Download Full counting statistics of incoherent multiple Andreev reflection

Full counting statistics of
incoherent multiple Andreev
reflection
Peter Samuelsson, Lund University,
Sweden
Sebastian Pilgram, ETH Zurich,
Switzerland
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.
Josephson effect
Voltage biased superconducting tunnel junction
Josephson, Phys. Lett. 1, 251 (1962)
I
S
S
1
2
V
 Josephson current
 Dc-component
Cohen, Falicov, Philips, PRL 8, 316 (1962)
3
Subharmonic gap structure and
excess current
Additional features in IV-curve
Cooper pair
tunneling
Schrieffer, Wilkins,
PRL 10, 17 (1963)
Taylor, Burstein, PRL 10, 14 (1963)
 Subharmonic gap structures
 Excess current
Van der Post et al, PRL 73, 2611 (1994)
Multiple Andreev reflections
Boltzmann approach (incoherent), weak link
Klapwijk, Blonder, Tinkham, Physica B+C, 109-110 1657 (1982), Octavio, Tinkham,
Blonder, Klapwijk, PRB, 27 6739 (1983).
(1083).
S
S
e
h
V
Current
Gives
 subharmonics
 excess current
Quantum point contacts
Coherent transport, single mode contact, transparency D
Theory
Arnold, 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).
Atomic point contacts
Scheer et al, PRL 78, 3535 (1998).
Scheer et al, Nature 394, 154 (1998).
Ludoph et al, PRB 61, 8561 (2000).
Noise: multiple charges
Theory
Cuevas, Martin-Rodero, Levy-Yeyati, PRL 82, 4086 (1999),
Naveh, Averin, PRL 82, 4090 (1999).
Zero frequency noise
Fano factor
Experiment
Cron et al PRL 86, 4104 (1999).
Quanta of
multiple
charge
Dieleman et al,
PRL 79, 3486 (1997).
Full counting statistics
Full distribution of transported charge
 Long measurement
time
 Charge
Cumulant generating function
Cumulants
[
non Gaussian fluctuations]
Coherent transport
Theory
Johansson, Samuelsson, Ingerman, PRL 91, 187002 (2003),
Cuevas, Belzig, PRL 91, xxx (2003); PRB xx, xxx (2004).
Cumulant generating function
n-particle scattering probability
Incoherent transport
Strong phase breaking
S
S
ballistic
suppressed proximity effect
S
S
S
diffusive
S
chaotic
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).
No theory for full counting statistics!
Incoherent full counting statistics
Stochastic path integral approach, semiclassics
Pilgram 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,
f
fL
f
fR
t
generalized Boltzmann-Languevin approach
Related approaches: Kindermann, Beenakker, Nazarov, PRB, xxxx,
Bodineau, Derrida, PRL 92, 180601 (2004), Gutman, Mirlin, Gefen, xxxx.
Our work
Pilgram, Samuelsson, PRL 94, 086806 (2005)
Example: ballistic SNS-junction, interface barriers
Octavio, Tinkham, Blonder, Klapwijk, PRB, 27 6739 (1983).
S
N
S
h
e
Generating function, NS-interface
Muzukantskii, Khmelnitskii, PRB 50, 3982 (1994).
Andreev / normal reflection probability
Composed from elementary scattering probabilities
S
Stochastic path integral approach
 Formulate
as path integral over possible internal
charge configugurations
 Integrate out fast charge fluctuations
effective
generating function in slow variables
.
For
Saddle point equations
Semiclassical limit
path integral in saddle point approximation
Solution inserted back into
Cumulants
gives
OTBK
(No simple expression...)
.....
Cumulants
Numerical evaluation, differential cumulants
 Subharmonic gap structure

diverges at low
Probability distribution
Stationary phase approximation
With
from
Conditional
distribution
functions
Low voltage limit
Low voltage limit
, finite normal back scattering
E
t
Quasiparticle diffusion in energy space
Generating function
Diffusive wire with renormalized charge
Jordan, Sukhorukov, Pilgram, J. Math. Phys. 45, 4386 (2004).
Generating function, saddle point solution
Low voltage cumulants
, diverges for
......
Holds for large class of junctions, only different
general incoherent low voltage behavior
Coherent junctions,
diverges for
Naveh, Averin, PRL 82, 4090 (1999), Johansson, Samuelsson, Ingerman,
PRL 91, 187002 (2003), Cuevas, Belzig, PRB xxx
 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 ...
Diffusive wire
S
S
 Diffusive normal region
 Normal conductance
 Negligiable interface resistance
Recent experiments on third cumulant Reulet, Les Houches.
Arbitrary voltage approach
3e
For a voltage
Injection energies

electron charges transfered
 Effective conductance
Injection energies

electron charges transfered
 Effective conductance
2e
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
Conclusions