Phase diagram of carbon nanotube ropes

The zero-temperature phase diagram of carbon nanotube ropes is studied using a computational framework that incorporates the renormalization of intratube interactions and the effect of intertube Coulomb screening. This allows us to undertake a systematic analysis as a function of the number of metallic nanotubes in the rope and the effective strength of the phonon-mediated interaction. We find that there is in general a weak-coupling regime of the interactions, corresponding to Luttinger-liquid behavior in thin ropes and to superconducting behavior in sufficiently thick ropes. Furthermore, we show the existence of exotic phases in the strong-coupling regime, characterized by the appearance of charge instabilities that depend on the helicity and the doping level of the nanotubes in the rope. Our approach allows for the simultaneous analysis of the scaling of the Cooper-pair tunneling amplitude between metallic nanotubes, making it possible to discern the crossover from purely one-dimensional physics to the setting of three-dimensional Cooper-pair coherence. We provide then good estimates of the superconducting transition temperature and discuss the connection of our results with recent experiments.

Figure 1

Scheme of the low-energy linear branches of armchair nanotubes (upper figure) and metallic zigzag nanotubes (lower figure) crossing at opposite momenta $k_F$ and $-k_F$. The spinors correspond to the relative electron amplitudes in the two sublattices of graphene rolled up to form the nanotube. The labels 1 and 2 at each linear branch are used to identify the corresponding subband according to the symmetry of the spinors.

Figure 2

Small momentum-transfer processes corresponding to the couplings $g_4^{\left(4\right)}$ and $g_2^{\left(4\right)}$.

Figure 3

Small momentum-transfer processes corresponding to the couplings $g_2^{\left(2\right)}$ and $g_4^{\left(2\right)}$.

Figure 4

The thick solid (dashed) line corresponds to the plot of $l=\ln (E_c∕{\omega }_0)$, where ${\omega }_0$ is the energy scale at which the parameter $K_{-}$ diverges in doped armchair (zigzag) nanotubes, as a function of the effective strength $G=4\mid g\mid ∕\pi v_F$ of the phonon-mediated interaction. The lower solid (dashed) curve represents the behavior of the $K_{+}$ parameter in doped armchair (zigzag) nanotubes.

Figure 5

(a) Diagrams contributing to the screening of Coulomb interactions between currents $l_a$ within the same nanotube. The currents have well-defined chirality and spin. The last term takes into account the polarization of the rest of $n-1$ metallic nanotubes in the rope. The prime stands for the exclusion $c\ne a$ in the summation. (b) Diagrams contributing to the screening of Coulomb interactions between currents $l_a$ and $l_b$ belonging to different metallic nanotubes. The prime stands for the exclusion $c\ne a,b$ in the summation.

Figure 6

Scale dependence of the coupling constants for $4\mid g\mid ∕\pi v_F=0.3$ in a bundle with $n=200$ undoped metallic zigzag nanotubes. For these parameters the system is in the SC state with a critical frequency ${\omega }_c$ shown in the figure. At that point, the scaling stops due to the opening of the SC single-particle gap. Here we have continued the flow to show that the system at ${\omega }_c$ is well inside the weak-coupling regime. The crossover to the strong-coupling regime would take place at a lower frequency ${\omega }_{xo}$.

Figure 7

Dependence of the critical frequency in a bundle with $n$-undoped metallic zigzag nanotubes for $4\mid g\mid ∕\pi v_F=0.3$ (solid), 0.4 (dashed), and 0.5 (dotted). The increase of ${\omega }_c$ with $n$ marks the setting of the SC phase. That kind of behavior, also observed in experimental samples, is characteristic of weak-coupling superconductivity (Ref. 14).

Figure 8

Phase diagram of undoped (a) and doped (b) armchair nanotubes as a function of the number of metallic nanotubes $n$ in the rope and the strength of the attractive interaction $G=4\mid g\mid ∕\pi v_F$. Thick lines represent the phase boundaries between a 3D phase-coherent SC phase and the 1D phases characterized by the breaking of the parameters $K_{+}$ and $K_{-}$. Thin lines are contours of constant critical frequency ${\omega }_c$, starting from above with $\ln (E_c∕{\omega }_c)=2,4,6,\dots$.

Figure 9

Phase diagram of undoped (a) and doped (b) zigzag nanotubes as a function of the number of metallic nanotubes $n$ in the rope and the strength of the attractive interaction $G=4\mid g\mid ∕\pi v_F$. Thick and thin lines have the same meaning as in Fig. 8. In the case of doped nanotubes, the factor $\delta$ of reduction of the effective interactions in Table has been set to 0.1. The narrow phase to the left of diagram (b) corresponds to the divergence of the $K_{-}$ parameter.