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.