1. Capri Spring School, April 8, 2006
Signatures of Tomonaga-Luttinger liquid behavior in shot noise of a carbon nanotube
Patrik Recher, Na Young Kim, and Yoshihisa Yamamoto
E.L. Ginzton Lab, Stanford University, USA
Institute of Industrial Science, University of Tokyo, Japan

2. Outline good contact to electrodes Theory and experimental results
Brief overview of single-walled carbon nanotubes (SWNTs)
Luttinger-liquid model for a metallic carbon nanotube in
good contact to electrodes
The transport problem: Keldysh functional approach
Conductance and low-frequency noise properties:
Theory and experimental results
Finite frequency noise (theory only)
Conclusion

3. Overview of carbon nanotubes
wrapped graphene sheets with diameter of only few nanometer
Ideal (ballistic) one-dimensional conductor up to length
scales of and energies of ~1 eV
exists as semiconductor or metal with depending on the wrapping condition
Wildoer et al., Nature 391, 59 (1998)
High geometric aspect ration. Can have length of several mm but diameter of only a few nm.

4. Density of states
Metallic SWNT: constant DOS around E=0, van Hove singularities at opening of new subbands
Semiconducting tube: gap around E=0
Energy scale in SWNTs is about 1 eV, effective field theories
valid for all relevant temperatures

5. Predicted Tomonaga-Luttinger liquid behavior in metallic tubes at
energies : => crucial deviations from Fermi liquid
- spin-charge separation (decoupled movements of charge and spin)
and charge fractionalization
- Power-law energy density of states (probed by tunneling)
- Smearing of the Fermi surface
Tomonaga-Luttinger liquid parameter
quantifies strength of electron-electron
interaction, for repulsive interaction

6. Electron transport through metallic single-walled carbon nanotubes
bad contacts to tube (tunneling regime):
Mention CB at low T in QDs, two charge states degenerate-> transport, otherwise CB valley. Above CB, power-law scaling in Vds.
Differential conductance as function
of gate voltage : Crossover from CB behavior
to metallic behavior with increasing
Differential conductance as function of
bias voltage at different temperatures
Dashed line shows power-law ~ which gives
averaged over gate voltage

7. Well-contacted tubes:
tube lengths 530 nm (a) – 220 nm (b)
Liang et al., Nature 411, 665 (2001)
Conductance as function of bias voltage and gate voltage at temperature 4K.
Unlike in Coulomb blockade regime, here, wide high conductance peaks are separated
by small valleys. The peak-to-peak spacing determined by and not by charging energy

8. Electron transport through SWNT in good contact to reservoirs
Gate
SiO2
Drain
Source Vg
Vds
two-bands (transverse channels)
cross Fermi energy
Effective low-energy physics (up to 1 eV) in metallic carbon nanotubes:
C. Kane, L. Balents and M.P.A. Fisher, PRL 79, 5086 (1997)
R. Egger and A. Gogolin, PRL 79, 5082 (1997)
length of tube: nm. Will refer to specific sample with L=360nm
Metal contacts are :Ti/Au, Ti only, Pd (Palladium); SiO2 layer of 0.5 um, Heavily doped Si substrate as gate.
including e-e interactions channel Luttinger liquid with spin
For reflectionless (ohmic) contacts :
non-interacting value
(Landauer Formula applies)

9. Theory of metallic carbon nanotubes
Hamiltonian density for nanotube:
band indices i=1,2 ;
is long-wavelength component of Coulomb interaction
Interaction couples to the total charge density :
 Only forward interactions are retained : good approximation for nanotubes if r large
Forward scattering where e stay in same branch. Amplitude for backscattering is down by 1/N, N is # atoms around circumference. Use bosonization description of one-dimensional electrons where the electron creation and annihilation operators are written as an exponential of bosonic fields \phi and \theta.
bosonization dictionary for right (R) and left (L) moving electrons:
Cut-off length due to finite bandwidth 

10. 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 :
Free field theory
with decoupled
degrees of freedom
Mention that charge mode has velocity vF/g whereas other modes have v_F
Luttinger liquid parameter
 strong correlations can be expected

11. Physical meaning of the phase-fields :
Using:
It follows immediately that :
total charge density
total current density
total spin density
n=number operator, j=current operator
total spin current density
It also holds that :
which follows from the continuity eq. for charge : or

12. backscattering and modeling of contacts
inhomogeneous
Luttinger-liquid model:
Safi and Schulz ’95
Maslov and Stone ’95
Ponomarenko ‘95
Combination of all flavor fields in the exponent, not only the charge field. This leads to interference in transport involving correlation of all flavors.
are the bare backscattering amplitudes
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

13. Including a gate voltage
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

14. Keldysh generating functional
source field;
Keldysh form of current :
Action for the system without barrier :
and similar for
Keldysh rotation:
Keldysh generating functional
Keldysh contour :
Keldysh generating functional
Keldysh contour :
Keldysh generating functional
Keldysh contour :
Definition of current
Definition of current
Definition of current
Action for the system without barrier
Action for the system without barrier
Action for the system without barrier
Action for the system without barrier
source field
source field
source field
correlation function :
correlation function :
correlation function :
retarded function :
retarded function :
retarded function :
Keldysh rotation
Keldysh rotation
Keldysh rotation
and similar for
and similar for
and similar for

15. correlation function :
Green’s function matrix is composed out of equilibrium correlators
correlation function :
Correlation function :
Retarded Green’s function :
these functions describe the clean system without barriers and in equilibrium ( =0)

16. Conductance In leading order backscattering where with
without barriers
backscattered current
In leading order backscattering
[see also Peca et al., PRB 68, (2003)]
Conductance cannot be calculated analytically in general for g<1 and we have to refer to numerical calculation of integral. Still we can get inside into the physics by
looking at the ret. and correlation functions to be discussed next.
sum of 1 interacting (I) and 3 non-interacting (F) functions,
and similar for
describes the incoherent addition of two barriers
describes the interference of two barriers
voltage in dimension of non-interacting level spacing

17. Retarded Green’s functions
The retarded functions are temperature independent
sum indicates the multiple reflection at inhomogeneity of
smeared step function :
reflection coefficient of charge :
I. Safi and H. Schulz, Phys. Rev. B, (1995)
cut-off parameter associated
with bandwidth :
non-interacting functions
obtained with =1

18. Correlation functions
Relation to retarded functions via
fluctuation dissipation theorem:
correlation at
finite temperature correction

19. for
=> exponential suppression of backscattering
for

20. conductance plots
bias difference between
minimas (or maximas)
Only at special Vg values we see the charge mode. Oscillation is probably hard to extract since oscillation amplitude is small
(tuned by gate voltage)
main effect of interaction: power-law renormalization

21. Differential conductance: Theory versus Experiment
0.38
0.36
0.34
0.32
0.30
dI/dV ds
-20
-10 10 20 V
(mV)
measurement
@ 4K
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 extract

22. Current noise In terms of the generating functional: symmetric noise:
We think that noise is much better suited than conductance to see interaction effects, since in the weak backsacttering regime, the conductance is dominated by the
the larger value of transmission and the backscattered correction is modified bu g. But noise is shown to be directly proportional to the backscatterd component.

23. Low-frequency limit of noise:
for
renormalization of charge absent due to finite size effect of interaction * !
What kind of signatures of interaction can we still see ?
Fano Factor:
* The same conclusion for single impurity in a spinless TLL:
B. Trauzettel, R. Egger, and H. Grabert, Phys. Rev. Lett. 88, (2002)
B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, (2004)
F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, (2005)

24. Asymptotic form of backscattered current
reflection coefficient of charge :
I. Safi and H. Schulz ’95
g=0.23
shot noise is well suited
to extract power-laws
in the weak backscattering regime
Say that shot noise is directly proportional to the weak backscattered component of current which contains the information about g. Therefore, shot noise is well suited to extract g through power-law in the weak backscattering regime.

25. Experimental Setup and ProceduresGv
( ) 2
Lock-In
Resonant
Circuit
RPD>>RCNT
Signal DC VG
RCNT
-20V
LED
Vdc
Vac
CNT +
Cparasitic * #
Parallel circuit of two noise sources. LED/PD pair which is weakly coupled (PD) ~0.4 % => serves as an ideal full shot noise source: S=2eI.
Resonance circuit filters frequency \omega~15-20 MHZ. Measures voltage noise via full modulation technique 22 Hz frequency)->get rid of thermal noise and enhances signal to noise ratio. Fano factor: F=S(I)/S_PD(I). This technique gets rid of noise from circuit which is assumed to be the same for both noise measurements.
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
Key point :

26. Comparison with experiments on low frequency shot noise
Power-law scaling
PD=Shot noise of a photo diode
light emitting diode pair exhibiting full
shot noise serving as a standard shot
noise source.
Experimental Fano factor F (blue) compared
with theory for g~0.25 (red) and g~1(yellow).
F is compared with power-law scaling
Here we display noise and Fano factor in a log-log Plot. We determine the power-law and compare it with the theoretical exponent, which leads to g.
( red dashed line) giving
g~0.18 for this particular gate voltage.
In average over many gate voltages we have g~0.22
with g~0.16 for
particular gate voltage shown and g~0.25 if
we average over many gate voltages.

27. Device : 13A2426 Vg = - 7.9V Blue: Exp Yellow: g = 1 Red: g = 0.25
T = 4 K
Fano factor in linear scale at temperature 4K with thermal noise subtracted. We use the same backscattering strengths U1 and U2 as we have used for the fitting of conductance.
Fano factor can be fitted quantitatively in the range of the validity of the weak backscattering theory.

28. Finite frequency impurity noise
Incoherent
part
frequency dependent conductivity
of clean wire
dominant at
large voltages
In addition to the usual beat note of shot noise we also see an additional modulation with frequency containing only the charge mode described by the conductivity of the clean wire. At large voltage, the incoherent part is dominant due to the monotonic increase with voltage and will be strongly modified by sigma^2.
In addition we see that also the interference oscillation amplitudes will be modified with frequency, therefore the strength of Fabry-Perot oscillations is also modified by frequency dependent quantities.
coherent
part
depends on point of measurement

29. Frequency dependent conductance of clean SWNT+reservoirs
related to retarded function of total charge only !
is assumed to be in the right lead and
see also:
B. Trauzettel, I. Safi, F. Dolcini, and H. Grabert, Phys. Rev. Lett. 92, (2004)
F. Dolcini, B. Trauzettel, I. Safi, and H. Grabert, Phys. Rev. B 71, (2005)
independent of
not true for real part and
imaginary part of
(in units of )
oscillations are due to backscattering of partial charges arising from inhomogeneous

30. Finite frequency excess noise for the non-interacting system
T=4K
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 )
Excess noise is zero for large frequencies in agreement with LB-transport. Further we see only one frequency scale. Oscillations in Omega and V are factor 2 different
since in frequency epsilon+omega interferes with epsilon-omega
Excess noise as a function of
at =35 for
Excess noise as a function of at

31. Signatures of spin-charge separation in the interacting system
Interacting levelspacing
and non-interacting levelspacing
clearly distinguished in excess noise !
from oscillation periods without any
fitting parameter
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 )
charge roundtrip
time
Excess noise as a function of
at =35 for
Excess noise as a function of at

32. Dependence of excess noise on measurement point
=35
g=0.23
T=4K
d=0.14
d=0.3
d=0.6
=35
g=1

33. 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