Dataset and article "Imaging Orbital Vortex Lines in Three-Dimensional Momentum Space"

Supplementary Information: Imaging Orbital Vortex Lines in Three-Dimensional Momentum Space T. Figgemeier,∗M. ¨ Unzelmann,†J. Schusser, B. Geldiyev, P. Kagerer, and F. Reinert Experimentelle Physik VII and W¨ urzburg-Dresden Cluster of Excellence ct.qmat, Universit¨ at W¨ urzburg, Am Hubland, D-97074 W¨ urzburg, Germany P. Eck,∗L. Crippa, and G. Sangiovanni ITPA and W¨ urzburg-Dresden Cluster of Excellence ct.qmat, Universit¨ at W¨ urzburg, Am Hubland, D-97074 W¨ urzburg, Germany J. N. Neu and T. Siegrist Department of Chemical and Biomedical Engineering, FAMU-FSU College of Engineering Tallahassee, FL 32310, USA and National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA J. Buck and M. Kall¨ ane Institut f¨ ur Experimentelle und Angewandte Physik, Christian-Albrechts-Universit¨ at zu Kiel, D-24098 Kiel, Germany and Ruprecht Haensel Laboratory, Kiel University and DESY, Germany M. Hoesch Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany K. Rossnagel Institut f¨ ur Experimentelle und Angewandte Physik, Christian-Albrechts-Universit¨ at zu Kiel, D-24098 Kiel, Germany Ruprecht Haensel Laboratory, Kiel University and DESY, Germany and Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany L.-K. Lim Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, 1 Zhejiang 310027, People’s Republic of China R. Moessner Max Planck Institute for the Physics of Complex Systems and W¨ urzburg-Dresden Cluster of Excellence ct.qmat, Noethnitzer Strasse 38, D-01187 Dresden, Germany D. Di Sante Department of Physics and Astronomy, Univerity of Bologna, I-40136 Bologna, Italy H. Bentmann Experimentelle Physik VII and W¨ urzburg-Dresden Cluster of Excellence ct.qmat, Universit¨ at W¨ urzburg, Am Hubland, D-97074 W¨ urzburg, Germany and Center for Quantum Spintronics, Department of Physics, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: October 10, 2024) ∗These authors contributed equally to the present work. †These authors contributed equally to the present work.; [email protected]urg.de 2 CONTENTS Supplementary Note 1: Additional data analysis of the nodal line 4 Supplementary Note 2: Complementary dichroism data 8 Supplementary Note 3: Model for the orbital vortex line 10 Supplementary Note 5: In-plane OAM texture: DFT and CD 13 References 15 3 SUPPLEMENTARY NOTE 1: ADDITIONAL DATA ANALYSIS OF THE NODAL LINE Detailed analysis of the band dispersion and energy splitting around the nodal point In this section, we provide an additional analysis of the almost movable Weyl nodal line (amWNL). Fig. S1a shows a high-resolution SX-ARPES spectrum taken at a photon energy of hν = 618 eV, which corresponds to a ΓΣ cut in the 3D bulk Brillouin zone of TaAs. Several energy distribution curves around the crossing point were fitted by the sum of two Gaussians including a Fermi-Dirac distribution and a linear background. The resulting peak positions are plotted as green and orange dots in Fig. S1a. From this, we can calculate the energy splitting ∆E of the two spin branches which is plotted in Fig. S1b as a function of kx. For wave vectors in the range of |kx| ≈ 0.6...0.9˚ A−1we find ∆E∝ |kx|allowing for linear interpolation of the a detailed position of the crossing point which yields q0= (0.76 ±0.05) ˚ A−1. Figure S1. Detailed analysis of the high-resolution SX-ARPES spectra. a SX-ARPES spectrum recorded at hν = 618 eV corresponding to an kz= 0 plane. The green and orange dots indicate the peak position of the two spin branches received through fitting EDCs. bEnergy splitting of the spin branches obtained from a. Open and filled markers represent the splitting for positive and negative kx, respectively. The energy splitting is linear in karound the nodal point as shown by the fit corresponding fit (solid line). 4 Determination of the kzand energy-dependent undulation of the nodal point In order to trace the nodal line within the kzkxmirror planes (see Fig. 2d in the main text), we recorded SX-ARPES spectra using different photon energies hν = 618...673 eV in steps of ∆hν = 5 eV. Given the long acquisition time needed to obtain data in high resolution and sufficient statistics allowing for a detailed fitting procedure explained in the previous section, we determined the in-plane wave vector q0,x of the band crossing points in three different manners. This is shown in Fig. S2 for two exemplary photon energies of hν = 638 eV and hν = 663 eV. First, we analyzed the kx-dependent modulation of the photoemission intensity I, which is likely induced by the change of orbital character at the crossing point, as outlined in the main text. In Fig. S2a,d we show the symmetrized data sets I(+kx) + I(−kx), which clearly reveal abrupt changes by following the intensity as a function of kx. Taking the center height of the step-like function I(kx), we determine the wave vectors q0,x marked by the dashed red lines. In the second case (Fig. S2b,d), we extracted the crossing point from the second derivative of the photoemission intensity I2nd ∝∂2I(kx,E) ∂k2 x +∂2I(kx,E) ∂E2, which can enhance the visibility of fine structures even in more broad features of ARPES-spectra. Again, the estimated wave vectors are indicated by the red dashed lines. In the last case, we were considering the Linear Dichroism (LD), which is sensitive to the kxdependent complex phase iγ(kx)(see main text) and accordingly orbital angular momentum (OAM) Lyin the initial state. In particular, Lychanges its sign at the position of the nodal line and we use this to determine its kz(kx)evolution as exemplary shown in Fig. S2c,f. Taken together, we determined the undulation of the nodal line in three-dimensional momentum space using three distinct methods. The result is shown in Fig. 2d in the main text proofing a reasonable agreement for all three approaches and overall excellent agreement with the DFT calculation. Interestingly, we find that the nodal line does not only undulate as a function of momentum but — given the kz-dispersion of the bulk bands in TaAs — also the energy of the crossing point changes. In Fig. S3, we show the experimentally determined energy as a function of kzin comparison to the corresponding DFT calculation. The reasonable agreement between experimental data and calculation further underpins the reliability of our data analysis. 5 Figure S2. Exemplary data sets for the determination of the undulation of the nodal point. Spectra in a-c are taken at hν = 638 eV (kz≈0.38 2π/c) and in d-f at hν = 663 eV (kz≈0.84 2π/c). For each photon energy, the position of the nodal point has been determined in three different ways (see text) as shown in (a,d), (b,e), and (c,f). In extracted wave vector q0is marked by the dashed line in each case. 6 Figure S3. Binding energy undulation of the nodal point. Data points represent the experimental values extracted from the photon energy series (partially) shown in Fig. 2a of the main part. The solid line represents the binding energy undulation obtained by our DFT calculation (see Fig. 2b in the main part). 7 SUPPLEMENTARY NOTE 2: COMPLEMENTARY DICHROISM DATA In Fig. S4, we compare the LD (both measured and for the one-step calculation) at different photon energies which correspond to an equivalent out-of-plane momentum, i.e. kz≈0. For both hν we find an overall reasonable agreement and, in particular, the important sign change at the WNL. As pointed out in the main manuscript, a deviation between theory and experiment is the appearance of a surface feature in the calculation, denoted as “S” in Fig. S4(c) and (e). Owing to the existence of several approximations to shape the surface barrier [1], all of which fulfill the expected 1/z asymptotics, the energetics and momentum dependence of the surface states emerging from its existence can take different shapes [2]. The choice of a modeled surface barrier with given parameters should then be done so that it reflects the real physical experiment as well as possible. Surface states in addition to surface resonances are sensitive to the shape of the potential barrier and their description requires detailed comparison to experiment within our computational approach. However, surface states (and resonances) play a negligible role concerning the bulk states discussed in our manuscript. This stems from the fact that the corresponding observables of the photoemission experiment of the bulk states remain unaltered by the given shape of the surface potential. This consequently implies that as long as the corresponding bulk state is present, the calculation correctly captures the underlying wavefunction properties. The high reliability of SPR-KKR in modeling physical systems and their initial state properties of bulk states has been demonstrated elsewhere, see, e.g., Ref. [3]. 8 Figure S4. Linear dichroism at different photon energies. a,c Experimental and calculated LD at hν = 618 eV corresponding to kz≈0(see main text). d,e Equivalent equi-kzcut measured and calculated using a photon energy of 433 eV. 9