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: Superconductor

• TB: Tunneling Barrier

• HI: Hopping Insulator

SC TB HI

Typical situation: studying a hopping insulator using superconducting contacts

Superconductor

• Cooper pairs – electrons dancing the Viennese Waltz

• Energy gap prevents single electron transport if > kBT and >eV

• Coherence length, • Fermi wave number, kF

• Anomalous Greens Function: 2 2

2 2

sin( )R

F

F

k Re

k R

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:

1 2| |/ | |/

1 2 3

1( , , )

s sr r a r r a

s s

e eG r r

a i

Theoretical approach(for the specially interested)

• Kubo Linear Response Theory=[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, Vinokur

Phys. Rev. Letters 96, 107004 (2006)

The Problem

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

SC TB HI

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:

TBSC HI

152.6689693731328496919146125035145839725143192401392 -152.6689693731328496919146125035145839725143192027575

1

2 2

(| | | |) /2 21 1

/00

sin | |

sin ( )

l sr r r r aF

r z aF sl F

d rd r k r r e

C k rdrJ k r e

=

z=a, kFa=100 HI

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

2 2 2 2/ ( ) /0 00 0( ) ( )r z a r zF FrdrJ k r e rdrJ k r e

Deep Barrier Minimum

• Gaussian behaviour near barrier minimum

• Barrier variation rather than impurity variation determines transport

• Back to e-z/a

SC HITB

Localisation length under barrier

a

Shallow Barrier Minimum

• r<a, positive accumulation• R>a, negative accumulation• Assume barrier T+ (r-a)• One part proportional to Te-kFz

• Other part proportional to e-z/a

SC HITB

a

Conclusions Barriers and Conduction

GaussianHydrogen-like

Perfect barrier

Deep minimum (of width ’w’)

Shallow minimumof length ’a’

NORMAL

LOW (w/a)

NORMAL

VERY LOW (e-ka)

LOW (w/a)

LOW (/T)

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.

SC TB I

QD

QD