Indirect Exchange Coupling in Carbon Nanotubes

Nanostructured magnetic composites based on carbon nanotubes (CNTs) and ferromagnetic nanoparticles (FNPs) are of great interest both from an applied and fundamental point of view. In particular, one of the features of CNTs with FNPs is the possibility of magnetic interaction of nanoparticles through the conducting medium of CNTs. For a detailed description of this special type of interaction, which is the indirect exchange coupling, it is necessary to establish the relationship between the macroscopic and microscopic parameters of the physical system. In nanostructured ferromagnets, these dependences are described within the framework of a random magnetization model in which the spin system and, consequently, the main macroscopic characteristics (coercivity, susceptibility, and saturation magnetization) are determined by such microscopic parameters as the exchange interaction constant, the FNP magnetization, the local magnetic anisotropy constant, and the grain size. In this paper, on the basis of the previously obtained microscopic parameters of CNT – FNP nanocomposites, the possibility of obtaining long-range magnetic correlations through the indirect exchange coupling (IEC) between FNP embedded inside a multi-wall CNT (MWCNT) is considered. A model Hamiltonian is used that takes into account the diameter, chirality, chemical potential and spin-orbit interaction (SOI) in the system. The reason for the appearance of a noticeable SOI in CNTs is the curvature of the tubes, which significantly increases the SOI compared to graphene, as well as possible defects and the presence of FNP. IEC is realized by means of p-electrons of the inner wall of the MWCNT. The propagation of the spin susceptibility along the MWCNT axis is calculated and it is shown that a long-range magnetic order is realized under the condition that the chemical potential enters the gap opened by the SOI. Coherence is realized at distances up to micrometers. The proposed approach also made it possible to estimate the energy of the exchange interaction between the FNP belonging to one CNT. The results obtained indicate the prospects for the use of CNT– FNP nanocomposites in carbon spintronics.

1. Weller D., Moser A., Folks L., Best M.E., Lee W., Toney M.F., Schwickert M., Thiele J.-U., Doerner M.F. High KU materials approach to 100 Gbit/in2. IEEE Trans. Magn. 2000;36(1):10-15.

2. Socoliuc V., Peddis D., Petrenko V.I., Avdeev M.V., Susan-Resiga D., Szabó T., Turcu R., Tombácz E., Vékás L. Magnetic nanoparticle systems for nanomedicine – a materials science perspective. Magnetochemistry. 2020;6(1):36.

3. Liu J.P. Ferromagnetic nanoparticles: synthesis, processing, and characterization. JOM. 2010;62(4):56-61.

4. Danilyuk A.L., Prudnikava A.L., Komissarov I.V., Yanushkevich K.I., Derory A., Le Normand F., Labunov V.A., Prischepa S.L. Interplay between exchange interaction and magnetic anisotropy for iron based nanoparticles in aligned carbon nanotube arrays. Carbon. 2014;68:337-345.

5. Chudnovsky E.M. Magnetic properties of amorphous ferromagnets. J. Appl. Phys. 1988;64(11):5770-5775.

6. Chudnovsky E.M., Saslow W.M., Serota R. A. Ordering in ferromagnets with random anisotropy. Phys. Rev. B. 1986;33(1);251-261.

7. Danilyuk A.L., Komissarov I.V., Labunov V.A., Le Normand F., Derory A., Hernandez J.M., Tejada J., Prischepa S.L. Manifestation of coherent magnetic anisotropy in a carbon nanotube matrix with low ferromagnetic nanoparticle content. New J. Phys. 2015;17(2):023073.

8. Danilyuk A.L., Kukharev A.V., Cojocaru C.S., Le Normand F., Prischepa S.L. Impact of aligned carbon nanotubes array on the magnetostatic isolation of closely packed ferromagnetic nanoparticles. Carbon. 2018;139:1104-1116.

9. Prischepa S.L., Danilyuk A.L., Kukharev A.V., Le Normand F., Cojocaru C.S. Self-assembled magnetically isolated co nanoparticles embedded inside carbon nanotubes. IEEE Trans. Magn. 2019;55(2):2300304.

10. Danilyuk A.L., Komissarov I.V., Kukharev A.V., Le Normand F., Hernandez J.M., Tejada J., Prischepa S.L. Impact of CNT medium on the interaction between ferromagnetic nanoparticles. Europhys. Lett. 2017;117(2):27007.

11. Schulz A., De Martino A., Ingenhoven P., Egger R. Low-energy theory and RKKY interaction for interacting quantum wires with Rashba spin-orbit coupling. Phys. Rev. B. 2009;79(20):205432.

12. Klinovaja J., Loss D. RKKY interaction in carbon nanotubes and graphene nanoribbons. Phys. Rev. B. 2013;87(4);045422.

13. Klinovaja J., Schmidt M.J., Braunecker B., Loss D. Helical modes in carbon nanotubes generated by strong electric fields. Phys. Rev. Lett. 2011;106(15):156809.

14. Stetter A., Vancea J., Back C.H. Determination of the intershell conductance in a multiwall carbon nanotube. Appl. Phys. Lett. 2008;93(17):172103.

15. Kogan E. RKKY interaction in graphene. Phys. Rev. B. 2011;84(11):115119.

16. Ciechan A., Bogusławski P. Theory of the s, p–d coupling of transition metal impurities with free carriers in ZnO. Sci. Rep. 2021;11(2):3848.

17. Leon A.O., d`Albuquerque e Castro J., Retamal J.C., Cahaya A.B., Altbir D. Manipulation of the RKKY exchange by voltages. Phys. Rev. B. 2019;100(1):014403.

18. Havlicek M., Jantsch W., Wilamowski Z., Yanagi K., Kataura H., Rümmeli M.H., Malissa H., Tyryshkin A., Lyon S., Chernov A., Kuzmany H. Indirect exchange interaction in fully metal-semiconductor separated single-walled carbon nanotubes revealed by electron spin resonance. Phys. Rev. B. 2012;86(4):045402.