1. Charge pumping in mesoscopic systems coupled to a superconducting lead
In collaboration with:
E.J. Heller (Harvard)
Yu.V. Nazarov (Delft)
Workshop on “Mesoscopic Physics and Electron Interaction”,
Trieste, 1 July 2002

2. Outline • Pumping • Pumping of charge
• Pumping of charge in presence of superconductivity
Application: Nearly-closed quantum dot
Conclusions

3. Adiabatic pumping of particles
Idea behind pumping : to generate motion of particles by slow periodic modulations
of their environment, e.g. their confining potential or a
magnetic field.
U(x) = U sin(2x/a)
Thouless pump, 1983 : adiabatic transport
of electrons in 1D periodic potential
For moving potential, every minimum is
shifted by a after each period T:
U(x+vt) = U sin(2x/a) cos(2t/T)
+ U cos(2x/a) sin(2t/T)
Superposition of two standing waves with
a phase difference can produce pumping

4. Archimedean screw More than 2000 years old, used for pumping water
‘s Hertogenbosch, Netherlands

5. Pumping of charge through quantum dots
2DEG
gates
leads
Marcus group webpage
Coulomb blockade turnstile
Electrons transported one by
one, pumped charge is quantized
“classical” pumping
Kouwenhoven et al., PRL 67, 1626 (1992)
Pothier et al., Europhys. Lett. 17, 249 (1992)

6. Open quantum dots Spivak et al, PRL 51, 13226 (1995)
Physical picture : a small change of system
parameters X during a time t leads to a
redistribution of charge Q within the system,
due to changing electrostatic landscape.
This produces electron flows I = Q /t i
The pumped charge depends on the interference of electron wavefunctions
in the system.

7. Theory of quantum pumping
[ Brouwer, PRB 58, R10135 (1998); Aleiner et al., PRL 81, 1286 (1998);
Zhou, PRL 82, 608 (1999) ]
Idea : view as transmission problem, and describe current in terms of the
scattering matrix S of the system
Conductance : Landauer formula
Pumping :
X (t) = X sin(t)
X (t) = X sin(t + ) 1 2
Brouwer, PRB 58, R10135 (1998)

8. Quantum pumping experiment
Switkes et al., Science 283, 1905 (1999)
Experimental set-up, open quantum dot
Red gates control the conductance
of the point contacts
Black gates are used for pumping
Pumped current vs. phase difference 

9. Quantum pumping in the presence of superconductivity
Presence of a superconductor introduces
Andreev reflection : electron-to-hole
reflection at the interface between a
normal metal and a superconductor
Phase coherent reflection: hole travels back
along (nearly) the same path where the electron
came from
Assume : constant pair potential (r) =  e
2. ideal NS interface, i.e. no specular reflection
for energies 0 <  <  i

10. Pumped current into the normal lead :
Blaauboer, PRB 65, (2002)

11. Applications : 1. Nearly-closed quantum dot
Energy landscape
Δ : level spacing
 : level broadening
T ,T « 1 and k T <  « : transport via
resonant transmission 1 2 B
Conductance :
two normal leads
Breit-Wigner formula
one normal and one superconducting lead
Beenakker, PRB 46, (1992)

12. Doubling of conductance due to presence of the holes G /G = 2
Pumped current : 1 2
V : shape changing
voltages ,
G /G NS N
Comparison
vs.
I / I
at resonance for T = T 1 2
Doubling of conductance due to
presence of the holes
G /G = 2 NS N
Quadrupling of the pumped current
due to presence of holes + absence of
bias
I / I = 4 NS N

13. « 1 for T « T Comparison G / G vs. I / I at resonance for T  TNS
at resonance for T  T NS N 1 2
G /G < 2 NS N
for T = 1.26 T (maximum) 1 2
« 1 for T « T
I /I = it depends NS N
Pumped current peak heights at higher temperatures,
 « k T «  B
I / I = (maximum)
NS,peak
N,peak

14. Conclusions Presence of Andreev reflection enhances or reduces
the pumped current through quantum dots by a factor
varying from ~ 4.23 to « 1
For dots with symmetric tunnel coupling to the leads
the enhancement is a factor of 4