1. 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

2. 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 ???

3. 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

4. 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.

5. 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).

6. 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: 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)

7. 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)

8. 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.

9. 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.

10. Arrays without ground plane: more detailed look at the SITB
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

11. 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)

12. 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.

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

14. “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

15. Array II (4x4 supercells)
“Insulating” Regime
Array I (8x8 supercells)
R (2K)= k
Array II (4x4 supercells)
R (2K)= 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)

16. 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 

17. 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.

18. 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.

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

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

21. 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.

22. 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)

23. 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.