Electron energy loss spectroscopy of bulk gold with ultrasoft pseudopotentials and the Liouville-Lanczos method

Oleksandr Motornyi, Nathalie Vast, Iurii Timrov, Oscar Baseggio, Stefano Baroni, and Andrea Dal Corso

Phys. Rev. B 102, 035156 – Published 28 July 2020

Abstract

The implementation of ultrasoft pseudopotentials into time-dependent density-functional perturbation theory is detailed for both the Sternheimer approach and the Liouville-Lanczos (LL) method, and equations are presented in the scalar relativistic approximation for periodic solids with finite momentum transfer q. The LL method is applied to calculations of the electron energy loss (EEL) spectrum of face-centered cubic bulk Au both at vanishing and finite q. Our study reveals the richness of the physics underlying the various contributions to the density fluctuation in gold. In particular, our calculations suggest the existence in gold of two quasiseparate $5 d$ and $6 s$ electron gasses, each one oscillating with its own frequency at, respectively, 5.1 eV and 10.2 eV. We find that the contribution near 2.2 eV comes from $5 d \to 6 s$ interband transitions modified by the intraband contribution to the real part of the dielectric function, which we call a mixed excitation.

Received 2 December 2019
Revised 14 May 2020
Accepted 15 June 2020

DOI:https://doi.org/10.1103/PhysRevB.102.035156

Physics Subject Headings (PhySH)

First-principles calculations Plasmons

Electron energy loss spectroscopy Pseudopotentials Time-dependent DFT

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Oleksandr Motornyi^1,*, Nathalie Vast^1,†, Iurii Timrov ², Oscar Baseggio ³, Stefano Baroni ^3,4, and Andrea Dal Corso ^3,4

¹Laboratoire des Solides Irradiés, CEA/DRF/IRAMIS, École Polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau cédex, France
²Theory and Simulation of Materials (THEOS), and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
³Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, IT-34136 Trieste, Italy
⁴CNR-IOM DEMOCRITOS, Via Bonomea 265, IT-34136 Trieste, Italy

^*motorny.sasha@gmail.com
^†nathalie.vast@polytechnique.edu

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Issue

Vol. 102, Iss. 3 — 15 July 2020

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Images

Figure 1
EEL spectrum of bulk Au computed with 19 electrons in the valence region, with ultrasoft and norm-conserving pseudopotentials, for $| q | \approx 0.03 Å^{- 1}$ in the [100] direction. Previous FP-LAPW data [48] is reported for $q = 0$ . Peaks are labeled according to Ref. [48] and we have added labels $1'$ and $2'$ . The horizontal arrow marks the frequency domain of pure interband transitions. Computations performed in TDDFPT-GGA.
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Figure 2
EEL spectrum of bulk Au computed with US-PP and 11 valence electrons (black dashed line) or 19 valence electrons (blue solid line) for $| q | \approx 0.03 Å^{- 1}$ in the [100] direction. Solid black line: From REELS experiment [55] at q = 0. Peak labels as in Fig. 1. Computations performed in TDDFPT-LDA.
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Figure 3
Dielectric function (DF) and loss function (LF) of bulk Au with 19 electron PPs. Real part (bottom panel) and imaginary part (center panel) of the DF and LF (top panel) up to 15 eV for $| q | \approx 0.03 Å^{- 1}$ in the [100] direction. Solid lines: Liouville-Lanczos's (LL) method as implemented in the turboEELS code. Dashed lines: SIMPLE code at q = 0 with both interband and intraband contributions. Dotted line: Interband contributions only to the imaginary part of the DF at q = 0. The vertical bars show the zeros of the real part (all panels). Slights differences that are observed in the imaginary part and real part of the DF below 10 eV when comparing the calculation at q = 0 and the calculation at $| q | \approx 0.03 Å^{- 1}$ comes from the diverging behavior of the intraband contribution at q = 0. Differences that are observed above 10 eV are due to the neglect of local fields in the calculation with the SIMPLE code at q = 0. Computations in TDDFPT-LDA.
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Figure 4
Toy-model EEL spectra computed using modified NC-PP pseudopotentials, without the $6 s$ electron (black dashed line) or without $5 d$ electrons (red dashed line). They are compared to the EEL spectrum of the 19-electron US-PP that contains both $5 d$ and $6 s$ electrons (blue solid line, same as in Fig. 2). In the legend, w/o stands for without. Results computed using the LL method. Computation performed within TDDFPT-LDA.
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Figure 5
Loss function of bulk Au with the 19-electron US-PP. Loss function as a function of energy and transferred momentum $q$ in the [100] direction (black solid lines). Energy below 12 eV (top) and above 12 eV (bottom panel). The thin dotted lines in the top panel are guides for the eyes to highlight the peak dispersion. Results computed with the LL method within TDDFPT-LDA.
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