Transcript Slide 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
Outline
• 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
Overview of carbon nanotubes
•
wrapped graphene sheets with diameter of only few nanometer
Wildoer et al., Nature 391, 59 (1998)
• Ideal (ballistic) one-dimensional conductor up to length
scales of 1-10  m and energies of ~1 eV
•
exists as semiconductor or metal with vF ~ 8 10^5 m/s
depending on the wrapping condition
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    0 ~ 1eV : => crucial deviations from Fermi liquid
- spin-charge separation (decoupled movements of charge and spin)
and charge fractionalization
- Power-law energy density of states n(   
( g 1 1) / 4
(probed by tunneling)
- Smearing of the Fermi surface
nk  k F   k  k F
( g 1 / g  2 ) / 8
g
Tomonaga-Luttinger liquid parameter
quantifies strength of electron-electron
interaction, g  1 for repulsive interaction
Electron transport through metallic single-walled carbon nanotubes
bad contacts to tube (tunneling regime):
G ~ 0.01  0.1 e 2 / h
T  15 K
T  10 K
T  7K
T  3K
T  1.5K
Differential conductance dI / dV as function of
bias voltage V at different temperatures T
0.7
Dashed line shows power-law ~ V
which gives
g ~ 0.28 averaged over gate voltage
Differential conductance dI / dV as function
of gate voltage VG : Crossover from CB behavior
to metallic behavior with increasing T
Well-contacted tubes:
G ~ 2  3 e2 / h
• 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 hvF / Le and not by charging energy
Electron transport through SWNT in good contact to reservoirs
P. Recher, N.Y. Kim, and Y. Yamamoto, cond-mat/0604613
Vds
Source
Drain
F
SiO2
Gate
 kF
 kF
• two-bands (transverse channels)
cross Fermi energy
Vg
• 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)
• including e-e interactions
 2-channel Luttinger liquid with spin
• For reflectionless (ohmic) contacts :
__
I  (4e 2 / h)Vds  non-interacting value
(Landauer Formula applies)
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
bosonization dictionary for right (R) and left (L) moving electrons:
 : Cut-off length due to
finite bandwidth

It is advantageous to introduce new fields (and similar for i) :
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
Luttinger liquid parameter g ~ 0.2
 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
inhomogeneous
Luttinger-liquid model:
Safi and Schulz ’95
Maslov and Stone ’95
Ponomarenko ‘95
are the bare backscattering amplitudes
m =1,2 denotes the two positions x   L / 2
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
Including a gate voltage
In the simplest configuration, the electrons couple to a gate voltage
(backgate) via the term :
H '  Vg n   Vg
2

 x 1
This term can be accounted for by making the linear shift
in the backscattering term H bs
1  1  Vg x
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
kF
Keldysh generating functional
 ( x, t ) 
source field;
Keldysh form of current :
Action for the system without barrier :
and similar for
Keldysh rotation:
a ( x, )
• 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 (V =0)
Conductance
I  G0 V  I B
where
with
G0  (4e2 / h)
without barriers
backscattered current
In leading order backscattering
[see also Peca et al., PRB 68, 205423 (2003)]
I
F
sum of 1 interacting (I) and 3 non-interacting (F) functions,
Rmm '    Rmm
(

)

3
R
'
mm ' ( )
and similar for Cmm ' ( )
U1  [(u1ij )2  ( u2ij )2 ] describes the incoherent addition of two barriers
ij
U 2  2 u1ij u2ij cos[Vg L  2ij ]
describes the interference of two barriers
ij
v  eV / L voltage V in dimension of non-interacting level spacing L  vF / L
Retarded Green’s functions
The retarded functions are temperature independent
g eff
sum indicates the multiple reflection at inhomogeneity of
smeared step function :
reflection coefficient of charge :
I. Safi and H. Schulz, Phys. Rev. B, 52 17040 (1995)
cut-off parameter associated
with bandwidth  0 :
  L /  0
non-interacting functions
F
rmm
' (t )
obtained with
=1
Correlation functions
Relation to retarded functions via
fluctuation dissipation theorem:
  k BT
correlation at
T 0
finite temperature correction
ln(sinh[])  2ln   for   
=> exponential suppression of backscattering
sinh[]  1 for   0
conductance plots
U1  U 2  0.2
G
g  0.25
  k BT / L

V : bias difference between
minimas (or maximas)
v
2
4
g  0.25
eV  hL
G
g 1

U1  U 2  0.07
g  0.25
1
0.9
eV  hL / 2 g
0.8
0
T 0
main effect of interaction: power-law renormalization
5
10
15
V
U1  0.07 U 2  0
T 0
20
25
(tuned by gate voltage)
Differential conductance: Theory versus Experiment
0.38
Vg  9V
Vg  8.3V
dI/dVds
0.36
Vg  7.7V
0.34
0.32
0.30
measurement @ 4K
-20
-10
0
10
20
V ds (mV)
1.00
U 2  0.1 • damping of Fabry-Perot oscillation
amplitude at high bias voltage observed
0.95
U2  0
dI/dVds
0.90
U 2  0.1
0.85
U1  0.14
g  0.25
T  4K
0.80
0.75
0.70
-20
-10
0
Vds (mV)
10
20
L  360 nm
• clear gate voltage dependence of
FP-oscillation frequency
• From the first valley-to-valley distance
around Vds  0 we extract g ~ 0.22
Current noise
symmetric noise:
^
^
^
 I  I I
In terms of the generating functional:
Low-frequency limit of noise:
S  e | IB |
for eV  kBT
renormalization of charge absent due to finite size effect of interaction * !
What kind of signatures of interaction can we still see ?
Fano Factor: F  (S  2kBT G) / eI
* 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)
Asymptotic form of backscattered current
eV  L / g , kBT
reflection coefficient of charge :
I. Safi and H. Schulz ’95
• shot noise is well suited
to extract power-laws
in the weak backscattering regime
T 0
U1  0.1 g=0.23
U 2  0.1
Experimental Setup and Procedures
Vdc
Vac
+
*
-20V
#
RPD>>RCNT
LED
Signal
Gv
RCNT
Vdc
Vac
+
Cparasitic
CNT
VG
Resonant
Circuit
(
)2
Lock-In
DC
• 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 : F  S SWNT ( I ) / S PD ( I )
Comparison with experiments on low frequency shot noise
Power-law scaling
-0.4
log (Fano factor)
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
0.0
• PD=Shot noise of a photo diode
light emitting diode pair exhibiting full
shot noise serving as a standard shot
noise source.
S SP  Vds
1 / 2
S SWNT  V ds
  (1  g ) /(1  g )
with g~0.16 for
particular gate voltage shown and g~0.25 if
we average over many gate voltages.
0.5
1.0
1.5
log ( Vds )
Experimental Fano factor F (blue) compared
with theory for g~0.25 (red) and g~1(yellow).
F is compared with power-law scaling
F  Vds
 / 2
( red dashed line) giving
g~0.18 for this particular gate voltage.
In average over many gate voltages
we have g~0.22
0.35
Device : 13A2426
0.30
0.25
0.20
0.15
0.10
0
10
20
30
40
Vds (mV)
-0.4
-0.5
Blue: Exp
Yellow: g = 1
Red: g = 0.25
T=4K
U1  0.14 U 2  0.1
log (Fano factor)
Fano factor
Vg = - 7.9V
-0.6
-0.7
-0.8
-0.9
-1.0
0.0
0.5
log ( Vds )
1.0
1.5
Finite frequency impurity noise
Incoherent
part
frequency dependent conductivity
of clean wire

V

coherent
part
• depends on point of measurement
x
1 / 2
dominant at
large voltages
Frequency dependent conductance of clean SWNT+reservoirs
related to retarded function of total charge only !
•
xis assumed to be in the right lead
t F  L / vF and tv  t F g
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)
2


m 1, 2
0
( x, xm ;  )
2
g  0.23
1.5
independent of
x
not true for real part and
imaginary part of  0 ( x, xm ; )
1
0.5
5
10
15
20
25
30
35

(in units of
tF
1
)
• oscillations are due to backscattering of partial charges arising from inhomogeneous
g
Finite frequency excess noise for the non-interacting system
g=1
T=4K
3D plot of excess noise Se  S (V , )  S (0, ) in units of G0 L at T=4K for g=1 measured at barrier
as function of bias
(in units of L / e ) and frequency
(in units of  L )
v

eV  hvF / L
Excess noise as a function of 
at
=35 for U1  0.12, U 2  0.1
v
Excess noise as a function of
at   0
v
Signatures of spin-charge separation in the interacting system
g=0.23
Interacting levelspacing vF / Lg
and non-interacting levelspacing  vF / L
clearly distinguished in excess noise !
g from oscillation periods without any
fitting parameter
3D plot of excess noise Se  S (V , )  S (0, ) in units of G0 L at T=4K for g=0.23 measured at barrier
2 as function of bias
(in units of L / e ) and frequency
(in units of  L )

v
  2 / Tc
eV  hvF / L
Tc  2Lg / vF
charge roundtrip
time

Excess noise as a function of
at =35 for U1  0.12, U 2  0.1
v
Excess noise as a function of
at   0
Dependence of excess noise on measurement point
v =35
g=0.23
T=4K
d=0.14
d=0.3
v =35
b)
g=1
d  ( x / L)  0.5
d=0.6
Se

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