RKKY interaction in carbon nanotubes and graphene nanoribbons

We study Rudermann-Kittel-Kasuya-Yosida (RKKY) interaction in carbon nanotubes (CNTs) and graphene nanoribbons in the presence of spin-orbit interactions and magnetic fields. For this, we evaluate the static spin susceptibility tensor in real space in various regimes at zero temperature. In metallic CNTs, the RKKY interaction depends strongly on the sublattice and, at the Dirac point, is purely ferromagnetic (antiferromagnetic) for the localized spins on the same (different) sublattice, whereas in semiconducting CNTs, the spin susceptibility depends only weakly on the sublattice and is dominantly ferromagnetic. The spin-orbit interactions break the SU(2) spin symmetry of the system, leading to an anisotropic RKKY interaction of Ising and Dzyaloshinskii-Moriya form, aside from the usual isotropic Heisenberg interaction. All these RKKY terms can be made of comparable magnitude by tuning the Fermi level close to the gap induced by the spin-orbit interaction. We further calculate the spin susceptibility also at finite frequencies and thereby obtain the spin noise in real space via the fluctuation-dissipation theorem.

Figure 1

The Dirac spectrum of a metallic nanotube. Each branch is characterized by the isospin value $\sigma$, which is an eigenvalue of the Pauli matrix ${\sigma }_2$. We note that the Kramers partners at $K$ and $K^{\prime }$, respectively, are characterized by the same value of the isospin, and two partner states at the same cone are characterized by opposite isospins.

Figure 2

The spin susceptibility ${\chi }_{\alpha \beta }$, given in Eqs. (44, 45, 46), plotted as a function of distance between localized spins $z$, for a semiconducting (11,1) CNT in the presence of spin-orbit interaction and at zero B field. Here, ${\chi }_0=a^2{k}_G/2\pi \hslash {\upsilon }_F$. For clarity, we suppress the fast oscillating factors and plot only the slowly varying envelopes at small [(a), (c)] and large scales [(b), (d)]. The chemical potential $\mu =472\mathrm{meV}$ ($\delta \mu \equiv \mu -\hslash {\upsilon }_F{k}_G+{\beta }_{+}=1\mathrm{meV}$) is tuned inside the gap opened by SOI with corresponding value $2{\beta }_{+}=1.7\mathrm{meV}$ for a (11,1) CNT (Ref. 34). The diagonal components ${\chi }_{zz}$ [(a), (b) blue full curve] and ${\chi }_{xx}$ [(c), (d)] oscillate with period $2\pi /k_{+,-}\approx 2\pi /k_F$. In contrast to that, the off-diagonal component ${\chi }_{xy}$ [(a), (b) red dashed curve] oscillates with period $2\pi /k_{+,+}\approx \pi /k_F$. All components decay as $1/z$. Note that the diagonal and off-diagonal components are of comparable size.

Figure 3

The same as Figs. 2a and 2b but with the chemical potential $\mu =474\mathrm{meV}$ being tuned above the gap opened by SOI. The spin susceptibility decays as $1/z$ and exhibits beatings, with period determined by the SOI parameters.

Figure 4

The same as Figs. 2a and 2b but with the chemical potential $\mu =472.7\mathrm{meV}$ being tuned in such a way that there are three pairs of states at the Fermi level and with a magnetic field $B=1\mathrm{T}$. The spin susceptibility decays as $1/z$ and exhibits beatings, with period determined by the SOI parameters and the magnetic field.