Energy band for single wall carbon nanotubes with spin-orbit interaction is calculated using nonorthogonal tight-binding method. A Bloch function with spin degree of freedom is introduced to adapt the screw symmetry of nanotubes. The energy gap opened by spin-orbit interaction for armchair nanotubes, and the energy band splitting for chiral and zigzag nanotubes are evaluated quantitatively. Spin polarization direction for each split band is shown to be parallel to the nanotube axis. The energy gap and the energy splitting depend on the diameter and chirality in an energy scale of sub-milli-electron volt. An effective model for reproducing the low energy band structure shows that the two mechanism of the band modification, shift of the energy band in two dimensional reciprocal lattice space, and, effective Zeeman energy shift, are relevant. The effective model explains well the energy gap and splitting for more than 300 nanotubes within the diameter between 0.7 to 2.5 nm.

Tunneling conductance through two quantum dots, which are connected in series to left and right leads, is calculated by using the numerical renormalization group method. As the hopping between the dots increases from very small value, the following states continuously appear; (i) Kondo singlet state of each dot with its adjacent-site lead, (ii) singlet state between the local spins on the dots, and (iii) double occupancy in the bonding orbital of the two dots. The conductance shows peaks at the transition regions between these states. Especially, the peak at the boundary between (i) and (ii) has the unitarity limit value of 2e 2 /h because of coherent connection through the lead-dot-dotlead. For the strongly correlated cases, the characteristic energy scale of the coherent peak shows anomalous decrease relating to the quantum critical transition known for the two-impurity Kondo effect. The two dots systems give the new realization of the two-impurity Kondo problem.

The tunneling conductance is calculated as a function of the gate voltage in wide temperature range for the single quantum dot systems with Coulomb interaction. We assume that two orbitals are active for the tunneling process. We show that the Kondo temperature for each orbital channel can be largely different. The tunneling through the Kondo resonance almost fully develops in the region T < ∼ 0.1T * K ∼ 0.2T * K , where T * K is the lowest Kondo temperature when the gate voltage is varied. At high temperatures the conductance changes to the usual Coulomb oscillations type. In the intermediate temperature region, the degree of the coherency of each orbital channel is different, so strange behaviors of the conductance can appear. For example, the conductance once increases and then decreases with temperature decreasing when it is suppressed at T = 0 by the interference cancellation between different channels. The interaction effects in the quantum dot systems lead the sensitivities of the conductance to the temperature and to the gate voltage. KEYWORDS: quantum dot, tunneling, Kondo effect, Coulomb oscillations §1. IntroductionCoulomb oscillations behavior in quantum dot systems is well known as one of the typical phenomena due to the Coulomb repulsion between electrons in the dot. But it is one aspect of the electron-electron interaction effects. Local spin moment is induced by the Coulomb repulsion in the dot sometime. It will fluctuate by exchanging spin moment with leads and thus cause the strong inelastic scattering. But the local spin disappears in the lowest temperature due to the spin singlet formation between leads and the dot. These effects have been investigated as the Kondo effect for the magnetic impurity systems in many years, 1) and for the quantum dot systems recently. 2,

We investigate the effect of mechanical strain on graphene synthesized by chemical vapor deposition (CVD) transferred onto flexible polymer substrates by observing the change in the Raman spectrum and then compare this to the behavior of exfoliated graphene. Previous studies into the effect of strain on graphene have focused on mechanically exfoliated graphene, which consists of large single domains. However, for wide scale applications CVD produced films are more applicable, and these differ in morphology, instead consisting of a patchwork of smaller domains separated by domain boundaries. We find that under strain the Raman spectra of CVD graphene transferred onto a silicone elastomer exhibits unusual behavior, with the G and 2D band frequencies decreasing and increasing respectively with applied strain. This unusual Raman behavior is attributed to the presence of domain boundaries in polycrystalline graphene causing unexpected shifts in the electronic structure. This was confirmed by the lack of such behavior in mechanically exfoliated large domain graphene and also in large single-crystal graphene domains grown by CVD. Theoretical calculation of G band for a given large shear strain may explain the unexpected shifts while the shift of the Dirac points from the K point explain the conventional behavior of a 2D band under the strain.