Deterministic phase slips in mesoscopic superconducting rings

Abstract

The properties of one-dimensional superconductors are strongly influenced by topological fluctuations of the order parameter, known as phase slips, which cause the decay of persistent current in superconducting rings and the appearance of resistance in superconducting wires. Despite extensive work, quantitative studies of phase slips have been limited by uncertainty regarding the order parameter's free-energy landscape. Here we show detailed agreement between measurements of the persistent current in isolated flux-biased rings and Ginzburg-Landau theory over a wide range of temperature, magnetic field and ring size; this agreement provides a quantitative picture of the free-energy landscape. We also demonstrate that phase slips occur deterministically as the barrier separating two competing order parameter configurations vanishes. These results will enable studies of quantum and thermal phase slips in a well-characterized system and will provide access to outstanding questions regarding the nature of one-dimensional superconductivity.

ARTICLE

Received 5 Jun 2016 | Accepted 14 Oct 2016 | Published 24 Nov 2016

The properties of one-dimensional superconductors are strongly inuenced by topological uctuations of the order parameter, known as phase slips, which cause the decay of persistent current in superconducting rings and the appearance of resistance in superconducting wires. Despite extensive work, quantitative studies of phase slips have been limited by uncertainty regarding the order parameters free-energy landscape. Here we show detailed agreement between measurements of the persistent current in isolated ux-biased rings and GinzburgLandau theory over a wide range of temperature, magnetic eld and ring size; this agreement provides a quantitative picture of the free-energy landscape. We also demonstrate that phase slips occur deterministically as the barrier separating two competing order parameter congurations vanishes. These results will enable studies of quantum and thermal phase slips in a well-characterized system and will provide access to outstanding questions regarding the nature of one-dimensional superconductivity.

DOI: 10.1038/ncomms13551 OPEN

Deterministic phase slips in mesoscopic superconducting rings

I. Petkovi1, A. Lollo1, L.I. Glazman1,2 & J.G.E. Harris1,2

1 Department of Physics, Yale University, 217 Prospect Street, New Haven, Connecticut 06520, USA. 2 Department of Applied Physics, Yale University, 15 Prospect Street, New Haven, Connecticut 06520, USA. Correspondence and requests for materials should be addressed to I.P. (email: mailto:[email protected]

Web End [email protected] ).

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Phase slips are topological uctuations of the order parameter in one-dimensional superconductors1. They are responsible for the emergence of nite resistance in

the superconducting state and for the decay of supercurrent in a closed loop24. Despite extensive research and a good understanding of their basic features, there remain a number of open questions related to their dynamics5. One of the conceptually simplest systems in which to study phase slips is an isolated, ux-biased ring. Such a system can access several metastable states and undergoes a phase slip when it passes from one of these states to another2. Tuning the free-energy barrier between the states to zero with the applied ux F will result in a deterministic phase slip from the state that has become unstable6, whereas tuning the barrier to a small but non-zero value will lead to a stochastic phase slip via thermal activation2,3 or quantum tunnelling714.

The interpretation of measurements of stochastic phase slips7,1525 has been complicated by these processes strong dependence on the systems details, such as the form of the free-energy landscape, the damping of the order parameter and the noise driving its uctuations. Of particular importance is accurate knowledge of the barrier between metastable states, which enters exponentially into the rate of stochastic phase slips5. In contrast, deterministic phase slips are governed solely by the form of the free-energy landscape: they occur when the barrier is tuned to zero. For a strictly one-dimensional ring (in which the order parameter only varies along the rings circumference), GinzburgLandau (GL) theory can be used to analytically calculate the barrier height, the ux at which the deterministic phase slips occur2,3 and the measurable properties of the metastable states, for example, their persistent current2,3,26 and heat capacity27. As a result, measurements of these properties that demonstrate precise agreement with theory are important for benchmarking a system in which to study thermal and quantum stochastic phase slips. Previous measurements of persistent current I(F) in isolated superconducting rings have found quantitative agreement with theory only at low magnetic eld and very close to the critical temperature Tc, where metastability is absent or nearly absent26,28,29. However, at lower temperatures, where metastability is well-established, only qualitative agreement with theory has been demonstrated3032.

Here we present measurements of I(F) in isolated superconducting rings for temperatures spanning Tc/2oToTc.

The results, over the full range of magnetic eld, show quantitative agreement with the GL theory augmented by the empirical two-uid model4; the latter states the temperature dependence of the input parameters of GL theory in a broad temperature domain. The combination of the GL theory, nominally valid only at T-Tc, with the two-uid model has been shown to accurately represent the results of microscopic theory down to TETc/2 and was successfully used, for example, in explaining measurement of the parallel critical eld of thin Al lms33 in this temperature range. We nd that phase slips occur at the ux values predicted by GL theory, even to the point of demonstrating a small correction due to the rings nite circumference34,35. In addition, we nd that the dynamics of the phase slips is strongly damped, so that the disappearance of a barrier leads the system to relax to the adjacent local minimum. The measurement described here employs cantilever torque magnetometry, which has been shown to be a minimally invasive probe of persistent current in isolated metal rings36 and is capable of resolving individual phase slips in a single ring37. As a result, these measurements demonstrate the essential features for studying stochastic phase slips: samples with a well-characterized free-energy landscape and a detection scheme suitable for measuring their intrinsic dynamics.

ResultsDescription of the system. In this experiment, four separate samples were measured. Each sample consists of an array of 1001,000 nominally identical aluminum rings. Arrays were used to get a better signal-to-noise ratio. Ring radii of the four samples are R 288780 nm, with nominal widths of

w 6580 nm and thickness d 90 nm. Detailed sample

properties are listed in the Methods section and in the Supplementary Table 1. Scanning electron microscopy photos of the sample are shown in Fig. 1a.

The measurement setup is shown in Fig. 1b. A uniform magnetic eld of magnitude B is applied normal to the rings equilibrium orientation. As the cantilever oscillates, current circulating in the rings experiences a torque gradient, which shifts the cantilevers resonant frequency by an amount df, monitored by driving the cantilever in a phase-locked loop. More details on the measurement setup are given elsewhere37,38. In the conguration used here, df k I F, where F BpR2 and k

is a constant depending on the cantilever parameters, inversely proportional to the spring constant36,37. A detailed description of the conversion of data from df to I is given in the Supplementary Notes 1 and 2, and Supplementary Fig. 1.

Metastable states and hysteresis. A superconducting ring is considered one dimensional if its lateral dimensions are smaller than the coherence length x and the penetration depth l. The equilibrium properties of such a ring have three distinct temperature regimes, which are set by R/x(T). For temperature

T only slightly below Tc such that 2Rox, the ring is in a superconducting state for some values of F, whereas for the other values it is in the normal state39,40, due to competition between the superconducting condensation energy and the ux-imposed kinetic energy of the supercurrent. At slightly lower T (such that xo2Ro

3

p x), the condensation energy is slightly larger and for each value of F the ring has exactly one superconducting state. Finally, at even lower T such that 2R4

3

p x, the condensation energy is high enough to allow for several equilibrium states at a given F. Depending on the rings circumference, these three regimes may occur in the vicinity of Tc described by the GL theory or may extend to lower temperatures, prompting the use of the empirical two-uid model along with GL.

Figure 1ce shows I(B) for the sample with R 538 nm as T is

varied. The red points show measurements taken while B is increasing and the blue points while B is decreasing. All the measurements exhibit sawtooth-like oscillations whose period is inversely proportional to the ring area pR2. The smooth parts of the sawtooth represent current In in equilibrium states characterized by the order parameter winding number n and the jumps correspond to phase slips between these states. The jumps occur with ux spacing equal to the superconducting ux quantum F0 h/2e, indicating that n changes by unity at each jump.

Measured I(B) curves for all other temperatures and ring sizes are given in Supplementary Fig. 2. The three qualitative regimes described previously are accessed by varying either T or B, as they both diminish the condensation energy. For low T and B the data are hysteretic, indicating the presence of multiple equilibrium states. At sufciently high T or B the hysteresis vanishes, indicating that only one superconducting state is available. For the highest values of B and T there are ranges of B over which I 0 (to within the resolution of the measurement),

corresponding to the rings re-entry into the normal state. In this so-called LittleParks regime we observe the expected features: the persistent current goes through zero when the ux bias equals an integer number of ux quanta, whereas the winding number changes at half-integer values39,40. This is

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b

a

PIEZO

B

PLL

PD

c

R = 538 nm T = 758 mK

T = 1,061 mK

T = 1,228 mK

100

0

100

0.00

50

0

0

I(A)I(A)I(A)

0.02 0.04 0.06

d

50

10

0.00

0.00

0.01

0.02

0.03 0.04

e

10

0.01

B (T)

0.02

Figure 1 | Measured I(B). (a) Scanning electron microscopy photos of the sample. Left to right: several free-standing cantilevers, an array of rings on a single cantilever, a single Al ring from the array. The scale bars left to right are: 100 mm, 2 mm and 500 nm. (b) Measurement setup. A cantilever supporting rings is placed in a perpendicular magnetic eld B. The cantilevers position is monitored by a laser interferometer (red). The signal from the photodiode (PD) is sent to a phase-locked loop (PLL), which drives a piezoelectric element (green) under the cantilever. The current in the rings is determined from the frequency of the PLL drive. (ce) Supercurrent per ring I as a function of magnetic eld B for rings with radius R 538 nm at different temperatures

T (marked on each panel). Points are data; thick curves are the ts described in the text. Red (blue) corresponds to increasing (decreasing) B.

described in more detail in Supplementary Note 3 and shown in Supplementary Figs 3 and 4.

Fit to theory. To compare these measurements with theory, we rst identify the winding number n of each smooth portion of I(B). Next, we simultaneously t all of the smooth portions of I(B) using the analytic expression derived from the GL theory for one-dimensional rings26. This expression includes the rings nite

width w, which accounts for the magnetic eld penetration into the ring volume and is crucial for reproducing the overall decay of I at large B. At each value of T, the tting parameters are x and the Pearl penetration depth lP l2/d, appropriate when the

bulk penetration depth l4d (ref. 4; ref. 41) which holds. The cantilever spring constant is assumed to be temperature independent and is used as a global t parameter for each sample, along with the ring dimensions w and R. The resulting ts are shown as thick curves in Fig. 1ce. The full set of ts to

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measured I(B) for all R and T is shown in Supplementary Figs 5 and 6, along with a more detailed description of the tting procedure given in Supplementary Note 4.

In each data set we identify the rings critical eld Bc3, which

we take to be the value of B at which I becomes indistinguishable from 0 and remains so for all B4Bc3. It is noteworthy that the identication of Bc3 is independent of any theoretical model. Next, we use the GL result for one-dimensional rings31

Bc3 3.67F0/(2pwx(T)) to extract x(T) (the t parameters are

Bc3,0 3.67F0/(2pwx0) for each sample and Tc common to all the

samples). The coherence lengths x(T) extracted from the ts of I(B) and from the Bc3(T) data agree with each other in the entire temperature interval and are approximated remarkably well by x T

x0

1 t2

= 1 t2

p

, where t T/Tc. The same relation

inspired by the two-uid model4 was used successfully to treat the thin-lm upper critical eld33,42. Along with x(T), ts of I(B) yield the temperature dependence of the Pearl penetration depth, which agrees well with the two-uid model, lP(T) lP0/(1 t4). Figure 2 shows the best-t parameters x and

lP, as well as Bc3, all as a function of T. The best-t values of x0 (B200 nm), lP0 (B100 nm), Tc (B1.32 K) and Bc3,0, along with more details, are given in Supplementary Note 5 and Supplementary Table 1. Lastly, we note that Bc3(T) should be

independent of R and proportional to 1/w, consistent with the data in Fig. 2c.

Criterion for deterministic phase slip. Figure 1ce shows that on each branch In, the values of current at which the phase slips occur for increasing and decreasing B are located nearly symmetrically around zero current. To examine the locations of

these phase slips quantitatively, we dene Df n f n fmin;n.

Here f n is the experimental value of the normalized ux f F/F0 at which the transition n$n 1 occurs and fmin,n is

the value of f at which In reaches zero. Flux fmin,n is either directly measured or obtained by extrapolation between sweep-up and sweep-down branches. As dened, Dfn are positive (increasing B, for which n-n 1) and Df n are negative

(decreasing B, for which n-n 1). (In the following we nor

malize all ux values by F0 and denote them by the character f.)

Our next step is to compare the experimental values of switching ux Df n with theory. In the LangerAmbegaokar picture, valid for a current-biased wire much longer than x, the barrier between states n and n 1 vanishes when the bias current

reaches the critical current Ic (ref. 2). In the case of a ux-biased ring, still for R44x, the barrier between states n and n1 goes to zero at ux values

f c;nfmin;n

R

3

p x O

w

R 2

; 1

where fmin;n

n 1

w

2R

. In the case R\x, which corresponds to

our experimental situation, it was shown that the system remains stable beyond f c;n and loses stability at a ux34,35

f f;nfmin;n

R

3

2

p x

x2 2R2

s O

w

R 2

: 2

From these expressions we see that the switching ux is set by the ratio R/x and therefore the precise determination of x is crucial for quantitative comparison with theory. To simplify this

1

a

b

1,000

500

0.10

[afii9841] (nm)

[afii9838] P (nm)

300

100

0.8 1.0 1.2 0.8 1.0 1.2 T (K)

T (K)

T (K)

c

B c3(T)

0.05 R (nm)780538406288 80

65

w (nm)

65

80

0.000.0 0.5 1.0

Figure 2 | Coherence length, penetration depth and rings critical eld. (a,b) Coherence length x and Pearl penetration depth lP as a function of temperature. The squares are the best-t values from the GL ts described in the text. (c) Rings critical eld Bc3 as function of temperature. The squares are determined from measurements of I(B). The lines in all panels are the ts described in the text.

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comparison, it is convenient to refer all quantities not to zero eld, but to the zero current eld of each winding number, so we dene Df c;n f c;n fmin,n and Df f;n f f;n fmin,n.

Additional details on the free energy landscape close to the phase slip points are given in Supplementary Note 6 and Supplementary Fig. 7.

Figure 3 shows the measured Df n as a function of n. The vertical axis in Fig. 3 is normalized to Dff;0. The horizontal axis is normalized to the experimentally observed maximum winding number nmax, where nmaxE

3

p R2wx . The

ratio n/nmax is very close to B/Bc3. There is a symmetry Dfn Df n for nmax r n r nmax; thus, it sufces to

consider nZ0. Figure 3a shows the data for R 288 nm. The bars

represent the width of the steep portion of the sawtooth oscillations, primarily due to the small size inhomogeneities in the array (see Supplementary Figs 8 and 9, and Supplementary Notes 7 and 8). In Fig. 3b we show the data for all four samples, normalized such that all the data collapse together. Supplementary Fig. 8 shows the same data separated into four panels by ring size for a more detailed comparison.

The solid lines in Fig. 3 show the predicted Df f;n/Dff;0 (see equation (2)), whereas dotted lines in Fig. 3a show

Df c;n/Dff;0 (equation (1)). The difference between the solid and dotted lines increases with the ratio x(T)/R and is therefore

the most pronounced for small rings (Fig. 3a) or at high temperature due to the increase of x(T). We see that the prediction Df f;n/Dff;0 , which includes the nite-circumference effect (R\x), agrees well with the measured switching locations over the full range of T, B and R.

The nite-circumference effect can also be seen directly in Fig. 4, which shows I(B) over a narrow range of B for the smallest rings. For both increasing B (red) and decreasing B (blue), each sawtooth oscillation reaches a maximum current and then starts to diminish before the switching occurs, as seen in the regions indicated by the black arrows.

Damping. For T well below Tc once x is exceeded sufciently by the circumference of the ring, there are typically multiple free-energy minima into which the system may relax. Despite this freedom, we nd that the winding number always changes as |Dn| 1. This is seen for all measured rings and all T down to the

lowest value T 460 mK. In contrast, previous experiments30,31

with Al rings at To400 mK have found |Dn|41.

We expect the tendency for |Dn|41 to increase with lowering T. Indeed, a circulating current of almost-critical value and temperature T close to Tc result, respectively, in the suppression of the BCS (Bardeen-Cooper-Schrieffer) singularity in the electron

a

R = 288 nm

1.0

0.5 T (mK)

T (mK)

758

800

900

1,000

1,100

1,200 288

+

f,0

+

n

[p10] /[p10]

861

921

993

0.0

0.5

1.0

1.0

0.5

0.0

0.5

1.0

0.0 0.2 0.4 0.6 n/nmax

b

R (nm)

+

f,0

780

+

n

[p10] /[p10]

538

406

0.8 1.0

Figure 3 | Phase slip ux as a function of winding number. Dots: experimental values; bars in a: observed width of each jump due to size inhomogeneities in the array; full lines: prediction for the phase slip ux Df f;n; dotted lines in a: prediction for the phase slip ux Df c;n (see text). Colours represent temperature. (a) The sample with R 288 nm and (b) data from all the samples. The normalization of the axes is explained in the text.

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I(A)

100

50

0

50

100

0.00 0.01

B (T)

0.02

Figure 4 | Direct observation of the nite-length correction to the phase slip criterion. Supercurrent per ring I as function of magnetic eld B for rings with radius R 288 nm and temperature T 861 mK. Red (blue) points: increasing (decreasing) B. The regions over which I(B) diminishes at xed winding

number are indicated by black arrows. Diminishing of current after having reached a maximum but before the phase slip event is due to the nite-length correction to the phase slip criterion.

density of states and high density of Bogoliubov quasiparticles in a superconductor4. These are the two conditions making the dynamics of the order parameter dissipative and well described43 by the time-dependent GL equation. In the context of phase slips3, it determines a viscous motion of the phase difference across the phase slip, j t

(t being time), down the monotonic part of the

effective potential relief V j

, and this viscous motion results in

|Dn| 1. In the opposite limit of low temperatures, the quasi-

particle density is low and we may try considering the phase slip dynamics in terms of the Andreev levels associated with the phase slip. Their time evolution caused by the variation of j t

results in

LandauZener tunnelling between the occupied and empty levels, thus leading to dissipation44 of the kinetic energy of the condensate (the energy is irreversibly spent on the production of quasi-particles). Our estimate (Supplementary Note 9) of the energy lost in this way is Ediss S=e2rx

D, where r and S are, respectively,

the normal state resistivity and cross-section of the aluminum wire forming the ring, and D is the superconducting gap; a numerical proportionality factor is beyond the accuracy of the estimate.

The condensate energy difference between the two metastable states involved in a |Dn| 1 transition is EDn1 =e

jcS

S=e2rx

D; here, jc D= erx

is the critical current density.

Furthermore, the lower of the two states is protected by a barrier dFDn1 x=R

5=2EDn1 (the estimate is easily obtained from the

LangerAmbegaokar2 scaling, dFp(1 j/jc)5/4, of the barrier with

the current density j, see Supplementary Note 9). The height of the barrier is smaller for larger rings.

We nd the irreversibly lost energy Ediss to be of the order of the

energy difference between the two metastable states EDn 1. The

above estimates, given their limited accuracy, allow (but do not guarantee) the condensate to have a sufcient excess of kinetic energy to overcome a small barrier out of the metastable state with Dn 1. In addition to higher temperatures, in a notable difference

from the previous experiments the rings studied here had smaller R, providing a better protection of the metastable states.

DiscussionWe have studied the persistent current in arrays of ux-biased uniform one-dimensional superconducting Al rings. We found detailed agreement with GL theory, including the location of deterministic phase slips, which are predicted to occur when the

barrier conning the metastable state occupied by the ring goes to zero. In one dimension, GL theory has a relatively simple, analytic form and, due to their small width, our rings are strictly in the one-dimensional limit, in contrast to those studied previously3032. As a result, GL theory provides detailed knowledge of the free-energy landscape in these samples. This should enable systematic study of thermal and quantum phase slips in isolated rings and progress towards the quantitative understanding of coherent quantum phase slips45,46, one of the outstanding goals in the eld14,47,48.

Methods

Sample fabrication. Ring radii of the four measured samples are R 288, 406, 538

and 780 nm, nominal widths are w 65 nm (for R 406, 780 nm) and 80 nm

(for R 288, 538 nm), and thickness d 90 nm. Further details on sample properties

are listed in the Supplementary Table 1. Each array is fabricated on a Si cantilever of length B400 mm, thickness 100 nm and width B60 mm, with resonant frequency fB2 kHz, spring constant kB1 mN m 1 and quality factor QB105. Cantilevers are fabricated out of a silicon-on-insulator wafer. They are patterned out of the top silicon layer by means of optical lithography followed by a reactive ion etch. Rings are then fabricated on top of patterned cantilevers using standard e-beam lithography with a polymethyl methacrylate (PMMA) mask, into which Al is evaporated in a high-vacuum thermal evaporator. After lift-off, the top of the wafer is protected and the backing silicon layer is etched in KOH, followed by a BOE etch of the SiO2 layer and drying in a critical point dryer. This results in cantilevers being fully suspended. Further details on the the fabrication process are given elsewhere36,37.

Data availability. The data that support the ndings of this study are available from the corresponding author upon request.

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Acknowledgements

We thank Amnon Aharony, Richard Brierley, Michel Devoret, Ora Entin-Wohlman, Alex Kamenev, Konrad Lehnert, Hendrik Meier and Zoran Radovi for useful discussions, and Ania Jayich and Will Shanks for fabricating the samples.

We acknowledge support from the National Science Foundation (NSF) Grant Number 1106110 and the US-Israel Binational Science Foundation (BSF). L.G. was supported by DOE contract DEFG02-08ER46482.

Author contributions

A.L. and I.P. performed the measurement. All authors conducted the analysis. I.P., J.G.E.H. and L.I.G. wrote the manuscript. All authors discussed the results and commented on the manuscript.

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How to cite this article: Petkovi, I. et al. Deterministic phase slips in mesoscopic superconducting rings. Nat. Commun. 7, 13551 doi: 10.1038/ncomms13551 (2016).

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NATURE COMMUNICATIONS | 7:13551 | DOI: 10.1038/ncomms13551 | http://www.nature.com/naturecommunications

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