Indirect exchange coupling between magnetic adatoms in carbon nanotubes

The long-range character of the exchange coupling between localized magnetic moments indirectly mediated by the conduction electrons of metallic hosts can play a significant role in determining the magnetic order of low-dimensional structures. Here we consider how this indirect coupling influences the magnetic alignment of adatoms attached to the walls of carbon nanotubes. A general expression for the indirect coupling in terms of single-particle Green functions is presented. Contrary to the general property that magnetic moments embedded in a metal display Friedel-like oscillations in their magnetic response, calculated values for the coupling across metallic zigzag nanotubes show monotonic behavior as a function of the adatom separation. Rather than an intrinsic property, the monotonicity is shown to reflect a commensurability effect in which the coupling oscillates with periods that coincide with the lattice parameter of the nanotube host. Such a commensurability effect does not dominate the coupling across semiconducting zigzag or metallic armchair nanotubes. We argue that such a long-range character in the magnetic interaction can be used in future spintronic devices.

Figure 1

(Color online) Schematic representation of the magnetic adatoms on a carbon nanotube.

Figure 2

Exchange coupling between Co adatoms as a function of the adatom separation on a zigzag (6,0) nanotube (solid line). The dashed line is the coupling when the nanotube is slightly doped, corresponding here to a small shift in its Fermi energy. The inset shows schematically the Fermi wave vector for pure (solid arrow) and doped (dashed arrow) tubes. All coupling values are given in units of the nearest-neighbor electronic hopping $\gamma$.

Figure 3

Exchange coupling as a function of longitudinal separation $D$ between Ni adatoms on (6,0) (solid line), (9,0) (dashed line), and (12,0) (dot-dashed line) nanotubes. The inset shows how the coupling amplitude decays as a function of tube diameter. All coupling values are given in units of the nearest-neighbor electronic hopping $\gamma$.

Figure 4

Exponentially decaying exchange coupling between Co adatoms on a semiconducting (31,0) nanotube (solid line). A small amount of doping is enough to populate the tube’s conduction band and give origin to an oscillating, long-ranged exchange coupling (dashed line). All coupling values are given in units of the nearest-neighbor electronic hopping $\gamma$.

Figure 5

(a) Exchange coupling as a function of longitudinal separation $D$ between two Co adatoms on a (6,6) armchair nanotube. The inset shows the wave vectors contributing to the coupling with different oscillation periods. (b) Amplitude of the exchange coupling as a function of temperature. All coupling values are given in units of the nearest-neighbor electronic hopping $\gamma$.

Figure 6

Exchange coupling $\Delta \mathcal{E}\left(\pi \right)$ multiplied by the distance between impurities $D$, as a function of $D$, for (a) (6,6), (b) (24,24), and (c) (48,48) armchair nanotubes. It is evident that in the (6,6) and (24,24) nanotubes the coupling decays as $1∕D$, whereas in the (48,48) tube it decays faster. All coupling values are given in units of the nearest-neighbor electronic hopping $\gamma$.