![Page 1: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/1.jpg)
Full counting statistics of incoherent multiple Andreev
reflection
Peter Samuelsson, Lund University, Sweden
Sebastian Pilgram, ETH Zurich,
Switzerland
![Page 2: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/2.jpg)
Outline
Voltage biased Josephson junctions, multiple Andreev reflections.
Coherent and incoherent transport. Noise and full counting statistics,
stochastic path integral approach. Examples: double barrier and diffusive
wire junctions. Low voltage - energy space diffusion. Conclusions.
![Page 3: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/3.jpg)
Josephson effectVoltage biased superconducting tunnel junctionJosephson, Phys. Lett. 1, 251 (1962)
Josephson current
Dc-component Cohen, Falicov, Philips, PRL 8, 316 (1962)
S S
I
V1 2 3
![Page 4: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/4.jpg)
Subharmonic gap structure and excess current
Additional features in IV-curve
Taylor, Burstein, PRL 10, 14 (1963)
Schrieffer, Wilkins, PRL 10, 17 (1963)
Subharmonic gap structures
Excess current
Cooper pair tunneling
Van der Post et al, PRL 73, 2611 (1994)
![Page 5: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/5.jpg)
Multiple Andreev reflectionsBoltzmann approach (incoherent), weak linkKlapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1083).
Gives subharmonics excess current
Current
S S
V
Klapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983).
e
h
![Page 6: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/6.jpg)
Quantum point contactsCoherent transport, single mode contact, transparency D
Atomic point contacts
TheoryArnold, J. Low. Temp. Phys. 68 1 (1987).Bratus et al, PRL 74, 2110 (1995).Averin, Bardas, PRL 75, 1831 (1995).Cuevas, Martin-Rodero, Levy-Yeyati,PRB 74, xxxx (1996).
Scheer et al, PRL 78, 3535 (1998).Scheer et al, Nature 394, 154 (1998).Ludoph et al, PRB 61, 8561 (2000).
![Page 7: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/7.jpg)
Noise: multiple chargesTheory Cuevas, Martin-Rodero, Levy-Yeyati, PRL 82, 4086 (1999),
Naveh, Averin, PRL 82, 4090 (1999).
Experiment Cron et al PRL 86, 4104 (1999).
Quanta of multiple charge
Zero frequency noise
Fano factor
Dieleman et al, PRL 79, 3486 (1997).
![Page 8: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/8.jpg)
Full counting statisticsFull distribution of transported charge
Long measurement time Charge
Cumulant generating function
Cumulants
[ non Gaussian fluctuations]
![Page 9: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/9.jpg)
Coherent transportTheoryCuevas, Belzig, PRL 91, xxx (2003); PRB xx, xxx (2004).
Johansson, Samuelsson, Ingerman, PRL 91, 187002 (2003),
Cumulant generating function
n-particle scattering probability
![Page 10: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/10.jpg)
Incoherent transport Strong phase breaking suppressed proximity effect
S S S S S S
Experimentally important regime (noise)Jehl et al, PRL 83, 1660 (1999), Hoss et al, PRB 62 4079 (2000), Roche et al Physica C 352, 73 (2001), Hoffmann, Lefloch, Sanquer, EPJB 29 629 (2002).
Current and noise theory (incoherent)Bezuglyi et al, PRL 83, 2050 (1999), Nagaev, PRL 86, 3112 (2001), Bezuglyi et al, PRB 63, 100501 (2001), Samuelsson et al, PRB 65, 180514 (2002).
ballistic diffusive chaotic
No theory for full counting statistics!
![Page 11: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/11.jpg)
Incoherent full counting statisticsStochastic path integral approach, semiclassicsPilgram et al, PRL 90, 206801 (2003), Jordan, Sukhorukov, Pilgram, J. Math. Phys. 45, 4386 (2004).
Separation of time scales: Nagaev, xxxx. fast quasiparticle scattering, slow dynamics of distribution functions,
generalized Boltzmann-Languevin approach
fL fRf
t
f
Related approaches: Kindermann, Beenakker, Nazarov, PRB, xxxx, Bodineau, Derrida, PRL 92, 180601 (2004), Gutman, Mirlin, Gefen, xxxx.
![Page 12: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/12.jpg)
Our work
Generating function, NS-interface
S S
Example: ballistic SNS-junction, interface barriersOctavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983).
e
h
Muzukantskii, Khmelnitskii, PRB 50, 3982 (1994).
S
Composed from elementary scattering probabilities
Andreev / normal reflection probability
N
Pilgram, Samuelsson, PRL 94, 086806 (2005)
![Page 13: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/13.jpg)
Formulate as path integral over possible internal charge configugurations Integrate out fast charge fluctuations effective generating function in slow variables .
Stochastic path integral approach
For
![Page 14: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/14.jpg)
Saddle point equationsSemiclassical limit path integral in saddle point approximation
Solution inserted back into
Cumulants givesOTBK
.....
(No simple expression...)
![Page 15: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/15.jpg)
CumulantsNumerical evaluation, differential cumulants
Subharmonic gap structure diverges at low
![Page 16: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/16.jpg)
Probability distributionStationary phase approximation
With from
Conditional distributionfunctions
![Page 17: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/17.jpg)
Low voltage limitLow voltage limit , finite normal back scattering
E
t
Quasiparticle diffusion in energy space
Generating function
Diffusive wire with renormalized chargeJordan, Sukhorukov, Pilgram, J. Math. Phys. 45, 4386 (2004).
![Page 18: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/18.jpg)
general incoherent low voltage behavior
Low voltage cumulants , diverges for
Holds for large class of junctions, only different
Generating function, saddle point solution
......
Theory breaks down at , inelastic scattering cuts off divergence. Effect of environment not considered. Reulet et al, PRL xx, xx (xxxx), Kindermann, Beenakker, Nazarov PRL ...
Coherent junctions, diverges for Naveh, Averin, PRL 82, 4090 (1999), Johansson, Samuelsson, Ingerman, PRL 91, 187002 (2003), Cuevas, Belzig, PRB xxx
![Page 19: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/19.jpg)
Diffusive wire
S S Diffusive normal region Normal conductance Negligiable interface resistance
Recent experiments on third cumulant Reulet, Les Houches.
![Page 20: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/20.jpg)
electron charges transfered
Arbitrary voltage approach
For a voltage
Injection energies electron charges transfered Effective conductance
Injection energies
Effective conductance
3e 2e
![Page 21: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/21.jpg)
Generating function – adding up the two processes
First cumulants
Nagaev, PRL 86, 3112 (2001), Bezuglyi et al, PRB 63, 100501 (2001).
shows subharmonic gap structure
Excess generating function
![Page 22: Full counting statistics of incoherent multiple Andreev reflection Peter Samuelsson, Lund University, Sweden Sebastian Pilgram, ETH Zurich, Switzerland](https://reader030.vdocuments.mx/reader030/viewer/2022032800/56649d395503460f94a12967/html5/thumbnails/22.jpg)
Conclusions