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Electron Transport over Superconductor - Hopping Insulator Interface

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

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Electron Transport over Superconductor - Hopping Insulator Interface

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  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 > kBT and D >eV • 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 thickness&height • 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 a>>aH • IMPORTANT QUANTITY: kFa ~ 100 • Resistance in insulator lower than in barrier • Greens Function:

  7. 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)

  8. The Problem • What is the conductivity of such a barrier, if this is the dominant channel? SC TB HI

  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 • BUT:Transport 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 • Positive area: • Negative area: = TB HI SC HI z=a, kFa=100 152.6689693731328496919146125035145839725143192401392 -152.6689693731328496919146125035145839725143192027575

  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. SC 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

  18. 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

  19. 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...

  20. 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

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