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Measuring Currents in Mesoscopic Rings. From femtoscience to nanoscience, INT, Seattle 8/3/09. H a. F a. Classical conducting rings. The current through a classical conducting loop decays with time as:. I. If R is very small, the current I can persist for a long time:

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slide1

Measuring Currents in Mesoscopic Rings

From femtoscience to nanoscience, INT, Seattle 8/3/09

classical conducting rings

Ha

Fa

Classical conducting rings

The current through a classical conducting loop decays with time as:

I

If R is very small,

the current I can persist for a long time:

not what we call “persistent” currents.

persistent currents in mesoscopic rings
persistent currents in mesoscopic rings
  • require phase coherence and therefore directly reflect the quantum nature of the electrons
  • are a thermodynamic property of the ground state (for mesoscopic metallic rings, nonequilibrium currents decay with a decay time of L/R ~ picosecond)
  • can have flux periodicity h/e and higher harmonics

Fa

I

measuring currents in mesoscopic rings
Measuring Currents in Mesoscopic Rings
  • technique
  • dirty aluminum rings: fluxoids
    • 1 order parameter
    • 2 order parameters
  • cleaner aluminum rings: fluctuations
  • gold rings: h/e-periodic persistent currents in normal metals
  • surprising spins
measuring currents in mesoscopic rings1
Measuring Currents in Mesoscopic Rings

Measurements

Hendrik Bluhm

Nick Koshnick

Julie Bert

SQUIDs

Martin Huber

Funded by NSF, CPN, and Packard

measuring persistent currents
Measuring persistent currents

Ha

Fa

Apply field 

Measure current ?

measuring persistent currents1
Measuring persistent currents

Ha

Fa

Apply field 

  • Measure magnetic field
  • Difficulties:
  • Small signal
  • Large background
scanning magnetic measurements
Scanning Magnetic Measurements

SQUID

2 mm

Location of

pickup loop

Sample Substrate

  • Advantages
  • Background measurements
  • Measure many samples in one cooldown
  • Measure samples made on any substrate
squid susceptometer
SQUID susceptometer

OUT

DC

feedback

Front end

SQUID

IN

I-V

R ~ 50 m

100-SQUID

array preamp

(NIST)

field coil

pickup loop

shielding

field coil

feedback

12 m

bias

1 mm

substrate polished to create

a corner at the pickup loop

Low inductance “linear coaxial” shields allow for:

• optimized junctions

- noise best when LI0 = 0/2

• low field environment near susceptometer core

• reduced noise n ~ L3/2

• independent tip design

performance
Performance

2 mm

5 mm

susceptometry of a ring

10

white noise floor
White noise floor

flux

0.2 0/Hz

ring current

0.2 nA /Hz

spin

200 B/Hz

ring current sensitivity

S1/2I = MS1/2F where M = mutual inductance ~ 0.1 - 1 F0/mA

spin sensitivity

(conventional but optimistic conversion)

Real experiments limited by

1/f noise

background

S1/2s (in mB) = S1/2F (in F0) x a/re

where a = pickup loop radius = 2 mm

and re = classical electron radius = 2.8x10-9mm

background elimination
Background elimination

SQUID

SQUID

Sample substrate

Sample substrate

  • Ring signal:  0.1 mF0
  • White noise: 0.50/Hz
  • Applied field: 45 0 in each pickup loop
  • => need to eliminate background to
    • 1 part in 109

background measurement

measurement

background measurement
Background measurement

Record complete nonlinear response by averaging over many sinusoidal field sweeps at each position.

Susceptibility scans

(In-phase linear response)

Raw signal after tuning Icomp (step 2)

mF0

Measurement positions:

+ background + signal

o background

  • Compute (+) - (o)
  • Subtract ellipse (linear response)
measuring currents in mesoscopic rings2
Measuring Currents in Mesoscopic Rings
  • Technique
  • Dirty aluminum rings: fluxoids
    • 1 order parameter
    • 2 order parameters
  • Cleaner aluminum rings: fluctuations
  • Gold rings: h/e-periodic persistent currents in normal metals
  • Surprising spins
mesoscopic superconducting rings

Superfluid Density

= Fluxoid #

Superconducting Coherence Length

mesoscopic superconducting rings

Energy

Y = |Y|eif

n=0

n=1

n=2

GL:

0

1

2

F/F0

phase

gradient

magnetic

vector potential

Current

R

F/F0

w

R = 0.5 – 2 m

d = 45 nm

w = 30 – 350 nm linewidth

d

sample structure

R

w

d

w = line width

oxide

Sample structure
  • Fabrication:
  • - PMMA e-beam lithography.
  • - E-beam evaporation of d = 40 nm Al:
  • Background pressure 10-6 mBar
  • Deposition rate ~1 Angstrom/sec
  • ~10 min interrupt during deposition
  • - Liftoff

Deduced film structure:

a i data and models

Superfluid Density

= Fluxoid #

Superconducting Coherence Length

Fit

Data

a-I data and models

0.40 K

1.00 K

1.35 K

n = 3

n = 0

Hysteretic Response Described by Rate Equation

n = -3

1.49 K

High Temperature Response Well Described by Boltzmann Distributed Fluxoid States

1.524 K

D = 4 micron, w = 90 nm, t = 40 nm, le = 4 nm

anomalous a i curves of 190 nm rings

-

-

-

-

-

-

-

-

Anomalous Φa-Icurves of 190 nm rings
  • Reentrant hysteresis
  • Transitions not periodic in Φa/Φ0
  • Branches of Φa-Icurves shifted by less than one Φ0.
  • Unusual shape of non-hysteretic Φa-Icurves.
  • Not an effect of averaging over many cycles.
  • Motivation for 2-OP model

R = 1.2 mm

Two order parameters

Single order parameter

n

n2

only one (monotonic) transition path connects two different metastable states.

n1

multiple transition paths exist

anomalous a i curves of 190 nm rings1

-

-

-

-

-

-

-

-

Anomalous Φa-Icurves of 190 nm rings
  • Reentrant hysteresis
  • Transitions not periodic in Φa/Φ0
  • Branches of Φa-Icurves shifted by less than one Φ0.
  • Unusual shape of non-hysteretic Φa-Icurves.
  • Not an effect of averaging over many cycles.

Two order parameters

Single order parameter

n

n2

only one (monotonic) transition path connects two different metastable states.

n1

multiple transition paths exist

anomalous a i curves of 190 nm rings2

-

-

-

-

-

-

-

-

Anomalous Φa-Icurves of 190 nm rings
  • Reentrant hysteresis
  • Transitions not periodic in Φa/Φ0
  • Branches of Φa-Icurves shifted by less than one Φ0.
  • Unusual shape of non-hysteretic Φa-Icurves.
  • Not an effect of averaging over many cycles.
  • Motivation for 2-OP model

Two order parameters

Single order parameter

n

n2

only one (monotonic) transition path connects two different metastable states.

n1

multiple transition paths exist

anomalous a i curves of 190 nm rings3

-

-

-

-

-

-

-

-

Anomalous Φa-Icurves of 190 nm rings
  • Reentrant hysteresis
  • Transitions not periodic in Φa/Φ0
  • Branches of Φa-Icurves shifted by less than one Φ0.
  • Unusual shape of non-hysteretic Φa-Icurves.
  • Not an effect of averaging over many cycles.
  • Motivation for 2-OP model

Two order parameters

Single order parameter

n

n2

only one (monotonic) transition path connects two different metastable states.

n1

multiple transition paths exist

two order parameter gl fits

w (nm) # rings 2-OP features

100 2

120 6

190 7

250 14

320 1

370 5

coupling g increases with w

=> stronger

proximitization

None

Soliton states

only manifest in T-dep

Tc,1

Tc,2

oxide

Two-order-parameter GL - fits

Fits to representative datasets.

Summary of all data:

summary on 2 op rings
Summary on 2-OP rings
  • "Textbook" single-OP behavior observed for many Al rings.
  • Bilayer rings form a model system for two coupled order parameters with the following features: - metastable states with two different phase winding numbers, manifest in unusual Φa-I curves and reentrant hysteresis. - unusual T-dependence of xandl-2.
  • Extracted parameters for two-order-parameter Ginzburg-Landau model with little a priori knowledge.
measuring currents in mesoscopic rings3
Measuring Currents in Mesoscopic Rings
  • Technique
  • Dirty aluminum rings: fluxoids
    • 1 order parameter
    • 2 order parameters
  • Cleaner aluminum rings: fluctuations
  • Gold rings: h/e-periodic persistent currents in normal metals
  • Surprising spins
little parks effect
Little-Parks effect

Energy

n=0

n=1

n=2

In a thin-walled sample near Tc,

kinetic energy can exceed the condensation energy:

well-known “Little-Parks Effect”

0

1

2

F/F0

tin cylinder

~1 micron diameter

37.5 nm wall thickness

previously observed anomalous resistance in little parks regime liu et al science 2001
Previously observed anomalous resistance in Little-Parks regime: Liu et al. Science 2001

150 nm diameter Al cylinder

wall thickness 30 nmreported x(T) = 161 nm at T = 20 mK from Hc||(T)

R=0 => global phase coherence

regions separated by finite-R regions

predicted by deGennes, 1981

previously observed anomalous diamagnetic susceptibility zhang and price 1997
Previously observed anomalousdiamagnetic susceptibility(Zhang and Price, 1997)

Zhang and Price, 1997

(1 ring, zero-field response only)

ring fabrication
Ring fabrication

PMMA

PMMA

Al

silicon oxide

Samples

silicon substrate

e-beam evaporation and liftoff

2nd generation:

Background pressure <10-7 mBar

Deposition rate ~3.5 nm/sec

le = 30 nm on unpatterned film

le ~ 19 nm small features with PMMA (inferred)

R

w

d

R = 0.5 – 2 m

d = 70 nm

w = 30 – 350 nm linewidth

1st generation samples le = 4 nm

+ accidental layered structure for w > 150nm

model system for 2 coupled order parameters.

Bluhm et al, PRL 2006.

applied flux dependence
Applied Flux Dependence

d = 60 nm

w = 110 nm

A-C:

R = 350 nm

Tc = 1.247 K (fitted)

D:

R = 2,000 nm

Tc = 1.252 K (fitted)

In von Oppen and Riedel, the geometrical factors enter only through Ec and

our results
Our Results
  • disagree with previous results
  • agree with GL-based theory (von Oppen and Riedel)

Zhang and Price, 1997

(1 ring)

Present Work

(15 rings measured, 4 rings shown)

comparison of large and small rings
Comparison of “Large” and “Small” Rings

Blue: Data

Red: Theory

Green: Mean field

The Little-Parks Effect is washed out by fluctuations when >1

summary on fluctuations in superconducting rings
Summary on Fluctuations in Superconducting Rings
  • Agreement with fluctuation theory developed by Riedel and von Oppen.
    • Contrary to previous results, we find no anomalously large susceptibility at zero field.
    • Fluctuations in the Little-Parks regime

( ) are large.

  • No evidence for inhomogeneous states, but they could be contributing to the fluctuation response.
  • Rings with largest fluctuation regimes could not be compared to theory in the LP regime due to numerical intractability.
  • Little-Parks Effect washed out by fluctuations when >1
measuring currents in mesoscopic rings4
Measuring Currents in Mesoscopic Rings
  • Technique
  • Dirty aluminum rings: fluxoids
    • 1 order parameter
    • 2 order parameters
  • Cleaner aluminum rings: fluctuations
  • Gold rings: h/e-periodic persistent currents in normal metals
  • Surprising spins
pure 1 dimensional ring

 = 0

- k +k

T = 0, disorder = 0

I

T > 0

/0

Pure 1-Dimensional Ring

E

EF

Typical current

Büttiker et al.,

Phys. Lett. 96A (1983)

Cheung et al.,

PRB 37 (1988)

periodic in h/e, including higher harmonics

ensembles vs single rings
Ensembles vs. single rings

Idea: Measure many (N) rings at once to enhance signal.

h/2e

h/e

  • Previous measurements: (Levy, Deblock, Reulet)
  • Magnitude ~Ec/f0 - factor of a few larger than expected
  • Sign not well understood
  • Temperature dependence as expected

Need to measure

severalindividual rings

diffusive rings mean free path ring circumference
Diffusive ringsmean free path << ring circumference

Response depends on disorder configuration

Ih/e has a distribution of magnitudes and signs

consider ensemble averages ….

Thouless energy:

Riedel and v. Oppen

PRB 47 (1993)

Related contributions:

Cheung and Riedel.,

PRL 66 (1989)

Determined by interactions

previous measurement ballistic

Gates

Calibration

coil

Junctions

2DEG

Pickup

Previous measurement - ballistic

Single ballistic GaAs ring: (L > le )

Mailly et al., PRL 70 (1993)

  • Magnitude of h/e signal agrees with theoretical expectation
  • Gates allow background characterization.
previous measurement diffusive
Previous measurement - diffusive

Observed periodic component in 3 rings:

60 Ec /f0

12 Ec /f0

220 Ec /f0

Background not always well behaved.

Chandrasekhar et al., PRL 67 (1991)

Raw signal

The result of the only previous measurement of individual diffusive rings (in 1991) was two orders of magnitude larger than expected!

Fitted background subtracted.

sample
Sample

R

I ~ 10 mA, 10 GHz

w

d

Fac

Pring ~ 10-14 W

0.5 mm

Fabrication

Optical and e-beam lithography,

e-beam evaporation (6N source), liftoff

Diffusivity: D = 0.09 m2/s

Mean free path: le = 190 nm

Dephasing length Lj = 16 mm

d = 140 nm

w = 350 nm

R = 0.57 - 1 mm

Grid for navigating sample

optical image magnetic scan

expected signal
Expected signal

(excludes factor 2 for spin because of spin-orbit coupling)

Riedel and v. Oppen

PRB 47 (1993)

Ourexpected T = 0 SQUID signal is independent of L:

ring - SQUID inductance

response from 15 rings
Response from 15 rings

R = 0.67 mm

linear component subtracted (in- and out of phase)

mean as background
Mean as background

Assume: Signal = background-response + persistent current

similar for all rings:

suspect spin response

Ih/e  = 0

=

-1

0

1

-1

0

1

-

=….

variations in ring response
Variations in ring response

data - data =

Sine-fits:

fixed period

fitted period

Ih/e 21/2 M

= 0.12 mF0

= 0.9 nA M

data

Expected: Ih/e 21/2 M = 0.1 mF0 (Tel = 150 mK)

temperature dependence
Temperature dependence
  • Difference of signals from two rings with a large and opposite response
  • Any common background is eliminated
  • Fair agreement with theory:
is the flux periodic signal from persistent currents
Is the flux-periodic signal from persistent currents?

see also recent results by A. Bleszynski-Jayich, J. Harris, and coauthors

Consistency Checks:

  • Expected distribution of magnitudes
  • Expected temperature dependence
  • Periodic signal does not appear in larger (R = 1 m) rings
    • 6 rings measured
    • larger Ec => steeper falloff with temperature
    • better coupling to SQUID => larger electron temperature
  • Periodic signal does not depend on frequency (in 2 rings)
  • Amplitude of periodic signal does not depend on sweep amplitude.

Causes for Doubt:

  • Zero-field anomaly (from spins?) not fully understood
  • Electron temperature of isolated rings
measuring currents in mesoscopic rings5
Measuring Currents in Mesoscopic Rings

Technique

Dirty aluminum rings: fluxoids

1 order parameter

2 order parameters

Cleaner aluminum rings: fluctuations

Gold rings: h/e-periodic persistent currents in normal metals

Surprising spins

anomalously large spin response
Susceptibility signal suggest an area spin density of s = 4 x105mm-2

Observed on every film studied: even on gold films with no native oxide

Similar to excess flux noise observed in SQUIDs and superconducting qubits

45 m

Anomalously Large Spin Response

Susceptibility Image

(Linear in-phase term)

Optical Image

electron temperature

heatsunk ring

isolated ring

Tel150 mK

0.03

0.5

0.1

Electron temperature

Linear susceptibility

I ~ 10 mA, ~10 GHz

Fac

Pring ~ 10-14 W

Expect Tel~ 150 mK

  • 1/T dependence of paramagnetic susceptibility => spins
  • heat sinking effective => spins equilibrate with electrons
  • origin of spin signal not understood
  • Likely related to aperiodic component in nonlinear response(subtracted mean)
comparative magnitude and t dependence
Comparative Magnitude and T-dependence

Linear Paramagnetic Susceptibility

  • Bare Si has no paramagnetic response (from height dependence).
  • Gold films have a larger response than AlOx films
  • Response from layered structures not additive.
  • 140 nm thick e-beam defined Au rings and heatsink wires, evaporated 1.2nm/s on Si with native oxide, 6N purity source
spin interaction with conduction electrons

5 mm

Spin Interaction with Conduction Electrons
  • Spins do not cause electronic decoherence in the ring
    • Weak localization measurements show long coherence times, suggesting ~0.1 ppm or less for concentration of spins causing decoherence.
  • Spins are well enough coupled that they are thermalized with the conduction electrons from the ring
    • Josephson oscillations from the SQUID heats isolated rings, and poor electron-phonon coupling prevents electrons from cooling
    • Response from isolated rings saturates at ~150mK: calculated electron temperature based on Josephson heating

0.5 mm

Heat Sunk Ring

Isolated Ring

out of phase and nonlinear susceptibility
Out of phase component f2 is ~two orders of magnitude smaller than in phase component

Existence of out-of-phase component implies magnetic noise from spins

Nonlinear component should provide clues to spin dynamics

Out of Phase and Nonlinear Susceptibility

Linear Out of Phase

spin density inferred from magnitude
Spin Density Inferred from Magnitude
  • Areal density: For g = 2 and J = 1/2, the signal of the purest gold film corresponds to an area density
  • 4 · 1017 spins/m2 or 4 · 105 spins/micron2
  • Volume density, if in gold rather than surface or interface:
    • About 60 ppm if in the gold itself
    • 3 ppm for g2J(J+1) = 35
comparison with 1 f noise

E

}

hw

Comparison with 1/f Noise

Koch, DiVincenzo and Clarke Model

  • Trapping energies have broad distribution compared to kBT
  • Uncorrelated changes in spin direction yield a 1/f power spectrum
  • Expected defect density 5x105mm-2
  • 1/f noise is generated by the magnetic moments of electrons trapped in defect states
  • Electron spin is locked while it occupies the trap trap (Kramer Degenerate Ground State)

Koch, DiVincenzo and Clarke PRL 98, 267003 (2007)

measuring currents in mesoscopic rings6
Measuring Currents in Mesoscopic Rings

Technique

RSI79, 053704 (2008).

APL93, 243101 (2009).

Dirty aluminum rings: fluxoids in 2-OP ring

PRL97, 237002 (2006).

Cleaner aluminum rings: fluctuations in LP regime

Science318 , 1440 (2007).

Gold rings: h/e-periodic persistent currents

PRL102, 136802 (2009).

Surprising spins

PRL103, 026805 (2009).

slide55

10 mm

next generation pickup loops: 500 nm

spin sensitivity < 100 B/rt-Hz

slide56

10 mm

next generation pickup loops: 500 nm

spin sensitivity < 100 B/rt-Hz

fabrication deposition sample i
(A) 80 nm e-beam defined Au wire grid and bond pads

Evaporated on Si with native oxide, source purity unknown

50 nm thick AlOx patterned using optical lithography

Rings and wires e-beam evaporated at a rate of 1.2nm/s from 6N Au

7

6

mF0/mA

5

4

3

2

1

0

Fabrication & Deposition: Sample I

(C)

(A)

(B)

Flux detected by pick up loop

Applied Excitation by field coil

fabrication deposition sample ii
Redesigned after (Sample I) to have smaller spin susceptibility

140 nm thick e-beam defined Au rings and heatsink wires

Evaporated 1.2nm/s on Si with native oxide, 6N purity source

100 nm thick optically defined heatbanks and current grid

7nm Ti sticking layer

mF0/mA

10

5

0

Fabrication & Deposition: Sample II

15mm

Flux detected by pick up loop

Applied Excitation by field coil

conclusions and outlook
Conclusions and Outlook
  • In mesoscopic gold rings, we observe an h/e-periodic magnetic signal whose magnitude and temperature are consistent with theoretical expectations for persistent currents.
  • We also observe what appears to be an unexpectedly high density of nearly free spins in gold as well as in other samples.

0.2K

0.1K

0.035K

0.035K

slide60

Observation of persistent currents in thirty metal rings, one at a time

*see also recent results by A. Bleszynski-Jayich, J. Harris, and coauthors

smaller rings
Smaller rings

R = 0.57 mm

Raw nonlinear response

Mean

Raw data

- mean

Ih/e 21/2 M

= 0.07 mF0

signal from heatsunk rings
Signal from heatsunk rings
  • Linear response
  • paramagnetic
  • ~1/T dependence
  • => spins
  • Nonlinear response
  • Likely due to relaxation effects
  • spatial dependence same as for linear

(+) - (o) - linear component (~ 120 mF0)

heatsunk rings
Heatsunk rings
  • Measured 4 (R = 0.8, 1 mm)
  • Found no periodic,but large aperiodic response
  • Flux captured in heatsink might break periodicity
  • Largest plausible amplitude 0.2 mF0

Raw signal (linear in-phase subtracted)

– ellipse (linear out-of-phase subtracted)

– phenomenological “step”

pure 1 dimensional ring1
Pure 1-Dimensional Ring

F > fo/2

 = 0

0 < F<fo/2

I

- k +k

0

1

/f0

T = 0

I

T > 0

/f0

E

EF

=> Typical current

Büttiker et al.,

Phys. Lett. 96A (1983)

Cheung et al.,

PRB 37 (1988)

Effect of temperature,

disorder:

what is the background
What is the background?
  • Hysteretic
  • Frequency dependent(10 – 300 Hz)
  • Decreases at higher T
  • Also seen in other metal structures
  • => Suspect nonequilibrium spin response

-1

0

1

frequency and amplitude dependence
Frequency and amplitude dependence
  • Nonlinear signals from two rings with a large and opposite response at different field sweep frequencies.
  • Frequency dependent background
  • Pair wise difference is
  • h/e periodic and frequency independent

Difference signal at different sweep amplitudes

conclusion on spins
Conclusion on Spins
  • Spin susceptibility with 1/T dependence measured in micropatterened thin films
    • Corresponds to an area spin density of ~4x105mm-2
    • Agreement with what’s inferred in SQUIDs
  • Strong metallic response
    • Spins related to silver and gold rather than silicon or native silicon oxide
    • (Spins observed on other insulators, eg AlOx, thermal silicon oxide)
  • Signals from layered structures are not additive
    • Possible interactions between layers
  • Increase of out of phase response with frequency
    • Flux noise varies slower than 1/w
  • Probable connection with superconducting films
  • Scanning SQUID susceptometry excellent technique for further investigation of flux noise
conclusion and outlook
Conclusion and outlook
  • Measured magnetic response of 33 mesoscopic goldrings, one ring at a time.
  • Observed oscillatory component with period h/e and different sign and amplitude in different rings.
  • Typical magnitude and temperature dependence are consistent with expected typical persistent current, Ih/e21/2.
  • Also find a background response that is most likely due to unpaired spins.
slide70

10 mm

next generation pickup loops: 500 nm

spin sensitivity < 100 B/rt-Hz

squid
SQUID

Gradiometer

I0

I0

I0

Susceptometer

I0

Magnetometer

Low inductance “linear coaxial” shields allow for:

• optimized junctions

- noise best when LI0 = 0/2

• low field environment near susceptometer core

• reduced noise n ~ L3/2

• independent tip design

Applied field ~ 10s of 0

Desired signal ~0.1 0

Requires background elimination to 1 part in 108

typical images
Typical Images

2 mm

5 mm

susceptometry of a ring

magnetometry of a vortex

in a bulk superconductor

72

comparison of l e 4 nm and l e 19 nm

Fit

Data

Comparison of le = 4 nm and le = 19 nm

0.40 K

1.00 K

1.35 K

n = 3

n = 0

n = -3

1.49 K

le = 19 nm

D = 1 micron

w = 75 nm

t = 70 nm

1.524 K

two order parameter gl model
Two-order-parameter GL - model

T <Tc1=> x1large, strong pair breaking.

Fluxoid transition inhibited by coupling

to other component.

~

For n1 = n2 = 0: i(x) = const.

=> solve numerically to get fit model

hysteretic curves data and model
Hysteretic curves - data and model

T <Tc1=> x1large.

1 transitions earlierthan 2 if coupling

weak enough.

=> formation of

metastable states

with n1 n2

~

Assume transition occurs when activation energy <~ 5 kBT.

Data Model

Simple Explanation

summary on 2 op rings1
Summary on 2-OP rings
  • "Textbook" single-OP behavior observed for many Al rings.
  • Bilayer rings form a model system for two coupled order parameters with the following features: - metastable states with two different phase winding numbers, manifest in unusual Φa-I curves and reentrant hysteresis. - unusual T-dependence of xandl-2.
  • Extracted parameters for two-order-parameter Ginzburg-Landau model with little a priori knowledge.
few ring experiments
Few-ring experiments
  • 30 Au rings
  • Reasonable amplitude
  • I or I2 ?

16 connected GaAs rings

Rabaud et al., PRL 86 (2001)

Jariwala et al., PRL 86 (2001)