Magnetic-Field-Driven in Unconventional Josephson Arrays Phase Transitions in Unconventional Josephson Arrays Joshua Paramanandam, Matthew Bell, and Michael Gershenson Department of Physics and Astronomy, Rutgers University, New Jersey, USA Theoretical encouragement: Lev Ioffe (Rutgers) and Misha Feigelman (Landau Inst.) “Strongly Disordered Superconductors and Electronic Segregation” Lorentz Center, Leiden, 26 Aug. 2011

Outline: Several long-standing (~20 years) issues: - magnetic-field-induced “metallicity” in Josephson arrays; - dissipation mechanisms; - transport in the insulating regime. Our weapon of choice: Josephson arrays with a large number of nearest-neighbor islands. “S-I” transition at EJ/Ec ~ 1, the “critical” resistance varies by three orders of magnitude depending on screening. “Metallicity”: several alternating “S” and “I” phases (commensurability) with very small (  T) characteristic energies. Insulating regime (no traces of emergent inhomogeneity…): - “Arrhenius” activation energy correlates with the “offset” voltage across the whole array ??? - the power threshold of quasiparticle generation is “universal” and scales with the array area ???

Bosonic Model of SIT (preformed Cooper pairs) Efetov et al., ‘80 Ma, Lee ‘85 Kapitulnik, Kotliar ‘85 Fisher ‘90 Wen and Zee ‘90 ≫𝑜𝑡ℎ𝑒𝑟 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠 Only phase fluctuations Josephson energy The SIT is driven by the competition between Cooper pair hopping and Coulomb repulsion: Charging energy Charge-vortex duality (M. Fisher, ’90) R T Insulator RQ superconductor van der Zant et al, ‘96 B=0

Magnetic-field-driven SIT in Josephson Arrays Chen et al., (’94) T (K) At odds with the “dirty boson” model, a T-independent (“metallic”) resistivity was observed over a wide range of R. f = /0 Potential complications: Random charges in the environment (static and fluctuating) Flux noise Random scatter of Josephson energies and its fluctuations Static and dynamic disorder ? disorder + B-induced frustrations emergent inhomogeneity, glassines, etc.

JJ arrays with large number of nearest-neighbor islands Characteristic energies per island (no gate electrode, CJ>>Cg ): 𝐸 𝐽 ∗=𝑁 𝐸 𝐽 𝐸 𝐶 ∗= 𝐸 𝐶 /𝑁 𝐸 𝐽 𝐸 𝐶 ∗=𝑁2 𝐸 𝐽 𝐸 𝐶 J Potential advantages of large N: better averaging of the fluctuations of the parameters of individual JJs. the effect of magnetic field is expected to be stronger (NEJ EJN in B>0/A); exploration of a much wider range of the JJ parameters (e.g., junctions with RN >>RQ). The characteristic energies are 2-3 times smaller than that for the conventional arrays (still exceed the temperature of the quasiparticle “freeze-out”, ~0.2K).

B0/Aarray Array Fabrication Experimental realization: “Manhattan pattern” nanolithography Multi-angle deposition of Al Typical normal-state R of individual junctions: no ground plane: 30-200 k with ground plane: up to1 M Aarray~ 100100m2 B0/Aarray N=10 array IC (nA) B (G) - in line with numerical simulations (Sadovskyy)

Arrays without ground plane Array A R (2K)=15.2 k RJ =133 k EC = 1.8K EJ = 0.06 K N2(EJ/EC) = 3.3 Array B R (2K)= 5.0 k RJ = 43 k EC = 1.2 K EJ = 0.18 K N2(EJ/EC) = 15 Arrays: 8x8 “supercells” (100×100 m2) C (per island) ~ 5 fF, EC (per island) ~ 0.2 K C/Cg ~ 100 Incoherent transport of Cooper pairs Quasiparticle freeze-out A The “critical” R ~ 3-20 k for the arrays without a ground plane. R (k) NEJ Mag. field B T (K)

Arrays with conducting ground plane Rarray(2K) kΩ RJ NEJ K ECisland NEJ/Ecisland (B = 0) 1 17.3 150 0.5 0.035 14 2 39 345 0.23 0.024 10 3 124 1,100 0.07 Al2O3 3 nm Al 20 nm resistances at 2K 1 2 3 The “S-I” transition at NEJ /Ecisland ~1. NEJ The “critical” R ~1 M for this array with a ground plane.

Probably, the first experiment which shows that (EJ/EC)island is the only relevant parameter, the critical resistance Rcr can vary a great deal depending on the capacitance matrix.

Arrays without ground plane: more detailed look at the SIT B R (k) f =/0 – normalized flux per 10 unit cells R (k) f f Multiple SITs (commensurate structure) at different R ~ 3-20 k. R (k) R (k) alternating “S” and “I” phases f f van der Zant et al, ‘96

Finite-Bias Transport Rarray (4K)= 18.9 k RJ = 160 k EC ~ 2K, EJ ~ 0.05K N2(EJ/EC) ~ 2.5 Finite-Bias Transport Color-coded differential resistance dV/dI(I,B) f I (nA)

Direct “S”  “I ” Transitions Array B “insulator”: R (k) 𝑇0= 2𝑒 𝑘𝐵 𝑑𝑉 𝑑𝐼 𝐼 − 𝑑𝑉 𝑑𝐼 𝐼 ∗ 𝑑𝐼 “superconductor”: T (K) 𝑇0= ħ 2𝑒𝑘𝐵 𝑑𝐼 𝑑𝑉 𝑉 − 𝑑𝐼 𝑑𝑉 𝑉 ∗ 𝑑𝑉 20 -20 Low Rcr (< 10 k): direct “S” – “I” transitions.

Lack of Duality at High Rcr Array A A R (k) f T (K) I (nA) High Rcr (>10 k): Lack of “duality”.

“Metallicity”: At least partially due to alternating S and I phases (commensurability) with very small activation energies. The phase transitions observed at low “critical” R < 10k follow the “dirty boson” scenario (direct SIT). However, the duality is lacking for the transitions observed at larger R > 10k. f=0 f=0.27 Chen et al., (’94) T (K) f = /0

Array II (4x4 supercells) “Insulating” Regime Array I (8x8 supercells) R (2K)= 16.63 k Array II (4x4 supercells) R (2K)= 16.47 k RJ = 156 k EC = 2.5 K EJ = 0.05 K N2(EJ/EC) = 2 Sub-pA bias is required in the “insulating” regime. B V* V (V) V* is the voltage drop across the whole array I I (nA) R (k) B 500 Lines: 1/T (1/K) I R(T) ~ exp[2eV*/kBT] II 2eV*(B)/kB (mK) II 250 R (k) B 0.5 1.0 1.5 B (G) 1/T (1/K)

Insulating Regime in N = 4 Array Rarray (300K)= 37.5 k EC ~ 1.2K, EJ ~ 0.23K EJ/EC ~ 0.2 N2(EJ/EC) ~ 3 2eV*(B) ~ kBT0(B) f = /0 Arrhenius: R(T)=R0exp(T0/T) T0= T0(B) R0  104 

Possible Explanations? 2eV*(B)~kBT0(B) could be signatures of a collective process. Emergent inhomogeneity? Cooper pair hopping along the chain of islands with an effective charge close to (2n+1)e (costs no energy to add/subtract a Cooper pair). The “bottleneck” is the island with a larger deviation of its q from (2n+1)e. - The voltage drops across the most resistive link with the largest local T0. 2eV*(B)=kBT0(B) However, the same values of the resistance observed for two halves of the array seem to rule out the latter option.

Macroscopic Homogeneity in the “Insulating” Regime Solid curves: total array Dashed curves: one half No significant difference in the resistance and T0 for two halves of the array was observed.

System-size dependence of T0 and VT in thin films T0 ~ lnL VT, mV 2eVT (L) ~ (10100) kBT0 (L)

Threshold of Quasiparticle Generation The “threshold” power does not depend on the zero-bias resistance. For all studied arrays Pth  10-14 -10-13 W.

Threshold Power V *I * N = 11 array Rarray (4K)= 15.4 k RJJ ~ 150 k EC ~ 0.7K, EJ ~ 0.06K EJ/EC ~ 0.08 N2(EJ/EC) ~ 10 Pth is T-independent below ~ 0.2K, whereas R(I=0) and Ith still depend on T.

Scaling with Array Area Two arrays on the same chip: The “threshold” power is proportional to the array’s area (the total number of junctions)

Summary: Unconventional Josephson arrays with a large number of nearest-neighbor islands have been fabricated. Multiple “S-I” transitions (due to commensurate effects) over a wide range of critical resistances R ~ 3-20 k were observed. “Metallisity” – due to alternating “S” and “I” phases with very low (typically < 100 mK) characteristic energies. The phase transitions observed for these arrays resemble the “dirty boson” SIT at low “critical” Rcr ~ few k, however the duality is lacking for the transitions observed at larger Rcr . On the “insulating” side of the SIT, the R(T) dependences can be fitted with the Arrhenius law R(T)~exp(T0/T), where kBT0 is close to the “Coulomb” gap 2eV* (V* is the offset voltage across the whole array). The threshold for quasiparticle generation at high bias currents is surprisingly universal for samples with vastly different zero-bias resistances. This power scales with the array area.