Dependence of the magnitude and direction of the persistent current on the magnetic flux in...

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2007, Vol. 105, No. 6, pp. 1157–1173. © Pleiades Publishing, Inc., 2007.Original Russian Text © V.L. Gurtovoi, S.V. Dubonos, A.V. Nikulov, N.N. Osipov, V.A. Tulin, 2007, published in Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, 2007, Vol. 132,No. 6, pp. 1320–1339.

1157

1. INTRODUCTION

It is known that the current

I

=

E

/

R

l

induced by

the Faraday electromotive force

E

= –

d

Φ

/

dt

, flow-

ing along a conducting loop

l

s

, should cause the poten-tial difference

on the segment

l

s

with the average resistance

,

different from the average resistance

R

l

/

l

=

⟨ρ

/

s

⟩

l

=

/

sl

along the whole circle.

Under certain conditions, the potential difference

V

dc

(

Φ

/

Φ

0

) is observed on an asymmetric superconduct-ing loop with [1] and without Josephson junctions [2,3]. The sign and magnitude of the potential differenceperiodically depend on the magnetic flux

Φ

inside theloop; the period is equal to the flux quantum

Φ

0

=

π

/

e

.It is known [4] that the averaged value

⟨

I

p

⟩

of the circu-lating persistent current

I

p

(

Φ

/

Φ

0

) flowing in such loopsshould exhibit a similar dependence on the magneticflux. Thus, there is a certain analogy between the per-

ld∫°ld∫°

Rls

ls

------ρs---

ls

lρdsls

--------

s

∫= =

Vρs---

ls

ρs---

l–⎝ ⎠

⎛ ⎞ lsl Rls

Rlls

l--------–⎝ ⎠

⎛ ⎞ I RasymI= = =

lρd∫°

sistent current [4], i.e., the direct current existing due toBohr quantization under equilibrium conditions in thedc magnetic field

Φ

≠

n

Φ

0

and the ordinary circulatingcurrent existing due to the Faraday electromotive force,i.e., at

d

Φ

/

dt

≠

0. Since the observed dependences

V

dc

(

Φ

/

Φ

0

) [1–3] are proportional to

⟨

I

p

⟩

(

Φ

/

Φ

0

), we canwrite

introducing the quantity

R

asym

by analogy with theresistance difference (

R

ls

–

R

l

l

s

/

l

) defining the relationbetween the potential difference and circulating currentin an ordinary asymmetric loop. We emphasize thatobservation of oscillations

V

dc

(

Φ

/

Φ

0

) on superconduct-ing structures both with [1] and without [2, 3] Joseph-son junctions confirms the analogy between a one-dimensional ring and the superconducting quantuminterferometer, noted in some papers [5].

The quantity

R

asym

in a superconducting ring, unlikean ordinary ring, is not constant, but depends on variousfactors, first of all, on temperature. It is clear that

R

asym

≠

0 and the potential difference

can be observed only in the resistive state. The ringresistance in an equilibrium state is nonzero,

R

l

> 0, attemperatures higher than

T

>

T

c

or close,

T

≈

T

c

, to thesuperconducting transition temperature

T

c

. Observa-

Vdc Φ/Φ0( ) Rasym I p⟨ ⟩ Φ/Φ0( ),=

Vdc Φ/Φ0( ) Rasym I p⟨ ⟩ Φ/Φ0( )=

Dependence of the Magnitude and Direction of the Persistent Current on the Magnetic Flux in Superconducting Rings

V. L. Gurtovoi, S. V. Dubonos, A. V. Nikulov, N. N. Osipov, and V. A. TulinInstitute of Microelectronic Technology and High-Purity Materials, Russian Academy of Sciences,

Institutskaya ul. 6, Chernogolovka, Moscow oblast, 142432 Russiae-mail: [email protected]

Received January 25, 2006

Abstract—The obtained periodic magnetic-field dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) of the critical currentmeasured in opposite directions on asymmetric superconducting aluminum rings has made it possible to explainpreviously observed quantum oscillations of dc voltage as a result of alternating current rectification. It wasfound that a higher rectification efficiency of both single rings and ring systems is caused by hysteresis of thecurrent–voltage characteristics. The asymmetry of current–voltage characteristics providing the rectificationeffect is due to the relative shifts of the magnetic dependences Ic–(Φ/Φ0) = Ic+(Φ/Φ0 + ∆φ) of the critical currentmeasured in opposite directions. This shift means that the position of Ic+(Φ/Φ0) and Ic–(Φ/Φ0) minima does notcorrespond to n + 0.5 magnetic flux Φ quanta, which is in direct contradiction to measured Little–Parks resis-tance oscillations. Despite this contradiction, the amplitude Ic, an(Φ/Φ0) = Ic+(Φ/Φ0) – Ic–(Φ/Φ0) of critical cur-rent anisotropy oscillations and its variations with temperature correspond to the expected amplitude of persis-tent current oscillations and its variations with temperature.

PACS numbers: 74.78.Na

DOI: 10.1134/S1063776107120072

ORDER, DISORDER, AND PHASE TRANSITIONIN CONDENSED SYSTEMS

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JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 105 No. 6 2007

GURTOVOI et al.

tions by Little and Parks [6] of resistance oscillations∆R(Φ/Φ0) of a thin-walled superconducting cylinder ina magnetic field showed that not only Rl > 0, but alsoIp(Φ/Φ0) ≠ 0 at temperatures T ≈ Tc corresponding to theresistive transition Rln > Rl > 0. According to the con-ventional explanation [4], the periodic variation in theresistance R(Φ/Φ0) of a thin-walled superconductingcylinder [6] or a ring [7], measured at a temperaturecorresponding to the resistive transition Rln > Rl > 0, iscaused by the change in the critical temperature,

The decrease in Tc at Φ ≠ nΦ0,

is related to an increase in the superconducting state

energy in proportion to (Φ/Φ0), when the allowedsuperconducting pair velocities vs(Φ/Φ0) ∝ (n – Φ/Φ0)cannot be zero [4, 5]. Little–Parks oscillations∆R(Φ/Φ0) = ∆V(Φ/Φ0)/Iext are observed at the measuredcurrent Iext, which can differ by hundreds of times [7];there is no reason to doubt that they should also beobserved in the limit of infinitesimal values of Iext , i.e.,at thermodynamic equilibrium. The persistent currentIp = sjp = s2ensvs ∝ n – Φ/Φ0 observed due to the quan-tization of the velocity circulation of superconductingpairs [4],

(1)

is an equilibrium phenomenon. At nonzero resistanceRl > 0, the persistent current can be observed due tothermal fluctuations [8], which switch the ring betweensuperconducting states with different wavefunctioncoherences.

At T < Tc , where Rl = 0 in equilibrium state, thesephenomena are observed at an external current exceed-ing the critical superconducting current Ic(T). In thecase of Little–Parks oscillation, this corresponds to theshift of the resistive transition Tc(Iext) by direct measur-ing current Iext [7]. Quantum oscillations Vdc(Φ/Φ0) ofthe dc voltage are induced at T < Tc by alternating cur-rent [3] or uncontrollable electrical noise [1, 2]. Theresults of the measurements [3] on asymmetric alumi-num rings showed that quantum oscillations Vdc(Φ/Φ0)of the dc voltage at T < Tc are induced by the externalalternating current, e.g., Iext = I0sin(2πft), when itsamplitude I0 exceeds the critical value I0c close to thecritical current Ic(T). It was shown that the result isindependent of the alternating current frequency f in thestudied frequency range f = 100 Hz–1 MHz, and theamplitude VA of Vdc(Φ/Φ0) quantum oscillation non-monotonically depends on the current amplitude I0 [3].As the Vdc(Φ/Φ0) amplitude reaches the maximum VA =VA, max at I0 = I0, max, it decreases as the external currentamplitude I0 further increases (see Figs. 11 and 12).

∆R Φ/Φ0( ) ∆Tc Φ/Φ0( ).–∝

∆Tc Φ/Φ0( )– v s2 Φ/Φ0( ) n Φ/Φ0–( )2,∝ ∝

v s2

lv sd∫°2πm

---------- nΦΦ0------–⎝ ⎠

⎛ ⎞ ,=

This result can be explained in two consistent ways.One explanation proceeds from the relationVdc(Φ/Φ0) = Rasym⟨Ip⟩(Φ/Φ0) obtained above by theanalogy between persistent and ordinary currents. Thenonmonotonic dependence of the amplitude VA ofquantum oscillations Vdc(Φ/Φ0) on the external currentamplitude I0 can be explained as a result of the super-conducting transition shift by the current Iext. Thepotential difference is nonzero, Vdc(Φ/Φ0) ≠ 0 atRasym ≠ 0 and ⟨Ip⟩(Φ/Φ0) ≠ 0. Above the superconduct-ing transition temperature, the resistance is nonzero,Rasym ≠ 0, but the persistent current is zero,Ip(Φ/Φ0) = 0; while Ip(Φ/Φ0) ≠ 0, but Rasym = 0 belowthe superconducting transition. At small current ampli-tude I0, the shift of the superconducting transition tem-perature Tc(Iext) does not reach the temperature T < Tc atwhich measurements are performed; hence,Vdc(Φ/Φ0) = 0. Quantum oscillations Vdc(Φ/Φ0)become observable, when I0 lowers the superconduct-ing transition temperature to the measurement temper-ature Tc(I0) ≈ T < Tc(0). In this case, they initiallybecome observable under strong magnetic fields whichpromote the superconducting transition shift to lowtemperatures (see Fig. 11). The amplitude VA reaches amaximum at I0 corresponding to the current whichshifts the transition by T – Tc(0) and decreases as I0 fur-ther increases due to the persistent current disappear-ance into a normal state.

The second explanation proposed in [3] interpretsquantum oscillations Vdc(Φ/Φ0) as a result of rectifica-tion of the external alternating current due to the asym-metry of current–voltage characteristic (CVCs) of theasymmetric ring. Such a ring with the geometry used inboth [3] and this study is shown in Fig. 1. To explain theobserved oscillations Vdc(Φ/Φ0), the CVC asymmetrysign and magnitude should be periodic functions of themagnetic field. Such periodic variations in the asymme-try are explained in [3] as a result of the superpositionof external Iext and persistent Ip(Φ/Φ0) currents.According to this explanation, the persistent current ina ring with unequal cross sections of halves (see Fig. 1)can be found as a quantity proportional to the differenceof critical currents measured in opposite directions.

The goal of this study is to test the model explainingquantum oscillations Vdc(Φ/Φ0) as a result of alternat-ing current rectification and to study in detail the corre-lation of the CVC behavior and asymmetry with therectification efficiency in structures with differentgeometries. The temperature dependences of the maxi-mum amplitude VA, max of quantum oscillations, theamplitude I0, max of the current Iext at which oscillationsare observed, and the critical current amplitude I0c werestudied in detail. The dependences of the persistent cur-rent as a quantity proportional to the critical currentanisotropy on the magnetic field and temperature wereobtained.

To solve the posed problem, first of all, CVCs of thesystems under consideration were measured at various


DEPENDENCE OF THE MAGNITUDE AND DIRECTION OF THE PERSISTENT CURRENT 1159

temperatures. The results of these measurements aregiven in Section 3.1. The magnetic dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0) of the critical current measuredin opposite directions of the external current Iext are pre-sented in Section 3.2. In Section 3.3, it is shown that thevoltage Vdc(Φ/Φ0) is a result of alternating current rec-tification on asymmetric rings whose CVC asymmetryvaries periodically in a magnetic field due to periodicvariations of the anisotropy of critical current Ic+(Φ/Φ0)

and Ic–(Φ/Φ0). It is emphasized that the rectificationmechanism near Tc where CVCs are reversible, i.e., nohysteresis is observed, can differ from that at lowertemperatures where a significant CVC hysteresis isobserved. The measured temperature dependences ofthe maximum amplitude VA, max of quantum oscillationsVdc(Φ/Φ0), the amplitude I0, max of the current inducingthese oscillations, and the rectification efficiencydefined as the ratio of these parameters are given in

1 µm

0.4 µm

0.2 µm

+Ip

+Iext

+Ip

+Iext

wcon

Fig. 1. Micrograph of the structure used in measurements. The width of ring halves is ww = 0.4 µm and wn = 0.2 µm; the width ofcontacts supplying the external current is wcon = 0.4 µm.

0

–500–2.5

V, µV

–Ic–

2.5 I, µA

Ic+

1

2

3

500

Fig. 2. CVC of the system of 18 asymmetric rings con-nected in series with wcon = 0.4 µm at temperatures (1) T =1.261 K ≈ 0.995Tc, (2) T = 1.252 K ≈ 0.988Tc, and (3) T =1.247 K ≈ 0.984Tc.

–500

–10

V, µV

–Ic–

10I, µA

Ic+

1

2

3

500

0 20–20

Fig. 3. CVC of the systems of 20 asymmetric rings con-nected in series with wcon = 1 µm at temperatures (1) T > Tc,(2) T = 1.222 K ≈ 0.988Tc, and (3) T = 1.199 K ≈ 0.969Tc.

1160


GURTOVOI et al.

Section 3.4. The observed amplitude of oscillations ofthe critical current and its anisotropy are compared withthe expected amplitude of persistent current oscilla-tions in Section 3.5. The qualitative differences ofobserved critical current oscillations from the expectedones are considered in Section 3.6. In the Conclusions,we note that the results obtained allow us to explain thepreviously observed phenomena; however, pose newproblems requiring further study.

2. EXPERIMENTAL DETAILS

The study was conducted in the temperature rangeT = 1.19–1.3 K on aluminum film nanostructures withthe superconducting transition temperature Tc ≈ 1.23–

1.27 K, the resistance per square of ~0.5 Ω at 4.2 K, andthe resistance ratio R(300 K)/R(4.2 K) ≈ 3. The diame-ter of all rings was 2r = 4 µm, the widths of ring halveslw and ln were ww = 0.4 µm and wn = 0.2 µm, respec-tively (see Fig. 1), the thickness was d = 40–50 nm; thecross sections were, respectively, sw = wwd ≈ 0.016–0.02 µm2 and sn = wnd ≈ 0.008–0.01 µm2. The measure-ments were performed on four samples. The first tworepresented a system of 20 rings connected in serieswith current contacts of width wcon = 0.4 µm (in thisgeometry, the signal was picked up from 18 rings) andwcon = 1 µm between them. The second pair consistedof single rings; in this case, the width of current con-tacts was wcon = 0.6 µm and wcon = 0.7 µm. The micro-structures were fabricated on Si substrates using a

–10

–2

V, µV

2 I, µA

1

2

30

0

10

20

Fig. 4. CVC of the single asymmetric ring with wcon =0.7 µm at temperatures (1) T = 1.243 K ≈ 0.999Tc, (2) T =1.241 K ≈ 0.997Tc, and (3) T = 1.239 K ≈ 0.995Tc.

–2

Ic, µA

Ic+

2

–10

–5

0

5

10

–Ic–

1

3

45

54321

–3 –1 0 1 2 3Φ/Φ0

Fig. 5. Dependences of critical currents on the magneticfield, Ic+(Φ/Φ0) and Ic–(Φ/Φ0), measured on the system of18 asymmetric rings at temperatures (1) T = 1.234 K ≈0.972 Tc, (2) T = 1.238 K ≈ 0.975Tc, (3) T = 1.241 K ≈0.977Tc, (4) T = 1.245 K ≈ 0.981Tc, and (5) T = 1.250 K ≈0.985Tc.

–8

Ic, µA

Ic+

2

–10

–5

0

5

10

–Ic–

1

3

–12 –4 0 4 8 12Φ/Φ0

3

2

Fig. 6. Critical current oscillations Ic+(Φ/Φ0) and Ic–(Φ/Φ0)in a magnetic field, measured (1) on the system of 18 ringswith wcon = 0.4 µm at T = 1.245 K = 0.981Tc; (2) on the 20-ring system with wcon = 1 µm at T = 1.209 K = 0.984Tc; and(3) on the single ring with wcon = 0.6 µm at T = 1.222 K =0.991Tc .

–2

Ic, an, µA

2–2.5

0

2.5

1

–3 –1 0 1 2 3Φ/Φ0

–4 4

Fig. 7. Critical current anisotropy oscillations Ic, an(Φ/Φ0)measured (1) on the system of 18 rings with wcon = 0.4 µmat T = 1.247 K = 0.982Tc and (2) on the single ring withwcon = 0.6 µm at T = 1.221 K = 0.991Tc.



JEOL-840A scanning electron microscope trans-formed into a laboratory electron lithograph using theNANOMAKER software package.

Measurements were performed by the four-probemethod in a glass helium cryostat using He4 as a cool-ant whose pumping allowed a temperature decrease to1.19 K. Direct, sawtooth, or sinusoidal current from aKeithley 6221 precision source was applied to currentcontacts. The direct current was used to measure thedependences of the sample resistance on the magneticfield R(B) (Little–Parks oscillations) and temperatures

R(T). The sawtooth current was used to record CVCsand the dependences of the critical current on the tem-perature Ic(T) and magnetic field Ic(B). The sinusoidalcurrent, along with the sawtooth current, was used tostudy the rectification effect and to measure the depen-dences of the rectified voltage on the magnetic fieldVdc(B). The voltage was measured on potential contactsby an instrumental amplifier with a gain of 1000 and anoise level (reduced to the input) of 20 nV in a fre-quency band from zero to 1 Hz. Then the signal wasamplified by an SR560 (Stanford Research Systems)preamplifier used for additional amplification and the

–2

Ic, µA

Ic+

2

–5

0

5

10

–Ic–

1

–5 –1 0 1 2 3Φ/Φ0

2

1

–3–4 4 5

Ic– – 3 µA

–Ic+ + 1.5 µA

Fig. 8. Relative shift of oscillations Ic+(Φ/Φ0) andIc−(Φ/Φ0) along the Φ/Φ0 axis, measured at opposite direc-tions of the measuring current (1) on the 20-ring systemwith wcon = 1 µm at T = 1.209 K = 0.984Tc and (2) on the18-ring system with wcon = 0.4 µm at T = 1.241 K =0.978Tc.

Ic, µA; Vdc, µV

–20

0

20Ic+

–10 –5 0 5 10Φ/Φ0

–15 15

–Ic–

Vdc–10

10

Fig. 9. Comparison of oscillations of the critical currentIc+(Φ/Φ0), Ic−(Φ/Φ0) and the dc voltage Vdc(Φ/Φ0) inducedby alternating current with frequency f = 400 Hz and anamplitude of 17.6 µA on the single ring at T = 1.216 K ≈0.986Tc.

–2

Ic, an, µA; Vdc/20, µV

–5.0

0

2.5

5.0

–4 –1 0 1 2 3Φ/Φ0

–3 4

–2.5

Vdc/20

Fig. 10. Comparison of oscillations of the critical currentanisotropy Ic, an(Φ/Φ0) and the dc voltage Vdc(Φ/Φ0)induced by alternating current with frequency f = 500 Hzand an amplitude of 10.6 µA in a magnetic field on the 20-ring system at T = 1.209 K ≈ 0.985Tc.

Vdc, µV

–200

0

600

–8 –4 0 4 8Φ/Φ0

–12 12

200

400

I0 = 30.0 µA

12.0

10.5

9.9

9.6

8.85

Fig. 11. Oscillations of dc voltage Vdc(Φ/Φ0) induced byalternating current with frequency f = 40 kHz and variousamplitudes I0 = 8.85, 9.6, 9.9, 10.5, 12, and 30 µA on the18-ring system at T = 1.234 K = 0.972Tc. All the depen-dences, except for I0 = 9.6 µA, are shifted vertically.

1162


GURTOVOI et al.

formation of a required signal band by low- and high-frequency filters. The temperature was measured usinga calibrated thermistor (R(300 K) = 1.5 kΩ) with a mea-suring current of 0.1 µA. The amplified voltage fromthe sample and the signals proportional to the samplecurrent, magnetic field, and temperature were simulta-neously digitized by a 16-bit ADC with eight differen-tial inputs.

The magnetic field B perpendicular to the sampleplane was induced by a copper solenoid placed outsidethe cryostat. The dependences of the measured valueson the solenoid current Isol were recorded. The mag-netic field induced by the solenoid current was deter-mined by the calibration Bsol = ksolIsol performed usinga Hall sensor probe, ksol = 129 G/A. The period B0 =Φ0/S = 1.4–1.6 G of all periodic dependences R(B),Vdc(B), and Ic(B) corresponded to the ring area S = πr2 =14.8–13.0 µm2 used in measurements, where r = 2.2–2.0 µm is the quantity close to the inner radius of agiven ring. To reduce the geomagnetic field, the cry-ostat region with a sample was shielded by a Permalloycylinder. The residual magnetic field was Bres ≈ 0.15 G,i.e., about one-tenth of the period B0. Due to partialshielding, the resistance minimum R(Bsol) and the zerovalue of the rectified voltage Vdc(Bsol) were observed atBsol = –Bres ≈ –0.15 G. In this paper, we present all themeasured dependences as functions of the magneticflux inside the ring, Φ = SB = S(Bsol + Bres), induced byexternal fields Bsol + Bres. The exact value of the area Swas chosen from the condition of the equality of theperiod of oscillations R(Φ), Vdc(Φ), and Ic(Φ) to themagnetic flux quantum Φ0. The exact value of the resid-ual magnetic field Bres was chosen from the condition of

R(Φ) minimum at Φ = 0, Vdc(Φ) = 0 at Φ = 0, and thecondition Ic–(Φ) = Ic+(–Φ). In all cases, the value of Bresis about the same. When plotting the dependencesR(Φ), Vdc(Φ), and Ic(Φ), the magnetic flux ΦI = ΦIext +ΦIp induced by the external Iext and persistent Ip currentswas neglected. We used the approximation Φ = Φext +ΦI ≈ Φext = S(Bsol + Bres), since the flux induced by theexternal ΦIext and persistent ΦIp currents does notexceed a few hundredths of the flux quantum in thestudied temperature range at the inductance L ≈ 2 ×10−11 H of rings used in this study and their low criticaland persistent currents. This problem is considered inmore detail in Section 3.6.

The dependences R(B), Vdc(B), and Ic(B) wererecorded during sawtooth temporal variations of thesolenoid current Isol(t) with a frequency of 0.01–0.1 Hz.The amplitude of magnetic field variations did notexceed 100 G. Little–Parks oscillations were recordedusing the direct current Iext = 0.1–2 µA. The depen-dences of the rectified voltage Vdc(Φ) were measuredusing the sinusoidal or sawtooth current Iext(t) with anamplitude up to 50 µA and a frequency of 0.5–5 kHz.The width of the rectified voltage spectrum caused bymagnetic field variations did not exceed 1 Hz; there-fore, the rectified voltage was amplified in a frequencyband from zero to 10–30 Hz. On the one hand, this didnot result in signal shape distortions; on the other hand,this prevented ac voltage appearance at the amplifieroutput.

The dependences of the critical current on the mag-netic field, Ic+(B) and Ic–(B), were measured from CVCsperiodically (10 Hz) repeated in a slowly varying mag-netic field (≈0.01 Hz) by the following algorithm:(i) the condition of the structure existence in the super-conducting state was tested and then (ii), as the thresh-old voltage (set above interferences and noises of themeasuring system and controlling the minimum mea-surable critical current) was exceeded, the magneticfield and critical current were determined with a delayof ~30 µs. Thus, the critical was sequentially measuredin positive Ic+ and negative Ic– directions of the measur-ing current Iext . The measurement of a single depen-dence Ic+(B), Ic–(B) containing 1000 points took ~100 s.

3. RESULTS AND DISCUSSION

To test the hypothesis about observation of quantumoscillations of the direct voltage Vdc(Φ/Φ0) as a result ofalternating current rectification, CVCs and the depen-dence of their anisotropy on the magnetic field shouldbe studied for the structures under consideration.

3.1. Current–Voltage Characteristics

Measurements showed three CVC types whichshould be distinguished to describe the rectificationprocess. We attribute smooth reversible CVCs to thefirst type (Figs. 2–4). This type is observed for all struc-tures near the superconducting transition: in the mea-

60

0 10

VA, µV

I0, µA5 15 20 25

180

120

VA, maxI0, max

1

2

3

Fig. 12. Dependences of the amplitude VA of rectified volt-age oscillations Vdc(Φ/Φ0) on the external current ampli-tude I0 for the 18-ring system with wcon = 0.4 µm at (1)sinusoidal and (2) sawtooth currents with f = 40 kHz, T =1.234 K ≈ 0.972Tc and (3) the 20-ring system with wcon =1 µm at sinusoidal current with f = 1 kHz, T = 1.214 K ≈0.989Tc.



surements on 18 rings with wcon = 0.4 µm at T > 0.993Tc(Fig. 2), on 20 rings with wcon = 1 µm at T > 0.985Tc(Fig. 3), and on single rings at T > 0.997Tc (Fig. 4). Thesecond and third types include CVCs (measured in thecurrent source mode) with hysteresis (Figs. 2–4). Thethird type is characterized by a sharp transition of theentire structure to the resistive state (Figs. 2, 4). In con-trast to this, in the second-type CVC, this transition isobserved in a certain range of the current Iext (Figs. 2, 3).The second-type CVCs were observed for the 20-ringsystem in an intermediate temperature range, 0.993Tc >T > 0.985Tc, on a system with wcon = 0.4 µm (Fig. 2),and in the entire range below T < 0.985Tc on a systemwith wcon = 1 µm (Fig. 3). In the measurements on a sin-gle ring, no second-type CVCs were observed (Fig. 4).The third-type CVCs are observed at low temperatures,T < 0.985Tc , in the measurements on 18 rings withwcon = 0.4 µm and T < 0.995Tc on a single ring.

The difference of the 20-ring structure with wcon =1 µm from the structure with wcon = 0.4 µm is theabsence of the third-type CVCs down to the lowesttemperatures at which the measurements were per-formed (Fig. 3). In contrast to this, CVCs of single ringsbelong to the third type in almost the entire temperaturerange up to T = 0.995Tc (Fig. 4). In this case, a sharpchange in the resistance is observed not only during thetransition of the single ring to the resistive state, butalso during the reverse transition from the resistive tothe superconducting state (Fig. 4), which is a distinctivefeature of single rings in comparison with a ring system(see Figs. 2 and 3). The difference is also observed inthe shape of resistive transitions: the lower part of theresistive transition of the single ring is narrower thanthat of the ring system (see Fig. 15). We relate thesequalitative differences between resistive characteristicsof the single ring and a long chain of rings to the prox-imity effect of wide superconducting areas arranged at

a distance of several µm from the single ring. The prox-imity effect on resistive characteristics of aluminumstructures appears in an anomalous increase in theresistance before the superconducting transition (seeFig. 15). Such an anomaly was observed by manyauthors [9] in measurements on aluminum structureswith sizes comparable to the correlation length ξ(T).This anomaly, as well as others [9], has not yet beenreasonably explained and requires further study. In thispaper, the features of resistive characteristics of alumi-num structures, including the anomaly, are consideredonly for better understanding of alternating current rec-tification resulting in Vdc(Φ/Φ0).

The third-type CVCs (curves in Figs. 2 and 4) aretypical of superconducting nanostructures with closecritical currents of their elements, when the transitionto the resistive state occurs due to exceeding the depair-ing current density [4],

For aluminum with the thermodynamic critical fieldextrapolated to the zero temperature, Hc(0) = 105 Oe,the London penetration depth λL(0) = 50 nm, and thedepairing current density is estimated as jc =jc(0)(1 − T/Tc)3/2 ≈ 9 × 106 A/cm2 (1 – T/Tc)3/2. Theexperimental temperature dependences of the criticalcurrent in the studied range T = 0.965 – 0.995Tc can bedescribed as Ic(T) = Ic(0)(1 – T/Tc)3/2, where Ic(0) ≈1.4 mA for an 18-ring system with wcon = 0.4 µm (theminimum cross section is scon ≈ 0.016 µm2), Ic(0) ≈3 mA for the 20-ring system with wcon = 1 µm (the min-imum cross section is sw + sn ≈ 0.03 µm2), Ic(0) ≈5.5 mA and Ic(0) ≈ 4.3 mA for single rings with wcon =0.7 µm and wcon = 0.6 µm, respectively. The critical cur-rent density (jc(0) ≈ 9 × 106 A/cm2 for the 18-ring sys-

jc

Hc T( )3 6πλL T( )-----------------------------

Hc 0( )3 6πλL 0( )---------------------------- 1 T /Tc–( )3/2.= =

–2

V, µV

–5

0

5

–4 0 2 4Φ/Φ0

Fig. 13. Quantum oscillations of the dc voltage induced bysawtooth alternating current with frequency f = 40 kHz andan amplitude of 3.2 µA on the 18-ring system at tempera-ture T = 1.263 K ≈ 0.996Tc, corresponding to the first-typeCVC, see Fig. 2.

5

1.20

Ic, µA; I0, max, µA

1.23

10

1.22

15

25

1.21 1.24T, K

0

30

20

35

10

20

30

40

VA, max, µV

0

Fig. 14. Temperature dependences of the critical current Ic(squares), the maximum amplitude VA, max of quantumoscillations of dc voltage, and the amplitude I0, max (closedtriangles) of the external current inducing VA, max (open tri-angles), measured on the single ring.

1164


GURTOVOI et al.

tem and jc(0) ≈ 107 A/cm2 for the 20-ring system) isclose to theoretical values. The critical current densityof single rings (jc(0) ≈ 2.3 × 107 A/cm2 for a ring withwcon = 0.7 µm and jc(0) ≈ 1.8 × 107 A/cm2 for a ring withwcon = 0.6 µm) is somewhat higher. The increased crit-ical current in single-ring structures can be caused bythe effect of wide superconducting banks due to a rela-tively short length of the structure with a minimumcross section of ≈0.024 µm2 in comparison with thecorrelation length ξ(T).

The sharp transition to the resistive state of thewhole structure at once and the significant CVC hyster-esis are caused by the nonmonotonic dependence of thesuperconducting current density js = 2ensvs on thesuperconducting pair velocity vs at velocities compara-

ble to the depairing velocity vsc = /m ξ(T) [4]. Thetransition to the normal state, ns = 0, as the criticalvelocity vs = vsc of superconducting pairs is reached byany structure region, results in a decrease in the pairdensity ns in neighboring regions at distances of theorder of the correlation length ξ(T), hence, in anincrease in the pair velocity at a given current Iext = Is =sjs = s2ensvs. An increase in the velocity above the crit-ical value results in the transition of these neighboringregions to the normal state, if their cross section areaslightly exceeds the least cross section area. Therefore,when the current density reaches the depairing currentdensity in one of segments, the whole structure transitsto the resistive state, even a long 18-ring system con-nected in series with a full length of 176 µm (seeFig. 2). The proximity effect on the pair velocity inneighboring regions of the structure should weaken asthe cross section of isolated regions increases and the

3

equilibrium pair density decreases. Therefore, thethird-type CVCs are not observed in the 20-ring struc-ture with wcon = 1 µm (Fig. 3); in the 18-ring systemwith wcon = 0.4 µm, the transition from the third to thesecond CVC type as Tc is approached is observed(Fig. 2). The hysteresis is observed because of theappearance of the nonzero density ns of superconduct-ing pairs at a given current Iext can be stable only ifns > Iext/2sevsc. Otherwise, pairs should be acceleratedin an electric field to the depairing velocity and disap-pear. The formation of the high pair density on suffi-ciently large length scales has a low probability; there-fore, the return to the superconducting state occurs atlower Iext than the transition to the resistive state(Figs. 2–4). The CVC hysteresis is observed if a changein the superconducting pair density due to fluctuationsdoes not exceed ns = Ic/s2evsc . This takes place at lowtemperatures, when the equilibrium pair density is sig-nificant and the kinetic energy of pairs,

,

corresponding to the depairing current Ic, is muchhigher than the thermal fluctuation energy kBT in a vol-ume equal to the product of the superconductor crosssection s and the correlation length ξ(T). Near Tc, thekinetic energy corresponding to the depairing current Ic

∝ (1 − T/Tc)3/2 becomes comparable to the thermal fluc-tuation energy,

Using the values λL(0) = 50 nm, ξ(T) = 170 nm,s = 0.016 µm2, Ic(T) = Ic(0)(1 – T/Tc)3/2 = 1.4 mA ×(1 − T/Tc)3/2, we find that

at T ≈ 0.993Tc; this corresponds to the temperature atwhich the CVC hysteresis of the 18-ring system (Fig. 2)disappears.

3.2. Magnetic-Field Dependenceof Critical Currents Ic+ and Ic–

Our measurements showed that the CVC portionwith hysteresis, corresponding to the current decrease,cannot be the cause of the CVC asymmetry and peri-odic variation with the magnetic field. This portion ischaracterized by the absence of systematic asymmetrywith respect to zero and appreciable variation with themagnetic field. In particular, the current Ic, s at which thereturn to the superconducting state occurs is indepen-dent of both the measuring current direction and mag-netic field. In contrast to this, the critical current corre-

Ekin

λL2 T( )ξ T( )

2s--------------------------µ0Ic

2 T( )=

Ekin

λL2 T( )ξ T( )

2s--------------------------µ0Ic

2 T( ) kBT .≈=

Ekin

λL2 0( )ξ 0( )

2s-------------------------µ0Ic

2 0( ) 1 TTc

-----–⎝ ⎠⎛ ⎞ 3/2

kBT≈=

EffRe, R/5Rn

1.021.00

0.1

0.98 1.04T/Tc

00.96

0.2 1

23

4

5

Fig. 15. Temperature dependences of the rectification effi-ciency EffRe = VA, max/I0, maxRn, measured on (1) the singlering and (2) the 18-ring system with Rn = 92 Ω. The resistivetransitions of (3) the single ring with Rn = 3.3 Ω, (4) the 20-ring system with Rn = 58 Ω , and (5) the 2-ring system withRn = 5.0 Ω .



sponding to the transition to the resistive state dependson both the measuring current direction and magneticfield value. We denote these quantities measured atopposite Iext directions as Ic+ and Ic– (Figs. 2 and 3). Forthe third type of CVC, Ic+ and Ic– correspond to a sharptransition of the whole system to the resistive state(Fig. 2); for the second type of CVC, Ic+ and Ic– corre-spond to the transition onset (Fig. 3). The technique forconstructing the dependences of these quantities on themagnetic field Ic+(B) and Ic–(B) is described inSection 2.

Measurements performed at various temperatureson both single rings and systems of rings showed peri-odic dependences of the critical current Ic+(B) andIc−(B) on the magnetic field. In all cases, the periodB0 = 1.4–1.6 G corresponded to the flux quantum B0S =Φ0 ≈ 20.7 G µm2 inside a ring with an area of S = 14.8–13.0 µm2. Since simultaneous changes in the magneticfield B and external current Iext directions are equivalentto a system rotation by 180°, which should not causechanges in the measured values, the equality Ic+(B) =Ic−(–B) should be satisfied for all values of B, where B =Bsol + Bres (Bsol = ksolIsol) is the magnetic field induced bythe copper solenoid, which was measured by the cur-rent Isol and the residual magnetic field Bres. For themeasured dependences, the equality Ic+(B) = Ic–(–B)was satisfied at Bres ≈ 0.15 G, which corresponded to theresidual field obtained from other measurements. Allfigures show the dependences of the critical current andother quantities on Φ/Φ0 = S(Bsol + Bres)/Φ0.

The noise level allowed us to reliably observe theperiodic dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) at a crit-ical current level exceeding 3 µA (Fig. 5). Near Tc ,where the first type of CVCs are observed, the depen-dences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) were not studiedbecause of the small critical current Ic < 3 µA. At thesame time, the constraint on the minimum temperatureof about 1.19 K (which could be achieve on the setupused) did not allow us to observe oscillations at a criti-cal current larger than 30 µA. In addition to the periodicdependence, a monotonic decrease in the critical cur-rent with the magnetic field is observed (Fig. 6).Despite this, more than 25 oscillation periods wereobserved (Fig. 6). Measurements performed on systemsof 20 rings connected in series with a current contactwidth wcon = 0.4 µm and wcon = 1 µm and on single ringswith wcon = 0.6 µm and wcon = 0.7 µm showed that thereis no qualitative difference between the dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0) obtained on the ring systemand the single ring (Fig. 6). At the same time, a certaineffect of the current contact width wcon is observed. Thecritical current decreases with increasing magnetic fieldto a greater extent in a structure with wider current con-tacts (Fig. 6). We recall here that other parameters, i.e.,the width of ring halves ww = 0.4 µm, wn = 0.2 µm, andthe ring diameter 2r = 4 µm, were identical in all struc-tures. We should emphasize the qualitative differencebetween the dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) mea-sured on structures with current contact widths smaller

and larger than (or equal to) the total width of ringhalves, wcon = 0.4 µm < ww + wn = 0.6 µm and wcon = 1,0.7, 0.6 µm ≥ ww + wn = 0.6 µm (Fig. 6). In the formercase, there are minima and plateaus in the dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0) (Figs. 5 and 6); in the lattercase, there are minima and maxima (Figs. 6 and 9).

The critical current anisotropy

is a periodic sign-alternating function of the magneticfield (Fig. 7). In all cases studied, this dependencecrosses the zero value at Φ = nΦ0 and Φ = (n + 0.5)Φ0(Fig. 7). We detected that the dependences of the criti-cal current, measured in opposite directions, are similaror almost similar, i.e., coinciding,

when one of the dependences is shifted along the mag-netic field axis by ∆φ = ∆Φ/Φ0 ≤ 0.5 (Fig. 8). The crit-ical current anisotropy

CVC anisotropy, and all associated effects are due tothis shift. The periodicity of the dependences Ic+(Φ/Φ0)and Ic–(Φ/Φ0) and a large number of periods, which canbe observed, allow sufficiently accurate determinationof ∆φ by superposing the measured dependencesIc−(Φ/Φ0) and Ic+(Φ/Φ0 + ∆φ). Since these dependenceswere measured simultaneously, i.e., Ic+ and Ic− weremeasured sequentially for each value of Φ/Φ0, theuncertainty in the Φ/Φ0 magnitude cannot affect thedetermination accuracy of the relative shift ∆φ. There-fore, a ∆φ determination accuracy of up to 0.02Φ0 canbe guaranteed.

The dependences Ic+(Φ/Φ0) and Ic−(Φ/Φ0) measuredat various temperatures are similar (see Figs. 5 and 18),and the shift ∆φ is independent of temperature. Theshift of the dependences measured on single rings isequal to half the flux quantum, ∆φ = 0.50 ± 0.02 for aring with wcon = 0.6 µm and ∆φ = 0.48 ± 0.05 for a ringwith wcon = 0.7 µm. A smaller shift is observed for thedependences measured on 20-ring systems. We nowcannot argue with certainty that there is a principal dif-ference between specifically the single ring and ringsystem. On the system with wcon = 1 µm ≥ ww + wn =0.6 µm, for which the dependences Ic+(Φ/Φ0) andIc−(Φ/Φ0) qualitatively does not differ from the depen-dences obtained for single rings, we performed only asingle measurement which yielded ∆φ = 0.30 ± 0.02.On the system with wcon = 0.4 µm, we carried outdetailed measurements at various temperatures (Figs. 5and 8), which showed ∆φ = 0.36 ± 0.02. However, inthis case, it is impossible with certainty to call this thecause in the difference in the shift from ∆φ = 0.50; i.e.,is it associated with the number of rings or with thequalitative difference of the dependences Ic+(Φ/Φ0) and

Ic an, Φ/Φ0( ) Ic+ Φ/Φ0( ) Ic– Φ/Φ0( )–=

Ic– Φ/Φ0( ) Ic+ Φ/Φ0 ∆Φ+( ),=

Ic an, Φ/Φ0( ) Ic+ Φ/Φ0( ) Ic– Φ/Φ0( )–=

= Ic+ Φ/Φ0( ) Ic+ Φ/Φ0 ∆φ+( ),–

1166


GURTOVOI et al.

Ic−(Φ/Φ0) measured on structures with wcon < ww + wnand wcon ≥ ww + wn (Fig. 6).

It is clear that the observed periodicity of the depen-dences Ic+(Φ/Φ0) and Ic−(Φ/Φ0) of the critical current onthe magnetic field is associated with quantization of thevelocity of superconducting pairs in the ring (see (1)).As the ring returns to the superconducting state, thequantum number n defining the momentum circulationof superconducting pairs takes an integer value. Thequantity n – Φ/Φ0 nonzero at Φ ≠ nΦ0 defines the rela-tion between the velocities vsn and vsw in ring halves

according to (1). According to this relation, at the mea-suring current

the velocities should be equal;

in the narrow ring half and

in the wide ring half,

The measured critical currents given in Section 3.1 sug-gest that these values correspond to the depairing veloc-ities vsc, which would be expected for a narrow super-conducting channel [4]. Therefore, the critical currentIc+(Φ/Φ0), Ic–(Φ/Φ0) should correspond to the measur-ing current |Iext | at which the superconducting pairvelocity reaches the depairing velocity in one of struc-tural elements. If the velocity reaches a critical value inthe narrow ring half, |vsn | = vsc ,

(2a)

if this occurs in the wide half, |vsw | = vsc ,

lnv sn lwv sw–l2--- v sn v sw–( ) 2π

m---------- n

ΦΦ0------–⎝ ⎠

⎛ ⎞= =

Iext In Iw+ sn jn sw jw+ 2ens snv sn swv sw+( ),= = =

v sn

Iext

2ens sn sw+( )--------------------------------

2mr-------

sw

sn sw+--------------- n

ΦΦ0------–⎝ ⎠

⎛ ⎞+=

v sw

Iext

2ens sn sw+( )--------------------------------

2mr-------

sn

sn sw+--------------- n

ΦΦ0------–⎝ ⎠

⎛ ⎞ .–=

Ic+ Ic–, Ic0 2I p A, nΦΦ0------– 1

sw

sn

-----+⎝ ⎠⎛ ⎞ ;–=

–2

Ic, an, µA

–5

0

5

–4 0 2 4Φ/Φ0

–3 –1 1 3

Fig. 16. Critical current anisotropy oscillations Ic, an(Φ/Φ0),measured on the single ring with wcon = 0.7 µm at varioustemperatures T = 1.211 K ≈ 0.972Tc, T = 1.227 K ≈ 0.983Tc,and T = 1.235 K ≈ 0.992Tc.

T, K

Ip, A, µA; R, Ω

1.25

2

1.2301.21

4

1

2

3

4

6

Fig. 17. Temperature dependences of IA, an/1.5 and Ic, pl/2 –Ic+/3, which should correspond to the persistent currentamplitude oscillations, measured on (1) the single ring withwcon = 0.7 µm, (2) the single ring with wcon = 0.6 µm, and(3) the 18-ring system with wcon = 0.4 µm. Symbols areexperimental data. Lines are the dependences Ip, A(T) =Ip, A(0)(1 – T/Tc) for (1) Ip, A(0) = 0.227 mA and Tc =1.246 K, (2) Ip, A(0) = 0.179 mA and Tc = 1.236 K,(3) Ip, A(0) = 0.068 mA and Tc = 1.263 K, and (4) the resis-tive transition of the ring with wcon = 0.7 µm.

–4

Ic+, –Ic–, µA; ∆R, Ω

–20

0

20

–8 0 4Φ/Φ0

–6 –2 2 6

12

3456

7

65432

1

Fig. 18. Oscillations of the critical current Ic+(Φ/Φ0),Ic−(Φ/Φ0) and resistance R(Φ/Φ0) measured on the singlering at various temperatures: (1) T = 1.2106 K = 0.9716Tc,(2) T = 1.2161 K = 0.9760Tc, (3) T = 1.2217 K ≈ 0.9805Tc,(4) T = 1.2268 K ≈ 0.9846Tc, (5) Tc = 1.2315 K ≈ 0.9883Tc,(6) T = 1.2346 K ≈ 0.9909Tc; (7) ∆R(Φ/Φ0) measured atIext = 1 µA and T = 1.232 K ≈ 0.999Tc. The dependenceR(Φ/Φ0) is shifted vertically.



(2b)

where Ic0 = 2ens(sn + sw)vsc is the critical current atΦ/Φ0 = n; Ip, A = 2ens(/mr)snsw/(sn + sw) is the valuecorresponding to the persistent current amplitude asn − Φ/Φ0 varies between –0.5 and 0.5. According to thegeometry shown in Fig. 1, expression (2a) is applicablefor Ic+ at positive n – Φ/Φ0 and for Ic– at negative values;this is vice versa for expression (2b). Relations (2) wereobtained on then basis of the conventional descriptionof the superconducting state as a macroscopic quantumphenomenon in the approximation of small width and

cross section of ring halves, wn, ww 2r; sn, sw < (T).We proceeded from velocity quantization expression(1) which follows from the Bohr quantization of themomentum circulation of superconducting pairs

Expressions (2) can also be derived from the Ginzburg–Landau theory, starting from the expression for thesuperconducting current density js = (2e/m)ns(∇ϕ –2eA) [4], where ϕ is the wavefunction phase and A is thevector potential.

Let us consider those aspects of the measureddependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) obtained accord-ing to the generally accepted concepts on the supercon-ducting state, which are consistent and inconsistentwith relations (2). The first and main consistency is theperiodicity of expected (2) and observed dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0). Since the energy differencebetween allowed levels defined by the quantum numbern is large (much larger than the thermal fluctuationenergy kBT), the lowest level, i.e., the level with thesmallest value of (n – Φ/Φ0)2, has an overwhelmingprobability. Therefore, the equilibrium quantity |n –Φ/Φ0 | in expressions (2) is a periodic function of themagnetic field Φ = SB with period Φ0. Therefore, theperiodic dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) (Figs. 5and 6) are observed.

The second consistency is the qualitative differencebetween the dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) mea-sured on structures with wcon < ww + wn and wcon ≥ ww +wn (Figs. 5 and 6). In the structure with cross sectionscon = wcond ≥ (sn + sw) = (ww + wn)d, when Iext/(sn + sw) ≥Iext/scon, the current density js, con = Iext/scon in currentcontacts cannot exceed the current density simulta-neously in both halves, since Iext = snjn + swjw . In thiscase, the velocity reaches a critical value initially in oneof them; then, one of expressions (2) is valid. In con-trast to this case, in the structure with scon < (sn + sw), thecritical velocity can be reached initially in current con-tacts at |Iext | = 2enssconvsc < Ic0 = 2ens(sn + sw)vsc . Thisquantity is independent of |n – Φ/Φ0 |; therefore, themeasured critical current should not depend on themagnetic field, when |Iext | = 2enssconvsc is smaller thanthe critical current (2) controlled by the velocity in one

Ic+ Ic–, Ic0 2I p A, nΦΦ0------– 1

sn

sw

-----+⎝ ⎠⎛ ⎞ ,–=

λL2

dlp∫° dl mv 2eA+( )∫° m dlv∫° 2eΦ+ n2π.= = =

of ring halves. This corresponds to the existence of theplateau with Ic+ = Ic– = Ic, pl = 2enssconvsc in the depen-dences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) measured on the struc-ture with wcon = 0.4 µm < ww + wn = 0.6 µm (Figs. 5and 6) and the absence of plateau for the structures withwcon = 1; 0.7; 0.6 µm ≥ ww + wn = 0.6 µm (Fig. 6).

A decrease in the critical current with increasingmagnetic field (Fig. 6) is also consistent with com-monly accepted concepts. This is associated with adecrease in the superconducting pair density ns infinite-width strips [4]. The observed decrease not onlyin Ic+(Φ/Φ0) and Ic−(Φ/Φ0), but also in their oscillationamplitude at large Φ/Φ0 (Fig. 6) is consistent with (2),since not only the critical Ic0 current, but also the persis-tent Ip, A current depend on the pair density ns . Theeffect of the current contact width wcon on decreases inIc0 at large Φ/Φ0 in structures with wcon ≥ ww + wn(Fig. 6) is a manifestation of the proximity effect:stronger suppression of ns in wider current contactsresults in ns suppression in rings through the proximityeffect. The results of our measurements suggest that theproximity effect appears at distances of severalmicrometers. In a structure with contact areas widerthan 4 µm arranged at a distance of 2 µm from the ring,the superconductivity is suppressed even in weak fieldsand no more than five oscillation periods can beobserved. More than 25 periods of critical current oscil-lations can be observed on such a structure, but withareas spaced from the ring at a distance longer than10 µm (Fig. 6).

The fourth consistency is the periodic sign-alternat-ing dependence of the critical current anisotropy on themagnetic field. According to (2), for structures withwcon ≥ ww + wn, the anisotropy is given by

(2c)

The sign of the anisotropy is positive at positive n –Φ/Φ0, according to the geometry shown in Fig. 1. Aspositive, we take the direction from right to left for Iextand clockwise for the persistent current,

The amplitude IA, an of observed anisotropy oscillations

linearly increases as it recedes from Tc,

according to

The value of IA, an is close to the expected value of(sw/sn – sn/sw)Ip, A (see Section 3.5 for more detail).

Ic an, Ic+ Ic–– 2I p A, nΦΦ0------–⎝ ⎠

⎛ ⎞ sw

sn

-----sn

sw

-----–⎝ ⎠⎛ ⎞ .= =

I p 2I p A, n Φ/Φ0–( ).=

Ic an, T Φ/Φ0,( ) IA an, T( ) f an Φ/Φ0( )=

IA an, T( ) 1 T /Tc–( ),∝

I p A, ns 2esnsw

sn sw+---------------

mr-------⋅ ns 1 T

Tc

-----–⎝ ⎠⎛ ⎞ .∝ ∝=

1168


GURTOVOI et al.

Having regard for such important consistencies, themeasured dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) differessentially from the expected dependences (2). Onaccount of the importance of these inconsistencies, weconsider them in separate Section 3.6. A certain para-dox is that, despite these fundamental differences, bothexpected (2c) and observed (Fig. 7), critical currentanisotropy oscillations Ic, an(Φ/Φ0) explain the rectifiedvoltage oscillations Vdc(Φ/Φ0) observed in [3]. In theSection 3.3, we shall consider the consistency betweenmeasured oscillations Ic+(Φ/Φ0), Ic−(Φ/Φ0) andVdc(Φ/Φ0). In Section 3.4, we present the measuredtemperature dependences of quantities characterizingthe rectification efficiency. It should be emphasized thatthese dependences are consistent with the observedcritical current oscillations with temperature. To moreemphasize the strangeness of the inconsistencybetween expected and observed oscillations Ic+(Φ/Φ0),Ic–(Φ/Φ0), we show in Section 3.5 that the observed val-ues and temperature dependences of the amplitudeIc, an(Φ/Φ0) are consistent with expected dependencesof persistent current.

3.3. Quantum Oscillations of ac Voltageas a Result of Alternating Current Rectification

The observed asymmetry of the critical current andits periodic dependence on the magnetic flux,Ic, an(Φ/Φ0) = Ic+(Φ/Φ0) – Ic–(Φ/Φ0), allow us to explainthe rectification effect and the dependence Vdc(Φ/Φ0) ofthe rectified voltage on the magnetic flux. A compari-son of the dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) withdependences Vdc(Φ/Φ0) caused by both the sinusoidal,Iext(t) = I0sin(2πft), and sawtooth currents showed that|Vdc|(Φ/Φ0) maxima are close to Ic+(Φ/Φ0) and Ic–(Φ/Φ0)minima in all cases (Fig. 9). The position of |Vdc |(Φ/Φ0)maxima shifts from the values closer to integer fluxquanta, Φ = (n + 0.14)Φ0 and Φ = (n + 0.86)Φ0, at smallalternating current amplitudes to the values Φ = (n +0.25)Φ0 and Φ = (n + 0.75)Φ0 corresponding to themidpoint between the flux quantum and half the quan-tum at large amplitudes Iext . Good similarity isobserved between quantum oscillations of the dc volt-age and quantum oscillations of the critical currentanisotropy Vdc(Φ/Φ0) ∝ –Ic, an(Φ/Φ0) (Fig. 10).

The rectified voltage, i.e., the time-averaged voltage

,

can be calculated from the V(Iext) CVC (see Figs. 2–4)and the time dependence of the external current, e.g.,Iext(t) = I0sin(2πft). The measurement results of theCVC and the magnetic dependences of the critical cur-rent anisotropy, presented in Sections 3.1 and 3.5,respectively, allow quantitative description of the recti-fication effect resulting in observation of quantumoscillations Vdc(Φ/Φ0). The rectified voltage oscillationamplitude depends on several factors, including the

Vdc Θ 1– tV Iext t( )( )d

Θ∫=

CVC type, but the main one is the relation between theexternal current amplitude I0 and the critical currentsIc+(Φ/Φ0) and Ic–(Φ/Φ0).

The dc voltage Vdc(Φ/Φ0) appears when the externalcurrent amplitude I0 exceeds the least Ic, min among crit-ical values of Ic+(Φ/Φ0) and Ic–(Φ/Φ0). At larger Φ/Φ0,this occurs at smaller amplitudes I0 (Fig. 11), since thecritical current is here suppressed by the magnetic field(see Figs. 6 and 9). A decrease in the amplitude VA withincreasing I0 immediately after reaching the maximumVA, max at I0 = I0, max (Figs. 11 and 12) is caused by theopposite-sign voltage appeared as I0 reaches the largervalue Ic, max among Ic+(Φ/Φ0) and Ic–(Φ/Φ0). The peakwidth in the dependence VA(I0) (Fig. 12) corresponds tothe Ic+(Φ/Φ0), Ic–(Φ/Φ0) amplitude (Figs. 5 and 6) at thethird-type CVC. For this CVC type with stepwise vari-ation and strong hysteresis (Figs. 2 and 4), a stepwiseincrease in the rectified voltage would be expected atI0 = Ic+(Φ/Φ0), Ic–(Φ/Φ0). The absence of such a step isprobably associated with uncontrollable noises Inoise ,due to which the actual external current Iext(t) =I0sin(2πft) + Inoise(t) and its actual amplitude vary intime and not always coincides with I0.

The dependence VA(I0) shape is unchanged with theexternal current Iext(t) frequency and shape (sinusoidalor sawtooth) and the CVC type (Fig. 12). However, theVA(I0) peak height and width depend on both the currentIext(t) shape and CVC type (Fig. 12). As expected, themaximum amplitude VA, max is observed at the third-type CVC and sinusoidal current (Fig. 12). For the sec-ond-type CVC, the VA(I0) peak is wider than that for thethird-type CVC (Fig. 12).

In the presence of the CVC hysteresis, the voltageV(Iext(t)) has a nonzero value for a relatively large frac-tion of the Iext(t) period, even if the current amplitude I0only insignificantly exceeds the critical currentIc+(Φ/Φ0), Ic–(Φ/Φ0). Therefore, the hysteresisincreases the rectification efficiency. In the approxima-tion V ≈ RnIext(t), as the sawtooth current Iext(t)decreases down to the superconducting transition atIext = Ic, s (Figs. 2 and 4), the maximum amplitude of therectified voltage, observed at I0 ≥ I0, max, can be esti-mated by the formula

For all the measured structures with the third-typeCVC, the hysteresis value is larger than the anisotropyvalue, Ic, max – Ic, s > IA, an, even taking into account thatIc, s should be considered as an effective value corre-sponding to the hysteresis area (Fig. 2). The anisotropyamplitude of the critical current, measured, e.g., at T =1.234 K ≈ 0.972Tc on the 18-ring system with normal-state resistance Rn = 92 Ω, is IA, an ≈ 1 µA. This corre-sponds to

V A max,Rn

4----- 2Ic max, Ic min,– Ic s,–( )≈

≈Rn

4----- Ic max, Ic s,– IA an,+( ).



in the absence of hysteresis, which is several timessmaller than the maximum amplitude VA, max ≈ 100 µVof quantum oscillations which induce a sawtooth cur-rent at this temperature. Estimations taking intoaccount the hysteresis Ic, max – Ic, s yield VA, max valuesclose to the measured values.

In addition to the quantitative difference, there isalso an important qualitative difference between quan-tum oscillations Vdc(Φ/Φ0) observed in the presence(Fig. 11) and in the absence of CVC hysteresis(Fig. 13). The existence of the hysteresis means that notonly structure regions in which the current densityexceeds the critical value (j > jc) can be in the resistivestate, but also neighboring regions in which j < jc . Inthis case, the persistent current promotes the transitionto the resistive state of not only that ring half in whichthe persistent current increases the current density, e.g.,jn = Iext/(sn + sw) + Ip/sn > jc , but also in that ring half inwhich it decreases the current density, jw = Iext/(sn + sw) –Ip/sw < jc . The absence of CVC hysteresis means thatthe transition to the resistive state of one ring half doesnot result in a similar transition of the second half untilthe current density in it reaches the critical value.

3.4. Temperature Dependence of the Maximum Amplitude VA, max of Quantum Oscillations.

Rectification Efficiency

It is known that the critical Ic and persistent Ip cur-rents decrease when approaching the critical tempera-ture Tc . It can be expected that the critical currentamplitude I0c ≈ Ic(T) and the maximum amplitudeVA, max of quantum oscillations V(Φ/Φ0) = RasymIp(Φ/Φ0)should decrease when approaching Tc similarly to Ic(T)and Ip(T). A decrease in the critical amplitude I0c to zeronear Tc means that noises with arbitrarily low intensitycan induce quantum oscillations Vdc(Φ/Φ0) in this tem-perature range.

Here we present the first results of measurements ofthe temperature dependences of the critical amplitudeI0c(T), maximum amplitude VA, max(T) of quantum oscil-lations, and the external current amplitude I0, max(T) atwhich VA, max(T) is observed (Fig. 12). The resultsobtained show the difference of rectification mecha-nisms in the immediate vicinity of Tc and at lower tem-peratures and make it possible to more reliably estimatethe possibility to observe noises of arbitrarily low inten-sity.

The measurements performed on single ringsshowed that, as expected, I0c(T) and I0, max(T) are closeto the dependence of the critical current Ic(T) (Fig. 14).In this figure, only I0, max(T) is shown, since I0c(T) isanalogous to it. The maximum amplitude VA, max(T) ofquantum oscillations decreases with increasing T simi-

V A max,RnIA an,

4---------------- 23 µV≈ ≈

larly to I0, max(T) at temperatures not close to the resis-tive transition. In the range T = 0.95–0.99Tc , the ratioVA, max/I0, max = 0.79 ± 0.03 Ω (at the normal-state ringresistance Rn = 3.3 Ω) is independent of temperature towithin the measurement error and sharply decreasesbefore the resistive transition (Fig. 15). Our measure-ments confirmed the increase in the amplitude of quan-tum oscillations with the number of rings, observed in [3].

To compare the rectification efficiency in systemswith different numbers of rings N, the ratio Reff =VA, max/I0, maxN can be used, as well as EffRe =VA, max/I0, maxRn, since the normal-state resistance Rn

increases with the number of rings. Our measurementsshowed that Reff and EffRe do not significantly differ fora single ring and a system of rings at low temperatures.For the single ring, Reff ≈ 0.79 Ω and EffRe ≈ 0.24 at T =(1.18–1.23) K = (0.95–0.99)Tc; for the system of18 rings, Reff ≈ 1.1 Ω and EffRe ≈ 0.22 at T = 1.240 K =0.977Tc (Fig. 15); for the 20-ring system, Reff ≈ 0.45 Ωand EffRe ≈ 0.16 at T = 1.210 K = 0.978Tc. The high rec-tification efficiency EffRe is observed at low tempera-tures due to the CVC hysteresis. For the second- andthird-type CVCs with hysteresis, a higher efficiency isobserved in the latter case. This is particularly clearwhen comparing the values of EffRe measured on thesystem of 18 rings (wcon = 0.4 µm) with the third-typeCVC at low temperatures (Fig. 2) and the 20-ring sys-tem (wcon = 1 µm) with the second-type CVC (Fig. 3).The sharp decrease in the rectification efficiency EffRewith increasing temperature (Fig. 15) coincides withthe change in the CVC type (Figs. 2–4). Despite thedecrease in the rectification efficiency near Tc, theobserved increase in NReff = VA, max/I0, maxwith the num-ber of rings N confirms the possibility of using systemswith a large number of asymmetric superconductingrings as highly sensitive noise sensors [10].

3.5. Amplitudes of Critical and Persistent Current Anisotropy Oscillation

According to expressions (2) for the critical current,obtained in Section 3.2 on the basis of conventionalconcepts, the critical current anisotropy of a structurewith sw/sn = 2 and scon ≥ sn + sw should be given by

According to this relation, anisotropy oscillationsIc, an(Φ/Φ0) measured at various temperatures should besimilar, Ic, an(Φ/Φ0) = IA, an(T)fan(Φ/Φ0), and their ampli-tude should linearly decrease when approaching thecritical temperature, IA, an = IA, an(0)(1 – T/Tc). Our mea-surements confirmed similarities of anisotropy oscilla-tions of the critical current measured at various temper-

Ic an,sw

sn

-----sn

sw

-----–⎝ ⎠⎛ ⎞ I p

sw

sn

-----sn

sw

-----–⎝ ⎠⎛ ⎞ I p A, T( )= =

× 2 nΦΦ0------–⎝ ⎠

⎛ ⎞ 1.5I p A, T( ) 2 nΦΦ0------–⎝ ⎠

⎛ ⎞ .⋅=

1170


GURTOVOI et al.

atures (Fig. 16). This allows us to compare the temper-ature dependences and amplitudes of observed andexpected oscillations despite their qualitative differ-ence.

The expected amplitude of the critical currentanisotropy of a structure with scon ≥ sn + sw should begiven by

Based on this relation, we can estimate the persistentcurrent amplitude from measured amplitudes of thecritical current anisotropy. For structures with scon <sn + sw, there is no simple relation between Ic, an(T, Φ/Φ0)and Ip(T, Φ/Φ0), since the critical current at an insuffi-ciently large persistent current is controlled by its valuein current contacts, Ic+ = Ic– = Ic, pl = scon jc . The persis-tent current can be determined by the value by which itdecreases the critical current Ic+ (or Ic–) (Fig. 5),

The persistent current amplitude calculated by theserelations from experimentally determined IA, an(T) andIc, pl/2 – Ic+/3 linearly decreases when approaching thecritical temperature, Ip, A = Ip, A(0)(1 – T/Tc) (Fig. 17).This is consistent with the linear decrease in the persis-tent current amplitude

proportional to the pair density ns ∝ (1 – T/Tc). Usingthe expression for the critical current density jc = ns ×2e(/m ξ(T)) [4], the persistent current amplitudecan be expressed in terms of the critical current as

for scon ≥ sn + sw and

for (sn + sw)/scon = 1.5.This allows us to compare the theoretical persistent

currents and their values obtained from measurementsof critical current oscillations, using the experimentaltemperature dependences of the critical current given inSection 3.1,

,

with Ic(0) = 5.5 and 4.3 mA for single rings with wcon =0.7 and 0.6 µm, respectively, and Ic(0) ≈ 1.4 mA for thesystem of 18 rings with wcon = 0.4 µm.

IA an, T( )sw

sn

-----sn

sw

-----–⎝ ⎠⎛ ⎞ I p A, T( ) 1.5I p A, T( ).= =

I p

Ic pl, sn

scon-------------

Ic+sn

sn sw+---------------–

Ic pl,

2---------

Ic+

3------.–= =

I p A, T( ) ns 2× esnsw

sn sw+---------------

mr------- ns 2× e

2 sn sw+( )9

------------------------ mr-------= =

3

I p A, T( ) Ic T( )29--- 3ξ T( )

r------------------- Ic 0( )2

9--- 3ξ 0( )

r------------------ 1 T

Tc

-----–⎝ ⎠⎛ ⎞= =

I p A, T( ) Ic 0( )13--- 3ξ 0( )

r------------------ 1 T

Tc

-----–⎝ ⎠⎛ ⎞=

Ic T( ) Ic 0( ) 1 T /Tc–( )3/2=

The values Ip, A(0) = 0.23, 0.18, and 0.07 mA obtainedfrom Ic+(Φ/Φ0) and Ic–(Φ/Φ0) measurements on ringswith wcon = 0.7 and 0.6 µm and the system of 18 ringswith wcon = 0.4 µm (Fig. 17) are close to the correspond-

ing values Ic(0)(2/9)( ξ(0)/r) = 2.1 mA(ξ(0)/r),

Ic(0)(2/9)( ξ(0)/r) = 1.6 mA(ξ(0)/r), and

Ic(0)(1/3)( ξ(0)/r) = 0.8 mA(ξ(0)/r) at ξ(0)/r ≈ 0.1.At a radius r = 2 µm of rings used in the measurements,this means that the theoretical amplitude of the persistentcurrent and the amplitude obtained from measurementsof critical current oscillations are in agreement if the cor-relation length ξ(T) = ξ(0)(1 – T/Tc)–1/2 extrapolated tothe zero temperature is ξ(0) ≈ 0.2 µm. This value doesnot contradict that expected for the aluminum filmstructures used in this study, ξ(0) = 0.13 µm in a struc-ture with a smaller ratio of resistances [7] and ξ(0) =1.6 µm for pure aluminum.

Thus, we can conclude that the expected andobserved amplitudes of critical current oscillations andtheir temperature dependences are in agreement, evenwhile oscillations are qualitatively different.

3.6. Qualitative Difference between Expectedand Observed Oscillations of the Critical Current

of Asymmetric Superconducting Rings

The main difference of the observed dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0) (Figs. 5, 6, and 8) fromexpected ones (2) is the similarity of the dependencesof the critical current measured in opposite directionsand the positions of extrema (Fig. 8). It follows fromthe condition of the quantization of the velocity ofsuperconducting pairs (see (1)) that the critical current,independently of the measurement direction and ringasymmetry, should have maximum and minimum val-ues at the total flux inside the ring Φ = nΦ0 and Φ =(n + 0.5)Φ0, respectively. The total magnetic flux Φ =Φext + ΦI is induced by the external magnetic fieldΦext = BextS (in our measurements, Bext = Bsol + Bres, seeSection 2) and currents Iw, In flowing in ring halves.ΦI = LnIn – LwIw , Lw + Ln = L is the ring inductance, andLw ≈ Ln ≈ L/2 are the inductances of halves. Accordingto the quantization condition (1), the currents in halvesare given by

In a symmetric ring with sw = sn , an additional flux isinduced only by the persistent current ΦIp = LIp; in anasymmetric ring with sw = 2sn, it is also induced by theexternal current

3

3

3

In

Iextsn

sw sn+--------------- I p+

Iext

3------- I p,+= =

Iw

Iextsw

sw sn+--------------- I p–

2Iext

3---------- I p.–= =



We recall that the left-to-right direction corresponds topositive values of Iext, Iw , and In; the clockwise directioncorresponds to Ip. Therefore, for the geometry shown inFig. 1 (used in this study), ΦIext and ΦIp have the samesign, while Iext and Ip have opposite signs.

The existence of the additional flux ΦI can shift thepositions of extrema by ΦI in the dependences on theexternal magnetic field. The persistent current does notaffect the positions of extrema of critical current oscil-lations Ic(Φext/Φ0) in symmetric rings. Numerous stud-ies of symmetric superconducting quantum interferom-eters, i.e., the same superconducting loops, but withtwo Josephson junctions being a basis of the DCSQUID (Superconducting Quantum InterferenceDevice) [11], suggest that critical current maxima andminima are observed at Φ = nΦ0 and Φ = (n + 0.5)Φ0,respectively. The same is observed for a symmetric alu-minum ring without Josephson junctions [12]. Asym-metric rings were not studied before our experiments;however, there are papers on oscillations of the criticalcurrent of asymmetric interferometers [11, 13]. The dif-ference of the positions of extrema in the magneticdependences of the critical current from Φ = nΦ0 andΦ = (n + 0.5)Φ0 observed in these studies is associatedwith the difference of the total flux Φ = Φext + ΦI in theinterferometer from the measured Φext . Due to interfer-ometer asymmetry, the measuring current Iext inducesthe additional flux ΦIext, whose sign varies with the Iextsign. Therefore, the extrema of Ic+(Φext/Φ0) andIc−(Φext/Φ0) are observed at Φext = Φ – |ΦIext | =nΦ0 − |ΦIext |, Φext = (n + 0.5)Φ0 – |ΦIext | and Φext =nΦ0 + |ΦIext |, Φext = (n + 0.5)Φ0 + |ΦIext |, respectively;i.e., the extrema of the dependences of the critical cur-rent measured in opposite directions are shifted withrespect to each other by 2|ΦIext |.

However, this explanation is inapplicable to the shift∆φ observed in this study. We obtained the dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0) at various temperatures forcritical currents Ic from 3 µA (Fig. 5) to 30 µA(Fig. 18), which corresponds to the values of |ΦIext | =LIc/6, differing by an order of magnitude. If theobserved shift ∆φ would be determined by the value of2ΦIext , as is the case in [13], its value should change byan order of magnitude as Ic changes from 3 µA to30 µA. However, all our measurements showed that ∆φis independent of Ic (Fig. 18). At the inductance L =1.7 × 10–11 H of the ring used in this study, the addi-tional flux is ΦIext = LIext/6 = 0.04Φ0 at Iext = 30 µA andΦIext = 0.004Φ0 at Iext = 3 µA. This estimate confirmsthe impossibility to explain the observed shift by thedifference of the total flux Φ from the measured fluxΦext . The estimates suggest that the used approximationΦ ≈ Φext = BextS is valid with an accuracy of a few per-

ΦIextLnIextsn

sw sn+------------------

LwIextsw

sw sn+-------------------–

LIext

6-----------,–= =

ΦI LnIn LwIw– ΦIext ΦIp.+= =

cent of Φ0. The magnetic flux ΦIp = LIp induced by thepersistent current with maximum amplitude Ip = 7 µA(Fig. 17) does not exceed 0.06Φ0.

The impossibility of explaining the observed shiftby the difference of the total flux in the ring from themeasured flux makes it an extraordinarily mysteriousphenomenon. The periodicity of the dependencesIc+(Φ/Φ0) and Ic–(Φ/Φ0) leaves no doubt that thedecrease in the critical current is associated with thevelocity quantization (1). According to the quantizationcondition (1), the equilibrium (corresponding to theenergy minimum) velocity of superconducting pairshas zero and maximum values at Φ = nΦ0 and Φ =(n + 0.5)Φ0, respectively, in both symmetric and asym-metric rings. Therefore, it seems impossible to explainthe shift of extrema by ±∆φ/2 as asymmetry appears inthe ring. The results of our measurements of Little–Parks oscillations ∆R(Φ/Φ0) in the studied rings makethe situation quite strange. Measurements showed the

resistance extrema ∆R(Φ/Φ0) ∝ (Φ/Φ0) of both theasymmetric and symmetric rings are observed atΦ = nΦ0 and Φ = (n + 0.5)Φ0 (Fig. 18). According tothe conventional explanation [4], which relates resis-tance oscillations and the squared velocity of pairs,

∆R(Φ/Φ0) ∝ (Φ/Φ0), this means that is zero atΦ = nΦ0 and is maximum at Φ = (n + 0.5)Φ0. However,if critical current oscillations Ic+(Φ/Φ0) and Ic–(Φ/Φ0)are also related to pair velocity |vs | oscillations, theirmaxima and minima should be observed at Φ = nΦ0 andΦ = (n + 0.5)Φ0, respectively, as in the symmetric ring.For the observed similarity of the dependencesIc−(Φ/Φ0) = Ic+(Φ/Φ0) + ∆φ, this would imply theabsence of CVC asymmetry, Ic–(Φ/Φ0) = Ic+(Φ/Φ0) at∆φ = 0, and the absence of rectified voltage oscillationsVdc(Φ/Φ0). Thus, it should be concluded that the resultsof measurements of resistance and critical currentoscillations on the same asymmetric ring contradicteach other based on conventional concepts.

The critical current anisotropy in the expecteddependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0) (2) is caused bythe absence of similarity (Fig. 19), rather than by theirrelative shift. In the intervals nΦ0 – (n + 0.5)Φ0 and(n + 0.5)Φ0 – nΦ0, these dependences have differentslopes due to the difference between the cross sectionsof ring halves, sw = 2sn; at (n + 0.5)Φ0, there is a discon-tinuity due to a change in the persistent current direc-tion (Fig. 19). The absence of such a discontinuity inthe observed dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0)(Fig. 19) is one more important difference of theobtained result from the expected one. According toexpressions (2a) and (2b), which are based on conven-tional views, the critical current of the asymmetric ringshould depend not only on the magnitude, but also onthe direction of the persistent current. According tovelocity quantization condition (1), two allowed stateswith minimum energy, n and n + 1, should exist at

v s2

v s2

v s2

1172


GURTOVOI et al.

Φ = (n + 0.5)Φ0. States with equal in magnitude butoppositely directed persistent currents Ip ≈ n – Φ/Φ0 =±0.5 should correspond to different values of the quan-tum number n. A measurement of the critical currentcorresponds to a single measurement of the quantumstate. When the ring transits to the superconductingstate at Φ = (n + 0.5)Φ0, the quantum number takes thevalue of either n or n + 1, which cannot change beforethe ring transition to the normal state at |Iext | = Ic+ or|Iext | = Ic–. Since the states n and n + 1 have oppositedirections of the persistent current Ip, they should cor-respond to different critical currents,

,

of the rings under study with the ratio sw/sn = 2 (Fig. 19).Since all states, except for Ip = Ip, A(n – Φ/Φ0) = ±0.5Ip, A,are forbidden, the dependences Ic+(Φ/Φ0) and Ic–(Φ/Φ0)should contain a discontinuity whose value 1.5/Ip, Ashould be equal (according to (2)) to half the amplitudeof the critical current oscillations (Fig. 19). However,we observe oscillations Ic+(Φ/Φ0) and Ic–(Φ/Φ0)(Figs. 6, 8, and 18) which, in contrast to the expectedones (2), contain no discontinuity at Φ = (n + 0.5)Φ0(Fig. 19).

This result of a single measurement is even stranger,since multiple measurements show two allowed statesat Φ = (n + 0.5)Φ0. Here it is noteworthy that the depen-dences of the rectified voltage Vdc(Φ/Φ0) cross zero atΦ = nΦ0 and Φ = (n + 0.5)Φ0 (Figs. 9–11), while resis-tance oscillations ∆R(Φ/Φ0) have a minimum and amaximum at Φ = nΦ0 and Φ = (n + 0.5)Φ0, respectively(Fig. 18). The rectified voltage is proportional to the

Ic+ Ic0 I p A, 1sw

sn

-----+⎝ ⎠⎛ ⎞– Ic0 3I p A, ,–= =

Ic+ Ic0 I p A, 1sn

sw

-----+⎝ ⎠⎛ ⎞– Ic0 1.5I p A,–= =

multiply measured pair velocity, Vdc(Φ/Φ0) ∝ ⟨vs⟩ ∝⟨n – Φ/Φ0⟩, while the resistance is proportional to the

squared velocity, ∆R(Φ/Φ0) ∝ ⟨ ⟩ ∝ ⟨(n – Φ/Φ0)2⟩. The

∆R(Φ/Φ0) ∝ ⟨ ⟩ minimum and Vdc(Φ/Φ0) ∝ ⟨vs⟩ = 0at Φ = nΦ0 correspond to the contribution of only one

level. However, the ∆R(Φ/Φ0) ∝ ⟨ ⟩ maximum atVdc(Φ/Φ0) ∝ ⟨vs⟩ = 0 cannot be explained without theexistence of two states n and n + 1 at Φ = (n + 0.5)Φ0with equal and oppositely directed velocities ⟨vs⟩ ∝⟨n − Φ/Φ0⟩ ∝ (1/2) + (–1/2) = 0, while ⟨ ⟩ ∝⟨(n − Φ/Φ0)2⟩ ∝ (1/2)2 + (–1/2)2 = 1/2. The detected dis-agreement between the results of single and multiplemeasurements of states of the macroscopic quantumsystem can be of great importance for the problem ofquantum calculations [14], in particular, the problem ofdeveloping the quantum bit on the basis of supercon-ductor nanostructures [15].

4. CONCLUSIONS

The goal of this study was to test the quite obviousassumptions that (i) quantum oscillations of the recti-fied voltage Vdc(Φ/Φ0) are caused by periodic variationsin the CVC asymmetry of asymmetric superconductingrings in a magnetic field; (ii) the CVC asymmetry iscaused by the superposition of the measuring and per-sistent currents, and the periodic CVC variation iscaused by periodic variations of the persistent current,Ip(Φ/Φ0). The results obtained not only confirmed thefirst assumption, but also allowed us to explain thechange in the rectification efficiency with the currentamplitude and temperature. As for the second assump-tion, the periodic dependence of the critical current onthe magnetic field leaves no doubt that the CVC asym-metry is caused by quantization of the superconductingpair velocity, i.e., the periodic dependence of the persis-tent current. Many results confirm the assumption onthe CVC asymmetry as a consequence of the superpo-sition of measuring and persistent currents. However,the similarity of the dependences Ic+(Φ/Φ0) andIc−(Φ/Φ0), their shift with respect to each other, and theabsence of discontinuity at Φ = (n + 0.5)Φ0 are not onlyinconsistent with the expected dependence, but alsocontradict the measurements of Little–Parks resistanceoscillations. The cause of this contradiction should beclarified in further studies.

ACKNOWLEDGMENTS

This study was supported by the Russian Founda-tion for Basic Research, project no. 04-02-17068, thebasic research program of the Department of Informa-tion Technologies and Computing Systems of the Rus-sian Academy of Sciences “Organization of Calcula-tions Using New Physical Principles” within the project“Quantum Bit Based on Micro- and Nanostructures

v s2

v s2

v s2

v s2

Ic+; –Ic–, µA

–10

0

5

–3 0 2Φ/Φ0

–2 –1 1 3

–5

10

Fig. 19. Comparison of the dependences of the critical cur-rent: the expected from (2) at Ic(0) = 7 µA, Ip, A = 1 µA andthe observed on the single ring at T = 1.225 K ≈ 0.991Tc.



with Metallic Conductivity”, and the program of thePresidium of the Russian Academy of Sciences “Low-Dimensional Quantum Structures.”

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Translated by A. Kazantsev

Dependence of the magnitude and direction of the persistent current on the magnetic flux in...

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