Circular-polarized-light-induced spin polarization characterized for the Dirac-cone surface state at W(110) with C2v symmetry

The C2v surface symmetry of W(110) strongly influences a spin-orbit-induced Dirac-cone-like surface state and its characterization by spin- and angle-resolved photoelectron spectroscopy. In particular, using circular polarized light, a distinctive k-dependent spin texture is observed along the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{{\boldsymbol{\Gamma }}{\boldsymbol{H}}}$$\end{document}ΓH¯ direction of the surface Brillouin zone. For all spin components Px, Py, and Pz, non-zero values are detected, while the initial-state spin polarization has only a Py 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 set-up.

In this work, we use spin-resolved ARPES with left and right circular polarized light to measure the three spin-polarization components of photoelectrons emitted from the Dirac-cone like surface state along ΓH on W(110) with C2v symmetry. We show that the observed complex spin-polarization texture as a function of momentum can be explained by multiple contributions of the dipole-transition matrix element for the given highly symmetric experimental geometry. The observed spin polarization of the photo-emitted 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 Dirac-like surface state on W(110), we show how the spin polarization of photo-emitted electrons can be controlled experimentally.

Results and Discussion
Figure 1(a) shows spin-integrated ARPES data of W(110) along H Γ , 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 Dirac-cone-like dispersion and a crossing point at E B = 1.25 eV at Γ. Figure 1(b,c) show spin-ARPES data as energy distribution curves (EDC's) for selected emission angles θ e along H Γ , excited by right (C + ) and left (C − ) circular polarized light. Left, middle and right columns present spin-ARPES results for P x , P y and P z , respectively. Spin-up and spin-down intensities are plotted as red and blue solid lines, respectively.
At first, we will discuss spin-EDC's for P y in the middle panel of Fig. 1(b). A sharp peak S 1 in the spin-up (spin-down) 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 spin-dependent intensities. This behavior is reminiscent of our previous results obtained for p-polarized 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 spin-up intensities always exceed the spin-down 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 spin-ARPES results are summarized as E vs k || plots, where the spin-polarized photoemission intensities and their differences are given as colored contours. In the following, we focus on the spin differences for the three spin-polarization components in the right columns of Fig. 2(a) for C + light and Fig. 2 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 ΓH, due to the m xz mirror plane as shown in Fig. 1(d), the initial-state spin polarization is restricted to P y . As a consequence, the observed non-zero P x and P z spin components are caused by the photoemission process itself, which is described by the full dipole-transition matrix element including the final state. The following discussion focuses on the photoemission-induced 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 ΓH on W(110).
Second, observation (ii) can be explained in the framework of a group-theoretical analysis of spin-dependent photoemission on the basis of dipole-transition matrix elements including final states of Σ 1 representation (without spin-orbit interaction) 21 . In general, an initial state consisting of a mixture of spin-up and spin-down states may result in a spin polarization of the photo-emitted electron, which is different from the initial-state spin polarization.
Following the approach of ref. 21 , we can derive the total photoemission intensity I and the intensity differences The orbital-dependent 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 spin-orbit interaction at the detector (vacuum), spin-orbit coupling is very weak for electronic states with energies much higher than the Fermi level, and only even states can be detected.
) 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 p-polarized (s-polarized) 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 Γ 31 . From these equations, P y originates from a mixing of Σ 1 and Σ 3 even-symmetry orbitals with respect to the m xz mirror plane. This photoemission-induced effect of P y is caused by only the p-polarized component of the light. The intensity difference for P x (P z ) is generated by a mixing of Σ 1 (Σ 3 ) even-symmetry orbitals and Σ 4 (Σ 4 ) odd-symmetry 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 set-up, 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 || , and M s (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 double-group representation, only one representation (Σ 5 ) exists which can be decomposed into the four single-group 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, and || , 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 even-odd properties in k-space as the orbital itself. Therefore, M p . 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 first-principles calculations within the relativistic one-step model calculation for W(110) along H Γ . The calculation includes the full photoemission process for excitation with circular polarized light and the geometrical set-up 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 spin-dependent intensities for P y at θ e = 0°, are well reproduced.
In conclusion, we have clarified the origin of circular-polarized-light-induced spin signals in photoemission results from the Dirac-cone-like surface state on W(110) with C 2v crystal symmetry. We observed non-zero spin polarization in all spin components P x , P y , and P z . In our case, for photoelectrons from the Dirac-like state on W(110) with hν = 43 eV, we obtained spin-polarization 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 photo-emitted 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 spin-polarization 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 spin-selective transport, the general idea may be applied to optical-induced spin-polarized transport in optospintronic devices (see, e.g., 35 ). One may even speculate about spin-selective 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 spin-ARPES experiments were performed with synchrotron radiation generated by a quasi-periodic variable polarizing undulator at BL-9B at Hiroshima Synchrotron Radiation Center (HiSOR), equipped with highly efficient three-dimensional spin-polarization analysis of the ESPRESSO machine 36,37 . At BL-9B, 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 spin-ARPES system with high angular-and energy-resolution can resolve all three spin polarization components: out-of-plane (P z ) and in-plane (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 (spin-ARPES) at BL-9B 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 first-principles calculations include the photoemission process in the one-step 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.