Abstract
Spinsplit Rashba bands have been exploited to efficiently control the spin degree of freedom of moving electrons, which possesses a great potential in frontier applications of designing spintronic devices and processing spinbased information. Given an intrinsic breaking of inversion symmetry and sizeable spin–orbit interaction, twodimensional (2D) surface alloys formed by heavy metal elements exhibit a pronounced Rashbatype spin splitting of the surface states. Here, we have revealed the essential role of atomic orbital symmetry in the hexagonally warped Rashba spinsplit surface state of the \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} monatomic alloy by scanning tunneling spectroscopy (STS) and density functional theory (DFT). From dI/dU spectra and calculated band structures, three holelike Rashbasplit bands hybridized from distinct orbital symmetries have been identified in the unoccupied energy region. Because of the hexagonally deformed Fermi surface, quasiparticle interference (QPI) mappings have resolved scattering channels opened from interband transitions of p_{x},p_{y} (m_{j} = 1/2) band. In contrast to the s,p_{z}derived band, the hexagonal warping is predominately accompanied by substantial outofplane spin polarization S_{z} up to 24% in the dispersion of p_{x},p_{y} (m_{j} = 1/2) band with an inplane orbital symmetry.
Introduction
Hexagonal warping, i.e., a strong deformation of the Fermi surface in topological surface states due to interaction with neighboring (bulk) states, has been observed in a wide variety of topological insulators (TIs)^{1,2,3,4}. In the case of Bi_{2}Te_{3}, it has a rhombohedral crystalline structure of space group \(R3\bar m\), resulting in C_{3} rotation symmetry and three M mirror reflections on the [111] surface. According to \(k \cdot p\) theory, the warping term from cubic spin–orbit coupling together with C_{3} and M symmetries can lead to an anisotropic Fermi surface in Bi_{2}Te_{3}^{5}. Such a hexagonally warped Fermi surface gives rise to several interesting effects on the topological surface state, including outofplane spin polarization, an energy band gap opened at Dirac point, the emergence of Friedel oscillation on the local density of states (LDOS), and a possible spin density wave (SDW) phase^{6,7,8,9,10}.
With sizeable spin–orbit coupling and broken inversion symmetry, the spin degeneracy of the electronic band structure can be intrinsically lifted in twodimensional (2D) systems to create a Rashba spin splitting of surface states^{11,12,13,14}. In recent years, a class of singleatomiclayer surface alloys has been successfully fabricated by means of evaporating heavy atoms onto the noble metal surfaces, for example, Pb, Bi, and Sb grown on Ag(111) and Cu(111)^{15,16,17,18,19,20,21,22,23}. The gigantic Rashba spin splitting characterized by the Rashba parameter α_{R} of 3.05 has been reported for the \(\sqrt 3 \times \sqrt 3 R30^0\) BiAg_{2} binary alloy^{16,24}.
In view of the \(\sqrt 3 \times \sqrt 3 R30^0\) surface reconstruction, the system presents a C_{3v} point group, including the C_{3} rotational symmetry and the M mirror plane in the Γ–K direction. Since significant spin–orbit interaction is inherently rooted in heavy elements, one could therefore expect the presence of a hexagonal warping effect in Rashbatype spinsplit surface states with \(\sqrt 3 \times \sqrt 3 R30^0\) structure^{25}. In addition, atomic orbital symmetries from elements of Rashba binary alloys, e.g., Ag 5 s, Bi 6p_{x}, 6p_{y}, and 6p_{z} in BiAg_{2}, are responsible for the hybridization of different Rashbasplit surface states. These states locate separately in occupied and unoccupied energy regions, which in principle will offer an opportunity to investigate the mutual interplay between orbital symmetry and hexagonal warping.
In this work, we study the hexagonal warping effect in unoccupied Rashbasplit surface states of the \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} surface alloy by scanning tunneling spectroscopy (STS) and DFT. Differential conductance dI/dU spectra have resolved three holelike Rashbasplit bands, and their corresponding spectroscopic features locate separately at 0.23 eV (s,p_{z}), 1.4 eV (p_{x},p_{y} (m_{j} = 1/2)), and 1.6 eV (p_{x},p_{y} (m_{j} = 3/2)) in the unoccupied energy region, which is consistent with calculated band structures. From energydependent QPI mappings on Fermi surface with hexagonal deformations, we have observed standing wave patterns developed by interband spin–flip scatterings in the dispersion of the p_{x},p_{y} (m_{j} = 1/2) band. Due to the inplane symmetry of the Bi 6p_{x},6p_{y} orbitals, they are effectively coupled to inplane potential gradients on the \(\sqrt 3 \times \sqrt 3 R30^0\) surface. A substantial outofplane spin polarization S_{z} reaching 24% maximally has been revealed in the considerably warped p_{x},p_{y} (m_{j} = 1/2)derived Rashba spinsplit band.
Results
Characterization of monatomic bismuth surface alloy
According to structure models obtained from previous Xray diffraction and STM studies, the surface alloys of Bi grown on Cu(111) have Bi coverages of 1/3 ML and 1/2 ML for \(\sqrt 3 \times \sqrt 3 R30^0\) and [2012] phases, respectively^{26,27}. This, in principle, provides a way how we can calibrate the amount of Bi in order to prepare the sample with a welldefined and extended \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} alloy. The overview STM topography of about 0.40 ML Bi on Cu(111) has been shown in Fig. 1a and the magnified image of Fig. 1b from the square area framed by black dashed lines in Fig. 1a displays one antiphase domain boundary (blue arrow) with few Bi vacancies (red arrow) in the BiCu_{2} phase. These two types of surface defects can be served as scattering centers enabling us to study the dispersion of Rashbasplit bands from energydependent QPI measurements^{22}. From the atomic resolution image in Fig. 1c, the surface reconstruction exhibits a \(\sqrt 3 \times \sqrt 3\) periodicity with a 30° rotation toward the high symmetry axis, e.g., \([1\bar 10]\) of Cu(111), which is in agreement with the corresponding structure model of BiCu_{2} alloy shown in the inset^{26,27}. Since the Bi coverage of this sample is slightly above 1/3 ML, a few isolated patches of [2012] phase also coexist with the \(\sqrt 3 \times \sqrt 3 R30^0\) surface alloy. For example, the atomic resolution image of Bi[2012] shown in Fig. 1d has been resolved from the blue square frame in Fig. 1a, and structure model in the inset of Fig. 1d explains the zigzag chainlike feature along the \([1\bar 10]\) direction of Cu(111) surface with threefold rotational symmetry.
Electronic structures of unoccupied Rashba spinsplit bands
In order to determine the energy positions of different Rashbasplit bands of Bi/Cu(111), tunneling spectra have been acquired on both \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} and Bi[2012] phases. Figure 2a displays the conductance dI/dU curve taken in modest bias range around the Fermi level, e.g., from 0.7 to −0.4 V. On BiCu_{2} (black line), an asymmetric peak at 0.23 V originates from the singularity of the LDOS at the band edge of s,p_{z}derived Rashba band^{22}. The Rashba splitting, Rashba energy, and Rashba parameter of this band along the high symmetry direction \(\overline {{\it{{\Gamma}{\rm{M}}}}}\) \(\left( {\overline {{\it{{\Gamma}K}}} } \right)\) derived from our firstprinciples calculations are k_{0} = 0.0347(0.0359), E_{R} = \(\hbar ^2k_0^2/2m^ \ast\) = 0.0156(0.0169), and α_{R} = \(\hbar ^2k_0/2m^ \ast\) = 0.899(0.941), respectively. Note that our DFT Rashba energy E_{R} = 16 meV is similar to ~ 20 meV obtained from photoemission studies^{28}.
To explore Rashbasplit bands hybridized from Bi 6p_{x},p_{y} orbitals of BiCu_{2}, we have measured the dI/dU curve at a larger bias range from 2.0 to −0.4 V and the results are shown in Fig. 2b. Interestingly, a pronounced peak at 1.6 V (red arrow) accompanied by a shoulder at 1.4 V (blue arrow) have been observed on BiCu_{2} (black line), but no characteristic features can be observed from dI/dU curve taken at Bi[2012] (blue line). Note that there are surface states at 2.5 eV and 2.8 eV above the Fermi level reported from previous 2PPE measurements and calculations^{29}. Since the intensity of conductance peak at 1.6 V is unusually stronger than the Rashba peak at 0.23 V (black arrow), we have doublechecked the normalized dI/dU/(I/U) from the simultaneously measured I/U curve. The intensity of conductance peak at 1.6 V is indeed about six times stronger than that of Rashba peak at 0.23 V (see Supplementary Fig. S1 for more details), but less pronounced as compared to the resulting output directly from lockin amplifier.
Apart from tunneling spectroscopy measurements, we also performed firstprinciple electronic structure calculations to reveal the origin of spectra peaks observed at different energy positions. The orbitaldecomposed band structure of the BiCu_{2} surface is shown in Fig. 2c. It can be seen that two sets of Rashbatype bands cross the \({\it{\bar {\Gamma}}}\) point with band edges at about 0.2 eV and 1.4 eV, yielding the observed minor peak and the shoulder in our STS in Fig. 2a, b. In addition, the projected DOS from another Rashba band (p_{x},p_{y} (m_{j} = 1/2)) crossing the \({\it{\bar {\Gamma}}}\) near 1.8 eV contributes to the major peak observed in STS of Fig. 2b. For a direct comparison with STS measurement, we further integrate the DOS around the \({\it{\bar {\Gamma}}}\) point with a broadening ∆E as shown in Fig. 2d. Note that the overall features of the DOS remain similar after integration at different heights above the surface. Our DFT calculations fairly support the minor peak at about 0.2 eV, and the major peak at ~1.6 eV as well as the shoulder at ~1.4 eV observed in our STS measurement. Note that the exact position of these states depends sensitively on the Bi relaxations^{30}.
QPI mappings on hexagonally warped Rashbasplit surface states
To uncover the dispersion of the p_{x},p_{y} (m_{j} = 1/2)derived Rashbasplit band and compare with the s,p_{z}derived Rashba band, we have carried out the QPI measurements covering both occupied and unoccupied energy regions on BiCu_{2}^{22}. According to the theoretical calculation by Mirhosseini et al., the p_{x},p_{y} (m_{j} = 1/2)derived Rashba band exhibits an unconventional spin topology, the overall feature has been illustrated in Fig. 3a with intra and interband scattering vectors (q_{intra} and q_{inter}), respectively^{31}. Note that the energydependent scattering vector q_{n}(E) is defined by q_{n}(E) = k_{f}(E) − k_{i}(E) in the framework of elastic scattering, which links initial and final momentum eigenstates for mapping out the dispersion relation of surface bands^{22,32,33}.
Another feature worth being mentioned is the hexagonal shape of constant energy contour (CEC) on p_{x},p_{y} (m_{j} = 1/2)derived Rashba band, which has been observed in the occupied states of both BiAg_{2} and BiCu_{2} surface alloys by photoemission studies before^{24,30}. As illustrated in Fig. 3b, the CEC of Rashba band in general consists of two closed contours, and the outer one with a larger k value is hexagonally deformed due to strong warping effects. The outofplane spin polarization can be stabilized at all corner points lying on the \(\overline {{\it{{\Gamma}{\rm{M}}}}}\) direction and vanish due to mirror symmetry along \(\overline {{\it{{\Gamma}K}}}\) on \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} surface. With such kind of spin topology and multiple stationary k points, one could therefore expect additional scattering channels opened from the outer contour of the hexagonal CEC^{5,34,35,36}. The multiple pairs of stationary k points, as well as the density of states, are mainly distributed on the extremal curvature, i.e., in particular, the corners and the midpoints, of the hexagonal CEC, giving rise to the dominant intensity in the QPI pattern^{5,35}. For instance, possible interband spin–flip scatterings (solid light and dark gray arrow lines) have been depicted in Fig. 3b for the outer hexagon, and these scattering vectors as well as their symmetric counterparts, i.e., rotating every 60°, will construct two circlelike features in the reciprocal q space as shown in Fig. 3c (solid light and dark gray dots). On the contrary, if we simply consider scattering vectors from intraband transitions (dashed light and dark gray arrow lines in Fig. 3b), they will merely result in a single closed contour (formed by empty light and dark gray dots) in Fig. 3d as reported before^{32,33}. Note that the expected QPI pattern from the hexagonally deformed CECs in Fig. 3b needs to take both the inter and the intraband transitions into account, i.e., overlapping Fig. 3c with the Fig. 3d, which has been graphically illustrated in Supplementary Fig. S2.
From Fig. 3e to g, representative dI/dU mappings clearly show spatial LDOS modulations in unoccupied of p_{x},p_{y} (m_{j} = 1/2)derived Rashba band^{37}. Note that the standing waves become barely visible when bias voltages are above 1.4 V (see Supplementary Movie for details). The electron interference pattern results in two circlelike features in the Fouriertransformed dI/dU (FTdI/dU) image as indicated by red and blue arrows in the middle of Fig. 3e to g. Note that the two circlelike features in FTdI/dU images are a result of the superposition of the inter and the intraband transitions, which indicates the presence of hexagonally deformed CECs as conceptually illustrated in the Fig. 3b–d. In order to exclude possible defect induced artifacts, we have inverted the FTdI/dU signals from individual circlelike features to restore the dI/dU mappings separately and found that electronstanding waves are indeed developed with a longrange coherence (see Supplementary Fig. S3 for details). According to the line profiles along \(\overline {{\it{{\Gamma}{\rm{M}}}}}\) and \(\overline {{\it{{\Gamma}K}}}\) of FTdI/dU images with an exponential background removed at the bottom of Fig. 3e–g, two scattering vectors have been extracted by employing the Gaussian fits (red/blue lines)^{15,22,32,33} (see Supplementary Fig. S4 for line profiles without removing the exponential background). Although spinconserving scattering, i.e., k_{f}(E) and k_{i}(E) with the same direction of spin polarization, is typically considered, we are also aware of spin–flip scattering on nonmagnetic scattering centers that need to be taken into account for the case of band structures with a spinpolarization inversion^{15,22,32,33,38}. In contrast to the simple selection rule that the scatterings between exact antiparallel spins are forbidden, spinflip scattering process permits the transitions from a 180°. reversal of spin direction accompanied by a 90°. rotation of orbital moment. Due to the spin–orbit entanglement, the spinflip scattering transitions between surface states with different orbital parities, i.e., (sp_{z}), (p_{x} − ip_{y}), and (p_{x} + ip_{y}), also become allowed as long as the total angular momentum remains conserved^{15,33} (see Supplementary Fig. S5 for the spin–orbit entanglement in the orbitaldependent spinresolved band structures). However, we denote that either spinconserving or spinflip scattering along would give rise to only a single scattering vector in the p_{x},p_{y} (m_{j} = 1/2)derived Rashba band, if only the intraband transitions are involved.
Discussion
Figure 4a summarizes the dispersions, D_{n}, i.e., the energydependent scattering vectors q_{n}(E) along \(\overline {{\it{{\Gamma}{\rm{M}}}}}\) within bias voltage range from 1.5 V to −0.45 V (for q_{n}(E) along \(\overline {{\it{{\Gamma}K}}}\) see Supplementary Fig. S6). Figure 4b is the spindecomposed band structure with inplane spin polarization S_{x,y} of the BiCu_{2} surface states. For the scattering vector dispersions of D_{1} (black dots) and D_{2} (green dots), they come from intra and interband scattering processes of s,p_{z} and p_{x},p_{y} (m_{j} = 1/2) Rashba bands^{22}. The black and green lines are the parabolic fits to obtain the effective masses \(m_{\overline {{\Gamma}{\rm{M}}} }^ \ast\) of (−0.34 ± 0.02)m_{e} and (−0.39 ± 0.01)m_{e}, respectively. These \(m_{\overline {{\Gamma}{\rm{M}}} }^ \ast\) values are consistent with those reported from photoemission work^{28}. Besides D_{1} and D_{2}, QPI results also reveal the band dispersions of D_{3} (gray dots) in the occupied energy region. We associate D_{3} with the scattering between the outer branch of s,p_{z} and inner one of the p_{x},p_{y} (m_{j} = 1/2) bands since it starts appearing at about 0.1 eV well above the projected bulk band gap of the Cu(111) substrate, whereas the strong hybridization between the outer branch of s,p_{z} band and bulk states does not play a significant role in forbidding the emergence of this scattering vector q_{3}^{22,23,30}.
Interestingly, two dispersions of D_{4} and D_{5} have been resolved from the p_{x},p_{y} (m_{j} = 1/2) band in the unoccupied energy region, and they correspond to scattering vectors q_{4} and q_{5}, respectively. In order to explain q_{4} and q_{5} in terms of interband scattering events in the hexagonally deformed CEC as illustrated in Fig. 3c, we have further calculated spindependent CEC with outofplane spin polarization S_{z} in Fig. 4c and the spindecomposed band structure with S_{z} in Fig. 4d. According to the CEC at 0.6 eV in Fig. 4c, there is an inner ringlike circle accompanied by an outer circle with a hexagonal shape. They both are colored by red and blue for positive and negative S_{z} components, which are the most important characteristics that come along with the warping deformation of the p_{x},p_{y} (m_{j} = 1/2) band (see Supplementary Fig. S7 for details). Typically, the Rashba band consists of two concentric circles on the CEC with an inplane chiral spin structure perpendicularly locked to the momentum vector, leading to prohibited backscattering as a consequence of timereversal symmetry. However, this scenario is not perfectly true if there is a warping distortion on the CEC, where not only a single pair of stationary points can be stabilized but also new scattering channels can be opened^{5,8,9,10,35}. To confirm our observation, we have also performed simulations on QPI structures based on generic \(k \cdot p\) theory with a cubic warping term included (see Supplementary Note for details)^{5,35}. Given that the effective scattering potentials in Tmatrix formalism are predominately developed from the corners and midpoints on the hexagonal CEC, as shown in Fig. 4e, two circlelike features (red and blue arrows) are in line with FTdI/dU image in Fig. 3d, where the inner circle (red arrow) arises essentially from interband transitions permitted in the hexagonal CEC^{35}. On the other hand, the outer circle (blue arrow) is attributed to not only interband but also to extra input from intraband scattering channels as described in Fig. 3b, c. Note that the quantitative values in Fig. 4e are slightly smaller than FTdI/dU in Fig. 3d, which could be a deviation of the offset k_{0} in DFT calculations on k_{} of the highlying, unoccupied Rashba band.
When the energy approaches the p_{x},p_{y} (m_{j} = 1/2) band edge, the CEC turns into a circular shape with a reduced S_{z} component because hexagonal warping interaction is less influential on smaller k values. With the plot in Fig. 4f, in contrast to the s,p_{z}derived band, we have found substantial outofplane S_{z} up to 24% in the dispersion of p_{x},p_{y} (m_{j} = 1/2) band. This result indicates that, due to the inplane symmetry of atomic orbitals, they are more effectively coupled to inplane potential gradient from \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} surface, leading also to a significant outofplane S_{z} polarization when spin–orbit coupling is involved. For comparison, we have also calculated the S_{z} of BiAg_{2} and found that BiCu_{2} can have S_{z} about three times larger than that of BiAg_{2} because of the stronger mixing of the pstates caused by the larger outward protrusion of Bi atoms ~1.0 Å on BiCu_{2} surface (see Supplementary Fig. S8 for details).
In conclusion, we have investigated the hexagonal warping effect in unoccupied Rashba spinsplit surface states of the \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} surface alloy. From dI/dU spectra as well as calculated band structures, three holelike Rashbasplit bands have been resolved. Their spectroscopic features locate separately at 0.23 eV (s,p_{z}), 1.4 eV (p_{x},p_{y} (m_{j} = 1/2)), and 1.6 eV (p_{x},p_{y} (m_{j} = 3/2)). Caused by hexagonally deformed Fermi surface, the energydependent QPI mappings have visualized standing wave patterns from interband scattering transitions of p_{x},p_{y} (m_{j} = 1/2) band. Due to a strong inplane potential gradient on BiCu_{2} surface, hexagonal warping is accompanied by outofplane S_{z} spin polarization up to 24% in the dispersion of p_{x},p_{y} (m_{j} = 1/2) band. This strong effect is enabled by the inplane symmetry of atomic orbitals. In light of such an orbitalselective hexagonal warping effect, our results, therefore, demonstrate an important and generic aspect of engineering relevant spindependent properties on lowdimensional Rashba materials by exploiting atomic orbital symmetry.
Methods
Experiment
The Bi/Cu(111) surfaces were prepared in an ultrahigh vacuum (UHV) chamber with the base pressure below P ≤ 2 × 10^{−10} mbar. The clean Cu(111) surface was first prepared by cycles of Ar^{+} ion sputtering with an ion energy of 500 eV at room temperature and subsequent annealing up to 800 K. The Bi source with the purity of 99.999% (Goodfellow) was ebeam sublimated onto Cu(111) surface at elevated temperature of 400 K at which the welldefined and extended \(\sqrt 3 \times \sqrt 3 R30^0\) BiCu_{2} binary alloy can be grown. After preparation, the sample was immediately transferred into a lowtemperature scanning tunneling microscope (LTSTM) from Unisoku Co. Ltd. (operation temperature T ≈ 4.5 K). The topography images were obtained from the constantcurrent mode with the bias voltage U applied to the sample. For STS measurements, a small bias voltage modulation (U_{mod} = 20–50 mV) was added to U (frequency v = 3991 Hz), such that tunneling differential conductance dI/dU spectra, as well as dI/dU maps, can be acquired by detecting the first harmonic signal by means of a lockin amplifier.
Theory
Firstprinciples calculations are performed using the Vienna Ab initio Simulation Package (VASP) based on the DFT^{39,40,41}. The projectoraugmentedwavetype pseudopotential with the Ceperley–Alder and Perdew–Zunger (CAPZ) type exchangecorrelation functional are adopted in the local density approximation (LDA) calculations^{42,43,44,45,46}. We consider the BiCu_{2} monolayer on top of a ninelayer Cu(111) \(\sqrt 3 \times \sqrt 3\) substrate to simulate our experimental system. The ion positions of the top three layers are optimized until the residue force is smaller than 0.02 eVÅ^{−1}. After the geometrical optimization, the buckling height of Bi ion of the BiCu_{2} alloy is fixed at 1.0 Å as deduced from our experimental spectroscopy result and from ref. ^{26} where Kaminski et al. give 1.05 Å. Spin–orbit coupling is included in the selfconsistent calculations with the energy cutoff of 400 eV over the 12 × 12 × 1 kmesh. The DOS is integrated over the zone center near the \({\it{\bar {\Gamma}}}\) point with an additional broadening of 0.01 and 0.1 eV. The twodimension energy contours of the Rashba bands around the \({\it{\bar {\Gamma}}}\) point are calculated over the 40 × 40 kmesh and then interpolated over the 360 × 360 kmesh.
Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
References
Hsieh, D. et al. A tunable topological insulator in the spin helical Dirac transport regime. Nature 460, 1101 (2009).
Chen, Y. L. et al. Experimental realization of a threedimensional topological insulator, Bi_{2}Te_{3}. Science 325, 178 (2009).
Kuroda, K. et al. Hexagonally deformed Fermi surface of the 3D topological insulator Bi_{2}Se_{3}. Phys. Rev. Lett. 105, 076802 (2010).
Arakane, T. et al. Tunable Dirac cone in the topological insulator Bi_{2x}Sb_{x}Te_{3y}Se_{y}. Nat. Commun. 3, 636 (2012).
Fu, L. Hexagonal warping effects in the surface states of the topological insulator Bi_{2}Te_{3}. Phys. Rev. Lett. 103, 266801 (2009).
Souma, S. et al. Direct measurement of the outofplane spin texture in the Diraccone surface state of a topological insulator. Phys. Rev. Lett. 106, 216803 (2011).
Herdt, A. et al. Spinpolarization limit in Bi_{2}Te_{3} Dirac cone studied by angle and spinresolved photoemission experiments and ab initio calculations. Phys. Rev. B 87, 035127 (2013).
Zhang, T. et al. Experimental demonstration of topological surface states protected by timereversal symmetry. Phys. Rev. Lett. 103, 266803 (2009).
Roushan, P. et al. Topological surface states protected from backscattering by chiral spin texture. Nature 460, 1106–1109 (2009).
Alpichshev, Z. et al. STM imaging of electronic waves on the surface of Bi_{2}Te_{3}: topologically protected surface states and hexagonal warping effects. Phys. Rev. Lett. 104, 016401 (2010).
Bychkov, Y. A. & Rashba, E. I. Properties of a 2D electron gas with lifted spectral degeneracy. JETP Lett. 39, 78–81 (1984).
Bihlmayer, G., Rader, O. & Winkler, R. Focus on the Rashba effect. N. J. Phys. 17, 050202 (2015).
Gambardella, P. & Miron, I. M. Currentinduced spin–orbit torques. Philos. Trans. R. Soc. A 369, 3175–3197 (2011).
Noguchi, R. et al. Direct mapping of spin and orbital entangled wave functions under interband spinorbit coupling of giant Rashba spinsplit surface states. Phys. Rev. B 95, 041111(R) (2017).
ElKareh, L. et al. A combined experimental and theoretical study of Rashbasplit surface states on the \(\sqrt 3 \times \sqrt 3 s\) Pb/Ag(111)\(\sqrt 3 \times \sqrt 3 s\) surface. N. J. Phys. 16, 045017 (2014).
Ast, C. R. et al. Local detection of spinorbit splitting by scanning tunneling spectroscopy. Phys. Rev. B 75, 201401(R) (2007).
Bian, G., Zhang, L., Liu, Y., Miller, T. & Chiang, C. T. Illuminating the surface spin texture of the giantRashba quantumwell system Bi/Ag(111) by circularly polarized photoemission. Phys. Rev. Lett. 108, 186403 (2012).
Moreschini, L. et al. Assessing the atomic contribution to the Rashba spinorbit splitting in surface alloys: Sb/Ag(111). Phys. Rev. B 79, 075424 (2009).
Wissing, S. N. P., Ritter, K. T., Krüger, P., Schmidt, A. B. & Donath, M. Spindependent size of interband hybridization gap: The interplay of adlayer and substrate states in Pb/Cu(111). Phys. Rev. B 91, 201403(R) (2015).
Bentmann, H. & Reinert, F. Enhancing and reducing the Rashbasplitting at surfaces by adsorbates: Na and Xe on Bi/Cu(111). N. J. Phys. 15, 115011 (2013).
Bentmann, H. et al. Spin orientation and sign of the Rashba splitting in Bi/Cu(111). Phys. Rev. B 84, 115426 (2011).
Steinbrecher, M., Harutyunyan, H., Ast, C. R. & Wegner, D. Rashbatype spin splitting from interband scattering in quasiparticle interference maps. Phys. Rev. B 87, 245436 (2013).
Moreschini, L. et al. Influence of the substrate on the spinorbit splitting in surface alloys on (111) noblemetal surfaces. Phys. Rev. B 80, 035438 (2009).
Ast, C. R. et al. Giant spin splitting through surface alloying. Phys. Rev. Lett. 98, 186807 (2007).
Meier, F., Dil, H., LoboCheca, J., Patthey, L. & Osterwalder, J. Quantitative vectorial spin analysis in angleresolved photoemission: Bi/Ag(111) and Pb/Ag(111). Phys. Rev. B 77, 165431 (2008).
Kaminski, D., Poodt, P., Aret, E., Radenovic, N. & Vlieg, E. Surface alloys, overlayer and incommensurate structures of Bi on Cu(111). Surf. Sci. 575, 233–246 (2005).
Girard, Y. et al. Growth of Bi on Cu(111): alloying and dealloying transitions. Surf. Sci. 617, 118–123 (2013).
Ünal, A. A. et al. Polarization dependence and surface sensitivity of linear and nonlinear photoemission from Bi/Cu(111). Phys. Rev. B 86, 125447 (2012).
Mathias, S. et al. Quantumwellinduced giant spinorbit splitting. Phys. Rev. Lett. 104, 066802 (2010).
Bentmann, H. et al. Origin and manipulation of the Rashba splitting in surface alloys. EPL 87, 37003 (2009).
Mirhosseini, H. et al. Unconventional spin topology in surface alloys with Rashbatype spin splitting. Phys. Rev. B 79, 245428 (2009).
ElKareh, L., Sessi, P., Bathon, T. & Bode, M. Quantum interference mapping of Rashbasplit Bloch states in Bi/Ag(111). Phys. Rev. Lett. 110, 176803 (2013).
Schirone, S. et al. Spinflip and elementsensitive electron scattering in the BiAg_{2} surface alloy. Phys. Rev. Lett. 114, 166801 (2015).
Sessi, P. et al. Visualizing spindependent bulk scattering and breakdown of the linear dispersion relation in Bi_{2}Te_{3}. Phys. Rev. B 88, 161407(R) (2013).
Lee, W. C., Wu, C. J., Arovas, D. P. & Zhang, S. C. Quasiparticle interference on the surface of the topological insulator Bi_{2}Te_{3}. Phys. Rev. B 80, 245439 (2009).
Rakyta, P., Pályi, A. & Cserti, J. Electronic standing waves on the surface of the topological insulator Bi_{2}Te_{3}. Phys. Rev. B 86, 085456 (2012).
Friedel, J. Metallic alloys. Nuovo Cim. Suppl. 7, 287–311 (1958).
Hirayama, H., Aoki, Y. & Kato, C. Quantum interference of Rashbatype spinsplit surface state electrons. Phys. Rev. Lett. 107, 027204 (2011).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for openshell transition metals. Phys. Rev. B 48, 13115 (1993).
Kresse, G. & Furthmüller, J. E. Efficiency of ab initio total energy calculations for metals and semiconductors using a planewave basis set. Computational Mater. Sci. 6, 15 (1996).
Kresse, G. & Furthmüller, J. E. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758 (1999).
Ceperley, D. M. & Alder, B. J. Ground state of the electron gas by a stochastic method. Phys. Rev. Lett. 45, 566 (1980).
Perdew, J. P. & Zunger, A. Selfinteraction correction to densityfunctional approximations for manyelectron systems. Phys. Rev. B 23, 5048–5079 (1981).
Kohn, W. & Sham, L. J. Selfconsistent equations including exchange and correlation effects. Phys. Rev. 140, A1133 (1965).
Acknowledgements
D.S.L. and P.J.H. acknowledge support from the competitive research funding and the helium liquefier system from instrumentation center in National Tsing Hua University, Ministry of Science and Technology of Taiwan under Grants No. MOST1102731M007001, MOST1082636M007002 and MOST1072112M007001MY3, and center for quantum technology from the featured areas research center program within the framework of the higher education sprout project by the Ministry of Education (MOE) in Taiwan. H.T.J. also acknowledges support from NCHC, CINCNTU, ASiMATE10913, Taiwan.
Author information
Authors and Affiliations
Contributions
G.Y.C. and A.H. contributed equally to this work. G.Y.C., Y.H.L., and C.J.C. carried out the STM/STS experiments and analyzed the data. A.H., D.S.L., P.Y.C., G.B., and H.T.J. performed the theoretical calculations. H.T.J. and P.J.H. coordinated and supervised the project. All authors discussed the results and contributed to the paper.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, GY., Huang, A., Lin, YH. et al. Orbitalenhanced warping effect in p_{x},p_{y}derived Rashba spin splitting of monatomic bismuth surface alloy. npj Quantum Mater. 5, 89 (2020). https://doi.org/10.1038/s41535020002933
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41535020002933
This article is cited by

Reversal of spinpolarization near the Fermi level of the Rashba semiconductor BiTeCl
npj Quantum Materials (2023)

Intrinsic spin Hall resonance in Bibased Janus monolayers
Nano Research (2023)