Abstract
A quantitative understanding of the electromagnetic response of materials is essential for the precise engineering of maximal, versatile, and controllable light–matter interactions. Material surfaces, in particular, are prominent platforms for enhancing electromagnetic interactions and for tailoring chemical processes. However, at the deep nanoscale, the electromagnetic response of electron systems is significantly impacted by quantum surfaceresponse at material interfaces, which is challenging to probe using standard optical techniques. Here, we show how ultraconfined acoustic graphene plasmons in graphene–dielectric–metal structures can be used to probe the quantum surfaceresponse functions of nearby metals, here encoded through the socalled Feibelman dparameters. Based on our theoretical formalism, we introduce a concrete proposal for experimentally inferring the lowfrequency quantum response of metals from quantum shifts of the acoustic graphene plasmons dispersion, and demonstrate that the high field confinement of acoustic graphene plasmons can resolve intrinsically quantum mechanical electronic lengthscales with subnanometer resolution. Our findings reveal a promising scheme to probe the quantum response of metals, and further suggest the utilization of acoustic graphene plasmons as plasmon rulers with ångströmscale accuracy.
Introduction
Light is a prominent tool to probe the properties of materials and their electronic structure, as evidenced by the widespread use of lightbased spectroscopies across the physical sciences^{1,2}. Among these tools, farfield optical techniques are particularly prevalent, but are constrained by the diffraction limit and the mismatch between optical and electronic length scales to probe the response of materials only at large length scales (or, equivalently, at small momenta). Plasmon polaritons—hybrid excitations of light and free carriers—provide a mean to overcome these constraints through their ability to confine electromagnetic radiation to the nanoscale^{3}.
Graphene, in particular, supports gatetunable plasmons characterized by an unprecedentedly strong confinement of light^{4,5,6}. When placed near a metal, graphene plasmons (GPs) are strongly screened and acquire a nearly linear (acousticlike) dispersion^{7,8,9,10} (contrasting with the squareroottype dispersion of conventional GPs). Crucially, such acoustic graphene plasmons (AGPs) in graphene–dielectric–metal (GDM) structures have been shown to exhibit even higher field confinement than conventional GPs with the same frequency, effectively squeezing light into the fewnanometer regime^{8,9,10,11}. Recently, using scanning nearfield optical microscopy, these features were exploited to experimentally measure the conductivity of graphene, σ(q,ω), across its frequency (ω) and momentum (q) dependence simultaneously^{8}. The observation of momentum dependence implies a nonlocal response (i.e., response contributions at position r from perturbations at \({\bf{r}}^{\prime}\)), whose origin is inherently quantum mechanical. Incidentally, traditional optical spectroscopic tools cannot resolve nonlocal response in extended systems due to the intrinsically small momenta k_{0} ≡ ω/c carried by farfield photons. Acoustic graphene plasmons, on the other hand, can carry large momenta—up to a significant fraction of the electronic Fermi momentum k_{F} and with group velocities asymptotically approaching the electron’s Fermi velocity v_{F}—and so can facilitate explorations of nonlocal (i.e., qdependent) response not only in graphene itself but also, as we detail in this Article, in nearby materials. So far, however, only aspects related to the quantum response of graphene have been addressed^{8}, leaving any quantum nonlocal aspects of the adjacent metal’s response unattended, despite their potentially substantial impact at nanometric graphene–metal separations^{12,13,14,15,16}.
Here, we present a theoretical framework that simultaneously incorporates quantum nonlocal effects in the response of both the graphene and of the metal substrate for AGPs in GDM heterostructures. Further, our approach establishes a concrete proposal for experimentally measuring the lowfrequency nonlocal electrodynamic response of metals. Our model treats graphene at the level of the nonlocal randomphase approximation (RPA)^{4,9,17,18,19} and describes the quantum aspects of the metal’s response—including nonlocality, electronic spillout/spillin, and surfaceenabled Landau damping—using a set of microscopic surfaceresponse functions known as the Feibelman dparameters^{12,13,15,16,20,21}. These parameters, d_{⊥} and d_{∥}, measure the frequencydependent centroids of the induced charge density and of the normal derivative of the tangential current density, respectively (Supplementary Note 1). Using a combination of numerics and perturbation theory, we show that the AGPs are spectrally shifted by the quantum surfaceresponse of the metal: toward the red for \({\rm{Re}} {\,}{d}_{\perp } > \,0\) (associated with electronic spillout of the induced charge density) and toward the blue for \({\rm{Re}} {\,}{d}_{\perp } < \,0\) (signaling an inward shift, or “spillin”). Interestingly, these shifts are not accompanied by a commensurately large quantum broadening nor by a reduction of the AGP’s quality factor, thereby providing the theoretical support explaining recent experimental observations^{11}. Finally, we discuss how stateoftheart measurements of AGPs could be leveraged to map out the lowfrequency quantum nonlocal surface response of metals experimentally. Our findings have significant implications for our ability to optimize photonic designs that interface far and midinfrared optical excitations—such as AGPs—with metals all the way down to the nanoscale, with pursuant applications in, e.g., ultracompact nanophotonic devices, nanometrology, and in the surface sciences more broadly.
Results
Theory
We consider a GDM heterostructure (see Fig. 1) composed of a graphene sheet with a surface conductivity σ ≡ σ(q,ω) separated from a metal substrate by a thin dielectric slab of thickness t and relative permittivity ϵ_{2} ≡ ϵ_{2}(ω); finally, the device is covered by a superstrate of relative permittivity ϵ_{1} ≡ ϵ_{1}(ω). While the metal substrate may, in principle, be represented by a nonlocal and spatially nonuniform (near the interface) dielectric function, here we abstract its contributions into two parts: a bulk, local contribution via \({\epsilon }_{\text{m}}\equiv {\epsilon }_{\text{m}}(\omega )={\epsilon }_{\infty }(\omega ){\omega }_{\text{p}}^{2}/({\omega }^{2}+\text{i}\omega {\gamma }_{\text{m}})\), and a surface, quantum contribution included through the dparameters. These parameters are quantummechanical surfaceresponse functions, defined by the first moments of the microscopic induced charge (d_{⊥}) and of the normal derivative of the tangential current (d_{∥}); see Fig. 1 (Supplementary Note 1 gives a concise introduction). They allow the leadingorder corrections to classicality to be conveniently incorporated via a surface dipole density (∝ d_{⊥}) and a surface current density (∝ d_{∥})^{9,15,16}, and can be obtained either by firstprinciples computation^{20,21}, semiclassical models, or experiments^{15}.
The electromagnetic excitations of any system can be obtained by analyzing the poles of the (composite) system’s scattering coefficients. For the AGPs of a GDM structure, the relevant coefficient is the ppolarized reflection (or transmission) coefficient, whose poles are given by \(1\ \ {r}_{p}^{2{\mathrm{g}} 1}\ {r}_{p}^{2{\mathrm{m}}}\ {\text{e}}^{{\text{i}}2{k}_{z,2}t}=0\) (ref. ^{22}). Here, \({r}_{p}^{2 \text{g} 1}\) and \({r}_{p}^{2 \text{m}}\) denote the ppolarized reflection coefficients for the dielectric–graphene–dielectric and the dielectric–metal interface (detailed in Supplementary Note 2), respectively. Each coefficient yields a materialspecific contribution to the overall quantum response: \({r}_{p}^{2\text{g} 1}\) incorporates graphene’s via σ(q,ω) and \({r}_{p}^{2 \text{m}}\) incorporates the metal’s via the dparameters (see Supplementary Note 2). The complex exponential [with \({k}_{z,2}\equiv {({\epsilon }_{2}{k}_{0}^{2}{q}^{2})}^{1/2}\), where q denotes the inplane wavevector] incorporates the effects of multiple reflections within the slab. Thus, using the abovenoted reflection coefficients (defined explicitly in Supplementary Note 2), we obtain a quantumcorrected AGP dispersion equation:
for inplane AGP wavevector q and outofplane confinement factors \({\kappa }_{j}\equiv ( {q}^{2}{\epsilon }_{j}{k}_{0}^{2} )^{1/2}\) for j ∈ {1, 2, m}.
Since AGPs are exceptionally subwavelength (with confinement factors up to almost 300)^{8,10,11}, the nonretarded limit (wherein κ_{j} → q) constitutes an excellent approximation. In this regime, and for encapsulated graphene, i.e., where ϵ_{d} ≡ ϵ_{1} = ϵ_{2}, Eq. (1) simplifies to
For simplicity and concreteness, we will consider a simple jellium treatment of the metal such that d_{∥} vanishes due to charge neutrality^{21,23}, leaving only d_{⊥} nonzero. Next, we exploit the fact that AGPs typically span frequencies across the terahertz (THz) and midinfrared (midIR) spectral ranges, i.e., well below the plasma frequency ω_{p} of the metal. In this lowfrequency regime, ω ≪ ω_{p}, the frequency dependence of d_{⊥} (and d_{∥}) has the universal, asymptotic dependence
as shown by Persson et al.^{24,25} by exploiting Kramers–Kronig relations. Here, ζ is the socalled static imageplane position, i.e., the centroid of induced charge under a static, external field^{26}, and ξ defines a phasespace coefficient for lowfrequency electron–hole pair creation, whose rate is ∝ qωξ^{21}: both are groundstate quantities. In the jellium approximation of the interacting electron liquid, the constants ζ ≡ ζ(r_{s}) and ξ ≡ ξ(r_{s}) depend solely on the carrier density n_{e}, here parameterized by the Wigner–Seitz radius \({r}_{s}{a}_{\text{B}}\equiv {(3{n}_{\text{e}}/4\pi )}^{1/3}\) (Bohr radius, a_{B}). In the following, we exploit the simple asymptotic relation in Eq. (3) to calculate the dispersion of AGPs with metallic (in addition to graphene’s) quantum response included.
Quantum corrections in AGPs due to metallic quantum surfaceresponse
The spectrum of AGPs calculated classically and with quantum corrections is shown in Fig. 2. Three models are considered: one, a completely classical, localresponse approximation treatment of both the graphene and the metal; and two others, in which graphene’s response is treated by the nonlocal RPA^{4,9,17,18,19} while the metal’s response is treated either classically or with quantum surfaceresponse included (via the d_{⊥}parameter). As noted previously, we adopt a jellium approximation for the d_{⊥}parameter. Figure 2a shows that—for a fixed wavevector—the AGP’s resonance blueshifts upon inclusion of graphene’s quantum response, followed by a redshift due to the quantum surfaceresponse of the metal (since \({\rm{Re}} {\,}{d}_{\perp } > \,0\) for jellium metals; electronic spillout)^{13,15,16,21,27,28}. This redshifting due to the metal’s quantum surfaceresponse is opposite to that predicted by the semiclassical hydrodynamic model (HDM) where the result is always a blueshift^{14} (corresponding to \({\rm{Re}}{\,}{d}_{\perp }^{\text{HDM}} < \,0\); electronic “spillin”) due to the neglect of spillout effects^{29}. The imaginary part of the AGP’s wavevector (that characterizes the mode’s propagation length) is shown in Fig. 2b: the net effect of the inclusion of d_{⊥} is a small, albeit consistent, increase of this imaginary component. Notwithstanding this, the modification of \({\rm{Im}}\, q\) is not independent of the shift in \({\rm{Re}}{\,}q\); as a result, an increase in \({\rm{Im}}{\,}q\) does not necessarily imply the presence of a significant quantum decay channel [e.g., an increase of \({\rm{Im}}{\,}q\) can simply result from increased classical loss (i.e., arising from local response alone) at the newly shifted \({\rm{Re}}\, q\) position]. Because of this, we inspect the quality factor \(Q\equiv {\rm{Re}}{\,}q/{\rm{Im}}{\,}q\) (or “inverse damping ratio”^{30,31}) instead^{32} (Fig. 2c), which provides a complementary perspective that emphasizes the effective (or normalized) propagation length rather than the absolute length. The incorporation of quantum mechanical effects, first in graphene alone, and then in both graphene and metal, reduces the AGP’s quality factor. Still, the impact of metalrelated quantum losses in the latter is negligible, as evidenced by the nearly overlapping black and red curves in Fig. 2c.
To better understand these observations, we treat the AGP’s qshift due to the metal’s quantum surfaceresponse as a perturbation: writing q = q_{0} + q_{1}, we find that the quantum correction from the metal is q_{1} ≃ q_{0}d_{⊥}/(2t), for a jellium adjacent to vacuum in the \({\omega }^{2}/{\omega }_{\text{p}}^{2}\ll {q}_{0}t\ll 1\) limit (Supplementary Note 3). This simple result, together with Eq. (3), provides a nearquantitative account of the AGP dispersion shifts due to metallic quantum surfaceresponse: for ω ≪ ω_{p}, (i) \({\rm{Re}}\, {d}_{\perp }\) tends to a finite value, ζ, which increases (decreases) \({\rm{Re}}\, q\) for ζ > 0 (ζ < 0); and (ii) \({\rm{Im}}{\,}{d}_{\perp }\) is \(\mathop{\propto}\omega\) and therefore asymptotically vanishing as ω/ω_{p} → 0 and so only negligibly increases \({\rm{Im}}\, q\). Moreover, the preceding perturbative analysis warrants \({\rm{Re}}{\,}{q}_{1}/{\rm{Re}}{\,}{q}_{0} \approx {\rm{Im}}{\,}{q}_{1}/{\rm{Im}}{\,}{q}_{0}\) (Supplementary Note 3), which elucidates the reason why the AGP’s quality factor remains essentially unaffected by the inclusion of metallic quantum surfaceresponse. Notably, these results explain recent experimental observations that found appreciable spectral shifts but negligible additional broadening due to metallic quantum response^{10,11}.
Next, by considering the separation between graphene and the metallic interface as a renormalizable parameter, we find a complementary and instructive perspective on the impact of metallic quantum surfaceresponse. Specifically, within the spectral range of interest for AGPs (i.e., ω ≪ ω_{p}), we find that the “bare” graphene–metal separation t is effectively renormalized due to the metal’s quantum surfaceresponse from t to \(\tilde{t}\equiv ts\), where s ≃ d_{⊥} ≃ ζ (see Supplementary Note 4), corresponding to a physical picture where the metal’s interface lies at the centroid of its induced density (i.e., \({\rm{Re}}{\,}{d}_{\perp }\)) rather than at its “classical” jellium edge. With this approach, the form of the dispersion equation is unchanged but references the renormalized separation \(\tilde{t}\) instead of its bare counterpart t, i.e.:
This perspective, for instance, has substantial implications for the analysis and understanding of plasmon rulers^{33,34,35} at nanometric scales.
Furthermore, our findings additionally suggest an interesting experimental opportunity: as all other experimental parameters can be wellcharacterized by independent means (including the nonlocal conductivity of graphene), highprecision measurements of the AGP’s dispersion can enable the characterization of the lowfrequency metallic quantum response—a regime that has otherwise been inaccessible in conventional metalonly plasmonics. The underlying idea is illustrated in Fig. 3; depending on the sign of the static asymptote ζ ≡ d_{⊥}(0), the AGP’s dispersion shifts toward larger q (smaller ω; redshift) for ζ > 0 and toward smaller q (larger ω; blueshift) for ζ < 0. As noted above, the qshift is ~q_{0}ζ/(2t). Crucially, despite the ångströmscale of ζ, this shift can be sizable: the inverse scaling with the spacer thickness t effectively amplifies the attainable shifts in q, reaching up to several μm^{−1} for fewnanometer t. We stress that these regimes are well within current stateoftheart experimental capabilities^{8,10,11}, suggesting a new path toward the systematic exploration of the static quantum response of metals.
Probing the quantum surfaceresponse of metals with AGPs
The key parameter that regulates the impact of quantum surface corrections stemming from the metal is the graphene–metal separation, t (analogously to the observations of nonclassical effects in conventional plasmons at narrow metal gaps^{13,36,37}); see Fig. 4. For the experimentally representative parameters indicated in Fig. 4, these come into effect for t ≲ 5 nm, growing rapidly upon decreasing the graphene–metal separation further. Chiefly, ignoring the nonlocal response of the metal leads to a consistent overestimation (underestimation) of AGP’s wavevector (group velocity) for d_{⊥} < 0, and vice versa for d_{⊥} > 0 (Fig. 4a); this behavior is consistent with the effective renormalization of the graphene–metal separation mentioned earlier (Fig. 4b). Finally, we analyze the interplay of both t and E_{F} and their joint influence on the magnitude of the quantum corrections from the metal (we take d_{⊥} = −4 Å, which is reasonable for the Au substrate used in recent AGP experiments^{7,8,11}); in Fig. 4c we show the relative wavevector quantum shift (excited at λ_{0} = 11.28 μm^{32}). In the fewnanometer regime, the quantum corrections to the AGP wavevector approach 5%, increasing further as t decreases—for instance, in the extreme, oneatomthick limit (t ≈ 0.7 nm^{11}, which also approximately coincides with edge of the validity of the dparameter framework, i.e., t ≳ 1 nm^{15}) the AGP’s wavevector can change by as much as 10% for moderate graphene doping. The pronounced Fermi level dependence exhibited in Fig. 4c also suggests a complementary approach for measuring the metal’s quantum surfaceresponse even if an experimental parameter is unknown (although, as previously noted, all relevant experimental parameters can in fact be characterized using currently available techniques^{8,10,11,15}): such an unknown variable can be fitted at low E_{F} using the “classical” theory (i.e., with d_{⊥} = d_{∥} = 0), since the impact of metallic quantum response is negligible in that regime. A parameterfree assessment of the metal’s quantum surfaceresponse can then be carried out subsequently by increasing E_{F} (and with it, the metalinduced quantum shift). We emphasize that this can be accomplished in the same device by doping graphene using standard electrostatic gating^{8,10,11}.
Discussion
In this Article, we have presented a theoretical account that establishes and quantifies the influence of the metal’s quantum response for AGPs in hybrid GDM structures. We have demonstrated that the nanoscale confinement of electromagnetic fields inherent to AGPs can be harnessed to determine the quantum surfaceresponse of metals in the THz and midIR spectral ranges (which is typically inaccessible with traditional metalbased plasmonics). Additionally, our findings elucidate and contextualize recent experiments^{10,11} that have reported the observation of nonclassical spectral shifting of AGPs due to metallic quantum response but without a clear concomitant increase of damping, even for atomically thin graphene–metal separations. Our results also demonstrate that the metal’s quantum surfaceresponse needs to be rigorously accounted for—e.g., using the framework developed here—when searching for signatures of manybody effects in the graphene electron liquid imprinted in the spectrum of AGPs in GDM systems^{8}, since the metal’s quantumsurface response can lead to qualitatively similar dispersion shifts, as shown here. In passing, we emphasize that our framework can be readily generalized to more complex graphene–metal hybrid structures either by semianalytical approaches (e.g., the Fourier modal method^{38} for periodically nanopatterned systems) or by direct implementation in commercially available numerical solvers (see refs. ^{15,39}), simply by adopting dparametercorrected boundary conditions^{15,16}.
Further, our formalism provides a transparent theoretical foundation for guiding experimental measurements of the quantum surfaceresponse of metals using AGPs. The quantitative knowledge of the metal’s lowfrequency, static quantum response is of practical utility in a plethora of scenarios, enabling, for instance, the incorporation of leadingorder quantum corrections to the classical electrostatic image theory of particle–surface interaction^{20} as well as to the van der Waals interaction^{21,25,40} affecting atoms or molecules near metal surfaces. Another prospect suggested by our findings is the experimental determination of ζ ≡ d_{⊥}(0) through measurements of the AGP’s spectrum. This highlights a new metric for comparing the fidelity of firstprinciple calculations of different metals (inasmuch as ab initio methods can yield disparate results depending on the chosen scheme or functional)^{41,42} with explicit measurements.
Our results also highlight that AGPs can be extremely sensitive probes for nanometrology as plasmon rulers, while simultaneously underscoring the importance of incorporating quantum response in the characterization of such rulers at (sub)nanometric scales. Finally, the theory introduced here further suggests additional directions for exploiting AGP’s highsensitivity, e.g., to explore the physics governing the complex electron dynamics at the surfaces of superconductors^{43} and other strongly correlated systems.
Data availability
The data that underlie the findings of this study are available from the corresponding authors upon reasonable request.
Code availability
The codes developed in this study are available from the corresponding authors upon reasonable request.
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Acknowledgements
N.A.M. is a VILLUM Investigator supported by VILLUM FONDEN (Grant No. 16498) and Independent Research Fund Denmark (Grant No. 702600117B). The Center for Nano Optics is financially supported by the University of Southern Denmark (SDU 2020 funding). The Center for Nanostructured Graphene (CNG) is sponsored by the Danish National Research Foundation (Project No. DNRF103). This work was partly supported by the Army Research Office through the Institute for Soldier Nanotechnologies under Contract No. W911NF1820048. N.M.R.P. acknowledges support from the European Commission through the project “GrapheneDriven Revolutions in ICT and Beyond” (No. 881603, Core 3), COMPETE 2020, PORTUGAL 2020, FEDER and the Portuguese Foundation for Science and Technology (FCT) through project POCI010145FEDER028114 and through the framework of the Strategic Financing UID/FIS/04650/2019. F.H.L.K. acknowledges financial support from the Government of Catalonia through the SGR grant and from the Spanish Ministry of Economy and Competitiveness (MINECO) through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV20150522), support by Fundació Cellex Barcelona, Generalitat de Catalunya through the CERCA program, and the MINECO grants Plan Nacional (FIS201681044P) and the Agency for Management of University and Research Grants (AGAUR) 2017 SGR 1656. Furthermore, the research leading to these results has received funding from the European Union’s Horizon 2020 program under the Graphene Flagship Grant Agreements No. 785219 (Core 2) and no. 881603 (Core 3), and the Quantum Flagship Grant No. 820378. This work was also supported by the ERC TOPONANOP (Grant No. 726001).
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P.A.D.G., T.C., N.M.R.P., I.E., F.H.L.K., M.S. and N.A.M. conceived the research. P.A.D.G. and T.C. spearheaded the research and drafted the manuscript. N.M.R.P., A.P.J., M.S., and N.A.M. supervised the work. All authors provided significant contributions to all aspects of this work and to the writing of the manuscript.
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Gonçalves, P.A.D., Christensen, T., Peres, N.M.R. et al. Quantum surfaceresponse of metals revealed by acoustic graphene plasmons. Nat Commun 12, 3271 (2021). https://doi.org/10.1038/s41467021230618
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DOI: https://doi.org/10.1038/s41467021230618
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