Enhanced NMR Relaxation of Tomonaga-Luttinger Liquids and the Magnitude of the Carbon Hyperfine Coupling in Single-Wall Carbon Nanotubes

Recent transport measurements [Churchill et al. Nature Phys. 5, 321 (2009)] found a surprisingly large, 2–3 orders of magnitude larger than usual ${}^{13}\mathrm{C}$ hyperfine coupling (HFC) in ${}^{13}\mathrm{C}$ enriched single-wall carbon nanotubes. We formulate the theory of the nuclear relaxation time in the framework of the Tomonaga-Luttinger liquid theory to enable the determination of the HFC from recent data by Ihara et al. [Europhys. Lett. 90, 17 004 (2010)]. Though we find that $1/T_1$ is orders of magnitude enhanced with respect to a Fermi-liquid behavior, the HFC has its usual, small value. Then, we reexamine the theoretical description used to extract the HFC from transport experiments and show that similar features could be obtained with HFC-independent system parameters.

Figure 1

The temperature normalized NMR relaxation rate in SWCNTs at 3 T (symbols) from Ref. 17 fitted with the TLL model (solid line). The relaxation rate calculated in a Fermi-liquid picture, $(T_{1}T{)}_{\mathrm{FL}}^{-1}$ (dashed line), is shown multiplied by a factor of 250.

Figure 2

(a) Schematics of a DQD; the contact leads (grey areas) and the internal hyperfine fields in the left and right dots are indicated. (b) Energy levels for the DQD. Note that the tunneling parameter couples the on-site $S$ and two-site $S$ states and results in the singlet-triplet splitting.