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									 Coherent Transport Through
   Andreev Interferometers

         Philippe Jacquod
            U of Arizona


M. Goorden (Delft)
J. Weiss (Arizona)
The two-slit experiment (textbook version)




                      1




                      2




                                
The two-slit experiment (non-textbook version)




                                       ?
The two-slit experiment (non-textbook version)




                                   Regular cavity
                                   Chaotic cavity
IS THIS DECOHERENCE ?

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

DO CHAOTIC / COMPLEX SYSTEMS DECOHERE
 WHEREAS REGULAR / INTEGRABLE
 SYSTEMS DO NOT ?




           NO!
  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
But what about multiple
     scattering ?
     The two-slit experiment (non-textbook version)

“Despite multiple chaotic scattering
the Gaussian envelope still exhibits
(small) modulations !”




A.k.a. weak localization                Regular cavity
                                        Chaotic cavity
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)
Measurement sample      Magnetoresistance
                        ~Aharonov-Bohm oscillations




           Amplitude of oscillations decreases
           with temperature ~ decoherence
From the mesoscopic AB effect
to Andreev interferometers

Giant enhancement of
oscillations amplitude!
                        “house”



                      thermal
                      charge




S

    “parallelogram”
     <G> =1600
      dG =70

     <G> =7700
      dG =300
(in units of e2/h)
From the mesoscopic AB
effect to Andreev
interferometers

Giant enhancement
  of oscillations!
Symmetries of multiterminal
  transport !?
Symmetry vs. antisymmetry
  of thermopower !?
OUR MOTIVATION
          “These effects cannot be all
          explained by the scattering
          approach to transport.”




               -V. Chandrasekhar
                Tucson, Feb 24, 2006
Outline
Mesoscopic superconductivity - Andreev reflection

Density of states in ballistic Andreev billiards

Transport through ballistic Andreev interferometers

Transport through diffusive Andreev interferometers
Outline
Mesoscopic superconductivity - Andreev reflection

Density of states in ballistic Andreev billiards

Transport through ballistic Andreev interferometers

Transport through diffusive Andreev interferometers
Mesoscopic Superconductivity
                 Mesoscopic metal (N) in contact
                 with superconductors (S)

      << L
                       S invades N
                 “Mesoscopic proximity effect”


S       N

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

      << L
                     S invades N

S       N             But how ??

             S
Mesoscopic Superconductivity


                 Effect of S in N depends on:
      << L
                 (i) Electronic dynamics in N
                 (ii) Symmetry of S state
                      (s- or d-wave; S phases…)
                 (iii) E/D
S       N

             S
 What is Andreev reflection
     • low-energy electron quasiparticle approaches
       superconductor from normal region

       “Charge-reversing retro-reflection”

                                             Supercond.
                                             pair potential




Andreev, „64; sidenote : Andreev reflection ~ Hawking radiation
Andreev reflection

 (e,EF+  )           (h, EF-)

 Reflection phase :               (fig taken from Wikipedia)




 Angle mismatch : Snell‟s law

                                                 S phase
                                                 + : h->e
                                                 - : e->h
  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
  Andreev billiards: classical dynamics
                                                   At NI interface:
   superconductor


                                                   Normal reflection

                                Note #1: Billiard is chaotic
                                  all trajectories
                                     become periodic!


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


                                                   Normal reflection

                                Note #1: Billiard is chaotic
                                  all trajectories
                                     become periodic!


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


                           Normal reflection

                     Note #2: Action on P.O.




                       Andreev reflection
At NS interface:
                       phase
Andreev reflection
  Andreev billiards: semiclassical quantization

                                All orbits are periodic
                                -> Bohr-Sommerfeld
  S           N
                                       Andreev reflection phase


                                       x


           Distribution of return times to S
           chaos-> exp. Suppression at E=0
           regular->algebraic / others
See also: Melsen et al. „96;Ihra et al. „01; Zaitsev „06
 Andreev billiards: semiclassical quantization

                          All orbits are periodic
                          -> Bohr-Sommerfeld
 S        N
=0                         x
              
 : DoS has peak at E=0 !!
All trajs touching both
contribute to n=0 term
                                   Goorden, PJ, Weiss „08
 Andreev billiards: random matrix theory
N = MxM RMT Hamiltonians
S -> 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; Altland+Zirnbauer „97
  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 ?
     What is the Ehrenfest* time?
  •Classical chaos ~ local exponential divergence


  •Quantum chaos




 Deep semiclassical limit              
                                         E    tn


*Why the name ?
For larger times, breakdown of Ehrenfest‟s 1927 thm
that « a wavepacket follows Newton‟s laws on average »

           Larkin and Ovchinnikov „69
           Bermann and Zaslavsky ‟78
   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
   Andreev billiards - Solution to the “gap problem”



   Universal, RMT regime:                        Deep semiclassical regime:
   Gap at Thouless energy                        Gap at Ehrenfest 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
   Andreev billiards: DoS at 


   Universal, RMT regime:     Deep semiclassical regime:
   Minigap at level spacing   Large peak around E=0 !




Goorden, PJ and Weiss „08.
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
Scattering approach to coherent transport
A.k.a. “Landauer-Buttiker”




                      “Conductance as transmission”
                      (in units of e2/h)
       Random matrix theory of transport
 Going from diffusive to ballistic systems
                  Scattering approach (Landauer-Büttiker)




“Chaotic cavity”  S as a         Random Matrix
 Distr. of T‟s

 Conductance                             symmetry index
                                          1 with TRS
                                          2 without TRS
 UCF
                                          4 without SRS, with TRS


Note: Scattering approach equivalent to Kubo (Fisher & Lee „81)
           Ray optics for the 21st century
 Scattering approach




  Entrance / exit points




  Classical…       trajectories, stability and action

R. Whitney and PJ, PRL/PRB „05/06
Transport through Andreev interferometers

               Lambert „93 formula



               Average conductance for NL=NR




              New, Andreev reflection term
              Gives classically large
               interference contributions
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   
Transport through Andreev interferometers

No tunnel barrier :
Coherent backscattering is
-O(N)
-positive, increases G
                                              QuickTime™ and a
                                    TIFF (Uncompressed) decompre ssor
                                       are neede d to see this picture.




This is (obviously) not
related to the DoS in
the Andreev billiard


 !! INTRODUCE TUNNEL BARRIERS
 TUNNELING CONDUCTANCE ~ DOS !!

Beenakker, Melsen and Brouwer „95
 Tunneling transport through Andreev interferometers




Plan a) : extend circuit theory
          to tunneling




Goorden, PJ and Weiss „08; inspired by : Nazarov „94; Argaman „97.
 Tunneling transport through Andreev interferometers




Plan a) : extend circuit theory
          to tunneling




Goorden, PJ and Weiss „08; inspired by : Nazarov „94; Argaman „97.
 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.
 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.
 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.
 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
Stability of trajectory      Barriers (Whitney „07)
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
Tunneling transport through Andreev interferometers

Plan b) : semiclassics

“Macroscopic Resonant Tunneling”

Calculate transmission




->Pair all trajs. (w. different p‟s) on    1+  3
->Substitute



               Determine B as for normal transport
               ~classical transmission probabilities
Tunneling transport through Andreev interferometers

Plan b) : semiclassics

“Macroscopic Resonant Tunneling”


                                     Measure of trajs.

                                     Resonant tunneling




                                     Measure of trajs.

                                     Resonant tunneling
 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.
 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.
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
 Symmetry of multi-terminal transport

 NORMAL METAL:
 Two-terminal measurement      G(H)=G(-H)

 Four-terminal measurement   Gij;kl(H)= Gkl;ij(-H)

                O(e2/h)




Onsager, Casimir…
Buttiker „86
Benoit et al „86
                        “house”



                      thermal
                      charge




S

    “parallelogram”
Symmetry of multi-terminal transport
    with superconductivity
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.: <G>=1500 / 7700
       G= 60 / 300

Unreachable numerically - use circuit theory!
 Symmetry of multi-terminal transport
     with superconductivity

Nazarov‟s circuit theory:
Valid for N>>1
Neglects “weak loc”
 effects

symmetric 4-terminal
  “charge” conductance
AB oscillations O(N)
Minimum at =0
Ratio R/<R> is in
 good agreement with exps

  C.Th.: Nazarov „94; Argaman „97.
Symmetry of multi-terminal transport
    with superconductivity




Nazarov „94; Argaman „97.
          <G> =1600
           dG =70

          <G> =7700
           dG =300


<G> =18
 dG < 1

								
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