Measurements of persistent currents (flux-periodic equilibrium currents) in individual diffusive normal-metal rings by scanning SQUID magnetometry agreed with theory for the typical h/eh/e-periodic current, but were limited by unexplained spin-related backgrounds: a ubiquitous zero-field steplike nonlinear anomaly of unknown microscopic origin, a paramagnetic defect-spin population of areal density about 4×10174\times 10^{17} spins/m2^2 whose coupling to electrons and to the nonlinear response was uncharacterized, and statistics too small to extract ensemble-averaged currents. The problem is to identify the microscopic origin of these spin backgrounds and model them quantitatively, so that genuine equilibrium persistent currents — including ensemble averages such as ⟨Ih/2e⟩\langle I_{h/2e}\rangle — can be separated from spin-induced artifacts in single-ring magnetometry at low magnetic field.

Background

A conducting ring threaded by a magnetic flux Φa\Phi_a carries a persistent current: because the flux enters the electron wave functions as a phase factor in the boundary condition, the free energy and its derivative I=−∂F/∂ΦaI = -\partial F/\partial \Phi_a oscillate with period h/eh/e in the flux (hh is Planck's constant, ee the electron charge). Elastic disorder does not destroy the effect. In a diffusive ring, where the mean free path is shorter than the circumference LL, the typical (root-mean-square) first-harmonic current is ⟨Ih/e2⟩1/2=(Ec/ϕ0)e−kBT/Ec\langle I_{h/e}^2\rangle^{1/2} = (E_c/\phi_0)e^{-k_B T/E_c} with the correlation energy Ec=ℏπ2D/L2E_c = \hbar\pi^2 D/L^2 (DD is the diffusion constant, ϕ0=h/e\phi_0 = h/e). Because the sign of this h/eh/e harmonic fluctuates randomly between nominally identical rings, its ensemble average ⟨Ih/e⟩\langle I_{h/e}\rangle vanishes, apart from a small predicted contribution that survives disorder averaging but oscillates with kFLk_F L and is usually negligible for metallic rings; the second harmonic, of period h/2eh/2e, can instead have a nonzero ensemble average ⟨Ih/2e⟩\langle I_{h/2e}\rangle due to electron-electron interactions and to canonical versus grand-canonical ensemble differences.

Experiments have long been in tension with this theory. Ensemble measurements on copper and silver rings (Lévy et al. 1990; Jariwala et al. 2001; Deblock et al. 2002) found average h/2eh/2e currents generally a factor of a few larger than most theoretical predictions, while the single diffusive gold ring measured by Chandrasekhar et al. (1991) showed a h/eh/e response 10–200 times larger than predicted; a ballistic GaAs-AlGaAs ring (Mailly et al. 1993) agreed with theory. Bluhm, Koshnick, Bert, Huber, and Moler (Phys. Rev. Lett. 102, 136802 (2009)) measured the magnetic response of 33 individual diffusive gold rings, one ring at a time, using a scanning SQUID susceptometer — a superconducting quantum interference device pickup loop with a concentric field coil that is scanned over the sample, allowing in situ measurement of the sensor background. The amplitude distribution and temperature dependence of the h/eh/e-periodic component agreed with theory, resolving the contradiction with the earlier single-ring result. However, the raw response of nearly every ring was dominated by: (i) a paramagnetic linear susceptibility corresponding, assuming spin 1/2, to an areal spin density of about 4×10174\times 10^{17} spins/m2^2; and (ii) an unexpected, poorly understood steplike anomaly near zero applied field, whose frequency dependence (the response changed when the field sweep frequency was varied between 13 and 333 Hz) suggested nonequilibrium effects in the spin response, and which could mask a persistent-current-like signal. Because a mean background had to be subtracted from each ring's data and only a few tens of rings were measured, no ensemble-averaged quantity could be extracted, and several rings were excluded for poor reproducibility or anomalously large zero-field features.

A companion study by the same group (Bluhm et al., Phys. Rev. Lett. 103, 026805 (2009)) showed that a similar paramagnetic, roughly 1/T1/T susceptibility appears even in oxide-free gold and in insulating films, with an out-of-phase component implying 1/f1/f magnetic noise through the fluctuation-dissipation theorem; the inferred density of about 0.4 spins/nm2^2 matches the spin density needed to explain the low-frequency flux noise that limits superconducting quantum circuits (Koch et al. 2007; Sendelbach et al. 2008). These 'dilute surface spins' proved to be weakly coupled to conduction electrons, which explains why they produce little electron dephasing; but their microscopic identity, and the mechanism of the nonlinear zero-field response, remained unexplained. Later single-ring and array measurements with micromechanical cantilever magnetometry on aluminum rings (Bleszynski-Jayich et al. 2009; Castellanos-Beltran et al. 2013) confirmed the independent-electron theory for the typical current and measured the full current distribution, but only at multi-tesla fields where time-reversal symmetry is broken and the interaction-sensitive average current is suppressed. The open question left by the gold-ring measurements is therefore: what are the defect spins and the zero-field anomaly in these structures, and how can the equilibrium persistent current be separated from them?

Problem Statement

Resolve the spin-related backgrounds that the scanning-SQUID measurements of Bluhm et al. (Phys. Rev. Lett. 102, 136802 (2009)) left unexplained, so that genuine equilibrium persistent currents in individual diffusive normal-metal rings can be separated from spin-induced artifacts in single-ring magnetometry at low magnetic field. The open objectives concern the two backgrounds that dominated the raw response of nearly every ring — a poorly understood zero-field anomaly and a paramagnetic defect-spin population of unknown identity — together with the separation their resolution enables. Objective 1 owns the phenomenology and microscopic origin of the observed zero-field step (its field, frequency, and temperature dependence), while objective 2 owns the defect population itself and its driven dynamics under the ac field and the SQUID back-action. All three objectives listed below are jointly required — modeling the two backgrounds (objectives 1 and 2) is necessary but not sufficient, and a complete resolution must also specify how the equilibrium persistent-current signal is separated from the modeled backgrounds (objective 3):

  1. Origin of the zero-field anomaly. Identify the microscopic origin of the ubiquitous zero-field, steplike nonlinear anomaly observed in nearly all rings, and provide a quantitative model of it. The anomaly was reported with the following observations: the step near zero applied field; the change of the step feature when the field sweep frequency is varied between 13 and 333 Hz, with little effect on the oscillatory component; the response at temperatures below about 500 mK; and the different behavior of isolated versus heatsunk rings (radii from 0.57 to 1 µm). Its frequency dependence suggests nonequilibrium effects in the spin response, and it might mask a persistent-current-like effect, but its microscopic mechanism has not been established.

  2. Nature of the paramagnetic spins. Identify the spin population responsible for the linear susceptibility (areal density about 4×10174\times 10^{17} spins/m2^2 assuming spin 1/2), determine its coupling to conduction electrons (spin relaxation, electron dephasing, and why the measured dephasing bound is not violated), and account for its nonequilibrium nonlinear response under the ac drive and the SQUID's own back-action (the sensor's pickup loop carries a 10 µA, 10 GHz Josephson current that heats isolated-ring electrons, whose temperature saturates near 150 mK).

  3. Equilibrium signal separation. With these backgrounds understood, determine how the persistent-current content of single-ring measurements can be separated from the modeled spin backgrounds at low magnetic field, where time-reversal symmetry holds and interaction effects on the average current are not suppressed — including the ensemble-averaged quantities such as ⟨Ih/2e⟩\langle I_{h/2e}\rangle that such a separation would make accessible. The source experiment could not perform this separation, because a mean background had to be subtracted from each ring's data and only a few tens of rings were measured.

References

  1. Hendrik Bluhm, Nicholas C. Koshnick, Julie A. Bert, Martin E. Huber, Kathryn A. Moler, "Persistent Currents in Normal Metal Rings", Phys. Rev. Lett. 102, 136802 (2009). DOI: 10.1103/PhysRevLett.102.136802, https://doi.org/10.1103/PhysRevLett.102.136802; arXiv:0810.4384, https://arxiv.org/abs/0810.4384
  2. Hendrik Bluhm, Julie A. Bert, Nicholas C. Koshnick, Martin E. Huber, Kathryn A. Moler, "Spinlike Susceptibility of Metallic and Insulating Thin Films at Low Temperature", Phys. Rev. Lett. 103, 026805 (2009). DOI: 10.1103/PhysRevLett.103.026805, https://doi.org/10.1103/PhysRevLett.103.026805; arXiv:0903.3748, https://arxiv.org/abs/0903.3748
  3. A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf, E. Ginossar, F. von Oppen, L. Glazman, J. G. E. Harris, "Persistent Currents in Normal Metal Rings", Science 326, 272 (2009). DOI: 10.1126/science.1178139, https://doi.org/10.1126/science.1178139; arXiv:0906.4780, https://arxiv.org/abs/0906.4780
  4. M. A. Castellanos-Beltran, D. Q. Ngo, W. E. Shanks, A. B. Jayich, J. G. E. Harris, "Measurement of the Full Distribution of Persistent Current in Normal-Metal Rings", Phys. Rev. Lett. 110, 156801 (2013). DOI: 10.1103/PhysRevLett.110.156801, https://doi.org/10.1103/PhysRevLett.110.156801; arXiv:1204.3821, https://arxiv.org/abs/1204.3821
  5. Germain M. Souche, Julien Huillery, H. Pothier, Philippe Gandit, Jérôme I. Mars, Sergey E. Skipetrov, Olivier Bourgeois, "Searching for thermal signatures of persistent currents in normal-metal rings", Phys. Rev. B 87, 115120 (2013). DOI: 10.1103/PhysRevB.87.115120, https://doi.org/10.1103/PhysRevB.87.115120; arXiv:1301.4401, https://arxiv.org/abs/1301.4401
  6. S. E. de Graaf, A. A. Adamyan, T. Lindström, D. Erts, S. E. Kubatkin, A. Ya. Tzalenchuk, A. V. Danilov, "Direct Identification of Dilute Surface Spins on Al2O3: Origin of Flux Noise in Quantum Circuits", Phys. Rev. Lett. 118, 057703 (2017). DOI: 10.1103/PhysRevLett.118.057703, https://doi.org/10.1103/PhysRevLett.118.057703; arXiv:1609.04562, https://arxiv.org/abs/1609.04562
  7. Hui Wang, Chuntai Shi, Jun Hu, Sungho Han, Clare C. Yu, R. Q. Wu, "Candidate Source of Flux Noise in SQUIDs: Adsorbed Oxygen Molecules", Phys. Rev. Lett. 115, 077002 (2015). DOI: 10.1103/PhysRevLett.115.077002, https://doi.org/10.1103/PhysRevLett.115.077002; arXiv:1504.08075, https://arxiv.org/abs/1504.08075
  8. Zhe Wang, Hui Wang, Clare C. Yu, R. Q. Wu, "Hydrogen as a source of flux noise in SQUIDs", Phys. Rev. B 98, 020403(R) (2018). DOI: 10.1103/PhysRevB.98.020403, https://doi.org/10.1103/PhysRevB.98.020403; arXiv:1709.06208, https://arxiv.org/abs/1709.06208
  9. S. Sendelbach, D. Hover, A. Kittel, M. Mück, John M. Martinis, R. McDermott, "Magnetism in SQUIDs at Millikelvin Temperatures", Phys. Rev. Lett. 100, 227006 (2008). DOI: 10.1103/PhysRevLett.100.227006, https://doi.org/10.1103/PhysRevLett.100.227006
  10. Roger H. Koch, David P. DiVincenzo, John Clarke, "Model for 1/f Flux Noise in SQUIDs and Qubits", Phys. Rev. Lett. 98, 267003 (2007). DOI: 10.1103/PhysRevLett.98.267003, https://doi.org/10.1103/PhysRevLett.98.267003
  11. L. P. Lévy, G. Dolan, J. Dunsmuir, H. Bouchiat, "Magnetization of mesoscopic copper rings: Evidence for persistent currents", Phys. Rev. Lett. 64, 2074 (1990). DOI: 10.1103/PhysRevLett.64.2074, https://doi.org/10.1103/PhysRevLett.64.2074
  12. V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gallagher, A. Kleinsasser, "Magnetic response of a single, isolated gold loop", Phys. Rev. Lett. 67, 3578 (1991). DOI: 10.1103/PhysRevLett.67.3578, https://doi.org/10.1103/PhysRevLett.67.3578
  13. D. Mailly, C. Chapelier, A. Benoit, "Experimental observation of persistent currents in a GaAs-AlGaAs single loop", Phys. Rev. Lett. 70, 2020 (1993). DOI: 10.1103/PhysRevLett.70.2020, https://doi.org/10.1103/PhysRevLett.70.2020
  14. E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, R. A. Webb, "Diamagnetic Persistent Current in Diffusive Normal-Metal Rings", Phys. Rev. Lett. 86, 1594 (2001). DOI: 10.1103/PhysRevLett.86.1594, https://doi.org/10.1103/PhysRevLett.86.1594
  15. R. Deblock, R. Bel, B. Reulet, H. Bouchiat, D. Mailly, "Diamagnetic Orbital Response of Mesoscopic Silver Rings", Phys. Rev. Lett. 89, 206803 (2002). DOI: 10.1103/PhysRevLett.89.206803, https://doi.org/10.1103/PhysRevLett.89.206803
  16. H. Bary-Soroker, O. Entin-Wohlman, Y. Imry, "Effect of Pair Breaking on Mesoscopic Persistent Currents Well above the Superconducting Transition Temperature", Phys. Rev. Lett. 101, 057001 (2008). DOI: 10.1103/PhysRevLett.101.057001, https://doi.org/10.1103/PhysRevLett.101.057001
  17. Norman O. Birge, "Sensing a Small But Persistent Current", Science 326, 244 (2009). DOI: 10.1126/science.1180577, https://doi.org/10.1126/science.1180577
  18. S. Günzler, J. Beck, D. Rieger, N. Gosling, N. Zapata, M. Field, S. Geisert, A. Bacher, J. K. Hohmann, M. Spiecker, W. Wernsdorfer, I. M. Pop, "Spin Environment of a Superconducting Qubit in High Magnetic Fields", Nat. Commun. 16, 9564 (2025). DOI: 10.1038/s41467-025-65528-y, https://doi.org/10.1038/s41467-025-65528-y; arXiv:2501.03661, https://arxiv.org/abs/2501.03661