Joule spectroscopy of hybrid superconductor-semiconductor nanodevices

Abstract

Hybrid superconductor-semiconductor devices offer highly tunable platforms, potentially suitable for quantum technology applications, that have been intensively studied in the past decade. Here we establish that measurements of the superconductor-to-normal transition originating from Joule heating provide a powerful spectroscopical tool to characterize such hybrid devices. Concretely, we apply this technique to junctions in full-shell Al-InAs nanowires in the Little-Parks regime and obtain detailed information of each lead independently and in a single measurement, including differences in the superconducting coherence lengths of the leads, inhomogeneous covering of the epitaxial shell, and the inverse superconducting proximity effect; all-in-all constituting a unique fingerprint of each device with applications in the interpretation of low-bias data, the optimization of device geometries, and the uncovering of disorder in these systems. Besides the practical uses, our work also underscores the importance of heating in hybrid devices, an effect that is often overlooked.

Fig. 1. Principle of Joule spectroscopy.

a Schematics of the device geometry. A Josephson junction is formed by etching a 200-nm segment of a full-shell Al-InAs nanowire (NW). Voltage applied to a side gate, V_g, tunes the junction resistance, R_J. The balance between the Joule heat dissipated at the nanowire junction (equal to the product of the voltage, V, and current, I) and the cooling power from the superconducting leads 1 and 2 (P₁ and P₂) results in a temperature gradient along the device, T(x). At a critical value of Joule dissipation, the temperature of the leads, T_0,1 and T_0,2, exceed the superconducting critical temperature and the leads turn normal. Each lead can display different superconducting gaps Δ₁ and Δ₂. An external magnetic field, B, is applied with an angle θ to the NW axis. T_bath is the cryostat temperature. b I (solid black line) and differential conductance, dI/dV (solid blue line), as a function of V measured at V_g = 80 V in device A. For V < 2Δ/e, transport is dominated by Josephson and Andreev processes. By extrapolating the I–V curve just above V = 2Δ/e, an excess current of I_exs ≈ 200 nA is estimated (dashed black line). Upon further increasing V, the Joule-mediated transition of the superconducting leads to the normal state manifests as two dI/dV dips (V_dip,1 and V_dip,2). These transitions fully suppress I_exs (dashed red line). c The nanowire is modeled as a quasi-ballistic conductor with N conduction channels with transmissions τ. We assume that the energy of the quasiparticles injected in the superconductors is fully converted into heat. d Keldysh-Floquet calculations of I(V) and dI/dV(V) using device A parameters (see Supplementary information for more information), reproducing the main features in (b).

Fig. 2. Characterization of the superconductor-to-normal metal transition of the epitaxial Al leads.

a Gate voltage dependence of the dI/dV for device A. The data is plotted both as a function of V (top panel) and of I (bottom panel). Enhanced dI/dV features at low V and I can be attributed to Josephson and Andreev processes. Two dI/dV dips, which signal the superconductor-to-normal metal transition of the leads, can be identified in each of the panels (V_dip,i and I_dip,i). The presence of the two dips is shown in greater detail in the inset of the top panel. The white and red dashed lines are fits to Eq. (3) with a single free fitting parameter per lead (R_lead,1 and R_lead,2). b dI/dV as a function of V and of T_bath. A faint dip at V_dip,lith is attributed to the Ti/Al contacts to the NW. c P_dip,1 = V_dip,1I_dip,1/2 (blue squares) and P_dip,2 = V_dip,2I_dip,2/2 (yellow squares) as a function of T_bath. The solid lines are fits to the power law in Eq. (4), yielding an exponent γ = 3.4.

Fig. 3. Joule effect as a spectroscopical tool.

a Oscillations of V_dip,1 and V_dip,2 with applied magnetic field, which result from the modulation of T_c,i by the Little Parks (LP) effect. The dashed lines are fits to the Abrikosov–Gor'kov (AG) theory, from which we conclude that the primary cause for the different LP oscillations are differences in the superconducting coherence lengths of the leads. b Keldysh–Floquet calculations of the Andreev conductance at low V and of the dI/dV dips at high V as a function of B using device A parameters (Supplementary information to: Joule spectroscopy of hybrid superconductor-semiconductor nanodevices (2022).), capturing the main experimental observations. Panels (c) and (d) demonstrate the spectroscopical potential of the technique. c Zero-bias dV/dI normalized by the normal state resistance of the device. The dashed lines correspond to T_c,i(B) calculated with the AG parameters extracted by fitting the dips in panel (a). d Low-V transport characterization of device A as a function of B. The dashed lines show the spectral gaps, Ω₁(B)/e (white) and Ω₂(B)/e (green), and their sum, (Ω₁(B) + Ω₂(B))/e (black), obtained from V_dip,i(B).

Fig. 4. Application of Joule spectroscopy to different NW devices.

a Low-bias transport characterization of device B as a function of magnetic field. Dashed lines show fittings of the spectral gaps, Ω₁(B)/e (white) and Ω₂(B)/e (green), and their sum, (Ω₁(B) + Ω₂(B))/e (black), obtained from V_dip,i(B). b Joule spectroscopy as a function of B clearly identifies that one of the superconducting leads is not doubly-connected, i.e., it behaves as a partial-shell lead. Dashed lines are fits to the AG theory. c Schematics of device B, as concluded from the Joule spectroscopy characterization (not to scale). d (dV/dI)/R_n as a function of T and B for device C. The dashed lines correspond to T_c,i obtained from T_c,i(B = 0) and the AG fits to V_dip,i(B) (not shown, see SI ). e T-dependence of V_dip,1 and V_dip,2 in device C. Lead 1 displays a lower critical temperature owing to its closer proximity to the lithographic Cr/Au contacts, as depicted in the schematics in panel f (not to scale).