Abstract
Exciting phenomena may emerge in non-centrosymmetric two-dimensional electronic systems when spin–orbit coupling (SOC)1 interplays dynamically with Coulomb interactions2,3, band topology4,5 and external modulating forces6,7,8. Here we report synergetic effects between SOC and the Stark effect in centrosymmetric few-layer black arsenic, which manifest as particle–hole asymmetric Rashba valley formation and exotic quantum Hall states that are reversibly controlled by electrostatic gating. The unusual findings are rooted in the puckering square lattice of black arsenic, in which heavy 4p orbitals form a Brillouin zone-centred Γ valley with pz symmetry, coexisting with doubly degenerate D valleys of px origin near the time-reversal-invariant momenta of the X points. When a perpendicular electric field breaks the structure inversion symmetry, strong Rashba SOC is activated for the px bands, which produces spin–valley-flavoured D± valleys paired by time-reversal symmetry, whereas Rashba splitting of the Γ valley is constrained by the pz symmetry. Intriguingly, the giant Stark effect shows the same px-orbital selectiveness, collectively shifting the valence band maximum of the D± Rashba valleys to exceed the Γ Rashba top. Such an orchestrating effect allows us to realize gate-tunable Rashba valley manipulations for two-dimensional hole gases, hallmarked by unconventional even-to-odd transitions in quantum Hall states due to the formation of a flavour-dependent Landau level spectrum. For two-dimensional electron gases, the quantization of the Γ Rashba valley is characterized by peculiar density-dependent transitions in the band topology from trivial parabolic pockets to helical Dirac fermions.
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The authors declare that the main data supporting the findings of this study are available within the paper and the Supplementary Information file. Extra data are available from the corresponding authors upon reasonable request. Source data are provided with this paper.
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Acknowledgements
This work is supported by the National Key R&D Program of the MOST of China (grant numbers 2017YFA0303002, 2016YFA0300204 and 2019YFA0308602), the National Science Foundation of China (grant numbers 11790313, 11574264, 11774305 and 61904205), Zhejiang Provincial Natural Science Foundation (grant numbers D19A040001 and R21A040006) and the Natural Science Foundation of Hunan Province (grant number 2020JJ4677). Y.Z. acknowledges the funding support from the Fundamental Research Funds for the Central Universities. A portion of this work was performed on the Steady High Magnetic Field Facilities, High Magnetic Field Laboratory, CAS. We thank Z. H. Liu of Longde Group Ltd for providing natural BAs crystals.
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Q.X. and Y.Z. initiated and supervised the project. F.S. fabricated BAs devices and carried out all the measurements, assisted by M.C., J.H. and X.S. C.H. and Y.L. did the DFT calculations. K.W. and T.T. prepared high-quality boron nitride single crystals. F.S., C.H., M.C., Q.X. and Y.Z. analysed the data and wrote the paper with inputs from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Projected band structures of BP, BAs and SnSe.
a–c, Energy bands of ML BP (a), BAs (b) and SnSe (c). According to orbital analysis, the Γ valley consists of pz electrons, whereas along Γ–X, px orbitals dominate the TRS-paired D valleys. d–f, Projected band structures of bulk BP (d), BAs (e) and SnSe (f), which are consistent with the ML results. Note that SnSe has an extra set of py-symmetry valleys along the armchair direction17, which become merged with Γ valleys in BP and BAs11,16.
Extended Data Fig. 2 Electric-field-dependent valley energy gaps and GSE coefficients of bilayer, trilayer and quadruple-layer BAs.
a, Energy gaps versus Ez for the Γ and D valleys in bilayer (BL) BAs. The GSE coefficients are the linear slopes of the gap curves. b, c, Energy gap evolutions of the Γ and D valleys in trilayer (TL) (b) and quadruple-layer (QL) (c) BAs. For different layer numbers, the D-valley GSE coefficient SD is consistently larger than the Γ-valley GSE coefficient SΓ. This is mainly due to Ez-dependent s–p hybridization, which contributes negatively to the GSE due to the antibonding VBM of the Γ valley (see Supplementary Note 5).
Extended Data Fig. 3 Energy bands of QL BAs under an electric field.
a, Pristine band structure of QL BAs without an electric field. All bands are spin doubly degenerate. b, Electronic band structure of QL BAs under Ez = 0.3 V nm−1, which activates the synergetic Rashba and Stark effects, and thus creates the spin–valley-flavoured D± valleys in the VB and the conventional Γ Rashba valleys in the CB. Near the D± VBM, the energy splitting is about 6.85 meV. Layer-dependent Rashba splitting is summarized in Supplementary Note 6. Note that under the realistic experimental conditions (Ez > 0.1 V nm−1), gate-induced charge carriers in BAs are quantum confined within four interfacial MLs to form the 2DEG or the 2DHG for positive and negative Vg, respectively. Therefore, our DFT calculations mainly focus on QL BAs, compared with ML, BL and TL BAs.
Extended Data Fig. 4 Ez-dependent Rashba splitting energy ESO of the D± and Γ valleys in QL BAs.
The DFT-calculated ESO of the D± valleys is larger than that of the Γ Rashba valleys within the whole Ez range. Using the LK formula fitting (Supplementary Note 9), we can also deduce experimental ESO from Vg-dependent SdHOs, as shown by the black curve. Although DFT overestimates ESO by a factor of two, it is clear that both DFT and experimental results show the same trend in the linear growth of ESO as a function of Ez. Equally important, both results prove that ESO is substantial enough to be probed by quantum transport below liquid helium temperature. Note that Ez is calculated by interfacial Gauss’s law (Supplementary Note 6).
Extended Data Fig. 5 FET transfer curves and Vg-dependent current–voltage (I–V) characteristics of high-mobility BAs-S9.
a, Transfer curves of BAs-S9 at 0.27 K (black line) and 300 K (red line), respectively. At 0.27 K, the device shows ambipolar FET characteristics with a well defined bandgap for Vg between 1.4 V and −1.1 V. b, Ambipolar I–V characteristics of BAs-S9 for varying Vg at 0.27 K, proving ohmic contacts achieved by Pd/Au electrodes for both the 2DHG and the 2DEG when \(|{V}_{{\rm{g}}}|\) ≥ 2 V. The data were measured in a two-terminal configuration using a Keithley 2400 source-measure unit.
Extended Data Fig. 6 Angle-dependent SdHOs of the 2DHG and the 2DEG in BAs devices.
a, Angle-dependent Rxx versus B⊥ in the 2DHG with p = 7.14 × 1012 cm−2 (BAs-S2, Vg = −7 V), which barely shows any changes in the positions of the SdHO extrema. b, Angle-dependent Rxx versus B⊥ in the 2DEG with n = 3.95 × 1012 cm−2 (BAs-S9, Vg = 5 V), which also shows high agreement in SdHO peak positions. These results confirm that no Zeeman-splitting-induced LL crossing occurs within the experimental maximum B of 14 T. Note that Rxx data for different θ are offset for clarity. The measurement T is 1.6 K. See Supplementary Note 2 for detailed discussions.
Extended Data Fig. 7 Three-dimensional illustration of particle–hole asymmetric Rashba bands in BAs with B = 15 T.
a, The spin–valley-flavoured Rashba bands in the VB of QL BAs. b, The Γ Rashba bands in the CB of QL BAs. The three-dimensional band structures of QL BAs are based on the calculation results in Extended Data Fig. 3 (Ez = 0.3 V nm−1). For the 2DHG, low, negative Vg creates two single-flavoured FSs with τzsz = 1, whereas high, negative Vg leads to the formation of inner–outer-nested FSs when EF intersects oppositely flavoured spin channels. For the 2DEG, the FS topology is dependent on the relative position of EF to the Zeeman energy gap ES. Note that the high in-plane band anisotropy in BAs has profound effects on the D± Rashba formation, by pushing py orbitals to much higher energy. Without the unique puckering square lattice, the D± valley would split into helical inner–outer Rashba bands centring on X. For the 2DEG, we can also see the square lattice warping of the Γ Rashba bands.
Extended Data Fig. 8 LL spectrum of the spin–valley-flavoured D± valleys in QL BAs.
a, Spin–valley-flavoured D± valleys in QL BAs under Ez = 0.3 V nm−1. Here, the D− valley is placed on the left side of D+ to highlight the fact that the D± Rashba splitting is centred on the TRIM points of X. b, Flavour-dependent LL spectrum of the D± Rashba valleys, in which LLs are divided into two groups (nI and nII) with opposite τzsz signs. Note that the nI = 0 and nII = 0 LLs are spin non-degenerate, whereas all ni ≠ 0 LLs are doubly degenerate as required by TRS (see Supplementary Note 3).
Extended Data Fig. 9 DFT-calculated evolution of EF versus Vg for the 2DHG and the 2DEG.
a, EF positions of the 2DHG for Vg = −2 V and Vg = −8 V, represented by blue and red lines, respectively, in the D± Rashba valleys of QL BAs under Ez = 0.2 V nm−1. b, EF positions of the 2DEG for Vg = 2 V and Vg = 8.5 V, represented by blue and red lines, respectively, in the Γ Rashba valley of QL BAs under Ez = 0.2 V nm−1. As discussed in Supplementary Note 5, the Vg-induced charge carriers are mainly confined within four interfacial MLs with Ez = 0.2 V nm−1. By calculating the QL BAs band structure, the FS size and effective Fermi wavevector \({k}_{{\rm{F}}}^{{\prime} }\) can be determined for different chemical potentials measured from the VBM/CBM. It is also standard to extract the experimental kF at different Vg for both the 2DHG and the 2DEG, by the LK formula fitting to the beating patterns of SdHOs. By matching the effective \({k}_{{\rm{F}}}^{{\prime} }\) to the experimental kF, we can quantitatively get the positions of Vg-dependent EF in the electronic bands of BAs. It is clear that the DFT calculation results are highly consistent with the SdHO results.
Extended Data Fig. 10 Berry phase analysis by the LK formula fitting.
a, The black curve is a representative SdHO curve for the Vg = +9 V 2DEG measured at 0.27 K, compared with the red curve representing the LK fitting result. The fitting processes reveal a π phase change in high-Vg SdHOs at around 5 T, when cyclotron orbiting electrons of the inner FS are allowed to acquire a helical geometric phase without adverse inter LL scattering (the inset of a). b, The phase offsets versus Vg curve of the 2DEG obtained from the LK fitting. The reference line corresponds to the π Berry phase. For the 2DEG, at low Vg (<5 V), both inner and outer pockets are trivial, consistent with the quantum Hall measurements in Fig. 3. As Vg increases above 7 V, the inner pocket acquires a Berry phase of π, whereas the outer FS remains trivial. The different band topology between the inner and outer FSs of the Γ Rashba valleys may be induced by square lattice warping or suggest the existence of hierarchical Rashba band splitting40. c, The black curve is the SdHOs for the Vg = −7 V 2DHG, which is compared with the red curve from the LK fitting. d, Vg-dependent phase offsets of the 2DHG obtained from the LK fitting. For the 2DHG, the LK formula fitting confirms that both FSs, either two single flavoured pockets (low, positive Vg) or inner–outer-nested FSs, have trivial geometric phase offsets.
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This file contains Supplementary Notes 1-9, Supplementary References, and Supplementary Figures 1-13.
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Sheng, F., Hua, C., Cheng, M. et al. Rashba valleys and quantum Hall states in few-layer black arsenic. Nature 593, 56–60 (2021). https://doi.org/10.1038/s41586-021-03449-8
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DOI: https://doi.org/10.1038/s41586-021-03449-8
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