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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Measuring Currents in Mesoscopic Rings
From femtoscience to nanoscience, INT, Seattle 8/3/09
Fa
Classical conducting ringsThe 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.
Fa
I
Measurements
Hendrik Bluhm
Nick Koshnick
Julie Bert
SQUIDs
Martin Huber
Funded by NSF, CPN, and Packard
Ha
Fa
Apply field
SQUID
2 mm
Location of
pickup loop
Sample Substrate
OUT
DC
feedback
Front end
SQUID
IN
IV
R ~ 50 m
100SQUID
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
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 = 2.8x109mm
SQUID
SQUID
Sample substrate
Sample substrate
background measurement
measurement
Record complete nonlinear response by averaging over many sinusoidal field sweeps at each position.
Susceptibility scans
(Inphase linear response)
Raw signal after tuning Icomp (step 2)
mF0
Measurement positions:
+ background + signal
o background
= Fluxoid #
Superconducting Coherence Length
mesoscopic superconducting ringsEnergy
Y = Yeif
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
w
d
w = line width
oxide
Sample structureDeduced film structure:
= Fluxoid #
Superconducting Coherence Length
Fit
Data
aI data and models0.40 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 ΦaIcurves of 190 nm ringsR = 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 ΦaIcurves of 190 nm ringsTwo order parameters
Single order parameter
n
n2
only one (monotonic) transition path connects two different metastable states.
n1
multiple transition paths exist







Anomalous ΦaIcurves of 190 nm ringsTwo order parameters
Single order parameter
n
n2
only one (monotonic) transition path connects two different metastable states.
n1
multiple transition paths exist







Anomalous ΦaIcurves of 190 nm ringsTwo order parameters
Single order parameter
n
n2
only one (monotonic) transition path connects two different metastable states.
n1
multiple transition paths exist
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 Tdep
Tc,1
Tc,2
oxide
Twoorderparameter GL  fitsFits to representative datasets.
Summary of all data:
Energy
n=0
n=1
n=2
In a thinwalled sample near Tc,
kinetic energy can exceed the condensation energy:
wellknown “LittleParks Effect”
0
1
2
F/F0
tin cylinder
~1 micron diameter
37.5 nm wall thickness
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 finiteR regions
predicted by deGennes, 1981
Zhang and Price, 1997
(1 ring, zerofield response only)
PMMA
PMMA
Al
silicon oxide
Samples
silicon substrate
ebeam evaporation and liftoff
2nd generation:
Background pressure <107 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.
d = 60 nm
w = 110 nm
AC:
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
Zhang and Price, 1997
(1 ring)
Present Work
(15 rings measured, 4 rings shown)
Blue: Data
Red: Theory
Green: Mean field
The LittleParks Effect is washed out by fluctuations when >1
( ) are large.
 k +k
T = 0, disorder = 0
I
T > 0
/0
Pure 1Dimensional RingE
EF
Typical current
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.
h/2e
h/e
Need to measure
severalindividual rings
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
Calibration
coil
Junctions
2DEG
Pickup
Previous measurement  ballisticSingle ballistic GaAs ring: (L > le )
Mailly et al., PRL 70 (1993)
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.
R
I ~ 10 mA, 10 GHz
w
d
Fac
Pring ~ 1014 W
0.5 mm
Fabrication
Optical and ebeam lithography,
ebeam 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
(excludes factor 2 for spin because of spinorbit coupling)
Riedel and v. Oppen
PRB 47 (1993)
Ourexpected T = 0 SQUID signal is independent of L:
ring  SQUID inductance
Assume: Signal = backgroundresponse + persistent current
similar for all rings:
suspect spin response
Ih/e = 0
=
1
0
1
1
0
1

=….
data  data =
Sinefits:
fixed period
fitted period
Ih/e 21/2 M
= 0.12 mF0
= 0.9 nA M
data
Expected: Ih/e 21/2 M = 0.1 mF0 (Tel = 150 mK)
see also recent results by A. BleszynskiJayich, J. Harris, and coauthors
Consistency Checks:
Causes for Doubt:
Technique
Dirty aluminum rings: fluxoids
1 order parameter
2 order parameters
Cleaner aluminum rings: fluctuations
Gold rings: h/eperiodic persistent currents in normal metals
Surprising spins
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 ResponseSusceptibility Image
(Linear inphase term)
Optical Image
isolated ring
Tel150 mK
0.03
0.5
0.1
Electron temperatureLinear susceptibility
I ~ 10 mA, ~10 GHz
Fac
Pring ~ 1014 W
Expect Tel~ 150 mK
Linear Paramagnetic Susceptibility
0.5 mm
Heat Sunk Ring
Isolated Ring
Existence of outofphase component implies magnetic noise from spins
Nonlinear component should provide clues to spin dynamics
Out of Phase and Nonlinear SusceptibilityLinear Out of Phase
}
hw
Comparison with 1/f NoiseKoch, DiVincenzo and Clarke Model
Koch, DiVincenzo and Clarke PRL 98, 267003 (2007)
Technique
RSI79, 053704 (2008).
APL93, 243101 (2009).
Dirty aluminum rings: fluxoids in 2OP ring
PRL97, 237002 (2006).
Cleaner aluminum rings: fluctuations in LP regime
Science318 , 1440 (2007).
Gold rings: h/eperiodic persistent currents
PRL102, 136802 (2009).
Surprising spins
PRL103, 026805 (2009).
Evaporated on Si with native oxide, source purity unknown
50 nm thick AlOx patterned using optical lithography
Rings and wires ebeam 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
140 nm thick ebeam 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 II15mm
Flux detected by pick up loop
Applied Excitation by field coil
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. BleszynskiJayich, J. Harris, and coauthors
(+)  (o)  linear component (~ 120 mF0)
Raw signal (linear inphase subtracted)
– ellipse (linear outofphase subtracted)
– phenomenological “step”
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:
1
0
1
Difference signal at different sweep amplitudes
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
Data
Comparison of le = 4 nm and le = 19 nm0.40 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
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
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
16 connected GaAs rings
Rabaud et al., PRL 86 (2001)
Jariwala et al., PRL 86 (2001)