Broadband microwave conductance across the T=0 superconductor-resistive magnetic field tuned transition in InO x! N. Peter Armitage! Dept. of Physics and Astronomy! The Johns Hopkins University!
Lidong Pan (JHU) Wei Liu (JHU) Sambandamurthy Ganapathy (UB) Minsoo Kim (UB) and hopefully soon Danny Shahar and Maoz Ovadia (Weizmann)
In this `paperʼ we...! 1.) demonstrate a new(ish) technique to perform broadband low ω (0.002 20 GHz) microwave conductivity measurements at low T (~ 300 mk)! 2.) get dynamical information about superconducting fluctuations in disordered InO films! 3.) find evidence for critical slowing down along the T axis and a transition of KTB type at B=0! 4.) investigate the 2D SIT and find complete suppression of superconducting response and anomalous dependence of the fluctuation frequency scales (speeding up) on approach to nominal T=0 transition.! Does it imply that true transition is at a lower field well below nominal transition at B cr?!
Superconducting fluctuations:! The conventional view!
Fluctuations of an ordered state above T c! Consider ferromagnetism! U ij = - T cos Δθ ιϕ T < T c! T > T c! ξ { Below T c spins point in well-defined direction. Order parameter is magnetization! Above T c there is no long range order, but still short range fluctuations of length ξ, which persist for time τ Critical slowing down! General phenomena. Correlations of an ordered state above T c of a 2 nd order transition (magnetism, liquid crystals, superconductivity).! Usually significant precursor effects above T c enhanced magnetization, structure factor S(q,ω), conductivity etc.! Typical is power law dependencies τ = ξ z = ξ 0 (1 T/T c ) zν (KTB is different faster than power law )!
Superconducting fluctuations! Different regimes of superconducting fluctuations Ψ = Δe ιφ Order parameter Resistance Ω/ " Superconductivity Phase Fluctuations! Amplitude Fluctuations Temperature (Kelvin) Τ ΚΤΒ Τ c0 Normal State Amplitude (Δ) fluctuations; Ginzburg-Landau theory; Δ 0 Transverse phase fluctuations Vortices x e iφ 0 Longitudinal phase fluctuations; spin waves ;. e iφ 0 (in neutral superfluid) Size set by phase `stifness ~T θ
Schematic of the Broadband Microwave Corbino Spectrometer Measure complex optical response σ = σ 1 + i σ 2 10 MHz - 40 GHz (100 MHz -16GHz) T > 290 mk B ~ 8 T Measure complex reflection coefficient S11 to get complex impedance at sample Very involved calibration procedure (3 standards) to account for numerous reflections and losses
Superconductor AC Conductance @ ω << 2Δ 0.35 Conductivity 0.30 0.25 0.20 0.15 Real Conductivity! Imaginary Conductivity! 0.10 0.05 0 0 20 40 60 80 Frequency
Superconductor AC Conductance @ ω << 2Δ 30 nm amorphous InO x measured in `Corbinoʼ geometry! broadband microwave system! R N ~ 1400 Ohms; T c ~ 2.4 K 2Δ > 170 GHz! Normal state scattering 1/τ > 10 THz!
Superfluid (Phase) Stiffness! Many of the different kinds of superconducting fluctuations can be viewed as disturbance in phase field! Energy for deformation of any continuous elastic medium (spring, rubber, concrete, etc.) has a form that goes like square of generalized coordinate! e.g. Hookeʼs law!!!!u = ½ kx 2!
Superfluid (Phase) Stiffness! Superfluid density can be parameterized as a phase stiffness:! Energy scale to twist superconducting phase Ψ = Δ e ιθ θ 1 θ 2 θ 3 θ 4 θ 5 θ 6 Uij = - T cos Δθ ιϕ (Spin stiffness in discrete model. Proportional to Josephson coupling)! Energy for deformation has this form in any continuous elastic medium.! T is a stiffness, a spring constant.!
Superconductor AC Conductance @ ω << 2Δ 0.35 Conductivity 0.30 0.25 0.20 0.15 Real Conductivity! Imaginary Conductivity! 0.10 0.05 0 0 20 40 60 80 Frequency
Kosterlitz-Thouless-Berezinskii Transition KTB showed power-law ordered phase can exist at low T <φ(0) φ(r)> 1/r ν Since high T phase is exponentially correlated <φ(0) φ(r)> ~ e -r/ξ a finite temperature transition exists Transition happens by proliferation (unbinding) of topological defects (vortex - antivortex) Coulomb gas Superfluid Density N s bare stiffness ~ superfluid density T KTB = π/2 Ν s! Ν s BCS T KTB Temperature T c0 Superfluid stiffness falls discontinuously to zero at universal value of Ν s /T
Kosterlitz-Thouless-Berezenskii Transition! Superfluid Density McQueeny et al. (1984) He3-He4 mixtures of different proportions! Superfluid stiffness proportional to superfluid density! Measured via torsion oscillator! Superfluid stiffness falls discontinuously to zero at temperature set by superfluid stiffness itself.!
Frequency Dependent Superfluid Stiffness! Kosterlitz Thouless Berezenskii Transition! T KTB = π/2 ρ s! Superfluid stiffnes! bare superfluid! density! ω=0! increasing ω! ω=inf! Probing length set by diffusion relation.! T KTB! T m! Temperature! In 2D static superfluid stiffness survives at finite frequency (amplitude is still well defined). Finite frequency probes short length scale. If ω>ω then system looks superconducting. Approaches ʻbareʼ stiffness as ω gets big.!
Frequency Dependent Superfluid Stiffness! Superfluid Stiffness(Kelvin)!
Superconductor AC Conductance ΓΗζ Close to transition scaling forms for the conductivity are expected *. σ 1 (10 3 Ω 1 ) Data collapse in terms of a characteristic relaxation frequency Ω(T) = 1/τ Temperature (Kelvin) * Fisher, Fisher, Huse PRB, 1991
Fisher-Widom Scaling Hypothesis Close to continuous transition, diverging length and time scales dominate response functions. All other lengths should be compared to these Scaling Analysis
Scaling in superconductors! Close to transition scaling forms are expected.! Data collapse with characteristic relaxation frequency Ω(T) = 1/τ! Functional form may look unusual, but it is not. Drude model obeys this form.! Important! Since prefactors are real, phase of S is also phase of σ!! With φ = tan -1 (σ 2 /σ 1 ). φ should collapse with one parameter scaling.!
Scaling in 2D superconductors! All temperature dependencies enter through extracted Ω and T θ 0 from scaling!
What is T dependence of Ω? Power-law or KTB?! Ω 0 181 GHz and Tʼ = 0.21Κ Ω 0 90 GHz and zν = 1.58!One expects Ω 0 to be of order inverse time to diffuse coherence length! Prefactors and exponents are reasonable in both cases, so fitting cannot distinguish between scenarios!
Activation gives evidence for vortex plasma regime! Well above T KTB one expects activated forms! Ansatz Core Vortex Interaction Within BCS one expects that:! µ ~ T θ0 /8! T* ~ 0.27K @ High T ξ is small interactions donʼt matter (vortex plasma screening)! High T limit of T* ~ 0.27K, which compares embarrasingly well with BCS estimate from T θ 0 ~ 0.3 K!
Quantum Phase Transitions in 2D superconductors!
Conventional Wisdom for 2DSIT Dirty Bosons! Sheet Resistance (Ω) Macroscopic QM wavefunction! ψ = Δ e ιθ(r) Uncertainty principle phase and density; Δφ Δn > 1 Conventional Wisdom (Fisher and others)! SIT is direct quantum phase transition from phase eigenstate to number eigenstate (insulator).! Like in conventional phase transition, in QPT diverging length and time scales! Universal dependencies critical slowing down! Temperature (Kelvin) Haviland et al. 1988 Resistance @ critical point R Q ~ h/4e 2
Phase Diagram Expectations Thermal T 0 T KTB Phase defined Amplitude Phase Dominated Transition: Transition Dirty Bosons Amplitude defined Ψ = Δ(x,t) e i φ (x,t) Superconducting B c Insulating B c2 Quantum
Degree of disorder changes character of transition! Weak! Disorder! Strong! Steiner and Kapitulnik 2008!
Non-universal behavior is observed Critical resistance can be far from R Q Intervening metal states are observed. Are these real? Are they a lack of refrigeration? Amorphous MoGe Yazdani and Kapitulnik, Physics Today 1998
2D superconductor- insulator transition 1200 Resistance per Square (Ohms) 1000 800 600 400 200 B=0T B=1T B=2T B=3.5T B=4T B=5T B=6T B=7T B=7.5T B=8T B=11T B=14T 75mK 150mK 0 0 1 2 3 4 4 6 8 10 12 14 Temperature (K) Magnetic Field (Tesla) 2D SIT with exceptionally weak insulating state 30 nm InO x!!t c ~ 2.35! B cr ~ 7.5 Tesla!k F l ~ 21!
T 300 mk
Field Dependence of Superfluid Stiffness! T 300 mk B cr ~ 7.5T
Field dependence of ω dependent phase stiffness! 10 B cr Temperature (Kelvin) 1 0.1 0.01 Phase Stiffness @ 50MHz Phase Stiffness @ 100MHz Phase Stiffness @ 300MHz Phase Stiffness @ 1GHz Phase Stiffness @ 3GHz Phase Stiffness @ 15GHz 0.001 0 2 4 6 8 Field (Tesla)
Strongly insulating InO x has different response! 10 5 Resistance (Ohm) 10 4 10 3 10 2 H SIT = 3.69 T T=0.236 K T=0.499 K T=0.749 K T=0.999 K T=1.250 K T=1.497 K T! (Kelvin) 6 4 2 T = 0.5 K T = 0.68 K T = 1 K T = 1.25 K T = 1.75 K T = 2.0 K T = 2.25 K T = 2.5 K T = 2.75 K T = 3.25 K T = 4 K T = 4.5 K B c = 3.69 T! 10 1 0 2 4 Field (Tesla) 6 8 0 0 2 4 Magnetic field (Tesla) 6 8 Resistance Phase stiffness Single ω measurement @ 20 GHz.! Appreciable superfluid response on insulating side of transition even when R ~ 10 6 Ohms and increasing! R. Crane, NPA et al. PRB 2007!
Phase Diagram Expectations Thermal T 0 T KTB Phase defined Amplitude Phase Dominated Transition: Transition Dirty Bosons Amplitude defined Ψ = Δ(x,t) e i φ (x,t) Superconducting B c Insulating B c2 Quantum
Phase Diagram Expectations Thermal T 0 T KTB Phase defined Amplitude Phase Dominated Transition: Transition Dirty Bosons Amplitude defined Ψ = Δ(x,t) e i φ (x,t) Superconducting B c Insulating B c2 Quantum
T 300 mk
Temperature dependence of fluctuation rate! Very narrow Drude -like peak! Width ~ 4 GHz! 100 80 Fluctuation rate (GHz) 60 40 Fluctuation Rate @ 5 Tesla Temperature in GHz 20 0 0.0 0.2 0.4 0.6 0.8 1.0 Temperature (Kelvin) 1.2 1.4 Normal state scattering rate ~ 100 THz!
Characteristic Fluctuation Rate 12 10 Fluctuation rate (GHz) 8 6 4 What is! happening here?! B cr! 2 0 2 3 4 5 Field (Tesla) 6 7 8 T = 300 mk!
Phase Diagram 10 B cr Temperature (Kelvin) 1 0.1 Temperature R DC is 0.3% R N Fluctuation Rate in Kelvin Phase Stiffness @ 50MHz Phase Stiffness @ 15GHz 0.01 0 2 4 6 8 Field (Tesla) (Caution: some of these `temperaturesʼ are frequencies and some are energy scales)
Conclusions Evidence for a significant thermal fluctuation regime that exhibits evidence for KTB-like transition, critical slowing down, vortex plasma regime! Along quantum axis on weakly disorder sample, DC transport is consistent with previous measurements of SIT with seperatrix field B cr, but! No evidence of phase stiffness past this field unlike in strongly disordered samples! More anomalously we see speeding up of fluctuations as we approach field B cr that is nominal QCP for 2D SIT Looks like thermal transition! Does it imply that true transition is at a lower field well below field B cr?! We need strongly disordered InOx!! R. Crane et al. PRB 2007! W. Liu et al., PRB 2011! W. Liu et al., to be submitted 2012!