A spin−rotation mechanism of Einstein–de Haas effect based on a ferromagnetic disk

doi:10.1007/s11467-023-1389-9

Front. Phys.

2024, Vol. 19

Issue (5) : 53201 https://doi.org/10.1007/s11467-023-1389-9

A spin−rotation mechanism of Einstein–de Haas effect based on a ferromagnetic disk

Xin Nie¹, Jun Li¹, Trinanjan Datta^2,³(

), Dao-Xin Yao^1,⁴(

)

¹. Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State Key Laboratory of Optoelectronic Materials and Technologies, Center for Neutron Science and Technology, School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
². Department of Physics and Biophysics, Augusta University, 1120 15th Street, Augusta, Georgia 30912, USA
³. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
⁴. International Quantum Academy, Shenzhen 518048, China

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Abstract

Spin−rotation coupling (SRC) is a fundamental interaction that connects electronic spins with the rotational motion of a medium. We elucidate the Einstein−de Haas (EdH) effect and its inverse with SRC as the microscopic mechanism using the dynamic spin−lattice equations derived by elasticity theory and Lagrangian formalism. By applying the coupling equations to an iron disk in a magnetic field, we exhibit the transfer of angular momentum and energy between spins and lattice, with or without damping. The timescale of the angular momentum transfer from spins to the entire lattice is estimated by our theory to be on the order of 0.01 ns, for the disk with a radius of 100 nm. Moreover, we discover a linear relationship between the magnetic field strength and the rotation frequency, which is also enhanced by a higher ratio of Young’s modulus to Poisson’s coefficient. In the presence of damping, we notice that the spin−lattice relaxation time is nearly inversely proportional to the magnetic field. Our explorations will contribute to a better understanding of the EdH effect and provide valuable insights for magneto-mechanical manufacturing.

Keywords Einstein−de Haas effect spin−rotation coupling

Corresponding Author(s): Trinanjan Datta,Dao-Xin Yao

Issue Date: 09 April 2024

Cite this article:

Xin Nie,Jun Li,Trinanjan Datta, et al. A spin−rotation mechanism of Einstein–de Haas effect based on a ferromagnetic disk[J]. Front. Phys. , 2024, 19(5): 53201.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1389-9
https://academic.hep.com.cn/fop/EN/Y2024/V19/I5/53201

Fig.1 Schematic diagram of SRC mechanism in an iron disk. (a) Initial ferromagnetic configuration. (b) After the application of an external magnetic field

B

in the

z

-direction, the change in spin angular momentum causes atomic motion, resulting in local rotation of the disk. (c) The microscopic rotational “message” is transmitted throughout the disk at the speed of sound, leading to the macroscopic EdH rotation, as discussed in Refs. [1, 2].

Fig.2 Evolution of the time derivative of angular momentum in the

z

-direction.

J ˙ S

corresponds to the spin angular momentum,

J ˙ L

to the mechanical angular momentum, and

J ˙ d

to the angular momentum loss. (a) Undamped case,

J ˙ t o t o a l = J ˙ S + J ˙ L

. (b) Damped case,

J ˙ t o t a l = J ˙ S + J ˙ L + J ˙ d

. Damping factors

η = 0.5

and

ζ = 0.5

are used. The large damping coefficients are chosen to display the final state of the system.

Fig.3 Influence of the magnetic field on angular momentum transfer. (a) The oscillation frequency of angular momentum versus the magnetic field

B

. In the linear fitting

y = k 1 B + b 1

k 1 = 4.493

and

b 1 = 0.1367

. (b) Magnetization evolution under different magnetic fields. Damping factors

η = 0.5

and

ζ = 0.5

. The inset shows the case without damping, but with magnetic field

B = 1 T

. (c) Spin−lattice relaxation time versus

B

. The solid line (shown in red) represents an inverse proportional fitting with

a = 0.19

Fig.4 The oscillation frequency of angular momentum versus

E / (1 + ν)

and

I

. Here,

E 0 = 1.85 × 1011 P a

ν 0 = 0.32

I 0 = 4.29 × 10 − 20 J

, and

B = 0 T

Fig.5 Evolution of each part of energy.

E e x

represents the Heisenberg energy,

E u

includes the kinetic energy of the displacement field and elastic energy,

E Z

is the Zeeman energy,

E d

illustrates the effect of spin damping on the Zeeman term, and

E t o t a l

is their sum.

Δ

denotes their change. (a) Undamped case with

B = 2 T

. (b) Damped case with

B = 50 T

ζ = 0.5

, and

η = 0.5

B

is set to

50 T

to amplify the variation of

E Z

Fig.6 The oscillation frequency of energy versus the magnetic field

B

. In the linear fitting

y = k 2 B + b 2

k 2 = 4.5

and

b 2 = − 4.45

. The critical magnetic field (corresponding to the field at zero frequency) is about

1 T

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