### Idea

We think about a GDM heterostructure (see Fig. 1) composed of a graphene sheet with a floor conductivity *σ* ≡ *σ*(*q*,*ω*) separated from a metallic substrate by a skinny dielectric slab of thickness *t* and relative permittivity *ϵ*_{2} ≡ *ϵ*_{2}(*ω*); lastly, the system is roofed by a superstrate of relative permittivity *ϵ*_{1} ≡ *ϵ*_{1}(*ω*). Whereas the metallic substrate could, in precept, be represented by a nonlocal and spatially non-uniform (close to the interface) dielectric perform, right here we summary its contributions into two elements: a bulk, native contribution through ({epsilon }_{textual content{m}}equiv {epsilon }_{textual content{m}}(omega )={epsilon }_{infty }(omega )-{omega }_{textual content{p}}^{2}/({omega }^{2}+textual content{i}omega {gamma }_{textual content{m}})), and a floor, quantum contribution included via the *d*-parameters. These parameters are quantum-mechanical surface-response capabilities, outlined by the primary moments of the microscopic induced cost (*d*_{⊥}) and of the conventional by-product of the tangential present (*d*_{∥}); see Fig. 1 (Supplementary Be aware 1 offers a concise introduction). They permit the leading-order corrections to classicality to be conveniently integrated through a floor dipole density (∝ *d*_{⊥}) and a floor present density (∝ *d*_{∥})^{9,15,16}, and may be obtained both by first-principles computation^{20,21}, semiclassical fashions, or experiments^{15}.

The electromagnetic excitations of any system may be obtained by analyzing the poles of the (composite) system’s scattering coefficients. For the AGPs of a GDM construction, the related coefficient is the *p*-polarized reflection (or transmission) coefficient, whose poles are given by (1 – {r}_{p}^{2|{mathrm{g}}| 1} {r}_{p}^{2|{mathrm{m}}} {textual content{e}}^{{textual content{i}}2{okay}_{z,2}t}=0) (ref. ^{22}). Right here, ({r}_{p}^{2| textual content{g}| 1}) and ({r}_{p}^{2| textual content{m}}) denote the *p*-polarized reflection coefficients for the dielectric–graphene–dielectric and the dielectric–metallic interface (detailed in Supplementary Be aware 2), respectively. Every coefficient yields a material-specific contribution to the general quantum response: ({r}_{p}^{2|textual content{g}| 1}) incorporates graphene’s through *σ*(*q*,*ω*) and ({r}_{p}^{2| textual content{m}}) incorporates the metallic’s through the *d*-parameters (see Supplementary Be aware 2). The advanced exponential [with ({k}_{z,2}equiv {({epsilon }_{2}{k}_{0}^{2}-{q}^{2})}^{1/2}), where *q* denotes the in-plane wavevector] incorporates the results of a number of reflections inside the slab. Thus, utilizing the above-noted reflection coefficients (outlined explicitly in Supplementary Be aware 2), we acquire a quantum-corrected AGP dispersion equation:

$$ left[frac{{epsilon }_{1}}{{kappa }_{1}}+frac{{epsilon }_{2}}{{kappa }_{2}}+frac{,text{i},sigma }{omega {epsilon }_{0}}right]left[{epsilon }_{text{m}}{kappa }_{2}+{epsilon }_{2}{kappa }_{text{m}}-left(right.{epsilon }_{text{m}}-{epsilon }_{2}left)right.left(right.{q}^{2}{d}_{perp }-{kappa }_{2}{kappa }_{text{m}}{d}_{parallel }left)right.right] =left[frac{{epsilon }_{1}}{{kappa }_{1}}-frac{{epsilon }_{2}}{{kappa }_{2}}+frac{,text{i},sigma }{omega {epsilon }_{0}}right]left[{epsilon }_{text{m}}{kappa }_{2}-{epsilon }_{2}{kappa }_{text{m}}+left(right.{epsilon }_{text{m}}-{epsilon }_{2}left)right.left(right.{q}^{2}{d}_{perp }+{kappa }_{2}{kappa }_{text{m}}{d}_{parallel }left)right.right]{textual content{e}}^{-2{kappa }_{2}t},$$

(1)

for in-plane AGP wavevector *q* and out-of-plane confinement components ({kappa }_{j}equiv ( {q}^{2}-{epsilon }_{j}{okay}_{0}^{2} )^{1/2}) for *j* ∈ {1, 2, m}.

Since AGPs are exceptionally subwavelength (with confinement components as much as nearly 300)^{8,10,11}, the nonretarded restrict (whereby *κ*_{j} → *q*) constitutes a superb approximation. On this regime, and for encapsulated graphene, i.e., the place *ϵ*_{d} ≡ *ϵ*_{1} = *ϵ*_{2}, Eq. (1) simplifies to

$$left[1+frac{2{epsilon }_{text{d}}}{q}frac{omega {epsilon }_{0}}{text{i},sigma }right]left[frac{{epsilon }_{text{m}}+{epsilon }_{text{d}}}{{epsilon }_{text{m}}-{epsilon }_{text{d}}}-qleft(right.{d}_{perp }-{d}_{parallel }left)right.right]=left[right.1+qleft(right.{d}_{perp }+{d}_{parallel }left)right.left]proper.{textual content{e}}^{-2qt}.$$

(2)

For simplicity and concreteness, we are going to think about a easy jellium therapy of the metallic such that *d*_{∥} vanishes on account of cost neutrality^{21,23}, leaving solely *d*_{⊥} nonzero. Subsequent, we exploit the truth that AGPs usually span frequencies throughout the terahertz (THz) and mid-infrared (mid-IR) spectral ranges, i.e., properly beneath the plasma frequency *ω*_{p} of the metallic. On this low-frequency regime, *ω* ≪ *ω*_{p}, the frequency dependence of *d*_{⊥} (and *d*_{∥}) has the common, asymptotic dependence

$${d}_{perp }(omega )simeq zeta +,textual content{i}frac{omega }{{omega }_{textual content{p}}}xi ,qquad (textual content{for},,omega ll {omega }_{textual content{p}}),$$

(3)

as proven by Persson et al.^{24,25} by exploiting Kramers–Kronig relations. Right here, *ζ* is the so-called static image-plane place, i.e., the centroid of induced cost underneath a static, exterior discipline^{26}, and *ξ* defines a phase-space coefficient for low-frequency electron–gap pair creation, whose price is ∝ *q**ω**ξ*^{21}: each are ground-state portions. Within the jellium approximation of the interacting electron liquid, the constants *ζ* ≡ *ζ*(*r*_{s}) and *ξ* ≡ *ξ*(*r*_{s}) rely solely on the provider density *n*_{e}, right here parameterized by the Wigner–Seitz radius ({r}_{s}{a}_{textual content{B}}equiv {(3{n}_{textual content{e}}/4pi )}^{1/3}) (Bohr radius, *a*_{B}). Within the following, we exploit the straightforward asymptotic relation in Eq. (3) to calculate the dispersion of AGPs with metallic (along with graphene’s) quantum response included.

### Quantum corrections in AGPs on account of metallic quantum surface-response

The spectrum of AGPs calculated classically and with quantum corrections is proven in Fig. 2. Three fashions are thought of: one, a totally classical, local-response approximation therapy of each the graphene and the metallic; and two others, during which graphene’s response is handled by the nonlocal RPA^{4,9,17,18,19} whereas the metallic’s response is handled both classically or with quantum surface-response included (through the *d*_{⊥}-parameter). As famous beforehand, we undertake a jellium approximation for the *d*_{⊥}-parameter. Determine 2a reveals that—for a set wavevector—the AGP’s resonance blueshifts upon inclusion of graphene’s quantum response, adopted by a redshift because of the quantum surface-response of the metallic (since ({rm{Re}} {,}{d}_{perp } > ,0) for jellium metals; digital spill-out)^{13,15,16,21,27,28}. This redshifting because of the metallic’s quantum surface-response is reverse to that predicted by the semiclassical hydrodynamic mannequin (HDM) the place the result’s at all times a blueshift^{14} (comparable to ({rm{Re}}{,}{d}_{perp }^{textual content{HDM}} < ,0); digital “spill-in”) because of the neglect of spill-out results^{29}. The imaginary a part of the AGP’s wavevector (that characterizes the mode’s propagation size) is proven in Fig. 2b: the online impact of the inclusion of *d*_{⊥} is a small, albeit constant, improve of this imaginary element. However this, the modification of ({rm{Im}}, q) just isn’t impartial of the shift in ({rm{Re}}{,}q); because of this, a rise in ({rm{Im}}{,}q) doesn’t essentially suggest the presence of a major 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]. Due to this, we examine the standard issue (Qequiv {rm{Re}}{,}q/{rm{Im}}{,}q) (or “inverse damping ratio”^{30,31}) as an alternative^{32} (Fig. 2c), which gives a complementary perspective that emphasizes the efficient (or normalized) propagation size somewhat than absolutely the size. The incorporation of quantum mechanical results, first in graphene alone, after which in each graphene and metallic, reduces the AGP’s high quality issue. Nonetheless, the affect of metal-related quantum losses within the latter is negligible, as evidenced by the practically overlapping black and pink curves in Fig. 2c.

To raised perceive these observations, we deal with the AGP’s *q*-shift because of the metallic’s quantum surface-response as a perturbation: writing *q* = *q*_{0} + *q*_{1}, we discover that the quantum correction from the metallic is *q*_{1} ≃ *q*_{0}*d*_{⊥}/(2*t*), for a jellium adjoining to hoover within the ({omega }^{2}/{omega }_{textual content{p}}^{2}ll {q}_{0}tll 1) restrict (Supplementary Be aware 3). This straightforward outcome, along with Eq. (3), gives a near-quantitative account of the AGP dispersion shifts on account of metallic quantum surface-response: for *ω* ≪ *ω*_{p}, (i) ({rm{Re}}, {d}_{perp }) tends to a finite worth, *ζ*, which will increase (decreases) ({rm{Re}}, q) for *ζ* > 0 (*ζ* < 0); and (ii) ({rm{Im}}{,}{d}_{perp }) is (mathop{propto}omega) and subsequently asymptotically vanishing as *ω*/*ω*_{p} → 0 and so solely negligibly will increase ({rm{Im}}, q). Furthermore, the previous perturbative evaluation warrants ({rm{Re}}{,}{q}_{1}/{rm{Re}}{,}{q}_{0} approx {rm{Im}}{,}{q}_{1}/{rm{Im}}{,}{q}_{0}) (Supplementary Be aware 3), which elucidates the rationale why the AGP’s high quality issue stays basically unaffected by the inclusion of metallic quantum surface-response. Notably, these outcomes clarify current experimental observations that discovered considerable spectral shifts however negligible extra broadening on account of metallic quantum response^{10,11}.

Subsequent, by contemplating the separation between graphene and the metallic interface as a renormalizable parameter, we discover a complementary and instructive perspective on the affect of metallic quantum surface-response. Particularly, inside the spectral vary of curiosity for AGPs (i.e., *ω* ≪ *ω*_{p}), we discover that the “naked” graphene–metallic separation *t* is successfully renormalized because of the metallic’s quantum surface-response from *t* to (tilde{t}equiv t-s), the place *s* ≃ *d*_{⊥} ≃ *ζ* (see Supplementary Be aware 4), comparable to a bodily image the place the metallic’s interface lies on the centroid of its induced density (i.e., ({rm{Re}}{,}{d}_{perp })) somewhat than at its “classical” jellium edge. With this method, the type of the dispersion equation is unchanged however references the renormalized separation (tilde{t}) as an alternative of its naked counterpart *t*, i.e.:

$$1+frac{2{epsilon }_{textual content{d}}}{q}frac{omega {epsilon }_{0}}{textual content{i}sigma }=frac{{epsilon }_{textual content{m}}-{epsilon }_{textual content{d}}}{{epsilon }_{textual content{m}}+{epsilon }_{textual content{d}}} {textual content{e}}^{-2qtilde{t}},$$

(4)

This angle, for example, has substantial implications for the evaluation and understanding of plasmon rulers^{33,34,35} at nanometric scales.

Moreover, our findings moreover counsel an fascinating experimental alternative: as all different experimental parameters may be well-characterized by impartial means (together with the nonlocal conductivity of graphene), high-precision measurements of the AGP’s dispersion can allow the characterization of the low-frequency metallic quantum response—a regime that has in any other case been inaccessible in standard metal-only plasmonics. The underlying concept is illustrated in Fig. 3; relying on the signal of the static asymptote *ζ* ≡ *d*_{⊥}(0), the AGP’s dispersion shifts towards bigger *q* (smaller *ω*; redshift) for *ζ* > 0 and towards smaller *q* (bigger *ω*; blueshift) for *ζ* < 0. As famous above, the *q*-shift is ~*q*_{0}*ζ*/(2*t*). Crucially, regardless of the ångström-scale of *ζ*, this shift may be sizable: the inverse scaling with the spacer thickness *t* successfully amplifies the attainable shifts in *q*, reaching as much as a number of μm^{−1} for few-nanometer *t*. We stress that these regimes are properly inside present state-of-the-art experimental capabilities^{8,10,11}, suggesting a brand new path towards the systematic exploration of the static quantum response of metals.

### Probing the quantum surface-response of metals with AGPs

The important thing parameter that regulates the affect of quantum floor corrections stemming from the metallic is the graphene–metallic separation, *t* (analogously to the observations of nonclassical results in standard plasmons at slender metallic gaps^{13,36,37}); see Fig. 4. For the experimentally consultant parameters indicated in Fig. 4, these come into impact for *t* ≲ 5 nm, rising quickly upon reducing the graphene–metallic separation additional. Mainly, ignoring the nonlocal response of the metallic results in a constant overestimation (underestimation) of AGP’s wavevector (group velocity) for *d*_{⊥} < 0, and vice versa for *d*_{⊥} > 0 (Fig. 4a); this conduct is per the efficient renormalization of the graphene–metallic separation talked about earlier (Fig. 4b). Lastly, we analyze the interaction of each *t* and *E*_{F} and their joint affect on the magnitude of the quantum corrections from the metallic (we take *d*_{⊥} = −4 Å, which is affordable for the Au substrate utilized in current AGP experiments^{7,8,11}); in Fig. 4c we present the relative wavevector quantum shift (excited at *λ*_{0} = 11.28 μm^{32}). Within the few-nanometer regime, the quantum corrections to the AGP wavevector method 5%, growing additional as *t* decreases—for example, within the excessive, one-atom-thick restrict (*t* ≈ 0.7 nm^{11}, which additionally roughly coincides with fringe of the validity of the *d*-parameter framework, i.e., *t* ≳ 1 nm^{15}) the AGP’s wavevector can change by as a lot as 10% for average graphene doping. The pronounced Fermi stage dependence exhibited in Fig. 4c additionally suggests a complementary method for measuring the metallic’s quantum surface-response even when an experimental parameter is unknown (though, as beforehand famous, all related experimental parameters can in reality be characterised utilizing presently obtainable strategies^{8,10,11,15}): such an unknown variable may be fitted at low *E*_{F} utilizing the “classical” idea (i.e., with *d*_{⊥} = *d*_{∥} = 0), because the affect of metallic quantum response is negligible in that regime. A parameter-free evaluation of the metallic’s quantum surface-response can then be carried out subsequently by growing *E*_{F} (and with it, the metal-induced quantum shift). We emphasize that this may be completed in the identical system by doping graphene utilizing normal electrostatic gating^{8,10,11}.