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Plasmon Dispersion in Graphite: a Comparison of Current Ab Initio Methods Sean Anderson, Bernardo Mendoza, Giorgia Fugallo, Francesco Sottile

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Plasmon Dispersion in Graphite: a Comparison of Current Ab Initio Methods Sean Anderson, Bernardo Mendoza, Giorgia Fugallo, Francesco Sottile Plasmon dispersion in graphite: A comparison of current ab initio methods Sean Anderson, Bernardo Mendoza, Giorgia Fugallo, Francesco Sottile To cite this version: Sean Anderson, Bernardo Mendoza, Giorgia Fugallo, Francesco Sottile. Plasmon dispersion in graphite: A comparison of current ab initio methods. Physical Review B: Condensed Matter and Mate- rials Physics (1998-2015), American Physical Society, 2019, 100 (4), 10.1103/PhysRevB.100.045205. hal-02191262 HAL Id: hal-02191262 https://hal.archives-ouvertes.fr/hal-02191262 Submitted on 23 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW B 100, 045205 (2019) Plasmon dispersion in graphite: A comparison of current ab initio methods Sean M. Anderson,1,* Bernardo S. Mendoza,1 Giorgia Fugallo,2 and Francesco Sottile3,4 1Centro de Investigaciones en Óptica, León, Guanajuato, México 2CNRS, UMR 6607, Laboratorie de Thermique et Energie de Nantes (LTeN) Polytech’Nantes, Université de Nantes, Rue Christian Pauc, F-44306 Nantes Cedex 3, France 3Laboratoire des Solides Irradiés, École Polytechnique, CNRS, CEA, Université Paris-Saclay, F-91128 Palaiseau, France 4European Theoretical Spectroscopy Facility (ETSF) (Received 10 December 2018; revised manuscript received 9 May 2019; published 22 July 2019) We perform a systematic study of the macroscopic dielectric function and electron energy loss (EEL) spectra for graphite. We obtain the dispersion behavior for the π plasmon, as a function of the momentum transfer q for two nonequivalent paths that traverse the first four Brillouin zones. We carry out these calculations within both time-dependent density functional theory (with two exchange-correlation functionals) and the Bethe-Salpeter equation. Additionally, we explore the effects of using the complete excitonic Hamiltonian (with all electron- hole pairs and antipairs), and within the Tamm-Dancoff approximation (neglecting antipairs). By analyzing the behavior of the macroscopic dielectric function, we are able to determine which peaks are predominantly from plasmonic behavior or only interband transitions. We compare the calculated spectra against several experiments that span almost five decades; our results present clear trends that follow the physical origins of the observed peaks. We carry out this study over a large range of momentum transfer in order to better evaluate the different theoretical methods compared to experiment, and predict the plasmonic behavior beyond available experimental data. Our results indicate that including the complete Hamiltonian with the exciton coupling included is essential for accurately describing the observed EEL spectra and plasmon dispersion of graphite, particularly for low values of momentum transfer. However, the solution of the Bethe-Salpeter equation is computationally intensive, so time-dependent density functional theory methods used in conjunction with the complete Hamiltonian may be an attractive alternative. DOI: 10.1103/PhysRevB.100.045205 I. INTRODUCTION functional [3] in TDDFT (the TDDFT equivalent of DFT- LDA). This functional does not consistently improve the The study of the optical spectra of solids yields important calculated optical spectra of solids with respect to the common insight into the underlying electronic and structural properties random phase approximation (RPA), where the exchange- of a material. When an incident photon is absorbed by a correlation functional, the main ingredient of TDDFT, is ne- material, the system is energetically excited from its ground glected. It is in fact well known [4] that today functionals and to an excited state, forming an electron-hole pair or exci- approximations to TDDFT are not well suited for accurately ton, a localized quantum of neutral electron-hole (e-h) pairs calculating the absorption spectra of most solids [4,5], where bound by the Coulomb attraction. From a theoretical stand- excitonic behavior predominates. point, ab initio calculations are essential tools for elucidating Thus, the behavior of complex electronic excitations can these excited-state electronic properties of solids, surfaces, only be accurately described by making use of many-body and nanostructures. Ground-state properties can be calculated perturbation theory (MBPT) [6]. In particular, using Hedin’s accurately using density functional theory (DFT) [1,2], but GW approximation [7] followed by solving the Bethe-Salpeter the aforementioned electronic excitations necessitate further equation (BSE) [4,8–11] allows for the accurate calculation theoretical developments. Time-dependent density functional of the optical properties of many physical systems, within theory (TDDFT) [3] is a natural extension of DFT, and is a completely ab initio framework [12–23]. In general, ab- able to successfully describe spectra like electron energy loss sorption and scattering spectroscopies (that concern neutral spectroscopy (EELS) or inelastic x-ray scattering of simple excitations) are very well described by the BSE [4,24–27]. semiconductors, as well as the photoabsorption cross section Within this GW/BSE approach, excitons are described as a of simple molecules. However, while the use of the local combination of e-h pairs of a noninteracting system; we can density approximation (LDA) in DFT yields qualitative (and convert the problem to an effective eigenvalue problem in more often than not, quantitative) agreement with experi- the e-h basis. However, nanoscale materials involve a huge ment for the ground-state properties of many materials, the number of e-h pairs, which makes solving the BSE extremely same cannot be said for using the adiabatic LDA (ALDA) costly from a computational standpoint. The Tamm-Dancoff approximation (TDA) [6] is an approximation that only con- siders positive-energy e-h pairs, effectively neglecting the in- *[email protected] teraction between e-h pairs at positive and negative (antipairs) 2469-9950/2019/100(4)/045205(9) 045205-1 ©2019 American Physical Society ANDERSON, MENDOZA, FUGALLO, AND SOTTILE PHYSICAL REVIEW B 100, 045205 (2019) energies. The non-Hermitian BS problem is thus reduced to a This paper is organized as follows. In Sec. II, we present Hermitian problem that can be solved with efficient and stable the theoretical framework that describes the aforementioned iterative methods. The Haydock recursion scheme [28–30] methods. In Sec. IV, we present our results for the q- is one of the most efficient and computationally inexpensive dependent plasmon dispersion over two separate, nonequiv- iterative methods used for dealing with the Hamiltonian. alent paths in the Brillouin zone, and show several detailed Thanks to this reduction in the computational complexity, the comparisons with experiment. We list our conclusions and TDA has been applied to a variety of systems [31–33]. final remarks in Sec. V. Excitations from plasmons, collective oscillations of the electronic density that induce a macroscopic polarization, can II. THEORY also feature very prominently in the measured EEL spectra for some materials. These oscillations involve the creation of In order to better explain the differences between the e-h antipairs; thus, these plasmonic features are inherently ill TDDFT and BSE frameworks, we will describe them in the described by the TDA [34–37]. Accurately describing both same formalism for clarity and ease of interpretation. The key plasmonic and excitonic features is essential if we wish to quantity measured in absorption and EELS is the macroscopic accurately characterize and analyze the system; therefore, we dielectric function, M; specifically, Im M is measured in − / require a suitable test case in order to carry out a complete absorption experiments, while Im (1 M ) is measured in study of these theoretical methods. Optimally, this reference EELS. The macroscopic dielectric function is connected to − material will display plasmonic behavior, be theoretically the inverse dielectric function 1 as and experimentally well characterized, and have continued 1 (ω) ≡ lim , relevance in current research. We consider that bulk graphite M → −1 q 0 [ (q,ω)]G=G=0 meets these requirements and can make for a very effective benchmark. Early work revealed accurate band structure cal- where G and G are reciprocal lattice vectors. We can express − culations [38,39] that continue to be experimentally studied by 1 for both TDDFT and the BSE as a Dyson-like equation, very precise photoemission and ARPES experiments [40,41]. D = D(0) + D(0)KD. (1) A great variety of EELS [42–47] and absorption [47–49] measurements have been carried out with relevant theoretical For TDDFT, D is the two-point polarizability χ, from which developments, including detailed analysis of the low-energy we can obtain the inverse dielectric function −1 = 1 − vχ (v (π) and high-energy (π + σ) plasmons. Concerning the actual is the Coulomb potential); for the BSE, D is the two-particle plasmon dispersion, there is very recent work [50,51] featur- correlation function L which yields χ by contracting two of ing very high
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