Abstract
Spin–orbit interaction entangles the orbitals with the different spins. The spin–orbitalentangled states were discovered in surface states of topological insulators. However, the spin–orbitalentanglement is not specialized in the topological surface states. Here, we show the spin–orbital texture in a surface state of Bi(111) by laserbased spin and angleresolved photoelectron spectroscopy (laserSARPES) and describe threedimensional spinrotation effect in photoemission resulting from spindependent quantum interference. Our model reveals that, in the spin–orbitcoupled systems, the spins pointing to the mutually opposite directions are independently locked to the orbital symmetries. Furthermore, direct detection of coherent spin phenomena by laserSARPES enables us to clarify the phase of the dipole transition matrix element responsible for the spin direction in photoexcited states. These results permit the tuning of the spin polarization of optically excited electrons in solids with strong spin–orbit interaction.
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Introduction
Strongly spin–orbitcoupled materials such as Rashba systems and topological insulators have been intensively studied not only because of fundamental scientific interest on unique spin textures of the surface states but also realizing spintronic devices^{1,2,3,4,5,6}. In a standard model of the spin texture on the spin–orbitcoupled materials, the spin is locked to the momentum of an electron, resulting in a singlechiral spin texture^{5,7}. However, this picture is incomplete to describe the spin texture of the real system. Remarkably, the entangled spin–orbital textures on a topological insulator, Bi_{2}Se_{3} (refs 8, 9, 10, 11, 12, 13), and a Rashbatype ternary alloy, BiTeI^{14,15}, were revealed experimentally and theoretically; the spin texture is locked to the orbital texture of the bands. The spin–orbitalentanglement is a general consequence of the strong spin–orbit coupling, and thus is important not only for surface states but also bulk states.
In this article, we report on the spin–orbital texture of a surface state of an elemental Bi(111), which was considered to show the singlechiral spin texture^{16,17,18}, investigated by spin and angleresolved photoelectron spectroscopy using a vacuum ultraviolet laser (laserSARPES). We establish a general description of the spin–orbital texture in even–odd parity symmetry. Moreover, we draw a new concept to determine the phase of the dipole transition matrix element of photoemission through the spindependent quantum interference, which relies on the spin–orbitalentanglement and the laser field. We elucidate that the phase governs the spin direction in the final spinor field. The spin–orbitalentangled systems are one of the promising candidates^{19} to realize the spin manipulation of optically excited electrons^{20,21,22}.
Results
Spin–orbital texture on a mirror plane
All of the angleresolved photoelectron spectroscopy (ARPES) and SARPES data were acquired with the fixed experimental geometry shown in Fig. 1a. Figure 1b displays an ARPES intensity image recorded in a mirror plane on the Bi(111) surface. The spinsplit surface states exhibit upward energy dispersions while the band dispersing downward from the point is attributed to a bulk state. The results agree well with previous reports^{16,17}. The laserSARPES measurements were performed at selected k cuts with the s and plightpolarizations as shown in Fig. 2a–h. In each lightpolarization condition, the y component of the spin polarization (P_{y}) is inverted with respect to the point and the absolute values of P_{y} at k_{1} and k_{4} are almost 100%. Moreover, we observed the P_{y} reversal at each fixed k point with switching the light polarization, whereas there was no spin polarization in the x and z directions (P_{x,z}) (Supplementary Fig. 1; Supplementary Note 1).
The wavefunction of the surface state can be decomposed into the symmetric and antisymmetric parts with respect to the mirror plane of the crystal. According to the dipole selection rule of photoemission, only the state is excited with the s(p)polarized light. The results of the laserSARPES indicate that each spinpolarized branch consists of the linear combination of and states or and states (Fig. 2i). Recent orbitalparitybased studies of the spinpolarized surface states on W(110) and Bi_{2}Se_{3} also came to essentially the same conclusions^{13,23,24}.
To understand the reversal spin polarization in the mirror plane, we establish a model based on spinors coupled to the and states. The initial states of the spinlifted wavefunctions are denoted by
Here, we introduce a mirrorreflection operator . Note, the spinquantization axis is defined as a direction perpendicular to the mirror plane, which corresponds to the y direction in the present system. As a consequence of the mirror operation of equation (1), we obtain the following equation;
Thus, the eigenfunctions of the mirror eigenvalues +i and −i are given by
From this simple calculation, we reveal that the spins pointing to the mutually opposite directions with respect to the mirror plane are locked to the even and odd parts of the spinlifted states. This concept not only clearly explains the present results but also is generally applicable for explaining the spin–orbital texture on the mirror plane. The spin expectation values of these states are calculated with the Pauli matrices σ_{x,y,z}: the y spin component can be finite while the x and z spin components are strictly 0 (Supplementary Note 2).
Now, we show the calculated band structure of the surface state on Bi(111) in Fig. 3a. Figure 3b exhibits spin expectation values of the lower surface band as a function of the wave number on : the spin expectation values of the and states are +1 and −1, respectively. Here, the spin polarization rapidly reduced near the point can arise from the hybridization with the bulk states. By contrast, for the upper surface band (Fig. 3c), the spin expectation values of the and states are fully reversed. These results agree with equation (3). The net spin polarizations of the surface states, represented in Fig. 3a, result from the summation of the spin expectation values with taking the weight of each spin–orbitalcoupled state.
Spindependent quantum interference of photoelectron
Even if the mirror symmetry governs the spin orientation in the initial states, rotating the electricfield vector of the incident linearly polarized light can break the mirror symmetry of the experimental geometry, which leads to the spin polarization of photoelectrons in the x and z directions (Supplementary Fig. 2; Supplementary Note 1). This is produced by the spindependent interference of the wavefunctions, resulting from simultaneous excitation of the and ( and ) states. When we consider the eigenstates , the spin polarization P_{x,y,z} of photoelectrons and the photoelectron intensity I_{total} are expressed as a function of the lightpolarization angle (θ) from the mirror plane (Fig. 1a) as (Supplementary Note 2)
with the following ratio between the dipole matrix elements in photoemission:
Here, A is the vector potential of the light, p the momentum operator and a finalstate wavefunction that is assumed to be spindegenerated for simplicity. Then, α represents a phase difference between the dipole matrix elements from the even and odd states, and u is an absolute value of the complex number. As a consequence of the equation (4), the signs of P_{x} and P_{z} can be classified into four classes depending on the value of α (Fig. 4a–d).
To demonstrate the above prospect, we show the observed θ dependence of P_{x,y,z} at k_{4} in Fig. 4e,f since the and states exhibit the 100% spin polarization in the initial states. The P_{x,y,z} oscillate as a function of θ as expected. The P_{y} is to be zero at θ∼60° and 120°, indicating that photoelectrons from the and states cancel out each other at these angles. By contrast, the P_{x,z} are almost zero at θ∼0°, 90° and 180°, and exhibit maximum values at θ∼60° and 120°. The signs of P_{x} and P_{z} are negative (positive) with 0<θ<90° (90°<θ<180°). Thus, we can immediately judge π<α<3π/2 at k_{4} using Fig. 4a–d. The experimental results of P_{x,y,z} and the intensity were well reproduced by the equation (4) with u=0.62 and α=1.3π. Here, we note that the θ dependence of P_{x,y,z} should be changed with changing the photon energy since the photoexcited states, that is, the spindependent matrix elements, are different.
Discussion
The electron–photon interaction Hamiltonian of photoemission is given by the three terms corresponding to the dipole transition, surface photoemission, and spin–orbit coupling^{25}. In the earlier theoretical work^{26}, the spin rotation effect in photoemission was discussed with both spinconserving and spinflipping transitions with employing the dipole transition and spin–orbit terms in the interaction Hamiltonian. Subsequently, Jozwiak et al.^{27} experimentally demonstrated that the spin polarization of photoelectrons from the surface state of Bi_{2}Se_{3} is largely changed compared with that of the initial state, which was explained by the spinflip transition in photoemission: they considered the average spin texture in the initial state, but not the spin–orbital texture. In the present study, we demonstrate that the spin polarization of the photoelectrons excited by the linearly polarized light is successfully explained only with the dipole transition term in the interaction Hamiltonian with taking the mirror symmetry and the spin–orbital texture into account. It has not yet been established how important the spin rotation contribution to photoexcitation arising from the spin–orbit term is. In fact, Wissing et al.^{28} pointed out that the relativistic corrections of the dipole operator would be negligibly small corrections to the spin polarization of the photoelectrons, while in the photoemission study using the circularly polarized light it was discussed that the spin–orbit term in the interaction Hamiltonian is generally strong for systems with heavy elements^{25}.
For another mechanism of the spin rotation, a layerdependent interference effect in photoemission process was proposed^{13}. This mechanism is only achievable in the system with the layerdependent spin–orbital texture, and realizes the spin control of photoelectrons only by varying photon energy. The present concept is essentially different from this scheme. The spin rotation over three dimension results from simultaneous optical excitation of the linearlycombined even and odd parts of the wavefunctions, and thus the spin direction of photoelectron can be readily controlled just by tuning linearpolarization axis of the light with the fixed photon energy. This concept is comprehensive and no longer needs the layerdependent interference picture to demonstrate the optical spin control.
Furthermore, the present concept is applicable not only to the present system but to the other and mixed systems. In the case of Bi_{2}Se_{3}, the sign of P_{x} is the same as Bi(111), but the sign of P_{z} opposite^{19}. Thus, the phase difference of the matrix elements should be in the range of 3π/2<α<2π in accordance with Fig. 4d: the P_{x,y,z} were well fitted with u=0.45 and α=1.6π. This indicates that the phase difference is a materialinherent variable.
So far, in the photoelectron spectroscopy, one has observed only the intensity of photoelectrons, meaning that the phase information of the dipole matrix element has been lost. By contrast, the threedimensional SARPES with varying linearpolarization angle provides the phase information that is essential to describe the nature of the spin polarization of the photoexcited electron. The combination of threedimensional SARPES and the linearpolarizationcontrolled laser is an innovative tool for quantummechanical understanding of the photoexcitation process.
The results offer opportunities for photocathodes as highly spinpolarized electron sources. The disadvantage of commonly used GaAs photocathodes as spinpolarized electron sources is that it is hard to tune the direction of the spin polarization and that the degree of spin polarization is only 50% (ref. 29). On the other hand, the present expermental results clearly show the 100% spin polarization of photoelectron (Fig. 4g), as theoretically predicted in the former report^{26}, and its direction readily controllable just by tuning the linear photon polarization. A technique using the quantummechanical phase degree of freedom opens new avenues for the optical spin control.
Methods
Sample preparation
The Bi sample was in situ prepared in a molecular beam epitaxy chamber connected to the analysis chamber. We used ntype Si(111) substrates. A clean Si(111) surface was prepared by flushing at 1,420 K. Then, Bi with the thickness of 100 bilayers (BL) was deposited onto the clean Si(111)7 × 7 surface at room temperature from a Knudsen cell^{30}. The deposition rate was calibrated by observing wellknown quantumwellstates on the Bi film by ARPES^{31}. The Bi film exhibits a sharp (1 × 1) lowenergy electrondiffraction pattern and an excellent Fermi surface image by ARPES.
LaserARPES and SARPES measurements
Our ARPES and SARPES measurements using an ultraviolet laser were performed at the Institute for Solid State Physics, The University of Tokyo^{32}. Our laser system provides 6.994eV photons^{33}. Photoelectrons were analysed with a combination of a ScientaOmicron DA30L analyzer and twin verylowenergyelectrondiffraction (VLEED) type spin detectors. The experimental geometry is represented in Fig. 1a. The light incident plane is in the x–z plane on the sample axis, which corresponds to the mirror plane. We used linearly polarized light, and the direction of its electricfield vector is arbitrarily adjustable between the p and spolarizations. Rotation angle of the electricfield vector is given by θ, where the light is of the p(s)polarization at θ=0° and 180 (90°). The energy and angular resolutions were set to 6 meV and 0.7°, respectively. The sample temperature was kept at 15 K during the laserSARPES measurements.
Electronic band structure calculation
The firstprinciples calculation was performed using the Vienna Ab initio Simulation Package (VASP)^{34}. The projector augmented wave (PAW) method^{35} was used in the planewave calculation. The generalized gradient approximation (GGA) by Perdew, Burke and Ernzerhof (PBE)^{36} was used for the exchangecorrelation potential. The spin–orbit interaction was included. The cutoff energy was 110 eV. The Bi film was modelled by a freestanding 30BL Bi(111) slab. The slabs in the repeated slab structure were separated by vaccums with a thickness more than 10 Å. Atom positions in the slab were taken from the experimental data shown in ref. 37.
Data availability
The data supporting the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Yaji, K. et al. Spindependent quantum interference in photoemission process from spinorbit coupled states. Nat. Commun. 8, 14588 doi: 10.1038/ncomms14588 (2017).
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Acknowledgements
Numerical calculations were performed using supercomputers at the Institute of Solid State Physics, The University of Tokyo. This work was supported by J.S.P.S. GrantinAid for Scientific Research (B), Grant No. 26287061, for Scientific Research (C), Grant No. 26390063 and for Young Scientists (B) Grant No. 15K17675.
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K.Y. and K.Ku. conceived the research with guidance from F.K. K.Y. and S.T. fabricated and characterized the sample. K.Y. and K.Ku. carried out ARPES and SARPES measurements under the support of S.T., A.H., Y.I., S.W., C.C., F.K. and S.S. K.Ko. carried out the theoretical calculation. K.Y., K.Ku., K.Ko. and F.K. wrote the manuscript. F.K. and S.S. supervised the project. All authors discussed the paper.
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Supplementary Figures 12, Supplementary Notes 12 and Supplementary References (PDF 172 kb)
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Yaji, K., Kuroda, K., Toyohisa, S. et al. Spindependent quantum interference in photoemission process from spinorbit coupled states. Nat Commun 8, 14588 (2017). https://doi.org/10.1038/ncomms14588
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DOI: https://doi.org/10.1038/ncomms14588
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