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arXiv:1007.5225 · Physical Review B

Metastability in a nanobridge-based hysteretic dc SQUID embedded in a superconducting microwave resonator

Eran Segev, Oren Suchoi, Oleg Shtempluck, Fei Xue, Eyal Buks

Published Mar 17, 2011• 4 citations
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Abstract

We study the metastable response of a highly hysteretic dc superconducting quantum interference device (SQUID) made of a niobium loop interrupted by two nanobridges. We excite the SQUID with an alternating current and with direct magnetic flux, and find different stability zones forming diamond-like structures in the measured voltage across the SQUID. When such a SQUID is embedded in a transmission line resonator, similar diamond structures are observed in the reflection pattern of the resonator. We calculate the dc-SQUID stability diagram in the plane of the exciting control parameters, both analytically and numerically. In addition, we obtain numerical simulations of the SQUID equations of motion, taking into account temperature variations and nonsinusoidal current-phase relation of the nanobridges. Good agreement is found between experimental and theoretical results.

DOI
10.1103/physrevb.83.104507

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We study the metastable response of a highly hysteretic dc superconducting quantum interference device (SQUID) made of a niobium loop interrupted by two nanobridges. We excite the SQUID with an alternating current and with direct magnetic flux, and find different stability zones forming diamond-like structures in the measured voltage across the SQUID. When such a SQUID is embedded in a transmission line resonator, similar diamond structures are observed in the reflection pattern of the resonator. We calculate the dc-SQUID stability diagram in the plane of the exciting control parameters, both analytically and numerically. In addition, we obtain numerical simulations of the SQUID equations of motion, taking into account temperature variations and nonsinusoidal current-phase relation of the nanobridges. Good agreement is found between experimental and theoretical results.
This study examined the voltage response of nanobridge-based direct-current superconducting quantum interference devices (dc-SQUIDs) fabricated on a Si3N4 membrane. Such a configuration may help in reducing 1/f noise, which possibly originates from substrate fluctuating defects. Results showed that the poor thermal coupling between the dc-SQUID and the substrate leads to a strong hysteretic response of the superconducting quantum interference device (SQUID) even though it is biased by an alternating current. In addition, when the dc-SQUID is biased near a threshold of spontaneous oscillations, the measured voltage has an intermittent pattern, which depends on the applied magnetic flux threading the loop of the SQUID.
We have measured the thermally-activated escape rate out of the zero-voltage state of hysteretic dc superconducting quantum interference devices (SQUID's) at 4.2 K. We found evidence that the zero-voltage state of these SQUID's is in general not unique, but can correspond instead to several metastable states. Each of these substates is associated with a given number of flux quanta trapped in the super-conducting loop of the SQUID's. The existence of this multiplicity is a direct consequence of the two-dimensional character of the dc-SQUID dynamics. For a fixed external magnetic flux, each zero-voltage substate is characterized by a particular value of the critical current, the expression of which can be derived from the double-cosine tilted two-dimensional SQUID potential. Experimentally, we observed multiple peaks in the switching-current distribution. Each peak was assigned to a given zero-voltage substate and we measured the dependence of the associated critical currents on the external magnetic flux. A procedure was developed in order to select a given substate among the multiplicity of the zero-voltage state. This allowed a precise measurement of the lifetime of one preselected zero-voltage substate as a function of the barrier height. Good agreement with two-dimensional transition-state theory was obtained.
Conventional rf SQUIDs (superconducting quantum interference devices) consist of a superconducting loop interrupted by one weak link, and when read out via a tank circuit they produce output signals in the order of a few tens of microvolts. When the single weak link is replaced by a dc SQUID (a superconducting loop with two junctions) a so-called double quantization loop SQUID is formed. In this paper it is shown theoretically, and supported experimentally with thin-film devices of three different designs, that the general behaviour can be explained by a simplified picture of the fixed critical current of the single junction of the rf SQUID being replaced by an effective critical current of a dc SQUID loop that is varied by the applied magnetic flux (i.e., we have a rf SQUID with a variable β parameter). A more detailed analysis of the experimental data and the theory of this novel device reveals its interesting nonlinear character and shows the deviations from this simple picture. It is demonstrated experimentally that output signals in the order of ≥1 mV can be obtained with a suitable read-out mode for the tank circuit, a clear advantage when compared to conventional rf SQUIDs. With respect to dc SQUIDs, the coupling to integrated planar input coils will be unproblematic since no microwave resonances should occur in the structures.
We carry out a quantum analysis of a dc superconducting quantum interference device (SQUID) mechanical displacement detector, comprising a SQUID with mechanically compliant loop segment, which is embedded in a microwave transmission line resonator. The SQUID is approximated as a nonlinear current-dependent inductance, inducing an external flux tunable nonlinear Duffing self-interaction term in the microwave resonator mode equation. Motion of the compliant SQUID loop segment is transduced inductively through changes in the external flux threading SQUID loop, giving a ponderomotive radiation pressure-type coupling between the microwave and mechanical resonator modes. Expressions are derived for the detector signal response and noise, and it is found that a soft-spring Duffing self-interaction enables a closer approach to the displacement detection standard quantum limit, as well as cooling closer to the ground state.
In the recent years, nanoscale superconducting quantum interference devices (nano-SQUIDs) played a fundamental role in the study of small spin systems. Nano-SQUIDs typically employ nano-Dayem bridges having dimensions (length LLL and/or width WWW) greater than coherence length $\ensuremath{\xi}$ of the superconducting film. They exhibit characteristics different from those of standard SQUIDs because the current-phase relationship (CPR) is nonsinusoidal, rendering most of the theoretical predictions based on the standard SQUID theory usually unreliable. Here, we present a noise theory of dc nano-SQUIDs based on Dayem nanobridges. We have computed the main characteristics of this quantum device including current-voltage and voltage-magnetic flux characteristics, magnetic flux-to-voltage transfer factor, and spectral densities of voltage and magnetic flux noise for LLL/$\ensuremath{\xi}$ ratios ranging from 1 (sinusoidal limit) to 3.5 (hysteretic limit). The CPRs have been computed by using the theory of Josephson weak links based on the Ginzburg--Landau equation. The results show a dependence of the magnetic flux noise spectral density on (LLL/$\ensuremath{\xi}$)4/3{}^{4/3}4/3 involving a degradation of about a factor of five between the two extreme cases and are consistent with experimentally measured magnetic flux noises reported in the literature. These results provide useful information for both device physics and their applications.
The dynamics and noise of a dc superconducting quantum interference device (SQUID) with the McCumber parameter βc=2πR2IcC/Φ0 close to the unity (where Ic, R, C are the critical current, the shunt resistance, and the capacitance of the Josephson junctions comprising the SQUID, respectively, and Φ0=2.07×10−15 Wb is the magnetic flux quantum) integrated with a planar spiral input coil have been experimentally studied. The length of the spiral input coil was chosen to match its λ/4 microwave resonance frequency to the plasma resonance frequency of the SQUID. The input coil resonance enhances the overall quality factor Q of the Josephson oscillations in the SQUID and, as a result, increases the dynamic resistance Rd and the gradient of the flux-to-voltage characteristics ∂V/∂Φ without hysteresis. This relaxes the tolerance for the βc parameter, simplifies the technological process, and improves the yield of devices. A dc SQUID with loop inductance L=31.4 pH, βc=0.72, and a six turn input coil has demonstrated a nondistorted quasisinusoidal flux-to-voltage transfer function with an exceptionally large modulation depth of approximately 140 μV peak-to-peak. A spectral density of the intrinsic magnetic flux noise as low as 3.5×10−7 Φ0/Hz1/2 has been measured in the double stage configuration at a temperature of 4.2 K using direct read-out electronics. This corresponds to the intrinsic energy resolution of ε=12.5h. In combination with an intermediary transformer, the current resolution of the SQUID is as low as 1.25 pA/Hz1/2 with an input coil inductance of 58 nH. The coupled energy resolution is εc=45h in the white noise region.