Superconducting Proximity Effect in Quantum-Dot Systems

Transcription

1 p. 1 Superconducting Proximity Effect in Quantum-Dot Systems Jürgen König Institut für Theoretische Physik Universität Duisburg-Essen

2 Transport through S-N and S-I-S Interfaces BCS superconductivity: pair potential = g c k c k quasiparticles with gap in density of states Cooper-pair condensate subgap transport Andreev reflection Josephson current S N S S V Cooper pair breaks/forms transfer of Cooper pairs J jos (Φ) = J c sinφ p. 2

3 QDs Coupled to Superconductors - Exps carbon nanotube quantum dot Buitelaar, Nussbaumer, Schönenberger, PRL 2; Cleziou, Wernsdorfer, Bouchiat, Ondarcuhu, Monthioux, Nature Nanotech. 6; Jarillo-Herrero, van Dam, Kouwenhoven, Nature 6; Jorgensen, Grove-Rasmussen, Novotny, Flensberg, Lindelof, PRL 6; Hermann, Portier, Roche, Levy Yeyati, Kontos, Strunk, PRL 1; Pillet, Quay, Morfin, Bena, Levy Yeyati, Joyez, Nature Phys. 1;... quantum dot in InAs nanowire van Dam, Nazarov, Bakkers, De Franceschi, Kouwenhoven, Nature 6; Hofstetter, Csonka, Nygard, Schönenberger, Nature 1;... InAs quantum dot with Al electrodes Buizert, Oiwa, Shibata, Hirakawa, Tarucha, PRL 7; Deacon, Tanaka, Oiwa, Sakano, Yoshida, Shibata, Hirakawa, Tarucha, PRL 1;... quantum dot in graphene Dirks, Hughes, Lal, Uchoa, Chen, Chialvo, Goldbart, Mason, Nature Phys. 11 p. 3

4 p. 4 QDs Coupled to Superconductors - Thy Andreev and multiple Andreev reflections through QDs Levy Yeyati, Cuevas, Lopez-Davalos, Martin-Rodero, PRB 97; Fazio, Raimondi, PRL 98, PRL 99; Kang, PRB 98; Schwab, Raimondi, PRB 99; Johansson, Bratus, Shumeiko, Wendin, PRB 99; Clerk, Ambegaokar, Hershfield, PRB ; Cuevas, Levy Yeyati, Martin-Rodero, PRB ;... transport in Kondo regime Clerk, Ambegaokar, PRB ; Avishai, Golub, Zaikin, PRB 3; Choi, Lee, Kang, Belzig, PRB 4; Siano, Egger, PRL 4; Sellier, Kopp, Kroha, Barash, PRB 5; Nussinov, Shnirman, Arovas, Balatsky, Zhu, PRB 5; Bergeret, Levy Yeyati, Martin-Rodero, PRB 6; Lopez, Choi, Aguado, PRB 7; Karrasch, Oguri, Meden, PRB 8; Karrasch, Meden, PRB 9;... Josephson current through QD with spin-orbit coupling Dell Anna, Zazunov, Egger, Martin, PRB 7; Zazunov, Egger, Jonckheere, Martin, PRL 9;...

5 p. 5 Nature 442, 667 (26) quantum dot in InAs nanowire

6 p. 5 Nature 442, 667 (26) Josephson coupling due to cotunneling J jos Γ 2 π-state for odd occupation theory: van Dam et al., Nature 6 Glazman, Matveev, JETP Lett. 89 Rozhkov, Arovas, Guinea, PRB

7 p. 6 How to Establish a Josephson Coupling? higher-order tunneling (cotunneling) J jos Γ 2 (equilibrium) proximity effect in single-level QD finite pair amplitude: d d requirement: E E < E = E < ǫ U/2 impossible for large charging energy, U k B T,Γ nonequilibrium proximity effect in single-level QD finite pair amplitude: d d third, normal electrode drives dot out of equilibrium

8 p. 7 Model: Anderson Hamiltonian N µ N Γ N Φ/2 Γ S QD Γ S Φ/2 E = ǫ ǫ 2ǫ+U J L J R quantum dot: H D = σ ǫd σd σ +Un n leads: H η = kσ ǫ kc ηkσ c ηkσ ) k ( η c ηk c η k +H.c. superconducting leads: phase biased with ±iφ/2 normal lead: voltage biased with µ N ( ) tunneling: H tunn,η = V η kσ c ηkσ d σ +H.c. charging energy, superconductivity, and nonequilibrium

9 Diagrammatic Transport Theory Pala, Governale, J.K., NJP 7; Governale, Pala, J.K., PRB 8 general idea: reduced density matrix for dot integrate out leads: contractions c ηkσ c ηkσ and c ηkσ c ηkσ expand in tunnel coupling, treat interaction exactly L ρ L R L R L R L changes due to superconductivity: anomalous contractions: c ηkσ c η k σ and c η k σc ηkσ finite pair amplitude on dot: d d A p. 8

10 p. 9 2S-dot-N in Nonequilibrium N µ N Governale, Pala, J.K., PRB 8 Γ N Φ/2 Γ S QD Γ S Φ/2 : all orders in Γ S J L J R first order in Γ N isospin representation I x = Re d d ; I y = Im d d ; I z = d d +d d 1 2 kinetic equation for isospin: di dt = A R I+I B how to proximize the dot: combined S-dot-N Andreev reflection: A (1) Andreev reflection at S-dot interface: I B ()

11 p. 1 Andreev bound states without SC: resonances at ǫ and ǫ+u with SC: Andreev bound-state energies (for ): E A,γ,γ = γ U (ǫ+u/2) 2 +γ 2 +Γ 2 S cos2 (Φ/2) with γ,γ = ± 1.5 E A,+,+ Ε Α,γ,γ / U 1.5 E A,+,- E A,-,+ E A,-,- four resonances avoided crossing at δ = 2ǫ+U = -.5 with gap 2Γ S cos(φ/2) ε / U

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14 p. 13 Andreev Bound States for Finite Futterer, Swiebodzinski, Governale, J.K., PRB 13 resummation scheme for finite exact for very nice agreement with NRG Martin Rodero, Levy Yeyati, JPCM 12 much better than Hartree Fock

15 Josephson Current J L J R Γ S /U =.1 Γ S /U =.5 Γ S /U = 1 µn/u hj jos /(2eΓ N ) (a) µn/u hj jos /(2eΓ N ) (a) µn/u hj jos /(2eΓ N ) (a) ǫ/u ǫ/u ǫ/u equilibrium Josephson current suppressed by charging nonequilibrium Josephson current for finite µ N π-transition driven by ǫ or µ N Andreev bound-state energies: E A,γ,γ = γ U 2 (ǫ+u/2) +γ 2 +Γ 2 S cos2 (Φ/2) with γ,γ = ± Governale, Pala, J.K., PRB 8 p. 14

16 p. 15 Andreev Current J L +J R Γ S /U =.1 Γ S /U =.5 Γ S /U = hj and /(2eΓ N ) (b) 1.5 hj and /(2eΓ N ) (b) 1.5 hj and /(2eΓ N ) (b) µn/u µn/u µn/u ǫ/u ǫ/u ǫ/u Andreev current probes nonequilibrium proximity effect maximal around ε = U/2 Andreev bound-state energies: E A,γ,γ = γ U 2 (ǫ+u/2) +γ 2 +Γ 2 S cos2 (Φ/2) with γ,γ = ± Governale, Pala, J.K., PRB 8

17 p. 16 Shot Noise Reveals Proximity Effect Braggio, Governale, Pala, J.K., SSC 11 N Γ N QD off resonance ( δ Γ S ) Cooper pair cotunneling Poissonian transfer of 2e Fano factor F = 2 on resonance ( δ Γ S ) Cooper pair oscillations, interrupted by SET Poissonian transfer of e Fano factor F = 1 Γ S S a) µ/u b) F 2 Fano factor Andreev current µ = 1.5U µ =.5U ǫ/u hκ 1 /(eγ N ) F 3 Skewness

18 p. 17 Proximity by AC Driving Moghaddam, Governale, J.K., PRB 12 off resonant: δ Γ S I z = 1 2 no bias voltage: µ = ac driving: Γ S (t) = Γ S +δγ S cosωt B(t) acts on I Rabi oscillations DC current into N lead resonance at Ω = δ IN[(e/ h)γn] 1..5 δ/u = 3 δ/u = 2 δ/u = Ω/U

19 Cooper Pair Splitter with Double Dots double dot in InAs quantum wire Hofstetter, Csonka, Nygard, Schönenberger, Nature 9 Das, Ronen, Heiblum, Mahalu, Kretinin, Shtrikman, Nat. Comm. 12 double dot in CNT Hermann, Portier, Levy Yeyati, Kontos, Strunk, PRL 1 spin-entangled electron pair in normal leads p. 18

20 p. 19 Nonlocal Proximity Effect in Double Dots Eldridge, Pala, Governale, J.K., PRB 1 S N L Γ SL Γ SR Γ NL Γ NR QD QD N R 1.5 hi S /(eγ N ) non-local pair amplitude: 1.6 d L d R d L d R negative bias µ S < : transport through singlet µ/u Cooper pair splitter positive bias µ S > : δ/u spin triplet occupied triplet blockade

21 p. 2 Unconventional Superconductivity pair amplitudes: F iσ,i σ (t) = Tc iσ(t)c i σ () symmetry: under exchange of spin/orbital/time even-singlet: odd/even/even, BCS even-triplet: even/odd/even, 3 He, Sr 2 RuO 4 odd-triplet: even/even/odd, FM-SC hybids odd-singlet: odd/odd/odd,? even/odd - singlet/triplet pairing in double dots?

22 .. p. 21 Unconventional SC in Double Dots Weiss, Sothmann, Governale, J.K., in preparation T + S µ S = B QD +µ µ B QD B T T B S Γ S large B field: T occupied supercond. lead: S inhom. B field: S T noncol. B field: T T

23 p. 22 Superconducting Proximity Effect in... quantum dot + superconductor -π-transitions for Josephson current local and non-local Andreev transport double dot + superconductor non-local Cooper pairs triplet blockade double dot + superconductor + magnetic field non-local unconventional pairing

24 Collaborators Alessandro Braggio James Eldridge Rosario Fazio David Futterer Michele Governale Bastian Hiltscher Ali Moghaddam Marco Pala Björn Sothmann Janine Splettstoesser Jacek Swiebodzinski Fabio Taddei Uni Genova, Italy VU Wellington, New Zealand SNS Pisa, Italy Uni Duisburg-Essen, Germany VU Wellington, New Zealand Uni Duisburg-Essen, Germany Uni Duisburg-Essen, Germany CNRS Grenoble, France Uni Geneva, Switzerland RWTH Aachen, Germany Uni Duisburg-Essen, Germany SNS Pisa, Italy Stephan Weiß Uni Duisburg-Essen, Germany p. 23

25 extra slides p. 24

26 p. 25 Proximity by AC Driving Moghaddam, Governale, J.K., PRB 12 off resonant: δ Γ S I z = 1 2 no bias voltage: µ = ac driving: Γ S (t) = Γ S +δγ S cosωt B(t) acts on I Rabi oscillations.25 δ/u = 3 δ/u = 2 δ/u = 1 ac-induced proximity resonances atω = δ±γ S QD Ω/U

27 p. 25 Proximity by AC Driving Moghaddam, Governale, J.K., PRB 12 off resonant: δ Γ S I z = 1 2 no bias voltage: µ = ac driving: Γ S (t) = Γ S +δγ S cosωt B(t) acts on I Rabi oscillations DC current into N lead resonance at Ω = δ IN[(e/ h)γn] 1..5 δ/u = 3 δ/u = 2 δ/u = Ω/U

28 .. p. 26 Quantum Dot + FM + SC Sothmann, Futterer, Governale, J.K., PRB 1 S µ = S F Γ L Γ S QD Γ R +µ µ F non-collinear fm leads spin accumulation on QD fm leads exchange field current into superconductor (symmetric in µ) very sensitive to exchange field indication of unconventional pairing