1. A New Understanding of the Tunneling Conductance Anomaly in Multi-Wall Carbon Nanotubes 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. G min does not occur strictly at zero-bias (deviation from zero-bias anomaly). V min is temperature-dependent. G vs.V curves exhibit asymmetry about V min. 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 ) 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

2. 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 one- dimensional  -orbital Hamiltonian supplemented by short-range/long- range 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), 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) Fermi-Liquid Caveat: A LL theory applies only to a true 1D system !!

3. Experimental Evidences A p ower-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)

4. 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  ~ 0.36 ‘Bulk-contacted’ Sample

5. Suppression of tunneling into multi-wall nanotubes Bachtold et al., PRL 87, 166801 (2001) Mceuen’s Group

6. Tunneling Conductance Results: MWCNT UofL Experiments – Chakraborty et al. Collapse of data onto A “ single curve” Power-law behavior G ~ T   ~ 0.2 VmVm V m shifts from 0.4 mV at 2.7 K to 1.2 mV at 20 K Asymmetry of the dip in G with respect to V min

7. Specific Features of the Experimental Results on MWCNTs – UofL experiments  G min 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 V min  V min depends on temperature. Do factors other than electron- electron correlations play a role in these observations ?

8. Tunneling Conductance Spectra of zigzag SWCNTs Ouyang et al. Science 292, 702 (2001) – Lieber’s Group Atomic Structure of “metallic” zigzag SWCNTs using STM Gap Calculated DOS A complete suppression of DOS Experiment Curvature effect ! Tunneling Conductance E g ~ 0.042 eV E g ~ 0.08 eV E g ~ 0.029 eV

9. 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 E g ~ 100 meV DOS suppressed but not reduced completely to zero at E f E g ~ 1/d “Pseudo-Gap” No Gap Calculated DOS of isolated ASWCNT An isolated tube has practically a constant DOS and no suppression at E f. (induced by tube-tube interaction?)

10. 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.

11. 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? An important clue from the experiment on ropes of ASWCNT

12. Lambin, Meunier, and Rubio – PRB 62, 5129 (2000) Intra-layer interactions Inter-layer interactions  -orbital tight-binding Hamiltonian for a MWCNT Theoretical Calculations

13. Tunneling Conductance

14. Numerically Fitted  s   s exhibits features that cannot be described by a power-law behavior in the vicinity of Fermi energy   s is asymmetric with respect to E F. DOS Calculated G (solid line) 2.7 K 20 K 4K 8K 12K 16K Tunneling conductance calculated (solid line) from numerically fitted  s is compared with the experimental G (points) The fitting of experimental conductance according to Eq. (1) can lead to a determination of the DOS of CNT samples of unknown compositions. Extracting the sample DOS

15. Calculation of DOS for a model MWCNT

16. 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.

17. 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 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 Outermost shell of the MWCNT (52,52) SWCNT

18. 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.

19. 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 inter- shell 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 (20- nm) as well as its composition.

20. 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

21.  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.

22. Numerical Fitting 1 2  s corresponding to different cases: A Summary Isolated SWCNT Outermost Shell 20 nm DOS of the Sample 7 nm MWNT Inter-shell interaction included Inter-shell interaction mimicked by scenario 1 and 2  s, respectively.

23. Schnonenberger Appl. Phys. (’99) Chakraborty et al. (UofL) Bachtold et al. PRL (2001) Scenarios #1 and #2 agree with the experiments of Schonenberger and Chakraborty, but disagree with that of Bachtold -- Why ?? Log-Log plots of G vs. T based on different scenarios for  s compared to 3-different experiments ( ) Scenario-1: Solid Scenario-2: dash Scenario-3: long-short dashes Scenario-2: dash Scenario-1: solid This discrepancy can be traced to the difference in the exponent  (0.2 vs. 0.36) The exponent and squeezing factors of pairs of vH peaks are related It depends on the composition of the MWCNT  exp ~ 0.2  exp ~ 0.36 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.

24. GT -  vs. eV/kT for different scenarios for  s Scenario -1 Scenario -2  = 0.19 Scenario-3 Collapse of all data into one universal curve, which is normally taken as the evidence for a Luttinger-Liquid behavior. However, we obtain such a result without invoking electron-electron correlations.  = 0.18  = 0.63

25. Inter-shell interaction seems to have provided the most consistent explanation for experimental observations on tunneling conductance anomaly in MWCNTs. Conclusion

26. Posters Energetics of Silicon Nanostructures on Si(111)-7x7 Surface using a Self-Consistent and Environment- Dependent 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

27. 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 tight- binding) by incorporating the inter-shell interactions in a MWCNT.