Magnetic-Field-Driven
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
1
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 ???
2
Bosonic Model of SIT (preformed Cooper pairs)
 ≫ 𝑜𝑡ℎ𝑒𝑟 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠
Only phase fluctuations
The SIT is driven by the competition between
Cooper pair hopping and Coulomb repulsion:
Charge-vortex duality (M. Fisher, ’90)
R
cos(i )  0
Insulator
Efetov et al., ‘80
Ma, Lee ‘85
Kapitulnik, Kotliar ‘85
Fisher
‘90
Wen and Zee ‘90
Josephson
energy
Charging
energy
EJ  EJ 0 cos 
e2
EC 
2C J
B=0
van der Zant
et al, ‘96
EJ / E C  1
EJ / E C  1
EJ / E C  1
RQ
superconductor
cos(i )  0
T
3
Magnetic-field-driven SIT in Josephson Arrays
T (K)
At odds with the “dirty boson” model,
a T-independent (“metallic”)
f = /0
f=0.27
resistivity was observed over
a wide range of R.
f=0
Chen et al., (’94)
Potential complications:
Random charges in the environment (static and fluctuating)
Flux noise
Random scatter of Josephson energies and its fluctuations
disorder +
B-induced frustrations
?
Static and
dynamic
disorder
emergent inhomogeneity,
glassines, etc.
4
JJ arrays with large number of nearest-neighbor islands
Characteristic energies
per island
(no gate electrode, CJ>>Cg ):
∗
𝐸𝐽 = 𝑁𝐸𝐽
∗
𝐸𝐶 = 𝐸𝐶 /𝑁
𝐸𝐽
𝐸𝐶
∗
=
𝑁2
𝐸𝐽
𝐸𝐶 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).
5
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
Aarray~ 100100m2
with ground plane: up to1 M
150
B0/Aarray
-1.050E-5
-9.063E-6
IICC (nA)
(nA)
N=10 array
-7.625E-6
100
-6.187E-6
-4.750E-6
-3.312E-6
-1.875E-6
-4.375E-7
50
0
1.000E-6
-0.2
0.0
B (G)
0.2
B (G)
- in line with numerical
simulations (Sadovskyy)
6
Arrays without ground plane
Array A
Array B
R (2K)=15.2 k
R (2K)= 5.0 k
RJ =133 k
RJ = 43 k
Arrays: 8x8 “supercells” (100×100 m2)
EC = 1.8K
EC = 1.2 K
C (per island) ~ 5 fF, EC (per island) ~ 0.2 K
EJ = 0.06 K
EJ = 0.18 K
N2(EJ/EC) = 3.3
N2(EJ/EC)
= 15
Incoherent
transport of
Cooper pairs
R
)
R (k(k)
100
C/Cg ~ 100
Quasiparticle
freeze-out
A
The “critical” R ~ 3-20 k
NEJ
Mag. field
10
for the arrays without a
ground plane.
B
1
0.0
0.2
0.4
T (K)
(K)
0.6
7
Arrays with conducting ground plane
Array
1
2
3
Al2O3 3 nm
Al 20 nm
3
NEJ
Rarray(2K)
RJ
NEJ
kΩ
kΩ
K
17.3
150
0.5
39
345 0.23
124
1,100 0.07
ECisland
K
0.035
0.024
0.035
NEJ/Ecisland
(B = 0)
14
10
2
The “S-I” transition
at NEJ /Ecisland ~1.
2
1
5
ance()
1x10
frustrated
B=0
resistances
at 2K
The
total “critical” R ~1 M for
this array with a ground plane.
4
8x10
right side
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.
9
Arrays without ground plane: more detailed look at the SIT
75
50
25
40mK
100mK
150mK
-5
RR (k)
(k)
50
0
ff
5
10
40mK
100mK
150mK
alternating
“S” and “I”
phases
0.00
4
2
0
-1
0
1
ff
2
3
Multiple SITs (commensurate
structure) at different R ~ 3-20 k.
25
0
B
R (k)
0
-10
f =/0 –
normalized
flux
per 10
unit cells
RR (k
(k)
)
A
100
R R(k)
(k)
6
f
f
0.25
0.50
f
van der Zant et al, ‘96
10
Rarray (4K)= 18.9 k
Finite-Bias
Transport
RJ = 160 k
EC ~ 2K, EJ ~ 0.05K
N2(EJ/EC) ~ 2.5
Color-coded differential
resistance dV/dI(I,B)
0.6
f
f
0.5
0.4
0.3
-2
-1
0
(nA)
II (nA)
1
2
11
R (k)
Direct “S”  “I ” Transitions
Array B
R (k)
10
“insulator”:
2𝑒
𝑇0 =
𝑘𝐵
Voltage Temp(mK)
Current Temp(mK)
1
T0 (mK) T0 (mK)
0.0
0.2
0
∗
𝑑𝐼
“superconductor”:
0.4 Temp(mK)0.6
Voltage
TTemp(mK)
(K)
(K)
CurrentT
𝑇0 =
20
𝑑𝑉
𝑑𝑉
𝐼 −
𝐼
𝑑𝐼
𝑑𝐼
ħ
2𝑒𝑘𝐵
𝑑𝐼
𝑑𝐼
𝑉 −
𝑉
𝑑𝑉
𝑑𝑉
∗
𝑑𝑉
0
-40
-20
-40
-80
0.7
Low Rcr (< 10 k):
direct “S” – “I” transitions.
0.8
B (G)
0.9
1.0
12
R (k
Lack of Duality at High Rcr
10
100
R
)
R(k(k)
Array A
0.4
0.3
10000
A
1
10
2.125E4
3.250E4
0.2
4.375E4
0.0
0.2
0.4
0.2
0.4
f
f
5.500E4
T (K)
(K)
T
0.6
6.625E4
0.15
7.750E4
8.875E4
1.000E5
1
0.1
0.0
-0.2
-0.1
0.0
0.1
T (K)
0.6
0.2
(nA)
II (nA)
High Rcr (>10 k):
Lack of “duality”.
13
“Metallicity”:
At least partially due to alternating S and
T (K)
I phases (commensurability) with very
f = /0
small activation energies.
f=0.27
The phase transitions observed at low
“critical” R < 10k
follow the “dirty
boson” scenario (direct SIT).
f=0
Chen et al., (’94)
However, the duality is lacking for the
transitions observed at larger R > 10k.
14
“Insulating” Regime
Array I (8x8
supercells)
Sub-pA bias is required
in the “insulating” regime.
R (2K)= 16.63 k
25
V V(V)
(V)
Array II (4x4
supercells)
R (2K)= 16.47 k
B
V*
0
V* is the voltage drop across
the whole array
RJ = 156 k
EC = 2.5 K
EJ = 0.05 K
-0.2
-0.1
0.0
0.1
0.2
I (nA)
I (nA)
N2(EJ/EC) = 2
R R(k)
(k)
-25
I
3
10
B
2
10
0
500
5
II
250
250
15
20
1/T (1/K)
II
3
10
R R(k)
(k)
R(T) ~
exp[2eV*/kBT]
I
10
1/T (1/K)
Lines:
T0 (mK)
2eV*(B)/kB (mK)
500
B
2
10
00
0
0
1
2
0.5 B (mA) 1.0
B (G)
3
1.5
0
5
10
15
1/T (1/K)
1/T (1/K)
20
15
Insulating Regime in N = 4 Array
N = 4 array
Rarray (300K)= 37.5 k
EC ~ 1.2K, EJ ~ 0.23K
2eV*(B) ~ kBT0(B)
EJ/EC ~ 0.2
N2(EJ/EC) ~ 3
f = /0
Arrhenius:
R(T)=R0exp(T0/T)
T0= T0(B)
R0  104 
16
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.
2eV*(B)=kBT0(B)
- The voltage drops across the most resistive link
with the largest local T0.
However, the same values of the resistance observed for two halves
of the array seem to rule out the latter option.
17
Macroscopic Homogeneity in the “Insulating” Regime
Solid curves: total array
Dashed curves: one half
5
1x10
frustrated
B=0
T=base , B=4mA
Total
Right half
Left half
110.24uV
Resistance()
70.0µ
Voltage(V)
65.24uV
44.98uV
0.0
4
8x10
No significant difference in the
resistance and T0 for two halves of
4
4x10was observed.
the array
-70.0µ
-100.0p
0.0
100.0p
Current(A)
18
System-size dependence of T0 and VT in thin films
VT, mV
T0 ~ lnL
2eVT (L) ~ (10100) kBT0 (L)
Threshold of Quasiparticle Generation
Pth  I th  Vth
The “threshold” power does not depend
on the zero-bias resistance.
For all studied arrays Pth  10-14 -10-13 W.
20
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
15
T=30mK
Resistance (k)
12
9
6
3
-5
0
5
10
15
20
25
Magnetic Field (G)
Pth is T-independent below ~ 0.2K,
whereas R(I=0) and Ith still depend on T.
21
Scaling with Array Area
B=1.3mA
B=3mA
Threshold Power(W)
Two arrays on the same chip:
4 4 supercells
1E-13
1E-14
Threshold Power(W)
1E-12
B=.3mA
B=1mA
B=2mA
0
100
200
300
400
500
Temperature(mK)
The “threshold” power is
1E-13
proportional to the array’s area
8 8 supercells
1E-14
200
Temperature(mK)
(the total number of junctions)
400
22
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.
23