π-plasmon dispersion in free-standing graphene by momentum-resolved electron energy-loss spectroscopy

Abstract

The π-plasmon dispersion in graphene was scrutinized by momentum(q)-resolved electron energy-loss spectroscopy with an improved q resolution and found to display the square root of q dispersion characteristic of the collective excitation of two-dimensional electron systems, in contrast with previous experimental and theoretical studies which reported a linear q dispersion. Our theoretical elaborations on the q-dependent spectra affirm this square root of q relation and further unveil an in-plane electronic anisotropy. The physical property of the π plasmon is thoroughly compared to that of the two-dimensional plasmon due to carriers of the Dirac fermions. A clear distinction between the π plasmon and the two-dimensional Dirac plasmon was demonstrated, clarifying the common notion on correlating the linearly-dispersed Dirac cones to the linear dispersion of the π plasmon previously reported.

Changing character of electronic transitions in graphene: From single-particle excitations to plasmons

“…Diamonds in Fig. 3(a) show the energies of π plasmon peaks in the measured spectra [12]. We see very nice agreement with our theoretical calculation throughout the whole energy region.…”

“…[12] the authors make a hasty conclusion about the √ Q dispersion of σ plasmon, also not taking into account that two kinds of σ → σ * excitations exist close in energy and influence each other. In experimental papers, Refs.…”

“…[12]. There the authors demonstrated that the graphene π plasmons represent the in-plane charge density oscillations which led them to the conclusion that they behave like plasmons in a 2D electron gas with ∼ √ Q dispersion relation.…”

“…Another group [12] found strong evidence for 2D plasmon character of π and σ electron excitations, based on the electron energy loss spectroscopy (EELS) experiment showing the √ Q dependent dispersion. Even taking into account possible uncertainties arising from experimental difficulties in EELS measurements for low Q values, it is obvious that this apparent controversy deserves to be analyzed and resolved.…”

Angle-resolved electron energy loss spectroscopy in hexagonal boron nitride

“…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.…”