Supercurrent in a Double Quantum Dot

J. C. Estrada Saldaña, A. Vekris, G. Steffensen, R. Žitko, P. Krogstrup, J. Paaske, K. Grove-Rasmussen, and J. Nygård

Phys. Rev. Lett. 121, 257701 – Published 20 December 2018

Abstract

We demonstrate the Josephson effect in a serial double quantum dot defined in a nanowire with epitaxial superconducting leads. The supercurrent stability diagram adopts a honeycomb pattern. We observe sharp discontinuities in the magnitude of the critical current, $I_{c}$ , as a function of dot occupation, related to doublet to singlet ground state transitions. Detuning of the energy levels offers a tuning knob for $I_{c}$ , which attains a maximum at zero detuning. The consistency between experiment and theory indicates that our device is a faithful realization of the two-impurity Anderson model.

Received 22 August 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.257701

Physics Subject Headings (PhySH)

Coulomb blockade Josephson effect Quantum transport

Josephson junctions Nanowires Quantum dots

Anderson impurity model Numerical Renormalization Group

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. C. Estrada Saldaña¹, A. Vekris¹, G. Steffensen¹, R. Žitko^2,3, P. Krogstrup^1,4, J. Paaske¹, K. Grove-Rasmussen¹, and J. Nygård^1,*

¹Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
²Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
³Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
⁴Microsoft Quantum Materials Lab Copenhagen, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark

^*Corresponding author. nygard@nbi.ku.dk

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Vol. 121, Iss. 25 — 21 December 2018

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Images

Figure 1
(a) Schematics of the device. Gates numbered 1 to 5 were used to electrostatically define two quantum dots in the channel of the bare InAs nanowire. A backgate voltage, $V_{b g}$ , was used for further fine-tuning of the coupling of the dots. (b) Scanning-electron micrograph of the measured device. Au contacts, epitaxial Al leads, and InAs nanowire are colored in yellow, blue, and red, respectively. (c) Schematics of two quantum dots in series, whose energy level is tuned by the voltage on gates $V_{g L}$ and $V_{g R}$ (for left and right dot), which in our device correspond to gates 2 and 4. (d) Energy diagram of the system, consisting of two single levels in series with energy levels $ε_{L}$ and $ε_{R}$ , coupled to the closest superconducting lead with tunneling rates $Γ_{L}$ and $Γ_{R}$ , respectively, and with interdot coupling $t_{d}$ . The charging energies, $U_{L} \approx 1.85 meV$ and $U_{R} \approx 1.62 meV$ , are much larger than the superconducting gap, $Δ = 0.265 meV$ . The large level spacings, $Δ E_{L} \approx 1.64 meV$ and $Δ E_{R} \approx 1.76 meV$ , of the order of $U_{L}$ and $U_{R}$ , respectively, make the single-level approximation valid. (e) Charge stability diagram $I_{c} (n_{L}, n_{R})$ calculated through fourth order perturbation theory [23], where $n_{L}$ and $n_{R}$ represent the gate-induced charges in the left and right dots, respectively.
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Figure 2
Charge stability diagram as a function of plunger gate voltages. Effective charge numbers $N_{L}$ , $N_{R}$ determined from the shell-filling pattern (see Ref. [26]) are indicated for each sector. Lines on the diagram indicate line cuts shown in subsequent figures. The gate range in the two axes has been adjusted so distances amount to approximately equal energies. $T = 15 mK$ , $V_{s d} = 0$ , $V_{g 1} = - 9.2 V$ , $V_{g 3} = - 9.3 V$ , $V_{g 5} = - 0.25 V$ , and $V_{b g} = 10.4 V$ .
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Figure 3
(a)–(d) Line cuts along the dashed lines in Fig. 2 for fixed $N_{R}$ (a) and (b) and for fixed $N_{L}$ (c) and (d). Overlays. Fitted $I_{c}$ (white curve) [52]. The notation used to name each sector is $GS (N_{L}, N_{R})$ , where $GS = D$ , S represents the ground state of the sector. (e) and (f) Line cuts of $| I_{c} |$ calculated by fourth order perturbation theory along trajectories in occupation number (proportional to gate voltage) given in Fig. 1. Red (blue) color indicates that $I_{c}$ is positive (negative). Line cut trajectories in Figs. 1 and 2 are symbolically indicated on each panel.
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Figure 4
(a) and (b) Detuning $ε_{L} - ε_{R}$ line cuts along the continuous lines in Fig. 2. Overlays. Fitted $I_{c}$ (white line). Note that the ground state, either doublet or singlet, does not change in these line cuts. (c) and (d) $| I_{c} |$ traces calculated by fourth order perturbation theory. Red (blue) accounts for positive (negative) $I_{c}$ . Green arrows indicate zero-detuning points.
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Figure 5
(a)–(d) Line cuts along the dotted lines in Fig. 2, through (a) and (c) $S_{B}$ and (b) and (d) $D_{B}$ molecular ground states. The levels are aligned per (a) $ε_{L} = ε_{R} + U_{R}$ , (c) $ε_{L} + U_{L} = ε_{R}$ , and (b) and (d) $ε_{L} = ε_{R}$ . The difference is due to an additional electron in one of the levels in (a) and (c). Overlays. Fitted $I_{c}$ (white line) [52]. (e) and (f) Line cuts of calculated $| I_{c} |$ through (e) $S_{B}$ and (f) $D_{B}$ ground states, done by fourth order perturbation theory. As before, red (blue) accounts for positive (negative) $I_{c}$ . Note that line cut (c) covers two additional charge sectors outside the range of the theory cut (e).
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