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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 (Caltech)
T. Senthil (MIT)
$$$: Research corporation, Packard Foundation, Sloan Foundation
Outline
• Experimental Motivation – 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.
V
I
InO
Ins
B
B
(Hebard, Paalanen, PRL 1990)
SC
Observation of Superconductor-insulator transition
• Thin amorphous films: B tunes a SC-Insulator transition.
InO:
(Sambandamurthy,
Engel,
(Steiner,
Kapitulnik)
Johansson, Shahar, PRL 2004)
• Saturation as T
0
• Insulating peak different from sample to sample, scaling different – log, activated.
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)
Vortex Paradigm
Correlated states in higher dimensions:
Superconductivity
• Pairs of electrons form Cooper pairs:

 
k  k 
 c c
(c’s are creation operators
for electrons)
• Cooper pairs are bosons, and can Bose-Einstein condense:
   GS   GS ~  exp[ i ]
H. K. Onnes,
Commun. Phys. Lab.12,120, (1911)
Meissner effect
Gap in tunneling density of states
http://dept.phy.bme.hu/research/sollab_nano.html
www.egglescliffe.org.uk
Free energy of a superconductor –
Landau-Ginzburg theory
• Most interesting, phase dynamics
 ~  exp[ i ] .
• Supercurrent:
• Vortices:
J ~ 
 I  I J sin(  ) 
(Josephson I)
• Vortex motion leads to a voltage drop:
V
V
 

2e
(Josephson II)
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”:


2
H       s cos( ji )  Unˆi 
i 
j

 S exp[ i(i  i 1 )]
Hopping:
Charging
energy:
Unˆ 2j
[nˆ ,  ]  i
• Large U - Mott insulator
(no charge fluctuations)
• Large
s -
Superfluid
(intense charge fluctuations,
no phase fluctuations)
insulator
superfluid
s /U
Vortex description of the SF-insulator transition
(Fisher, 1990)
• Vortex hopping: (result of charging effects)
ˆ
Vortex:
 tV cos( jˆi )
[nˆV , ]  i
ˆ
• Vortex-vortex interactions:
 
1
 s  nV i  nV j  ln | xi  x j |
2 i, j
Cooper-pairs:
insulator
s /U
Condensed vortices
= insulating CP’s
superfluid
Vortices:
V-superfluid
V- insulator
 s / tV
Universal (?) resistance at SF-insulator transition
Assume that vortices and Cooper-pairs
I
are self dual at transition point.
• Current due to CP hopping:
I
2e

• EMF due to vortex hopping:
 
  V
2e
• Resistance:
2e
 2
V 
2e 
V 2
R 
I 2 e
2e


h
 6.5k
2
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
nˆV  B
2e
• 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
insulator
Disorder localized
Electrons:
Normal (unpaired)
B
Magnetically tuned Superconductor-insulator transition
• Net vortex density:
h
nˆV  B
2e
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
Disorder induced Gapless QP’s
(electron channel)
(delocalized core states?)
Uncondensed vortices:
Cooper-pair channel
Finite conductivity:





jV  V FV   V zˆ  J CP




1
E  zˆ  jV
J CP 
E
V
Two channels in parallel:


Je   eE


1
J  ( e   V ) E
Transport properties of the vortex-metal
 eff   e   V1
Effective conductivity:
• Assume:
-  e grows from zero to
N
.
-
V
grows from zero to infinity.
V
e
N
Be
B
Vortex
metal
BV
Reff
 e 1
Weak insulators:
Ta, MoGe
Weak InO
Be  BV
V
Normal
(unpaired)
Be
BV
B
B
Transport properties of the vortex-metal
Effective conductivity:
 eff   e   V1
Chargless spinons contribute to conductivity!
• Assume:
-  e grows from zero to
 N.
-
V
grows from zero to infinity.
V
e
N
Be
B
Vortex
metal
BV
Reff
TiN, InO
Be  BV
V
Insulator
Strong insulators:
 e 1
Normal
(unpaired)
BV
Be
B
B
More physical properties of the vortex metal
Cooper pair tunneling
• A superconducting STM can tunnel Cooper
pairs to the film:
G  G2 e  GCP
(Naaman, Tyzer, Dynes, 2001).
More physical properties of the vortex metal
Cooper pair tunneling
• A superconducting STM can tunnel Cooper
pairs to the film:
G  G2e  GCP
Vortex metal phase:
GCP ~
Normal phase:
1
2
exp(


ln
T)
V
2
T
G2e ~  e
2
CP
CP
Vortex
metal
Reff
V
 e 1
Insulator
G  GCP
strongly T
dependent
e
G  G2 e
Normal
(unpaired)
BV
Be
B
~ 1 / ln 2 T
e
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
Pardigm II: superconducting vs. Normal regions percolation
• Strong disorder breaks the film into superconducting and normal regions.
NOR
SC
SC
SC
SC
SC
R
B
• Near percolation – thin channels of the disorder-localized normal phase.
Pardigm II: superconducting vs. Normal regions percolation
• Strong disorder breaks the film into superconducting and normal regions.
NOR
R
B
• Near percolation – thin channels of the disorder-localized normal phase.
• Far from percolation – normal electrons with disorder.
Magneto-resistance curves in the percolation picture
(Dubi, Meir, Avishai, 2006)
• Simulate film as a resistor network:
NOR island
SC island
Nor-SC links:
R ~ R0 e EG / T
Normal links:
R ~ R0e
Reuslting MR:
(|1 || 2 ||1  2 |) / kT
SC-SC links:
R ~ T
Drag in a bilayer system
Giaever transformer – Vortex drag
(Giaever, 1965)
Two type-II bulk superconductors:
V2
iD
I1
B
B
V1
Vortices tightly bound:
V2  V1
RD 
V2
I1
2DEG bilayers – Coulomb drag
V2
RD 
I1
V2
Two thin electron gases:
iD
I1
V1
• Coulomb force creates friction between the layers.
• Inversely proportional to density squared:
• Opposite sign to Giaever’s vortex drag.
1
RD  2
ne
2DEG bilayers – Coulomb drag
V2
RD 
I1
V2
Two thin electron gases:
iD
I1
V1
Example:
T  1
“Excitonic condensate”
(Kellogg, Eisenstein, Pfeiffer, West, 2002)
Thin film Giaever transformer
VD
I1
d Ins ~ 5nm
d SC ~ 20nm
Insulating layer,
Josephson tunneling:
J  0 or J  0
Amorphous (SC) thin films
Percolation paradigm
Vortex condensation paradigm
• Drag is due to inductive current
interactions, and Josephson coupling.
• Drag is due to coulomb interaction.
• Electron density:
3
n2 d ~ d SC 10 cm  2 10 cm
20
14
( QH bilayers: n2 d ~ 5 10 cm
10
Drag suppressed
2
2
)
?
• Vortex density:
nV ~
B
0
~ (1011  B[T ] )cm  2
Significant Drag

$
?
Vortex drag in thin films bilayers: interlayer interaction
• Vortex current suppressed.
e.g., Pearl penetration length:
eff
2 L

d SC
2
r
• Vortex attraction=interlayer induction.
Also suppressed due to thinness.
0 2
e  qa
U inter (q) 
2
1
2eff q[( q  eff ) 2  e 2 qa / 2eff ]
 d SC
2
0 2
~
ln( r ), r  eff
4eff
0 2 r 2
~
, ra
2
4eff a
Vortex drag in thin films within vortex metal theory
V2
V
• RD  2
I1
I1
• Perturbatively:
RD  GV
drag
• Expect:
~ [ j1 , j2 ]
GV
drag

Ue
A1
j1
V 1V 1 V 

2 V2
Ue


nVn1V 1 nV2nV 2
j2
R1 R2

B B
A2
(Kamenev, Oreg)
Drag generically
proportional to MR slope.
(Following von Oppen, Simon, Stern, PRL 2001)
• Answer:
e 20 R1 R2
3
2 Im 1 Im  2
 4
d

dq
q
|
U
|
8 T B B  
sinh 2 ( / 2T )
2
RD  GV
drag
U – screened inter-layer potential.

- Density response function (diffusive FL)
Vortex drag in thin films: Results
e 20 R1 R2
3
2 Im 1 Im  2
 4
d  dq q | U |

8 T B B
sinh 2 ( / 2T )
2
RD  GV
drag
• Our best chance (with no J tunneling) is the highly insulating InO:
• maximum drag:
RD
max
~ 0.1m
Note: similar analysis for SC-metal ‘bilayer’ using a ground plane.
Experiment: Mason, Kapitulnik (2001)
Theory: Michaeli, Finkel’stein, (2006)
Percolation picture: Coulomb drag
• Solve a 2-layer resistor network with drag.
- Normal
- SC
• Can neglect drag with the SC islands:
D
D
RSC
,
R
 SC
SC  NOR  0
• Normal-Normal drag – use results for disorder localized electron glass:
R
D
NOR  NOR
1 R1 R2
T2
1

ln
2
2
2
2
96  / e (e ne a d film )
2 x0
(Shimshoni, PRL 1987)
Percolation picture: Results
• Drag resistances:
D
SC  SC
R
D
SC  NOR
, R
0
D
RNOR
 NOR
1 R1 R2
T2
1

ln
96 2  / e 2 (e 2 ne a d film ) 2 2 x0
• 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:
RD
vortex
~ 0.1m
vs.
RD
percolation
~ 10 11 
• 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.