Dissipative phase fluctuations in superconducting wires capacitively coupled to diffusive metals

We study the screening of the Coulomb interaction in a quasi-one-dimensional superconductor by the presence of either a one- or a two-dimensional electron gas nearby. To that end, we derive an effective low-energy phase-only action, which amounts to treating the Coulomb and superconducting correlations in the random-phase approximation. We concentrate on the study of dissipation effects in the superconductor, induced by the effect of Coulomb coupling to the diffusive modes in the electron gas, and study its consequences on the behavior of the one-dimensional plasma mode, and the static and dynamical conductivity. Our results point toward the importance of the dimensionality of the screening metal in the behavior of the superconducting plasma mode of the wire at low energies. In absence of topological defects, and when the screening is given by a one-dimensional electron gas, the superconducting plasma mode is completely damped in the limit $k\to 0$, and consequently superconductivity is lost in the wire. In contrast, we recover a Drude-type response in the conductivity when the screening is provided by a two-dimensional electron gas.

Figure 1

Representation of the capacitively coupled superconducting-wire-normal-metal system. The metal placed at a distance $d$ screens the long-range Coulomb interaction in the superconducting wire. In (a) the metal is a diffusive wire and in (b) we consider a diffusive 2D electron gas.

Figure 2

Equivalent transmission-line circuit representing the superconducting wire capacitively coupled to the metallic environment. The kinetic inductance per unit of length $\Lambda$ represents the dissipationless superconducting nature of the wire while the effective admittance $Z^{-1}\equiv Z^{-1}\left(k_{\parallel },\omega \right)$ encodes all the capacitive and resistive effects arising from the coupling to the environment (cf. Fig. 1).

Figure 3

Screening regimes for a superconducting wire screened by a diffusive 1DEG. The curve $\omega =D{k}_{\parallel }^2$ (dashed line) separates the regime of static screening $\omega ⪡D{k}_{\parallel }^2$ (light gray area) from that of dynamical screening $\omega ⪢D{k}_{\parallel }^2$ (white area). The dispersion relation of the 1D plasma mode (thick solid line) is obtained from the numerical evaluation of Eq. (17), using Eq. (26) and the parameters in Table . Note that the dispersion relation crosses over from the static regime to the dynamical regime and eventually the plasmon mode is completely damped. In the regime of frequencies $\omega ⪢D{k}_{\parallel }^2\frac{2e^2}{{ϵ}_{\text{r}}}{\mathcal{N}}_{n,1\text{D}}^0\text{ln}\frac{2}{\left|k_{\parallel }{r}_0\right|}$ (dark gray area) the unscreened Mooij-Schön plasma mode is recovered (see inset).

Figure 4

(Color online) Real (solid line) and imaginary (dotted line) components of the 1D plasma mode $\omega \left(k_{\parallel }\right)$, obtained by numerical evaluation of the equation of motion Eq. (17), using Eq. (26). The real part (solid line) gives the dispersion relation while the imaginary part (dotted line) represents the damping of the mode. As in Fig. 3, the curves have been calculated for realistic experimental parameters (cf. Table ). The curve $\omega =D{k}_{\parallel }^2$ (dashed line), indicating the crossover between the static and dynamical screening regimes, is shown as a reference.

Figure 5

Screening regimes for a superconducting wire capacitively coupled to a diffusive 2DEG. The dispersion relation (solid line) results from Eq. (41), valid in the regime $\omega \left(k_{\parallel }\right)⪡D{k}_{\text{TF}}\left|k_{\parallel }\right|$. The dashed line $\omega =D{k}_{\text{TF}}\left|k_{\parallel }\right|$ separates the regime of static (light gray area) from that of dynamic (white area) screening. The parameters used in the calculations are shown in Table .

Figure 6

(Color online) Real and imaginary components of the 1D plasma mode $\omega \left(k_{\parallel }\right)$, obtained by numerical evaluation of the equation of motion Eq. (41), valid in the regime $\omega \left(k_{\parallel }\right)⪡D{k}_{\text{TF}}\left|k_{\parallel }\right|$. The blue dashed line $\omega =D{k}_{\text{TF}}{k}_{\parallel }$ separates the regime of static (light gray area) from that of dynamic (white area) screening. The parameters used in the calculations are in Table .

Figure 7

Dynamic conductivity $\text{Re}\sigma \left(k_{\parallel },\omega \right)$ of a superconducting wire dynamically screened by a diffusive 1DEG. The plasma mode, which is better defined at high energy and momentum becomes completely damped in the limit $\left\{\omega ,k_{\parallel }\right\}\to 0$ by the effects of the dissipative environment.

Figure 8

(Color online) dc conductivity $\sigma \left(\omega \right)=\text{Re}\left[\sigma \left(k_{\parallel }\to 0,\omega \right)\right]$ of a superconducting wire screened by a diffusive 1DEG. The values on the axis are normalized to ${\sigma }_{\text{dc}}=\sigma \left(\omega =0\right)={\left(\frac{2e}{c}\right)}^{2}2D{\mathcal{N}}_{n,1\text{D}}^0$ and ${\omega }_0\equiv \frac{{\mathcal{D}}_0}{2{\mathcal{N}}_{n,1\text{D}}^{0}D}$. The plasma mode is completely damped in the limit $k_{\parallel }\to 0$ (cf. Fig. 7) and the wire presents finite resistivity.

Figure 9

Dynamic conductivity $\text{Re}\left[\sigma \left(k_{\parallel },\omega \right)\right]$ of a superconducting wire dynamically screened by a diffusive 2DEG. The plasma mode is worse defined at high energy and momentum but in the limit $\left\{\omega ,k_{\parallel }\right\}\to 0$ the effects of dissipation vanish.