Spin Gap and Luttinger Liquid Description of the NMR Relaxation in Carbon Nanotubes

Abstract

Recent NMR experiments by Singer et al. [Singer et al. Phys. Rev. Lett. 95, 236403 (2005).] showed a deviation from Fermi-liquid behavior in carbon nanotubes with an energy gap evident at low temperatures. Here, a comprehensive theory for the magnetic field and temperature dependent NMR 13 C spin-lattice relaxation is given in the framework of the Tomonaga-Luttinger liquid. The low temperature properties are governed by a gapped relaxation due to a spin gap (∼ 30 K), which crosses over smoothly to the Luttinger liquid behaviour with increasing temperature.PACS numbers: 71.10. Pm,71.20.Tx, Low dimensional carbonaceous systems, fullerenes, carbon nanotubes (CNTs), and graphene display a rich variety of exotic states and strongly correlated phenomena. These include superconductivity in alkali doped fullerenes 1 , quantized transport in singlewall carbon nanotubes (SWCNTs), and massless Dirac quasi-particles showing a halve integer quantum Halleffect in graphene even at room temperature. A compelling correlated state of one-dimensional systems is the Tomonaga-Luttinger liquid (TLL) state. The TLL state has been suggested to describe the low energy properties of CNTs with a single shell, the single-wall carbon nanotubes 2,3,4,5,6 . Transport 7 and photoemission studies 8,9 provided evidence for the existence of the TLL state in SWCNTs. In these studies, power-law behavior of temperature and bias dependent conductivity and a power-law Fermi edge was observed, respectively.Nuclear magnetic resonance (NMR) is a powerful method to characterize correlated states of materials as it is sensitive to the density of states near the Fermi edge. For a material with a Fermi-liquid state, the temperature dependent spin-lattice relaxation time, T 1 follows the so-called Korringa temperature dependence for which 1/T 1 T is constant. Recently, 13 C enriched SWCNTs were grown inside carbon nanotubes from 13 C enriched fullerenes 10 . This allowed a high precision measurement of T 1 in small diameter SWCNTs by Singer et al. 11 . A tentative fit of the experiments with a gapped Fermi liquid type density of states (DOS) indicated overall agreement but obvious discrepancies in detail. When the magnetic field and temperature dependent data for T 1 were fitted with this phenomenological model a gap at the Fermi surface with 2∆ ≃ 40 K opened already above room temperature, i.e. its T c is larger than 300 K. This strongly violates the 2∆/k B T > 3.52 relation, thus simple mean field theories are not applicable 12 . Also, the phenomenological description can not account for the strong overshoot of 1/T 1 T when T approaches the gap.Here, we analyze the NMR results in the framework of the Luttinger liquid and Luther-Emery liquid pictures (interacting one-dimensional electrons without and with a gap, respectively). At high temperatures, the former dominates, while the latter accounts for the dominance of a spin gap at low temperatures. We show that the temperature and magnetic field dependent T 1 of 13 C can be explained in the TLL scenario wit...

Potassium-intercalated single-wall carbon nanotube bundles: Archetypes for semiconductor/metal hybrid systems

“…9 The TLL and its characteristic power-law scaling have been confirmed by transport studies, 10 VB PES, 8 and nuclear magnetic resonance ͑NMR͒. 11 Alkali-metal intercalation has been demonstrated to shift the Fermi level into the conduction band ͑CB͒ of graphite. 12 In this way the CB becomes accessible to VB methods such as angle-resolved PES.…”

Temperature dependence of the NMR spin-lattice relaxation rate for spin-12chains

“…(9) and (10)] the time integral of the real part of these correlations from t = 0 to t = +t 0 is required. Numerical results are taken at discrete times which are multiples of the time step δt chosen for the real-time evolution within tMPS.…”

Crossover from Peierls distortion to one-dimensional superconductivity in arrays of (5,0) carbon nanotubes

“…[24][25][26][27] For the case of ultrathin diameter 4-Angstrom metallic SWCNTs, experimental observation of the Meissner effect in 2001 1 has prompted some theoretical works exploring the possibility of superconductivity being the ground state of the system. While some authors predicted Peierls ground state, [28][29][30][31][32] others predicted superconductivity. [33][34][35][36][37][38][39][40] More recently, the observation of the superconducting electrical transition for arrays of aligned 4-Angstrom nanotubes (embedded in AFI zeolite crystals) has shown that the resistance drop is precipitated by a BerezinskiiKosterlitz-Thouless (BKT)-like transition [41][42][43] in the plane perpendicular to the c-axis of the nanotubes, thereby facilitating a 1D to 3D crossover that leads to bulk superconductivity [2][3][4] at low temperatures.…”