measuring currents in mesoscopic rings
DESCRIPTION
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: - PowerPoint PPT PresentationTRANSCRIPT
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
Measuring Currents in Mesoscopic Rings
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
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
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
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.
Two order parametersSingle order parameter
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
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
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.
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
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
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
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
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 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
Measuring Currents in Mesoscopic Rings
• 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
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)