Resonant decoherence due to electron-electron interactions in carbon nanotubes

Recent experiments [B. Stojetz et al., Phys. Rev. Lett. 94, 186802 (2005)] show that weak localization in multiwall carbon nanotubes is suppressed at some values of the gate voltage. Using the one-dimensional formula for the conductance corrections due to weak localization, the corresponding phase coherence length was determined there as a function of the gate voltage. We consider the decoherence due to electron-electron scattering and calculate numerically the corresponding coherence time. We show that the coherence length becomes particularly short—i.e., comparable to or even shorter than the elastic mean free path due to scattering on imperfections (defects and∕or inner-shell-induced disturbances) at gate voltages corresponding to the onset of new populated subbands of the outermost nanotube shell. Suppression of the phase coherence length at these points with increasing gate voltage leads to suppression of the corresponding weak-localization corrections, which is consistent with the experimental observations.

Figure 1

(Color online) Electronic subband spectrum in the vicinity of the point $K$. Only the four lowest electron subbands are indicated, and the energy is expressed in the units of hopping integral $t$. The formula describes dispersion of the subbands. $E_F^{\left(0\right)}$ is the Fermi level in the absence of gate voltage. Generally, the position of the Fermi level can be controlled by the gate voltage.

Figure 2

Feynman diagram for the self-energy associated with the inelastic scattering rate due to electron-electron scattering. The solid and dashed lines correspond to the Green functions of electrons from the subbands ${\epsilon }_{0,k}$ and ${\epsilon }_{i,k}$, respectively. The dotted line represents the Coulomb interaction. In the notation used here, $\epsilon$ stands for energy, $i$ is the subband index, and $k$ is the wave vector.

Figure 3

(Color online) Inelastic scattering rate for individual subbands as a function of energy. The elastic relaxation rate $1∕\tau$ is also shown by the dashed line.

Figure 4

(Color online) (a) Inelastic scattering rate for the lowest subband calculated for the indicated temperatures. (b) Temperature dependence of $1∕{\tau }_0^{ee}$ at the first dominant peak $\mathbf{I}$. The hatched area below $T=5\mathrm{K}$ is not properly described by the present method.