Electron Transport over Superconductor Hopping Insulator Interface

1
Electron Transport over Superconductor -Hopping
Insulator Interface

• A surprising and delicate interference-like
  cancellation phenomenon

Martin Kirkengen, Joakim Bergli, Yuri Galperin
2
Structure of presentation

• Model presentation/the physics
• Results what was expected, and what was not
  expected at all...
• Origin of unexpected cancellations
• Robustness of cancellations, three different
  attempts to avoid them
• Relevance of problem and results

3
The Model
SC
TB
HI

• SC Superconductor
• TB Tunneling Barrier
• HI Hopping Insulator

Typical situation studying a hopping insulator
using superconducting contacts
4
Superconductor

• Cooper pairs electrons dancing the Viennese
  Waltz
• Energy gap D prevents single electron transport
  if D gt kBT and D gteV
• Coherence length, x
• Fermi wave number, kF
• Anomalous Greens Function

5
Tunneling Barrier

• E.g. Shottky Barrier, due to band bending
• Simplest case- electrons enter and exit at same
  position- constant thicknessheight
• Various variations will be considered

SC
TB
HI
6
Hopping Insulator

• Localized electron states centered on impurities
  (surface states are ignored)
• Electrons may hop between impurities
• Hydrogen-like wavefunctions, but with radius
  agtgtaH
• IMPORTANT QUANTITY kFa 100
• Resistance in insulator lower than in barrier
• Greens Function

7
Theoretical approach(for the specially
interested)

• Kubo Linear Response TheoryCH,I/E
• Hamiltonian H I A
• Greens function formalism
• Matsubara technique
• Loads of contractions, complex integrations,
  Fourier transforms, analytical continuations
• Following Kozub, Zyuzin, Galperin, VinokurPhys.
  Rev. Letters 96, 107004 (2006)

8
The Problem
SC
TB
HI

• What is the conductivity of such a barrier, if
  this is the dominant channel?

9
Expected Behaviour

• Transport function of distance (z) of impurities
  from barrier, e-z/a
• Sufficient active impurities will allow us to
  ignore surface states contribution to transport
• Maximum distance between contributing impurities
  limited by coherence length
• Some fluctuation due to sin(kFr) from
  superconductor Greens function

10
Found Behaviour

• Maximum distance between contributing impurities
  limited by coherence length
• Some fluctuation due to sin(kFr) from
  superconductor propagator
• BUTTransport determined by distance (z) of
  impurities from barrier as e-kFz , not e-z/a!
• Only states VERY NEAR surface can contribute.

11
Where the Error Occured...

• Two sin(kFr) from the SC Greens function
• Replaced by average of sin2(kFr) when integrated
  over space.
• Integration extremely sensitive to phase

12
The Essential Integral

TB
SC
HI
HI
za, kFa100

• Positive area
• Negative area

152.6689693731328496919146125035145839725143192401
392
-152.668969373132849691914612503514583972514319202
7575
13
How to kill cancellations...

• Effect of finite width of barrier
• Different impurity wave function
• Strong barrier fluctuations
• Weak barrier fluctuations

14
Perfect Barrier Directional Sensitivity

• Allow entry/exit coordinates to differ Reduced
  transverse component of momentum
• Integration over TB/HI-interface introduces
  polynomial correction to impurity wave function
  seen from SC/TB-interface
• Essential behaviour remains e-kFz

SC
TB
HI
15
Importance of Impurity Shape

• Square potential hydrogen-like wave function
  Strong cancellations, e-kFz
• Parabolic potential gaussian wave function No
  cancellations, back to e-z/a

16
Deep Barrier Minimum

• Gaussian behaviour near barrier minimum
• Barrier variation rather than impurity variation
  determines transport
• Back to e-z/a

Localisation length under barrier
TB
SC
HI
a
17
Shallow Barrier Minimum

• rlta, positive accumulation
• Rgta, negative accumulation
• Assume barrier T dq(r-a)
• One part proportional to Te-kFz
• Other part proportional to d e-z/a

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Conclusions Barriers and Conduction
Gaussian
Hydrogen-like
Perfect barrier
NORMAL
VERY LOW (e-ka)
Deep minimum (of width w)
LOW (w/a)
LOW (w/a)
Shallow minimumof length a
NORMAL
LOW (d/T)
19
Macroscopic Consequences

• Impurity pairs where barrier defects allow
  transport will dominate
• Number of active impurities ltlt total number of
  impurities
• Surface states can maybe be ignored after all...

20
Possible Relevance The Quantum Entangler
QD
SC
TB
I
QD

• Idea a Cooper pair is split, with one electron
  going to each electrode, their spins being
  entangled.
• Choice of fabrication metod for quantum dots may
  be essential for success.