Dynamical vortex transitions in a gate-tunable two-dimensional Josephson junction array

Editors' Suggestion

Dynamical vortex transitions in a gate-tunable two-dimensional Josephson junction array

C. G. L. Bøttcher, F. Nichele, J. Shabani, C. J. Palmstrøm, and C. M. Marcus

Phys. Rev. B 108, 134517 – Published 27 October 2023

Abstract

We explore vortex dynamics in a two-dimensional Josephson junction array of micron-size superconducting islands fabricated from an epitaxial Al/InAs superconductor-semiconductor heterostructure, with a global top gate controlling Josephson coupling and vortex pinning strength. With applied dc current, minima of differential resistance undergo a transition, becoming local maxima at integer and half-integer flux quanta per plaquette, $f$ . The zero-field transition from the superconducting phase is split, but unsplit for the anomalous metal phase, suggesting that pinned vortices are absent or sparse in the superconducting phase, and abundant but frozen in the anomalous metal. The onset of the transition is symmetric around $f = 1 / 2$ but skewed around $f = 1$ , consistent with a picture of dilute vortices/antivortices on top of a checkerboard $(f = 1 / 2)$ or uniform array of vortices $(f = 1)$ . Transitions show good scaling but with exponents that differ from Mott values obtained earlier. Besides the skewing at $f = 1$ , transitions show an overall even-odd pattern of skewing around integer $f$ values, which we attribute to vortex commensuration in the square array leading to symmetries around half-integer $f$ .

Received 23 December 2022
Accepted 20 September 2023

DOI:https://doi.org/10.1103/PhysRevB.108.134517

Physics Subject Headings (PhySH)

Dynamical phase transitions Superconducting phase transition Vortex dynamics Vortices in superconductors

Heterostructures Josephson junctions

Resistivity measurements

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

C. G. L. Bøttcher ^1,*, F. Nichele^1,†, J. Shabani^2,‡, C. J. Palmstrøm ^2,3,4, and C. M. Marcus¹

¹Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
²California NanoSystems Institute, University of California, Santa Barbara, California 93106, USA
³Department of Electrical Engineering, University of California, Santa Barbara, California 93106, USA
⁴Materials Department, University of California, Santa Barbara, California 93106, USA

^*Present Address: Department of Applied Physics, Yale University, New Haven, Connecticut 06520, USA.
^†Present Address: IBM Research Laboratory, Zürich, Zürich, Switzerland.
^‡Present Address: New York University, New York, New York 10003, USA.

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 108, Iss. 13 — 1 October 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Hybrid array and frustration-dependent dynamical vortex transitions. (a), (b) Schematic of device with $W = 100$ by $L = 400$ epitaxial $a = 1 μ m$ Al squares (gray), separated by $b = 150$ nm strips (blue) of exposed InAs heterostructure, with ac $+$ dc current bias, $I$ , and voltage $V$ measured using side probes. False-color micrograph taken before top gate was deposited. Gate voltage, $V_{g}$ , controls carrier density in InAs strips between Al squares [15]. (c) Sheet resistance, $R_{s} \equiv (W / L) V / I$ , as a function of dc current, $I_{dc}$ , and perpendicular magnetic field, $B_{⊥}$ , shows $R_{s} = 0$ for small $I_{dc}$ with enhanced critical current, $I_{0}$ at $f = 0$ , 1/4, 1/3, 1/2, 2/3, and 1. (d) Line cuts of (c) shows dips in $R_{s}$ at $f = 0$ , 1/2, and 1, going to $R_{s} \sim 0$ for low $I_{dc}$ . (e), (f) Evolution of dips in differential resistance, $d V / d I_{s} \equiv (W / L) d V_{a c} / d I_{ac}$ , as a function of $B_{⊥}$ into a split peak at $f = 0$ , a symmetric peak at $f = 1 / 2$ , and an asymmetric peak at $f = 1$ . Each case is discussed in the text.
Reuse & Permissions
Figure 2
Zero-field dynamical transition from the anomalous metal phase. (a) Sheet resistance, $R_{s}$ , as a function of perpendicular magnetic field, $B_{⊥}$ over a range of dc currents, $I_{dc}$ . For all dc currents, zero field remains a resistance minimum, similar to the superconducting regime, Fig. 1. (b) Dip-to-peak transition at $f = 0$ over the same range of dc currents. This behavior contrasts the superconducting regime in Fig. 1, which shows a split peak at $B_{⊥} = 0$ . The dip-to-peak transition suggests that vortices and antivortices are absent at $f = 0$ in the superconducting phase but present in the anomalous metal phase.
Reuse & Permissions
Figure 3
Dynamical transitions, scaling, and critical exponents. (a), (b) Differential sheet resistance, $d V / d I_{s}$ , shows a dip-to-peak transition is a function of dc current, $I_{dc}$ , at commensurate frustration. ( $f = 1 / 2$ ) and full ( $f = 1$ ). Insets: the transitions on each side, denoted as the left and right branches of $f = 1 / 2$ and $f = 1$ , are represented as superconductor-insulator transitions; down-bending curves are transitions to the pinned vortex state, while upward-bending curves are transitions to a state of vortex flow. The horizontal field-independent curves separating each state are marked as separatrices used in the scaling analysis. (Left: black solid line. Right: black dotted line). Scaling plots of $f = 1 / 2$ (c), (d) and $f = 1$ (e), (f), showing that left and right branches yield different scaling exponents, i.e., there is an asymmetry around each critical frustration field. Exponents extracted for $f = 1 / 2$ are $ε = 2.1$ (left) and 1.2 (right), while at $f = 1$ we extract $ε = 1.5$ (left) and 2.5 (right).
Reuse & Permissions
Figure 4
Even, odd, and zero flux states. (a) Differential sheet resistance curves for different values of dc current as a function of perpendicular magnetic field, $B_{⊥}$ , at gate voltage $V_{g} = - 2.8 V$ , showing the evolution of vortex states for half-integer frustration $f = 1 / 2$ and integers $f = 1, 2, 3,$ and 4. Odd minima (black dots) move down in field with increasing dc current, while even minima (green dots) move up in field. Minima at $f = 1 / 2$ is roughly insensitive to dc current, consistent with the overall symmetry in $f$ of the $f = 1 / 2$ transition. (b) Differential sheet resistance (color) as a function of $B_{⊥}$ and $I_{dc}$ shows commensurate features at factional and integer frustration, $f$ . Bright features at the tips of the zero-resistance spikes are the dip-to-peak signature of the dynamical transition. Note the alternating, even-odd asymmetric bright features at integer $f$ values, the symmetric bright feature at $f = \pm 1 / 2$ , and the vanishing differential sheet resistance remaining at large $I_{dc}$ at $f = 0$ .
Reuse & Permissions
Figure 5
Extracting scaling exponents. (a)–(d) Log-log plots of the slope of the differential resistance, with the separatrix subtracted, constructed to extract scaling exponents, $ɛ$ , separately for left and right branches around $f = 1 / 2$ and $f = 1$ . Means of logs are calculated for each value of $b$ (solid black curve) then means fit to a linear function (dashed green line).
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics