Plasmons probe the quantum response Electronic systems are typically considered as classical Fermi liquids, and the quantum mechanical interactions and processes are usually only accessed at very low temperatures and high magnetic fields. Lundeberg et al. used tunable plasmons to probe the quantum response of the electron gas of graphene (see the Perspective by Basov and Fogler). They studied shape deformations of the Fermi surface during a plasmon oscillation, as well as many-body electronic effects. Science, this issue p. 187; see also p. 132

Abstract The response of electron systems to electrodynamic fields that change rapidly in space is endowed by unique features, including an exquisite spatial nonlocality. This can reveal much about the materials’ electronic structure that is invisible in standard probes that use gradually varying fields. Here, we use graphene plasmons, propagating at extremely slow velocities close to the electron Fermi velocity, to probe the nonlocal response of the graphene electron liquid. The near-field imaging experiments reveal a parameter-free match with the full quantum description of the massless Dirac electron gas, which involves three types of nonlocal quantum effects: single-particle velocity matching, interaction-enhanced Fermi velocity, and interaction-reduced compressibility. 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). Although optical techniques have been applied to study correlated electron materials (3), far-field optical probes can wash out purely short-range effects in disorder-free systems as they probe the response to electrical fields with long length scales (4). In contrast, when free electron systems are driven by electric fields varying rapidly in both time and space, the response pattern in dynamical current reveals complex short-range effects.

This aspect of electron response—known as nonlocality or spatial dispersion in conductivity—arises due to the internal spreading of energy via the moving electrons. Even in ambient conditions, 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. 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 forces—can act as a carrier of the spatiotemporal electric fields necessary to probe nonlocality. In general, electronic conductors exhibit nonlocal 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 (5–7). 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 (5) and possible complications by edge effects and tunneling (6–9).

In graphene, it is possible to access a different, velocity-based, type of nonlocality due to the ability to tune its plasmon phase velocity to low values, close to its Fermi velocity of v F ≈ c/300, where c is the speed of light in a vacuum. Here, we use heterostructures of high-quality graphene, hexagonal boron nitride (h-BN), and nearby metal (10) to confine the plasmons vertically down to 5 nm (see insets in Fig. 1A) and as a consequence slow down the propagation velocity (Fig. 1A). In addition, by tuning to relatively low carrier density, as accessible in high-quality graphene, we can further slow plasmons down to about c/250, approaching the Fermi velocity, such that nonlocal effects become appreciable. We probe propagating plasmons with near-field optical scattering scanning probe technique (11, 12) for frequencies as low as a few THz (10) to avoid interband plasmon losses. Importantly, our technique provides for an accurate determination of the plasmon wavelength, independent of edge effects, which enables us to experimentally determine the nonlocal dynamical conductivity of graphene, σ(ω,q), a function of angular frequency ω and wave vector q.

Fig. 1 Concept for enhancing nonlocal effects by slowing down the graphene plasmon velocity. (A) Effect on graphene plasmon velocity from changing separation d in the graphene-metal system. Insets show plasmon electric field distribution (red and blue colors) for large or small separation d. The separation d is controlled by the thickness of the dielectric h-BN. (B) Frequency–wave number dispersion of plasmon at various d; the solid lines are all computed with equal carrier density n s = 1012 cm–2, whereas the dashed line shows the smallest-d case with a factor-10-lower carrier density n s = 1011 cm–2. The horizontal dashed gray line indicates the frequency for which the experiment has been performed. (C) Plasmon velocity dependence on d and carrier density n s . Contours indicate discrepancy between local and nonlocal plasmon models.

To illustrate the tunability of the nonlocal effects, we consider the effect of the environment on the graphene plasmon properties. In general, graphene plasmons occur for ω,q values that satisfy the self-oscillation condition [see the supplementary materials (13)] (1)where C(ω,q) is the dynamical capacitance of the environment around the graphene.

For graphene-dielectric plasmons, the small capacitance values (C = 2εq, for permittivity ε) mean that nonlocal regimes are only accessible for short plasmon wavelengths on the order of the Fermi wavelength (14, 15). In such experiments, the high observed phase velocities, typically more than twice v F , have given negligible nonlocal effects (16). The addition of a metal film at distance d from the graphene increases capacitance up to C ≈ ε/d (the metal sheet conductivity vastly exceeds that of the graphene and can be taken as effectively infinite). As a consequence of Eq. 1 and as shown in Fig. 1B, this drives the plasmon to larger values of q, and thus smaller velocity. Because this capacitive effect also morphs the plasmon from the graphene-dielectric unscreened dispersion ω ∝ to a linear dispersion ω ∝ q, we take the plasmon phase velocity v p = ω/q as the key parameter characterizing plasmons in this system (10).

The graphene-metal system has two tuning parameters: the separation d fixed at fabrication, which controls capacitance C, and the graphene sheet carrier density n s , which gives in situ control of conductivity σ. This combined tunability (Fig. 1C) shows that the combination of the two tuning parameters n s and d allows the plasmon velocity v p to be lowered to small values near the Fermi velocity v F ≈ 1.0 × 106 m/s. Figure 1C also shows how the local approximation begins to break down in this regime (with >10% error for v p /v F < 2) as nonlocal effects increase.

For our experimental realization (Fig. 2), we encapsulated graphene in the h-BN dielectric, a combination that has shown the highest-quality graphene plasmons to date (16), and placed it on top of an AuPd metal layer. Three devices of this form were created, with distinct graphene-metal separation d controlled by the chosen thickness of the bottom h-BN layer [see the supplementary materials for further details (13)], and contacted with Au electrodes (17). To visualize plasmons and characterize their propagation, we used a scattering-type near-field optical microscope in photocurrent mode (10, 18, 19). We operated the microscope in the terahertz (THz) frequency range (we chose a laser frequency of ω/2π = 3.11 THz), because this allowed us to probe down to the lowest plasmon velocities while respecting the minimum-wavelength limitations of the near-field probe (20). To detect the photocurrent, we left a narrow split in the AuPd metal layer (Fig. 2A, inset) so that a graphene p-n junction could be formed by applying distinct gate voltages V L and V R . As a second purpose, this sharp split also served as a launching edge for graphene plasmons.

Fig. 2 Experimental setup and near-field imaging data. (A) A metallized tip (inverted pyramid) scans over a graphene sheet that has been encapsulated in h-BN and placed on a split metallic film. Terahertz laser light illuminates the entire device, launching plasmons (orange arrows) at the tip and split. Gate voltages V L and V R control the electron density and the junction photocurrent sensitivity. (B) Photocurrent traces in three different devices, each at n s = 1.0 × 1012 cm–2, showing interference fringes used to extract the plasmon wavelength λ [and hence velocity v p = λ(ω/2π)] via the indicated fits.

Plasmons appear as interference fringes in the dependence of photocurrent on tip position (Fig. 2B), due to tip-plasmon interference that modulates the absorbed power (10, 19). Besides the edge-reflection fringes examined in earlier works, we also observed fringes associated with plasmons launching from the split in the AuPd gate, particularly in the thinner-d devices. In either case, the fringes allowed a determination of the plasmon wavelength λ = 2π/q (Fig. 2B) by fitting to the photocurrent with an appropriately subtracted background [see the supplementary materials for further details (13)]. This, in combination with the known excitation frequency, directly yields the plasmon phase velocity v p .

In each of the three devices, we extracted the plasmon phase velocity from many scanning photocurrent maps, each taken with a different gate voltage (Fig. 3). The data have been collated into a common form by converting gate voltage to carrier density n s (13), allowing a direct comparison with theory. Qualitatively and consistent with the map in Fig. 1C, the smallest plasmon velocities are seen for the smallest n s and d. We compare the experimental v p values to two theories: The local approximation theory (dashed curve) shows a discrepancy with the data, whereas the full nonlocal theory (shaded curve) shows excellent agreement without any fitting parameters. We do observe a slight discrepancy in the d = 27 nm case, likely connected to the different type of fringe pattern that was used to measure λ in this device compared with the other two devices (13). The local approximation predicts plasmon velocities falling below v F for the 5.5-nm device, in contrast to the full theory, which is forbidden from this region (for reasons explained below).

Fig. 3 Tunable nonlocal effects. Extracted plasmon wavelength dependence on carrier density n s , for three devices of differing separation d, show parameter-free agreement with the full theory (color map) and considerable deviation from a local-response theory (dashed line). The red color map indicates the inverse of the left side of Eq. 1, so that plasmons appear as a red peak, the width being associated with propagation distance. The hatched region below the solid line indicates phase velocities below v F . The upper and lower rows show the same data, plotted differently.

Figure 4A depicts the three layers of our nonlocal theory, based on dominant effects known from electron liquid theory (1, 2). Including all nonlocal corrections, the conductivity takes the following convenient form (for frequency and wave vector below Fermi values, as in this experiment) (2)where f(z) is a dimensionless function that describes the nonlocal response (3)where the dimensionless δ will be used to introduce one of the corrections (see below). Using this functional form, we can gradually introduce the different layers of nonlocal response, which are plotted in Fig. 3B. The local approximation consists of ignoring the q-dependence [which amounts to setting f(z) = 1], yielding a Drude response σ ∝ i/ω from Eq. 2.

Fig. 4 Nonlocal conductivity of graphene. (A) Schematic representations of the three main mechanisms governing graphene response beyond the local approximation. (B) Experimentally extracted σ(ω,q) at n s = 1.0 × 1012 cm–2, compared with theoretical approximations for the interacting electron system in graphene: RPA, with added velocity renormalization (RPA+VR), and then with compressibility correction (RPA+VR+CC); local RPA appears as a horizontal line.

The first layer of nonlocal response is to consider the response of noninteracting electrons (14, 15) [via random phase approximation (RPA)], which is the case of Eq. 3 with δ = 0. In the RPA, conductivity σ increases with q, due to the change in Fermi surface deformations. This is closely related to Landau-Bohm-Gross dispersion (21) in classical plasma physics: Some of the electrons, those travelling with a velocity that nearly matches v p , can interact longer with each passing wavefront and thereby provide enhanced response (Fig. 4A). Classically, this nonlocal dispersion would come along with Landau damping due to fast thermal electrons that fully match the plasma velocity and dissipate energy; this does not occur in a quantum degenerate system due to the narrowly distributed electron velocity (the Fermi velocity), which instead yields a divergent intraband contribution to the conductivity as q → ω/v F (i.e., z → 1), and no Landau damping before this point. This divergence results in the nonlocal plasmon velocity never falling below the Fermi velocity (as can be seen in Fig. 3), in contrast to the prediction of a local approximation.

The second and third layers of our nonlocal theory involve microscopic electron-electron interactions (many-body effects). We have calculated these many-body corrections fully consistently, including the realistic screening by capacitance C(ω,q) (figs. S1 and S2) (13). The major many-body effect is renormalization of band structure, which in graphene amounts to an increase in Fermi velocity (Fig. 4B). Although the value v F = 1.0 × 106 m/s is nominally assumed in graphene plasmon studies, the Fermi velocity actually varies logarithmically with carrier density, from its bare value of 0.85 × 106 m/s up to as much as 3.0 × 106 m/s for very low carrier densities (22–25). Because our experiments enter a regime of relatively low electron densities, it is crucial to include this n s -dependent velocity renormalization. The secondary many-body effect has to do with electron liquid correlations, which produce a Pauli-Coulomb hole (1, 2) around each reference electron (Fig. 4C). We capture this by including a local field factor G(ω,q), via , that forces consistency between the dynamic response and the isothermal compressibility (2). In our experimental regime, this ultimately introduces a factor δ = 1 − (κ 0 /κ) into Eq. 3 [see the supplementary materials for further details (13)], where κ 0 is the RPA compressibility and κ is the proper isothermal compressibility (26, 27).

Figure 4B shows how we can isolate the graphene conductivity function σ(ω,q) to directly observe the nonlocality. This is possible due to Eq. 1, which implies that a determination of the plasmon wave vector, q p = q p (ω), produces a measurement of the dynamical conductivity at that wave vector: . We are able to exactly calculate C(ω,q) from Maxwell equations (13), and hence this approach of introducing variable d (causing variable C and variable q p ) allows us to map out as shown in Fig. 4. Each device (differing in d) thus provides a distinct probe of the functional dependence of the conductivity on wave vector q under otherwise-identical parameters (ω,n s ). It can be seen in Fig. 3B that our data are only matched by theory after taking into account all three layers of quantum corrections and that the measured conductivity of graphene shows considerable departures from the local theory (a horizontal line).

The recipe set forth in this work can be transferred to probe other electron systems with exotic physical properties. Not only does this technique reveal the collective excitation (plasmon), but we have also shown how one may isolate the electronic response from its environment, quantitatively mapping out the underlying response function (nonlocal conductivity) as a function of wavelength. This kind of spatial spectroscopy forms a valuable counterpart to the traditional temporal (frequency) spectroscopy, and the marriage of these two approaches into a precision spatiotemporal spectroscopy—a full determination of σ(ω,q)—would provide an unprecedented window into electron physics. This may allow a greatly enriched understanding of electron correlation physics such as those underlying fractional quantum Hall effects [e.g., magneto-rotons (28, 29)] and the binding mechanism in superconductors (30), as well as probing the nonlocality of Fermi-surface deformations in unusual band structures [e.g., Weyl fermions (31)].

Supplementary Materials www.sciencemag.org/content/357/6347/187/suppl/DC1 Materials and Methods Figs. S1 and S2 References (32–36)