Electrodynamics of Bardeen–Cooper–Schrieffer-Like Superconductors.

The electrodynamics of superconductors and other strongly correlated matter constitute a fertile research area (29, 30). In the following, we assume that the superconducting material is well described by the Bardeen–Cooper–Schrieffer (BCS) theory of superconductivity (21, 35, 36). Chiefly, the microscopically derived linear optical conductivity tensor of a superconductor requires a nonlocal framework due to the finiteness of the Cooper-pair wave function. For homogeneous superconducting media, the longitudinal and transverse components of the nonlocal optical conductivity tensor—while treating nonlocality to leading order—can be expressed as (21, 37, 38)*

In possession of the response functions epitomized by Eq. 1, we employ the semiclassical infinite barrier (SCIB) formalism (23, 39) to describe electromagnetic phenomena at a planar dielectric–superconductor interface (37, 38, 40). Within this framework, the corresponding reflection coefficient for $$ p$$-polarized waves is given by (SI Appendix) (23, 39)

In what follows, we assume a typical high-$$ T_c$$ superconductor, such as yttrium barium copper oxide (YBCO), with a normal state electron density of $$ n=6\hspace{0.17em}{\text{nm}}^{-3}$$ and a transition temperature of $$ T_c=93\hspace{0.17em}\text{K}$$ (yielding a superconducting gap of $$ {\mathrm{\Delta }}_0\left(0\right)\approx 14.2\hspace{0.17em}\text{meV}$$) (37, 38, 41).

Electrodynamics in Graphene–Dielectric–Superconductor Heterostructures.

With knowledge of the reflection coefficient for the dielectric–superconductor interface (2), the overall reflection coefficient, i.e., that associated with the dielectric–graphene–dielectric–superconductor heterostructure, follows from imposing Maxwell’s boundary conditions (42) at all of the interfaces that make up the layered system. At the interface defined by the two-dimensional graphene sheet, the presence of graphene enters via a surface current with a corresponding surface conductivity (22).

Signatures of the system’s collective excitations can then be found by analyzing the poles of the corresponding reflection coefficient, which are identifiable as features in the imaginary part of the (overall) reflection coefficient, $$ \text{Im}\hspace{0.17em}{r}_p$$ (SI Appendix).

Signatures of the Higgs Mode Probed by Graphene Plasmons.

Like ordinary conductors (44), superconductors can also sustain surface plasmon polaritons (SPPs) (45, 46). In turn, these collective excitations can couple to the superconductor’s Higgs mode (37, 38). Typically such interaction is extremely weak due to the large mismatch between the superconductor’s plasma frequency, $$ {\omega }_{\text{p}}$$, and that of its Higgs mode, $$ {\omega }_{\text{H}}=2{\mathrm{\Delta }}_0/\hslash $$; for instance, $$ {\omega }_{\text{H}}/{\omega }_{\text{p}}\sim 10^{-2}$$, with $$ {\omega }_{\text{p}}$$ and $$ {\omega }_{\text{H}}$$ falling, respectively, in the visible and terahertz spectral ranges. As a result, at frequencies around $$ {\omega }_{\text{H}}$$ the SPP resembles light in free space and thus the SPP–Higgs coupling is essentially as weak as when using far-field optics (Fig. 2A).

Fig. 2.

Spectra of surface electromagnetic waves in superconductors (A) and graphene–superconductor (B) structures, obtained from the calculation of the corresponding $$ \text{Im}\hspace{0.17em}{r}_p$$. (A) Dispersion diagram of SPPs supported by a vacuum–superconductor interface (the hatched area indicates the light cone in vacuum). Inset shows a closeup of an extremely small region (notice the change of scale) where the SPP dispersion crosses the energy associated with the superconductor’s Higgs mode; here, $$ \mathrm{\Delta }E=E-\hslash {\omega }_{\text{H}}$$ and $$ \mathrm{\Delta }{q}_{\parallel }=q_{\parallel }-{\omega }_{\text{H}}/c$$. (B) Dispersion relation of GPs exhibiting an anticrossing feature that signals their interaction with the Higgs mode of the nearby superconductor. The graphene–superconductor separation is $$ t=5\hspace{0.17em}\text{nm}$$. Setup parameters: We take $$ T=1\hspace{0.17em}\text{K}$$; moreover, $$ n=6\times 10^{21}\hspace{0.17em}{\text{cm}}^{-3}$$ (so that $$ E_{\text{F}}\approx 1.20\hspace{0.17em}\text{eV}$$ and $$ \hslash {\omega }_{\text{p}}\approx 2.88\hspace{0.17em}\text{eV}$$), $$ \hslash \gamma =1\hspace{0.17em}\mathrm{\mu }\text{eV}$$, and $$ T_c=93\hspace{0.17em}\text{K}$$ for the superconductor (38, 40, 41), and $$ E_{\text{F}}^{\text{gr}}=0.3\hspace{0.17em}\text{eV}$$ and $$ \hslash {\gamma }^{\text{gr}}=1\hspace{0.17em}\text{meV}$$, for graphene’s Drude-like optical conductivity (43).

On the other hand, graphene plasmons not only span the terahertz regime but also attain sizable plasmon wave vectors at those frequencies (22, 23). Moreover, when the graphene sheet is near a metal—or a superconductor for that matter—graphene’s plasmons become screened and acquire a nearly linear (acoustic) dispersion, pushing their spectrum further toward lower frequencies (i.e., a few terahertz) and larger wave vectors (23242526–27, 32). Therefore, these properties of acoustic-like GPs can be harnessed by placing a graphene monolayer near a superconducting surface, thereby allowing the interaction of graphene’s plasmons with the Higgs mode of the underlying superconductor (Fig. 2B). In this case the plasmon–Higgs interaction is substantially enhanced, a fact that is reflected in the observation of a clear anticrossing in the GP’s dispersion near $$ {\omega }_{\text{H}}$$, which, crucially, is orders of magnitude larger than that observed in the absence of graphene (Fig. 2 A and B).

Furthermore, the use of graphene plasmons for probing the superconductor’s Higgs mode comes with the added benefit of control over the plasmon–Higgs coupling by tuning graphene’s Fermi energy electrostatically (22, 23, 4748–49). This is explicitly shown in Fig. 3A, for a vacuum–hexagonal boron nitride (hBN)–graphene–hBN–superconductor heterostructure; as before, the coupling of GPs with the superconductor’s Higgs mode manifests itself through the appearance of an avoided crossing in the vicinity of $$ {\omega }_{\text{H}}$$, which occurs at successively larger wave vectors upon decreasing $$ E_{\text{F}}^{\text{gr}}$$. Another source of tunability is the graphene–superconductor distance, $$ t$$ (which, in the present configuration, corresponds to the thickness of the bottommost hBN slab). Strikingly, current experimental capabilities allow the latter to be controlled with atomic precision (24, 25, 32). We exploit this fact in Fig. 3B, where we have considered the same heterostructure, but now we have varied $$ t$$ instead, while keeping $$ E_{\text{F}}^{\text{gr}}$$ fixed. Naturally, the manifestation of the GP–Higgs mode interaction seems to be more pronounced for smaller $$ t$$, reducing to a faint feature at large $$ t$$ (see the result for $$ t=50\hspace{0.17em}\text{nm}$$). Finally, it should be noted that the net effect of decreasing the graphene–superconductor separation $$ t$$ is the outcome of two intertwined contributions: The graphene–superconductor interaction is evidently stronger when the materials lie close together, but equally important is the fact that the (group) velocity of plasmons in the graphene sheet gets continuously reduced as $$ t$$ diminishes due to the screening exercised by the nearby superconductor [and, consequently, the GP’s dispersion shifts toward higher wave vectors, eventually reaching the nonlocal regime (23, 24, 27)].

Fig. 3.

Tuning the hybridization of acoustic-like plasmons in graphene with the Higgs mode of a superconductor in air–hBN–graphene–hBN–superconductor heterostructures. The colormap indicates the loss function via $$ \text{Im}\hspace{0.17em}{r}_p$$. (A and B) Spectral dependence upon varying the Fermi energy of graphene (A) and the graphene–superconductor distance (B). Setup parameters: The parameters of the superconductor are the same as in Fig. 2, and the same goes for graphene’s Drude damping. The thickness of the bottom hBN slab is given by $$ t$$, whereas the thickness of the top hBN slab, $$ t^{\prime }$$, has been kept constant ($$ t^{\prime }=10\hspace{0.17em}\text{nm}$$). Here, we have modeled hBN’s optical properties using a dielectric tensor of the form $$ {\overleftrightarrow{\mathit{ϵ}}}_{\text{hBN}}=\text{diag}\left[{ϵ}_{xx},{ϵ}_{yy},{ϵ}_{zz}\right]$$ with $$ {ϵ}_{xx}={ϵ}_{yy}=6.7$$ and $$ {ϵ}_{zz}=3.6$$ (24, 49, 50).

Higgs Mode Visibility through the Purcell Effect.

One way to overcome the momentum mismatch and investigate the presence of electromagnetic surface modes is to place a quantum emitter (22, 5152–53) (herein modeled as a point-like electric dipole) in the proximity of an interface and study its decay rate as a function of the emitter–surface distance. With the advent of atomically thin materials, and hBN in particular, all of the relevant distances, i.e., emitter–superconductor, emitter–graphene, and graphene–superconductor, can be tailored with nanometric precision [e.g., by controlling the number of stacked hBN layers (each $$ \sim 0.7\hspace{0.17em}\text{nm}$$ thick) (25, 32) or using atomic layer deposition (54, 55)]. Although the availability of good emitters in the terahertz range is unarguably limited, semiconductor quantum dots with intersublevel transitions in this range and with relatively long relaxation times do exist (56). The modification of the spontaneous decay rate of an emitter is a repercussion of a change in the electromagnetic LDOS, $$ \rho \left(\mathbf{r}\right)$$, and it is known as the Purcell effect (23, 33, 34). Specifically, the Purcell factor—defined as the ratio $$ \frac{\rho \left(\mathbf{r}\right)}{{\rho }_0\left(\mathbf{r}\right)}$$, where $$ {\rho }_0\left(\mathbf{r}\right)$$ is the LDOS experienced by an emitter in free space—can be greatly enhanced by positioning the emitter near material interfaces supporting electromagnetic modes (which are responsible for augmenting the LDOS). In passing, we note that this LDOS enhancement does not strictly require an “emitter,” since it can also be probed through the interaction of the sample with the illuminated tip of a near-field optical microscope (which may be modeled as an electric dipole in a first approximation)—in fact, most tip-enhanced spectroscopies rely on this principle (5758–59).

Since in the near-field region the overall LDOS is dominated by contributions from $$ p$$-polarization (and since plasmons possess $$ p$$-polarization), in the following we neglect $$ s$$-polarization contributions coming from the scattered fields. Then, the orientation-averaged Purcell factor—or, equivalently, the LDOS enhancement—can be determined via (34)

Fig. 4 shows the LDOS enhancement experienced by an emitter (or a nanosized tip) in the proximity of a superconductor; Fig. 4 A, B, D, and E refers to the case in the presence of graphene (located between the superconductor and the emitter), whereas Fig. 4C depicts a scenario where the graphene sheet is absent. The graphene sheet modifies the LDOS, affecting not only the absolute Purcell factor but also the peak/dip feature around the energy of the Higgs mode, $$ \hslash {\omega }_h=2{\mathrm{\Delta }}_0$$. Such modification depends strongly on the emitter–graphene separation $$ d$$ (Fig. 4 A and B). Fig. 4D shows the LDOS enhancement for $$ T>T_c$$ (i.e., above the superconductor’s transition temperature) and thus the feature associated with the Higgs mode vanishes; all that remains is a relatively broad feature related to the excitation of graphene plasmons.

Fig. 4.

Purcell factor near a vacuum–hBN–graphene–hBN–superconductor heterostructure. In A and B the graphene Fermi energy has been set at $$ E_{\text{F}}^{\text{gr}}=0.25\hspace{0.17em}\text{eV}$$; here, $$ T=1\hspace{0.17em}\text{K}$$ for the solid curves and $$ T=94\hspace{0.17em}\text{K}$$ (above $$ T_c$$) for the dashed curves, and the graphene sheet is placed 4 nm above the superconductor surface. We show results for two emitter–graphene distances: 13 nm (A) and 36 nm (B). In C we show the case without graphene, at $$ T=1\hspace{0.17em}\text{K}$$. The red curve corresponds to an emitter–superconductor separation of 17 nm and the blue curve to that of 40 nm. In D we show results for the same distances as in A (red curve) and B (blue curve), but for $$ T=94\hspace{0.17em}\text{K}$$. In E we show how the Purcell factor depends on the graphene–superconductor distance $$ t$$ at the energy of the Higgs mode, $$ \hslash {\omega }_h=2{\mathrm{\Delta }}_0\approx 28.32\hspace{0.17em}\text{meV}$$. The other parameters are kept fixed: $$ E_{\text{F}}^{\text{gr}}=0.5\hspace{0.17em}\text{eV}$$, $$ T=1\hspace{0.17em}\text{K}$$, and emitter–graphene distance of $$ d=13\hspace{0.17em}\text{nm}$$. Here, graphene’s conductivity has been modeled using the nonlocal random-phase approximation (23, 60).

Finally, Fig. 5 depicts the LDOS enhancement for different values of graphene’s Fermi energy (which can be tuned electrostatically), for two fixed emitter–graphene distances: $$ d=13\hspace{0.17em}\text{nm}$$ (Fig. 5, Top row) and $$ d=2\hspace{0.17em}\text{nm}$$ (Fig. 5, Middle row). For weakly doped graphene and the larger $$ d$$ the sharp feature associated with the hybrid GPs–Higgs mode dominates the Purcell factor, being eventually overtaken by the broader background with increasing $$ E_{\text{F}}^{\text{gr}}$$. To unveil the mechanisms underpinning the LDOS enhancement, we plot in Fig. 5, Bottom row the $$ q_{\parallel }$$-space differential LDOS enhancement (tantamount to the so-called $$ q_{\parallel }$$-space power spectrum, 39), which amounts to the integrand of Eq. 3. In the near field (well realized for the chosen setup and parameters), there are two contributions (34, 39): one from a resonant channel, corresponding to the excitation of the coupled Higgs–GP mode, and a broad, nonresonant contribution at larger $$ q_{\parallel }$$ due to lossy channels (phenomenologically incorporated through the relaxation rates $$ \gamma ,{\gamma }^{\text{gr}}$$). Mathematically, the polariton (Higgs–GP mode) resonant contribution arises from the pole in $$ \text{Im}\hspace{0.17em}{r}_p$$, occurring at $$ q_{\parallel }\simeq \text{Re}{q}_{GP}\left(\omega \right)$$ (where $$ q_{GP}\left(\omega \right)$$ is the wave vector of the Higgs–GP mode at frequency $$ \omega $$ that satisfies the dispersion relation) (Fig. 3). Consistent with this, the peak associated with the Higgs–GP polariton contribution to the $$ q_{\parallel }$$-space differential LDOS occurs at a larger wave vector in the $$ E_{\text{F}}^{\text{gr}}=50\hspace{0.17em}\text{meV}$$ case, since, for the same frequency, the Higgs–GP dispersion shifts toward larger wave vectors upon decreasing $$ E_{\text{F}}^{\text{gr}}$$ (24, 28). Ultimately, the amplitude of the resonant contribution depends on the specifics of the dispersion relation (i.e., $$ q_{GP}\left(\omega \right)=\text{Re}{q}_{GP}\left(\omega \right)+\text{iIm}{q}_{GP}\left(\omega \right)$$) and is further weighted by the $$ q_{\parallel }^2\exp \left(-2q_{\parallel }z\right)$$ factor that depends not only on the peak’s location, $$ q_{\parallel }\left(\omega \right)\simeq \text{Re}{q}_{GP}\left(\omega \right)$$ (and whose width $$ \propto \text{Im}{q}_{GP}\left(\omega \right)$$), but also on the emitter’s position $$ z=d-t^{\prime }$$ (Eq. 3). Finally, we stress that the relative contribution of each of the above-noted decay channels is strongly dependent on the emitter–graphene distance $$ d$$ (with the nonresonant, lossy contribution eventually dominating at sufficiently small emitter–graphene separations—quenching) (34, 39).

Fig. 5.

Purcell factor as a function of graphene’s Fermi energy. Here we show the effect of changing graphene’s Fermi energy (indicated at the top of each column) while keeping all other parameters fixed: $$ T=1\hspace{0.17em}\text{K}$$, emitter–graphene distance ($$ d=13\hspace{0.17em}\text{nm}$$ for Top row and $$ d=2\hspace{0.17em}\text{nm}$$ for Middle row), and graphene–superconductor distance $$ t=4\hspace{0.17em}\text{nm}$$. Here, graphene’s conductivity has been modeled using the nonlocal random-phase approximation (23, 60). For $$ d=13\hspace{0.17em}\text{nm}$$, the dependence of the decay rate on the emitter’s frequency changes quantitatively from low ($$ E_{\text{F}}^{\text{gr}}=50\hspace{0.17em}\text{meV}$$) to high ($$ E_{\text{F}}^{\text{gr}}=250\hspace{0.17em}\text{meV}$$) graphene doping. In Bottom panel we depict the $$ q_{\parallel }$$-space differential LDOS given by the integration kernel of Eq. 3; the energy has been fixed at the value $$ \hslash {\omega }_{\mathrm{H}}$$.