Supercurrent through carbon-nanotube-based quantum dots Tomáš Novotný Department of Condensed Matter Physics, MFF UK In collaboration with: K. Flensberg, H. I. Jørgensen, K. Grove-Rasmussen, P. E. Lindelof, and A. Rossini Nano-Science Center, University of Copenhagen Phys. Rev. B 72, (2005); Phys. Rev. Lett. 96, (2006); preprint

2 Outline of the talk 1.Brief introduction into the (normal) mesoscopic/nanoscopic quantum transport –Closed regime (Coulomb blockade) –Open regime (Fabry-Perot, scattering theory) 2.Superconducting transport - concepts –Josephson current –Andreev reflections –0-π transition –Phase dynamics 3.Experiments on S-CNT-S structures –Fabry-Perot regime (Josephson transistor) –Coulomb blockade regime (0-π transition)

3 1. Introduction

4 Mesoscopic systems in 2D electron gases

5 Aharonov-Bohm ring Quantum dots formed by using several gates (artificial atoms/molecules) More mesoscopic structures

6 Nanotube Nygård, Cobden, PRL Carbon nanotubes

7 gate V bias VgVg Single (!) molecule transistors

8 Park et al. Nature 407, 57 (2000) Park et al. Nature 417, 722 (2002) Co(tpy-(CH2)5-SH)2 C 60 Examples of single molecular devices

9 Electron lifetime on the molecule is long. Transport happens by independent tunneling events on and off the molecule. CASE 1 (weak coupling): Electron lifetime on the molecule is short. Molecule acts as a scattering for the electrons. CASE 2 (infinite coupling): Two limits (at least) CASE 1.5 (intermediate coupling): Stronly correlated regime. Kondo effect for odd occupation. Screening of the localized spin by lead electrons. Generally very difficult!

10  + V g U+2  +2 V g gate V g left contact right contact ”0” ”1” ”2” -V g V sd  + V g U+2  +2 V g ”0””1””2” E ”0” =E ”1” E ”1” =E ”2” CASE 1: Coulomb blockade spectroscopy

11 V sd Current kBTkBT Degeneracy Current through a single level

12 Sapmaz et al., Phys. Rev. B 67, (2003) A particular beatiful case – spectroscopy on nanotube

13 gate V g left contact right contact V sd N electrons N-1 electrons DOS Energy width =  Distance = U Not so weak coupling Hybridization with leads

14 DOS Energy width =  U  À U  ¿ U Effectively no interactions The electron transfer happens as independent event. Current can be calculated as a simple transmission problem of independent electrons The electron transfer is correlated = Coulomb blockade Electrons strongly interact but also transmit as waves Non-interacting particle or not ? Open or closed molecule ?

15 Maximum time before energy credit runs out: Minimum time required to make the deal: Relation to uncertainty principle ¿ l i f e = ~ ¡

16 Incoming wave Outgoing wave Reflected wave CURRENT: From transmission coefficients to conductance

17 Coulomb blockade  = I ~ /e = 0.5 meV  · 0.1 meV Fabry-Perrot From Fabry-Perot to Coulomb blockade

18 2. Superconducting transport – basic concepts

19 Josephson effect Cooper pair tunneling (in equilibrium) SS I J ( Á L ¡ Á R )

20 Andreev reflection (non-interacting limit) Multiple Andreev Reflections (MARs) – seen at finite bias (subharmonic gap structure) Bound Andreev States – carry the supercurrent

21 Andreev reflection (non-interacting limit) I exc ( g ) = e ~ ¢ g 2 h ( 4 ¡ g ) · 1 ¡ g 2 4 p 4 ¡ g ( 8 ¡ g ) l og 2 + p 4 ¡ g 2 ¡ p 4 ¡ g ¸ ; g ´ G h = e 2 If  >>  we can use theory for SC quantum point contacts: J. C. Cuevas et al., PRB 54, 7366 (1996) and V. S. Shumeiko, Low Temp. Phys. 23, 181 (1997) For a 4-fold degenerate SWCNT: I c ( g ) = e ~ ¢ gs i n' max 4 ~ p 1 ¡ g 4 s i n 2 ( ' max 2 ) t an h ~ ¢ p 1 ¡ g 4 s i n 2 ( ' max 2 ) 2 k B T Supercurrent (Josephson current) Excess current I exc ´ I ¢ ( V ! 1 ) ¡ I ¢ = 0 ( V ! 1 )

22 0-π transition (Coulomb blockade limit) F k ® ( ¿ ) = ¡ D T ¿ ³ c y ¡ k ® # ( ¿ ) c y k ® " ( 0 ) ´E 0 B ( ¿ 1 ; ¿ 2 ; ¿ 3 ) = D T ¿ ³ d y # ( ¿ 1 ) d y " ( ¿ 2 ) d # ( ¿ 3 ) d " ( 0 ) ´E 0 Graphical representation of this term Current in the lowest order in Γ 2 (cotuneling)

23  junction behavior gate 0-π transition (Coulomb blockade limit)

24 Phase dynamics When the junction is put into a circuit, the superconducting phase difference φ is actually a dynamical variable, moreover quite difficult to control. In principle the junction + environment compose a complicated quantum, nonlinear, stochastic, hysteretic, etc. dynamical system → many regimes of behavior. RCSJ model: C Ä ' + _ ' r + 2 e ~ ( I J ( ' ) ¡ I B ) = 2 e ~ i n ( t ) V = ~ _ ' 2 e “First Josephson relation” Second Josephson relation

25 3. Experiments

26 Josephson transistor (Fabry-Perot regime) P. Jarillo-Herrero, J. A. van Dam, and L. P. Kowenhoven, Nature 439, 953 (2006) Measured supercurrent largely influenced by the phase dynamics (underdamped junction): I cm / I 3 = 2 c

27 Josephson transistor (Fabry-Perot regime) H. Ingerslev Jørgensen et al., PRL 96, (2006)

28 0-π transition (Coulomb blockade regime) J. A van Dam et al., Nature 442, 467 (2006) Semiconducting nanowires (InAs) in a SQUID setup – enables the direct determination of the supercurrent sign

29 0-π transition (Coulomb blockade regime) J.-P. Cleuziou, W. Wernsdorfer et al., Nature Nanotechnology 1, 53 (2006) SWCNT SQUID setup direct observation of the transition

30 0-π transition (Coulomb blockade regime) H. Ingerslev Jørgensen et al., preprint (2007) Non-SQUID measurement, designed (controlled) fluctuations, true I c

31 Conclusions Josephson transistor demonstrated both theoretically and experimentally (i.e., gate voltage control of the Josephson current) Both open and closed regimes attainable Theoretical challenges: —Microscopic determination of I J (φ) for intermediate cases —Effects of dissipation, more realistic modeling of other degrees of freedom (more levels, oscillations, etc.) —Phase dynamics for non-sinusoidal current- phase relationships