Abstract
The C_{2v} surface symmetry of W(110) strongly influences a spinorbitinduced Diracconelike surface state and its characterization by spin and angleresolved photoelectron spectroscopy. In particular, using circular polarized light, a distinctive kdependent spin texture is observed along the \(\overline{{\boldsymbol{\Gamma }}{\boldsymbol{H}}}\) direction of the surface Brillouin zone. For all spin components P_{x}, P_{y}, and P_{z}, nonzero values are detected, while the initialstate spin polarization has only a P_{y} component due to mirror symmetry. The observed complex spin texture of the surface state is controlled by transition matrix element effects, which include orbital symmetries of the involved electron states as well as the geometry of the experimental setup.
Introduction
Topological insulators (TI) and Rashba systems attract great attention with regard to manipulation and generation of spin currents without magnetic field^{1,2,3}. The lack of spaceinversion symmetry at the surface of these materials leads to spinpolarized surface states driven by the spinorbit interaction. The electron spin is locked to its crystal momentum, forming a unique helical spin texture. As a consequence of spinorbit interaction, the spinpolarized surface state, by hybridization, consists of both spinup and spindown components with different weight of their partial wave functions^{4,5,6,7}, except for the special case of wavevectors with C_{3} symmetry^{8,9,10,11}.
The spin mixture causes spin entanglement of photoelectrons in topological materials as well as Rashba systems. Consequently, when spinup and spindown components in such initial bands are simultaneously excited by photons with a coherent mixture of s and ppolarized light or by circular polarized light, a spin component other than that in the initial state is caused by the superposition of complex transition matrix elements.
Recently, by utilizing the abovementioned effect, a fascinating idea was proposed: the manipulation and control of spin polarization of the photoelectron signal by a proper selection of the light polarization, experimental geometry and photon energy^{5,6,12,13,14,15,16,17,18,19}. These phenomena are promising in view of the materiallightspin relationship for potential applications in optospintronics devices with multiple functionalities. So far, most studies of the photoelectron spin are restricted to surfaces with C_{3v} symmetry or to cases with normal electron emission^{20,21,22,23,24}.
A surface state on W(110) within a spinorbitinduced symmetry gap^{25,26,27,28,29} shows Diracconelike dispersion with a spin texture reminiscent of a topological surface state (TSS)^{30}. The surface state is strongly influenced by the twofold symmetry (C2v) of the crystal surface: it shows a flattened dispersion behavior along the \(\overline{{\rm{\Gamma }}N}\) line of the surface Brillouin zone and a linear dispersion along \(\overline{{\rm{\Gamma }}H}\) and \(\overline{{\rm{\Gamma }}S}\)^{31,32,33}. Only \(\overline{{\rm{\Gamma }}N}\) and \(\overline{{\rm{\Gamma }}H}\) possess mirror planes. Based on several theoretical and experimental studies, along \(\overline{{\rm{\Gamma }}H}\), the surface state exhibits predominant Σ_{1} (\({d}_{{z}^{2}}\)) and Σ_{3} (d_{zx}) symmetry (singlegroup representation) with minor Σ_{2} (d_{xy}) and Σ_{4} (d_{yz}) contributions. We have demonstrated by angleresolved photoelectron spectroscopy (ARPES) using linear p and spolarized light^{4} that the spin polarization depends on the orbital symmetry in agreement with theoretical calculations^{7}. Moreover, recent theoretical research has shown that this surface state is topologically protected by mirror symmetry along \(\overline{{\rm{\Gamma }}H}\). Therefore, W(110) is called a topological crystalline transition metal^{34}.
In this work, we use spinresolved ARPES with left and right circular polarized light to measure the three spinpolarization components of photoelectrons emitted from the Diraccone like surface state along \(\overline{{\rm{\Gamma }}H}\) on W(110) with C2v symmetry. We show that the observed complex spinpolarization texture as a function of momentum can be explained by multiple contributions of the dipoletransition matrix element for the given highly symmetric experimental geometry. The observed spin polarization of the photoemitted electrons is not only determined by the intrinsic spin polarization, i.e. the spin polarization of the initial state under investigation, but also by extrinsic spin polarization, which is induced and controlled by experimental parameters. In detail, the latter is given by phase differences between complex partial transition matrix elements, which contain the orbital components of initial and final states as well as experimental parameters such as the crystal symmetry, light polarization and its angle of incidence. In short, using the example of the Diraclike surface state on W(110), we show how the spin polarization of photoemitted electrons can be controlled experimentally.
Results and Discussion
Figure 1(a) shows spinintegrated ARPES data of W(110) along \(\overline{{\rm{\Gamma }}H}\), obtained with right circular polarized light (C^{+}) of photon energy hν = 43 eV. In agreement with established results for linear polarized light^{4,31}, there are two characteristic surface states (S_{1} and S_{2}) and bulk continuum states at binding energies E_{B} higher than 1.45 eV. In this paper, we only discuss the surface state S_{1} with clear Diracconelike dispersion and a crossing point at E_{B} = 1.25 eV at \(\overline{{\rm{\Gamma }}}\).
Figure 1(b,c) show spinARPES data as energy distribution curves (EDC’s) for selected emission angles θ_{e} along \(\overline{{\rm{\Gamma }}H}\), excited by right (C^{+}) and left (C^{−}) circular polarized light. Left, middle and right columns present spinARPES results for P_{x}, P_{y} and P_{z}, respectively. Spinup and spindown intensities are plotted as red and blue solid lines, respectively.
At first, we will discuss spinEDC’s for P_{y} in the middle panel of Fig. 1(b). A sharp peak S_{1} in the spinup (spindown) channel is located at E_{B} = 1.42 eV (1.05 eV) for θ_{e} = −4° and moves to lower (higher) E_{B} with increasing θ_{e}. At θ_{e} = 0°, the bands cross, yet with spindependent intensities. This behavior is reminiscent of our previous results obtained for ppolarized light^{4}. Moreover, according to the middle panel of Fig. 1(c), the observed spin features remain unchanged upon switching the circular polarization.
For P_{z} in Fig. 1(b), the spin texture of S_{1} is similar to P_{y} but with reduced spin difference. For P_{x}, the spinup intensities always exceed the spindown intensities for both the upper and lower part of the Dirac cone. Note that, in contrast to P_{y}, P_{x} and P_{z} switch sign upon reversing the circular polarization of the light (see Fig. 1(c)).
In Fig. 2, our spinARPES results are summarized as E vs k_{} plots, where the spinpolarized photoemission intensities and their differences are given as colored contours. In the following, we focus on the spin differences for the three spinpolarization components in the right columns of Fig. 2(a) for C^{+} light and Fig. 2(b) for C^{−} light. Two major observations stand out and have to be explained:

(i)
By switching the circular polarization, the sign of P_{x} and P_{z} is reversed but P_{y} is unchanged.

(ii)
By reversing the sign of k_{} or θ_{e}, P_{y} and P_{z} change sign, while P_{x} is unchanged.
For measurements along \(\overline{{\rm{\Gamma }}H}\), due to the m_{xz} mirror plane as shown in Fig. 1(d), the initialstate spin polarization is restricted to P_{y}. As a consequence, the observed nonzero P_{x} and P_{z} spin components are caused by the photoemission process itself, which is described by the full dipoletransition matrix element including the final state. The following discussion focuses on the photoemissioninduced effect for P_{x} and P_{z}.
First, we discuss the origin of observation (i). It can be explained by considering the symmetry of the experimental geometry including the mirror plane m_{xz} as illustrated in Fig. 1(d). Upon reflection at the mirror plane, the circular polarization of the light is reversed, while the light incidence angle and the electron emission angle are unchanged. In addition, P_{y} is unchanged, while P_{x} and P_{z} change sign. As a consequence, P_{x} and P_{z} are expected to change sign upon switching the circular polarization of the light. This situation is generally realized in experiments, where the detection plane coincides with a mirror plane, which is the case for \(\overline{{\rm{\Gamma }}H}\) on W(110).
Second, observation (ii) can be explained in the framework of a grouptheoretical analysis of spindependent photoemission on the basis of dipoletransition matrix elements including final states of Σ_{1} representation (without spinorbit interaction)^{21}. In general, an initial state consisting of a mixture of spinup and spindown states may result in a spin polarization of the photoemitted electron, which is different from the initialstate spin polarization.
Following the approach of ref.^{21}, we can derive the total photoemission intensity I and the intensity differences \({I}_{{\rm{up}}}^{{\rm{x}},{\rm{y}},{\rm{z}}}{I}_{{\rm{down}}}^{{\rm{x}},{\rm{y}},{\rm{z}}}=I{{\rm{P}}}_{{\rm{x}},{\rm{y}},{\rm{z}}}\). The orbitaldependent spin polarization of the initial states is considered, while the final states have Σ_{1} symmetry. The latter follows from the fact that there is no spinorbit interaction at the detector (vacuum), spinorbit coupling is very weak for electronic states with energies much higher than the Fermi level, and only even states can be detected.
θ_{p} and ϕ_{p} represent polar and azimuthal angles of the incident photons, respectively. In our experimental geometry, ϕ_{p} = 0°. As a consequence, only the first terms of eqs (2–4) are relevant for the discussion of the observed spin polarization. \({M}_{p\perp }^{\mathrm{(1)}}\) and \({M}_{p}^{\mathrm{(3)}}\) (\({M}_{s}^{\mathrm{(4)}}\)) indicate the complex partial matrix elements for Σ_{1} and Σ_{3} (Σ_{4}) orbital contributions in the initial state, excited by normal and parallel electric field vector components of ppolarized (spolarized) light within circular polarized light, respectively. For normal electron emission and small θ_{e}, we neglect photoemission contributions of Σ_{2} states because they are forbidden by selection rules at \(\overline{{\rm{\Gamma }}}\)^{31}. From these equations, P_{y} originates from a mixing of Σ_{1} and Σ_{3} evensymmetry orbitals with respect to the m_{xz} mirror plane. This photoemissioninduced effect of P_{y} is caused by only the ppolarized component of the light. The intensity difference for P_{x} (P_{z}) is generated by a mixing of Σ_{1} (Σ_{3}) evensymmetry orbitals and Σ_{4} (Σ_{4}) oddsymmetry orbitals. The unexpected intensity differences for P_{x} and P_{z} are due to coherent superposition of p− and s−polarization components within the circular polarized light. In the experiment, we varied the photoelectron emission angle θ_{e} between −5° and +5°. Consequently, according to our setup, the light incidence angle θ_{p} varies between 45° and 55°, which will not strongly influence the intensity differences. The variation of θ_{e} is taken into account in the complex partial matrix elements \({M}_{p\perp }^{\mathrm{(1)}}\), \({M}_{p}^{\mathrm{(3)}}\), and \({M}_{s}^{\mathrm{(4)}}\).
Next, we discuss the symmetry of the partial matrix elements in more detail, as illustrated in Fig. 3. The partial matrix elements contain the wave function of the final state, the electric field vector of the photons, and the wave function of the initial state with C_{2v} symmetry, namely, Σ_{1}, Σ_{3}, and Σ_{4}. (In doublegroup representation, only one representation (Σ_{5}) exists which can be decomposed into the four singlegroup representations). The surface with C_{2v} symmetry includes two mirror planes: m_{xz} and m_{yz}. In our experimental geometry, final state, electric field vector components E_{z} and E_{y}, and Σ_{1} and Σ_{4} initial states have even symmetry with respect to m_{yz}. However, the photoelectron emission angle is reversed upon reflection at m_{yz}. As a consequence, \({M}_{p\perp }^{\mathrm{(1)}}({{\rm{\theta }}}_{e})={M}_{p\perp }^{\mathrm{(1)}}(\,\,{\theta }_{e})\) and \({M}_{s}^{\mathrm{(4)}}({{\rm{\theta }}}_{e})={M}_{s}^{\mathrm{(4)}}(\,\,{\theta }_{e})\). For \({M}_{p}^{\mathrm{(3)}}\), we can estimate the symmetry properties. In a simple model, a matrix element is the Fourier transform of the initial state’s orbitals; thus, it obeys the same evenodd properties in kspace as the orbital itself. Therefore, \({M}_{p}^{\mathrm{(3)}}\) has odd symmetry with respect to m_{yz}: \({M}_{p}^{\mathrm{(3)}}({{\rm{\theta }}}_{e})=\,{M}_{p}^{\mathrm{(3)}}(\,\,{\theta }_{e})\). By considering the θ_{e}dependent partial matrix elements, IP_{x} (IP_{y} and IP_{z}) show even (odd) symmetry, with respect to m_{yz}. Therefore, the intensity differences for P_{z} are reversed, when θ_{e} changes sign, while P_{x} does not change.
Figure 4 shows firstprinciples calculations within the relativistic onestep model calculation for W(110) along \(\overline{{\rm{\Gamma }}H}\). The calculation includes the full photoemission process for excitation with circular polarized light and the geometrical setup of our experiment. EDC’s for C^{+} and C^{−} light are presented for all three spin polarization components in Fig. 4(a,b), respectively. The intensity differences are shown in Fig. 4(c). We find good qualitative agreement between experimental and theoretical results, even details, such as spindependent intensities for P_{y} at θ_{e} = 0°, are well reproduced.
In conclusion, we have clarified the origin of circularpolarizedlightinduced spin signals in photoemission results from the Diracconelike surface state on W(110) with C_{2v} crystal symmetry. We observed nonzero spin polarization in all spin components P_{x}, P_{y}, and P_{z}. In our case, for photoelectrons from the Diraclike state on W(110) with hν = 43 eV, we obtained spinpolarization values of up to 20% for P_{x}, 90% for P_{y}, and 25% for P_{z}. While P_{y} is dominated by the intrinsic spin polarization of the initial states, P_{x} and P_{z} are exclusively extrinsic and originate from the photoemission process. Our detailed analysis of the complex partial transition matrix elements shows how the orbital contributions of the initial state and the geometrical details of the experiment, such as light incidence angle and light polarization, and the crystal symmetry lead to the experimentally observed spin texture. Within limits, this approach opens the way for manipulating and controlling the spin polarization of photoemitted electrons. The limits are given by the sample and its symmetry, the orbital characters of the respective electron states, and the experimental parameters. The challenge is to find optimum conditions for sample and experimental parameters that lead to maximum spinpolarization values.
The discussed mechanism of inducing spin polarization by circular polarized light is a general phenomenon depending on the specific orbital characters of the bands as well as geometrical parameters of the experiment. While the described test case of an occupied Dirac state at the metallic W(110) surface is not well suited for spinselective transport, the general idea may be applied to opticalinduced spinpolarized transport in optospintronic devices (see, e.g.,^{35}). One may even speculate about spinselective anisotropic transport phenomena by exploiting anisotropically dispersing states similar to the Dirac state on W(110)^{31}.
Methods
A clean surface of W(110) was obtained and evaluated by the same procedure as described elsewhere^{30,31}. The ARPES and spinARPES experiments were performed with synchrotron radiation generated by a quasiperiodic variable polarizing undulator at BL9B at Hiroshima Synchrotron Radiation Center (HiSOR), equipped with highly efficient threedimensional spinpolarization analysis of the ESPRESSO machine^{36,37}. At BL9B, the electric field vectors between left (C^{−}) and right circular (C^{+}) can be switched by changing the magnetic phase of the variable polarizing undulator. The angle of light incidence was 50° relative to the lens axis of the electron analyzer in all experiments as shown in Fig. 5. The spinARPES system with high angular and energyresolution can resolve all three spin polarization components: outofplane (P_{z}) and inplane (P_{x} and P_{y}). The positive (negative) sign of spin polarization is parallel (antiparallel) to the arrows of the x, y, and z axes in the sample coordinate system. The emission angle θ_{e} of the photoelectrons is defined as positive (negative), when the surface normal is moved away from (toward) the light propagation vector. The overall experimental energy and angular resolutions of ARPES (spinARPES) at BL9B were set to 50 meV (50 meV) and 0.3° (0.75°), respectively. All measurements have been performed at a sample temperature of 80 K.
The firstprinciples calculations include the photoemission process in the onestep model and are described in ref.^{7}.
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
This work was financially supported by KAKENHI (Grant Nos 23340105, 23244066, 25800179, 16K13823), GrantinAid for Scientic Research (A), (B) and for Young Scientists (B) of JSPS. K.M. gratefully acknowledges the hospitality of the Physikalisches Institut at the University of Münster and support by the Alexander von Humboldt foundation (Humboldt research fellowship for experienced researcher). The measurements were performed with the approval of the Proposal Assessing Committee of HSRC (Proposal No. 13B2). H.W. and M.D. gratefully acknowledge the hospitality of the Hiroshima Synchrotron Radiation Center.
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Affiliations
Hiroshima Synchrotron Radiation Center, Hiroshima University, 2313 Kagamiyama, HigashiHiroshima, 7390046, Japan
 K. Miyamoto
 & T. Okuda
Westfälische WilhelmsUniversität Münster, Physikalisches Institut, WilhelmKlemmStraße 10, 48149, Münster, Germany
 K. Miyamoto
 , H. Wortelen
 & M. Donath
MartinLutherUniversität HalleWittenberg, Institut für Physik, VonSeckendorffPlatz 1, 06120, Halle, Germany
 J. Henk
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Contributions
K.M. and M.D. conceived the experiments. K.M., H.W., T.O., and M.D. conducted the experiments. J.H. provided the SARPES calculations. K.M., M.D., and J.H. analyzed and discussed the results. K.M. and M.D. wrote the manuscript. All authors reviewed the manuscript.
Competing Interests
The authors declare no competing interests.
Corresponding author
Correspondence to K. Miyamoto.
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