In dc SQUIDs based on sub-micrometer (0.7–0.8 µm) Nb/HfTi/AlOx/Nb tunnel (SIS) Josephson junctions, the low-frequency excess flux noise shows a device-to-device bimodality of the spectral exponent at 4.2 K (α ≈ 1 in some devices, α ≈ 0.6 in others), an anomalous growth upon cooling, and a low-frequency plateau at intermediate temperatures (1–2 K). Storm, Kieler, and Körber (2020), who demonstrated this technology with energy sensitivities down to 0.65 h at 400 mK, stated that whether this behavior is intrinsic to the sub-micrometer SIS fabrication technology remains unclear and that resolving it requires further studies on a larger number of samples. The open problem addressed here is to identify the microscopic fluctuators and dynamics responsible, to account quantitatively for the observed spectra and their temperature dependence, and to determine whether the behavior is intrinsic to the sub-micrometer SIS junction technology. The broader question of low-frequency noise in superconducting quantum devices remains active, with competing candidate mechanisms (two-level fluctuators in the tunnel barrier causing critical-current fluctuations, surface/interface spin clusters, few-fluctuator statistics in nanoscale junction area), none of which has been established for these devices.

Background

A dc SQUID (superconducting quantum interference device) is a superconducting loop interrupted by two Josephson junctions — tunnel contacts between superconductors through a thin barrier, here of the SIS type (superconductor–insulator–superconductor, with Nb electrodes, an AlOx tunnel barrier, and a HfTi interlayer that provides self-shunting). The SQUID responds to magnetic flux threading the loop with a periodic voltage modulation, making it the most sensitive known detector of magnetic flux and current. Its performance is quantified by the energy sensitivity ε=SΦ/(2L)\varepsilon = S_\Phi/(2L), the flux-noise power spectral density SΦS_\Phi per loop inductance LL, commonly expressed in units of Planck's constant hh; thermal noise imposes a theoretical floor εth≈2(1+βL)Φ0kBT/(IcRN)\varepsilon_{th}\approx 2(1+\beta_L)\Phi_0 k_B T/(I_c R_N), where Φ0\Phi_0 is the flux quantum, IcI_c the junction critical current, RNR_N the normal resistance, and βL=2LIc/Φ0\beta_L = 2LI_c/\Phi_0 the screening parameter (the ratio of the loop inductance to the junction characteristic inductance).

Reducing the junction lateral size below one micrometer lowers the junction capacitance, which raises the voltage modulation depth and thereby lowers the achievable ε\varepsilon. Extending the junction technology of PTB (Physikalisch-Technische Bundesanstalt, the German national metrology institute, where these devices were developed) to a sub-micrometer SIS process, Storm, Kieler, and Körber achieved white energy sensitivities of about 5 h at 4.2 K and 0.65 h at 400 mK for uncoupled SQUIDs. However, below the white-noise regime the devices show excess noise that grows towards low frequency, conventionally modeled as a power spectrum SΦ∝1/fαS_\Phi \propto 1/f^{\alpha} with spectral exponent α\alpha (ideal 1/f noise has α=1\alpha = 1).

Low-frequency noise in SQUIDs and Josephson junctions has a long prehistory. In single tunnel junctions and SQUIDs, 1/f critical-current noise is conventionally attributed to atomic-scale two-level fluctuators (two-level systems, TLS — defects that switch between two configurations and modulate the barrier locally) in the amorphous tunnel barrier, a picture dating back to Rogers and Buhrman (1984). Wellstood, Urbina, and Clarke (1987) first reported strongly temperature-dependent low-frequency flux noise in dc SQUIDs below 1 K and left its microscopic origin open; later work by the same group showed the exponent depends on device geometry. At millikelvin temperatures Drung et al. (2011) observed 1/fα1/f^{\alpha}-type excess flux noise with α<1\alpha < 1 in PTB SQUIDs, and similar sub-unity exponents were seen in cross-type sub-micrometer-junction SQUIDs by Schmelz et al. (2017). Kempf, Ferring, and Enss (2016) analyzed 84 superconducting quantum devices below 1 K and found universal phenomenology — noise spectra at different temperatures crossing at a common frequency, and exponents depending systematically on device type and material — which they framed as 'noise engineering' without microscopic identification. On the theoretical side, Davis and Chamberlin (2018) showed that a model of interacting spin clusters at the metal–insulator interface, treated as finite entropy baths, reproduces α<1\alpha < 1 and the spectral-crossing phenomenology at millikelvin temperatures. For planar SNS (superconductor–normal metal–superconductor) sub-micrometer HfTi junctions, Beyer et al. (2015) attributed the excess noise to critical-current fluctuations and identified self-heating as the cause of low-temperature degradation; Morosh et al. (2020) confirmed self-heating-limited white voltage noise in sub-100 nm junctions at 4.2 K.

Against this backdrop, the 2020 sub-micrometer SIS results pose a sharpened, still-unanswered question. At 4.2 K the same fabrication process produced devices falling into two classes: one SQUID chip and the 2.5 µm-junction reference devices showed α≈1\alpha \approx 1, consistent with the classical picture of critical-current fluctuations, while the remaining sub-micrometer devices showed α≈0.6\alpha \approx 0.6; upon cooling from 4.2 K to 0.4 K the excess noise grew and the 1/f-noise corner shifted to higher frequency with α≈0.6\alpha \approx 0.6, but at the intermediate temperatures of 1 K and 2 K a low-frequency plateau emerged in the spectrum. The authors stated that whether this behavior is intrinsic to the fabrication technology remains unclear and that resolving it requires further studies on a larger number of samples.

Problem Statement

Identify the microscopic origin of the low-frequency excess flux noise and its temperature dependence in dc SQUIDs based on sub-micrometer (roughly 0.7–0.8 µm) Nb/HfTi/AlOx/Nb SIS Josephson junctions, as observed by Storm, Kieler, and Körber (2020). A complete answer should (i) specify the fluctuators and dynamics responsible — e.g. two-level fluctuators in the AlOx barrier modulating the critical current, surface or interface spin clusters, few-fluctuator statistics in the nanoscale junction area, self-heating-related effects, or another identified mechanism — and derive their predicted noise spectrum SΦ(f,T)S_\Phi(f,T); (ii) account quantitatively for the device-to-device bimodality of the spectral exponent at 4.2 K (α≈1\alpha\approx 1 versus α≈0.6\alpha\approx 0.6), for the emergence of a low-frequency plateau at the intermediate temperatures 1 K and 2 K between α≈0.6\alpha\approx 0.6 regimes at 0.4 K and 4.2 K, and for the growth of the excess noise with the 1/f-noise corner shifting to higher frequency upon cooling; and (iii) establish, by correlation with junction geometry, barrier properties, and fabrication parameters across a sufficiently large device ensemble, whether this behavior is intrinsic to the sub-micrometer SIS fabrication technology or sample-specific, together with predictions (parameter dependences, spectra, signature statistics such as telegraph-like switching) that are testable in further measurements. Parts (i) and (ii) together constitute a complete answer to the origin question; part (iii) is an additional objective that can be settled only by further measurements on a larger device ensemble, and is to be addressed by specifying the discriminating measurements and the predicted signatures they would test.

References

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