Josephson surface plasmons in spatially confined cuprate superconductors

In this work, we generalize the theory of localized surface plasmons to the case of high-$T_{\mathrm{c}}$ cuprate superconductors, spatially confined in the form of small spherical particles. At variance from ordinary metals, cuprate superconductors are characterized by a low-energy bulk excitation known as the Josephson plasma wave (JPW), arising from interlayer tunneling of the condensate along the $c$ axis. The effect of the JPW is revealed in a characteristic spectrum of surface excitations, which we call Josephson surface plasmons. Our results, which apply to any material with a strongly anisotropic electromagnetic response, are worked out in detail for the case of multilayered superconductors supporting both low-frequency (acoustic) and transverse-optical JPW. Spatial confinement of the Josephson plasma waves may represent a new degree of freedom to engineer their frequencies and to explore the link between interlayer tunneling and high-$T_{\mathrm{c}}$ superconductivity.

Figure 1

(a) Dispersion of the optical TM modes of a bilayered superconductor as a function of the $ab$-plane wave vector $k_x$, normalized to $\sqrt{{\varepsilon }_{\infty }}{\omega }_{\text{J1}}/c$. Each curve corresponds to a value of the $c$-axis wave vector $k_z$ (with the same normalization), indicated by the label. (b) Close-up of the low-frequency and long-wavelength region of the dispersion. Parameters are ${\omega }_{\text{T}}=9{\omega }_{\text{J1}}$, ${\omega }_{\text{J2}}=10{\omega }_{\text{J1}}$, and ${\omega }_{\text{P}}=100{\omega }_{\text{J1}}$.

Figure 2

Quasistatic Josephson surface plasmons in a spherical particle made of a $n=2$ multilayered superconductor. (a) The modal frequencies as a function of the azimuthal quantum number $l$ and for three values of the magnetic quantum number $m=0,1,2$. Notice the different scales in the frequency ranges of the $\omega$ axis. Parameters are ${\omega }_{\text{T}}=9{\omega }_{\text{J1}}$, ${\omega }_{\text{J2}}=10{\omega }_{\text{J1}}$, ${\omega }_{\text{P}}=100{\omega }_{\text{J1}},{\varepsilon }_{\infty }=10$, and ${\varepsilon }_{\text{d}}=1$. (b)–(e) Potential maps of the modes indicated by the dashed lines. Potential is shown on a cut along the $xz$ plane including the center of the sphere, with the $c$ axis oriented as $\widehat{\mathbf{z}}$.

Figure 3

The frequencies of the two $l=1$, $m=0$ JSPs for the same particle as in Fig. 2 vs the transverse-optical Josephson frequency ${\omega }_{\text{T}}$. The red dotted lines indicate the frequencies calculated with Eq. (14).

Figure 4

Black solid line: the EEL spectrum of a 300-eV electron moving along the $c$ axis at grazing incidence ($b=1.02R$) on a $R=400\mathrm{nm}$ sphere of a bilayered superconductor. Dashed line: the extinction cross section for a field polarized along the $c$ axis. Here, ${\omega }_{\text{J1}}=10$ meV, and the parameters are as in Fig. 2 (vertical dotted lines). In-plane dissipation is modeled assuming $\mathrm{Im}{\varepsilon }_{\text{ab}}=10{\omega }_{\text{J1}}{\omega }_{\text{P}}^2/{\omega }^3$, whereas inter- and intrabilayer tunneling are supplemented with the additional conductance $4\pi {\sigma }_{\text{c1}}={\omega }_{\text{J1}}/2$ and $4\pi {\sigma }_{\text{c2}}={\omega }_{\text{J2}}/4$, respectively. Inset: close-up around ${\omega }_{\text{T}}$ and ${\omega }_{\text{J2}}$.