Transcript Slide 1

A New Understanding of the Tunneling Conductance
Anomaly in Multi-Wall Carbon Nanotubes
L. Liu, S.Y. Wu, and C.S. Jayanthi
Dept. of Physics, University of Louisville
S. Chakraborty and B. Alphenaar
Dept. of Electrical and Computer Engineering
University of Louisville
This work was motivated by a recent experiment which reports subtle new features in
the suppression of the tunneling conductance (G) of multi-wall carbon nanotubes
(MWCNTs) in the vicinity of Fermi energy.
Gmin does not occur strictly at zero-bias (deviation from zero-bias anomaly).
Vmin is temperature-dependent.
G vs.V curves exhibit asymmetry about Vmin.
It is demonstrated that a theoretical calculation based on a π-orbital tight-binding
which includes inter-shell interaction can elucidate all the observed features of the
tunneling conductance anomaly in MWCNTs without invoking electron-electron
correlations.
Work Supported By the NSF and the U.S. DOE
(DMR-0112824, ECS-0224114, and DE-FG02-00ER45832)
Background
• Since carbon nanotubes are quasi-one-dimensional systems, it is
tempting to explain the anomalous transport properties of metallic carbon
nanotubes using a Luttinger-Liquid theory.
• A Luttinger-Liquid represents an interacting one-dimensional electron
system with a non-fermi liquid behavior, which is characterized by the
breakdown of the Landau quasi-particle picture, the opening of a small
charge/spin gap, and the suppression of electron tunneling density of
states with a power-law behavior.
• In fact, theoretical studies on isolated armchair SWCNTs based on a onedimensional -orbital Hamiltonian supplemented by short-range/longrange e-e interactions yield suppressed tunneling near the Fermi level
with a power law dependence of the conductance (G) on T at small bias
voltage V (eV<<kT) or on V at large biases (eV>>kT), a signature of the
Luttinger liquid (LL) behavior.
C. Kane, L. Balents, M. Fisher, PRL 79, 5086 (1997)
R. Egger and A.O. Gogolin, PRL 79, 5082 (1997)
Caveat: A LL theory applies only to a true 1D system !!
G  T
dI dV  V

Fermi-Liquid
 ~ 0.4
 0
Experimental Evidences
A power-law scaling of the conductance and differential conductance with
respect to T and V, respectively have been reported.
•
Ropes of SWCNTs - Bockrath et al., Nature 397, 598 (1999)
A suppression of G in the vicinity of zero-bias with power-law scaling of conductance
with respect to T (at zero-bias), or with respect to V at large biases (eV >> kT) have
been reported.
•
MWCNTs
(1) A. Bachtold et al., PRL 87, 166801-1 (2001)
(2) C. Schonenberger et al., Appl. Phys. A 69, 283 (1999)
(3) Chakraborty and Alphenaar (to be published)
Measured GT- vs. eV/kT for ropes of SWCNTs
Bockrath et al., Nature 397, 598 (1999)
• dI/dV at various
temperatures (1.6 K, 8K,
20K, 35 K)
• Power-law behavior at
large V
• Scaled conductance at
different
temperatures
fall onto a single curve
‘Bulk-contacted’
Sample
•  ~ 0.36
Suppression of tunneling into multi-wall nanotubes
Bachtold et al., PRL 87, 166801 (2001)
Mceuen’s Group
Tunneling Conductance Results: MWCNT
UofL Experiments – Chakraborty et al.
Asymmetry of the dip in G
with respect to Vmin
Power-law
behavior
Vm shifts from
0.4 mV at 2.7 K to
1.2 mV at 20 K
G ~ T
 ~ 0.2
Vm
Collapse of data onto
A “ single curve”
Specific Features of the Experimental
Results on MWCNTs – UofL experiments



Gmin does not exactly occur at zero-bias i.e.
there is deviation from the so-called zero-bias
anomaly (ZBA)
G vs. V curves are asymmetric about Vmin
Vmin depends on temperature.
Do factors other than electronelectron correlations play a role in
these observations ?
Tunneling Conductance Spectra
of zigzag SWCNTs
Ouyang et al. Science 292, 702 (2001) – Lieber’s Group
Atomic Structure of
“metallic” zigzag
SWCNTs using STM
A complete suppression
of DOS
Tunneling Conductance
Experiment
Eg ~ 0.08 eV
Gap
Calculated DOS
Eg ~ 0.042 eV
Eg ~ 0.029 eV
Eg  1 / d 2
Curvature effect !
Energy Gap of a (8,8) armchair SWNT in a rope/isolated tube
Atomically
resolved
images of an
(8,8) SWCNT
in a bundle
Atomically
resolved image
of an isolated
(8,8) tube on a
Au(111)
substrate
“Pseudo-Gap”
(induced by tube-tube
interaction?)
No Gap
Calculated DOS of
isolated ASWCNT
Eg ~ 100 meV
DOS suppressed but not
reduced completely to zero
at Ef
An isolated tube has
practically a constant
DOS and no
suppression at Ef .
Eg ~ 1/d
A Summary of all experimental evidences
• Metallic zigzag SWCNTs have energy gaps which vary
inversely proportional to the square of the radius, an
indication of the curvature effect.
• Isolated armchair SWCNTs do not have energy gaps.
• Armchair SWCNTs in ropes have pseudogaps.
• SWCNTs in ropes exhibit a suppression in the tunneling
density of states near the Fermi level.
• MWCNTs also exhibit a suppression in the tunneling
density of states near the Fermi level.
An important clue from the
experiment on ropes of ASWCNT
• Experimental evidences point to the fact that inter-tube
interactions is probably the reason for the appearance of
the pseudogap for the armchair SWCNT in a bundle
(mixing of π-π* bands due to breaking of rotational
symmetry in a bundle).
• The question we would like to pose is whether inter-shell
interactions can cause the suppression of the tunneling
density of states or tunneling conductance in MWCNTs?
Theoretical Calculations
-orbital tight-binding Hamiltonian for a MWCNT
 int ra layer  2.75eV
Intra-layer interactions
pp  HoppingT erm
 int er shell  W cos exp[( d   ) / L]
  Angle bet ween   orbitals
d  Distancebet ween coupled atoms
WAA / BB  0.36 eV ; WAB  0.16 eV
  0.334nm; L  0.045nm
Lambin, Meunier, and Rubio – PRB 62, 5129 (2000)
Inter-layer
interactions
Tunneling Conductance
dI
df ( E   eV )
   S ( E )
dE
dV
dE
 V  voltagedrop across thehighest impedancejunction
 S ( E )  sample DOS
Eq.(1)
Extracting the sample DOS
Numerically Fitted s
DOS
Calculated G (solid line)
20 K
16K
12K
8K
4K
 s exhibits features that cannot
2.7 K
be described by a power-law behavior
in the vicinity of Fermi energy
 s is asymmetric with respect to EF.
The fitting of experimental conductance
according to Eq. (1) can lead to a
determination of the DOS of CNT samples
of unknown compositions.
Tunneling conductance
calculated (solid line) from
numerically fitted s is
compared with the
experimental G (points)
Calculation of DOS for a model MWCNT
Calculation of DOS for a model MWCNT
•
A typical MWCNT of diameter 20 nm will be composed of 30 ~ SWCNT
shells (~ one third of them will be metals)
•
However, we will consider a 10-wall MWCNT with its configuration given
by: (7,7)@(12,12)@….(47,47)@(52,52) with a diameter of ~ 7 nm.
•
The MWCNT thus constructed is commensurate along the tube axis.
•
However, there is no commensurability along the circumferencial direction
of MWCNTs, thus allowing disorder in that direction.
• We calculate the local density of states (LDOS) using the -orbital
Hamiltonians with intra-layer as well as inter-shell interactions.
•
Examine the LDOS for the outermost shell.
DOS Results for the outermost shell of the MWCNT compared
with an isolated SWCNT of the same type as the outer shell
(7,7)@(12,12)@......@(52,52)
Diameter ~ 7 nm
Outermost
shell of the MWCNT
(52,52) SWCNT
• This comparison highlights the effect of inter-shell interaction
• When the inter-shell interaction is turned-on, the level-level repulsion pushes the
pairs of vH peaks above and below the Fermi-level closer together, leading to
squeezing of vH pairs and fine structures in the DOS.
• The asymmetric squeezing of vH pairs is due to different degree of squeezing
for the bonding and anti-bonding states
Effect of Inter-Shell Interaction
The first pair of vH peaks is squeezed by a factor of ~ 7,
the second pair by a factor of ~ 3, the third pair by a factor
of ~ 2.5, etc for the outermost (52,52) shell of the 7 nm
MWCNT with respect to the corresponding vH pairs of the
isolated SWCNT.
Modeling the DOS of a MWCNT of diameter ~ 20 nm :
Scenario 1
Since it is impossible to calculate the LDOS of the outermost
shell of a typical MWCNTof diameters ~ 20 nm once the intershell interaction is turned on, we design different schemes to
capture the effect of inter-shell interaction, which place
emphasis on different aspects of inter-shell interactions.
Scenario #1: The DOS of scenario-1 is constructed based on
the LDOS of the outermost shell of the 10-shell MWCNT (d ~ 7
nm) but scaled down by a factor of ~10 to reflect the
experimental sample both in terms of its larger diameter (20nm) as well as its composition.
Modeling the DOS of MWCNTs of Diameters ~
20 nm : Scenario 2
Construct the LDOS of the outermost shell using the average DOSs
of three SWCNTs (151,144),(150,145), and (149,146) with diameters
of ~ 20 nm
To mimic the effect of inter-shell interaction, apply the same squeeze
factors to vH pairs, namely, the first pair by a factor of 7, the second
pair by a factor of 3, and so on .., as obtained for the 10-wall
MWCNT.
However, such a scaling-down of the vH-pair separations will not
capture the asymmetric shift of vH peaks associated with different
degrees of squeezing for bonding and anti-bonding states
s for different Scenarios
Scenario #1
Scenario 2
s for scenario # 1 is asymmetric while that for
scenario #2 is symmetric. This is because there is
no explicit inclusion of inter-shell interaction in
scenario #2.
s corresponding to different cases: A
Summary
Numerical Fitting
DOS of the
Sample
2
Inter-shell interaction
mimicked by scenario 1
and 2 s, respectively.
20 nm
1
Isolated SWCNT
Outermost Shell
7 nm MWNT
Inter-shell
interaction
included
Log-Log plots of G vs. T based on different scenarios for
s compared to 3-different experiments ( )
Scenario-1: Solid
Scenario-2: dash
Schnonenberger
Appl. Phys. (’99)
 exp ~ 0.2
This discrepancy can be traced to
the difference in the exponent 
(0.2 vs. 0.36)
Scenario-2: dash
Scenario-1: solid
Chakraborty
et al. (UofL)
exp~ 0.2
Scenario-3:
long-short dashes
Scenarios #1 and #2 agree with
the experiments of
Schonenberger and Chakraborty,
but disagree with that of
Bachtold -- Why ??
Bachtold et al.
PRL (2001)
exp~ 0.36
The exponent and squeezing
factors of pairs of vH peaks are
related
It depends on the composition of
the MWCNT
Scenario #3: It is obtained by squeezing
the vH pairs of scenario 2 DOS by a
factor of 12 to account for a different
composition of the MWCNT sample.
GT- vs. eV/kT for different scenarios for s
Scenario -2
• Collapse of all
data into one
universal curve,
which is normally
taken as the
evidence for a
Luttinger-Liquid
behavior.
Scenario-3
• However, we
obtain such a
result without
invoking
electron-electron
correlations.
Scenario -1
 = 0.18
 = 0.19
 = 0.63
Conclusion
Inter-shell interaction seems to have
provided the most consistent explanation
for
experimental
observations
on
tunneling conductance anomaly in
MWCNTs.
Posters
• Energetics of Silicon Nanostructures on Si(111)-7x7
Surface using a Self-Consistent and EnvironmentDependent Hamiltonian
M.Yu, S.Y. Wu, and C.S. Jayanthi
• First-Principles calculation of the electronic
properties of Potassium-covered Carbon Nanotubes
Alex Tchernatinsky, G. Sumanasekera, S.Y. Wu, and
C.S. Jayanthi
A new and alternative understanding of the
tunneling conductance anomaly in MWCNTs
We will demonstrate that all the features associated with
the
suppression
of
tunneling
conductance,
those
previously reported as well as the new features observed
by Chakraborty et al., may be succinctly explained within
the framework of a one-electron theory (π-orbital tightbinding) by incorporating the inter-shell interactions in a
MWCNT.