The angular-momentum conversion phenomena between spin and mechanical rotation are recognized as the gyromagnetic effects discovered in the early 20th century by S. J. Barnett, A. Einstein, and W. J. de Haas. They observed magnetization induced by mechanical rotation [1] and mechanical rotation induced by magnetization [2] known as the Einstein-de Haas (EdH) effect, revealing the origin of magnetism is the angular momentum. The gyromagnetic effects are universal phenomena that appeared even in nonmagnetic materials ranging from macroscopic to microscopic scales in various branches of physics, including ultra-cold atoms, spintronics, nuclear spin physics, and quark-hadron physics.

In our study [3], we developed a quantum theory to describe the microscopic mechanism of a nanorotor driven by electron spin injection via the EdH effect. As a feasible setup, we considered the double-wall carbon nanotube rotor as shown in Fig. 1 (a). For the double-well carbon nanotube, mechanical rotational motion of the inner carbon nanotube was already studied both experimentally [4] and theoretically [5]. However, efficient driving forces for the rotational motion are still lacking.

In this system, the rotor (the inner carbon nanotube) is driven by four transitions as shown in Fig. 1 (b). By applying a bias voltage between two half-metallic ferromagnetic electrodes, an electron with a spin s_z = ℏ/2 tunnels from the left electrode into the nanorotor (from (b-1) to (b-2)). This electron cannot tunnel to the drain electrode as long as its spin state remains s_z = ℏ/2 because there is no electronic state for the spin s_z = ℏ/2 in the half-metallic ferromagnetic drain. Electron accumulation by additional electron tunneling is also forbidden because of the Coulomb blockade in the nanorotor that acts like a quantum dot. Therefore, only spin flipping of the injected electron enables continuous current flow through the rotor (from (b-2) to (b-3)). Simultaneously with this spin flipping, angular momentum ℏ is eventually transferred from the injected electron to the mechanical rotational motion. After this angular momentum transfer, the electron with the spin s_z = -ℏ/2 can leave the nanotube into the right electrode. Therefore, the current through the nanorotor is expected to be a driving force of the rotational motion of the nanorotor in this system. In our study, we clarified that the precession of the nanorotor as shown in Fig. 1 (b-3) always occurs in the angular momentum transfer from the electron spin to the nanorotor. Therefore, in order to drive the nanorotor efficiently, the state of the nanoroter is needed to be relaxed from a precession state into a sleeping top state (from (b-3) to (b-4)).

The transition into the precession state is explained as follows. When the nanoroter is assumed to be a rigid body, its quantum states are specified by three quantum numbers, L, M, and k, where L is the quantum number corresponding to the magnitude of the angular momentum, M is the z component of the angular momentum, and k is the angular momentum around the axis of the rotor. In Fig. 2, we show the eigenenergies of the roter for the sleeping top state (M = k, the lower inset of Fig. 2) and the precession state (M = k ± 1, the upper inset of Fig. 2). We showed that the spin-rotation coupling between electron spins and the nanoroter induces only the transition from (A) to (B) in Fig. 2 because the spin-rotation coupling only changes M without changing k. Therefore, we need to introduce a phonon heat bath which causes relaxation from the precession state (B) to the sleeping top state (C). In our work [3], we discussed a detailed condition for efficient and stable driving of the rotational motion of the nanoroter.

The present driving mechanism is not restricted to carbon nanotubes; it applies to other materials as well. The process discussed here would be a minimal model for the momentum transfer from an electron to rotor that effectively includes the other possible processes.

This study has been performed as a joint study with Mamoru Matsuo, who was a visiting professor of ISSP in the academic year 2016.

References

• [1] S. J. Barnett, Phys. Rev. 56, 239 (1915).
• [2] A. Einstein and W. J. de Haas, KNAW proc. 18 I, 696 (1915).
• [3] W. Izumida, R. Okuyama, K. Sato, T. Kato, and M. Matsuo, Phys. Rev. Lett. 128, 017701 (2022).
• [4] A. M. Fennimore, T. D. Yuzvinsky, W.-Q. Han, M. S. Fuhrer, J. Cumings, and A. Zettl, Nature 424, 408 (2003).
• [5] S. W. D. Bailey, I. Amanatidis, and C. J. Lambert, Phys. Rev. Lett. 100, 256802 (2008).

Authors

• W. Izumida^a, R. Okuyama^b, K. Sato^c, T. Kato, and M. Matsuo^d
• ^aTohoku University
• ^bMeiji University
• ^cNational Institute of Technology, Sendai College
• ^dKavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences