Probing Electron-Phonon Interactions Away from the Fermi Level with Resonant Inelastic X-Ray Scattering

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Probing Electron-Phonon Interactions Away from the Fermi Level with Resonant Inelastic X-Ray Scattering

C. D. Dashwood, A. Geondzhian, J. G. Vale, A. C. Pakpour-Tabrizi, C. A. Howard, Q. Faure, L. S. I. Veiga, D. Meyers, S. G. Chiuzbăian, A. Nicolaou, N. Jaouen, R. B. Jackman, A. Nag, M. García-Fernández, Ke-Jin Zhou, A. C. Walters, K. Gilmore, D. F. McMorrow, and M. P. M. Dean

Phys. Rev. X 11, 041052 – Published 15 December 2021

Abstract

Interactions between electrons and lattice vibrations are responsible for a wide range of material properties and applications. Recently, there has been considerable interest in the development of resonant inelastic x-ray scattering (RIXS) as a tool for measuring electron-phonon ( $e$ -ph) interactions. Here, we demonstrate the ability of RIXS to probe the interaction between phonons and specific electronic states both near to, and away from, the Fermi level. We perform carbon $K$ -edge RIXS measurements on graphite, tuning the incident x-ray energy to separately probe the interactions of the $π^{*}$ and $σ^{*}$ electronic states. Our high-resolution data reveal detailed structure in the multiphonon RIXS features that directly encodes the momentum dependence of the $e$ -ph interaction strength. We develop a Green’s-function method to model this structure, which naturally accounts for the phonon and interaction-strength dispersions, as well as the mixing of phonon momenta in the intermediate state. This model shows that the differences between the spectra can be fully explained by contrasting trends of the $e$ -ph interaction through the Brillouin zone, being concentrated at the $Γ$ and $K$ points for the $π^{*}$ states while being significant at all momenta for the $σ^{*}$ states. Our results advance the interpretation of phonon excitations in RIXS and extend its applicability as a probe of $e$ -ph interactions to a new range of out-of-equilibrium situations.

Received 10 May 2021
Revised 18 October 2021
Accepted 20 October 2021

DOI:https://doi.org/10.1103/PhysRevX.11.041052

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 Phonons

Graphite

Resonant inelastic x-ray scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. D. Dashwood ^1,*,‡, A. Geondzhian^2,3,‡, J. G. Vale¹, A. C. Pakpour-Tabrizi⁴, C. A. Howard ¹, Q. Faure¹, L. S. I. Veiga¹, D. Meyers^5,6, S. G. Chiuzbăian ^7,8, A. Nicolaou ⁷, N. Jaouen ⁷, R. B. Jackman ⁴, A. Nag ⁹, M. García-Fernández⁹, Ke-Jin Zhou⁹, A. C. Walters ⁹, K. Gilmore ^5,10,11, D. F. McMorrow¹, and M. P. M. Dean ^5,†

¹London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, London, WC1E 6BT, United Kingdom
²Max Planck POSTECH/KOREA Research Initiative, 37673 Pohang, South Korea
³Max Planck Institute for the Structure and Dynamics of Matter, Luruper Chaussee 149, 22761 Hamburg, Germany
⁴London Centre for Nanotechnology and Department of Electronic and Electrical Engineering, University College London, London, WC1E 6BT, United Kingdom
⁵Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York 11973, USA
⁶Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
⁷Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, B.P. 48, 91192 Gif-sur-Yvette, France
⁸Sorbonne Université, CNRS, Laboratoire de Chimie Physique-Matiére et Rayonnement, UMR 7614, 4 place Jussieu, 75252 Paris Cedex 05, France
⁹Diamond Light Source, Didcot, Oxfordshire, OX11 0DE, United Kingdom
¹⁰Physics Department and IRIS Adlershof, Humboldt-Universität zu Berlin, Zum Großen Windkanal 2, 12489 Berlin, Germany
¹¹European Theoretical Spectroscopy Facility (ETSF)

^*cameron.dashwood.17@ucl.ac.uk
^†mdean@bnl.gov
^‡These authors contributed equally to this work.

Popular Summary

Interactions between electrons and vibrations (phonons) of a crystalline lattice underpin a range of phenomena, from the temperature dependence of resistivity in metals to the binding of electrons into Cooper pairs in conventional superconductors. Given their ubiquity and importance, these interactions are the subject of much experimental focus. Currently available techniques, however, can investigate only electrons close to their equilibrium configuration, precluding their use in situations such as charge transport in photovoltaics, which involve highly excited states. Here, we demonstrate how a state-of-the-art technique—resonant inelastic x-ray scattering (RIXS)—can measure phonon interactions with excited electrons.

In RIXS, the energy and momenta of scattered x-ray photons are used to deduce information about a material. We perform high-resolution RIXS on graphite and develop an advanced theoretical approach to study how electrons and phonons interact during the measurement. For low-energy electrons, we confirm that only a subset of phonons with specific momenta are involved in the interactions. By contrast, we find that high-energy excited electrons interact with phonons of a wide range of momenta. This analysis highlights the importance of considering the phonon momentum when interpreting RIXS data, which is neglected in the most common theoretical model.

Our work offers a significantly improved understanding of RIXS measurements of phonons and opens up a new range of technologically important processes to which RIXS can be applied.

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Issue

Vol. 11, Iss. 4 — October - December 2021

Subject Areas

Condensed Matter Physics

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Images

Figure 1
Contrasting RIXS response at the $π^{*}$ and $σ^{*}$ resonances. (a) RIXS maps around both resonances, with x-ray absorption spectroscopy (XAS) above for reference (the dashed vertical lines in the XAS mark the peaks of the resonances at 285.6 and 298.1 eV, respectively). Both maps show a series of phonon features above the elastic line, with the contrasting resonance behavior most apparent for the two-phonon feature between 0.3 and 0.4 eV. The intense feature above approximately 1 eV in the $π^{*}$ map arises from electronic transitions. (b) Phonon dispersion of graphite from Ref. [36], with the transverse ( $T$ ), longitudinal ( $L$ ), and out-of-plane ( $Z$ ) acoustic ( $A$ ) and optical ( $O$ ) modes marked. The TO mode with significant coupling is marked in red. (c) The 2D-projected Brillouin zone of graphite, with the momenta accessible in our RIXS measurement indicated by the turquoise circle and high-symmetry positions $Γ$ , $K$ , and $M$ marked.
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Figure 2
Phonon generation in RIXS. (a) Schematic of the RIXS process at the carbon $K$ edge. An incident photon of energy $ℏ ω_{i}$ and wave vector $k_{i}$ excites an electron from the $1 s$ to the $π^{*}$ or $σ^{*}$ bands, leaving a core hole. In the intermediate state, the altered charge density perturbs the positions of surrounding ions and generates $n_{q}$ phonon modes. The excited electron then relaxes via the emission of a photon of energy $ℏ ω_{f}$ , leaving $n_{q}^{'}$ phonons with total energy $ℏ (ω_{i} - ω_{f})$ in the final state (the zone-center TO mode is depicted [36]). (b) Electronic band structure of graphite from Ref. [37], with the low-energy $π^{*}$ bands shaded in orange and the high-energy $σ^{*}$ bands in blue.
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Figure 3
$e$ -ph interactions for low-energy $π^{*}$ excitations. (a) Modeled momentum dependence of $G$ through the 2D Brillouin zone (orange), compared to $g$ determined by IXS (blue) [5], Raman spectroscopy (yellow) [6], and time-resolved ARPES (purple) [35]. (b) Normalized experimental (black points with error bars) and calculated (orange line, including experimental broadening) RIXS spectra on resonance (285.6 eV). The contribution from each phonon generation process is indicated by the vertical orange lines, grouped above by the number of phonons. (c) Incident-energy dependence of the experimental (black points with error bars) and calculated (red to orange lines) spectra, plotted alongside the XAS with arrows indicating the energies.
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Figure 4
$e$ -ph interactions for high-energy $σ^{*}$ excitations. (a) Best-fit momentum dependence of $G$ (blue line) and that with $G (M) = 0$ (orange line) and $G (K) = 0$ (yellow line). (b) Normalized experimental (black points with error bars) and calculated [blue, orange, and yellow lines, corresponding to those in (a)] RIXS spectra on resonance (291.8 eV). The contribution from each $n$ -phonon process is labeled above. (c) Incident-energy dependence of the experimental (black points with error bars) and calculated (blue to green lines) spectra, plotted alongside the XAS with arrows indicating the energies.
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