Long-Range Exchange Interaction between Ferromagnetic Nanoparticles Embedded in Carbon Nanotubes

Abstract

Investigation of magnetic properties of nanostructured ferromagnets, such as oriented arrays of carbon nanotubes (CNTs) containing ferromagnetic nanoparticles (FNPs), is still relevant. In addition to attractive applications in magnetoelectronics, CNTs with embedded FNPs are also a very useful model object for studying magnetic interaction of the latter through a conducting medium. For this, it is important to establish a relation between macroscopic and microscopic parameters of the system. In nanostructured ferromagnets, this dependence is described within the random magnetic anisotropy model (RAM), in which the spin system and, consequently, the main macroscopic characteristics (coercivity, susceptibility, saturation magnetization) are determined by such microscopic parameters as the exchange interaction constant, FNP magnetization, local magnetic anisotropy constant and grain size 2Rc [1].Recently, magnetic parameters like the exchange and anisotropy fields, effective magnetic anisotropy constant, Bloch and exchange constants in aligned CNT arrays containing FNPs were determined within the RAM by analyzing law of the approach to saturation magnetization and corresponding modeling of the correlation functions of the magnetic anisotropy axes of the FNPs [2,3]. Presence of the interplay between the exchange interaction and magnetic anisotropy, as well as the contribution of not only random, but also coherent anisotropy, was established [3]. In addition, the important role of magnetoelastic anisotropy in the case when a single FNP is localized inside a CNT has been revealed. [4].It was shown also that aligned CNT arrays with a low concentration of FNPs embedded only inside CNT have relatively high values of exchange fields, random and coherent anisotropy. They are manifested in CNT arrays, in which the average distance between the FNPs significantly exceeds the size of the latter, reaching hundreds of nanometers. These effects, which do not currently have a convincing explanation, are mainly associated with the presence of the exchange interaction between FNPs through the CNT matrix. However, there is still no reasonable mechanism of the long-range exchange interaction in CNT arrays. In our work [5], it was assumed that these effects are related to the indirect exchange coupling of the RKKY type via the conducting electrons of CNTs. The obtained preliminary estimations have shown that the RKKY exchange interaction is enhanced by the presence of spin-orbit coupling (SOC) and could propagate up to a micrometer scale [5].In this contribution a multiwall CNT (MWCNT) with embedded FNPs is considered. We present the results of modeling of the RKKY interaction in MWCNT depending on its diameter, chirality, chemical potential and SOC constant within the Klinovaya-Loss model [6]. The influence of the external longitudinal magnetic field is also studied. It is assumed that the main contribution to the RKKY interaction is caused by the conduction electrons of the inner wall of CNTs which is in contact with the FNP. In addition, spin-orbit interaction (SOI) in CNT is also considered. The SOI in a nanotube can occur due to the curvature effect, which significantly increases its contribution in comparison with a flat graphene. It can also be enhanced by FNP, CNT defects or impurity states. The spin susceptibility of CNT conduction electrons χ/χ0 is evaluated (χ0=a2kG/hυF, where υF is the Fermi velocity, kG is the circumferential direction, a is the lattice constant). Fast oscillations are excluded and only slowly changing envelopes of spin susceptibility are considered. The chemical potential is tuned inside the gap opened by SOI. It is shown that the decay of the amplitude of the χ/χ0 oscillations depends strongly on the chemical potential: the higher the Fermi energy of the CNT (εF), the more significant is the decay. The frequency of spin susceptibility oscillations increases with the increase of the εF (Fig. 1). This is due to the fact that with the increase of the Fermi energy of CNTs, the discrepancy between the gap opened by SOI and the value of chemical potential increases. Finally, the proposed approach allows evaluating the energy of the exchange interaction between FNPs belonging to the same CNT.The work is supported by the COST Action CA19118. **