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Electron Transport over Superconductor - Hopping Insulator InterfacePowerPoint Presentation

Electron Transport over Superconductor - Hopping Insulator Interface

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Electron Transport over Superconductor - Hopping Insulator Interface

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Electron Transport over Superconductor - Hopping Insulator Interface

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Electron Transport over Superconductor -Hopping Insulator Interface

A surprising and delicate interference-like cancellation phenomenon

Martin Kirkengen, Joakim Bergli, Yuri Galperin

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

The Model

SC

TB

HI

- SC: Superconductor
- TB: Tunneling Barrier
- HI: Hopping Insulator

Typical situation: studying a hopping insulator using superconducting contacts

Superconductor

- Cooper pairs – electrons dancing the Viennese Waltz
- Energy gap D prevents single electron transport if D > kBT and D >eV
- Coherence length, x
- Fermi wave number, kF
- Anomalous Greens Function:

Tunneling Barrier

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

SC

TB

HI

Hopping Insulator

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

Theoretical approach(for the specially interested)

- Kubo Linear Response TheoryC=[H,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)

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

Found Behaviour

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

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

The Essential Integral

- Positive area:
- Negative area:

=

TB

HI

SC

HI

z=a, kFa=100

152.6689693731328496919146125035145839725143192401392

-152.6689693731328496919146125035145839725143192027575

How to kill cancellations...

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

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

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

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

TB

HI

d

a

Shallow Barrier Minimum- r<a, positive accumulation
- R>a, negative accumulation
- Assume barrier T+ dq(r-a)
- One part proportional to Te-kFz
- Other part proportional tode-z/a

Conclusions Barriers and Conduction

Hydrogen-like

Gaussian

Perfect barrier

VERY LOW (e-ka)

NORMAL

Deep minimum (of width ’w’)

LOW (w/a)

LOW (w/a)

Shallow minimumof length ’a’

LOW (d/T)

NORMAL

Macroscopic Consequences

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

Possible Relevance – The Quantum Entangler

- 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.

QD

SC

TB

I

QD