Giant spin-valley polarization and multiple Hall effect in functionalized bismuth monolayers

Valleytronic materials, characterized by local extrema (valleys) 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 appear simultaneously in the bismuth monolayers decorated with hydrogen/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 spin-orbit coupling of bismuth 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 bismuth monolayers in electronics, spintronics, and valleytronics. Valley-related phenomena emerge in bismuth monolayers decorated with hydrogen/halogen or transition metal elements. A team led by Zhongqin Yang at Fudan University developed a tight-binding model aided by first-principles calculations to gain insight into the electronic band structures and the resulting valley-related properties of Bi2XY monolayers, where X and Y are H, F, Cl, Br, or I atoms. The appearance of topologically nontrivial band gaps at the K and K’ valleys, in the 0.8–1.2 eV range, provides a platform for generation of topological valleytronics. A staggered exchange field can be used to break the degeneracy of the valleys, leading to the emergence of time-reversal-symmetry-broken Quantum Spin Hall states, accompanied by spin-valley polarization. With electron or hole doping, a spin-valley polarized net transverse current can be generated by spin-down electrons or spin-up holes, respectively.


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) transitionmetal 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 behavior and valley polarizations in the valleytronic materials may give rise to new physics and applications. However, the widely studied valley-polarized material TMDs have topologically trivial band gaps. [16][17][18][19][20][21][22] 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.
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 valleyrelated 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 With the hydrogen (H), 30,31 halogens (F, Cl, Br, I) 31 or methyl (CH 3 ) 32 saturating p z orbitals of Bi atoms, the buckled Bi (111) ML forms a hexagonal flat geometry Bi 2 X 2 (X = H, F, Cl, Br, I, or CH 3 ), of which atomic structures are predicted to be stable at up to 600 K. 31 A giant SOC strength, producing global nontrivial band gaps up to 1.03 eV, was reported in the Bi 2 X 2 system. 31,32 These p x and p y orbital-induced large SOC should be beneficial for coupled spin and valley physics. Due to the presence of inversion symmetry in the crystal structure of Bi 2 X 2 , no valley-related phenomenon emerges in the previous work. 28,[30][31][32] In this work, by focusing on strong SOC in ML Bi (111)-based systems, we use tight-binding (TB) model and ab-initio studies to demonstrate materials design of topological effects which rely on valley-dependent Berry curvature. Large nontrivial band gaps (from 0.891 eV to 1.256 eV) are obtained at the two valleys in ML Bi (111) with different chemical decorations on the two surfaces of the MLs, forming Bi 2 XY (X, Y = H, F, Cl, Br, or I and X ≠ Y) structures, which provide a platform for fabricating the wide-frequency valley-light devices. To break the degeneracy of the K and K' valleys, we induce a staggered exchange field ΔM = M A -M B into the ML Bi 2 XY with the assumption of M A > M B , where M A is the magnetic exchange field in the A sublattice of the ML Bi and M B is the magnetic exchange field in the B sublattice. The TB model calculations show that when SOC is larger than both M A and M B , the time-reversal symmetry-broken (TRSB) QSH states and obvious spin-valley polarizations emerge in the system. With electron (hole) doping, the spin-down electrons (spin-up holes) produce a spin-valley polarized net transverse current, giving rise to spinvalley polarized anomalous valley Hall effects (AVHEs), also called anomalous charge/spin/valley Hall effects. Thus, with gate control, multiple Hall effects including QSHEs and anomalous charge/spin/ valley Hall effects can be manipulated in the single system. To carry out these effects in experiments, we propose two possible schemes based on ab-initio calculations. (1) A functionalized ML Bi 2 XY with transition-metal (TM = Cr, Mo, or W) atoms doped, in which the valley splitting can be giant, with a maximum value of 513 meV. (2) A heterostructure of ML Bi 2 H deposited on a LaFeO 3 (111) surface, where a valley splitting is about 78 meV. The predicted multiple Hall effects associated with multiple degrees of freedom of electrons in functionalized ML Bi pave a brand new way to electronics, spintronics, and valleytronics of twodimensional materials.

RESULTS AND DISCUSSION
The structure of ML Bi 2 XY (X, Y = H, F, Cl, Br, or I) is shown in Fig.  1a, 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 Bi 2 XY structures form the ML Bi 2 X 2 (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 Bi 2 X 2 has inversion symmetry with point group of D 3d . 31 The Bi p z orbitals are saturated by the later added X atoms. The strong SOC from Bi p x and p y orbitals near the Fermi level (E F ), thus, opens large nontrivial band gaps at the K and K' valleys. [30][31][32] The calculated local band gaps of Bi 2 X 2 (X = H, F, Cl, Br, or I) are found in the range from 1.160 eV to 1.306 eV as shown in Supplementary  Table S1 and Fig. S1 in Supplementary Information. A large SOC in the ML Bi 2 X 2 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 Bi 2 XY. We adopt the spherical harmonic functions ϕ þ ¼ À 1 ffiffi 2 p p x þip y À Á and ϕ À j i¼ 1 ffiffi 2 p p x À ip y À Á together with the spin {↑,↓} as the basis. The TB Hamiltonian of the ML Bi 2 X 2 is first discussed. Under the basis of Φ i ¼f ϕ þ ; ϕ À j igf"; #g, the Hamiltonian (H 1 ) can be written in a sum of the nearest-neighbor hopping (H 0 ) and on-site intrinsic SOC terms (H SO ) 33 : In Eqs.  with z¼ expð 2 3 iπÞ. t 1 and t 2 in Eq. (4) represent the hopping amplitudes. For the ML Bi 2 X 2 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 2λ so 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 p x and p y 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 Bi 2 X 2 system, we induce a staggered potential (H U ) between the sublattices A and B, equivalently forming the Bi 2 XY system. Simultaneously, the Rashba SOC 12 exists in the system due to the different decorations of X and Y. The H U can be written as for the A (B) sublattice and both σ 0 and τ 0 are 2 × 2 unitary matrices. The Rashba SOC H R can be written as 33 The resulting TB Hamiltonian for the ML Bi 2 XY is thus: where with z = exp(2/3iπ). λ R , λ 0 R reflect the Rashba SOC between different orbitals of nearest-neighbor sites. In general, the relation of λ R and λ 0 R follows tendency of the t 1 and t 2 . We assume λ 0 R ¼ λ R t 2 =t 1 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 be shown in the following band fitting. Without the SOC, the bands of the Hamiltonian H 0 + H U are plotted in Fig. 1e. The spin-up and spin-down bands are degenerate at the K and K' points. Because of the staggered potential, the Dirac points disappear and two trivial band gaps with the value of 2U emerge at the K and K' points. With the consideration of the SOC and λ SO > U, the topologically nontrivial band gaps are opened at the K and K' points (Fig. 1f). The SOC together with staggered potential lifts the spin degeneracy of energy bands and makes the system have a peculiar spin-valley coupling, as shown in Fig. 1f. An obvious spin splitting appears at both the valence and conduction bands with opposite spin moments in the two valleys. An obvious spin splitting appears at both the valence and conduction bands with opposite spin moments in the two valleys. The calculated Berry curvatures are opposite at different valleys (Fig. 1f), providing an effective magnetic field. Such a magnetic field not only defines the optical selection rules, but also generates an anomalous velocity for the charge carriers. Thus, the valley Hall effect exists in the system. However, in the presence of time-reversal symmetry, the energy bands are still valley degenerate when SOC is considered. The opposite Berry curvatures and spin moments at the two valleys give rise to both the valley and spin Hall effects, without net transverse charge Hall current. 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 Bi 2 XY (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 to 1.256 eV (see Table S1 and Fig. S3 in Supplementary Information) are opened in the two valleys, giving rise to quantum spin Hall effect and valley Hall effect. These band gaps of the ML Bi 2 XY not only give us an opportunity to generate valley polarizations with different wavelength of light, 23-25 but also provide a platform for fabricating the wide-frequency valleylight emitting diodes. 36 As an example, we now analyze the case of the ML Bi 2 HF, where the H atoms bond with the lower Bi atoms and the F atoms bond with the upper Bi atoms. We find that the lattice constant of this ML Bi 2 HF is 5.49 Å after a full relaxation. The calculated bond lengths of Bi-H and Bi-F are 1.84 and 2.08 Å, respectively. The energy bands for the ML Bi 2 HF without SOC are displayed in Fig. 2a. The band gaps at the K and K' points are found to be 141 meV. Comparing with Fig. 1c, it can be inferred that the staggered potential 2U in the ML Bi 2 HF is about 141 meV, large enough to break the inversion symmetry in the system. When the SOC is turned on, very large nontrivial band gaps of 1.16 eV are opened around the E F at the K and K' points. The orbital-resolved band structures for Bi 2 HF are shown in Supplementary Fig. S4. It is clear that the bands around the E F at the two valleys are dominated by the p x and p y orbitals of Bi atoms, indicating the constructed TB model can describe the Bi 2 HF well. By fitting TB bands to the ab-initio results (shown in Fig. S4), the obtained TB parameters for the ML Bi 2 HF are t 1 = 0.73 eV, t 2 = 1.06 eV, 2U = 0.14 eV, λ SO = 0.61 eV, and λ R = 0.022 eV. Very obvious spin polarization is also observed in the highest valence bands (HVBs) and the lowest conduction bands (LCBs).
To identify the QSH effect and VHE emerging in the ML Bi 2 HF, the spin Berry curvature Ω s (k) 14 and Berry curvature Ω(k) 37 are calculated as follows: Ω s n ðkÞ¼ À 2Im X m≠n <ψ nk jj s x jψ mk ><ψ mk jυ y jψ nk > h ðE m À E n Þ 2 ; ΩðkÞ ¼ X n f n Ω n ðkÞ; Ω n ðkÞ ¼ À2Im X m≠n <ψ nk jυ x jψ mk ><ψ mk jυ y jψ nk > h 2 ðE m À E n Þ 2 : In Eqs. (7)- (9), E n is the eigenvalue of the Bloch functions |ψ nk >, f n is the Fermi-Dirac distribution function at zero temperature, and j s x is the spin current operator defined as (s z υ x + υ x s z )/2, where υ x and υ y are the velocity operators and s z is the spin operator. The obtained spin Berry curvature Ω s (k) along the high-symmetry lines is plotted in Fig. 2c. By integrating Ω s (k) over the first Brillouin zone (BZ), we obtain the spin Chern number Cs = 1, proving that the ML Bi 2 HF is a QSH insulator. As shown in Fig. 2d, the calculated Ω(k) is sharply peaked in the valley region, with opposite signs for K and K', in agreement with Fig. 1f. The distribution of Ω(k) the indicates the valley Hall effect occurring in the system. Namely, when the ML Bi 2 XY channel is biased, electrons from different valleys experience opposite Lorentz-like forces and so move in opposite directions perpendicular to the drift current.
To employ the valley degree of freedom, valley splitting (Δ V KK0 ) needs to be introduced, which can be quantified by the energy difference between the topmost valence bands at K (E V K ) and K' ( [16][17][18][19][20][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, 23-25 electric field, 9 and magnetic field 26,27 are supposed to induce the is the local magnetic exchange field in the A (B) sublattice. Such a staggered exchange field has been proven to transform the bands at K and K' points in the silicence 40 and functionalized ML Sb, 33,41 leading to so-called quantum spin quantum anomalous Hall (QSQAH) effects. 33,40,41 The staggered exchange field is thus suitable to produce the desired spin-valley polarizations in the ML Bi 2 XY. The TB Hamiltonian of the ML Bi 2 XY with staggered exchange field can be written as: The schematic bands for H 3 without and with SOC are plotted as Fig. 3a, b, (Fig. 3b). To quantitatively analyze the sequence of the energy levels around the E F at the K and K' points, the Rashba SOC is neglected because it is small compared to intrinsic SOC and magnetic field. Analyzing the sequence of the energy levels around the E F at the two valleys from Fig. 3b (also see Fig. S5 and more details in Supplementary Information), we can obtain the local band gaps at the K and K' points with the values of 2λ so -2M A and 2λ so -2M B , respectively. These band gaps give rise to an interesting time-reversal-symmetry-broken (TRSB) QSH state. 42 The calculated spin Berry curvatures (Fig. 3c) and edge states (Fig. 3d) are displayed to identify the interesting TRSB-QSH states. Thus, when the E F is in the nontrivial band gap (E C K > E F > E V K ), TRSB-QSH edge states could be observed as shown in Fig.  3e.
For applications, static and large valley polarization is desirable. In the ML Bi 2 XY, the charge carriers in the two valleys have opposite transverse velocities due to the valley degeneracy and opposite signs of the Berry curvatures and thus the total Hall conductivity vanishes because of time-reversal symmetry. With the staggered exchange field induced, the valley splitting is obtained. Considering the energy levels at two valleys (see more details in Supplementary Information) , the up-spin holes at the K valley produce a transverse current under an in-plane electric field, as shown in Fig. 3f. Remarkably, the transverse current is 100% spin polarized. Thus, the flux of the spin holes carries three observable quantities: charge, spin, and valley-dependent orbital magnetic moments corresponding, respectively, to anomalous charge, spin, and valley Hall effects. Similarly, when E C K0 > E F > E C K , spin-down electrons will produce a net transverse charge/spin/ valley current, as shown in Fig. 3f. Therefore, we create intrinsic and robust valley polarization, instead of the using dynamic methods. [23][24][25] Considering QSHE also exists in the system whenE C K > E F > E V K , multiple Hall effects including TRSB-QSHE, anomalous charge/spin/valley Hall effects can be manipulated by gate voltage control in one single system as shown in Fig. 3e, f. Based on this multiple Hall effects control, we can flexibly manipulate the charge, spin, and valley degrees for transport, which is crucial for spintronics and valleytronics. In experiments, the staggered exchange fields may be induced by replacing the Y atoms with TM (Cr, Mo, W) atoms, forming Bi 2 XTM (X = H, F, Cl, Br, or I; TM = Cr, Mo, or W) structures. To realize this proposal, we calculate the electronic structures of Bi 2 HTM (TM = Cr, Mo, or W) MLs by using ab-initio methods. The geometry and calculation details of Bi 2 HTM (TM = Cr, Mo, or W) MLs are given in Fig. S6 in Supplementary Information. For convenience, here we give an example results about Bi 2 HMo. As shown in 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 and 710 meV are opened at the K and K' points, respectively (Fig. 4b), as well as a significant valley polarization emerges in the system. The orbital-resolved band structures of Bi 2 HMo (Fig. S7a) Fig. S9), respectively, both of which are record values and much larger than the previous reported maximum valley splitting. 19,21,22 With these giant valley splittings, we can readily create valley polarization with hole doping in the Bi 2 HTM (TM = Cr, Mo, or W) MLs. When E F is tuned to move down in energy within the range of Δ V KK0 (up to 513 meV) shown in Fig. 4b, the spin-up holes at K valley will produce a transversal current under a longitudinal in-plane electric field, giving rise to the anomalous charge/spin/valley Hall effects. Considering doping the TM atoms in the freestanding Bi MLs may be not easy in experiments, we also explore the possibility of depositing TM atoms on the heterostructures of Bi MLs on a SiC substrate (Bi-SiC), which has been fabricated very recently. 28 The calculated adsorption energy of Mo atom on the Bi-SiC heterostructure is about 2.0 eV and the bands of Mo@Bi-SiC are similar to those of Mo@BiH (see Supplementary Fig. S10), indicating TM doping can be a realistic way to induce the spin-valley splitting in the Bi MLs.
Besides doping TM atoms, proximity effects may be more realistic and effective way 43,44 to induce a staggered exchange field in the Bi 2 XY system and then give rise to spin-valley polarizations. LaFeO 3 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 Bi 2 H 2 lattice with the mismatch of~1%, 31,46 which has been fabricated with atomic-scale control and has a very high crystallographic quality. Therefore, the LaFeO 3 (111) film is a very promising substrate for the ML Bi 2 XY to produce the spin-valley splitting. A heterostructure of the ML Bi 2 H on a (111) surface of a LaFeO 3 thin film is designed as displayed in Fig. S6c in Supplementary Information. The calculated large adsorption energy (3.6 eV) for the configuration indicates a very strong interaction between the ML Bi 2 H and the substrate. Similar to the effect of doping TM atoms, the substrate also induces a staggered exchange field into the hydrogenated MLs. As shown in Fig. 4c, the Dirac bands of the ML Bi 2 H 2 are strikingly spin polarized, located just inside the bulk band gap (2.1 eV) of the LaFeO 3 film. With the SOC considered, local band gaps of about 310 and 420 meV are opened around the K and K' points (Fig. 4d). The scale of Δ V KK0 is found to be about 78 meV in the heterostructure, which may be tuned further by strain or an electric field. We also calculate the orbital-resolved band structures for the Bi 2 H-LaFeO 3 heterostructure as shown in Fig. S7c in Supplementary Information. In this case, the HVBs at the two valleys are still dominated by the p x and p y orbitals of Bi atoms but the bands of substrate (LaFeO 3 ) exist in the nontrivial band gap from p x and p y orbitals, making the conduction bands of p x and p y orbitals higher than LCBs of the system. Fitting TB bands to the ab-initio results (shown in Supplementary Fig. S7d) In summary, we systematically investigated the topological properties and valleytronic behaviors in the functionalized Bi MLs based on TB models and ab-initio calculations. The topologically nontrivial band gaps at the two valleys in the ML Bi 2 XY (X, Y = H, F, Cl, Br, or I) are found in the range from 0.891 to 1.256 eV. These band gaps not only give us an opportunity to generate valley polarizations with different wavelength of light, but also provide a platform for fabricating valley-light devices. Spin-valley polarizations can be generated with a staggered exchanged field introduced, which together with the staggered potential is found determining the strength of the valley splitting. The calculated spin Berry curvatures and edge states indicate TRSB-QSHEs could be observed when the E F is located in the nontrivial gap. The calculated Berry curvatures are nonzero and opposite at different valleys, driving opposite anomalous velocities of Bloch electrons. With electron (hole) doping, the spin-down electrons (spin-up holes) produce a spin-valley polarized net transverse current. Thus, with gate control, multiple Hall effects including QSHEs, anomalous charge, spin and valley Hall effects can be manipulated in the single system. Based on ab-initio calculations, we predict these large spin-valley polarizations and multiple Hall effects can be realized in the Bi 2 HTM (TM = Cr, Mo, or W) MLs (with valley splitting of up to maximum 513 meV) or Bi 2 H/LaFeO 3 heterostructures (with valley splitting of 78 meV). Our results not only extend the properties of known valleytronic materials, but provide new paths to realize emerging applications in electronics, spintronics, and valleytronics.

METHODS
The geometry optimization and electronic structure calculations were performed by using the first-principles method based on densityfunctional 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 Bi 2 HF, Bi 2 HMo and the Bi 2 H/LaFeO 3 heterostructure are calculated in Wannier function bases. 49 The geometry structures and more computational details about the ML Bi 2 HMo and the Bi 2 H/LaFeO 3 heterostructure are given in Supplementary Information.

Data availability
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 effecty T Zhou et al.