Electronic excitations in quasi-2D crystals: what theoretical quantities are relevant to experiment?

Nazarov, V. U.

doi:10.1088/1367-2630/17/7/073018

Cited by 46 publications

(69 citation statements)

References 29 publications

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“…Two energy ranges are generally distinguished, with a π plasmon peak in the 6-8 eV range and a σ + π peak at about 25 eV for bulk h-BN and also for graphite [43][44][45][46][47], the position and intensity of the latter peak being strongly dependent on the number of sheets in thin samples. The position of some structures can also be associated with specific interband transitions, particularly if they are correlated with the behavior of ε(q,ω) itself through Kramers-Kronig analyses [43], but some controversy has appeared recently between these two interpretations concerning the nature of the observed signals in 2D systems such as graphene [48][49][50][51]. Actually, deriving well-defined dispersion relations and deciding between the two possibilities is not obvious.…”

Section: B Low-loss Regionmentioning

confidence: 99%

Angle-resolved electron energy loss spectroscopy in hexagonal boron nitride

et al. 2017

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Electron energy loss spectra were measured on hexagonal boron nitride single crystals employing an electron energy loss spectroscopic setup composed of an electron microscope equipped with a monochromator and an in-column filter. This setup provides high-quality energy-loss spectra and allows also for the imaging of energy-filtered diffraction patterns. These two acquisition modes provide complementary pieces of information, offering a global view of excitations in reciprocal space. As an example of the capabilities of the method we show how easily the core loss spectra at the K edges of boron and nitrogen can be measured and imaged. Low losses associated with interband and/or plasmon excitations are also measured. This energy range allows us to illustrate that our method provides results whose quality is comparable to that obtained from nonresonant x-ray inelastic scattering but with advantageous specificities such as an enhanced sensitivity at low q and a much greater simplicity and versatility that make it well adapted to the study of two-dimensional materials and related heterostructures. Finally, by comparing theoretical calculations to our measures, we are able to relate the range of applicability of ab initio calculations to the anisotropy of the sample and assess the level of approximation required for a proper simulation of our acquisition method.

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Section: B Low-loss Regionmentioning

confidence: 99%

Angle-resolved electron energy loss spectroscopy in hexagonal boron nitride

et al. 2017

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“…Accordingly, KK relations for the permittivity do not hold in this case. The inverse permittivity, on the contrary, remains causal and does satisfy KK relations.We start by writing the permittivity of a Q2D crystal [11] (atomic units e 2 = = m e = 1 are used throughout unless otherwise indicated) …”

mentioning

confidence: 99%

“…2 and 3 for q = 0.049 and 0.152 a.u., respectively, along the ΓM direction. It is, however, known that ε 3D (q, ω; d), calculated in the super-cell geometry, is a quantity completely different from the permittivity ε(q, ω) of a single layer [11][12][13][14], as can be also immediately appreciated from the d-dependence of the former. Our second step consists, therefore, in finding the density-response function χ of the single-layer system from that of the array of those layersχ.…”

mentioning

confidence: 99%

“…Our second step consists, therefore, in finding the density-response function χ of the single-layer system from that of the array of those layersχ. This can be conveniently done by virtue of the matrix relation [11] …”

mentioning

confidence: 99%

“…6 this is shown as an intersection, in the case of q > q c , with the straight horizontal line representing the right-hand side of Eq. (11). (11), the permittivity of a single-layer graphene may become singular.…”

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Negative static permittivity and violation of Kramers-Kronig relations in quasi-two-dimensional crystals

Nazarov

2015

Phys. Rev. B

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We investigate the wave-vector and frequency-dependent screening of the electric field in atomically thin (quasi-two-dimensional) crystals. For graphene and hexagonal boron nitride we find that, above a critical wave-vector qc, the static permittivity ε(q > qc, ω = 0) becomes negative and the Kramers-Kronig relations do not hold for ε(q > qc, ω). Thus, in quasi-two-dimensional crystals, we reveal the physical confirmation of a proposition put forward decades ago (Kirzhnits, 1976), allowing for the breakdown of Kramers-Kronig relations and for the negative static permittivity. In the vicinity of the critical wave-vector, we find a giant growth of the permittivity. Our results, obtained in the ab initio calculations using both the random-phase approximation and the adiabatic time-dependent local-density approximation, and further confirmed with a simple slab model, allow us to argue that the above properties, being exceptional in the three-dimensional case, are common to quasi-two-dimensional systems. The concept of causality plays one of the central roles in contemporary science [1]. It is well known that causality in the time-domain (the impossibility for an effect to precede the cause in time) leads to the analyticity of a causal response-function in a complex half-plane in the frequency-domain, which, in turn, leads to KramersKronig (KK) relations between the real and the imaginary parts of the response function [2].It must be, however, recognized that the causality assumes that the response-function is applied to a cause and it produces an effect. In the case of the longitudinal electric field in a translationally invarient or a periodic system, the definition of the permittivity ε(q, ω) reads φ tot (q, ω) = φ ext (q, ω)/ε(q, ω), where φ ext and φ tot are the scalar potentials of the externally applied and the total electric fields, respectively. Since the cause is φ ext and the effect is φ tot , not vice versa, this is 1/ε that is guaranteed to be causal, but not ε itself [3]. Accordingly, KK relations must be satisfied by 1/ε, but may or may not be satisfied by ε. For |q| > 0, this leaves ε(q, ω = 0) a freedom to be negative without violating the causality or destroying the stability of the system [4,5]. If this happens, then the inverse permittivity has zeros in the upper half of the complex ω-plane, making the permittivity itself a non-analytic function.In the three-dimensional world the realizations of such negative static permittivity are scarce and they mostly concern exotic non-crystalline systems [6][7][8][9][10]. In this work we show that, above a critical wave-vector q > q c in the first Brillouin zone, the permittivity ε(q, ω) of the quasi-two-dimensional (Q2D) systems of the monolayer graphene and boron nitride is negative in the static limit. Accordingly, KK relations for the permittivity do not hold in this case. The inverse permittivity, on the contrary, remains causal and does satisfy KK relations.We start by writing the permittivity of a Q2D crystal [11] (atomic units e 2 = = m e = 1 are use...

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Atomic‐Scale Spectroscopic Imaging of the Extreme‐UV Optical Response of B‐ and N‐Doped Graphene

Hage

Kapetanakis

Idrobo

et al. 2019

Adv Funct Materials

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Substitutional doping of graphene by impurity atoms such as boron and nitrogen, followed by atom‐by‐atom manipulation via scanning transmission electron microscopy, can allow for accurate tailoring of its electronic structure, plasmonic response, and even the creation of single atom devices. Beyond the identification of individual dopant atoms by means of “Z contrast” imaging, spectroscopic characterization is needed to understand the modifications induced in the electronic structure and plasmonic response. Here, atomic scale spectroscopic imaging in the extreme UV‐frequency band is demonstrated. Characteristic and energy‐loss‐dependent contrast changes centered on individual dopant atoms are highlighted. These effects are attributed to local dopant‐induced modifications of the electronic structure and are shown to be in excellent agreement with calculations of the associated densities of states.

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Electronic excitations in quasi-2D crystals: what theoretical quantities are relevant to experiment?

Cited by 46 publications

References 29 publications

Angle-resolved electron energy loss spectroscopy in hexagonal boron nitride

Angle-resolved electron energy loss spectroscopy in hexagonal boron nitride

Negative static permittivity and violation of Kramers-Kronig relations in quasi-two-dimensional crystals

Atomic‐Scale Spectroscopic Imaging of the Extreme‐UV Optical Response of B‐ and N‐Doped Graphene

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