Coherent Transport Through Andreev Interferometers

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Coherent Transport Through Andreev
Interferometers

• Philippe Jacquod
• U of Arizona

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The two-slit experiment (textbook version)
1
2
?????????????
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The two-slit experiment (non-textbook version)
?
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The two-slit experiment (non-textbook version)
Regular cavity Chaotic cavity
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• IS THIS DECOHERENCE ?
• DO CHAOTIC / COMPLEX SYSTEMS DECOHERE
• WHEREAS REGULAR / INTEGRABLE
• SYSTEMS DO NOT ?

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• IS THIS DECOHERENCE ?
• DO CHAOTIC / COMPLEX SYSTEMS DECOHERE
• WHEREAS REGULAR / INTEGRABLE
• SYSTEMS DO NOT ?

NO!
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What is coherence ?
Coherent systems those which keep memory of the
phase
?I.e. Schroedinger equation gives a good
description even when V is very complicated -
random - but not time-dependent ! ? Phase
of the wavefunction evolves deterministically
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But what about multiple scattering ?
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The two-slit experiment (non-textbook version)
Despite multiple chaotic scattering the Gaussian
envelope still exhibits (small) modulations !
Regular cavity Chaotic cavity
A.k.a. weak localization
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Magnetoresistance Aharonov-Bohm oscillations
Measurement sample diam 1mm, width 0.04mm
Amplitude of oscillations decreases with
increasing temperature decoherence Maximal
amplitude e2/h (for conductance)
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Magnetoresistance Aharonov-Bohm oscillations
Measurement sample
Amplitude of oscillations decreases with
temperature decoherence
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From the mesoscopic AB effect to Andreev
interferometers Giant enhancement of
oscillations amplitude!
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house
thermal
charge
S
parallelogram
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ltGgt 1600 dG 70
ltGgt 7700 dG 300
(in units of e2/h)
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From the mesoscopic AB effect to Andreev
interferometers Giant enhancement of
oscillations! Symmetries of multiterminal
transport !? Symmetry vs. antisymmetry of
thermopower !?
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OUR MOTIVATION
These effects cannot be all explained by the
scattering approach to transport.

• V. Chandrasekhar
• Tucson, Feb 24, 2006

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Outline

• Mesoscopic superconductivity - Andreev reflection
• Density of states in ballistic Andreev billiards
• Transport through ballistic Andreev
  interferometers
• Transport through diffusive Andreev
  interferometers

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Outline

• Mesoscopic superconductivity - Andreev reflection
• Density of states in ballistic Andreev billiards
• Transport through ballistic Andreev
  interferometers
• Transport through diffusive Andreev
  interferometers

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Mesoscopic Superconductivity
Mesoscopic metal (N) in contact with
superconductors (S)
ltlt L
S invades N Mesoscopic proximity effect

S
N
S
Device by AT Filip, Groningen
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Mesoscopic Superconductivity
Mesoscopic metal (N) in contact with
superconductors (S)
ltlt L
S invades N

But how ??
S
N
S
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Mesoscopic Superconductivity

• Effect of S in N depends on
• Electronic dynamics in N
• Symmetry of S state
• (s- or d-wave S phases)
• (iii) ?E/?D

ltlt L

S
N
S
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What is Andreev reflection

• low-energy electron quasiparticle approaches
  superconductor from normal region

Charge-reversing retro-reflection
Supercond. pair potential

• Incoming electron

• retro-reflected hole

• (and vice versa)

Andreev, 64 sidenote Andreev reflection
Hawking radiation
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Andreev reflection

• (e,EF?) (h, EF-?)
• Reflection phase
• Angle mismatch Snells law

(fig taken from Wikipedia)
S phase h-gte - e-gth
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Outline

• Mesoscopic superconductivity - Andreev reflection
• Density of states in ballistic Andreev billiards
• Transport through ballistic Andreev
  interferometers
• Transport through diffusive Andreev
  interferometers

PJ, H. Schomerus, and C. Beenakker, PRL 03 M.
Goorden, PJ, and C. Beenakker, PRB 03 PRB 05
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Andreev billiards classical dynamics
At NI interface Normal reflection

superconductor
Note 1 Billiard is chaotic ? all trajectories
become periodic!

e
superconductor
At NS interface Andreev reflection
Kosztin, Maslov, Goldbart 95
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Andreev billiards classical dynamics
At NI interface Normal reflection

superconductor
Note 1 Billiard is chaotic ? all trajectories
become periodic!

h
superconductor
At NS interface Andreev reflection
Kosztin, Maslov, Goldbart 95
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Andreev billiards classical dynamics
At NI interface Normal reflection

superconductor
Note 2 Action on P.O.
Andreev reflection phase
At NS interface Andreev reflection
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Andreev billiards semiclassical quantization
All orbits are periodic -gt Bohr-Sommerfeld

S
N
Andreev reflection phase
x
Distribution of return times to S chaos-gt exp.
Suppression at E0 regular-gtalgebraic / others
See also Melsen et al. 96Ihra et al. 01
Zaitsev 06
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Andreev billiards semiclassical quantization
All orbits are periodic -gt Bohr-Sommerfeld

S
N
?0
x
?
????? DoS has peak at E0 !! All trajs
touching both contribute to n0 term
Goorden, PJ, Weiss 08
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Andreev billiards random matrix theory
N MxM RMT Hamiltonians S -gt particle-converting
projectors
CONSTANT DOS EXCEPT ? hard gap at 0.6 ET for
?0 ??linear gap of size ? for ?
? ?????(class CI with DoS
)
Melsen et al. 96, 97 AltlandZirnbauer 97
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Andreev billiards RMT vs. B-Sommerfeld
At ??? the gap problem ? which theory is
right ? ? which theory is wrong ?
At ??? macroscopic peak (semiclassics) vs.
minigap (RMT) ? which theory is right ? ? which
theory is wrong ?
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What is the Ehrenfest time?

• Classical chaos local exponential divergence
• Quantum chaos

Deep semiclassical limit ? ?????t n?????????
E
Why the name ? For larger times, breakdown of
Ehrenfests 1927 thm that a wavepacket follows
Newtons laws on average
Larkin and Ovchinnikov 69 Bermann and Zaslavsky
78
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Andreev billiards - Solution to the gap problem
Universal, RMT regime
Deep semiclassical regime
Note numerics on Andreev kicked rotator, PJ
Schomerus and Beenakker 03 See also Lodder and
Nazarov 98 Adagideli and Beenakker 02
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Andreev billiards - Solution to the gap problem
Deep semiclassical regime Gap at Ehrenfest energy
Universal, RMT regime Gap at Thouless energy
Note numerics on Andreev kicked rotator, PJ
Schomerus and Beenakker 03 See also Lodder and
Nazarov 98 Adagideli and Beenakker 02 Vavilov
and Larkin 03
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Andreev billiards DoS at ???
Deep semiclassical regime Large peak around E0 !
Universal, RMT regime Minigap at level spacing
Goorden, PJ and Weiss 08.
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Outline

• Mesoscopic superconductivity - Andreev reflection
• Density of states in ballistic Andreev billiards
• Transport through ballistic Andreev
  interferometers
• Symmetries of charge transport in presence
• of superconductivity

M. Goorden, PJ, and J. Weiss, PRL 08,
Nanotechnology 08
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Scattering approach to coherent transport

• A.k.a. Landauer-Buttiker

Conductance as transmission (in units of e2/h)
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Random matrix theory of transport

• Going from diffusive to ballistic systems
• Scattering approach (Landauer-Büttiker)

Chaotic cavity ? S as a Random Matrix
symmetry index 1 with TRS
2 without TRS 4 without SRS, with
TRS Note Scattering approach equivalent to
Kubo (Fisher Lee 81)

• Distr. of Ts
• Conductance
• UCF

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Ray optics for the 21st century
Scattering approach
Entrance / exit points
Classical trajectories, stability and action
R. Whitney and PJ, PRL/PRB 05/06
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Transport through Andreev interferometers
Lambert 93 formula Average conductance for
NLNR

• New, Andreev reflection term
• Gives classically large
• interference contributions

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Transport through Andreev interferometers
At ?0, any pair of Andreev reflected
trajectories contributes to in the sense of a
SPA ! These pairs give classically
large positive coherent backscattering at ?0,
vanishing for ???
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Transport through Andreev interferometers
No tunnel barrier Coherent backscattering
is -O(N) -positive, increases G This is
(obviously) not related to the DoS in the
Andreev billiard
!! INTRODUCE TUNNEL BARRIERS TUNNELING
CONDUCTANCE DOS !!
Beenakker, Melsen and Brouwer 95
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Tunneling transport through Andreev
interferometers
Plan a) extend circuit theory to
tunneling
Goorden, PJ and Weiss 08 inspired by Nazarov
94 Argaman 97.
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Tunneling transport through Andreev
interferometers
Plan a) extend circuit theory to
tunneling
Goorden, PJ and Weiss 08 inspired by Nazarov
94 Argaman 97.
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Tunneling transport through Andreev
interferometers
Plan b) semiclassics Macroscopic Resonant
Tunneling
contribution to contribution to
Why macroscopic ? A O(N) effect !
Goorden, PJ and Weiss 08.
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Tunneling transport through Andreev
interferometers
Plan b) semiclassics Macroscopic Resonant
Tunneling Calculate transmission on blue
trajectories (i.e. for )
primitive traj.
Andreev loop travelled p times
Goorden, PJ and Weiss 08.
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Tunneling transport through Andreev
interferometers
Plan b) semiclassics Macroscopic Resonant
Tunneling Calculate transmission on blue
trajectories (i.e. for )
primitive traj.
Andreev loop travelled p times
Goorden, PJ and Weiss 08.
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Tunneling transport through Andreev
interferometers
Plan b) semiclassics Macroscopic Resonant
Tunneling Calculate transmission on blue
trajectories with action phase and stability
Sequence of transmissions and reflections at
tunnel Barriers (Whitney 07)
Stability of trajectory
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Tunneling transport through Andreev
interferometers

• Plan b) semiclassics
• Macroscopic Resonant Tunneling
• Calculate transmission
• One key observation
• Andreev reflections refocus the dynamics
• for Andreev loops shorter than Ehrenfest time
• Stability does not depend on p !
• Stability is determined only by

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Tunneling transport through Andreev
interferometers
Plan b) semiclassics Macroscopic Resonant
Tunneling Calculate transmission -gtPair all
trajs. (w. different ps) on ?1 ?3 -gtSubstitute
Determine B? as for normal transport classical
transmission probabilities
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Tunneling transport through Andreev
interferometers
Plan b) semiclassics Macroscopic Resonant
Tunneling
Measure of trajs. Resonant tunneling
Measure of trajs. Resonant tunneling
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Tunneling transport through Andreev
interferometers
Plan c) numerics
Order of magnitude enhancement from universal
(green) to MRT (red)
Effect increases as kFL increases Peak-to-valley
ratio goes from ? to ?2
Goorden, PJ and Weiss PRL 08, Nanotechnology 08.
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Tunneling transport through Andreev
interferometers
Plan c) numerics
Tunneling through 10-15 levels i.e. half of
those in the peak in the DoS TUNNELING THROUGH
LEVELS AT ?0
Goorden, PJ and Weiss PRL 08, Nanotechnology 08.
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Outline

• Mesoscopic superconductivity - Andreev reflection
• Density of states in ballistic Andreev billiards
• Transport through ballistic Andreev
  interferometers
• Symmetries of charge transport in presence
• of superconductivity

J. Weiss and PJ, in progress
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Symmetry of multi-terminal transport
NORMAL METAL Two-terminal measurement
G(H)G(-H) Four-terminal measurement
Gijkl(H) Gklij(-H)
O(e2/h)
Onsager, Casimir Buttiker 86 Benoit et al 86
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house
thermal
charge
S
parallelogram
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Symmetry of multi-terminal transport with
superconductivity
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Symmetry of multi-terminal transport with
superconductivity

• Numerics
• No particular symmetry
• AB-Amplitude is O(N)
• G looks more and more
• symmetric as N grows
• Exps. ltGgt1500 / 7700
• ?G 60 / 300
• Unreachable numerically - use circuit theory!

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Symmetry of multi-terminal transport with
superconductivity

• Nazarovs circuit theory
• Valid for Ngtgt1
• Neglects weak loc
• effects
• symmetric 4-terminal
• charge conductance
• AB oscillations O(N)
• Minimum at ?0
• Ratio ?R/ltRgt is in
• good agreement with exps

C.Th. Nazarov 94 Argaman 97.
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Symmetry of multi-terminal transport with
superconductivity
Nazarov 94 Argaman 97.
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ltGgt 1600 dG 70
ltGgt 7700 dG 300
ltGgt 18 dG lt 1