measuring currents in mesoscopic rings

Measuring Currents in Mesoscopic RingsMeasuring Currents in Mesoscopic Rings

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

a

a

Classical conducting rings

I

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

I I0e tR / L

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

• 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

I F

I

a

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


SQUIDsMartin Huber

MeasurementsHendrik BluhmNick Koshnick

Julie Bert

Funded by NSF, CPN, and Packard

a

a

Measuring persistent currents

Apply field Measure current ?

a

a

Measuring persistent currents

Apply field Measure magnetic field

Difficulties:• Small signal• Large background

Advantages

•Background measurements

•Measure many samples in one cooldown

•Measure samples made on any substrate

Location ofpickup loop

Sample Substrate

SQUID

2 mm

Scanning Magnetic Measurements

feedback

1 mm

shielding field coil

bias

SQUID susceptometer

field coil

pickup loop

12 m

substrate polished to create a corner at the pickup loop

I-V

100-SQUID array preamp(NIST)

DC feedback

R ~

50

m

Front end SQUID IN

OUT

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

10

Performance

2 mm

5 m

susceptometry of a ring

White noise floor

flux

0.2 0/Hz ring current0.2 nA /Hz

spin

200 B/Hz

S1/2s (in B) = S1/2

(in 0) x a/rewhere a = pickup loop radius = 2 m

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

spin sensitivity

ring current sensitivity

S1/2I = MS1/2

where M = mutual inductance ~ 0.1 - 1 0/mA

(conventional but optimistic conversion)

Real experiments limited by1/f noise

background

Background elimination

• Ring signal: 0.1 0

• White noise: 0.5 0/Hz • Applied field: 45 0 in each pickup loop

=> need to eliminate background to

1 part in 109

Step Elimination procedure Residual backg.relative absolute

1Symmetric sensor design with counterwound pickup loops.

10-2 0.5 0

2 compensation using center tap 10-4 5 m0

3In-situ background measurement

10-8 0.50

4 Data processing 2x10-9 0.1 0

Raw signal after tuning Icomp (step 2)

Background measurement

Susceptibility scans(In-phase linear response)

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

Measurement positions:+ background + signalo background

• Compute (+) - (o)• Subtract ellipse (linear response)

0


• 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

Agd

rl

rBgd

ra

Energy

Current

n=0 n=1 n=2

0 1 2

2

202 2

0

12n

wsI n n

R R

0/

n= Fluxoid #

2

1

Superfluid

Density

Superconducting Coherence Length

GL:

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

oxide

Deduced film structure:

w = line width

R

dw

2

202 2

0

12n

wsI n n

R R

FitData

0.40 K

1.00 K

1.35 K

1.49 K

n = 0 n = 3

n = -3

1.524 K

0

/

1

kTEep

dt

dp

( ) exp ( ) /( )

exp ( ) /

n n Bn

n Bn

I E k TI

E k T

0 2

202

00 4n n

AE d I n

R

a-I data and models

0/

n= Fluxoid #2

1

Superfluid

Density

Hysteretic Response Described by Rate Equation

High Temperature Response Well Described by Boltzmann Distributed Fluxoid States

Superconducting Coherence Length

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

Anomalous Φa-I curves of 190 nm rings

-

-

-

--

-

- -

n1

n2n

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

• Reentrant hysteresis • Transitions not periodic in Φa/ Φ0

• Branches of Φa-I curves shifted by less than one Φ0.

• Unusual shape of non-hysteretic Φa-I curves.

• Not an effect of averaging over many cycles.

Motivation for 2-OP modelTwo order parametersSingle order parameter

R = 1.2 m


-

-

-

--

-

- -

n1

n2n


• Reentrant hysteresis• Transitions not periodic in Φa/ Φ0




Two order parametersSingle order parameter


-

-

-

--

-

- -

n1

n2n


• Reentrant hysteresis• Transitions not periodic in Φa/ Φ0




Motivation for 2-OP modelTwo order parametersSingle order parameter

Two-order-parameter GL - fits

Summary of all data:

Tc,1 Tc,2

coupling increases with w=> stronger proximitization

oxide

w (nm) # rings 2-OP features100 2120 6 190 7 250 14320 1370 5

None

Soliton states

only manifest in T-dep

Fits to representative datasets.

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 and -2.

• Extracted parameters for two-order-parameter Ginzburg-Landau model with little a priori knowledge.


• Technique





Little-Parks effect

Energyn=0 n=1 n=2

0 1 2

In a thin-walled sample near Tc,kinetic energy can exceed the condensation energy:

well-known “Little-Parks Effect”

tin cylinder ~1 micron diameter

37.5 nm wall thickness

Previously observed anomalous resistance in Little-Parks regime: Liu et al. Science

2001

150 nm diameter Al cylinder wall thickness 30 nmreported (T) = 161 nm at T = 20 mK from Hc||(T)

R=0 => global phase coherenceregions separated by finite-R regionspredicted by deGennes, 1981

Previously observed anomalousdiamagnetic susceptibility (Zhang and Price, 1997)

Zhang and Price, 1997(1 ring, zero-field response only)

Ring fabrication

Samples

silicon substrate

silicon oxide

PMMA PMMAAl

e-beam evaporation and liftoff

2nd generation:Background pressure <10-7 mBarDeposition rate ~3.5 nm/secle = 30 nm on unpatterned filmle ~ 19 nm small features with PMMA (inferred)

R = 0.5 – 2 md = 70 nm w = 30 – 350 nm linewidth

R

dw

1st generation samples le = 4 nm + accidental layered structure for w > 150nmmodel system for 2 coupled order parameters.Bluhm et al, PRL 2006.

Applied Flux Dependence

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

d = 60 nmw = 110 nm

A-C: R = 350 nmTc = 1.247 K (fitted)

D:R = 2,000 nmTc = 1.252 K (fitted)

Our Results

Zhang and Price, 1997(1 ring)

Present Work(15 rings measured, 4 rings shown)

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

Comparison of “Large” and “Small” Rings

16

M eff

Tc

Ec

Blue: DataRed: TheoryGreen: Mean field

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

Summary on Fluctuations in Superconducting Rings

16

M eff

Tc

Ec

0.87

M eff

L2

0le

dI

da

A

0

• 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


• Technique





Typical currentround

f

t

e

L

evI 0

E

Pure 1-Dimensional Ring

22

1Aep

mH

)( AekmL

eI nn EF

- k +k

= 0

/0

I T = 0, disorder = 0

T > 0 Büttiker et al., Phys. Lett. 96A (1983)

Cheung et al.,PRB 37 (1988)

periodic in h/e, including higher harmonics

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

Ih / e 0

Ih / e

N

N Ih / e2

1

2

Ih / 2e 0

Ih / 2e

N

N Ih / 2e

Ensembles vs. single rings

Need to measure several individual rings

h/2e h/e

Previous measurements: (Levy, Deblock, Reulet)• Magnitude ~ Ec/0 - factor of a few larger than expected• Sign not well understood• Temperature dependence as expected

Diffusive ringsmean free path << ring circumference

Ityp Ih / e2 1/ 2

e

D

ev f

L

le

L

E c

0

Thouless energy:2

2

L

DEc

02

exp0

/

eeff

ceh l

LM

EI

02/

ceh

EI Determined by interactions

Response depends on disorder configurationIh/e has a distribution of magnitudes and signsconsider ensemble averages ….

Riedel and v. Oppen PRB 47 (1993)

Cheung and Riedel.,PRL 66 (1989)

c

Bceh E

TkEI exp

0

2/12/

Related contributions:

Previous measurement - ballistic

Gates

2DEG

Calibrationcoil

Pickup

Junctions

Mailly et al., PRL 70 (1993)Single ballistic GaAs ring: (L > le )

• Magnitude of h/e signal agrees with theoretical expectation• Gates allow background characterization.

Previous measurement - diffusive

Fitted background subtracted.

Raw signal

Observed periodic component in 3 rings:

60 Ec /0

12 Ec /0

220 Ec /0

Background not always well behaved.

Chandrasekhar et al., PRL 67 (1991)

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

Sample

FabricationOptical and e-beam lithography,e-beam evaporation (6N source), liftoff

Diffusivity: D = 0.09 m2/sMean free path: le = 190 nmDephasing length L = 16 m

Pring ~ 10-14 W

I ~ 10 A, 10 GHz

ac

R

dw

d = 140 nmw = 350 nmR = 0.57 - 1 m

0.5 m

Grid for navigating sample

optical image magnetic scan

c

Bceh E

TkEI exp

0

2/12/

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

nA 1 ~2 2

0 L

eDEc

mK 400~3 2

2

2

2

BeF

c kL

lv

L

DE

Our expected T = 0 SQUID signal is independent of L:

2LM

Riedel and v. Oppen PRB 47 (1993)

Expected signal

00

0.15

cEM

ring - SQUID inductance

c

elBceh E

TkEM exp

0

2/12/

Response from 15 rings

R = 0.67 m

linear component subtracted (in- and out of phase)

Mean as background

=

- =….

Assume: Signal = background-response + persistent current

similar for all rings: suspect spin response

-1 0 1

-1 0 1

Ih/e = 0

Variations in ring response

Sine-fits: data - data =

dataIh/e

21/2 M= 0.12 0

= 0.9 nA M

fixed period

fitted period

Expected: Ih/e 21/2 M = 0.1 0 (Tel = 150 mK)

Temperature dependence

Difference of signals from two rings with a large and opposite response

Any common background is eliminated

Fair agreement with theory:

c

elBeh E

Tkexp

2/12/

Is the flux-periodic signal from persistent currents?

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

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


• Technique





Anomalously Large Spin Response

• Susceptibility signal suggest an area spin density of s = 4 x105 m-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

Optical Image

Susceptibility Image

(Linear in-phase term)

Electron temperature

heatsunk ring

isolated ringTel150 mK

0.1 0.50.03

Pring ~ 10-14 W

Expect Tel ~ 150 mK

I ~ 10 A, ~10 GHz

ac

Linear susceptibility

• 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

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

Heat Sunk Ring Isolated Ring

5/1

52

T

V

RITel

0.5 m5 m

1. 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.

2. 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

Out of Phase and Nonlinear Susceptibility

Out of phase component 2 is ~two orders of magnitude smaller than in phase componentExistence of out-of-phase component implies magnetic noise from spinsNonlinear component should provide clues to spin dynamics

Linear Out of Phase

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

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

Koch, DiVincenzo and Clarke Model

• 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)

• Trapping energies have broad distribution compared to kBT

• Uncorrelated changes in spin direction yield a 1/f power spectrum

• Expected defect density 5x105 m-2

E

} h


• Technique – RSI 79, 053704 (2008). – APL 93, 243101 (2009).

• Dirty aluminum rings: fluxoids in 2-OP ring– PRL 97, 237002 (2006).

• Cleaner aluminum rings: fluctuations in LP regime– Science 318 , 1440 (2007).

• Gold rings: h/e-periodic persistent currents– PRL 102, 136802 (2009).

• Surprising spins – PRL 103, 026805 (2009).

10 m

next generation pickup loops: 500 nmspin sensitivity < 100 B/rt-Hz

Fabrication & Deposition: Sample I

0

1

2

3

4

5

6

7

0/m

A

(A) 80 nm e-beam defined Au wire grid and bond pads

– Evaporated on Si with native oxide, source purity unknown

(B) 50 nm thick AlOx patterned using optical lithography

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

(A)(B)

(C)

mA

“ 0 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 layer10

5

0

0/m

A

15 m

mA

“ 0 Flux detected by pick up loop

Applied Excitation by field coil

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

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

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

Samples measured

Sample structure

Ag on AlOx Au on AlOx Au on SiOx Au on Si

# rings measured

8 7 2 33

Biggest problem

Variations in susceptibility of metal,

high base T

Transient response from AlOx,

nonlinear response

Bad film adhesion, high base T

Nonlinear response

Smaller rings

Mean

Raw data - mean

R = 0.57 mRaw nonlinear response

Ih/e 21/2 M

= 0.07 0

Signal from heatsunk rings

(+) - (o) - linear component (~ 120 0)

Linear response• paramagnetic• ~1/T dependence=> spins

Nonlinear response• Likely due to relaxation effects• spatial dependence same as for linear

Heatsunk rings

Raw signal (linear in-phase subtracted)

– phenomenological “step”

• Measured 4 (R = 0.8, 1 m)

• Found no periodic,but large aperiodic response

• Flux captured in heatsink might break periodicity

• Largest plausible amplitude 0.2 0

– ellipse (linear out-of-phase subtracted)

=> Typical currentround

f

t

e

L

evI 0

I

/0

0 1

E

Pure 1-Dimensional Ring

22

1Aep

mH

)( AekmL

eI nn EF

- k +k

= 00 < <o/2 o/2

Effect of temperature,disorder:

/0

I T = 0

T > 0

Büttiker et al., Phys. Lett. 96A (1983)

Cheung et al.,PRB 37 (1988)

What is the background?

-1 0 1

• Hysteretic• Frequency dependent

(10 – 300 Hz)• Decreases at higher T• Also seen in other metal structures

=> Suspect nonequilibrium spin response

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 ish/e periodic and frequency independent

Difference signal at different sweep amplitudes

Conclusion on Spins

• Spin susceptibility with 1/T dependence measured in micropatterened thin films– Corresponds to an area spin density of ~4x105 m-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/

• Probable connection with superconducting films• Scanning SQUID susceptometry excellent technique for further

investigation of flux noise

Conclusion and outlook

• Measured magnetic response of 33 mesoscopic gold rings, 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/e

21/2.

• Also find a background response that is most likely due to unpaired spins.

10 m

next generation pickup loops: 500 nmspin sensitivity < 100 B/rt-Hz

SQUID

I0

I0

Applied field ~ 10s of 0

Desired signal ~0.1 0

Requires background elimination to 1 part in 108

I0

I0

GradiometerMagnetometer

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

Susceptometer

72

Typical Images

2 mm

5 m

magnetometry of a vortexin a bulk superconductor

susceptometry of a ring

SQUID sensitivity

2

202 2

0

12n

wsI n n

R R

FitData

0.40 K

1.00 K

1.35 K

1.49 K

n = 0 n = 3

n = -3

1.524 K

Comparison of le = 4 nm and le = 19 nm

le = 19 nmD = 1 micronw = 75 nmt = 70 nm

Two-order-parameter GL - model

For n1 = n2 = 0: i(x) = const.=> solve numerically to get fit model

T <Tc1=> 1 large, strong pair breaking.Fluxoid transition inhibited by coupling to other component.

~

Hysteretic curves - data and model

Data Model

R

njump

0

T <Tc1=> 1 large.1 transitions earlierthan 2 if coupling weak enough.

=> formation of metastable states with n1 n2

~

Simple Explanation

Assume transition occurs when activation energy < kBT

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 and -2.

• Extracted parameters for two-order-parameter Ginzburg-Landau model with little a priori knowledge.

Few-ring experiments

16 connected GaAs ringsRabaud et al., PRL 86 (2001)

• 30 Au rings• Reasonable amplitudeI or I2 ?

Jariwala et al., PRL 86 (2001)

measuring currents in mesoscopic rings

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