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Physical Review B

Noise theory of dc nano-SQUIDs based on Dayem nanobridges

C. Granata, Antonio Vettoliere, M. Russo, B. Ruggiero

Published Dec 27, 2011• 34 citations
Publisher

Abstract

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 $L$ and/or width $W$) 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 $L$/$\ensuremath{\xi}$ 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 ($L$/$\ensuremath{\xi}$)${}^{4/3}$ 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.

DOI
10.1103/physrevb.84.224516

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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 \ensuremathξ\ensuremath{\xi}\ensuremathξ 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/\ensuremathξ\ensuremath{\xi}\ensuremathξ 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/\ensuremathξ\ensuremath{\xi}\ensuremathξ)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.
We have fabricated two types of NbN nanobridge dc Superconducting QUantum Interference Devices (SQUIDs). From the analyses of the response to magnetic field, it is found that the current-phase relationship for the planar-type nanobridge and the nanobridge fabricated by field evaporation is close to sinusoidal. For V\gtrsim720 µV, The spectral densities of the voltage noise show linear dependence on the voltage for the SQUIDs with nanobridges whose length is about the superconducting coherence length of NbN. The SQUIDs with nanobridges with lengths of several times the NbN coherence length exhibit parabolic voltage dependence.
We report on the design, the fabrication and the performance of an integrated magnetic nano-sensor based on niobium dc-SQUID (Superconducting QUantum Interference Device) for nanoscale applications is presented. The nano-sensors are based on nanometric niobium constrictions (Dayem bridges) inserted in a square loop having a side length of 200 nm. Measurements of voltage-flux characteristic, flux to voltage transfer factor and noise performances are reported. In small signal mode, the sensors have shown a magnetic flux noise spectral density of 1.5 μΦ 0 /Hz 1/2 corresponding to a spin sensitivity in unit of Bohr magneton of 60 spin/Hz 1/2 . Supercurrent decay measurements of these devices are also reported. Such measurements provide useful information for applications which employ the SQUID as a trigger where the sensor works on the zero voltage state. The experimental data, have shown an intrinsic current fluctuation less than 0.2% of the critical current at liquid helium temperature, corresponding to an intrinsic sensor magnetic flux resolution of a few mΦ 0 . In view of the nano-SQUID employments in the detection of small spin populations, the authors calculated the spin sensitivity and the magnetic response relative to the single spin, as a function of its position within the SQUID hole. The results show that the SQUID response depends strongly on the spin position.
Nanoscale superconducting quantum interference devices (nano-SQUIDS) where the weak links are made from nanobridges, i.e., nanobridge SQUIDs (NBSs), are one of the most sensitive magnetometers for nanoscale magnetometry. Because of very strong nonlinearity in the nanobridge-electrode joints, the applied magnetic flux (${\mathrm{\ensuremath{\Phi}}}_{a}$)--critical current (Ic{I}_{c}Ic​) characteristics of NBSs differ very significantly from conventional tunnel-junction SQUIDs, especially when the nanobridges are long and/or the screening parameter is large. However, in most of the theoretical descriptions, NBSs have been treated as the conventional tunnel-junction SQUIDs, which are based on the dc Josephson effect. Here, I present a model demonstrating that for long nanobridges and/or large screening parameters the ${I}_{c}({\mathrm{\ensuremath{\Phi}}}_{a})$ of a NBS can be explained by merely considering the fluxoid quantization in the NBS loop and the energy of the NBS; it is not necessary to take the Josephson effect into consideration. I also demonstrate that using the model, one can derive useful expressions such as the modulation depth and transfer function. I discuss the role of the kinetic inductance fraction ($\ensuremath{\kappa}$) in determining ${I}_{c}({\mathrm{\ensuremath{\Phi}}}_{a})$. I compare the predictions of the present model with the experimental data already published by several groups.
Electron-beam lithography has been used to produce ultra low noise Nb dc SQUIDs (Superconducting QUantum Interference Devices) with Josephson elements consisting of either small area tunnel junctions or very narrow variable thickness bridges (nanobridges). Detailed voltage noise and transfer function measurements have been made as a function of temperature and flux and current bias. These measurements allow a computation of the intrinsic energy resolution ε vs bias parameters as well as comparison with models of intrinsic SQUID noise. The best ε obtained was of order Planck's constant h. Moreover, the detailed characteristics of the SQUID voltage noise are in agreement with a small signal analysis that depends only on the current noise in each of the Josephson elements and measurable parameters.
For the fabrication of nanoscale SQUIDs, for example having loop diameters of order 1 µm or smaller, the conventional trilayer window-type Josephson junctions are too large. Instead, we consider the use of nanobridge junctions with dimensions of order 100 nm. The problem of hysteresis in the junctions' current–voltage characteristic (IVC) is avoided by depositing a gold overlayer, which serves as both a thermal and electrical shunt. Noise measurement of dc SQUIDs which incorporate such Nb nanobridges as the Josephson junctions are reported. Devices having different IVCs were fabricated and characterized. The best device has a flux noise of 5 × 10 −6 Φ 0 Hz −1/2 at 4.2 K, and there is no significant increase in the noise level in a static magnetic field up to 2 mT. Techniques for increasing the voltage modulation depth, and hence improving the flux noise performance, are discussed.
A superconducting quantum interference device (SQUID) miniaturized into the nanoscale is promising in the inductive detection of a single electron spin. A nano-SQUID with a strong spin coupling coefficient, a low flux noise, and a wide working magnetic field range is highly desired in a single spin resonance measurement. Nano-SQUIDs with Dayem bridge junctions excel in a high working field range and in the direct coupling from spins to the bridge. However, the common planar structure of nano-SQUIDs is known for problems such as a shallow flux modulation depth and a troublesome hysteresis in current–voltage curves. Here, we developed a fabrication process for creating three-dimensional (3-D) niobium (Nb) nano-SQUIDs with nanobridge junctions that can be tuned independently. Characterization of the device shows up to 45.9% modulation depth with a reversible current–voltage curve. Owning to the large modulation depth, the measured flux noise is as low as 0.34 μΦ 0 /Hz 1/2 . The working field range of the SQUID is greater than 0.5 T parallel to the SQUID plane. We believe that 3-D Nb nano-SQUIDs provide a promising step toward effective single-spin inductive detection.