Modulation of Luttinger liquid exponents in multiwalled carbon nanotubes

We develop in this paper a theoretical framework that applies to the intermediate regime between the Coulomb blockade and the Luttinger liquid behavior in multiwalled carbon nanotubes. Our main goal is to confront the experimental observations of transport properties, under conditions in which the thermal energy is comparable to the spacing between the single-particle levels. For this purpose we have devised a many-body approach to the one-dimensional electron system, incorporating the effects of a discrete spectrum. We show that, in the crossover regime, the tunneling conductance follows a power-law behavior as a function of the temperature, with an exponent that oscillates with the gate voltage as observed in the experiments. Also in agreement with the experimental observations, a distinctive feature of our approach is the existence of an inflection point in the log-log plots of the conductance vs temperature, at gate voltages corresponding to peaks in the oscillation of the exponent. Moreover, we evaluate the effects of a transverse magnetic field on the transport properties of the multiwalled nanotubes. For fields of the order of $4\mathrm{T}$, we find changes in the band structure that may be already significant for the outer shells, leading to an appreciable variation in the power-law behavior of the conductance. We then foresee the appearance of sizeable modulations in the exponent of the conductance for higher magnetic fields, as the different subbands are shifted towards the development of flat Landau levels.

Figure 1

Plot of $N_T\left({\epsilon }_F\right)$ for three different values of $kT∕\Delta \epsilon$: $kT∕\Delta \epsilon =0.1$ (dashed), $=0.15$ (solid), $=0.3$ (dotted-dashed). The value of $\epsilon$ is given in units of $\Delta \epsilon$

Figure 2

Self-consistent diagrammatic equation for the dressed Coulomb interaction.

Figure 3

Set of diagrams for the computation of the electron self-energy, where each particle-hole bubble represents the full polarization operator.

Figure 4

Plot of the exponent $\alpha$ as a function of the product of the number of subbands $N_s$ and the weight $N_T$ in the single-particle spectrum.

Figure 5

Energy dispersion relation of a (280,0) zigzag nanotube with $B=0$ (solid lines) and $B=4\mathrm{T}$ (dashed lines). The position of the Fermi level for a hole-doped nanotube is also indicated. The energy appears in units of eV (we used a hopping parameter $t=2.5\mathrm{eV}$ between nearest-neighbor carbon atoms) and the momentum is given in ${\mathrm{\AA }}^{-1}$).

Figure 6

(Color online) Energy dispersion relation of a metallic, e.g., armchair, nanotube with radius $\sim 10\mathrm{nm}$, for $B=0$ (solid lines) and $B=4\mathrm{T}$ (dashed lines). The energy appears in units of eV and the momentum is given in ${\mathrm{\AA }}^{-1}$ and is relative to one of the two Fermi points.

Figure 7

Plot of the exponent $\alpha$ as a function of the Fermi energy ${\epsilon }_F$ (in units of $\Delta \epsilon$) for $B=0$ (solid line) and $B=4\mathrm{T}$ (dashed line). Here the temperature is $kT∕\Delta \epsilon =0.15$. The system is the same as in Fig. 5.