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.)
Magnetic-Field-Driven
Phase Transitions
in Unconventional
Josephson Arrays
“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)Efetov et al., ‘80Ma, Lee ‘85Kapitulnik, Kotliar ‘85Fisher ‘90Wen and Zee ‘90
Only phase fluctuations
R
T
/ 1CJE E
/ 1CJE E
/ 1CJE E
cos( ) 0i
cos( ) 0i
Insulator
RQ
superconductor
The SIT is driven by the competition betweenCooper pair hopping and Coulomb repulsion:
Charge-vortex duality (M. Fisher, ’90)
0 cosJ JE E Josephson energy
JC C
eE
2
2
Charging energy
3
≫ h𝑜𝑡 𝑒𝑟 𝑟𝑒𝑙𝑒𝑣𝑎𝑛𝑡 𝑒𝑛𝑒𝑟𝑔𝑖𝑒𝑠
van der Zant et al, ‘96B=0
Magnetic-field-driven SIT in Josephson Arrays
At odds with the “dirty boson” model,
a T-independent (“metallic”)
resistivity was observed over
a wide range of R.
f=0
f=0.27
Chen et al., (’94)
T (K)
f = /0
Random charges in the environment (static and fluctuating)
Flux noise
Random scatter of Josephson energies and its fluctuations
disorder + B-induced frustrations
emergent inhomogeneity,glassines, etc.
?
Static and dynamic disorder
Potential complications:
4
JJ arrays with large number of nearest-neighbor islands
better averaging of the fluctuations of the parameters of individual JJs.
the effect of magnetic field is expected to be stronger (NEJ EJN in B>0/A);
exploration of a much wider range of the JJ parameters (e.g., junctions with RN >>RQ).
Potential advantages of large N:
𝐸 𝐽∗=𝑁 𝐸 𝐽
J
𝐸𝐶∗=𝐸𝐶 /𝑁
Characteristic energiesper island
(no gate electrode, CJ>>Cg ):
5
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).
Array Fabrication
N=10 array
Experimental realization:
“Manhattan pattern” nanolithography
Multi-angle deposition of Al
-0.2 0.0 0.20
50
100
150
B (G)
I C (n
A)
-1.050E-5
-9.063E-6
-7.625E-6
-6.187E-6
-4.750E-6
-3.312E-6
-1.875E-6
-4.375E-7
1.000E-6
B0/Aarray
- in line with numerical simulations (Sadovskyy) 6
B (G)
I C (
nA
)
Typical normal-state R of individual junctions:
no ground plane: 30-200 k
with ground plane: up to1 MAarray~ 100100m2
Arrays without ground plane
7
Array B
R (2K)= 5.0 k
RJ = 43 k
EC = 1.2 K
EJ = 0.18 K
N2(EJ/EC) = 15
Array A
R (2K)=15.2 k
RJ =133 k
EC = 1.8K
EJ = 0.06 K
N2(EJ/EC) = 3.3
0.0 0.2 0.4 0.6
1
10
100
R (k
)
T (K)
B
A
R (
k)
T (K)
Incoherent transport of
Cooper pairs
NEJ
Arrays: 8x8 “supercells” (100×100 m2)
C (per island) ~ 5 fF, EC (per island) ~ 0.2 K
C/Cg ~ 100
The “critical” R ~ 3-20 k
for the arrays without a
ground plane. Mag. field
Quasiparticle freeze-out
Arrays with conducting ground plane
resistances at 2K 1
2
3
ArrayRarray(2K)
kΩ
RJ
kΩ
NEJ
K
ECisland
K
NEJ/Ecisland
(B = 0)
1 17.3 150 0.5 0.035 142 39 345 0.23 0.024 103 124 1,100 0.07 0.035 2
Al2O3 3 nm
Al 20 nm
200 400 6000
4x104
8x104
1x105
right side
left side
frustrated B=0
Resis
tance(
)
Temperature(mK)
total
NEJ
The “critical” R ~1 M for
this array with a ground plane.
The “S-I” transition
at NEJ /Ec
island ~1.
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.
Arrays without ground plane: more detailed look at the SIT
10
-10 -5 0 5 100
25
50
75
100
40mK 100mK 150mK
R (
k)
f
A
-1 0 1 2 30
2
4
6
R (
k)
f
B
0.00 0.25 0.500
25
50 40mK 100mK 150mK
R (
k)
f
Multiple SITs (commensurate
structure) at different R ~ 3-20 k.
van der Zant et al, ‘96
ff
R (
k)
R (
k)
R (
k)
R (
k)
f f
f =/0 – normalized
flux per 10
unit cells
alternating “S” and “I”
phases
11
Finite-Bias Transport
Rarray (4K)= 18.9 k
RJ = 160 k
EC ~ 2K, EJ ~ 0.05K
N2(EJ/EC) ~ 2.5
-2 -1 0 1 2
0.3
0.4
0.5
0.6
I (nA)
f
0
156.3
312.5
625.0
1250
1875
2500
2813
3125
3438
3594
3750
4375
5000
Color-coded differential
resistance dV/dI(I,B)
I (nA)
f
12
Direct “S” “I ” Transitions
𝑇 0= 2𝑒𝑘 𝐵∫𝑑𝑉𝑑𝐼 ( 𝐼 )−( 𝑑𝑉𝑑𝐼 ( 𝐼 ))∗𝑑𝐼
𝑇 0= ħ2𝑒𝑘𝐵∫ 𝑑𝐼𝑑𝑉 (𝑉 )−( 𝑑𝐼𝑑𝑉 (𝑉 ))∗𝑑𝑉
“insulator”:
“superconductor”:
Low Rcr (< 10 k):
direct “S” – “I” transitions. 0.7 0.8 0.9 1.0
-80
-40
0
Voltage Temp(mK) Current Temp(mK)
T0 (m
K)
B (G)
0.7 0.8 0.9 1.0-80
-40
0
Voltage Temp(mK) Current Temp(mK)
T
0
(m
K)
B (G)
20
-20
0.0 0.2 0.4 0.6
1
10
100
R (
k)
T (K)
Array B
R (
k)
T (K)
13
Lack of Duality at High Rcr
-0.2 -0.1 0.0 0.1 0.2I (nA)
f
10000
2.125E4
3.250E4
4.375E4
5.500E4
6.625E4
7.750E4
8.875E4
1.000E5
0.1
0.15
0.2
0.3
0.4
Array A
0.0 0.2 0.4 0.6
1
10
100
R (k)
T (K)
A
R (
k)
0.0 0.2 0.4 0.6
1
10
100
R (k)
T (K)T (K)f
I (nA)
High Rcr (>10 k):
Lack of “duality”.
14
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
“Metallicity”:
15
0 5 10 15 20
102
103
R (
k)
1/T (1/K)
-0.2 -0.1 0.0 0.1 0.2
-25
0
25
V (V
)I (nA)
“Insulating” RegimeArray 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
V* is the voltage drop across the whole array
B
Sub-pA bias is requiredin the “insulating” regime.
R(T) ~exp[2eV*/kBT]
V*
R (
k)
1/T (1/K)
V (V
) B
I (nA)
Lines:
0 1 2 30
250
500
T0 (
mK
)
B (mA)
B (G)
2eV
*(B
)/k B
(m
K)
0 0.5 1.0 1.5
0
250
500
III
0 5 10 15 20
102
103
R (
k)
1/T (1/K)
B
I
II
R (
k)
1/T (1/K)
Insulating Regime in N = 4 Array
f = /0
Arrhenius:
N = 4 array
Rarray (300K)= 37.5 k
EC ~ 1.2K, EJ ~ 0.23K
EJ/EC ~ 0.2
N2(EJ/EC) ~ 3
16
2eV*(B) ~ kBT0(B)
R(T)=R0exp(T0/T) T0= T0(B) R0 104
17
Possible Explanations?
- The voltage drops across the most resistive link with the largest local T0.
2eV*(B)=kBT0(B)
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) could be signatures of a collective process.
Emergent inhomogeneity?
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
200 400 6000
4x104
8x104
1x105
right side
left side
frustrated B=0
Resis
tance(
)
Temperature(mK)
total
Solid curves: total arrayDashed curves: one half
-100.0p 0.0 100.0p
-70.0µ
0.0
70.0µ
Total Right half Left half
Vol
tage
(V)
Current(A)
65.24uV
110.24uV
44.98uV
T=base , B=4mA
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
VT,
mV
T0 ~ lnL
2eVT (L) ~ (10100) kBT0 (L)
Threshold of Quasiparticle Generation
ththth VIP
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 *
-5 0 5 10 15 20 253
6
9
12
15
Re
sist
an
ce (
k)
Magnetic Field (G)
T=30mK
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
21
Pth is T-independent below ~ 0.2K,
whereas R(I=0) and Ith still depend on T.
22
200 400
1E-14
1E-13
1E-12
B=.3mA B=1mA B=2mA
Thr
esho
ld P
ower
(W)
Temperature(mK)
0 100 200 300 400 500
1E-14
1E-13
B=1.3mA B=3mA
Thr
esho
ld P
ower
(W)
Temperature(mK)
Scaling with Array Area
supercells 44
supercells 88
The “threshold” power is
proportional to the array’s area
(the total number of junctions)
Two arrays on the same chip:
Summary:
23
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.