Charge of a quasiparticle in a superconductor

Nonlinear charge transport in superconductor–insulator–superconductor (SIS) Josephson junctions has a unique signature in the shuttled charge quantum between the two superconductors. In the zero-bias limit Cooper pairs, each with twice the electron charge, carry the Josephson current. An applied bias V_SD leads to multiple Andreev reflections (MAR), which in the limit of weak tunneling probability should lead to integer multiples of the electron charge ne traversing the junction, with n integer larger than 2Δ/eV_SD and Δ the superconducting order parameter. Exceptionally, just above the gap eV_SD ≥ 2Δ, with Andreev reflections suppressed, one would expect the current to be carried by partitioned quasiparticles, each with energy-dependent charge, being a superposition of an electron and a hole. Using shot-noise measurements in an SIS junction induced in an InAs nanowire (with noise proportional to the partitioned charge), we first observed quantization of the partitioned charge q = e*/e = n, with n = 1–4, thus reaffirming the validity of our charge interpretation. Concentrating next on the bias region eV_SD ～ 2Δ, we found a reproducible and clear dip in the extracted charge to q? ～ 0.6, which, after excluding other possibilities, we attribute to the partitioned quasiparticle charge. Such dip is supported by numerical simulations of our SIS structure.Excitations in superconductors (Bogoliubov quasiparticles) can be described according to the Bardeen–Cooper–Schrieffer (BCS) theory (1) as an energy-dependent superposition of an electron with amplitude u(ε), and a hole with amplitude v(ε), where the energy ε is measured relative to the Fermi energy (2). Evidently, the expectation value of the charge operator (applied to the quasiparticle wave function), which we address as the quasiparticle charge e* = q(ε)e, is smaller than the charge of an electron, q(ε) = |u(ε)|² ? |ν(ε)|² (3). Solving the Bogoliubov–de Gennes equations, one finds that |u(ε)|2=1/21+(ε2−Δ2/ε)] and |v(ε)|2=1/21−(ε2−Δ2/ε)], with the expected charge evolving with energy according to q(ε)=ε2−Δ2/ε––vanishing altogether at the superconductor gap edges (3). Note, however, that the quasiparticle wave function is not an eigenfunction of the charge operator (3, 4). Properties of quasiparticles, such as the excitation spectra (5), lifetime (6–10), trapping (11), and capturing by Andreev bound states (12, 13), had already been studied extensively; however, studies of their charge are lagging. In the following we present sensitive shot-noise measurements in a Josephson junction, resulting in a clear observation of the quasiparticle charge being smaller than e, q(eV_SD∼2Δ) < 1, and evolving with energy, as expected from the BCS theory.To observe the BCS quasiparticles in transport we study a superconductor–insulator–superconductor (SIS) Josephson junction in the nonlinear regime. The overlap between the wave functions of the quasiparticles in the source and in the drain is expected to result in a tunneling current of their effective charge. This is in contrast with systems which are incoherent (14, 15) or with an isolated superconducting island, where charge conservation leads to traversal of multiples of e – Coulomb charge (16). As current transport in the nonlinear regime results from “multiple Andreev reflections” (MAR), it is prudent to make our measurements credible by first measuring the charge in this familiar regime.In short, the MAR process, described schematically in Fig. 1, carries a signature of the shuttled charge between the two superconductors (SCs), being a consequence of n traversals through the junction (as electron-like and hole-like quasiparticles), with n an integer larger than 2Δ/eV_SD. A low transmission probability t (via tunneling through a barrier) in the bias range 2Δ/n < eV_SD < 2Δ/(n ? 1) assures dominance of the lowest order MAR process (higher orders are suppressed as tⁿ), with the charge evolving in nearly integer multiples of the electron charge. Although there is already a substantial body of theoretical (3, 17–23) and experimental (24–29) studies of the MAR process, charge determination without adjustable parameters is still missing. An important work by Cron et al. (27) indeed showed a staircase-like behavior of the charge using “metallic break junctions;” however, limited sensitivity and the presence of numerous conductance channels some of which with relatively high transmission probabilities did not allow exact charge quantization. Our shot-noise measurements, performed on a quasi-1D Josephson junction (single-mode nanowire) allowed clear observation of charge quantization without adjustable parameters. To count a few advantages: (i) the transmission of the SIS junction could be accurately controlled using a back-gate; (ii) this, along with our high sensitivity in noise measurements, enabled us to pinch the junction strongly (thus suppressing higher MAR orders); and (iii) with the Fermi level located near the 1D channel van Hove singularity, a rather monoenergetic distribution could be injected (SI Appendix, section S7).Open in a separate windowFig. 1.MAR. Illustrations of the leading processes contributing to the current as function of bias. In general, for 2Δ/(n ? 1) > eV_SD > 2Δ/n the leading charge contribution to the current is ne. An electron-like quasiparticle is denoted by a full circle, whereas a hole-like quasiparticle is denoted by an empty circle. (A) When the bias is larger than the energy gap, eV_SD > 2Δ, the leading process is a single-path tunneling of single quasiparticles from the full states (Left) to the empty states (Right). This current is proportional to the transmission coefficient t. Higher-order MAR process (dashed box), being responsible for tunneling of Cooper pairs, is suppressed as t². (B) For 2Δ > eV_SD > Δ, the main charge contributing to the current is 2e with probability t². (C) For Δ > eV_SD > 2Δ/3, the main charge contributing to the current is 3e with probability t³.