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, 224502 (2005); Phys. Rev. Lett. 96, 207003 (2006); preprint

2

Outline of the talk1. 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)

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1. Introduction

4

Mesoscopic systems in 2D electron gasesMesoscopic systems in 2D electron gases

5

Aharonov-Bohm ringQuantum dots formed by using several gates (artificial atoms/molecules)

More mesoscopic structuresMore mesoscopic structures

6

Nanotube

Nygård, Cobden, PRL 2002.

Carbon nanotubesCarbon nanotubes

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gate

Vbias

Vg

Single (!) molecule transistors Single (!) molecule transistors

8

Park et al. Nature 407, 57 (2000)

Park et al. Nature 417, 722 (2002)

Co(tpy-(CH2)5-SH)2

C60

Examples of single molecular devicesExamples of single molecular devices

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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)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!

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+ Vg

U+2+2 Vg

gate Vg

left

con

tact

right contact

”0” ”1” ”2”

-Vg

Vsd

Vsd

+ Vg

U+2+2 Vg

”0” ”1” ”2”

E”0” =E”1”E”1” =E”2”

CASE 1: Coulomb blockade spectroscopyCASE 1: Coulomb blockade spectroscopy

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Vsd

Current

kBT

Degeneracy

Current through a single levelCurrent through a single level

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Sapmaz et al., Phys. Rev. B 67, 235414 (2003)

A particular beatiful case –spectroscopy on nanotubeA particular beatiful case –spectroscopy on nanotube

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gate Vg

left

con

tact

right contact

Vsd

N electrons

N-1 electrons

DOS

Energy

width =

Distance = U

Not so weak couplingHybridization with leadsNot so weak coupling

Hybridization with leads

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DOS

Energy

width =

U

À U

¿ U

Effectively no interactionsThe electron transfer happens as independentevent.

Current can be calculatedas a simple transmissionproblem 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 ?

Non-interacting particle or not ?Open or closed molecule ?

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Maximum time before energy credit runs out:

Minimum time required to make the deal:

Relation to uncertainty principleRelation to uncertainty principle

¿life = ~¡

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Incoming wave Outgoing wave

Reflected wave

CURRENT:

From transmission coefficients to conductanceFrom transmission coefficients to conductance

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Coulomb blockade

= I ~/e = 0.5 meV · 0.1 meV

Fabry-Perrot

From Fabry-Perot to Coulomb blockadeFrom Fabry-Perot to Coulomb blockade

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2. Superconducting transport – basic concepts

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Josephson effectCooper pair tunneling (in equilibrium)

S S

I J (ÁL ¡ ÁR )

H = HM +HT +H BCSL +H BCS

R

HM = »X

¾=";#

dy¾d¾+Un"n#

HT =X

®=L ;R ;k

(tk®cyk®¾d¾+t¤

k®dy¾ck®¾)

H BCS® =

X

k;¾=";#

"k®cyk®¾ck®¾¡

ÃX

k

¢ ®eiÁ®cyk®"cy

¡ k®#+h:c:

!

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Andreev reflection (non-interacting limit)

•Multiple Andreev Reflections (MARs) – seen at finite bias (subharmonic gap structure)

•Bound Andreev States – carry the supercurrent

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Andreev reflection (non-interacting limit)

I exc(g) = e~¢ g2

h(4¡ g)

·1¡ g2

4p

4¡ g(8¡ g)log 2+

p4¡ g

2¡p

4¡ g

¸;g ´ Gh=e2

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~¢ gsin ' max

4~p

1¡ g4 sin2( ' max

2 )tanh

~¢p

1¡ g4 sin2( ' max

2 )2kB T

Supercurrent (Josephson current)

Excess current I exc ´ I ¢ (V ! 1 ) ¡ I ¢ =0(V ! 1 )

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0-π transition (Coulomb blockade limit)

Fk®(¿) = ¡DT¿

³cy¡ k®#(¿)cy

k®" (0)´E

0

B(¿1;¿2;¿3) =DT¿

³dy

#(¿1)dy" (¿2)d#(¿3)d" (0)

´E

0

I ® = 4ImZ ¯

0d¿1

Z ¯

0d¿2

Z ¯

0d¿3

£DT¿

³~H +

T ¹®#(¿1) ~H +T ¹®" (¿2) ~H ¡

T ®#(¿3) ~H ¡T ®"

´E

0

= 4ImX

k

X

p

t¤p¹®t¤¡ p¹®t¡ k®tk®

Z ¯

0d¿1

Z ¯

0d¿2

£Z ¯

0d¿3F k®(¿3)F ¤

p¹®(¿1 ¡ ¿2)B(¿1;¿2;¿3)

Graphical representation of this term

Current in the lowest order in Γ2 (cotuneling)

23junction behavior

gate

0-π transition (Coulomb blockade limit)

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Phase dynamicsWhen 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

+ 2e~

(I J (' ) ¡ I B ) = 2e~

in(t)

V = ~_'2e

“First Josephson relation”

Second Josephson relation

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3. Experiments

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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=2c

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Josephson transistor (Fabry-Perot regime)H. Ingerslev Jørgensen et al., PRL 96, 207003 (2006)

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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

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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

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0-π transition (Coulomb blockade regime)H. Ingerslev Jørgensen et al., preprint (2007)

Non-SQUID measurement, designed (controlled) fluctuations, true Ic

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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 IJ(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