Binding and spontaneous condensation of excitons in narrow-gap carbon nanotubes

Abstract

Ultraclean, undoped carbon nanotubes are observed to be always insulating, even when the gap predicted by band theory is zero: the residual band gap is then thought to have a many-body origin. Here we theoretically show that the correlated insulator is excitonic in $all stable$ narrow-gap tubes irrespective of their size, thus extending our previous claim, limited to gapless (armchair) tubes [D.~Varsano, S.~Sorella, D.~Sangalli, M.~Barborini, S.~Corni, E.~Molinari, M.~Rontani, Nature Communications $\mathbf{8}$, 1461 (2017)]. We derive the scaling law of the exciton binding energy with the tube radius and chirality, and compute self-consistently the fundamental transport gap of the excitonic insulator, by enhancing the two-band model with an accurate treatment of screening validated from first principles. Our findings point to the broader connection between the exciton length scale, dictated by structure, and the stability of the excitonic phase.

Figures (5)

• Figure 1: Attractive Coulomb force and exciton wave function. (a) Statically screened Coulomb interaction, $W(k)$, vs momentum parallel to the axis, $k$ (in reciprocal lattice units, with $a=$ 2.46 Å). The dressed potential $W$ is computed within either an accurate model for the nanotube screening, validated from first principles and including local fields Sesti2022 (CNT, solid line), or the effective mass approximation (EM, dashed line). (b) Exciton wave function, $\left|\psi(k)\right|^2$. The axis origin is the Dirac point. (c) Interband coherence of the excitonic insulator, $\left|\Delta(k)\right|$, for the (18,0) zigzag and (10,10) armchair tubes, which are respectively gapped and gapless.
• Figure 2: Exciton binding energy, $E_B$, vs radius, $R$, for narrow-gap carbon tubes. The chiral angle, $\theta$, varies between $\theta=0$ (zigzag tubes) and $\theta=\pi/6$ (armchair). Armchair tubes are gapless due to symmetry. The log-log plot (inset) shows that $E_B\sim R^s$, with $s=-2.54$ and $-1.53$$(\pm 0.07)$ for chiral and armchair tubes, respectively. The calculations were done including only the regions of the Brillouin zone around the two Dirac points with an extension 0.1 $\times 2 \;\pi/a$, each region discretized with 41 $k$ points. The results have proved equivalent to selected calculations on the full Brillouin zone with 801 $k$ points.
• Figure 3: Ratio of exciton binding energy, $E_B$, to quasiparticle gap, $\varepsilon_{g}$, as a function of radius, $R$, for representative chiralities. The dashed line indicates the threshold for the excitonic instability.
• Figure 4: Contour map of (a) interband coherence at Dirac point, $\Delta(k=0)$ (b) fundamental gap of the excitonic insulator, $E_{g}$, in the $(R,\theta)$ space.
• Figure 5: Scaling properties of tubes of very large radius. (a) Ratio of the exciton binding energy, $E_B$, to quasiparticle gap, $\varepsilon_{g}$, as a function of radius, $R$, for chiral angle $\theta=0$ (dark dots) and $\theta=15^{\circ}$ (light dots). The dashed line indicates the threshold for excitonic instability. Inset: Scaling of $E_B$ (log-log plot). The lines correspond to the scaling exponent $s=-2.54$, with $E_B\sim R^{s}$. (b) Comparison in semilog scale of the renormalized, $\varepsilon_g$, and bare curvature gap, $\varepsilon_g^{\text{bare}}$, for zigzag tubes ($\theta=0$). The blue and orange dashed lines correspond to the scaling $R^{-2.5}$ and $R^{-2}$, respectively ($\varepsilon_g^{\text{bare}}\sim 1/R^2$).