Tuning quantum nonlocal effects in graphene plasmonics

Abstract

The response of an electron system to electromagnetic fields with sharp spatial variations is strongly dependent on quantum electronic properties, even in ambient conditions, but difficult to access experimentally. We use propagating graphene plasmons, together with an engineered dielectric-metallic environment, to probe the graphene electron liquid and unveil its detailed electronic response at short wavelengths. The near-field imaging experiments reveal a parameter-free match with the full theoretical quantum description of the massless Dirac electron gas, in which we identify three types of quantum effects as keys to understanding the experimental response of graphene to short-ranged terahertz electric fields. The first type is of single-particle nature and is related to shape deformations of the Fermi surface during a plasmon oscillation. The second and third types are a many-body effect controlled by the inertia and compressibility of the interacting electron liquid in graphene. We demonstrate how, in principle, our experimental approach can determine the full spatiotemporal response of an electron system.The quantum physics of electron systems involves complex short-distance interactions and motions that depend sensitively on electron correlations and Fermi surface deformations.1,2 These are often considered irrelevant in optical and transport measurements, which probe the response to electrical fields with long length scales. When free electron systems are driven by electric fields varying rapidly in both time and space, however, the response pattern in dynamical current reveals these complex shortrange effects. This aspect of electron response, known as non-locality or spatial dispersion in conductivity, arises due to the internal spreading of energy via the moving electrons. Even in ambient conditions (as Fermi liquid parameters depend on temperature weakly 1,2 ), the spatial dispersion in an electron system retains a detailed connection to Fermi-surface and electron-electron correlation effects, and hence it provides a unique window into quantum theories of electron systems without requiring extremes of low temperature or high magnetic field. Unfortunately, these quantum regimes cannot be accessed by standard optical and transport probes.Plasmons-electric waves resulting from an inertial electron conductivity combined with electric restoring * Marco.Polini@iit.it † frank.koppens@icfo.eu forces-can act as a carrier of the spatiotemporal electric fields necessary to probe non-locality. All systems exhibit non-local effects for plasmon wavelengths approaching the electronic Fermi wavelength λ F , which has been confirmed in experimental studies of metals and semiconductor two-dimensional (2D) electron gases. 3-5 Such experiments have however led to challenges in quantitative interpretation, due to strong interactions that go beyond standard (e.g. random phase approximation) theoretical treatments, 3,4 and possible complications by edge effects and tunneling. 5-9In graphene, it is possible to access a diff...

Probing nonlocal effects in metals with graphene plasmons

“…The combination of graphene with two-dimensional (2D) insulators-such as hexagonal Boron Nitride (hBN)-has allowed an unprecedented control of the distance that a 2D material can be placed in the vicinity of metallic films or nanostructures. In fact, the placement of a graphene sheet at a distance of a few nanometers away from a metal surface has recently been experimentally demonstrated [9], in a similar setting as depicted in Fig. 1.…”

“…This is possible due to the interplay of graphene plasmons and the screening exerted by the nearby metal film. This approach has revealed that the nonlocal properties of graphene's conductivity have to be included in a proper account of the experimental data [9,37]. In graphene, it is more useful to describe the nonlocal response via the material's nonlocal dynamical conductivity, which within linear-response theory relates the surface current, K s (r,ω), and the in-plane electric field, E (r,ω), via [3] …”

“…Here, ↔ σ (r,r ,ω) refers to the nonlocal (surface) conductivity tensor of graphene. This quantity can be probed experimentally using graphene plasmons when the material is in the proximity of a metallic surface, and varying the graphene-metal distance, thereby retrieving the Fourier transform of ↔ σ (r,r ,ω), that is, ↔ σ (q,ω) [9]. In the absence of strain, graphene is isotropic and thus ↔ σ (q,ω) = σ (q,ω)1, where q labels the in-plane wave vector.…”

“…Throughout the manuscript, graphene's conductivity is taken as being nonlocal [3]. The aim of this work is therefore to use graphene plasmonics to probe nonlocal effects within the metal, in contrast to the study of nonlocality in graphene performed in a recent publication [9]. Specifically, we study the impact of nonlocal effects due to the metal as a function of the graphene-metal separation, and discuss its implications on the field distribution and plasmonic losses.…”

Compacted dimensions and singular plasmonic surfaces

“…We have assumed a local form for ε , ignoring the fact that at very short distances this assumption breaks down (12,13,14) which will remove the possibility of a perfect singularity as postulated here. Systems also have resistive loss as we have discussed and to some extent the two effects compensate for one another: non locality by removing the perfect singularity will tend to produce a discrete spectrum; on the other hand loss will smear a discrete but still dense spectrum into a continuum leaving the broad band absorption intact.…”