Josephson Junction $\ensuremath{\pi}\text{\ensuremath{-}}0$ Transition Induced by Orbital Hybridization in a Double Quantum Dot

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Josephson Junction $π − 0$ Transition Induced by Orbital Hybridization in a Double Quantum Dot

Rousan Debbarma, Athanasios Tsintzis, Markus Aspegren, Rubén Seoane Souto, Sebastian Lehmann, Kimberly Dick, Martin Leijnse, and Claes Thelander

Phys. Rev. Lett. 131, 256001 – Published 21 December 2023

Abstract

In this Letter, we manipulate the phase shift of a Josephson junction using a parallel double quantum dot (QD). By employing a superconducting quantum interference device, we determine how orbital hybridization and detuning affect the current-phase relation in the Coulomb blockade regime. For weak hybridization between the QDs, we find $π$ junction characteristics if at least one QD has an unpaired electron. Notably the critical current is higher when both QDs have an odd electron occupation. By increasing the inter-QD hybridization the critical current is reduced, until eventually a $π − 0$ transition occurs. A similar transition appears when detuning the QD levels at finite hybridization. Based on a zero-bandwidth model, we argue that both cases of phase-shift transitions can be understood considering an increased weight of states with a double occupancy in the ground state and with the Cooper pair transport dominated by local Andreev reflection.

Received 6 July 2023
Accepted 20 November 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.256001

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Coulomb blockade Josephson effect Quantum transport

Double quantum dots Josephson junctions Nanowires SQUID

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Rousan Debbarma ^1,*, Athanasios Tsintzis ¹, Markus Aspegren¹, Rubén Seoane Souto ^3,4, Sebastian Lehmann¹, Kimberly Dick^1,2, Martin Leijnse¹, and Claes Thelander ^1,†

¹Division of Solid State Physics and NanoLund, Lund University, S-221 00 Lund, Sweden
²Center for Analysis and Synthesis, Lund University, S-221 00 Lund, Sweden
³Departamento de Física Teórica de la Materia Condensada, Condensed Matter Physics Center (IFIMAC) and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain
⁴Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC), Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain

^*Corresponding author: rousan@chemical.iitd.ac.in Present address: Department of Chemical Engineering, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, Delhi 110016, India.
^†Corresponding author: claes.thelander@ftf.lth.se

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Vol. 131, Iss. 25 — 22 December 2023

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Figure 1
(a) SEM image of the nanowire. (b) SQUID with two JJs, where one contains a nanowire section with two wurtzite (WZ) barrriers (labelled DQD) and the other a zinc blende (ZB) segment (labelled REF) of the same nanowire. The magnetic field, $B$ , is applied perpendicular to the plane of the device for CPR measurements. (c) Schematic circuit diagram of the four-probe setup used for supercurrent measurements. (d) Schematics of the nanowire device and the conduction band energies of the crystal phases. (e) Cross-section schematic of the nanowire device showing the top-gate and back-gate controlled DQD. (f) Charge stability diagram as a function of the DQD top-gate ( $V_{TG}$ ) and back gate ( $V_{BG}$ ) for a source-drain voltage ( $V_{SD}$ ) of 0.43 mV, showing three different DQD crossings. Orbital crossings I, II, III occur when an orbital with strong electrostatic coupling to the top-gate interacts with orbitals with strong coupling to the back gate. The reference channel is in the off state ( $V_{TG - ref} < 0$ ).
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Figure 2
(a) Differential resistance ( $d V / d I$ ) across the SQUID for a spin- $1 / 2$ state (* in panel (c)) as a function of bias current and $B$ -field. The switching current ( $I_{SW}$ ) is given by the boundary of the white zero-resistance supercurrent region. The red curve is a sinusoidal fit for the extracted $I_{SW}$ . (b) $I_{SW}$ and the SQUID equation fit for spin-0 (blue) and spin- $1 / 2$ (red) states in orbital-1 measured by alternating $V_{TG}$ for each $B$ . The spin states are away from the DQD crossings ( $V_{BG} = 0 V$ in Fig. 1). (c) Charge stability diagram for the weakly coupled orbital crossing I at $V_{SD} = 0.2 3 mV$ . The numbers in parentheses indicate the charge states of the DQD. (d) CPR for various points inside the (1,1) honeycomb (positions indicated by the symbols in panel c). (e) Extracted $I_{SW}$ along the red vector in panel c with the reference channel in off-state ( $V_{TG - ref} < 0$ ).
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Figure 3
(a) Charge stability diagram for orbital crossing II with intermediate interdot tunnel coupling at $V_{SD} = 0.2 3 mV$ . (b) CPR for various points inside the (1,1) honeycomb showing the dependence of the phase-shift on level detuning. (c) Charge stability diagram for the strongly coupled orbital crossing III ( $V_{SD} = 0.2 3 mV$ ) and (d) CPR at the center of the (1,1) honeycomb with no phase-shift.
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Figure 4
(a) ZBW model calculation of Josephson current $(I_{J}^{π / 2})$ as a function of QD level energies for $t = 0$ showing higher negative $I_{J}^{π / 2}$ inside the (1,1) honeycomb compared to (1,0) and (1,2). (b) $I_{J}^{π / 2}$ plot for $t = 0.2 E_{C}$ showing a negative value with low magnitude at the center of the honeycomb. (c) Schematics showing two separate modes in the proximitized SC and local tunneling of Cooper pairs. (d) $I_{J}^{π / 2}$ as a function of $δ$ and $t$ (dotted line in panel a). (e) $I_{J}^{π / 2}$ and double occupancy probability as a function of $t$ at the center of the (1,1) honeycomb for $δ = 0$ . The increasing double occupancy weight leads to a $π − 0$ transition. (f) CPR at the center of the honeycomb for orbital crossings I and II after the subtraction of reference segment $I_{SW}$ .
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Josephson Junction π−0 Transition Induced by Orbital Hybridization in a Double Quantum Dot

Rousan Debbarma, Athanasios Tsintzis, Markus Aspegren, Rubén Seoane Souto, Sebastian Lehmann, Kimberly Dick, Martin Leijnse, and Claes Thelander

Phys. Rev. Lett. 131, 256001 – Published 21 December 2023

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Josephson Junction $π − 0$ Transition Induced by Orbital Hybridization in a Double Quantum Dot