Simple extension of the plane-wave final state in photoemission: Bringing understanding to the photon-energy dependence of two-dimensional materials

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Simple extension of the plane-wave final state in photoemission: Bringing understanding to the photon-energy dependence of two-dimensional materials

Christian S. Kern, Anja Haags, Larissa Egger, Xiaosheng Yang, Hans Kirschner, Susanne Wolff, Thomas Seyller, Alexander Gottwald, Mathias Richter, Umberto De Giovannini, Angel Rubio, Michael G. Ramsey, François C. Bocquet, Serguei Soubatch, F. Stefan Tautz, Peter Puschnig, and Simon Moser

Phys. Rev. Research 5, 033075 – Published 3 August 2023

Abstract

Angle-resolved photoemission spectroscopy (ARPES) is a method that measures orbital and band structure contrast through the momentum distribution of photoelectrons. Its simplest interpretation is obtained in the plane-wave approximation, according to which photoelectrons propagate freely to the detector. The photoelectron momentum distribution is then essentially given by the Fourier transform of the real-space orbital. While the plane-wave approximation is remarkably successful in describing the momentum distributions of aromatic compounds, it generally fails to capture kinetic-energy-dependent final-state interference and dichroism effects. Focusing our present study on quasi-freestanding monolayer graphene as the archetypical two-dimensional (2D) material, we observe an exemplary $E_{kin}$ -dependent modulation of, and a redistribution of spectral weight within, its characteristic horseshoe signature around the $\bar{K}$ and ${\bar{K}}^{'}$ points: both effects indeed cannot be rationalized by the plane-wave final state. Our data are, however, in remarkable agreement with ab initio time-dependent density functional simulations of a freestanding graphene layer and can be explained by a simple extension of the plane-wave final state, permitting the two dipole-allowed partial waves emitted from the C $2 p_{z}$ orbitals to scatter in the potential of their immediate surroundings. Exploiting the absolute photon flux calibration of the Metrology Light Source, this scattered-wave approximation allows us to extract $E_{kin}$ -dependent amplitudes and phases of both partial waves directly from photoemission data. The scattered-wave approximation thus represents a powerful yet intuitive refinement of the plane-wave final state in photoemission of 2D materials and beyond.

Received 25 April 2023
Accepted 23 June 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033075

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Electronic structure Total cross sections

Graphene Synchrotron radiation facilities

Angle-resolved photoemission spectroscopy Photoelectron diffraction Photoelectron techniques Time-dependent DFT

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Christian S. Kern ¹, Anja Haags^2,3,4, Larissa Egger¹, Xiaosheng Yang ^2,3,4, Hans Kirschner ⁵, Susanne Wolff ^6,7, Thomas Seyller ^6,7, Alexander Gottwald ⁵, Mathias Richter⁵, Umberto De Giovannini ^8,9, Angel Rubio ^8,10, Michael G. Ramsey¹, François C. Bocquet ^2,3, Serguei Soubatch ^2,3, F. Stefan Tautz ^2,3,4,*, Peter Puschnig ^1,†, and Simon Moser ^11,‡

¹Institute of Physics, NAWI Graz, University of Graz, 8010 Graz, Austria
²Peter Grünberg Institut (PGI-3), Forschungszentrum Jülich, 52425 Jülich, Germany
³Jülich Aachen Research Alliance (JARA), Fundamentals of Future Information Technology, 52425 Jülich, Germany
⁴Experimental Physics IV A, RWTH Aachen University, 52074 Aachen, Germany
⁵Physikalisch-Technische Bundesanstalt (PTB), 10587 Berlin, Germany
⁶Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
⁷Center for Materials, Architectures and Integration of Nanomembranes (MAIN), Chemnitz University of Technology, 09107 Chemnitz, Germany
⁸Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany
⁹Dipartimento di Fisica e Chimica-Emilio Segrè, Università degli Studi di Palermo, 90123 Palermo, Italy
¹⁰Center for Computational Quantum Physics (CCQ), Flatiron Institute, New York, New York 10010, USA
¹¹Physikalisches Institut and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany

^*s.tautz@fz-juelich.de
^†peter.puschnig@uni-graz.at
^‡simon.moser@physik.uni-wuerzburg.de

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Issue

Vol. 5, Iss. 3 — August - October 2023

Subject Areas

Condensed Matter Physics

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Images

Figure 1
Angle-resolved photoemission spectroscopy of graphene. Overview (a) and close-up (b) of the typical horseshoe pattern arising in momentum maps close to the $\bar{K}$ and ${\bar{K}}^{'}$ points, recorded at initial-state energy 1.35 eV below the Dirac point at $E_{kin} = 30$ eV. Panel (b) also displays the contour (green line) and angle $β$ along which the experimental data are plotted in Figs. 2–2. (c) Amplitude (bottom) and phase (top) of the initial-state structure factor that gives rise to the horseshoe. (d) Nearest-neighbor scattering factor that gives rise to intensity redistributions around the horseshoe. (e) Experimental horseshoe patterns around $\bar{K}$ for seven representative kinetic energies measured in normal incidence geometry (top), compared to TDDFT calculations in precisely the same geometry (bottom).
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Figure 2
Kinetic-energy-dependent photoemission intensities around the horseshoe. (a) Normal incidence (NI) geometry. (b) Oblique incidence (OI) geometry. The incident light polarization is shown by the yellow sine wave. Angle-resolved detection in $d Ω$ around inclination $θ$ and azimuth $ϕ$ is illustrated in green. The red lobes visualize the angular distributions of the pure $d$ channels, $d_{NI}^{2}$ and $d_{OI}^{2}$ , respectively [cf. Eqs. (4) and (5)]. In our experiment only photoelectrons emitted into the $(x, z)$ plane, where $ϕ = 0$ , are detected. To obtain the momentum maps in Fig. 1, the sample is rotated around the $z$ axis by varying $φ$ . (c)–(e) Experimental photoemission intensities in three geometries as indicated, extracted along the green contour in Fig. 1 and compiled for densely sampled kinetic energies between 15 and 80 eV (first column), compared to predictions of the PWA (second column), TDDFT (third column, shifted by 3 eV), and SWA without nearest-neighbor scattering (fourth column). Intensities in each column are plotted to scale, with dark red corresponding to high intensity. Between the columns the scaling is arbitrary. The blue arrows mark an intensity minimum that arises from a node in the $d$ channel.
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Figure 3
Photoemission into $s$ and $d$ channels as a function of final-state kinetic energy. (a) Amplitudes $| \tilde{g} |$ and $| \tilde{f} |$ of the $s$ and $d$ channels, respectively; (b) their ratio in the OI geometry. The curves were extracted from experimental data at $β = 0$ in NI and OI geometries as indicated, using the SWA without nearest-neighbor scattering. Positive real values were obtained only for in-phase photoemission in the $s$ and $d$ channels, i.e., $Δ σ = 0$ .
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Figure 4
Influence of nearest-neighbor scattering. (a) Experimental photoemission intensity in the NI geometry reproduced from Fig. 2 with a different contrast scaling. (b) Corresponding TDDFT simulations from Fig. 2. (c) ${SWA}^{NN}$ prediction including nearest-neighbor final-state scattering. (d) $k_{f}$ -dependent scattering amplitude and phase of $u (k_{f})$ that best fit the data in (a). For a compact display we have plotted $Abs \equiv | u | \times sgn (arg u)$ and $Arg \equiv mod (arg u, π)$ .
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Figure 5
Atomic structure of graphene. Carbon atoms of the two sublattices A and B are displayed in red and orange, respectively. The primitive unit cell containing one atom each of both sublattices is shown in black. The three vectors $n_{0}, n_{1}$ , and $n_{2}$ from an atom in sublattice A to its nearest neighbors in sublattice B are indicated in green. Nonprimitive unit cells that contain all nearest neighbors of a sublattice representative are shown in yellow. In the ${SWA}^{NN}$ including nearest-neighbor scattering, the total photoemission intensity is given by the sum of identical contributions from the two yellow unit cells.
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Figure 6
Phase $ϑ_{k}$ of graphene. The black hexagon indicates the first Brillouin zone. $\bar{Γ}, \bar{K}$ , and ${\bar{K}}^{'}$ points are labeled.
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