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Measurement of the winding number instability

in mesoscopic superconducting rings

I. Petkovic1, A. Lollo1, L.I. Glazman1,2, and J.G.E. Harris1,21 Department of Physics, Yale University,

New Haven, Connecticut 06520, USA2 Department of Applied Physics, Yale University,

New Haven, Connecticut 06520, USA

We present measurements of the equilibrium current in superconducting mesoscopic rings threadedby a magnetic flux. As the flux is varied, the smooth evolution of the rings’ current is interruptedby abrupt transitions. These transitions correspond to discrete changes in the winding number ofthe superconducting order parameter. We find that the smooth evolution of the current and thelocations of the abrupt transitions are in quantitative agreement with Ginzburg-Landau theory inthe full magnetic field range and over a wide span of temperature and ring size. We also find thatthe transitions always correspond to unit changes in the winding number. The dynamics of thesetransitions is not well understood, but appears to reflect strong damping of the order parameter innucleating phase slips.

Superconductivity in reduced dimensions is stronglyinfluenced by fluctuations. One-dimensional supercon-ducting wires acquire finite resistance through the pro-cess of phase slips, in which the order parameter goestemporarily to zero at some point along the wire, accom-panied by a phase jump of 2π [1]. This type of fluctuationcan be thermal close to the critical temperature [2] orquantum at low temperature [3, 4]. In some cases quan-tum phase slips can be coherent [5]. Coherent quantumphase slips (CQPS) are a topic of fundamental interestas a dual to the Josephson effect [6] and have potentialapplications in quantum computing [7].

The majority of experiments on phase slips have fo-cused on thermal and incoherent quantum phase slipsand have used transport measurements of the subgap re-sistance of thin wires [3, 8–18]. Some experiments havestudied CQPS using microwave spectroscopy [19, 20], buthave left a number of open questions. In contrast tothese approaches, dispersively coupling a closed meso-scopic system to a high quality resonator can provide anon-invasive and well-characterized way to study thesephenomena, and allows access to the intrinsic behaviorof equilibrium states and their associated fluctuations.

We consider a uniform superconducting ring enclosingan externally imposed magnetic flux Φ. Within a mean-field picture, the state of the ring is described by thecomplex-valued order parameter Ψ(r), whose phase ac-quires a Φ-dependence. If the ring’s lateral dimensionsare smaller than the superconducting coherence length ξand penetration depth λ, then Ψ will be a function onlyof s, the coordinate along the ring’s circumference. Theboundary condition imposed by the ring requires thatΨ(s) = Ψ(s + 2nπR), where R is the ring’s radius andn ∈ Z is the winding number of the order parameter.In this one-dimensional limit and within the semiclassi-cal description the ring’s equilibrium states are uniquelyspecified by n [1, 4].

The equilibrium properties of such a system have threedistinct temperature regimes, which are set by the ratioof the ring’s radius R to the temperature-dependent ξ.For temperature T only slightly below the superconduct-ing critical temperature Tc such that 2R < ξ, the ring isin a superconducting state for some values of Φ while forthe other values it is in the normal state [22]. This is dueto competition between the superconducting condensa-tion energy and the flux-imposed kinetic energy of the su-percurrent. At slightly lower T such that ξ < 2R <

√3 ξ,

the condensation energy is always larger than (but of thesame order as) the kinetic energy and for each value of Φthe ring has exactly one superconducting state. Finally,at even lower T such that 2R >

√3 ξ, the condensation

energy is many times larger than the kinetic energy andat a given Φ several equilibrium states, each with differ-ent n, are available to the system.

These equilibrium states correspond to the local min-ima of the free energy functional F [Ψ(s)] and are sepa-rated from each other in the configuration space of Ψ(s)by barriers whose heights depend upon Φ [2, 6]. Thepresence of multiple local minima in the free energy land-scape means that the ring’s state depends upon its his-tory. Transitions between equilibria (i.e., changes in n, orphase slips), may occur via thermal activation over a bar-rier [2], quantum tunneling through a barrier [3, 4, 24], orby varying Φ to tune the barrier height to zero [25]. Thefirst two processes have been measured in rings that in-corporate weak links (such as Josephson junctions) [26–29], but are not readily observed in uniform supercon-ducting rings. The third process, in which the barrier ismade to vanish, is the focus of this paper. A detailedstudy of metastable current-carrying states and the bar-riers which separate them constitutes an important firststep towards the study of thermal and quantum phaseslips in uniform rings.

The Ginzburg-Landau (GL) expression for F [Ψ(s)] was

2

used to calculate the free energy of the equilibrium statesFn [2, 3, 6], as well as the values of Φ at which the barriervanishes and the corresponding state becomes unstable[2, 31, 32]. This calculation predicts that as Φ is varied,the ring’s equilibrium current In = −∂Fn/∂ Φ and heatcapacity CV,n = −T∂2Fn/∂T

2 undergo sawtooth-shapedoscillations in which the smooth portions correspond toevolution at fixed n, while the sharp jumps correspondto changes in n. Since stationary GL theory does notdescribe the dynamics of the transition, it does not pre-dict which state the ring will occupy after a transitionif there is more than one possibility (i.e., in the lowesttemperature regime described above). In principle thisquestion may be addressed by simulating the transitiondynamics via the time-dependent GL equation or othermethods [25, 33–37].

Measurements of I(Φ) [38–43] and C(Φ) [44, 45] havefound qualitative agreement with the features describedabove. However, quantitative agreement was found onlyfor T very close to Tc [3, 41–43, 45]. For lower T the de-tailed shape of I(Φ) and the values of Φ at which the irre-versible jumps occur were found to disagree with the pre-dictions of stationary GL theory [38–40], and the changein n at each transition has not been reproduced by dy-namical simulations [25, 33–37].

Here we describe measurements of I(Φ) in isolated one-dimensional aluminium rings. These measurements em-ploy cantilever torque magnetometry, and cover a widerange of ring dimensions, applied flux, and temperatureTc/2 < T < Tc (the range over which GL theory is ex-pected to be accurate). We find quantitative agreementbetween the data and GL theory regarding the form ofI(Φ) and the values of Φ at which the winding numbertransitions occur. The parameters extracted from thisanalysis agree well with the known properties of micro-fabricated Al, and with the temperature dependence ofξ and λ described by the commonly-used empirical ap-proximation [4].

Four separate samples were measured. All of the sam-ples were fabricated by depositing a single Al layer ofthickness d = 90 nm onto a lithographic pattern. In orderto increase the signal-to-noise ratio of the measurements,each sample consisted of an array of ∼ 103 nominallyidentical rings. The sample properties are listed in TableI. Each array is fabricated on a Si cantilever of length∼ 400 µm, thickness 100 nm and width ∼ 60 µm, withresonant frequency f ∼ 2 kHz, spring constant k ∼ 1mN/m and quality factor Q ∼ 105. The fabrication pro-cess is described elsewhere [1].

The measurement setup is also described elsewhere[47, 48]. A uniform magnetic field of magnitude B is ap-plied normal to the rings’ equilibrium orientation. As thecantilever oscillates, current circulating in the rings ex-periences a torque gradient, which shifts the cantilever’sresonant frequency by an amount df , monitored by driv-ing the cantilever in a phase-locked loop. In the configu-

ration used here, df = κ I Φ, where Φ = B πR2 and κ isa constant depending on the cantilever parameters andis proportional to the spring constant [1, 48]. A detaileddescription of the data analysis is given in the Supple-mental Material.

FIG. 1: Supercurrent per ring I as function of magnetic fieldB for T ∼ 900 mK and for various ring radii R (indicatedin each panel). Points are data; thick curves are the fits de-scribed in the text. Red corresponds to increasing B, andblue to decreasing B.

Figure 1 shows I(B) for each of the four samples atT ≈ 900 mK, while Figure 2 shows I(B) for the samplewith R = 538 nm as T is varied. The red points showmeasurements taken while B is increasing, and the bluepoints while B is decreasing. All the measurements ex-hibit sawtooth-like oscillations whose period is inverselyproportional to the ring area πR2. The jumps occurwith flux spacing equal to the superconducting flux quan-tum Φ0 = h/2e, indicating that the winding numbern changes by unity at each jump. The three qualita-tive regimes described previously are accessed by vary-ing either temperature or magnetic field. This is because

3

No Rnom (nm) RGL (nm ) wnom (nm) wGL (nm) N ξ0 (nm) λ0 (nm) λP0 (nm) Bc3,0 (T) B GLc3,0 (T)

1 250 288 80 65 1680 214(2) 97(1) 104(2) 0.0796(6) 0.087(1)

2 375 406 65 48 990 202(2) 95(1) 100(2) 0.1107(7) 0.125(1)

3 500 538 80 65 550 208(2) 95(1) 101(2) 0.0830(6) 0.089(1)

4 750 780 65 51 242 190(3) 98(1) 107(2) 0.1131(7) 0.125(2)

TABLE I: Summary of sample parameters. For each sample the table gives the nominal lithographic ring radius Rnom andwidth wnom, as well as the values RGL and wGL obtained as global fit parameters. The number of rings on each cantilever isN . ξ0, λP0, and Bc3,0 are the zero temperature values of the coherence length, Pearl penetration depth, and critical field Bc3

determined from the fits in Fig. 3. The penetration depth λ0 =√λP0 d. B GL

c3,0 is calculated using ξ0 and wGL, as described inthe text. The quoted error in the final digit of each fit value corresponds to the statistical uncertainty of the fit (one standarddeviation).

FIG. 2: Supercurrent per ring I as function of magnetic fieldB for rings with radius R = 538 nm at different temperaturesT (marked on each panel). Points are data; thick curves arethe fits described in the text. Red corresponds to increasingB, and blue to decreasing B.

T and B both diminish the condensation energy, whichcompetes with the flux-imposed kinetic energy. For lowT and B the data are hysteretic, indicating the presenceof multiple equilibrium states. At sufficiently high T orB the hysteresis vanishes, indicating that only one super-conducting state is available. For the highest values of Band T there are ranges of B over which I = 0 (to withinthe resolution of the measurement), corresponding to therings’ re-entry into the normal state.

FIG. 3: (a) Coherence length ξ as function of temperature.(b) Pearl penetration depth λP as function of temperature.In (a) and (b), the squares are the best-fit values from theGL fits described in the text, while the lines are fits to theexpressions given in the text. (c) Rings’ critical field Bc3 asfunction of temperature. The squares are determined frommeasurements of I(B), while the lines are the fits describedin the text.

In each data set we identify the rings’ critical field Bc3,which we take to be the value of B at which I becomesindistinguishable from 0 and remains so for all B > Bc3.As expected, Bc3 diminishes with increasing T (note thechange in horizontal scale between the panels of Fig. 2).

To compare these measurements with theory, we firstidentify the winding number of each smooth portion ofI(B). Then we fit the entire I(B) trace using the analyticexpression derived from GL theory for one-dimensional

4

FIG. 4: (a) The black curve shows the equilibrium free energy Fn as a function of B for all n, for the ring with R = 406 nm atT = 786 mK. In this panel, the Fn are calculated from the fit parameters determined in Fig. 1(c). The red (blue) curve showsthe path followed by the system as B is increased (decreased). (b) Free energy of an equilibrium state as a function of flux.(c) Absolute value of supercurrent as a function of flux. (d) Velocity of the superconducting condensate as a function of flux.The black dots in panels (b),(c) and (d) denote instability points φc, and red dots φ∗ (see text). Shaded regions are unstablein the sense specified in the text.

rings [3]. This expression includes the effect of the rings’finite width w, which is crucial for reproducing the over-all decay of I at large B. At each value of T , the fit-ting parameters are the coherence length ξ and the Pearlpenetration depth λP = λ2/d, appropriate for the regimewhere λ > d [4, 49]. The cantilever spring constant isassumed to be temperature independent, and is used asa global fit parameter for each sample along with the ringdimensions w and R. The portion of the data correspond-ing to the sharp jumps between states with different nis excluded from these fits (as described in the Supple-mental Material, the jumps are broadened by small in-homogeneities within the array). The resulting fits areshown as thick curves in Figures 1 and 2. Measurementsof I(B) and the corresponding fits for all R and T areshown in the Supplemental Material, along with a moredetailed description of the fitting procedure.Figure 3 shows the best-fit parameters ξ and λP, as

well as Bc3, all as function of T . The solid lines inFig. 3 (a) and (b) are fits to the expressions ξ(T ) =ξ0√

(1 + t2)/(1− t2) and λP(T ) = λP0/(1 − t4), wheret = T/Tc [4, 49]. In Fig. 3(a), the fit parameters areξ0 for each sample and Tc, taken to be the same for allthe samples; similarly, in Fig. 3(b) the fit parametersare λP0 for each sample and Tc for all the samples. Thesolid lines in Fig. 3(c) are fits to the expression valid forone-dimensional rings [39]

Bc3 = 3.67Φ0

2πwξ(T ). (1)

The fit parameters are Bc3,0(= 3.67Φ0/(2πwξ0)) for eachsample and Tc common to all the samples. The best-fit values of ξ0, λP0, and Bc3,0 (derived from the fits in

Figs. 3(a), (b), and (c), respectively) are given in TableI. The three values of Tc determined from these fits are1.316± 0.001, 1.391± 0.004 and 1.318± 0.002 K.

The values of ξ0 and Bc3,0 can be compared againsttwo separate estimates. First, we note that ξ0 can alsobe determined via transport measurements, using the re-lationship ξ0 = 0.855

√

ξb0le [4], where ξb0 = 1.6µm is thebulk Al coherence length and le the electron mean freepath. Transport measurements of Al wires that were co-deposited with the rings studied here give le = 35±5 nm[1]; this corresponds to ξ0 = 205± 15 nm, in close agree-ment with the values inferred from the measurements ofI(B). Second, we note that Bc3,0 can be calculated di-rectly from Eq. (1) using the values of ξ0 determinedfrom the fits in Fig. 3(a). The results of this approachare listed in Table I as B GL

c3,0. For each sample, B GLc3,0

and Bc3,0 agree to ≈ 10%. Lastly, we note that Eq.(1) indicates that Bc3(T ) should be independent of Rand proportional to 1/w, consistent with the data in Fig.3(c).

With values of ξ(T ) and λP(T ) obtained by fittingI(B), it is straightforward to calculate the free energyFn(B) of each equilibrium state [3]. The black line infigure 4(a) shows Fn(B) for the rings with R = 406 nmand T = 786 mK, corresponding to the measurement inFig. 1(c). In Fig. 4(a), the red (blue) curves show thepath taken by the rings as B is increased (decreased).The path is determined by using the values of n inferredfrom the data in Fig. 1(c). Equivalently, the red (blue)fit curves in Fig. 1 are related to the red (blue) curves inFig. 4(a) by I ∝ −∂F/∂B.

Figure 4(a) shows that the phase slips for increas-ing and decreasing B are located nearly symmetricallyaround the minima of Fn(B). Closer inspection shows

5

FIG. 5: Switching flux as function of winding number. Dots:experimental values; bars: observed width of each jump (asdiscussed in the Supplemental Material); full lines: predic-tion for the switching flux dφ∗

n; dotted lines: prediction forthe switching flux dφc,n (see text). Colors represent temper-ature. Top to bottom panels are for samples with ring radiiR = 288, 406, 538, 780 nm. The normalization of the axes isexplained in the text.

that the phase slips occur near the inflection points ofFn(B). To examine the location of these phase slipsquantitatively, we define dφn = φn − φmin,n, where φn isthe experimental value of the normalized flux φ = Φ/Φ0

at which the transition n ⇄ n + 1 occurs, and φmin,n isthe value of φ at which Fn reaches its minimum value. Asdefined, dφn are positive when B is increasing and nega-tive when B is decreasing. (In the following we normalize

all flux values by Φ0 and denote them by minuscule letterφ.)Our next step is to compare the experimental val-

ues of switching flux dφn with theory. In the Langer-Ambegaokar picture, valid for a current-biased wire muchlonger than ξ, the barrier between states n and n+1 goesto zero when bias current reaches critical current Ic [2].In the case of a flux-biased ring, still for R ≫ ξ, thisoccurs at normalized flux

φc,n = φmin,n +R√3 ξ

+O

(

(w

R

)2)

, (2)

where φmin,n = n

1+( w

2R )2 . In the case R & ξ, which cor-

responds to our experimental system, it was shown thatthe system remains stable beyond φc,n and loses stabilityat a higher flux [31, 32]

φ∗n = φmin,n +

R√3 ξ

√

1 +ξ2

2R2+O

(

(w

R

)2)

. (3)

These two switching fluxes are shown in Fig. 4(b)-(d) foran arbitrary n. To simplify comparisons with experimentwe define dφc,n = φc,n − φmin,n and dφ∗

n = φ∗n − φmin,n.

Figure 5 shows measured dφn as function of n. Thevertical axis in Fig. 5 is normalized to dφ∗ ≡ dφ∗

n=0.We chose this normalization because the limit n = 0 isequivalent to neglecting the rings’ width (i.e., taking w →0), which simplifies the expression for dφ∗

n. As a resultof this choice, the quantity shown in Figure 5 (dφn/dφ

∗)is expected to equal 1 for small B and to diminish as Bapproaches Bc3 (or as n → nmax).The horizontal axis in Fig. 5 is normalized to the ex-

perimentally observed maximum winding number nmax

(different for each sample), where nmax ≈√3R2

wξ. The

ratio n/nmax is very close to B/Bc3. Each panel showsdata for a different R at several temperatures. The datais given in four panels for clarity. In the SupplementalMaterial we show this data in a single panel and demon-strate that with this scaling all of the data collapses to-gether. The bars represent the width of the steep portionof the sawtooth oscillations in Figs. 1 and 2. As discussedin the Supplemental Material, this width is due to smallinhomogeneities within the arrays.The solid lines in Fig. 5 show the theory predic-

tion dφ∗n/dφ

∗ (see Eq. (3)), whereas dotted lines showdφc,n/dφ

∗ (Eq. (2)). The difference between the two in-creases with the ratio ξ(T )/R and is therefore the mostpronounced for small rings (top panel) or at high temper-ature due to the increase of ξ(T ) (all panels, high tem-perature). We see that the prediction dφ∗

n/dφ∗, which

includes the finite-circumference effect (R & ξ), agreeswell with the measured switching locations over the fullrange of T , B, and R. The largest disagreement occurs

6

for the largest rings (bottom panel) at low temperature;this discrepancy is likely due to the increased importanceof the rings’ self inductance (which is ignored in our anal-ysis) in this regime.

FIG. 6: Supercurrent per ring I as function of magnetic fieldB for rings with radius R = 288 nm and temperature T =861 mK. Red points: increasing B; blue points: decreasingB. The regions over which I(B) diminishes at fixed windingnumber are indicated by black arrows.

In addition to this quantitative impact, the finite-circumference effect can also be seen directly in Fig. 6,which shows measured I(B) over a narrow range of Bfor the smallest rings. For both increasing B (red) anddecreasing B (blue) each sawtooth oscillation reaches amaximum current, and then starts to diminish before theswitching occurs. The regions over which I(B) dimin-ishes (at fixed n) are indicated by black arrows.In these regions the velocity is super-critical (see Fig.

4(d)), but the diminishing density leads to the decreaseof current. This effect is only accessible in the the ringconfiguration. Both in a current-biased wire and in aflux-biased ring the phase of the order parameter at equi-librium is φ = ks, where k is a wave-vector, and currentis I ∝ k(1 − k2) [2, 3]. The boundary condition for thewire is kL = 2πn, where L is the wire length, and for aring kL + 2πφ = 2πn, where L = 2πR. When biasing awire with current I < Ic, k is not uniquely determinedsince I ∝ k(1−k2) has multiple solutions, and the systemwill always chose the value of k in the stable region (non-shaded area in Fig. 4(b)-(d)). (Here ”stable” refers tothe long wire/ring diameter limit). In contrast, when bi-asing a ring with flux, k is uniquely determined (throughthe boundary condition), and therefore it is possible tobias the system in the shaded region, which correspondsto the regions indicated by black arrows in Figure 6.The winding number transitions described above take

place when the barrier confining an equilibrium state van-

ishes. When this occurs, Ψ(s) evolves dynamically untilit relaxes to a new local minimum of F [Ψ(s)]. Unless Tor B is close to a critical value, there are typically mul-tiple minima into which the system may relax. Despitethis freedom, we measure that the winding number al-ways changes as |∆n| = 1. This is seen for all values ofR, B, and T (down to the lowest value T = 460 mK). Incontrast, previous experiments have all found |∆n| > 1for low B and low T [38–40]. One notable feature ofthese previous experiments is that the rings had larger Rand w than those studied here. Furthermore, the resultsof Refs. [38–40] together with the results presented hereshow a clear trend in which ∆n decreases as R and w aredecreased at low temperature.

We note that transitions with small ∆n would be ex-pected when the order parameter is strongly damped, asstrong damping will tend to cause the system to relaxinto a nearby equilibrium configuration. We speculatethat the observed trend indicates that the damping ofthe order parameter is greater for thinner rings. We alsospeculate that the rings studied here are overdamped, en-suring that when a local minimum disappears the systemrelaxes to the adjacent minimum, i.e., such that |∆n| = 1.

In conclusion, we have measured the transitions of thewinding number in flux-biased uniform one-dimensionalsuperconducting rings. We have shown that the insta-bility is reached as predicted by stationary Ginzburg-Landau theory. This detailed knowledge of the instabil-ity condition should enable systematic study of thermaland quantum phase slips near to the instability. Thisstudy can also be extended to rings formed from super-conductor/normal metal bilayers [50], where the dynam-ics of quasiparticles created by phase slips is of interestfor qubit decoherence [51].

We thank Michel Devoret, Zoran Radovic, Hen-drik Meier, Richard Brierley, Konrad Lehnert, AmnonAharony and Ora Entin-Wohlman for useful discussions,and Ania Jayich and Will Shanks for fabricating the sam-ples. We acknowledge support from the National Sci-ence Foundation (NSF) Grant No. 1106110 and the US-Israel Binational Science Foundation (BSF). L.G. wassupported by DOE contract DEFG02-08ER46482.

[1] W.A. Little, Phys. Rev. 156, 398 (1967).[2] J.S. Langer and V. Ambegaokar, Phys. Rev. 164, 498

(1967).[3] N. Giordano, Phys. Rev. Lett. 61, 2137 (1988).[4] J.M. Duan, Phys. Rev. Lett. 74, 5128 (1995).[5] H.P. Buchler, V.B. Geshkenbein, and G. Blatter, Phys.

Rev. Lett. 92, 067007 (2004).[6] J.E. Mooij and Yu.V. Nazarov, Nat. Phys. 2, 169 (2006).[7] J.E. Mooij and C.J.P.M. Harmans, New J. Phys. 7, 219

(2005).[8] R.S. Newbower, M.R. Beasley, and M. Tinkham, Phys.

7

Rev. B 5, 864 (1972).[9] N. Giordano and E.R. Schuler, Phys. Rev. Lett. 63 2417

(1989).[10] A. Bezryadin, C.N. Lau, and M. Tinkham, Nature 404,

971 (2000).[11] C.N. Lau, N. Markovic, M. Bockrath, A. Bezryadin, and

M. Tinkham, Phys. Rev. Lett. 87, 217003 (2001).[12] M.L. Tian, J. Wang, J.S. Kurtz, Y. Liu, M.H.W. Chan,

T.S. Mayer, and T.E. Mallouk, Phys. Rev. B 71, 104521(2005).

[13] M. Zgirski, K.-P. Riikonen, V. Touboltsev, and K. Aru-tyunov, Nano Lett. 5, 1029 (2005).

[14] A. Rogachev, A.T. Bollinger, and A. Bezryadin, Phys.

Rev. Lett. 94, 017004 (2005).[15] F. Altomare, A.M. Chang, M.R. Melloch, Y. Hong, and

C.W. Tu, Phys. Rev. Lett. 97, 017001 (2006).[16] M. Sahu, M.-H. Bae, A. Rogachev, D. Pekker, T.-C. Wei,

N. Shah, P.M. Goldbart, and A. Bezryadin, Nat. Phys. 5,503 (2009).

[17] P. Li, P.M. Wu, Y. Bomze, I.V. Borzenets, G. Finkelstein,and A.M. Chang, Phys. Rev. Lett. 107, 137004 (2011).

[18] T. Aref, A. Levchenko, V. Vakaryuk, and A. Bezryadin,Phys. Rev. B 86, 024507 (2012).

[19] O.V. Astafiev, L.B. Ioffe, S. Kafanov, Yu.A. Pashkin,K.Yu. Arutyunov, D. Shahar, O. Cohen, and J.S. Tsai,Nature 484, 355 (2012).

[20] J.T. Peltonen, O.V. Astafiev, Yu.P. Korneeva, B.M.Voronov, A.A. Korneev, I.M. Charaev, A.V. Semenov,G.N. Golt’sman, L.B. Ioffe, T.M. Klapwijk, and J.S. Tsai,Phys. Rev. B 88, 220506 (2013).

[21] M. Tinkham, ”Introduction to Superconductivity”,McGraw-Hill Book Co., NY (1975).

[22] W.A. Little and R.D. Parks, Phys. Rev. Lett. 9, 9 (1962);R.D. Parks and W.A. Little, Phys. Rev. 133A, 97 (1964).

[23] D.E. McCumber and B.I. Halperin, Phys. Rev. B 1, 1054(1970).

[24] K.A. Matveev, A.I. Larkin, and L.I. Glazman, Phys. Rev.Lett. 89, 096802 (2002).

[25] M.B. Tarlie and K.R. Elder, Phys. Rev. Lett. 81, 18(1998).

[26] J. Kurkijarvi, Phys. Rev. B 6, 832 (1972).[27] R. Rouse, S. Han, and J. E. Lukens, Phys. Rev. Lett. 75,

1614 (1995).[28] D.B. Schwartz, B. Sen, C.N. Archie, and J.E. Lukens,

Phys. Rev. Lett. 55, 1547 (1985).[29] V. Corato, S. Rombetto, C. Granata, E. Esposito, L.

Longobardi, M. Russo, R. Russo, B. Ruggiero, and P. Sil-vestrini, Phys. Rev. B 70, 172502 (2004).

[30] X. Zhang and J.C. Price, Phys. Rev. B 55, 3128 (1997).[31] L.S. Tuckerman and D. Barkley, Physica D 46, 57 (1990).[32] L. Kramer and W. Zimmermann, Physica D 16, 221

(1985).[33] M. Karttunen, K.R. Elder, M.B. Tarlie, and M. Grant,

Phys. Rev. E 66, 026115 (2002).[34] M.B. Tarlie, E. Shimshoni, and P.M. Goldbart, Phys.

Rev. B 49, 494 (1994).[35] D.Y. Vodolazov and F.M. Peeters, Phys. Rev. B 66,

054537 (2002).[36] M. Lu-Dac and V.V. Kabanov, Phys. Rev. B 79, 184521

(2009).[37] A. McKane and M. Tarlie, Phys. Rev. E 64, 026116

(2001).[38] S. Pedersen, G.R. Kofod, J.C. Hollingbery, C.B.

Sorensen, and P.E. Lindelof, Phys. Rev. B 64, 104522

(2001).[39] D.Y. Vodolazov, F.M. Peeters, S.V. Dubonos and A.K.

Geim, Phys. Rev. B 67, 054506 (2003).[40] H. Bluhm, N.C. Koshnick, M.E. Huber, and K.A. Moler,

arXiv:0709.1175v1 (2007).[41] J.A. Bert, N.C. Koshnick, H. Bluhm, and K.A. Moler,

Phys. Rev. B 84, 134523 (2011).[42] N.C. Koshnick, H. Bluhm, M.E. Huber, and K.A. Moler,

Science 318, 1440 (2007).[43] H. Bluhm, N.C. Koshnick, M.E. Huber, and K.A. Moler,

Phys. Rev. Lett. 97, 237002 (2006).[44] O. Bourgeois, S.E. Skipetrov, F. Ong, and J. Chaussy,

Phys. Rev. Lett. 94, 057007 (2005).[45] F.R. Ong, O. Bourgeois, S.E. Skipetrov, and J. Chaussy

Phys. Rev. B 74, 140503(R) (2006).[46] A.C. Bleszynski-Jayich, W.E. Shanks, B. Peaudecerf, E.

Ginossar, F. von Oppen, L. Glazman, and J.G.E. Harris,Science 326, 272 (2009).

[47] M.A. Castellanos-Beltran, D.Q. Ngo, W.E. Shanks, A.B.Jayich, and J.G.E. Harris, Phys. Rev. Lett. 110, 156801(2013).

[48] W.E. Shanks, Ph.D. thesis, Yale University, 2011.[49] J. Pearl, Appl. Phys. Lett. 5, 65 (1964).[50] O. Entin-Wohlman, H. Bary-Soroker, A. Aharony, Y.

Imry, and J.G.E. Harris, Phys. Rev. B 84, 184519 (2011).[51] C. Wang et al., Nat. Commun. 5, 5836 (2014).

8

Supplemental material

Measurement of the winding number instability

in mesoscopic superconducting rings

Data treatment

We measure the shift of the resonant frequency of the cantilever f as function of field B. An example of raw datataken for a ring with radius R = 406 nm at temperature T = 762 mK is shown in Figure 7(a). Red trace correspondsto the sweep up of bias field and blue to the sweep down. In addition to the sawtooth oscillations associated with therings’ superconductivity, we observe that f also undergoes a small drift as a function of time and of B. To removethis background, we fit the data above the rings’ critical field to a third-order polynomial, which is shown as the blackcurve in Fig. 7(a). We subtract this fit from f to obtain the frequency shift df due to the magnetic moment of therings (Figure 7(b)).

FIG. 7: (a) Raw data, cantilever frequency shift as function of field for increasing B (red) and decreasing B (blue) for the ringswith radius R = 406 nm and T = 762 mK. Third order polynomial background is shown as the black curve. (b) Cantileverfrequency shift after background substraction and averaging. The signal is due to the magnetic moment of the supercurrent.

Supercurrent as function of field

In our measurement configuration the magnetic field is perpendicular to the rings’ surface and the frequency shiftis related to supercurrent I as df(B) = κ I(B)BR2π, where κ is a cantilever-specific constant [1, 2]. This constantdepends on the cantilever’s resonant frequency, spring constant, length and the number of rings on it. The resonantfrequency is measured in the phase-locked loop, the length is measured by optical imaging, and the number of rings isknown from the lithography pattern. The spring constant k is obtained as a fitting parameter of the Ginzburg-Landaufit, as explained in the Main text and here in the following section. The best-fit value is within 20% of the nominalvalue computed as k = (2πf)2meff , where meff = m/4 is the effective mass of the cantilever and m is the cantilever’sactual mass. The rings’ radius is also obtained from the Ginzburg-Landau fits. It is highly constrained by the periodof Aharonov-Bohm oscillations, with the result that the statistical error on the best-fit value is ≈ 1 nm. The valuesreturned by this fit agree well with the values measured by SEM observations.The supercurrent obtained from the frequency shift as I(B) = df(B)/(2πκR2B) is shown in Figure 8. Every panel

shows data measured on a sample with a different ring size in the full available temperature range.The signal becomes very noisy close to zero field, and is not displayed for B very close to 0. This is because

I ∝ df/B and for fields close to zero, dividing the signal df by B leads to unreliable results.

Ginzburg-Landau fit for a one dimensional ring with finite width

We fit the data using one dimensional Ginzburg-Landau theory which includes the effects of finite ring width. Morespecifically, the expression for supercurrent is given as Eq. (7) in [3]. From this expression we compute the frequency

9

FIG. 8: Measured supercurrent per ring as function of field for different samples with R = 780, 538, 406, 288 nm (top panel tobottom panel), in the full temperature range. Lower curves of the same color correspond to increasing B and upper ones todecreasing B.

10

shift and fit it to the df data.

To do this, we first identify the winding number of each segment of df(B). For the measurements taken withincreasing B, we count the number of segments (i.e., the regions of smoothly varying df between jumps) between Bc3

and −Bc3. This number is 2nmax + 1, where nmax is the maximum winding number. We thus determine nmax. Thenwe start from Bc3 and count down from nmax to zero. We apply the equivalent process to measurements taken withdecreasing B.

FIG. 9: Supercurrent per ring as function of perpendicular field for different ring sizes (columns) and temperatures (marked oneach panel). Points and thin curves: data; thick curves: Ginzburg-Landau fit (see text). Red curves on each graph correspondto sweeping the field up, and the blue ones to sweeping down. Thin black dotted curves: the Ginzburg-Landau fit, extendedover the full field range of each winding number.

As explained in the Main text, it is a global fit which fits the entire I(B) measurement (i.e., for all winding numbersand for B increasing and decreasing). The fitting parameters are: superconducting coherence length ξ, penetrationdepth λ, ring radius R, ring width w and the spring constant k. Of these, we expect R, w and k to be fixed foreach sample (i.e., to not change with temperature), so we first undertake a preliminary fit to determine these threeparameters. In these preliminary fits, there is a degeneracy between λ and k, since they both set the amplitude of thesignal: λ affects the condensation energy, and therefore the amplitude of the current, while k affects the proportionalityconstant κ between current and frequency shift. Therefore we first set the starting value kin to its calculated nominalvalue (using the expression given in the preceding section) and λin such that Bc0 (the zero temperature bulk criticalfield, set by the product of ξ0 and λ0, where zeroes denote the zero temperature value) is 0.01 T, as expected foraluminium [4]. Then we run the fit for each of the I(B) measurements (i.e., at different T ) for that sample. We thenfix k to be the mean of the values returned by these preliminary fits. Values for R and w are fixed in the same way:by picking the mean of the values obtained from fits at different temperatures. The scatter between the obtainedvalues for k, R and w at different temperatures is rather small (a few percent for k and w and less than 1 nm for R).

In the second round of the fit only two fitting parameters remain, ξ and λ. Note that ξ also affects the condensationenergy, and therefore the amplitude of the signal, but it is not degenerate with k and λ since it is very accurately setby the rings’ critical field Bc3 ∼ ξ−1, as detailed in the Main text. This is in a sense lucky because our subsequentconclusions on the switching flux value hinge on the precise determination of ξ. This can be seen from Eqs. (1) and(2) in the Main text which show that the switching flux criteria depend only on R and ξ. Another remark is thathere we simply take λ as the fitting parameter (a number at each temperature) and the subsequent analysis of its

11

FIG. 10: Supercurrent per ring as function of perpendicular field for different ring sizes (columns) and temperatures (marked oneach panel). Points and thin curves: data; thick curves: Ginzburg-Landau fit (see text). Red curves on each graph correspondto sweeping the field up, and the blue ones to sweeping down. Thin black dotted curves: the Ginzburg-Landau fit, extendedover the full field range of each winding number.

temperature dependence reveals that it is in fact the Pearl penetration depth λP .We have made measurements for T > 400 mK. We have tried fitting below 750 mK (∼ Tc/2) but we have found that

the values of w, λ0 and ξ0 don’t converge to a fixed value like they do for T > 750 mK. We expect the Ginzburg-Landautheory to be valid roughly above Tc/2 and this is confirmed by the fit.The result of the measurement and the Ginzburg-Landau fit are shown in Figures 9 and 10, where data is shown

as points connected by thin curves and the fit is shown as thick curves. Three temperatures spanning the wholemeasured range are shown for each ring size. Red curves on each panel are for sweep up and blue for sweep down.The dotted black curves show the fit results extended over the full field range for each winding number; note that theportion of the dotted black curve occupied when B is increasing (red) is different from the part occupied for when Bis decreasing (blue) in the hysteretic part of I(B).The most pronounced discrepancy between the data and the fit is found for the biggest ring at the lowest temper-

atures, top left panel of Fig. 9, where the ring’s self-inductance L starts to play a role. In this regime, we estimateLI ∼ 0.13Φ0 in that case, which may lead to non-negligible skewing of the rings’ current-phase relation [5]. For therest of the measurements considered here, the effects of L are unimportant, i.e. LI ≪ Φ0. For smallest to largest ringsize we have computed the expected L = 0.5 − 2.3 pH. This gives LI ∼ 0.03, 0.05, 0.07, 0.13Φ0 respectively at 750mK. At higher temperatures LI is less since I decreases with temperature.

Transition width

We have also studied the width of the jumps from one winding number to another, which is non-zero since themeasurement is performed on an array of rings. The result is shown in Figure 11 where the transition widths aregiven as function of the winding number n, proportional to B, for all ring sizes and temperatures. We see that thetransition widths have a non-zero value at n = 0 and increase roughly linearly with B. The slope of the B-dependence

12

is independent of T and decreases with R. It is consistent with lithographic imprecision of 1 - 2 nm (the exact numberfor radius variation dR is given in the box of each graph) within each array of nominally identical rings.

FIG. 11: Observed transition width in units of normalized flux dΦ/Φ0 as function of winding number n, for different ring sizesand temperatures. Solid lines indicate a linear fit to the whole data set at each ring size. Value dR is the radius variationwhich would lead to the slope of the fit. Value Φn=0 is the extrapolated transition width at zero field, obtained from the fit.Radius sizes are: (a) R = 780 nm, (b) R = 538 nm, (c) R = 406 nm and (d) R = 288 nm.

The transition width at zero field also shows no discernible temperature dependence. It is a factor of 5− 10 largerthan expected from the rings’ mutual inductance, which results in rings at the middle of an array seeing a slightlydifferent field than those at the edge. Extrapolating the linear behavior of the transition width to zero field givesΦn=0, which is seen to increase with R.

We conclude from these observations the rings’ temperature does not influence the transition widths in measurementsof these arrays. This is consistent with the fact that the transition width expected for thermal switching across abarrier [6] is estimated to be several times less than the observed width.

Scaling of the phase slip flux

As mentioned in the Main text, Figure 5 is divided into four panels for clarity. Here it is given as a single figure,which demonstrates that the chosen normalization leads to the scaling of all data. The dotted lines from Figure5, which showed the prediction for the switching flux in the case of R ≫ ξ, are omitted here, as well as the barsindicating the transition width. Full lines show the prediction for the switching flux in the relevant case of R & ξ.

13

FIG. 12: Switching flux as function of the winding number. Dots: experimental values; full lines: prediction for the switchingflux dφ∗

n (see Main text). Colors represent temperature in the same way as in Figure 5 of the Main text. Ring sizes are denotedas: squares R = 780 nm, circles R = 538 nm, upward triangles R = 406 nm and downward triangles R = 288 nm.

[1] A.C. Bleszynski-Jayich, W.E. Shanks, B. Peaudecerf, E. Ginossar, F. von Oppen, L. Glazman, and J.G.E. Harris, Science326, 272 (2009).

[2] W.E. Shanks, PhD Thesis, Yale University (2011).[3] X. Zhang and J.C. Price, Phys. Rev. B 55, 3128 (1997).[4] M. Tinkham, ”Introduction to Superconductivity”, McGraw-Hill Book Co., NY (1975).[5] H.J. Fink and V. Grunfeld, Phys. Rev. B 33, 6088 (1986).[6] D.E. McCumber and B.I. Halperin, Phys. Rev. B 1, 1054 (1970).