Harnessing ultraconfined graphene plasmons to probe the electrodynamics of superconductors

Abstract

We show that the Higgs mode of a superconductor, which is usually challenging to observe by far-field optics, can be made clearly visible using near-field optics by harnessing ultraconfined graphene plasmons. As near-field sources we investigate two examples: graphene plasmons and quantum emitters. In both cases the coupling to the Higgs mode is clearly visible. In the case of the graphene plasmons, the coupling is signaled by a clear anticrossing stemming from the interaction of graphene plasmons with the Higgs mode of the superconductor. In the case of the quantum emitters, the Higgs mode is observable through the Purcell effect. When combining the superconductor, graphene, and the quantum emitters, a number of experimental knobs become available for unveiling and studying the electrodynamics of superconductors.

Keywords: graphene; near-field microscopy; plasmons; polaritons; superconductivity.

Fig. 1.

Schematic of the graphene–superconductor hybrid device considered here. Shown is an illustration of the heterostructure composed of a superconducting substrate, a few atomic layers of hexagonal boron nitride (hBN), a single sheet of graphene, and a capping layer of hBN. It should be noted that although here the hBN has been depicted in monolayer form, our model can accommodate any number of hBN layers. The red-blue sphere represents an electric dipole placed above the heterostructure.

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).

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).

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).

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 $\hslash {\omega }_{\mathrm{H}}$.