Signatures of Tomonaga-Luttinger liquid

behavior in shot noise of a carbon nanotube

Patrik Recher, Na Young Kim, and Yoshihisa Yamamoto

Institute of Industrial Science, University of Tokyo, Japan

E.L. Ginzton Lab, Stanford University, USA

Capri Spring School, April 8, 2006

Outline

• Brief overview of single-walled carbon nanotubes (SWNTs)

• The transport problem: Keldysh functional approach

• Conclusion

• Conductance and low-frequency noise properties: Theory and experimental results

• Finite frequency noise (theory only)

• Luttinger-liquid model for a metallic carbon nanotube in good contact to electrodes

Overview of carbon nanotubes

• wrapped graphene sheets with diameter of only few nanometer

• Ideal (ballistic) one-dimensional conductor up to length

scales of 1-10 and energies of ~1 eV

• exists as semiconductor or metal with depending on the wrapping condition

m

m/s5^108~ Fv

Wildoer et al., Nature 391, 59 (1998)

Density of states

a) Metallic SWNT: constant DOS around E=0, van Hove singularities at opening of new subbands

b) Semiconducting tube: gap around E=0

Energy scale in SWNTs is about 1 eV, effective field theories

valid for all relevant temperatures

• Predicted Tomonaga-Luttinger liquid behavior in metallic tubes at energies : => crucial deviations from Fermi liquid

eV1~0

- spin-charge separation (decoupled movements of charge and spin) and charge fractionalization

- Power-law energy density of states (probed by tunneling)4/)1( 1

(

gn

- Smearing of the Fermi surface

8/)2/1( gg

FF kkkkn

Tomonaga-Luttinger liquid parameterquantifies strength of electron-electroninteraction, for repulsive interaction1g

g

Electron transport through metallic single-walled carbon nanotubes

heG /1.001.0~ 2bad contacts to tube (tunneling regime):

KT 5.1

KT 3

KT 7

KT 10

KT 15

Differential conductance as functionof gate voltage : Crossover from CB behaviorto metallic behavior with increasing

dVdI /

GVTDifferential conductance as function of

bias voltage at different temperatures Dashed line shows power-law ~ which gives averaged over gate voltage

dVdI /

7.0VV T

28.0~g

heG /32~ 2Well-contacted tubes:

Conductance as function of bias voltage and gate voltage at temperature 4K.Unlike in Coulomb blockade regime, here, wide high conductance peaks are separatedby small valleys. The peak-to-peak spacing determined by and not by charging energyLehvF /

Liang et al., Nature 411, 665 (2001)

• tube lengths 530 nm (a) – 220 nm (b)

Electron transport through SWNT in good contact to reservoirs

• Effective low-energy physics (up to 1 eV) in metallic carbon nanotubes:

• two-bands (transverse channels) cross Fermi energy

• including e-e interactions 2-channel Luttinger liquid with spin

dsVheI )/4( 2__

non-interacting value (Landauer Formula applies)

Gate

SiO2

Drain Source

Vg

Vds

• For reflectionless (ohmic) contacts :

C. Kane, L. Balents and M.P.A. Fisher, PRL 79, 5086 (1997)R. Egger and A. Gogolin, PRL 79, 5082 (1997)

Fk Fk

F

P. Recher, N.Y. Kim, and Y. Yamamoto, cond-mat/0604613

Theory of metallic carbon nanotubes

Hamiltonian density for nanotube:

band indices i=1,2 ;

Interaction couples to the total charge density : Only forward interactions are retained : good approximation for nanotubes if r large

bosonization dictionary for right (R) and left (L) moving electrons:

: Cut-off length due to finite bandwidth

is long-wavelength component of Coulomb interaction

It is advantageous to introduce new fields (and similar for ) :

Where we have introduced the total and relative spin fields:

4 new flavors

In these new flavors :

i

Free field theory with decoupled degrees of freedom

Luttinger liquid parameter 2.0~g

strong correlations can be expected

Physical meaning of the phase-fields :

Using:

It follows immediately that :

total charge density

total current density

total spin density

total spin current density

It also holds that :

which follows from the continuity eq. for charge :

or

backscattering and modeling of contacts

2/Lx m =1,2 denotes the two positions of the delta scatterers

The contacts deposited at both sides of the nanotube are modeled by vanishing interaction ( g=1) in the reservoirs finite size effect

are the bare backscattering amplitudes

inhomogeneous Luttinger-liquid model:

Safi and Schulz ’95Maslov and Stone ’95Ponomarenko ‘95

Including a gate voltage

1

2'

xgg VnVH

xVg 11 bsH

In the simplest configuration, the electrons couple to a gate voltage (backgate) via the term :

This term can be accounted for by making the linear shift in the backscattering term

The electrostatic coupling to a gate voltage has the effect of shifting the energy of all electrons. It is equivalent of shifting the Fermi wave number Fk

),( tx

Keldysh generating functional

Action for the system without barrier :

),( xaand similar for

Keldysh rotation:

source field; Keldysh form of current :

• Green’s function matrix is composed out of equilibrium correlators

Correlation function :

Retarded Green’s function :

• these functions describe the clean system without barriers and in equilibrium ( =0) V

BIVGI 0

Conductance

)/4( 20 heG where with

without barriers

backscattered current

In leading order backscattering [see also Peca et al., PRB 68, 205423 (2003)]

)(3)( F'

I'' mmmmmm RR R sum of 1 interacting (I) and 3 non-interacting (F) functions,

and similar for )(' mmC

])()[( 22

211

ij

ijij uuU

]2cos[2 212 ijgij

ijij LVuuU

describes the incoherent addition of two barriers

describes the interference of two barriers

voltage in dimension of non-interacting level spacingLeVv / V LvFL /

0

Retarded Green’s functions

smeared step function : reflection coefficient of charge :

cut-off parameter associated with bandwidth :

The retarded functions are temperature independent

non-interacting functions

)(' tr Fmm obtained with =1

I. Safi and H. Schulz, Phys. Rev. B, 52 17040 (1995)

sum indicates the multiple reflection at inhomogeneity of

effg

0/ L

TkB

Correlation functions

Relation to retarded functions via fluctuation dissipation theorem:

correlation at finite temperature correction 0T

0

ln2])[sinhln( for

=> exponential suppression of backscattering

1][sinh for

2.021 UU

25.0g

1g

07.01 U 02 U

G

v

LBTk /

07.021 UU

25.0g25.0g

0 5 10 15 20 25V

0.8

0.9

1

G

0T0T

main effect of interaction: power-law renormalization

(tuned by gate voltage)

conductance plots

2

4

LhVe

:V bias difference between minimas (or maximas)

ghVe L 2/

Differential conductance: Theory versus Experiment

0.38

0.36

0.34

0.32

0.30

dI/d

Vd

s

-20 -10 0 10 20 Vds (mV)

1.00

0.95

0.90

0.85

0.80

0.75

0.70

dI/d

Vds

-20 -10 0 10 20 Vds (mV)

14.01 U

1.02 U

02 U

1.02 U

25.0g

KT 4

V9gV

V3.8gV

V7.7gV

@ 4Kmeasurement

• damping of Fabry-Perot oscillation amplitude at high bias voltage observed

• clear gate voltage dependence of FP-oscillation frequency

• From the first valley-to-valley distance around we extract0Vds 22.0~g

nm360L

Current noise

In terms of the generating functional:

symmetric noise:

^^^

III

Low-frequency limit of noise:

eIGTkSF B /)2( Fano Factor:

renormalization of charge absent due to finite size effect of interaction * !

What kind of signatures of interaction can we still see ?

* The same conclusion for single impurity in a spinless TLL:B. Trauzettel, R. Egger, and H. Grabert, Phys. Rev. Lett. 88, 116401 (2002)B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, 226405 (2004)F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, 165309 (2005)

|| BIeS for TkeV B

Asymptotic form of backscattered current

TkgeV BL ,/

• shot noise is well suited to extract power-laws in the weak backscattering regime

I. Safi and H. Schulz ’95

reflection coefficient of charge :

1.01 U1.02 U

g=0.23

0T

Gv ( ) 2 Lock-In

Resonant Circuit

RPD>>RCNT Signal

DC

VG

RCNT

-20V

LED

Vdc

Vac

CNT Vdc

Vac

+

+ Cparasitic

*

#

Experimental Setup and Procedures

Key point : )(/)( ISISF PDSWNT

• Parallel circuit of 2 noise sources: LED/PD pair (exhibiting full shot noise S=2eI) and CNT.

• Resonant Circuit filters frequency ~15-20 MHZ.

• Voltage noise measured via full modulation technique (@ 22 Hz) -> get rid of thermal noise

2/1 dsSWNT VS

dsSP VS

Comparison with experiments on low frequency shot noise

• PD=Shot noise of a photo diode light emitting diode pair exhibiting full shot noise serving as a standard shot noise source.

)1/()1( gg

Experimental Fano factor F (blue) compared with theory for g~0.25 (red) and g~1(yellow).F is compared with power-law scaling

with g~0.16 for particular gate voltage shown and g~0.25 if we average over many gate voltages.

2/ dsVF ( red dashed line) giving g~0.18 for this particular gate voltage.In average over many gate voltages we have g~0.22

Power-law scaling

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

lo

g (

Fan

o f

acto

r)

1.51.00.50.0

log ( Vds )

0.35

0.30

0.25

0.20

0.15

0.10

Fan

o f

acto

r

403020100Vds (mV)

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

lo

g (

Fan

o f

acto

r)

1.51.00.50.0

log ( Vds )

Blue: Exp

Yellow: g = 1

Red: g = 0.25

T = 4 K

Device : 13A2426

Vg = - 7.9V

14.01 U 1.02 U

Incoherent part

coherent part

dominant at large voltages

2/1 V

frequency dependent conductivity of clean wire

Finite frequency impurity noise

• depends on point of measurement x

5 10 15 20 25 30 35

0.5

1

1.5

2

2

2,10 );,(

m

mxx

Frequency dependent conductance of clean SWNT+reservoirs

related to retarded function of total charge only !

• is assumed to be in the right lead and

(in units of )1

Ft

FF vLt / gtt Fv

independent of

not true for real part andimaginary part of );,(0 mxx

• oscillations are due to backscattering of partial charges arising from inhomogeneous g

23.0g

see also:B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, 226405 (2004)F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, 165309 (2005)

x

x

3D plot of excess noise in units of at T=4K for g=1 measured at barrier as function of bias (in units of ) and frequency (in units of )

),0(),( SVSSe LG 0

eL / L

Excess noise as a function of at =35 for ,12.01 U

Excess noise as a function of at 0

g=1

LhvVe F /

Finite frequency excess noise for the non-interacting system

T=4K

1.02 Uvv

v

3D plot of excess noise in units of at T=4K for g=0.23 measured at barrier 2 as function of bias (in units of ) and frequency (in units of )

),0(),( SVSSe LG 0

eL / L

Excess noise as a function of at =35 for ,12.01 U

Excess noise as a function of at 0

g=0.23

LhvVe F /cT/2

Fc vLgT /2charge roundtriptime

Signatures of spin-charge separation in the interacting system

Interacting levelspacingand non-interacting levelspacingclearly distinguished in excess noise !

LvF /LgvF /

from oscillation periods without any fitting parameter

g

1.02 U

v

v

v

Se

b)

5.0)/( Lxd

Dependence of excess noise on measurement point

g=0.23

g=1

T=4K

d=0.14 d=0.3 d=0.6

=35

=35

v

v

Conclusions

• conductance and shot noise have been investigated in the inhomogeneous Luttinger-liquid model appropriate for the carbon nanotube (SWNT) and in the weak backscattering regime

• conductance and low-frequency shot noise show power-law scaling and Fabry-Perot oscillation damping at high bias voltage or temperature. The power-law behavior is consistent with recent experiments. The oscillation frequency is dominated by the non-interacting modes due to subband degeneracy.

• finite-frequency excess noise shows clear additional features of partial charge reflection at boundaries between SWNT and contacts due to inhomogeneous g. Shot noise as a function of bias voltage and frequency therefore allows a clear distinction between the two frequencies of transport modes g via oscillation frequencies and info about spin-charge separation