Metastable states and hidden phase slips in nanobridge SQUIDs

Lukas Nulens, Heleen Dausy, Michał J. Wyszyński, Bart Raes, Margriet J. Van Bael, Milorad V. Milošević, and Joris Van de Vondel

Phys. Rev. B 106, 134518 – Published 31 October 2022

Abstract

We fabricated an asymmetric nanoscale SQUID consisting of one nanobridge weak link and one Dayem bridge weak link. The current phase relation of these particular weak links is characterized by multivaluedness and linearity. While the latter is responsible for a particular magnetic field dependence of the critical current (so-called vorticity diamonds), the former enables the possibility of different vorticity states (phase winding numbers) existing at one magnetic field value. In experiments the observed critical current value is stochastic in nature, does not necessarily coincide with the current associated with the lowest energy state and critically depends on the measurement conditions. In this paper, we unravel the origin of the observed metastability as a result of the phase dynamics happening during the freezing process and while sweeping the current. Moreover, we employ special measurement protocols to prepare the desired vorticity state and identify the (hidden) phase slip dynamics ruling the detected state of these nanodevices. In order to gain insights into the dynamics of the condensate and, more specifically the hidden phase slips, we performed time-dependent Ginzburg-Landau simulations.

Received 29 April 2022
Revised 24 August 2022
Accepted 3 October 2022

DOI:https://doi.org/10.1103/PhysRevB.106.134518

Physics Subject Headings (PhySH)

Phase slips Quantum information processing Quantum sensing

Nanostructures SQUID Weak links

Time-dependent Ginzburg-Landau theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Lukas Nulens ^1,*, Heleen Dausy ¹, Michał J. Wyszyński ², Bart Raes ¹, Margriet J. Van Bael ¹, Milorad V. Milošević ², and Joris Van de Vondel ¹

¹Quantum Solid-State Physics, Department of Physics and Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
²Department of Physics, University of Antwerp, Groenenborgerlaan 171, B-2020 Antwerp, Belgium

^*Lukas.nulens@kuleuven.be

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Vol. 106, Iss. 13 — 1 October 2022

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Images

Figure 1
(a) The critical currents as a function of the applied magnetic field. The measured critical currents for positive (negative) bias current are shown in blue (red). The dashed lines represent the vorticity diamonds generated by the model discussed in the text. The vorticity number is indicated by a number in a rounded square. The $n_{v} = 0$ diamond is indicated by a light gray fill and a solid edge. The $n_{v} = 0$ unique vorticity diamond (UVD) has a dark gray fill and corresponds with the phase space where only the $n_{v} = 0$ state exists. (b) False colored scanning electron microscopy (SEM) image of a prototypical NBSQUID device as studied here. The top junction is a Dayem bridge, while the bottom junction indicated in green is a nanobridge with length $L$ and width $W$ . The white scale bar represents 200 nm. When measuring flux oscillations, the applied field $B$ is oriented as shown. [(c),(d)] A typical IV-curve at zero applied field for positive (negative) bias current sweep direction is shown in blue (red). The black dot corresponds with the critical current value shown in panel (a). (e) The energy of the different vorticity states of the SQUID under investigation as a function of the applied field (or the equivalent applied flux) at zero bias current. The size of the colored dots represents the probability to measure the critical current corresponding to the indicated vorticity state for positive bias currents, extracted from the experimental data in panel (a)
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Figure 2
(a) The time evolution of the applied current during one cycle of the preparation procedure under zero magnetic field. The current range between $I_{c 0}$ and $I_{c 1}$ , indicated in gray, corresponds to the current range of the $n_{v} = 0$ UVD. The retrapping current $I_{r}$ is field independent and approximately $\sim$ 21.5 $μ A$ . (b) The energy stored in the NBSQUID for vorticity states 0 (blue) and 1 (green) during one cycle of the preparation procedure according to Eq. (4). The probability to measure these specific vorticity states is denoted by $P_{0}$ and $P_{1}$ . At $I_{c 1}$ , vorticity state 1 ceases to exist and the SQUID switches to the normal state. As the applied current is decreased below the retrapping current the SQUID is frozen into vorticity state 0(1) with probability $P_{0 (1)}$ . The inset shows a zoom of the energy around the UVD current range. (c) The critical currents versus the applied magnetic field after first preparing the NBSQUID in vorticity state $n_{v} = 0$ at $B = 0$ mT. The dotted lines represent the vorticity diamonds generated by the model outlined in Sec. 2. At each field value, we collected seven critical currents. The $n_{v} = 0$ UVD is indicated as a shaded gray fill. The shaded areas indicate the field values where the critical current is multivalued. (d) The energy of the different vorticity states of the NBSQUID as a function of the applied field (or equivalently, flux) at zero bias current. The size of the colored dots represents the probability to measure the critical current corresponding to the indicated vorticity state, extracted from the experimental data in panel (c).
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Figure 3
(a) The critical currents vs the applied magnetic field after preparing the NBSQUID in vorticity state $n_{v} = 0$ are indicated in blue. The open-black circles with error bar correspond to the experimentally obtained edge of the $n_{v} = 0$ vorticity state using the measurement protocol as indicated by the black arrows and described in Sec. 4. (b) The probability to end up in vorticity state $n_{v} = 0$ after sweeping to a particular $I^{*}$ and $B^{*}$ . The curves correspond to bias currents $I^{*}$ ranging from $5 μ A$ to $75 μ A$ . The field locations of the sudden jumps from $P_{0} = 1$ to $P_{0} \neq 1$ for a specific $I^{*}$ correspond to the right edge of the $n_{v} = 0$ vorticity diamond in panel (a). (c) The time evolution of the applied current (black) and field (red) during the measurement protocol as described in Sec. 4, for the case $B^{*} > B_{c 0}$ . The gray area denotes the current range of the $n_{v} = 0$ UVD at $B$ = 0 mT. (d) The energy stored in the NBSQUID as a function of the time for the vorticity states $n_{v} = 0$ , 1, 2, and 3 after initializing in $n_{v} = 0$ and during the measurement protocol outlined in Sec. 4, for $I^{*} = 50 μ A$ and $B^{*} = 6$ mT $> B_{c 0}$ . As $B^{*} = 6$ mT exceeds $B_{c 0}$ the NBSQUID's vorticity is altered by the means of a $n^{'} \times 2 π$ phase slip in the Dayem bridge SQUID arm. During readout, some of these vorticity states $n^{'}$ cease to exist and the system switches to another vorticity state by a $n^{″} \times 2 π$ phase slip event in the nanobridge SQUID arm. (e) The energy stored in the NBSQUID as a function of the time for the vorticity states $n_{v} = 0$ , 1, and 2 after initializing in $n_{v} = 0$ and during the measurement protocol outlined in Sec. 4, for $I^{*} = 50 μ A$ and $B^{*} < B_{c 0}$ . As $B^{*}$ never exceeds the critical field value $B_{c 0}$ of vorticity state $n_{v} = 0$ , the system remains in vorticity state $n_{v} = 0$ .
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Figure 4
(a) The geometry of the simulated device and the orientation of the applied magnetic field (B) and current density ( $J_{ext}$ ). (b) Simulated critical currents as a function of the applied magnetic field (red dots). The surface area of each dot is proportional to the percentage of current sweeps at a given magnetic field for which this particular critical current value was obtained. The solid-blue lines represent vorticity diamonds from the simulation data. Increasing the width of the nanobridge by $\sim 10 %$ shifts the diamonds to a closer agreement with experimental data (dashed lines). [(c),(d)] Selected current-voltage characteristics (IVs) at the field marked by the dashed black line in panel (b) (–0.88 mT). Panel (c) shows two IVs without hidden phase slips, starting from vorticity states $n_{v} = - 1$ and $n_{v} = 0$ . While in panel (d), starting from initial vorticity state $n_{v} = 1$ , two hidden phase slip events occur. The spikes related to hidden phase slips HPS1 and HPS2 lead to the vorticity states indicated in green. The transition to the normal state (N) is preceded by a running phase slip (PS). (e) Cooper-pair densities of the states marked with red dots in panel (d). HPS1: Three antivortices entering via HPS in the bottom bridge. HPS2: Two antivortices leaving via HPS in the top bridge. PS: A continuously running phase slip, just before the transition to the normal state (N).
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