Analytical modeling of electron energy loss spectroscopy of graphene: Ab initio study versus extended hydrodynamic model
Introduction
With its unique electrical and optical properties (relatively low loss, high confinement, mechanical flexibility, and good tunability) graphene is an ideal material for plasmonic applications covering a wide frequency range from terahertz up to infrared, even stretching into the visible regime [1], [2], [3], [4]. Electron energy loss spectroscopy (EELS) is a commonly used experimental technique for investigating electronic and plasmonic properties of materials, including graphene sheets [1]. High-energy single-particle inter-band excitations in graphene, which are often misnomered as π and π + σ plasmons [5], have been studied recently by EELS experiments using high-energy electron beams (∼100 keV) in scanning transmission electron microscope (STEM) on samples consisting of free-standing, single-layer graphene (SLG) [5], [6], [7], [8], [9], [10], [11], [12], and multi-layer graphene (MLG) [8], [9], [10], [11], [12].
Theoretical modeling of the EELS data of SLG and MLG is an active field of research [13], [14], [15], [16], [17], [18], [19]. In our previous publication [20], we treated the MLG as layered electron gas with in-plane polarizability modeled by a two-dimensional (2D), two-fluid hydrodynamic (HD) model [21] for the inter-band transitions of π and σ electrons of SLG, yielding good agreement with the experimental EEL spectra [11] for N < 10 graphene layers in STEM. We have also used the same version of the HD model for graphene's π and σ electrons in conjunction with an empirical Drude-Lorentz model for metal substrate to reproduce the momentum-resolved experimental EELS data for low-energy electron reflection (∼10 eV) from monolayer graphene supported by Pt(111), Ru(0001), and Ni(111) substrates [19], as well as for high-quality graphene grown on peeled-off epitaxial Cu(111) foils [22].
It should be mentioned that, while the agreement of the HD model with the experimental EEL spectra for MLG from Ref. [11] covered the regions around the principal π and π + σ peaks, there was no experimental data for energy losses below ≈ 3 eV, which is a consequence of the subtraction of the zero-loss peak (ZLP). At the same time, the HD model does not incorporate the Dirac physics of low-energy excitations in graphene [20]. However, in the meantime, several EELS experiments were performed with high-energy electron beams in STEM, showing intriguing increase in spectral intensity as the energy loss decreases below ≈ 3 eV, even after the ZLP subtraction [5], [7], [12]. In that respect, we pose a question whether the low-energy, inter-band excitations of π electrons in intrinsic graphene play any detectable role in the low-loss range of EELS, and if so, whether the new generation of monochromators can open up possibility to explore the Dirac physics of graphene in STEM. In order to address this question, we attempt to reproduce the STEM-EELS data from those experiments [5], [7], [12] by formulating an extended HD (eHD) model, which includes a Dirac correction. This correction treats the low-energy contribution of graphene's π electron inter-band transitions in a manner consistent with the Dirac-cone approximation for graphene's π electron bands near the K point in the Brillouin zone (BZ).
On the other hand, Despoja et al. have used an ab initio method to calculate the energy-loss rate of a point blinking charge in the vicinity of a graphene monolayer [23] and have obtained values for the energies of π and π + σ peaks in the EELS spectra, which are in good agreement with the reported experimental values [11]. In addition, they have calculated the so-called loss function, , where ɛ(q, ω) is the dielectric function of SLG obtained by ab initio methods [24], and have obtained a very good agreement with the experimental STEM-EELS data for SLG [11]. Moreover, those authors were able to implement their ab initio method in the optical, or the long wavelength limit (q → 0), and hence compute a universal, frequency dependent 2D conductivity of SLG, σ(ω), in a broad range of energies of interest for EELS in STEM [25]. At the same time, it was observed in Ref. [5] that using a response function of graphene in the optical limit represents an excellent approximation for analytical modeling of the EELS data with the electron beam under normal incidence and for small collection angles. Taking advantage of that situation, we propose here an analytical expression for the optical conductivity σ(ω) of SLG within the eHD model, containing several free parameters which are fixed via direct comparison with the optical conductivity obtained by the ab initio method.
Moreover, taking further advantage of working in the limit of optical response of SLG, we derive an analytical expression for the probability density for losing energy ω, P(ω), of fast electrons traversing graphene under normal incidence, which takes frequency dependent conductivity σ(ω) as input. The resulting formula may be readily applied to model the EELS of any isotropic 2D material, which can be described by a scalar conductivity given in local form. Hence, we use both the eHD and ab initio results for σ(ω) of SLG to obtain probability densities P(ω) which are then directly compared with the experimental EELS data from three independent experiments.
Finally, using the eHD model with and without the Dirac correction, we explore the possible role of Dirac physics in the experimental STEM-EELS setup and its effects on the ZLP subtraction from those spectra.
Section snippets
Theoretical methods
In a typical (S)TEM-EELS experiment operating at the voltage on the order of several tens of kV (for example, 40 kV in Ref. [7], 60 kV in Ref. [5], and 100 kV in Ref. [11]) the momentum transfer of the incident electron is close to zero, so we shall use a straight-line trajectory while neglecting relativistic effects [20], [26]. We use a Cartesian coordinate system with and assume that SLG occupies the plane , where is the in-plane position and z the distance from it.
Results and discussion
As regards the ab initio study, the first part of the calculation consists of determining the KS ground state of SLG and the corresponding wave functions and energies . We use the experimental value of a.u. [29] for the graphene unit cell parameter in parallel direction. For the unit cell in the perpendicular (or z) direction (separation between periodically repeated graphene layers) we take a.u. For calculating KS wave functions and energies we use a
Conclusions
We have presented an analytical modeling of the EELS data for free-standing graphene using an ab initio method and the 2D, two-fluid eHD model. An analytical expression for the probability density for energy loss of fast electrons traversing graphene under normal incidence is derived in the long wavelength limit in terms of an arbitrary form of the conductivity of graphene given in the local, i.e., frequency-dependent form. We found a very good agreement between the results obtained by ab initio
Acknowledgements
This work is supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia (Project No. 45005). T.Dj. also acknowledges support from the Ministry of Education, Science and Technological Development of the Republic of Serbia (Project No. 171023). V.D. acknowledges support from the QuantiXLie Center of Excellence. T.Dj., I.R., V.D., and D.B. wish to acknowledge the support of the COST Action MP1306 “Modern Tools for Spectroscopy on Advanced Materials: a
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