AC CONDUCTANCE AND NON-SYMMETRIZED NOISE

AT FINITE FREQUENCY

IN QUANTUM WIRE AND CARBON NANOTUBE

Adeline CRÉPIEUX 1, Cristina BENA 2,3 and Inès SAFI 2

1 – Centre de Physique Théorique, CNRS Marseille, France

2 – Laboratoire de Physique des Solides, CNRS Orsay, France

3 – Institut de Physique Théorique, CEA Saclay, France

Physical Review B 78, 205422 (2008)

CURRENT FLUCTUATIONS

ORIGINS OF NOISE

High temperature : Johnson-Nyquist noise High voltage : Shot noise High frequency : Quantum noise We neglect the 1/f noise

ZERO FREQUENCY AND ZERO TEMPERATURE: SHOT NOISE

Schottky relation where jeS *0

HIGH FREQUENCY NOISE MEASUREMENTS

High-frequency measurement in a diffusive wire On-chip detection using SIS junction Direct measurement in a QPC

What is measured is

002

1jtjtjjFTS

symmetrized noise

tjjdteS ti 0

non-symmetrized noise

DEBLOCK et al., Science (2003)

ZAKKA-BAJJANI et al., PRL (2007)

SCHOELKOPF et al., PRL (1997)GHz201 GHz100GHz84

THE SYSTEM

GATE

WIRE

V1 V

3

V2

IMPURITY

xi

L

MODEL VB HHHH 0

22

20

1

2 xF

xgdx

vH

iFiB xktxH 2,4cos

WE CALCULATE PERTURBATIVELY

The non-symmetrized noise:

The AC conductance:

0

mVm

nnm tV

tIttG

tjjdteS nmti

nm 0

tvtV

tjtI

nn

nn

coswhere

RESULT

002

1nmnm

tinm jtjtjjdteS with symmetrized noise

nmnmnm GSS Re

GENERALIZED KUBO-TYPE FORMULA

nmnmnmnm GSSS Re2 SAFI AND SUKHORUKOV (unpublished)

where 3,2,1, mn

COULOMBINTERACTIONS

PARAMETER

BACKSCATTERINGAMPLITUDE

)(xg

0 2/L2/L

1

x

txxeVdx

H xV ,

AC CONDUCTANCE

WEAK-BACKSCATTERING LIMIT OF THE EXCESS AC CONDUCTANCE

0111111

V

GGG 12 VVV source-drain voltagewhere

05.0/

01.0/

0

0

eVg

eV

T

x

L

i

FOR g=1:

FOR g≠1:

011 G because of the linearity of the I-V caracteristic

gLvFL /oscillations with frequencywith a pseudo-period related to the wire frequency

The pseudo-perioddepends on L and g:

gLeV

hvFg

75.0

5.0

ZERO-FREQUENCY NOISE

IN THE WEAK-BACKSCATTERING LIMIT

where is the backscattering current

V

I

h

eTk

Tk

eVeIS B

BB

Bnm 222

coth02

BI

5/ LBTk

0/ LBTk

REGION A: short-wire limit linear variation with voltage qualitative agreement with experiments on carbon nanotubes

REGION B: long-wire limit oscillations whose envelope has a power-law dependence

REGION C: high temperature limit

behaves like the noise of an infinite length interacting wire: power-law variation

WU et al., PRL 99, 156803 (2007)HERRMANN et al., PRL 99, 156804 (2007)

LBTk

LeV

LeV

FINITE-FREQUENCY NON-SYMMETRIZED NOISE

SHORT-WIRE LIMIT 1/ eVg L

FOR g=1: the non-symmetrized excess noise is symmetric

FOR g≠1: the non-symmetrized excess noise becomes asymmetric

011 G nmnmnmnmnm SSGSS Re

WE CALCULATE 0

Vnmnmnm SSS

because when g = 1

EMISSIONABSORPTION

01.0/

0

0

eV

T

xi

FINITE-FREQUENCY NON-SYMMETRIZED NOISE

LONG-WIRE LIMIT

infinite lengthwire limit

V1 V2

infinite lengthnanotube limit

CARBON NANOTUBE

four channels of conduction

01.0/

01.0/

0

0

25.0

eVg

eV

T

x

g

L

i

1withsectorsothers3

1withsectorcharge

g

g

AVERAGE NON-SYMMETRIZED NOISE

EMISSION EXCESS NOISE AVERAGED FANO FACTOR

001.0/

01.0/

0

0

eVg

eV

T

x

L

i

NON-SYMMETRIZED NOISE

SYMMETRIZED NOISE

LF

WE CALCULATE THE AVERAGE OVER THE FIRST HALF PERIOD

we obtain

it should be possible to extract the value of the interaction parameter g

BeI

SF L

L

11 where L

LdSS

L

0 11111

gFL

gFL

CONCLUSION

ACKNOWLEDGMENTS TO: C. GLATTLI, H. BOUCHIAT, R. DEBLOCKT. KONTOS, B. REULET, E.

SUKHORUKOV

Simple relation between the AC conductance the non-symmetrized noise

In the presence of Coulomb interactions, the non-symmetrized noise is asymmetric:

Emission noise ( > 0) ≠ Absorption noise ( < 0)

At low-temperature and for a long wire or a long nanotube,we obtain oscillations with a period related to L and g

The average non-symmetrized excess noiseover the first half period gives the value of g

nmnmnm GSS Re

g

eI

S

B

L 11