curvature Ω(k) 37 are calculated as follows: In Eqs. (7) - (9), En is the eigenvalue of the Bloch functions |ψnk>, fn is the Fermi-Dirac distribution function at zero temperature, and s To employ the valley degree of freedom, valley splitting (∆ ′ ) needs to be introduced, which can be quantified by the energy difference between the topmost valence bands at K ( ) and K' ( ′ ) valleys, expressed as ∆ ′ = − ′ 16-21 . In this regard, the principle challenge of using ML Bi 2 XY as a valleytronic material is to break the degeneracy between the two prominent K and K' valleys, protected by time reversal symmetry. As in TMDs, some strategies of using external fields such as optical pumping [23][24][25] , electric field 9 , and magnetic field 26 can be written as: The schematic bands for H3 without and with SOC are plotted as Figs in-plane electric field, as shown in Fig. 3f. Remarkably, the transverse current is 100% spin polarized.  Fig. 4a, a distinct exchange field is induced from the Mo atoms, leading to large spin polarization of 369 meV for LCBs and 434 meV for HVBs, respectively.
With SOC, local energy gaps of about 210 meV and 710 meV are opened at the K and K' points, 11 respectively (Fig. 4b), as well as a significant valley polarization emerges in the system. The orbital-resolved band structures of Bi2HMo (Fig. S7a) S10), indicating TM doping can be a realistic way to induce the spin-valley splitting in the Bi monolayers. 12 Besides doping TM atoms, proximity effects may be more realistic and effective way 43,44 to induce a staggered exchange field in the Bi2XY system and then give rise to spin-valley polarizations. LaFeO3 is a G-type antiferromagnetic (AFM) insulator with the Fe sites forming alternating (111) ferromagnetic (FM) planes 45 . Its (111) surface lattice matches well the Bi2H2 lattice with the mismatch of ~1% 31,46 , which has been fabricated with atomic-scale control and has a very high crystallographic quality.

METHODS
The geometry optimization and electronic structure calculations were performed by using the first-principles method based on density-functional theory (DFT) with the projector-augmented-wave (PAW) formalism 47 , as implemented in the Vienna ab-initio simulation package (VASP) 48 . All calculations were carried out with a plane-wave cutoff energy of 550 eV and 12 × 12 × 1 Monkhorst-Pack grids were adopted for the first Brillouin zone integral. The Berry curvatures and spin Berry Curvatures for the ML Bi2HF, Bi2HMo and the Bi2H/LaFeO3 heterostructure are calculated in Wannier function bases 49 . The geometry structures and more computational details about the ML Bi2HMo and the Bi2H/LaFeO3 heterostructure are given in Supplementary Information. 14 The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Giant Spin-valley Polarization and Multiple Hall Effect in Functionalized Bi
Monolayers Tong Zhou 1,2 The energy levels at this K point can be obtained by diagonalizing above matrix. Around the E F , the energy levels can be analytically expressed as The energy levels near the E F are Because the Rashba term is much smaller than the M A (M B ), λ SO , and U terms in the functionalized Bi monolayers, we neglect the Rashba SOC in H 3 .