Vortex drag in a Thin-film Giaever transformer

Vortex drag in a 

Thin-film Giaever transformer 

Yue (Rick) Zou (Caltech) 

Gil Refael (Caltech) 

Jongsoo Yoon (UVA) 

Past collaboration: 

Victor Galitski (UMD) 

Matthew Fisher (station Q) 

T. Senthil (MIT)

Outline 

• Experimental background – SC-metal-insulator in InO, TiN, Ta and MoGe. 

• Two paradigms: 

- Vortex condensation: Vortex metal theory. 

- Percolation paradigm 

• Thin film Giaever transformer – amorphous thin-film bilayer. 

• Predictions for the no-tunneling regime of a thin-film bilayer 

• Conclusions

Quantum vortex physics 

SC-insulator transition 

• Thin films: B tunes a SC-Insulator transition. 

I 

InO 

Ins 

B 

B 

V 

SC 

(Hebard, Paalanen, PRL 1990)

Observation of Superconductor-insulator transition 

• Thin amorphous films: B tunes a SC-Insulator transition. 

InO: 

• Saturation as T 0 

(Sambandamurthy, 

(Steiner, 

Engel, 

Kapitulnik) 

Johansson, Shahar, PRL 2004) 

• Insulating peak different from sample to sample, scaling different – log, activated.

Vortex Paradigm

X-Y model for superconducting film: 

Cooper pairs as Bosons 

• When the superconducting order is strong – ignore electronic excitations. 

• Standard model for bosonic SF-Ins transition – Bose-Hubbard model: 

H 

= 

' 

!& 

)! 

i 

% 

j 

+ cos( ( * ) 

s 

j 

i 

+ Unˆ 

2 

i 

$ 

# 

" 

[ nˆ, 

φ] 

= −i 

ρ S 

exp[ i( 

φ i 

−φ 

i+ 1)] 

Hopping: 

Charging 

energy: 

2 

Unˆ j 

• Large U - Mott insulator 

(no charge fluctuations) 

• Large 

ρ s 

- Superfluid 

(intense charge fluctuations, 

no phase fluctuations) 

insulator 

superfluid 

ρ s 

/U

Vortex description of the SF-insulator transition 

• Vortex hopping: (result of charging effects) 

− 

tV 

cos( ∇ ˆ θ ) 

• Vortex-vortex interactions: 

1 

ρ ∑ s 

nV i 

⋅ nV j 

⋅ln | xi 

− x 

 

2 

i, 

j 

[ nˆ 

V 

, θ] 

= −i 

Condensed vortices 

= insulating CPs 

j 

i 

j 

| 

φˆ 

θˆ 

Cooper-pairs: 

Vortices: 


Vortex: 

(Fisher, 1990) 

superfluid 

ρ s 

/U 

V-superfluid 

V- insulator 

ρs / t V

Universal () resistance at SF-insulator transition 

Assume that vortices and Cooper-pairs 

are self dual at transition point. 

• Current due to CP hopping: 

• EMF due to vortex hopping: 

 

Δφ 

2π 

= ΔV 

ΔV 

= 

2e 

2e τ 

2e 

I = 

τ 

2e 

I 

• Resistance: 

V 2π 

2e 

h 

R = = = = 6. 5kΩ 

2 

I 2eτ 

τ 4e 

In reality superconducting films are not self dual: 

• vortices interact logarithmically, Cooper-pairs interact at most with power law. 

• Samples are very disordered and the disorder is different for cooper-pairs 

and vortices.

Magnetically tuned Superconductor-insulator transition 

• Net vortex density: 

h 

2e 

n 

ˆV 

= 

B 

• Disorder pins vortices for small field – superconducting phase. 

• Large fields some free vortices appear and condense – insulating phase. 

• Larger fields superconductivity is destroyed – normal (unpaired) phase. 

R 

Disorder 

pins vortices: 

Free vortices: 

Phase coherent 

SC 

Vortex SF 


Disorder localized 

Electrons: 

Normal (unpaired) 

B

Magnetically tuned Superconductor-insulator transition 

• Net vortex density: 

h 

2e 

n 

ˆV 

= 

B 

Problems 

• Saturation of the resistance – metallic phase 

• Non-universal insulating peak – completely different depending on disorder. 

R 

Metallic 

phase 

Disorder localized 

Electrons: 

Normal (unpaired) 

B

Two-fluid model for the SC-Metal-Insulator transition 

(Galitski, Refael, Fisher, Senthil, 2005) 

J CP 

+ J e 

Uncondensed vortices: 

Cooper-pair channel 

Finite conductivity: 

 

j V 

= σV 

FV 

 

FV 

= zˆ× 

J 

 

z ˆ× 

E = 

j V 

CP 

1 

! 

V 

! 

E = 

! 

J 

CP 

Two channels in parallel: 

Disorder induced Gapless QPs 

(electron channel) 

! 

J 

(delocalized core states) 

! 

J E 

! 

=! 

e 

e 

= (" 

+ " 

e 

!1 

V 

! 

) E

Transport properties of the vortex-metal 

Effective conductivity: 

σ 

eff 

−1 

= σ 

e 

+ σV 

• Assume: 

- grows from zero to . - grows from zero to infinity. 

σ 

e 

σ 

N 

σ V 

σ e 

σ V 

σ N 

B e 

B 

R eff 

Vortex 

metal 

B V 

B 

Weak insulators: 

−1 

σ e 

Ta, MoGe 

Weak InO 

B 

e 



σ V 

B e 

B V 

Normal 

(unpaired) 

B

Transport properties of the vortex-metal 

Effective conductivity: 

σ 

eff 

−1 

= σ 

e 

+ σV 

Chargless spinons contribute to conductivity! 

• Assume: 

- grows from zero to . - grows from zero to infinity. 

σ 

e 

σ 

N 

σ V 

σ e 

σ V 

σ N 

B e 

B 

R eff 


metal 

B V 

B 

Strong insulators: 

TiN, InO 

B 

e 

> B V 

σ V 

B V 

Insulator 

B e 

−1 

σ e 

Normal 

(unpaired) 

B

More physical properties of the vortex metal 

Cooper pair tunneling 

• A superconducting STM can tunnel Cooper 

pairs to the film: 

G = G 2e 

+ G CP 

(Naaman, Tyzer, Dynes, 2001).



G 

CP 



Vortex metal phase: 

1 

2 

~ exp( −σ 

ln T ) 

2 V 

T 

CP 

G = G 2e 

+ G CP 

Normal phase: 

G 

2 

2 e 

~ σ e 

CP 


metal 

R eff 

e 

e 

G ≈ G CP 

strongly T 

dependent 

σ V 

B V 

Insulator 

B e 

−1 

σ e 

Normal 

(unpaired) 

B 

G ≈ G 

~ 1/ ln 

2e 

2 

T

Percolation Paradigm 

(Trivedi, 

Dubi, Meir, Avishai, 

Spivak, Kivelson, 

et al.)

Pardigm II: superconducting vs. Normal regions percolation 

• Strong disorder breaks the film into superconducting and normal regions. 

SC 

NOR 

NOR 

R 

B



NOR 

SC 

SC 

SC 

SC 

SC 

R 

• Near percolation – thin channels of the disorder-localized normal phase. 

B



NOR 

R 

• Near percolation – thin channels of the disorder-localized normal phase. 

• Far from percolation – disordered localized normal electrons. 

B

Magneto-resistance curves in the percolation picture 

• Simulate film as a resistor network: 

(Dubi, Meir, Avishai, 2006) 

NOR island 

Nor-SC links: 

SC island 

R 

T 

R e E G / 

~ Normal links: 

SC-SC links: 

0 

R 

~ 

R e 

0 

(| ε | + | ε | + | ε −ε 

|)/ kT 

1 

2 

1 

2 

α 

R ~ T 

Reuslting MR:

Drag in a bilayer system

Giaever transformer – Vortex drag 

Two type-II bulk superconductors: 

(Giaever, 1965) 

V 2 

i D 

I 1 

B 

B 

V 1 

Vortices tightly bound: 

V 

2 

= V 1 

R D 

= 

V 

I 

2 

1

Two thin electron gases: 

2DEG bilayers – Coulomb drag 

V2 

R D 

= 

I 

V 2 

1 

i D 

I 1 

V 1 

• Coulomb force creates friction between the layers. 

• Inversely proportional to density squared: 

• Opposite sign to Giaevers vortex drag. 

R ∝ 

D 

n 

1 

2 

e



V2 

R D 

= 

I 

V 2 

1 

i D 

I 1 

Example: ν = 1 

T 

V 1 

Excitonic condensate 

(Kellogg, Eisenstein, Pfeiffer, West, 2002)

Thin film Giaever transformer 

V D 

d Ins 

~ 5nm 

Amorphous (SC) thin films 

Percolation paradigm 

d SC 

~ 20nm 

I 1 

Insulating layer, 

Josephson tunneling: 

J = 0 or J > 0 

Vortex condensation paradigm 

• Drag is due to coulomb interaction. 

• Electron density: 

n2d 

~ dSC 

⋅10 

( QH bilayers: 

cm 

→ 2⋅10 

20 −3 

14 −2 

10 −2 

n d 

~ 5⋅ 

cm ) 

2 

10 

Drag suppressed 

cm 

 

• Drag is due to inductive current 

interactions, and Josephson coupling. 

• Vortex density: 

n 

V 

~ 

B 

φ 

0 

~ (10 

11 

⋅ B 

[ T ] 

Significant Drag 

) cm 

−2

Vortex drag in thin films bilayers: interlayer interaction 

• Vortex current suppressed. 

e.g., Pearl penetration length: 

λ = 

eff 

2λ 

d 

2 

L 

SC 

r 

• Vortex attraction=interlayer induction. 

Also suppressed due to thinness. 

U 

inter 

2 

−qa 

( φ0 

e 

q) 

≈ 

2 

−1 

2 −2qa 

2πλ 

q[( 

q + λ ) − e / λ 

2 

eff 

2 

∝ d SC 

eff 

eff 

] 

~ 

2 

φ0 

4πλ 

~ 

eff 

φ 

4λ 

2 

0 

2 

eff 

ln( r), 

r 

a 

2 

, 

r 

r 

> λ 

< a 

eff

Vortex drag in thin films within vortex metal theory 

• R D 

= 

V 

I 

2 

1 

I 1 

V 2 

• Perturbatively: 

R 

D = 

G 

drag 

V 

~ [ j , j 1 2] 

A 1 

U e 

j1 

U e 

2 

j 

A 2 

(Kamenev, Oreg) 

• Expect: 

G 


V 

∂ σσ 

∝ 

σ∂σ 

⋅ 

∝ 

V1V 

1 ⋅ 

V 2 V 2 

n∂ 

Vn 

1V 

1 nV∂ 

2n 

V 2 

∂R1 

∂B 

∂R 

⋅ 

∂B 

∝ 

2 

Drag generically 

proportional to MR slope. 

(Following von Oppen, Simon, Stern, PRL 2001) 

• Answer: 

R 

D 

= 

G 


V 

= 

2 

2 

e $ 

0 

4 

8# 

T 

& R1 

& B 

& R 

& B 

2 

% 

d! 

% 

dq q 

3 

| U 

| 

2 

Im" 

1 

Im" 

2 

2 

sinh (! 

/ 2T 

) 

U – screened inter-layer potential. χ - Density response function (diffusive FL)

Vortex drag in thin films: Results 

R 

D 

= 

G 


V 

= 

2 

2 

e $ 

0 

4 

8# 

T 

& R1 

& B 

& R 

& B 

2 

% 

d! 

% 

dq q 

3 

| U 

| 

2 

Im" 

1 

Im" 

2 

2 

sinh (! 

/ 2T 

) 

• Our best chance (with no J tunneling) is the highly insulating InO: 

• maximum drag: 

max 

R D 

~ 0. 1m! 

Note: similar analysis for SC-metal bilayer using a ground plane. 

Experiment: Mason, Kapitulnik (2001) Theory: Michaeli, Finkelstein, (2006)

Percolation picture: Coulomb drag 

• Solve a 2-layer resistor network with drag. 

- Normal 

- SC 

• Can neglect drag with the SC islands: 

D 

D 

R SC 

, R ≈ 0 

−SC 

SC−NOR 

• Normal-Normal drag – use results for disorder localized electron glass: 

R 

D 

NOR −NOR 

= 

1 

2 

96π 

R1R 

/ e 

2 

2 

( e 

2 

n 

e 

2 

T 

a d 

film 

) 

2 

1 

ln 

2x 

0 

(Shimshoni, PRB 1994)

Percolation picture: Results 

• Solution of the random resistor network: 

Compare to vortex drag:

Conclusions 

• Vortex picture and the puddle picture: similar single layer predictions. 

• Giaever transformer bilayer geometry may qualitatively distinguish: 

Large drag for vortices, small drag for electrons, with opposite signs. 

• Drag in the limit of zero interlayer tunneling: 

R vortex 

D 

1 

percolation 

R D 

−11 

~ 0. mΩ vs. ~ 10 Ω 

• Intelayer Josephson should increase both values, and enhance the effect. 

(future theoretical work) 

• Amorphous thin-film bilayers will yield interesting complementary 

information about the SIT.

Conclusions 

• What induces the gigantic resistance and the SC-insulating transition 

• What is the nature of the insulating state Exotic vortex physics 

Phenomenology: 

• Vortex picture and the puddle picture: similar single layer predictions. 

Experimental suggestion: 

• Giaever transformer bilayer geometry may qualitatively distinguish: 

Large drag for vortices, small drag for electrons.

Observation of a metallic phase 

• MoGe: 

(Mason, Kapitulnik, PRB 1999)

Observation of a metallic phase 

• Ta: 

(Qin, Vicente, Yoon, 2006) 

• Saturation at ~100mK: New metalic phase (or saturation of electrons temperature)

Universal () resistance at SF-insulator transition 

Assume that vortices and Cooper-pairs 

are self dual at transition point. 

• Current due to CP hopping: 

• EMF due to vortex hopping: 

 

Δφ 

2π 

= ΔV 

ΔV 

= 

2e 

2e τ 

2e 

I = 

τ 

2e 

I 

• Resistance: 

V 2π 

2e 

h 

R = = = = 6. 5kΩ 

2 

I 2eτ 

τ 4e 

In reality superconducting films are not self dual: 

• vortices interact logarithmically, Cooper-pairs interact at most with power law. 

• Samples are very disordered and the disorder is different for cooper-pairs 

and vortices.





G = G 2e 

+ G CP 

(Naaman, Tyzer, Dynes, 2001).



G 

CP 



Vortex metal phase: 

1 

2 

~ exp( −σ 

ln T ) 

2 V 

T 

CP 

G = G 2e 

+ G CP 

Normal phase: 

G 

2 

2 e 

~ σ e 

CP 


metal 

R eff 

e 

e 

G ≈ G CP 

strongly T 

dependent 

σ V 

B V 

Insulator 

B e 

−1 

σ e 

Normal 

(unpaired) 

B 

G ≈ G 

~ 1/ ln 

2e 

2 

T



V2 

R D 

= 

I 

V 2 

1 

i D 

I 1 

Example: ν = 1 

T 

V 1 

Excitonic condensate 

(Kellogg, Eisenstein, Pfeiffer, West, 2002)

Vortex drag in a Thin-film Giaever transformer

Create successful ePaper yourself

Delete template?

Save as template?