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## Measuring Currents in Mesoscopic Rings

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**Measuring Currents in Mesoscopic Rings**From femtoscience to nanoscience, INT, Seattle 8/3/09**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**• 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**• 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**Measurements Hendrik Bluhm Nick Koshnick Julie Bert SQUIDs Martin Huber Funded by NSF, CPN, and Packard**Measuring persistent currents**Ha Fa Apply field Measure current ?**Measuring persistent currents**Ha Fa Apply field • Measure magnetic field • Difficulties: • Small signal • Large background**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**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**2 mm 5 mm susceptometry of a ring 10**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**SQUID SQUID Sample substrate Sample substrate • Ring signal: 0.1 mF0 • White noise: 0.50/Hz • Applied field: 45 0 in each pickup loop • => need to eliminate background to • 1 part in 109 background measurement 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 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**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**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:**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 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 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 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 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**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**• "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 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**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 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 anomalousdiamagnetic**susceptibility(Zhang and Price, 1997) Zhang and Price, 1997 (1 ring, zero-field response only)**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**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**• 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**Blue: Data Red: Theory Green: Mean field The Little-Parks Effect is washed out by fluctuations when >1**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 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** = 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**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 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**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**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**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**(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**R = 0.67 mm linear component subtracted (in- and out of phase)**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**data - data = Sine-fits: 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)**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?**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 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**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**heatsunk ring**isolated ring Tel150 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**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**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