RESONANT TUNNELING IN CARBON NANOTUBE QUANTUM DOTS

MILENA GRIFONI

M. THORWARTR. EGGERG. CUNIBERTI

H. POSTMAC. DEKKER

200 nmafter Postma et al. Science (2001)

25 nm

Discussions: Y. Nazarov

QUANTUM DOTS

eVTkB ,

addition energy

EC

eNN 2

)()1(2

dotdot

dotsourceV Vg

Cg drain

e e

001dot ),1( NNg EEVN

Coulomb blockade

)(dot N

a)

L R

single electron tunneling

)(dot N

b) ...1 NNN

L R

ORTHODOX SET THEORYC

ondu

ctan

ce

Gate voltage

hold woud)a(SWNTfor 8.0end

1MAX

end TGLuttinger leads SETs

Furusaki, Nagaosa PRB (1993)

1MAX

TG semiconducting dot + Fermi leads

Beenakker PRB (1993)

Sequential tunneling

Gate voltage C

ondu

ctan

ce T2 > T1(a)

TkE B

NANOTUBE DOT IS A SETPostma, Teepen, Yao, Grifoni, Dekker, Science 293 (2001)

Coulomb blockade in quantum regime

Ec = 41 meV, E = 38 meV > kBT up to 440 K

1MAX

TG1

MAXend TG

dI/dV d2I/dV2

Gate voltage (V)

Bia

s vo

ltage

(V)

30 K

unconventional

PUZZLE

Why nanotube SET not ? 1

MAXend TG

1MAX

endend TG endendend 2

Correlated sequential tunneling Gate voltage

T2 > T1(b)

Con

duct

ance

unscreened Coulomb interaction ? Maurey, Giamarchi, EPL (1997)

weak tunneling at metallic contacts ? Kleimann et al., PRB (2002)

asymmetric barriers ? Nazarov, Glazman, PRL (2003)

correlated tunneling ? Postma et al., Science (2001), Thorwart et al. PRL (2002)Hügle and Egger, EPL (2004)

OVERVIEW

METALLIC SINGLE-WALL NANOTUBES (SWNT)

SWNT LUTTINGER LIQUIDS SWNT WITH TWO BUCKLES

UNCOVENTIONAL RESONANT TUNNELING EXPONENT

1D DOT WITH LUTTINGER LEADS

CORRELATED TUNNELING MECHANISM

METALLIC SWNT MOLECULES

metallic 1D conductor with 2 linear bands

k

EF

Energy

LUTTINGER FEATURES

Let us focus on spinless LL case,

generalization to SWNT case later

DOUBLE-BUCKLED SWNT´s

after Rochefort et al. 1998

buckles act as tunneling barriers

Luttinger liquid with two impurities

50 x 50 nm2

WHAT IS A LUTTINGER LIQUID ?

Fk k

R

linear spectrum

bosonization identity )(/ ~ iLR e

charge density

xFkx

)(

example: spinless electrons in 1D

L

Fk

)(kE

L L

R Rq~0

)'()'()('1])()([2v

'22F

0 xxxVxdxdxxxdxH xxx

+ forward scattering )(ˆ qV

FE

LUTTINGER HAMILTONIAN

])(1)([2

v 220 x

gxgdxH x

2.0ln

v81

1

v1

12

0

RReV

gS

FF

)(rangedshort )( 0 xVdxVxV captures interactioneffects

nanotubes

g/vv F

TRANSPORT

extimptot HHHH 0

gCgVV

Luttinger liquids

localized impurities voltage sources

constkUxUxdxH impimpimp )(ˆ),()( backscattering

constxVxVxdxeH extextext )(),()( forward scattering

TRANSPORT

2/)()(2/)( LR xxtN

)()()( LR xxtn charge transferred across the dot

charge on the island

Brownian`particles´ n, Nin tilted washboard potentialCn

CCeV

NeVHtot

ggext

2

)cos()cos( nNUH impimp

)(2

lim tNeI t

continuity equation

) 0, , ; , , ( lim2,

i i f fn N

f tn Nt n N P N e

f f

reduced densitymatrix

11

nnNNe1 e2

nN

2

CURRENTExact trace over bosonic modes

L Rx x x x, ), (

]' , [ ]' , [ ]' [ ] [ ' ) 0, , ; , , (*

n n F N N F N A N A ND ND n Nt n N Pn N i i f f

bare actionext impH H bulk modes reduced

density matrix

) , (n N N

mass gap for ncharging energy

LINEAR TRANSPORTdynamics states dynamics states2

nonlocal in time coupling )' (' ), (t N t N

1 , 1 n n

n n,

1 , 1 n n

N N,

1 , 1 N N

1 , 1 N N

CORRELATIONS

tiTk

tJdtWB

sincoth)cos1()()(

02

W

,

0 /

) range finite for only ( 0 / 1

SWNT , 2. 0

g

g

correlations involving different/same barriers /WN N,

1 , 1 N N

1 , 1 N N

0 , ) ( g

J

gx

E F

0

v

) ( J

)()()()( 32212321 SSSSW = S+iR

dipole

dipole-dipole

FINITE RANGE?

FINITE RANGE?not needed

CORRELATIONS II

tiTk

tJdtWB

sincoth)cos1()()(

02

W

,

N N,

1 , 1 N N

1 , 1 N N

)()()()( 32212321 SSSS

W = S+iR

• zero range W : purely oscillatory WOhmic + oscillations

<cosh const, <sinh

0321 ,, Ohm

EFFECT OF THE CORRELATIONS ?

FIRST CONSIDER UNCORRELATED TUNNELING

• MASTER EQUATION APPROACH Ingold, Nazarov (1992) (g = 1), Furusaki PRB (1997)

• GENERATING FUNCTION METHOD (FROM PI SOLUTION) Grifoni, Thorwart, unpublished

MASTER EQUATION FOR UST

Rb Lb Rf Lf

tot

Rb Lb Rf Lfe

n N

N tt n P N e I,

) , ( lim2

linear regime: only n = 0,1 charges

end Lf

f LT e e dn iE W L

~ Re20

/ ) 1 ( ) (2

N N,

1 , 1 N N

1 , 1 N N

W

golden rule rateexample

Uncorrelated sequential tunneling:• only lowest order tunneling process • master equation for populations:

2/R L

Ingold, Nazarov (1992) (g = 1), Furusaki PRB (1997)

) , ( ) , 1 ( ) , 1 (

) , ( ) , ( ) ( ) , (

1 1 1

1

t n P t n P t n P

t n P t n P t n P

N Lb N Lf N Rb

N Rf N Rb Rf Lb Lf N

tot

MASTER EQUATION FOR UST II

) (b ftot

Rb Lb Rf Lfe e I

) , ( ) , ( ) , ( ) ( ) , (2 2t n P t n P t n P t n PN b N f N b f N

b

f

Note:

can also be obtained from the master eq.

Is there a simple diagrammatic interpretation of f/b ?

GENERATING FUNCTION METHOD

Different view from path integral approach

n N

Nn N

N tn P N t n P N I,

20

,

) , ( ˆ lim ) , ( lim

0

20) , ( lim

F

n N

NN

n P e F,

) , ( ˆ ) , (

generating function

exact series expression

2

) ( ) (0) ( ˆ ) ( ˆ lim

m

mb

mfI I I

contributions to the f/b current of order mExample: m = 2 (divergent!)

(c)2)N,2(N

N),(NN),(N

2)N,2(N

N),(N

2)N,2(N (a) (b)

W

cotunneling

GENERATING FUNCTION METHOD FOR ST

N N,

1 , 1 N N

1 , 1 N N Sequential tunneling approximation:

Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)

)(ˆlim2

)(0

m

mII

GENERATING FUNCTION METHOD FOR ST

N N,

1 , 1 N N

1 , 1 N N Sequential tunneling approximation:

Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)

)(ˆlim2

)(0

m

mII

GENERATING FUNCTION METHOD FOR ST

Sequential tunneling approximation:

Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)

)(ˆlim2

)(0

m

mII

bm

bfm

fm LTRRTLI 22)( )()()(ˆ

L

R

non trivial cancellationsamong contributionof different paths

GENERATING FUNCTION METHOD FOR CST

Sequential tunneling approximation:

Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)

)(ˆlim2

)(0

m

mII

bm

bfm

fm LTRRTLI 22)( )()()(ˆ

L

R

non trivial cancellationsamong contributionof different paths

Correlations!

GENERATING FUNCTION METHOD FOR UST

Sequential tunneling approximation:

Consider only (but all) paths which are back tothe diagonal after two steps(giustified for strong Coulomb interaction)

)(ˆlim2

)(0

m

mII

])()([)(ˆ 221

)(Lb

mtotRbRf

mtotLfm

m eI

L

R

b f e IRftot tot

Lf

... 1 1 lim

2

0

tot

Rb Lb Rf Lfe

0 lim

again

UST: onlyintra-dipoleCorrelations!

GENERATING FUNCTION METHOD FOR UST II

Interpretation:

Higher order paths provide a finite life-timefor intermediate dot state, which regularizes the divergent fourth-order paths

)(ˆlim2

)(0

m

mII

tot

LbRb

tot

RfLfbf ee )(

)](cos[)](cos[4 1111

)()(32

01

22231

REREeeeddd RfLfSSRL

ftot

)()()( iRStW

L

R

tot

2,lim 0

totecc RfLff

CST)()(22)( ˆˆ)()()(ˆ m

bm

fbm

bfm

fm IILTRRTLI

Let us lookorder by order:

)()()(32

01

22

0)2( 31321

4lim

SSRLf eeeddd

133132113 coscos),,(cosh RERE RfLf

133132113 sinsin),,(sinh RERE RfLf

m=2

/// RiSW

Short cut notation: RfLfRfLff sscceI )/()(ˆ )2(

cosh

sinh

divergent =0

exact!

CST II

)()(22)( ˆˆ)()(ˆ mb

mfb

mbf

mf

m IILTRRTLI

m=3

As for UST, sum up higher order terms to get a finite result

divergent

RfLfRfLfRfLfRfLff scsssccssccceI )/(ˆ )3(

RfLfRfLfRfLfRfLf ccccssccsssc

• Consider only diverging diagrams• Linearize in dipole-dipole interaction = 0

RbLbRfLf ccccc

Approximations:

CST III

summation over m Systematic expansion in

RfLfRfLfRfLfRfLff csscssccsscceI )/()(ˆ

tote

RfLff cc0lim

modified line width

end

end

TT

tot

RfLff const.

~~2

UST

at resonance

ss

tote

MASTER EQUATION FOR CST

2 / /,a b f L R n N

N tt n P N e I,

) , ( lim2

) , ( ) , (

) , ( ) , 1 ( ) , 1 (

) , ( ) , ( ) ( ) , (

2 2 2 2

1 1 1

1 2 2 1

t n P t n P

t n P t n P t n P

t n P t n P t n P

N f N b

N Lb N Lf N Rb

N Rf N f b tot N

transferthrough 1 barrier(irreducibile contributions ofsecond and higher order)

transfer trough dot (irreducibilecontributions at least of fourth order)

Thorwart et al. unpublished finite life-time due to higher orderpaths found self consistently

RESULTS GMAX

nanotubes:

spinless LL:

31

41, 1

effeff

gggg

112endend

g

endend

endend

*

1/

TG

TTGMAX

23.0g

4 bosonic fields

spin2 band2

Kane, Balents, Fisher PRL (1997), Egger, Gogolin, PRL (1997)

Thorwart et al., PRL (2002)

CONCLUSIONS & REMARKS

0endendend 1MAX )1( TgG

leads

dot

• UNCONVENTIONAL COULOMB BLOCKADE

0F /v xTkL BT LOW TEMPERATURES : BREAKDOWN OF UST IN LINEAR REGIME

NONINTERACTING ELECTRONS g =1REMARK