Toward a novel theoretical approach for determining the nature of electronic excitations in quasi-two-dimensional systems

In the last ten years, a variety of theoretical methods and experimental approaches has been intensively applied to graphene. In particular, the application capabilities of graphene-based plasmonic devices have stimulated research on plasmon modes in graphene and two-dimensional (2D) materials 'beyond graphene' [2–5].

The collective electronic excitation of 2D Dirac fermions (Dirac plasmon) forms a dispersive feature at 0–1 eV, with the characteristic $\surd q$ dependence [6–15], where q is the momentum. Such excitation, arising from intraband transitions involving electronic states in the Dirac cone, is observed only for doped graphene, where the doping may arise from gating potentials [1], the adsorption or the intercalation of chemical species [16], or just charge transfer from the metal substrate [17].

Although technological efforts have mainly focused in the spectral range from terahertz [18] to the visible [19], several potential applications of plasmon modes in the ultraviolet (UV) part of the electromagnetic spectrum exist. A possible advantage of UV plasmons is the matching of their high energy with the electronic transition energy of many organic molecules, thus paving the way for UV plasmonics [20], UV imaging, DNA sensing [21], UV absorbers [22], and metamaterials with UV plasmonic resonances [23].

Apart from these intriguing characteristics, the comprehension of UV plasmons has fundamental importance also for the detailed understanding of the elementary excitations of graphene, determined by the π and σ valence electrons lying outside the Dirac cone. Plasmon spectra of free-standing monolayer graphene were first obtained in [24], where the authors identified two distinct structures, attributed to the so-called π and π + σ plasmons. They observed that these two plasmonic modes were red-shifted as compared to the corresponding features in the bulk graphite [25–27], due to the reduction of macroscopic screening when going from graphite to graphene [28]. However, the plasmonic nature of these electronic excitations for the case of graphene is highly debated. Recently, it has been suggested [29] that the previously accepted attribution should be revised and that the π and π + σ plasmons are indeed single-particle π → π* and σ → σ* excitations, respectively, with a characteristic q² dependence of the energy. This affirmation is motivated by the finding that in graphene the value of the real part of the dielectric function ₁ does not go through zero for either the π → π* or the σ → π* transition, and, moreover, the imaginary part ₂ is not small in these energy regions to allow a plasmon excitation.

Similar debate has concerned the study of the dispersion relation of high-energy plasmons in free-standing monolayer and multilayer graphene. The early theoretical and experimental measurements observed linear dispersion of π plasmon in graphene [30–33], which differs from the q² dispersion observed in graphite [25, 26, 28, 34]. The reported linear dispersion has been correlated to transitions from the linearly dispersing Dirac cone and this claim was quickly widely accepted [35, 36]. Recently, the correlation of the reported linear dispersion of the π plasmon to the linear Dirac cone has been questioned. Strong evidence for 2D plasmon character of π and σ electron excitations has been demonstrated by means of energy loss spectroscopy (EELS) experiments, showing the $\surd q$ dependent dispersion [37]. Even taking into account possible uncertainties arising from experimental difficulties in EELS measurements for low q values, it is evident that this apparent controversy deserves to be analyzed and resolved.

This debate has been promptly solved by a theoretical study by Nazarov in this issue of New J. Phys. [1]. The starting point of Nazarov is to consider electronic excitations in quasi-two-dimensional (Q2D) samples, i.e. atomically thin crystals. Such systems represent real 2D films which are periodic and infinite in two-dimensions (x. y) but they have finite thickness in the z direction (perpendicular to the layer). They notably differ from both bulk (three-dimensional, 3D) and zero-thickness 2D crystals. Both 3D and 2D systems are usually treated by introducing the well-known dielectric function (q, ω). However, considering that the total (external plus induced) scalar potentials of the electric field φ^tot(q, ω) depend on the z coordinate, even when φ^ext is uniform in z direction, the definition of (q, ω) is not straightforwardly transferable to the Q2D case. Therefore, the dielectric function (and the related quantities as energy-loss function, conductivity, etc) of a Q2D crystal should be re-defined in accordance to the system's structure. In particular, Nazarov re-examines the problem of the correspondence between the theoretically calculated quantities and the observables in the measurements on Q2D crystals, finding that the energy-loss function $-{\rm{Im}}\frac{1}{\varepsilon \left({\bf{q}},\omega \right)},$ conventionally used for the interpretation of the EELS data, is not the right quantity to be compared with EELS experiments. Instead, in reflection EELS, the quantity, better characterizing the inelastic electron scattering, is the EELS-related energy-loss function, which is shown to be qualitatively and quantitatively different in the case of Q2D systems. Consequently, the use of an appropriate dielectric function proper of real 2D crystals with a finite z dimension is not straightforward to compare EELS measurements (both in reflection and transmission mode) with the so-called loss function.

A further limitation for theoretical approaches to plasmon modes in 2D materials is usually represented by the use of the 3D super-cell methods for calculating excitations in 2D materials to artificially replicate the periodicity of the system in the z direction, by choosing the interlayer distance d large enough to prevent the interlayer interactions. Clearly, whatever large is the interlayer separation d, at sufficiently small q the interlayer interaction persists and thus the super-cell calculation cannot be rightly transferred in the case of single-layer thickness. Nazarov [1] develops an appropriate method to get rid of this spurious contributions. The correct procedure allows concluding that the uncritical use of results of the super-cell calculations applied to Q2D systems has led to the misinterpretation of the π and π + σ peaks as single-particle interband transitions rather than plasmons [29].

As its practical application, Nazarov [1] calculates the dielectric function and the related excitation spectrum of single-layer graphene. By resolving the recent controversy in the interpretation of the π and π + σ peaks as plasmons or single-particle interband transitions, Nazarov's results [1] conclusively demonstrate that prominent π and π + σ collective excitations in graphene exist. They are also accompanied by interband transitions in a close energy range. Dispersing plasmon modes and non-dispersive single-particle interband transitions can be theoretically distinguished from each other by a momentum-resolved analysis.

The results obtained by Nazarov [1] are particularly suitable for describing the electronic excitations in Q2D crystals by correctly accounting the finite thickness of the investigated systems. These findings constitute an important milestone in the comprehension of collective electronics modes in low-dimensional systems. Nazarov's work [1] will facilitate the comparison between theoretical and experimental results, so as to improve the dialogue between experimentalists and theoreticians working on plasmons in Q2D systems.