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
Institut für Theoretische Physik, Freie Universität Berlin, Fachbereich Physik, 14195 Berlin, Germany
3
Department of Applied Physics, Yale University, New Haven CT, 06520 USA
2
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 manybody 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
"
# $ ( r 2 Inˆ is the magnetic moment of the PC, r is the ring radius, and n̂ 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
300 mK, corresponding to a magnetic moment sensitivity of ~ 11 #B/Hz1/2 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 I+,arises 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 I#typ ( f ! ) , the absolute value of the Fourier transform of the data in Figs 2A-
D (f! is the “flux frequency” in units of (h/e)-1). 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 I#typ ( 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
I#typ ( 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 halfwidth 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 I#typ ( 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
the ring at the Fermi energy, and hence should scale as 1/r2. In addition, Ih/e(T) is predicted[18] to
decrease on a temperature scale (known as the Thouless temperature) TT . 1/ r 2 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,
“average” current would scale as I +(avg) . N
rather than
N . Furthermore, the amplitude of
I +(avg) can be strongly enhanced by electron-electron interactions[4] and other effects[8,9].
However I +(avg) 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
suppresses I +(avg) by a factor ~ e /2( r /1.3$ B (where the magnetic length $ B $ h / eB ), which for this
experiment should render I +(avg) 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
predicted[18] to have a zero-temperature amplitude Ih/2e(0) = Ih/e(0)/23/2, 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
each independent realization of the disorder potential, Ih/pe (the pth harmonic of I(!)) is drawn
6
randomly from a distribution with a mean
general is non-zero. Here %
I h2 pe
1/2
D
D
Ih
pe D
$ 0 and an rms value
I h2 pe
1/2
D
which in
represents an average over disorder potentials. The quantity
can be calculated explicitly as a function of r, T, p, and the electrons’ diffusion constant
D for a variety of models.
To compare our data against these calculations, we make use of the fact that I h2 pe
1/2
D
can
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
under a peak in I#typ ( f ! ) (cf. Figs. 2E-H) at f! = p is simply related to I h2 pe
0
D
:
1/ 2
3 f!
4
2
5 9 I#typ ( f ! ) / b( f ! ) df ! 6
57 f!/
68
2
1/2
1
$ I h2 pe
1/ 2
D
.
(1)
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 !2 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 h2 pe
D
could only be determined from successive measurements of
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.
Measurements of I h2 e
1/ 2
D
for each sample and I h2 2 e
1/ 2
D
for the smallest rings are shown
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
that
I h2 2 e
1/ 2
D
decays more quickly than
I h2 e
1/ 2
D
, consistent with the discussion above. In
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.
The solid lines in Fig. 3 are fits to theoretical predictions in which I h2 pe
1/ 2
D
is calculated
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:
I h2/ pe (T )
where g ( x ) $
(6
3
D
; T <
$ g = p 2 > I h2/ pe (0)
? TT @
(2)
D
A
x 2 B n exp[ /(2( 3nx )1/ 2 ] , I h2/ pe (0)
n $1
1/2
D
$ 0.37 p /3/ 2
&( 2 D
3eD
T
,
and
$
.
T
k B (2( r ) 2
(2( r ) 2
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 highpurity 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! ' 2(r for nearly all the temperatures at which the PC
is observable. The closest approach between L! and 2(r 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 × (2(r). It is
conceivable that the more rapid decrease in I h2 e
1/ 2
D
observed in this sample above T = 2 K (Fig.
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 × (2(r). 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
Sample
308 nm array
418 nm array
793 nm array
418 nm ring
Wire (see SOM)
r (nm)
308
418
793
418
289,000 (length)
w (nm)
115
85
85
85
115
d (nm)
90
90
90
90
90
N
1680
990
242
1
1
D (cm2/s)
271 ± 2.6
214 ± 3.3
205 ± 6.5
215 ± 4.6
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 CB larger than shown in (A)-(D): CB = (I) 5.4 T, (J)
5.3 T, (K) 0.6 T, and (L) 1.1 T (full data shown in SOM).
12
,
1
,
308 nm array h/e
308 nm array h/2e
418 nm array
418 nm single ring
793 nm array
Current (nA)
0.1
0.01
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
indicates I h2 e
1/ 2
D
and
I h2 2 e
1/ 2
D
, the rms values of the Fourier amplitudes of the persistent current. In each data
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.
13
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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.
15
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3. Measurement diagnostics
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D'3e W+.('i'7 re/ev.(' @or -.<i(0
*o-2.ri%o(% 6i'3 '3eor7FJ .(d re%+/' i( '3e 2oi('% -.r<ed 97 .rro6% i( Si0. X o@ '3e -.i( 2.2er.
Sor '3e re-.i(der o@ '3e 2oi('% i( Si0. XJ 6e -.<e +%e o@ '3e @.*' '3.' IDBF i% @o+(d e-2iri*.//7 'o
de2e(d +2o( T o(/7 vi. .( over.// %*./i(0J .% %3o6( i( Si0. NKH. ?o 0e(er.'e '3e 2oi('% i( Si0. X
(o' -.r<ed 97 .rro6%J '3i% %*./i(0 i% .22/ied 'o '3e v./+e o@ I hG e
%6ee2%.
KX
Ka G
D
de'er-i(ed @ro- '3e /.r0e B
5. Supplementary Figures
Figure S1 NOP i-.0e o@ . GHI !- /o(0 6ire %i-i/.r 'o '3e o(e +%ed @or 'r.(%2or' -e.%+re-e('%.
R (!)
300
200
100
0
40
50
60 70
B (mT)
80
90
Figure S2 qe%i%'.(*e ver%+% -.0(e'i* @ie/d .' XME -\ @or '3e 'r.(%2or' 6ire. ?3e %+2er*o(d+*'i(0 'r.(%i'io( o**+r%
over . r.(0e o@ -.0(e'i* @ie/d 9e0i((i(0 .' EG -?. ?3e -.0(e'i* @ie/d 6.% %6e2' i( '3e dire*'io( o@ i(*re.%i(0 @ie/d
%'re(0'3.
Kd
20
HC (mT)
15
10
5
0
1.15
1.20
Temperature (K)
Figure S3 N+2er*o(d+*'i(0 *ri'i*./ @ie/d ver%+% 'e-2er.'+re @or '3e 'r.(%2or' 6ire. Sor '3e d.'. %3o6(J H*DTF 6.%
'.<e( 'o 9e '3e @ie/d .' 63i*3 '3e re%i%'.(*e 6.% 'e( 2er*e(' o@ '3e (or-./ %'.'e v./+e. ?3e d.'. .re @i' +%i(0 OW. NG
6i'3 DH* .(d T* .% @i''i(0 2.r.-e'er%. ee@i(i(0 H*DTF 'o o**+r .' . di@@ere(' 2er*e('.0e o@ '3e (or-./ %'.'e re%i%'.(*e
v./+e %3i@'% '3e @i''ed T* %/i03'/7 9+' doe% (o' .@@e*' DH*.
-4
"R/R ( x 10 )
3
2
2.9 K
4.7 K
12.6 K
1
0
-5
0
5
B (mT)
Figure S4 f3.(0e i( re%i%'.(*e ver%+% -.0(e'i* @ie/d 6i'3 @i'% 'o OW%. NX .(d NM @or '3ree di@@ere(' 'e-2er.'+re%. V(
order 'o .*3ieve .( .deW+.'e %i0(./ 'o (oi%e r.'ioJ '3e d.'. 6ere @i' over r.(0e% %3o6( de%2i'e '3e *o(di'io(% *i'ed i(
'3e 'e5' @or v./idi'7 o@ OW. NM.H
KE
A
5
L#$(%m)
4
3
2
1
0
0
2
4
6
8
10
12
Temperature (K)
B
6
5
4
L#$(%m)
3
2
1
.6
.5
2
3
4
5
6
7
8
9
10
Temperature (K)
Figure S5 O/e*'ro( 23.%e *o3ere(*e /e(0'3 LM ver%+% 'e-2er.'+re. ?3e do'% re2re%e('% v./+e% e5'r.*'ed @ro- @i'% 'o
'3e -.0(e'ore%i%'.(*e o@ '3e GHI !- 6ire. ?3e %.-e d.'. i% 2/o''ed o( . /i(e.r %*./e DRF .(d . /o04/o0 %*./e D:F. ?3e
%o/id /i(e i% '3e @i' 'o '3e @+(*'io(./ @or- de%*ri9ed i( '3e 'e5'J .(d '3e d.%3ed /i(e% i(di*.'e '3e %2e*i@i*
*o('ri9+'io(% 'o '3i% @i' @ro- e/e*'ro(4e/e*'ro( %*.''eri(0 .(d e/e*'ro(423o(o( %*.''eri(0J .% di%*+%%ed i( '3e 'e5'.
KM
Current (nA)
A
2
1
0
-1
-2
6.7
6.8
6.9
7.0
7.1
B (T)
Frequency Shift (mHz)
B
10
8
6
100
50
4
0
2
0.5
1.0
1.5
2.0
2.5
0
0.0
0.5
1.0
1.5
2.0
2.5
Flux amplitude through ring (#0)
Figure S6 (A) ee'.i/ o@ '3e 2er%i%'e(' *+rre(' ver%+% -.0(e'i* @ie/d @or '3e .rr.7 o@ XQH (- ri(0%. Sor '3e
-e.%+re-e(' %3o6(J '3e *.('i/ever 6.% orie('ed 6i'3 # ^ Ms. ?3e .rro6% i(di*.'e '6o @ie/d v./+e% .' 63i*3
-e.%+re-e('% o@ '3e *.('i/ever @reW+e(*7 %3i@' 6ere 2er@or-ed .% . @+(*'io( o@ *.('i/ever .-2/i'+de. (B) ei@@ere(*e
i( *.('i/ever @reW+e(*7 %3i@' @or '3e '6o @ie/d v./+e% i(di*.'ed i( (A) ver%+% *.('i/ever .-2/i'+de o@ o%*i//.'io(. ?3e
*.('i/ever 6.% drive( i( i'% %e*o(d @/e5+r./ -odeJ 63i*3 3.d . @reW+e(*7 o@ KXiKH Z; .(d . %2ri(0 *o(%'.(' o@ Q.QEX
na-. ?3e *.('i/ever 3.d . /e(0'3 o@ ddI !-. ?3e *.('i/ever .-2/i'+de i% 2/o''ed o( '3e x4.5i% i( 'er-% o@ '3e
.-2/i'+de o@ '3e @/+5 -od+/.'io(,M.* Di( +(i'% o@ '3e @/+5 W+.('+-F '3ro+03 '3e ri(0 2rod+*ed 97 '3e *.('i/ever
-o'io(. ?3e %o/id *+rve i% . @i' +%i(0 OW. NKG D6i'3 p ^ KF 6i'3 IK ^ E.E L Q.G (R .(d r ^ GME L G (-. ?3e i(%e'
%3o6% '3e @reW+e(*7 %3i@' -e.%+red .' '3e '6o 2oi('% i(di*.'ed i( (A) .(d 3.% '3e %.-e +(i'% .% '3e -.i( 2/o' i( (B).
Ki
Figure S7 f.('i/ever @reW+e(*7 %3i@' ver%+% -.0(e'i* @ie/d @or @ive di@@ere(' *.('i/ever o%*i//.'io( .-2/i'+de% q-.5
(-e.%+red .' '3e /o*.'io( o@ '3e ri(0%F. ?3e /o6er 2.(e/ %3o6% '3e %.-e d.'. .% '3e +22er 2.(e/J 9+' %*./ed %o .% 'o
0ive &' Dq-.5 $ QF +%i(0 OW. NKG 6i'3 p ^ K. ?3e 'r.*e% *o//.2%e o( 'o2 o@ e.*3 o'3erJ i(di*.'i(0 '3.' '3e7 .re d+e 'o
eW+i/i9ri+- 2er%i%'e(' *+rre('%.
KH
Figure S8 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d
-e.%+red 63e( o%*i//.'i(0 '3e *.('i/ever .' G.G <Z; D'3e *.('i/ever[% @ir%' @/e5+r./ re%o(.(*eF .(d .' KX.i <Z; D'3e
*.('i/ever[% %e*o(d @/e5+r./ re%o(.(*eF. ?3e 2er%i%'e(' *+rre(' doe% (o' de2e(d o( '3e *.('i/ever o%*i//.'io(
@reW+e(*7h '3e %/i03' di@@ere(*e i( '3e '6o *+rve%[ %-oo'3 9.*<0ro+(d% i% 2re%+-.9/7 d+e 'o di@@ere(' -e*3.(i*./
re%o(.(*e% 2re%e(' i( '3e %.-2/e 3o/der .' G.G .(d KX.i <Z;.
Figure S9 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' I´ Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or .
%erie% o@ /.%er 2o6er% i(*ide(' o( '3e *.('i/ever. Sor '3e d.'. %3o6( i( '3e -.i( 2.2erJ E (] o@ /.%er 2o6er 6.%
+%ed.
KI
Figure S10 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' I´ Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or
9o'3 -.0(e'i* @ie/d 2o/.ri'ie%. ?3e 54.5i% o@ '3e (e0.'ive -.0(e'i* @ie/d 'r.*e i% (e0.'ed.
Figure S11 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d +(der
di@@ere(' -ode% o@ o2er.'io( o@ '3e %+2er*o(d+*'i(0 %o/e(oid 2rod+*i(0 '3e -.0(e'i* @ie/d. ?3e o2er.'io(./ -ode%
o@ '3e -.0(e' .re i(di*.'ed i( '3e @i0+re. Sor '3e d.'. %3o6( .% 0ree( do'%J .// e/e*'ro(i*% De5*e2' @or '3e 2ie;o
driveF 6ere di%*o((e*'ed @ro- '3e *r7o%'.'.
GQ
Figure S12 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or .(
.rr.7 o@ KMHQ ri(0% 6i'3 r.di+% XQH (- .' T ^ XME -\. ?3e @+// %6ee2 i% %e2.r.'ed i('o '3ree 2.(e/% @or */.ri'7. ?3e
@ie/d i% .22/ied .' dEs 'o '3e 2/.(e o@ '3e ri(0%.
GK
Figure S13 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or .
%i(0/e ri(0 o@ r.di+% dKH (- .' T ^ XME -\. ?3e @ie/d i% .22/ied .' dEs 'o '3e 2/.(e o@ '3e ri(0. ?3e @+// %6ee2 i%
%e2.r.'ed i('o '6o *o('i0+o+% 2.(e/% @or */.ri'7.
GG
Figure S14 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF @or .( .rr.7 o@ IIQ ri(0% 6i'3
r ^ dKH (- .' T ^ XME -\. .(d ) ^ dEs 'o '3e 2/.(e o@ '3e ri(0%.
GX
Figure S15 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or .(
.rr.7 o@ GHG ri(0% 6i'3 r.di+% iIX (- .' T ^ XGX -\. ?3e @ie/d i% .22/ied .' Ms 'o '3e 2/.(e o@ '3e ri(0%.
Figure S16 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or .(
.rr.7 o@ IIQ ri(0% 6i'3 r.di+% dKH (- .' T ^ XGX -\. ?3e @ie/d i% .22/ied .' Ms 'o '3e 2/.(e o@ '3e ri(0.
Gd
Figure S17 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d @or .(
.rr.7 o@ KMHQ ri(0% 6i'3 r.di+% XQH (- .' T ^ XGX -\. ?3e @ie/d i% .22/ied .' Ms 'o '3e 2/.(e o@ '3e ri(0.
GE
Figure S18 ?3e deriv.'ive o@ '3e 2er%i%'e(' *+rre(' It Dderived @ro- OW%. NKE .(d NKMF ver%+% -.0(e'i* @ie/d D+22er
2/o'F @or '3e .rr.7 o@ r ^ XQH (- ri(0% -e.%+red 6i'3 '3e @ie/d .22/ied .' dEs 'o '3e 2/.(e o@ '3e ri(0%. ?3e /o6er 2/o'
%3o6% '3e So+rier 'r.(%@or- o@ '3e %.-e d.'.. ?r.*e% .re '.<e( .' di@@ere(' 'e-2er.'+re%J .% i(di*.'ed i( '3e @i0+re
/e0e(d. ?3e .-2/i'+de o@ *+rre(' o%*i//.'io(% de*re.%e% 6i'3 i(*re.%i(0 'e-2er.'+reJ 9+' '3e %3.2e o@ '3e o%*i//.'io(%
re-.i(% +(*3.(0ed.
GM
6. References and Notes
K
`. f3.(dr.%e<3.rJ P3.e. '3e%i%J u./e p(iver%i'7 DKIHIF.
G
?3e 9rid0e de%*ri9ed i( TKU 3.% . @o+r 'er-i(./ .rr.(0e-e('. R '3ree 'er-i(./ .rr.(0e-e(' 6.%
+%ed 3ere 9e*.+%e o(e /e.d 9ro<e .% '3e %.-2/e *oo/ed 'o d.G \ i( o+r *r7o%'.'. g( o(e %ide '3e
%.-2/e 6.% *o((e*'ed 'o '3e 9rid0e .(d '3e vo/'.0e 2ro9e '3ro+03 %e2.r.'e /e.d%J 63i/e o( '3e
o'3er %ide '3e %.-2/e 6.% *o((e*'ed 'o 0ro+(d .(d '3e 9rid0e '3ro+03 '3e %.-e /e.d. ?3e %.-2/e
re%i%'.(*e 6.% de*o+2/ed @ro- '3e /e.d re%i%'.(*e 97 '3e 2ro*ed+re de%*ri9ed i( '3e 'e5'.
X
n. R%3*ro@'J n. Per-i(J Solid State Physics DZ.r*o+r' :r.*eJ gr/.(doJ KIiMFJ 22. E4XH.
d
P. ?i(<3.-, Introduction to Superconductivity DP*br.6 Zi//J ne6 uor<J ed. GJ KIIMF.
E
P. c.S.r0eJ P. =o7e;J e. O%'eveJ f. pr9i(.J P. Z. eevore'J Nature 365J dGG DKIIXF.
M
P. N.('3.(.-J N. ]i(dJ e. Pro9erJ Phys. Rev. B 35J XKHH DKIHiF.
i
R. c.r<i(J JETP Lett. 31J GKI DKIHQF.
H
no'e '3e @ir%' *o(di'io( 9e*o-e% v./id @or T .% /o6 .% YK.XE \ D63ere '3e /.r0e 'er- i% Y KQQ
'i-e% '3e %-.// 'er-F. Zo6everJ '3e %e*o(d *o(di'io( re%'ri*'% '3e v./id r.(0e o@ -.0(e'i* @ie/d%
'o re/.'ive/7 %-.// v./+e% over '3e r.(0e o@ 'e-2er.'+re% re/ev.(' 'o o+r -e.%+re-e('%. :e*.+%e
'3e %i;e o@ '3e -.0(e'ore%i%'.(*e %i0(./ i% %-.// @or @ie/d r.(0e% o@ /e%% '3.( . @e6 -?J '3e d.'.
%3o6( i( Si0. NE 6ere @i' over . @ie/d r.(0e '3.' i(*re.%ed /i(e.r/7 6i'3 T @ro- E -? 'o KM -?.
Sor '3e /o6e%' 2/o''ed 'e-2er.'+re DK.i \FJ '3e %e*o(d *o(di'io( %3o+/d 9e v./id @or @ie/d% -+*3
/e%% '3.( Q.iE -?. :7 d \J '3e @ie/d r.(0e %3o+/d 9e re%'ri*'ed 'o -+*3 /e%% '3.( M -? .(d 97 KQ
\J GE -?. ?3e e5'r.*'ed LF doe% de2e(d 6e.</7 o( '3e %i;e o@ '3e @i' r.(0e. Sor e5.-2/eJ .' G.d
\ 63ere '3e e52re%%io( i% v./id @or @ie/d% -+*3 /e%% '3.( G -?J '3e @i''ed LF i(*re.%e% 97 GQo .%
'3e @i' r.(0e i% i(*re.%ed @ro- X -? 'o XQ -?.
I
]. c.6re(*eJ :. Pe.dorJ Phys. Rev. B 18J KKEd DKIiHF.
KQ
V. c. R/ei(erJ :. c. R/'%3+/erJ P. Z. ber%3e(%o(J Waves Rand. Compl. Media 9J GQK DKIIIF.
KK
Nee @or e5.-2/e Z. bo/d%'ei(J Classical Mechanics DRddi%o(4]e%/e7J ne6 uor<J ed. XJ
GQQGFJ 22. EdK4EdI
Gi