Persistent currents in normal metal rings: comparing high-precision experiment with theory

A. C. Bleszynski-Jayich,1 W. E. Shanks,1 B. Peaudecerf,1 E. Ginossar,1 F. von Oppen,2 L. Glazman,1,3 and J. G. E. Harris1,3

1 Department of Physics, Yale University, New Haven CT, 06520 USA 2 Institut für Theoretische Physik, Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany 3 Department of Applied Physics, Yale University, New Haven CT, 06520 USA

Quantum mechanics predicts that the equilibrium state of a resistive electrical circuit contains a dissipationless current. This persistent current has been the focus of considerable theoretical and experimental work, but its basic properties remain a topic of controversy. The main experimental challenges in studying persistent currents have been the small signals they produce and their exceptional sensitivity to their environment. To address these issues we have developed a new technique for detecting persistent currents which offers greatly improved sensitivity and reduced measurement back action. This allows us to measure the persistent current in metal rings over a wider range of temperature, ring size, and magnetic field than has been possible previously. We find that measurements of both a single ring and arrays of rings agree well with calculations based on a model of non-interacting electrons.

An electrical current induced in a resistive circuit will rapidly decay in the absence of an

applied voltage. This decay reflects the tendency of the circuit’s electrons to dissipate energy and

relax to their ground state. However quantum mechanics predicts that the electrons’ many-body

ground state (and, at finite temperature, their thermal equilibrium state) may itself contain a

“persistent” current which flows through the resistive circuit without dissipating energy or

decaying. A dissipationless equilibrium current flowing through a resistive circuit is highly

counterintuitive, but it has a familiar analog in atomic physics: some atomic species’ electronic

ground states possess non-zero orbital angular momentum, equivalent to a current circulating

around the atom.

Theoretical treatments of persistent currents (PC) in resistive metal rings have been

developed over a number of decades (see [1,2] and references therein). Calculations which take

1 into account the electrons’ inevitable coupling to a static disorder potential and a fluctuating

thermal bath predict several general features. A micron-diameter ring will support a PC of I ~ 1 nA at temperatures T 1 K. A magnetic flux threading the ring will break time-reversal symmetry, allowing the PC to flow in a particular direction around the ring. Furthermore, the

Aharonov-Bohm effect will require I to be periodic in with period 0 = h/e, thereby providing a clear-cut experimental signature of the PC.

These predictions have attracted considerable interest, but measuring the PC is challenging

for a number of reasons. For example, the PC flows only within the ring and so cannot be

measured using a conventional ammeter. Experiments to date[2,3] have mostly used SQUIDs to infer the PC from the magnetic field it produces. Interpretation of these measurements has been

complicated by the SQUIDs’ low signal-to-noise ratio (SNR) and the uncontrolled back action of

the SQUID’s ac Josephson oscillations, which may drive non-equilibrium currents in the rings.

In addition, SQUIDs perform optimally in low magnetic fields; this limits the maximum

which can be applied to the rings, allowing observation of only a few oscillations of I() and complicating the subtraction of background signals unrelated to the PC.

Experiments to date have produced a number of confusing results in apparent contradiction with theory and even amongst the experiments themselves[2,3]. These conflicts have remained without a clear resolution for nearly twenty years, suggesting that our understanding of how to

measure and/or calculate the ground state properties of as simple a system as an isolated metal

ring may be incomplete.

More recent theoretical work has predicted that the PC is highly sensitive to a variety of

subtle effects, including electron-electron interactions[4,5,6,7], the ring’s coupling to its

electromagnetic environment[8], and trace magnetic impurities within the ring[9]. These

2 theories have not explained the experimental results to date, but they do indicate that accurate

measurements of the PC would be able to address a number of interesting questions in many-

body condensed matter physics (in addition to resolving the long-standing controversy described

above).

Here we present measurements of the PC in resistive metal rings using a micromechanical

detector with orders of magnitude greater sensitivity and lower back-action than SQUID-based

detectors. This approach allows us to measure the PC in a single ring and arrays of rings as a

function of ring size, temperature, and the magnitude and orientation of the magnetic field over a

much broader range than has been possible previously. We find quantitative agreement between

these measurements and calculations based on a model of diffusive, non-interacting electrons.

This agreement is supported by independent measurements of the rings’ electrical properties.

Figures 1A-C show single-crystal Si cantilevers with integrated Al rings (their fabrication is described elsewhere[10]). All the PC measurements were made in magnetic fields well above the critical field of Al, ensuring the rings are in their normal (rather than superconducting) state.

The parameters of the four ring samples measured here are given in Table 1. In the presence of a magnetic field B , each ring’s current I produces a torque on the cantilever B as well as a shift in the cantilever’s resonant frequency . Here

r2 Inˆ is the magnetic moment of the PC, r is the ring radius, and nˆ is the unit vector normal to the ring. We infer I(B) from measurements of (B) ; the conversion between (B) and I(B) is described in the Supporting Online Material (SOM).

To monitor we drive the cantilever in a phase-locked loop. The cantilever is driven via a

piezoelectric element, and the cantilever’s displacement is monitored by a fiber-optic interferometer[11]. The cantilever’s thermally limited force sensitivity is ~ 2.9 aN/Hz1/2 at T =

3 1/2 300 mK, corresponding to a magnetic moment sensitivity of ~ 11 B/Hz and a current sensitivity of ~ 20 pA/Hz1/2 for a ring with r = 400 nm at B = 8 T. By comparison, SQUID

magnetometers achieve a current sensitivity 5 nA/Hz½ for a similar ring[12,13,14]. We have

shown previously that a cantilever’s noise temperature and the electron temperature of a metal

sample at the end of a cantilever equilibrate with the fridge temperature for the conditions used

here[11].

The frequency shift of a cantilever containing an array of N = 1680 nominally identical

rings with r = 308 nm at T = 323 mK is shown in Fig. 1D as a function of B. Oscillations with a

period 20 mT, corresponding to a flux h/e through each ring, are visible in the raw data. Depending upon r and (the angle between B and the plane of the ring) we observe as many as

450 oscillations over a 5.5 T range of B (data shown in the SOM).

Figure 1E shows the data from Fig. 1D after subtracting the smooth background and converting the data from (B) to I(B) using the expressions in the SOM. The left-hand axis in

Fig. 1E shows I, the total PC inferred from the measurement, which is the sum of the PC from each ring in the array. The right-hand axis shows the estimated typical single-ring PC: Ityp =

I/ N . This relationship between Ityp and Iarises because the PC in each ring is predicted to oscillate as a function of B with a phase which depends upon the ring’s microscopic disorder, and thus is assumed to be random from ring to ring. This assumption is verified below.

To establish that provides a reliable measure of the PC we measured Ityp(B) as a function of several experimental conditions: the laser power incident on the cantilever, the amplitude and frequency of the cantilever’s motion, the polarity and orientation of the magnetic field, and the presence or absence of room temperature electronics connected to the cryostat.

These data are shown in the SOM, and indicate that the measurements of Ityp(B) are independent

4 of these parameters (for the conditions of our experiment) and reflect the equilibrium persistent current in the rings.

Figures 2A-C show Ityp(B) for arrays of rings with three different radii: r = 308 nm, 418 nm, and 793 nm. We have also measured a single ring with r = 418 nm, shown in Fig. 2D.

Figures 2E-H show Ityp ( f ) , the absolute value of the Fourier transform of the data in Figs 2A-

-1 D (f is the “flux frequency” in units of (h/e) ). Figures 2I-L show Gtyp ( B) , the autocorrelation

of Ityp(B) for each of these samples. Gtyp ( B) is calculated from measurements of Ityp(B) taken over a much broader range of B than is shown in Figs. 2A-D; the complete data is shown in the

SOM.

We can draw a number of conclusions from a qualitative examination of this data. First, we note that Ityp(B) oscillates with a period h/e, but also contains an aperiodic modulation

which broadens the peaks in Ityp ( f ) and causes Gtyp ( B) to decay at large B . This modulation is due to the fact that we apply a uniform B to the sample, leading to magnetic flux inside the metal of each ring given by M = BAM where AM is the area of the metal projected along B . This leads to a new effective disorder potential (and hence a randomization of the phase of the I(B) oscillations) each time M changes by ~ 0[15]. As a result, the peaks in

Ityp ( f ) span a band of f roughly bounded by the rings’ inner and outer radii (the blue bars in

Figs. 2E-H), and the decay of Gtyp ( B) is found to occur on a field scale (defined as the half-

width half-max of Gtyp ( B) [16]) Bc = 0/AM. Here is a constant which is predicted[17] to be

1; we find 1 < < 3 in these samples. For the array samples, ring-to-ring variations in r

(estimated to be ~ 1%) should contribute negligibly to Bc and the peak widths in Ityp ( f ) . The

5 fact that the r = 418 nm array and the r = 418 nm single ring show similar peak width and Bc indicate that variations in r do not affect the signal appreciably.

It is clear from Fig. 2 that the PC is smaller in larger rings. This is consistent with the prediction[18] that the typical amplitude Ih/e(T = 0) of the h/e-periodic Fourier component of

I() at T = 0 corresponds roughly to the current produced by a single electron diffusing around

2 the ring at the Fermi energy, and hence should scale as 1/r . In addition, Ih/e(T) is predicted[18] to

2 decrease on a temperature scale (known as the Thouless temperature) TT 1/ r corresponding to the scale of disorder-induced correlations in the ring’s spectrum of single-electron states.

In Fig. 2E a small peak at f = 2 can be seen, corresponding to the second harmonic of

I(). This harmonic has attracted particular attention because under some conditions it has a component which is not random from ring to ring[4,19,20]. The signal from such a non-random,

(avg) “average” current would scale as I N rather than N . Furthermore, the amplitude of

(avg) I can be strongly enhanced by electron-electron interactions[4] and other effects[8,9].

(avg) However I arises from the cooperon contribution to the PC and so requires time-reversal symmetry within the metal, which in our experiments is broken by M. We calculate that M

(avg) 2 r /1.3B suppresses I by a factor ~ e (where the magnetic length B h / eB ), which for this

(avg) experiment should render I unobservably small. As a result, the peak in Fig. 2E at f = 2 presumably reflects the random component of the second harmonic of I(), which is

3/2 predicted[18] to have a zero-temperature amplitude Ih/2e(0) = Ih/e(0)/2 , to be suppressed on a temperature scale = TT/4, and to produce a signal with the same N scaling as Ih/e.

We now turn to a more quantitative analysis of the data. Theory predicts[18] that, for

th each independent realization of the disorder potential, Ih/pe (the p harmonic of I()) is drawn

6 1/2 randomly from a distribution with a mean I 0 and an rms value I 2 which in h pe D h pe D general is non-zero. Here represents an average over disorder potentials. The quantity D

1/2 I 2 can be calculated explicitly as a function of r, T, p, and the electrons’ diffusion constant h pe D

D for a variety of models.

1/2 To compare our data against these calculations, we make use of the fact that I 2 can h pe D be extracted from a measurement of I(B) when the measurement record spans many Bc. When this condition is satisfied, averages performed with respect to B are equivalent to averages performed with respect to disorder realizations, and it is straightforward to show that the area

2 2 1/2 under a peak in I ( f ) (cf. Figs. 2E-H) at f = p is simply related to I : typ h pe D

1/ 2 f 2 1/ 2 I ( f ) b( f ) df I 2 . (1) typ h pe D f

2 Here b is the noise floor in I typ ( f ) , and is estimated from the portions of the data away from

the peaks. We take the limits of integration f and f to be roughly the values of f corresponding to h/pe flux periodicity through the outer and inner radii of the ring, respectively.

In previous experiments, I 2 could only be determined from successive measurements of h pe D individual, nominally identical rings[21,3]. This approach was limited by the low SNR achieved in single-ring measurements and practical limits on the number of nominally identical rings

( 15) which could be measured.

1/ 2 1/ 2 Measurements of I 2 for each sample and I 2 for the smallest rings are shown h e D h 2e D in Fig. 3 as a function of T for = 45 (open symbols) and = 6 (closed symbols). From Fig. 3

7 it can be seen that the PC in larger rings decays more quickly with T than in smaller rings, and

1/ 2 1/ 2 that I 2 decays more quickly than I 2 , consistent with the discussion above. In h 2e D h e D addition, the agreement between the data for the r = 418 nm array and the r = 418 nm single ring indicates that the PC signal scales as N and hence that the PC is random from ring to ring.

1/ 2 The solid lines in Fig. 3 are fits to theoretical predictions in which I 2 is calculated h pe D for diffusive noninteracting electrons. This calculation closely follows that of Ref. [18] but takes into account the presence of the large magnetic field B inside the metal (which lifts the spin degeneracy and removes the cooperon contribution to the PC) as well as spin-orbit scattering (the rings’ circumference exceeds the spin-orbit scattering length, as discussed in the SOM). We find:

T I 2 (T ) g p2 I 2 (0) (2) h / pe D h / pe D TT

6 2 1/2 3eD D where g(x) x2 n exp[(2 3nx)1/ 2 ] , I 2 (0) 0.37 p3/ 2 , and T . h / pe D 2 T 2 3 n1 (2 r) kB (2 r)

The data from each sample in Fig. 3 was fit separately, in each case using D as the only fitting parameter. The best fit values of D are listed in Table 1. These values are typical for high- purity evaporated aluminum wires of the dimensions used here[22,23]; however, to further constrain the comparison between our data and theory we also independently determined D from the resistivity of a co-deposited wire (the wire’s properties are listed in Table I). This measurement is described in detail in the SOM and provides a value of D in good agreement with the values extracted from the persistent current measurements. We note that the values of D in Table 1 show a correlation with the samples’ linewidths which may reflect the increased contribution of surface scattering in the narrower samples.

8 The calculation leading to Eq. 2 assumes the phase coherent motion of free electrons around the ring. Measurements of the phase coherence length L(T) in the co-deposited wire are described in the SOM, and show that L 2r for nearly all the temperatures at which the PC is observable. The closest approach between L and 2r at a temperature where the PC can still be observed occurs in the 308 nm array at T = 3 K where we find L(3 K) = 1.86 × (2r). It is

1/ 2 conceivable that the more rapid decrease in I 2 observed in this sample above T = 2 K (Fig. h e D

3) is due to dephasing; however it is not possible to test this hypothesis in the other samples, as the larger rings’ PC is well below the noise floor when L(T) = 1.86 × (2r). To the best of our knowledge the effect of dephasing upon the PC has not been calculated.

In conclusion, we have measured the persistent current in normal metal rings over a wide range of temperature, ring size, array size, magnetic field magnitude, and magnetic field orientation with high signal-to-noise ratio, excellent background rejection, and low measurement back-action. These measurements indicate that the rings’ equilibrium state is well-described by the diffusive non-interacting electron model. In addition to providing a clear experimental picture of persistent currents in simple metallic rings, these results open the possibility of using measurements of the PC to search for ultra-low temperature phase transitions[6], or to study a variety of many-body and environmental effects relevant to quantum phase transitions and quantum coherence in solid state qubits[24,25]. Furthermore, the micromechanical detectors used here are well-suited to studying the PC in circuits driven out of equilibrium (e.g., by the controlled introduction of microwave radiation)[8]. The properties of persistent currents in these regimes have received relatively little attention to date but could offer new insights into the behavior of isolated nanoelectronic systems.26

9

2 Sample r (nm) w (nm) d (nm) N D (cm /s) 308 nm array 308 115 90 1680 271 ± 2.6 418 nm array 418 85 90 990 214 ± 3.3 793 nm array 793 85 90 242 205 ± 6.5 418 nm ring 418 85 90 1 215 ± 4.6 Wire (see SOM) 289,000 (length) 115 90 1 260 ± 12

Table 1. Sample parameters. For each of the four ring samples, the rings’ mean radius r, linewidth w, and thickness d are listed, along with the number N of rings in the sample. The electrons’ diffusion constant D, extracted from the fits in Fig. 3, is given. The stated errors are statistical errors in the fits. An additional 6% error in D is estimated for uncertainties in the overall calibration, as discussed in the SOM. The fifth sample is the co-deposited wire used in the transport measurements described in the SOM. For this sample D was determined from the wire’s resistivity.

10

Fig. 1. (A) Cantilever torque magnetometry schematic. An array of metal rings is integrated onto the end of a cantilever. The cantilever is mounted in a 3He refrigerator. A magnetic field B is applied at an angle from the plane of the rings. The out-of-plane component of B provides magnetic flux through the ring. The in-plane component of B exerts a torque on the rings’ magnetic moment and causes a shift in the cantilever’s resonant frequency . Laser interferometry is used to monitor the cantilever’s motion and to determine . (B) A scanning electron micrograph of several Si cantilevers similar to those used in the experiment. The light regions at the end of some of the cantilevers are arrays of Al rings. The scale bar is 100 m. The individual rings are visible in (C), which shows a magnified view of the region in (B) outlined in red. (D) Raw data showing as a function of B for an array of N = 1680 rings with r = 308 nm at T = 365 mK and = 45. (E) Persistent current inferred from the frequency shift data in (D) after subtracting a smooth background from the raw data. The left-hand axis shows the total current I in the array and the right-hand axis shows the estimated typical per-ring current Ityp = I/ N . Oscillations with a characteristic period of ~ 20 mT (corresponding to = h/e) are visible in (D) and (E).

11

Fig. 2. Persistent current versus magnetic field in: (A) the 308 nm array for T = 365 mK and = 45, (B) the 418 nm array for T = 365 mK, = 45, (C) the 793 nm array for T = 323 mK, = 6, and (D) the 418 nm ring for 365 mK, = 45. In each case a smooth background has been removed. (E)-(H) show Fourier transforms of the data in (A)-(D). The expected h/e and h/2e periodicities are indicated by the blue bar. The bars’ widths reflect the rings’ linewidth w. A small h/2e peak is present in (E) (visible in the log-scale graph, inset). (I)-(L) show the autocorrelation functions of the data shown in (A)-(D), but computed over a field range B larger than shown in (A)-(D): B = (I) 5.4 T, (J) 5.3 T, (K) 0.6 T, and (L) 1.1 T (full data shown in SOM).

12 , 308 nm array h/e 1 308 nm array h/2e , 418 nm array 418 nm single ring 793 nm array 0.1 ) A n (

t n e

r 0.01 r u C

0.001

0.0 0.5 1.0 1.5 2.0 2.5 Temperature (K)

Fig. 3. Temperature dependence of the h/e and h/2e Fourier components of the current per ring. The vertical axis

1/ 2 1/ 2 indicates I 2 and I 2 , the rms values of the Fourier amplitudes of the persistent current. In each data h e D h 2e D set, the solid points were taken with = 45 while for the hollow points = 6. The arrows indicate the data points derived from I(B) measurements taken over a magnetic field range much greater than Bc; other data points are derived from the scaling of I(B) measured over a smaller range of B. The lines (solid for array samples, dotted for the single ring) are fits to the prediction for noninteracting diffusive electrons. The electron diffusion constant D is the only fitting parameter, and is listed in Table I.

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26 We thank M. Devoret, R. Ilic, T. Ojanen, and J. C. Sankey for their assistance. A.C.B.-J.,

W.E.S. and J.G.E.H. are supported by NSF grants 0706380 and 0653377. F.v.O. is supported in part by DIP. J.G.E.H. acknowledges support from the Sloan Foundation. A.C.B.-J. acknowledges support from UNESCO-L’Oreal. L.G. is supported in part by DOE grant DE-FG02-08ER46482.

F.v.O and L.G. acknowledge the hospitality of KITP in the final stages of this work.

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ep AepT

ere Aep . or i6,9. or e odiio o or eree e

eeroeero e eri re ee ee oerved o e doied e roe o ie oiio i eer rer d o oo e or10

Re kB D ee AeeT T . w

e o eero e rei re i roie e o e o ee o re. e e ie i i. o i o e ered L T i . d d

e reio L D i Aep d D e ree reer. e ied ve re

Aep . . d D . e deoe i ve o D D . ie

D dier ro D d e ve o D ered ro e erie rre e oe L o deed o D d o rovide reive e ori o D. ddiio e oe . i derived io oideri e reee o erodi io i e ire. ee io er e eri oeiie i d ee e ered ve o D .

2. Calibration of the persistent current signals 2.1 Estimation of persistent current from cantilever frequency e ier e e erie rre I ro e i e reoe ree o e iever o i e e i oed. ere e derie e er i i I i ierred ro . e iever oei eer i o o er. e ir i e ei eer ored i e iever deorio i i ver er roi oei. e eod er rie ro e ierio eee e ri erie rre d e ied ei ied. i ei er i ore oied io o e iever diee o e erie rre d e rei ore deed o e e eee e ri d e ied ied. e rie e ei eer

B E B IB AidB

ere A i e re o e ri d i e e eee e e o e ri d e ied ei ied B i i e i er. e oe iever deeio ed o o rio d roio o e e e er e d ee i reoie or e oi eee e erie rre d e iever. e oider erie rre i i roie eriodi i e ro e ri ABi. reed i er o i orier ooe e rre i

BAi BAi IB Ih pe i p Ih pe o p p

i io o erie rre i i eriodi i e ro e ri i eer roiio or e roe o i io i e ered ro e ere o e erie rre d i e i er e.. i. e i er ide eriodi odio. i i ee e d i e i er i e ie vri e ied ei ied i dd o ro e ri d e e e er ed o eriodi odio o e rre died i e i er. oever e i e ree i given ied e re oered i odio o e de o e iever oio. i riorrd o o i oio redoi ode e ro e ri d o e ro e e.

ere p i eive o e vrie f i e i er d p orreod o eriod o he. e reee o ei iide e e o e ri ed o eriodi odio o e P oiio re e do o ori p o e ieer. i e odio eeie e oie er i . i re o ree i ree i e ie rever er i e e. e rie e orier erie over p rer ier i order o orreod ore oe i e d i oi o diree eree o IB d ee diree er I f . e oe e i i eeed d oerved o e doied i orier ooe i p . i i e ed i oe o e roiio eo.

e ei eer E e e eved ro . d

p BAi BAi E B Ih pe o p Ih pe i p p p

e oei oied i e erie rre . i ovio o roi d ee i ed o iever ree i i deed o e ide o e iever oio. or rirri ide e ree i i i

roorio o E . oever i roiio i o vid e e iever oio i eo e rei odio o e ro e ri ac . rie e

i e iever ide i ed o ac o e eed o derive or ide reev o or eree. i io i doe i oi errio eor i e iooi reor i i vid or d ee rorie or or eree. q e ei oi e e e rie i ere i e m L ve o e iever eiiri oiio L i e iever e qtip i e diee o e iever i ro eiiri d m i e rio eee e oe o

e iever e oiio o e ri d e or qtip L or er ode m. or ri

oed e iever i m .. or e ir o er ode.

e oiio o e iever e rerie i er o e ioe vrie

q J i ere k i e iever ei ri o J e io i k vrie d e oi oe e vrie. e i i iever reo ree i e ive 11

dE J J

AB m AB o mq o J p Ih pe o p Ih pe i p kLq p L

q ere e ve e o e oeorder er i e reer i d ed m L

i oe o . . J x i e ir ee io o e ir AB

id d q i e ide o oio o e iever i i i o die. e

redi veriied i e ii q . rede o E . ro i oi e e o roe o er IB ro or eree o B . e ir eod e er e orier ooe o IB i.e. e p ere I overi e ree i d io e derivive i B

Ih pe pAi m AB o mq AB o J p B kLq L

e roiio i oi ro . o . reie o e e re o e ee io vrie o i over ive d e. io i e iered

eri i ree o B o e IhpeB. IhpeB i e orier rored d o e orier ooe f p i e. i roie i reeed or e p rei i I f diviio N ive I f d ivere orier ror ive IB.

e dve o eod i i rovide re eie o e orier ooe over ide re o p d o i ie or d i i e erie rre

orier ooe er he d he e ever oerve i ier f. e didve o eod i ere re ve o p i ed o eroe i e ee io i . . d ei oie i e eree o i e overed o diveri I or ee p eive e eree i o eiive o ee orier ooe o I. e ve o p i ee diveree or e odiied vri q d e ed i o veri or e e o i e.. he ei ed i ee. rerore

or e e i ir re i e d rr q e o eo e diveree i e r o . orred i ve o p i.e. d o did o ierere i e ierred orier ooe he or he. re e I f or

ee e i. ed i eod . IB or ee e o i i. i e ivere orier ror o I f i e ver oe orier ooe orreodi o e oo rod viie i e r d e.. i i. dirded. e orier ooe ere e r o . divere ere o dirded i

i IB.

or e i eer i e ri d rr q od o e dereed r eo o e diveree i . eire o o e fd o iere

i.e. p . oever vri q e oired ere ever i he ove e oie oor. o ree e d ro ee e e e eod eod eod i e e o e erie rre i or ee e or eire er p i.e. o oerve ooe p d e re o e ee io i . vrie o e over ive d e. i eod e e p i e ee io d o rerie .

AB m AB o mq o J kLq L Ih pe o p Ih pe i p p

I i o o over e ered ree i dire o i i p B

I Ai m AB o mq AB o J B kLq L

i i i e iered eri d orier rored o e I f d e divided N o e I f . eod ed or e ri d rr i.

. od e eied e oie o q or i i d ed o ieiivi o orier ooe o IB orreodi o he eriod i.e. e eroe o

ee io i . oever vri q d ee vri e oio o e ieiive d i oer d r e oired ere o he eriodi ooe o IB ove e oie oor.

e d oi IB or ee e i. re e ivere orier ror o I f i e oe orier ooe orreodi o e oo rod viie i e r d e.. i i. dirded. o iree orier ooe ere dirded.

i eod d o ie d e i re i ve o I h pe D i dier . e ee o i eri i died eo i eio ..

2.2 Tests of the conversion between frequency shifts and persistent currents e eed e r o . i ever . dire orio eee

. d eree o ver q ie ve o B i dii o eror de o

e e iever ree deed o q eve i e ee o ei ied re ee o e e irii oieri o e iever ei

roerie. o reove i oei rod e ered io o q o iir ei ied idied i ire . er ri e o rve o ere i e ie o i. e reii ree i i doied e ooe de o e erie rre. e re i o i e i od o i. o i i o

eio i p i reoe reer or e ri ie d e ide o e he ooe o e rre. i. o o oer e i i e ierred rre oerved o e eed over ide re o iever ide d or eiio o o diere iever er ode. e i o e i i i. o i e d i i. iie or eod o eri e erie rre ro e ree i i re d rerore e iever oio i e ei ied doe o rode reie oeiiri ee.

2.3 Uncertainty estimates

eriie i e eree o I o i i. o e i er rie h pe D ro er o ore i e i ere. e eie or eerre eree ve eri o ed o e rer eiiio d orio i ied oi.

e ii error i or eiio o e rod bf re ro . or e rr o or e ri.

ie e i I i ie vrie o diriio e re eii h pe D ro iie d e e eri i or eie i e ive e drd rror o e rie . e i reed o e er o ideede reiio i or e i

er i e rio eee e o B over i e eree i de d B. or or

d e error i I re ro or e rr o or e h pe D ri. e eie e eri iroded e vrio roiio i eod d i . o ee ore o eri i re i error i vri deree o orreio eee e idivid d oi i i. . or ee e error de o e od e o or ive e . i. d ee ed o o o

i o e d ro e e ie e error i bf od e rdo or e

eree edi o er i i. . e eriie i D oed i e orreod o e ii error i e i o i. d ee ree e ore o error edi o er i i. . ed o drd error roio e eie eri i e over i o e rve i i. ed o ddiio eri i e ve o D.

3. Measurement diagnostics e erored vrie o dioi eree o rerie e ivivee o or ievered deeor. rir e ed o ere e eree doe o ide rio oeiiri ee ore or ii e P i. e ee o iever oiio ide d ree re died i e eio . ere e o e ee o redo er oer ei ied ori e erie ode d e reee o rooeerre eeroi oeed o e ro.

i. o e erie rre io o B or ive diere er oer Pi ere e er i ed or iereroeri deeio o e iever oiio. e i i ideede o er oer eee Pi . d d i o i eed Pi

. e d i e i er ere e i Pi . rir e d i i. idie e ee o ei e er oer ed i e eerie. e ered e erie rre o oiive d eive ei ied d e re re oed i i. i e xi eed or e eive B re. e i i eed e ori o e ied. e oo d i e e i diere oerio ode e e eried d e rre i e e ed red do o ero e e eried i rre oi i e e ed d e e o eried. o iiie eeroi oie i ode eeroi ere dioeed ro e der e.. eroer eer iid ei eve eer e. ee e P drive i ed ro rooeerre . oi o ier. e re oed i i. o o deedee o e erie rre i o e oeri ode o e e.

4. Magnetic Field Sweeps died i e i er eri e erie rre over re o B

i B o o deerie e diorder vered rre I . ire h e D

o d ro ee re B ee or e o e e ered. e o ee ire e i oed i IB ere

I IB Ai B

i i derived ro eree o e iever ree i i . . e oie o oriio i . e e oiio o IB i i. ve e e ide e oiio i IB.

ee re ee rovide dire ere o I e i reev or i h e D orio i eor d re i e oi red rro i i. o e i er. or e reider o e oi i i. e e e o e IB i od eiri o deed o T o vi over i o i i. . o eere e oi i i.

o red rro i i i ied o e ve o I deeried ro e re B h e D ee.

5. Supplementary Figures

Figure S1 ie o o ire iir o e oe ed or ror eree.

300

) 200 (

R 100

0 40 50 60 70 80 90

B (mT)

Figure S2 eie ver ei ied or e ror ire. e erodi riio or over re o ei ied eii . e ei ied e i e direio o irei ied re.

20 15 ) T m

( 10

C

H 5 0 1.15 1.20 Temperature (K)

Figure S3 erodi rii ied ver eerre or e ror ire. or e d o HT e o e e ied i e reie e ere o e or e ve. e d re i i .

i DH d T ii reer. eii HT o or diere eree o e or e reie ve i e ied T i doe o e DH.

3 )

4 - 2.9 K

10 2

x 4.7 K

( 12.6 K

R/R 1

0 -5 0 5

B (mT)

Figure S4 e i reie ver ei ied i i o . d or ree diere eerre. order o ieve dee i o oie rio e d ere i over re o deie e odiio ied i e e or vidi o . .

A 5

4 ) m

( 3 L 2

1

0 0 2 4 6 8 10 12 Temperature (K)

B 6 5 4 3 )

m 2 ( L 1

. 6 . 5 2 3 4 5 6 7 8 9 10 Temperature (K)

Figure S5 ero e oeree e L ver eerre. e do reree ve ered ro i o e eoreie o e ire. e e d i oed o ier e d oo e . e oid ie i e i o e io or deried i e e d e ded ie idie e eii oriio o i i ro eeroeero eri d eerooo eri died i e e.

A

) 2 A

n 1 (

t

n 0 e r r -1 u

C -2

6.7 6.8 6.9 7.0 7.1 B (T) B 10 ) z

H 8 100 m (

t f i

h 6 50 S

y

c 4 n 0 e

u 0.5 1.0 1.5 2.0 2.5 q 2 e r F 0

0.0 0.5 1.0 1.5 2.0 2.5

Flux amplitude through ring (0)

Figure S6 (A) ei o e erie rre ver ei ied or e rr o ri. or e eree o e iever orieed i . e rro idie o ied ve i eree o e iever ree i ere erored io o iever ide. (B) ieree i iever ree i or e o ied ve idied i (A) ver iever ide o oiio. e iever drive i i eod er ode i d ree o d ri o o . . e iever d e o . e iever ide i oed o e xi i er o e

ide o e odio i i o e ro e ri roded e iever

oio. e oid rve i i i . i p i I . . d r . e ie o e ree i ered e o oi idied i (A) d e e i e i o i (B).

Figure S7 iever ree i ver ei ied or ive diere iever oiio ide q (ered e oio o e ri. e oer e o e e d e er e ed o o

ive q i . i p . e re oe o o o e oer idii e re de o eiiri erie rre.

Figure S8 e derivive o e erie rre I derived ro . d ver ei ied ered e oii e iever . e iever ir er reoe d . e iever eod er reoe. e erie rre doe o deed o e iever oiio ree e i dieree i e o rve oo rod i re de o diere ei reoe ree i e e oder . d . .

Figure S9 e derivive o e erie rre I´ derived ro . d ver ei ied or erie o er oer iide o e iever. or e d o i e i er o er oer ed.

Figure S10 e derivive o e erie rre I´ derived ro . d ver ei ied or o ei ied oriie. e i o e eive ei ied re i eed.

Figure S11 e derivive o e erie rre I derived ro . d ver ei ied der diere ode o oerio o e erodi oeoid rodi e ei ied. e oerio ode o e e re idied i e ire. or e d o ree do eeroi ee or e ieo drive ere dioeed ro e ro.

Figure S12 e derivive o e erie rre I derived ro . d ver ei ied or rr o ri i rdi T . e ee i ered io ree e or ri. e ied i ied o e e o e ri.

Figure S13 e derivive o e erie rre I derived ro . d ver ei ied or ie ri o rdi T . e ied i ied o e e o e ri. e ee i ered io o oio e or ri.

Figure S14 e derivive o e erie rre I derived ro . d or rr o ri i r T . d o e e o e ri.

Figure S15 e derivive o e erie rre I derived ro . d ver ei ied or rr o ri i rdi T . e ied i ied o e e o e ri.

Figure S16 e derivive o e erie rre I derived ro . d ver ei ied or rr o ri i rdi T . e ied i ied o e e o e ri.

Figure S17 e derivive o e erie rre I derived ro . d ver ei ied or rr o ri i rdi T . e ied i ied o e e o e ri.

Figure S18 e derivive o e erie rre I derived ro . d ver ei ied er o or e rr o r ri ered i e ied ied o e e o e ri. e oer o o e orier ror o e e d. re re e diere eerre idied i e ire eed. e ide o rre oiio deree i irei eerre e e o e oiio rei ed.

6. References and Notes

. drer P.. ei e iveri . e ride deried i or eri rree. ree eri rree ed ere ee oe ed roe e e ooed o . i or ro. oe ide e e oeed o e ride d e voe roe ro ere ed ie o e oer ide e e oeed o rod d e ride ro e e ed. e e reie deoed ro e ed reie e roedre deried i e e. . ro . eri Solid State Physics ror re rdo . . . i, Introduction to Superconductivity r i e or ed. . P. re P. oe . eve . ri . . evore Nature 365 . P. . id . Proer Phys. Rev. B 35 . . ri JETP Lett. 31 . oe e ir odiio eoe vid or T o . ere e re er i ie e er. oever e eod odiio reri e vid re o ei ied o reive ve over e re o eerre reev o or eree. ee e ie o e eoreie i i or ied re o e e e d o i i. ere i over ied re ireed ier i T ro o . or e oe oed eerre . e eod odiio od e vid or ied e . . e ied re od e reried o e d

. e ered L doe deed e o e ie o e i re. or ee .

ere e ereio i vid or ied e e ied L iree e i re i ireed ro o . . ree . edor Phys. Rev. B 18 . . . eier . . er . . ereo Waves Rand. Compl. Media 9 . ee or ee . odei Classical Mechanics ddioee e or ed. .