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

Abstract

Recent transport measurements [Churchill et al. Nature Phys. 5, 321 (2009)] found a surprisingly large, 2-3 orders of magnitude larger than usual 13 C hyperfine coupling (HFC) in 13 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. DOI: 10.1103/PhysRevLett.107.187204 PACS numbers: 75.75.Àc, 71.10.Pm, 73.63.Àb, 74.25.nj Albeit small, the electron-nuclear hyperfine coupling (HFC) is the dominant mechanism in physical phenomena which are key to, e.g., nuclear quantum computing [1], magnetic resonance spectroscopy [2], and it plays a fundamental role in spintronics [3]. The HFC is due to the magnetic interaction between the nucleus and electrons, with a number of different mechanisms such as the Fermi contact, spin-dipole, core-polarization, orbital, and transferred HFC [4].It is generally accepted that the HFC does not change more than an order of magnitude for different environments of an atom [5]. Typical values for the 13 C HFC are 1-5 Â 10 À7 eV [6-8] with a largest known value of 1:8 Â 10 À6 eV in an organic free radical [9].It therefore came as a surprise that transport experiments [10] on a double quantum dot formed of 13 C enriched single-wall carbon nanotubes (SWCNTs) found a HFC,À4 eV, which is 2-3 orders of magnitude larger than measured for C 60 [7] or calculated for graphene [8], which are similar carbonaceous nanostructures. In Ref.[10], a theory developed for GaAs quantum dots [11] was used to analyze the data, which has some shortcomings. First, the HFC in GaAs is 2-3 orders of magnitude stronger than the usual value in carbon. Second, both Ga and As have a nearly isotropic HFC [12] whereas the anisotropic HFC usually dominates for carbon [7]. Third, SWCNTs possess an extra degree of freedom, the so-called valley degeneracy, which may lead to distinct behavior of the QD transport properties [13]. Fourth, the particular one dimensionality of SWCNTs may affect the analysis. Clearly, settling the issue calls for an analysis which yields the HFC directly from magnetic resonance spectroscopy.NMR measurement of the 13 C spin-lattice relaxation time, T 1 , in SWCNTs can give directly the HFC and such results were reported in Refs. [14][15][16][17]. However, the analysis requires care since the Fermi-liquid (FL) theory for T 1 does not apply in the SWCNTs as the low energy excitations in metallic SWCNTs are described by the Tomonaga-Luttinger liquid (TLL) framework [18][19][20][21]. The TLL is an exotic correlated state [22,23] and yet SWCNTs offer its best realization. So far the...

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“…37 ), Ξ(n, m) = √ 3/2 √ n 2 + nm + m 2 and C(K) = sin(πK)Γ(1 − K)Γ 2 (K/2)/2. The dotted lines in Fig.…”

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