Suppression of Nonequilibrium Quasiparticle Transport in Flat-Band Superconductors

We study nonequilibrium transport through a superconducting flat-band lattice in a two-terminal setup with the Schwinger-Keldysh method. We find that quasiparticle transport is suppressed and coherent pair transport dominates. For superconducting leads, the ac supercurrent overcomes the dc current, which relies on multiple Andreev reflections. With normal-normal and normal-superconducting leads, the Andreev reflection and normal currents vanish. Flat-band superconductivity is, thus, promising not only for high critical temperatures, but also for suppressing unwanted quasiparticle processes.

Figure 1

The model used in the calculations. (a) The two-terminal setup with a piece of sawtooth lattice (middle part $M$) connected to two leads modeled by semi-infinite chains, labeled left ($L$) and right ($R$). The figure also highlights the sawtooth lattice unit cell which has two sites, labeled $A$ and $B$, and the edge sites. The graph corresponds to the tight-binding model hopping amplitudes. Here, $U<0$, $V_g$, and $V_B$ correspond to interaction strength, gate potential, and boundary potential, respectively, in the middle part, and ${\mu }_{L/R}$, ${\mathrm{\Delta }}_{L/R}$, and $T_{L/R}$ are the chemical potential, superconducting order parameter, and temperature of the left and right leads, respectively. (b) Correspondence between the sawtooth lattice and the truncated piece. The flat-band states of the infinite system are also present in the finite size system, since the destructive interference remains. At the edges, the flat-band states are changed to edge states, which we tune to degeneracy with the flat-band states by a boundary potential. The dispersive band corresponds to the dispersive states.

Figure 2

Left lead current $I_L$ through the sawtooth lattice with three unit cells and an additional site between two superconducting leads at a varying gate potential $V_g$ with a constant bias $V$. The parameters are $U=-0.3t_{AA}$, $V=0.5t_{AA}$, $t_{LS}=t_{RS}=5.3t_{AA}$, ${\mathrm{\Delta }}_{L/R}=t_{AA}$, and $T_{L/R}=0$, and the leads are in the wideband limit with $t_{L/R}=30t_{AA}$. The dc current is dominated by the quasiparticle MAR processes, which are finite around the dispersive states located at gate potential $-4<V_g/t_{AA}<0$, where there are current peaks. The quasiparticle current around the flat band, located around $V_g/t_{AA}=2$, vanishes. In strong contrast, the ac Josephson current, that is, the ac sine component of the current at the Josephson frequency $2V$, has, in addition to the peaks at the dispersive states, a prominent peak corresponding to the flat-band states.

Figure 3

Left lead current $I_L$ through the sawtooth lattice with three unit cells and an additional site in an NN and NS lead configuration at a varying gate potential $V_g$ with a constant bias $V$. The dispersive states correspond to peaks in the current. Flat-band states correspond to only very small peaks, in the presence of interactions, in both cases. The parameters are otherwise the same as in Fig. 2, but the order parameters are zero for the normal leads. Coherent pair transport does not contribute to the transport: In the NS configuration, the Andreev reflection is the only means for transport, since the bias $V$ is smaller than the superconducting order parameter amplitude ${\mathrm{\Delta }}_R$, whereas in the NN case the usual transmission and reflection govern the transport. Even though the middle part acquires a finite order parameter due to the proximity effect, the superconducting order parameter phase is constant at the flat-band energy, and, thus, there is no coherent pair current. The small flat-band current contribution is due to the Hartree potential inhomogeneity.