Hyperfine interaction and electron-spin decoherence in graphene and carbon nanotube quantum dots

We analytically calculate the nuclear-spin interactions of a single electron confined to a carbon nanotube or graphene quantum dot. While the conduction-band states in graphene are p-type, the accordant states in a carbon nanotube are sp-hybridized due to curvature. This leads to an interesting interplay between isotropic and anisotropic hyperfine interactions. By using only analytical methods, we are able to show how the interaction strength depends on important physical parameters, such as curvature and isotope abundances. We show that for the investigated carbon structures, the ¹³C hyperfine coupling strength is less than 1 μeV, and that the associated electron-spin decoherence time can be expected to be several tens of microseconds or longer, depending on the abundance of spin-carrying ¹³C nuclei. Furthermore, we find that the hyperfine-induced Knight shift is highly anisotropic, both in graphene and in nanotubes of arbitrary chirality.