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

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

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

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

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)

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.

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)

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 

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 : 1. constant pair potential (r) =  e 2. ideal NS interface, i.e. no specular reflection for energies 0 <  <  i

Pumped current into the normal lead : Blaauboer, PRB 65, 235318 (2002)

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, 12841 (1992)

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

« 1 for T « T Comparison G / G vs. I / I at resonance for T  T NS at resonance for T  T NS N 1 2 G /G < 2 NS N 4.23 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 = 2.55 (maximum) NS,peak N,peak

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