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

Xin Nie  Jun Li  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  Trinanjan Datta
<EMAIL_ADDRESS>Department of Physics and Biophysics, Augusta University,
1120 15th Street, Augusta, Georgia 30912, USA  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, California 93106, USA  Dao-
Xin Yao<EMAIL_ADDRESS>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  International
Quantum Academy, Shenzhen 518048, China

###### 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.

##  I Introduction

Figure 1: Schematic diagram of SRC mechanism in an iron disk. (a) Initial
ferromagnetic configuration. (b) After the application of an external magnetic
field $\bm{B}$ 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].

Magnetization-rotation coupling is an intriguing and long-lasting topic.
Richardson [3] first proposed that the moment of momentum is proportional to
the magnetization using Ampère’s molecular currents. Soon after, the
gyromagnetic effect in macroscopic bodies was observed in the EdH experiment
[4] (change of magnetization induces mechanical rotation) and the inverse
Barnett experiment [5] (mechanical rotation triggers magnetization). In that
period, molecular orbital theory was used to explain the EdH effect—the
external magnetic field affected the angular momentum generated by electron
orbital motion, leading the iron cylinder in the experiment to produce a
mechanical angular momentum [6]. However, the discovery of electron spin
reveals that the gyromagnetic ratio (ratio of magnetic moment to angular
momentum) measured experimentally is close to the gyromagnetic ratio of the
spin, $e/m$, which is twice the value predicted by considering only electron
orbital motion [7]. It is now widely recognized that spin is an essential
origin of magnetism and that the spin-lattice coupling is indispensable for
the gyromagnetic effect. For transition metals like iron, the electronic
orbital angular momentum can be readily frozen by the surrounding crystal
field, resulting in a dominant contribution of spin to the magnetic moment. In
a contemporary interpretation of the EdH effect, the conservation of total
angular momentum is a basic principle, i.e., any change in the spin angular
momentum requires a corresponding compensation of the mechanical angular
momentum.

At the microscopic level, the mechanism via which angular momentum is
transferred from electrons to the entire body remains elusive, as it is
insufficient to explain the EdH effect by the conservation of angular momentum
alone. Researchers firstly pay attention to those EdH effects occurring at the
molecular scale due to finite degrees of freedom. For instance, the EdH effect
is studied for a system of two dysprosium atoms trapped in a spherically
symmetric harmonic potential [8]. A noncollinear tight-binding model capable
of simulating the EdH effect of an $\mathrm{O_{2}}$ dimer has been proposed
[9]. With the development of molecular-spintronics [10, 11, 12, 13], the
problem of single spin embedded within a macroscopic object has gained
attention [14, 15, 16]. The concept of phonon spin is used to explicate the
spin-phonon coupling involved [17]. The orbital part and the spin part of
phonon angular momentum (now considered pseudo-angular momentum [18]) are
clearly divided in Refs. [19] and [20], where their exchange with (real) spin
angular momentum are also discussed. Experiments have progressed to realize
the coupling of single-molecule magnets with nanomechanical resonators [21].
Nevertheless, the multiatomic EdH effect, which involves numerous degrees of
freedom, is inadequately comprehended despite the use of molecular dynamics
and micromagnetic simulations [22, 23, 24, 25]. The mechanical dynamics of a
magnetic cantilever resulting from changes in magnetization has also been
described [26, 27, 28], but only one-sidedly considering the effect of spin
evolution on lattice mechanics. Therefore, it is desired to develop a method
that treats spin and lattice as equally important and converts the microscopic
mechanism of the EdH effect into a macroscopic gyroscopic effect.

In this paper, we utilize Lagrangian formalism in classical field theory to
derive spin-lattice dynamical equations in an elastic ferromagnet, where spins
are coupled by exchange interactions. We assume that the magnetism of iron
atom originates from the electronic spin, $\bm{M}_{i}=\gamma\hbar\bm{S}_{i}$,
where $\bm{M}_{i}$ is magnetic moment of atom $i$, $\hbar$ is the Planck’s
constant, $\gamma=ge/(2m_{e})$ is the gyromagnetic ratio, and $g$ factor is 2
for spin $\bm{S}_{i}$. By means of the spin-rotation Hamiltonian arising from
the conservation of spin angular momentum and mechanical angular momentum, a
theoretical framework for the spin-lattice coupling is constructed—the change
of magnetization results in a force on the atom, whereas the evolution of
atomic displacement produces a torque acting on the spin. Then, we numerically
solve the spin-rotation dynamical equations in an iron disk to reveal the
transfers of angular momentum and energy, thereby validating the EdH effect.
The characteristics exhibited during the coupling process, such as the
transfer timescale and the dependence of the rotation frequency of the system
on magnetic field and material parameters, can enhance the comprehension for
laser-induced ultrafast demagnetization in solids [29, 30, 31, 32]. Exploring
the mutual transfer of mechanical angular momentum and spin angular momentum
can aid in creating new mechanical techniques for manipulating electron spins,
or alternatively, using spins to manipulate mechanical motion.

This work is organized as follows. The spin-rotation Hamiltonian and dynamics
equations merging spin and lattice are shown in Section II. The numerical
method of applying the coupling equations to a disk model is introduced in
Section III. And in Section IV, the pictures of angular momentum and energy
transfers with and without damping are presented. The effects of the magnetic
field on the rotation frequency and spin-relaxation time of the system are
also investigated. And the Barnett effect is briefly discussed through the SRC
mechanism. In Section V, we draw conclusions and propose some further
promising research ideas on spin-lattice coupling.

##  II spin-rotation Hamiltonian and coupling dynamical equations

Recently, a study using ultrafast electron diffraction to probe lattice
dynamics after femtosecond laser excitation has captured the transfer of
angular momentum from spins to the atoms, which induce circularly polarized
phonons as the atoms rotate around their equilibrium positions [2]. These
findings may shed light on the microscopic mechanism underlying the EdH
effect. For a stationary elastic magnet, if it is stimulated by an external
field like a magnetic field or heat bath, the spins within the magnet will
change their orientation. It is assumed here that the spins are formed by
strong exchange interactions that only alter their direction but not the
absolute value. As the system relaxes back to equilibrium, the magnet
generates mechanical rotation locally with atoms rotating around their
equilibrium positions, to preserve the total angular momentum of the system.
Subsequently, this local rotation transitions into an overall rotation (i.e.
the EdH macroscopic rotation), where all the atoms synchronously rotate around
the center of the lattice [1]. This interaction between spins and lattice is
called SRC, and a visual representation of it is depicted in Fig. 1.

Consider a magnetic system characterized by the magnetic moment $\bm{{M}}$ and
the mechanical angular momentum $\bm{{L}}$. Evidently, the magnetic moment
$\bm{{M}}$ corresponds to an internal angular momentum, $\gamma^{-1}\bm{{M}}$.
In the absence of external moments, $\bm{{L}}$ and $\gamma^{-1}\bm{{M}}$
should satisfy the conservation of total angular momentum,

$\bm{{L}}+\gamma^{-1}\bm{{M}}=\mathrm{{constant}}.$ (1)

Taking the time derivative on both sides of Eq. (1) yields
$\dot{\bm{L}}+\gamma^{-1}\dot{\bm{{M}}}=0$, where $\dot{\bm{L}}$ represents
the rate of change of mechanical angular momentum. The change in mechanical
angular momentum indicates that a mechanical torque must have been generated,

$\bm{{T}}=-\gamma^{-1}\dot{\bm{{M}}}.$ (2)

Eq. (2) informs us that a change in magnetization produces a mechanical torque
that rotates magnetic atoms. By denoting $\bm{\phi}_{j}$ as the rotation angle
at site $j$ and considering the relationship
$\bm{M}_{i}=\gamma\hbar\bm{S}_{i}$, the spin-rotation Hamiltonian in the
lattice can be generally expressed as

$H_{\mathrm{s-r}}=-\sum_{ij}A_{ij}^{ab}\hbar\dot{S}_{i}^{a}\phi_{j}^{b}.$ (3)

Here, $A_{ij}^{ab}$ is the coupling coefficient with $a,b\in{x,y,z}$.

Next, we introduce other interactions in the system. The spin $\bm{S}_{i}$ at
$i$th site can be decomposed into two mutually perpendicular components,

$\bm{S}_{i}=\bm{\mu}_{i}+\bm{n}_{i}\sqrt{S^{2}-\mu_{i}^{2}},$ (4)

where $\bm{\mu}_{i}\cdot\bm{n}_{i}=0$, but the orientations of themselves are
arbitrary. $S$ is the size of the spin. In terms of $\bm{\mu}_{i}$,
$\bm{n}_{i}$ and $\bm{u}_{i}$ (the displacement of the $i$th atom), the
kinetic energy of the system can be written as [33, 34]

$T=\frac{\hbar}{2S}\sum_{i=1}(\dot{\bm{\mu}}_{i}\times\bm{n}_{i})\cdot\bm{\mu}_{i}+\sum_{i=1}\frac{1}{2}m_{i}\dot{\bm{u}}^{2}_{i},$
(5)

with $m_{i}$ the mass of the $i$th atom. The first term represents the spin
kinetic energy and the second term represents the lattice kinetic energy.

The total potential energy contains the Heisenberg exchange interaction
$H_{\mathrm{ex}}$, the external Zeeman energy $H_{Z}$ and the crystal elastic
energy $H_{\mathrm{e}}$ [35],

$\displaystyle H_{\mathrm{ex}}$
$\displaystyle=-\frac{1}{2}\sum_{ij}I_{ij}\bm{S}_{i}\cdot\bm{S}_{j},$ (6)
$\displaystyle H_{Z}$ $\displaystyle=\sum_{i}g\mu_{B}\bm{B}\cdot\bm{S}_{i},$
(7) $\displaystyle H_{\mathrm{e}}$ $\displaystyle=\frac{1}{2}\int
d^{3}r~{}\sigma_{\alpha\beta}u_{\alpha\beta}.$ (8)

Here $I_{ij}$ is the ferromagnetic exchange coupling between $i$th and $j$th
sites, $\mu_{B}$ is the Bohr magneton, $\bm{B}$ is the external magnetic
field, and $\sigma_{\alpha\beta}$, $u_{\alpha\beta}$ are respectively the
$(\alpha,\beta)$ components of the stress tensor and strain tensor with
directional indexes $\alpha$, $\beta$.

Considering the large difference between the system size and interatomic
distance, as well as the relatively low-energy associated with the transfer in
comparison to the overall energy of the system, we treat the medium as
continuous and employ the displacement field $\bm{u}(\bm{r},t)$ and the spin
field $\bm{S}(\bm{r},t)$ at position $\bm{r}$ and time $t$ to describe the
degrees of freedom of lattice and spin, respectively. For linearly elastic
materials, the relationship between $\sigma_{\alpha\beta}$ and
$u_{\alpha\beta}$ conforms to Hooke’s law, where $u_{\alpha\beta}$ is given by
$u_{\alpha\beta}=(\nabla_{\alpha}u_{\beta}+\nabla_{\beta}u_{\alpha})/2$, and
$\sigma_{\alpha\beta}$ can be expressed as
$\sigma_{\alpha\beta}=C_{\alpha\beta\gamma\delta}u_{\gamma\delta}$. Here,
$C_{\alpha\beta\gamma\delta}=\mathcal{R}_{~{}\alpha}^{\mu}\mathcal{R}_{~{}\beta}^{\nu}\mathcal{R}_{~{}\gamma}^{\sigma}\mathcal{R}_{~{}\delta}^{\rho}C_{\mu\nu\sigma\rho}$,
with the coefficient tensor $C$, rotation transformation matrix $\mathcal{R}$,
and directional indexes $\gamma$, $\delta$.

We believe that local coupling accounts for the main contribution [27], so
this study focuses on the local coupling. According to the elastic theory,
$\bm{\phi}$ can be written as
$\bm{\phi}(\bm{r},t)=\nabla\times\bm{u}(\bm{r},t)/2$ [36]. We introduce the
spin density $\bm{S}^{\prime}$, defined as $\bm{S}^{\prime}=\bm{S}/V$, where
$V$ is the volume of the primitive cell and $\bm{S}$ is the total spin within
$V$. For simplicity, we assume that the SRC in the plane is isotropic.
Consequently, the continuous form of $H_{\mathrm{s-r}}$ can be given as

$\displaystyle H_{\mathrm{s-r}}=\frac{1}{2}\int
d^{3}r~{}\hbar\dot{\bm{S}}^{\prime}\cdot(\nabla\times\bm{u}).$ (9)

Eq. (9) incorporates the spins and the rotational motion of the lattice,
enabling the exchanges of angular momentum and energy between them.

For the nearest neighbor Heisenberg term, we approximate it by the Taylor
expansion, whereby
$\bm{S}_{j}=\bm{S}_{i+\delta_{j}}=\bm{S}_{i}+(\bm{\delta}_{j}\cdot\nabla)\bm{S}_{i}+\frac{1}{2}(\bm{\delta}_{j}\cdot\nabla)^{2}\bm{S}_{i}$
and $\bm{\delta}_{j}=\bm{S}_{j}-\bm{S}_{i}$. Since the Taylor expansion form
is restricted by the high rotational symmetry, other lattice choices introduce
nothing but the coefficient difference, which can be absorbed further into
parameter selection, thus we take the square lattice as an example. In order
to correctly obtain the spin dynamics equation, technically, we fix the
direction of $\bm{n}$, $\partial\bm{n}/{\partial t}=0$. The disk material is
considered isotropic with $I_{ij}=I$ for simplicity. We replace $\bm{S}_{i}$
and $\bm{u}_{i}$ with $\bm{S}(\bm{r},t)$ and $\bm{u}(\bm{r},t)$, respectively,
and introduce the definitions $\bm{\mu}^{\prime}=\bm{\mu}/V$, $S^{\prime}=S/V$
and $I^{\prime}=I\times V$. Under Eqs. (5)-(8), (9), the Lagrangian density of
the system reads,

$\displaystyle\mathcal{L}=$
$\displaystyle\frac{\hbar}{2S^{\prime}}(\dot{\bm{S}^{\prime}}\times\bm{n})\cdot\bm{S}^{\prime}+\frac{1}{2}\rho\dot{\bm{u}}^{2}+\frac{I^{\prime}}{2}\bm{S}^{\prime}\cdot[\bm{S}^{\prime}+a^{2}\nabla^{2}\bm{S}^{\prime}]$
$\displaystyle-g\mu_{B}\bm{B}\cdot\bm{S}^{\prime}-\frac{1}{2}C_{\alpha\beta\gamma\rho}u_{\gamma\rho}u_{\alpha\beta}-\frac{1}{2}\hbar\dot{\bm{S}}^{\prime}\cdot(\nabla\times\bm{u}),$
(10)

where $\rho$ is the mass density of the material and $a$ is the lattice
constant. Based on the Euler-Lagrange equations in classical field theory,

$\displaystyle\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{u}_{\alpha}}+\nabla\cdot\frac{\partial\mathcal{L}}{\partial(\nabla
u_{\alpha})}=\frac{\partial\mathcal{L}}{\partial u_{\alpha}},$ (11a)
$\displaystyle\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{S}_{\alpha}^{\prime}}+\nabla\cdot\frac{\partial\mathcal{L}}{\partial(\nabla
S_{\alpha}^{\prime})}=\frac{\partial\mathcal{L}}{\partial
S_{\alpha}^{\prime}},$ (11b)

with $\alpha=x$, $y$, and $z$, the spin-lattice dynamic equations can be
derived,

$\displaystyle\rho\frac{\partial^{2}\bm{u}}{\partial
t^{2}}-\nabla\cdot\bm{\sigma}+\frac{1}{2}\nabla\times\hbar\dot{\bm{S}^{\prime}}=0,$
(12)
$\displaystyle\hbar\dot{\bm{S}}=\bm{S}\times\left[Ia^{2}\nabla^{2}\bm{S}-g\mu_{B}\bm{B}+\frac{\hbar}{2}(\nabla\times\dot{\bm{u}})\right].$
(13)

Eqs. (12) and (13) tell us that the time evolution of the spin field acts as a
driving force $\bm{f}^{(R)}=-(\nabla\times\hbar\dot{\bm{S}}^{\prime})/2$ on
the displacement field, and in turn, the evolution of the displacement field
produces a torque $\hbar(\nabla\times\dot{\bm{u}})/2$ acting on the spin. Note
that the entire rotation contributes to $\nabla\times\dot{\bm{u}}$.

Let us now turn to angular momentum. By previous discussion [see Eq. (1)], the
total angular momentum (spin and mechanical) can be represented as

$\bm{J}=\int d^{3}r~{}[\hbar\bm{S}^{\prime}+\rho(\bm{r}\times\dot{\bm{u}})].$
(14)

For the time derivative of the $\alpha$-component of $\bm{J}$,

$\dot{J}_{\alpha}=\int d^{3}r~{}\hbar\dot{S}_{\alpha}^{\prime}+\int
d^{3}r~{}\epsilon_{\alpha\beta\gamma}r_{\beta}\rho\ddot{u}_{\gamma},$ (15)

the following expression can be obtained by using Eq. (12) and performing a
partial integration taking into account the symmetry of
$\sigma_{\alpha\beta}$,

$\dot{J}_{\alpha}=\int
dA_{\delta}~{}\epsilon_{\alpha\beta\gamma}r_{\beta}\sigma_{\gamma\delta}+\frac{1}{2}\int
dA_{\eta}~{}r_{\eta}\hbar\dot{S}_{\alpha}^{\prime}-\frac{1}{2}\int
dA_{\alpha}~{}r_{\beta}\hbar\dot{S}_{\beta}^{\prime}.$ (16)

Here $dA_{\delta}$, $dA_{m}$, and $dA_{\alpha}$ refer to the boundary surfaces
in the $\delta,m,\alpha=x,y,z$ directions.

##  III numerical solution for a disk model

A quasi-two-dimensional disk model depicted in Figure 1 is adopted to solve
Eqs. (12) and (13). In general, the coefficient tensor
$C_{\alpha\beta\gamma\delta}$ is a $9\times 9$ matrix. However, for an
isotropic object with symmetric stress and strain tensors,
$C_{\alpha\beta\gamma\delta}$ can be described only by two independent elastic
moduli $\lambda$ and $\mu$ [37],
$C_{\alpha\beta\gamma\delta}=\lambda\delta_{\alpha\beta}\delta_{\gamma\delta}+\mu(\delta_{\alpha\gamma}\delta_{\beta\delta}+\delta_{\alpha\delta}\delta_{\beta\gamma})$.
Hooke’s law relates it to the stress tensor,
$\sigma_{\alpha\beta}=E/(1+\nu)[u_{\alpha\beta}+\nu/(1-2\nu)u_{\gamma\gamma}\delta_{\alpha\beta}]$,
where $E$ is Young’s modulus given by
$E={\mu(3\lambda+2\mu)}/({\lambda+\mu})$, and $\nu$ is Poisson’s coefficient,
defined as $\nu={\lambda}/{2(\lambda+\mu)}$. Polar coordinates are more
convenient for the complanate and axisymmetric disk system (A detailed
derivation of coordinate transforation is provided in Appendix A). Assuming
plain strain and solely considering $\theta$-invariant solutions, we can
establish that $u_{az}=u_{za}=0$, $\nabla_{z}S_{a}=0$,
$\nabla_{\theta}u_{a}=0$ and $\nabla_{\theta}S_{a}=0$, where
$a\in\\{r,\theta,z\\}$.

To clearly demonstrate the influence of material properties on the evolution
of the system, we substitute all variables with dimensionless counterparts,
which are defined as follows,

$\displaystyle\overline{u}_{\theta}=\frac{u_{\theta}}{R},\overline{r}=\frac{r}{R},\overline{\bm{S}}=\frac{\bm{S}}{S},\overline{t}=tm,m\equiv\sqrt{\frac{E}{2(1+\nu)\rho
R^{2}}},$ (17)

where $R$ is the radius of the disk. The dynamical equations for
$\overline{u}_{\theta}$ (the displacement variable associated with the
rotation of the disk) and $\overline{\bm{S}}$ in polar coordinates can be
derived from Eqs. (12) and (13) (For further calculation details, please refer
to Appendix A),

$\frac{\partial^{2}\overline{u}_{\theta}}{\partial\overline{r}^{2}}+\frac{1}{\overline{r}}\frac{\partial\overline{u}_{\theta}}{\partial\overline{r}}-\frac{1}{\overline{r}^{2}}\overline{u}_{\theta}=\frac{\partial^{2}\overline{u}_{\theta}}{\partial\overline{t}^{2}}+K\overline{f}_{\theta},$
(18) $\displaystyle\frac{\partial\overline{\bm{S}}}{\partial\overline{t}}$
$\displaystyle=\overline{\bm{S}}\times[\alpha_{1}(\frac{\partial^{2}}{\partial\overline{r}^{2}}+\frac{1}{r}\frac{\partial}{\partial\overline{r}})\overline{\bm{S}}+\frac{1}{2}\left(\frac{\partial^{2}}{\partial\overline{r}\partial\overline{t}}+\frac{1}{\overline{r}}\frac{\partial}{\partial\overline{t}}\right)\overline{{u}}_{\theta}\bm{e}_{z}$
$\displaystyle\quad-\beta_{1}\bm{e}_{z}].$ (19)

The coupling coefficient $K=\hbar SS_{0}\sqrt{(1+\nu)/(2E\rho R^{2})}$ (with
$S_{0}$ representing the constant spin density [27]),
$\alpha_{1}=Ia^{2}S/(\hbar R^{2}m)$, $\beta_{1}=g\mu_{B}B/(\hbar m)$, and
$\overline{f}_{\theta}=\partial^{2}\overline{S}_{z}/(\partial\overline{t}\partial\overline{r})$.
In the present paradigm, the strength of the SRC exhibits correlations with
elastic coefficients ($E$ and $\nu$), density ($\rho$), and exchange
interaction ($I$).

Also, based on Eq. (16), the time derivative of angular momentum in polar
coordinates is given,

$\displaystyle\dot{J}_{r}$ $\displaystyle=0,$ (20a)
$\displaystyle\dot{J}_{\theta}$ $\displaystyle=\frac{1}{2}\int
dA_{r}~{}r\hbar\dot{S}_{\theta},$ (20b) $\displaystyle\dot{J}_{z}$
$\displaystyle=\int dA_{r}~{}r\sigma_{r\theta}+\frac{1}{2}\int
dA_{r}~{}r\hbar\dot{S}_{z}.$ (20c)

In this case, the angular momentum in the $r$-direction, $J_{r}$, is
automatically conserved.

Damping is ubiquitous in most situations. To account for this effect, we
introduce the first-order time derivative of $u_{\theta}$ into Eq. (18) and
integrate the Gilbert damping [38] into Eq. (III). As a result, a coupled
system with damping is obtained,

$\displaystyle\frac{\partial^{2}\overline{u}_{\theta}}{\partial\overline{r}^{2}}+\frac{1}{\overline{r}}\frac{\partial\overline{u}_{\theta}}{\partial\overline{r}}-\frac{1}{\overline{r}^{2}}\overline{u}_{\theta}=\frac{\partial^{2}\overline{u}_{\theta}}{\partial\overline{t}^{2}}+K\overline{f}_{\theta}+\eta\frac{\partial\overline{u}_{\theta}}{\partial\overline{t}},$
(21)
$\displaystyle\dot{\overline{\bm{S}}}=\overline{\bm{S}}\times\overline{\bm{\Gamma}}+\zeta(\dot{\overline{\bm{S}}}\times\overline{\bm{S}}),$
(22)

where

$\displaystyle\overline{\bm{\Gamma}}$
$\displaystyle=\alpha_{1}(\frac{\partial^{2}}{\partial\overline{r}^{2}}+\frac{1}{r}\frac{\partial}{\partial\overline{r}})\overline{\bm{S}}+\frac{1}{2}\left(\frac{\partial^{2}}{\partial\overline{r}\partial\overline{t}}+\frac{1}{\overline{r}}\frac{\partial}{\partial\overline{t}}\right)\overline{{u}}_{\theta}\bm{e}_{z}$
$\displaystyle\quad-\beta_{1}\bm{e}_{z},$ (23)

and $\eta$ (>0), $\zeta$ are dimensionless damping factors.

We combine the finite difference method [39, 40] and the fourth-order Runge-
Kutta method [41] to numerically solve the coupled differential Eqs. (21) and
(22). Under specific boundary and initial conditions (to be provided later),
the time evolutions of the displacement field and spin field can be
determined, enabling the subsequent calculation of the mechanical angular
momentum, the spin angular momentum and various energy terms.

\begin{overpic}[scale={0.15}]{angular_nodamping.png}
\put(-2.0,68.0){\large{(a)}} \end{overpic}

\begin{overpic}[scale={0.15}]{angular_damping.png}
\put(-2.0,68.0){\large{(b)}} \end{overpic}

Figure 2: Evolution of the time derivative of angular momentum in the
$z$-direction. $\dot{J}_{S}$ corresponds to the spin angular momentum,
$\dot{J}_{L}$ to the mechanical angular momentum, and $\dot{J}_{\mathrm{d}}$
to the angular momentum loss. (a) Undamped case,
$\dot{J}_{\mathrm{totoal}}=\dot{J}_{S}+\dot{J}_{L}$. (b) Damped case,
$\dot{J}_{\mathrm{total}}=\dot{J}_{S}+\dot{J}_{L}+\dot{J}_{\mathrm{d}}$.
Damping factors $\eta=0.5$ and $\zeta=0.5$ are used. The large damping
coefficients are chosen to display the final state of the system.

##  IV results and analysis

###  IV.1 Transfer of angular momentum

Our primary focus lies on the angular momentum in the $z$-direction, as it is
relevant to the rotation of the disk. Following from the $z$-component of Eq.
(15), the time derivative of the spin angular momentum is

$\displaystyle\dot{J}_{S}=\int d^{3}r~{}\hbar\dot{S}_{z},$ (24)

and the time derivative of the mechanical angular momentum is

$\displaystyle\dot{J}_{L}=\int d^{3}r~{}\rho r\ddot{u}_{\theta}.$ (25)

When solving Eqs. (21) and (22), we specify the boundary conditions as

$\left\\{\begin{aligned}
&\sigma_{r\theta}=0,~{}\sigma_{rr}=0\qquad\mathrm{when~{}}r\to 0\\\
&\sigma_{r\theta}=0,~{}\sigma_{rr}=0\qquad\mathrm{at~{}}r=R\end{aligned}\right.\qquad\mathrm{for~{}}u_{\theta},$
(26)

and

$\left\\{\begin{aligned}
&\dot{J}_{\theta}=0,~{}\dot{J}_{z}=0\qquad\mathrm{when~{}}r\to 0\\\
&\dot{J}_{\theta}=0,~{}\dot{J}_{z}=0\qquad\mathrm{at~{}}r=R\end{aligned}\right.\qquad\mathrm{for~{}}\bm{S}.$
(27)

The selection of $r\to 0$ as a boundary instead of $r=0$ stems from the
requirement to ensure the validity of the formula $1/r$. The explicit
formulations of $\sigma_{r\theta}$ and $\sigma_{rr}$ can be found in Appendix
A. The first two conditions on $u_{\theta}$ correspond to the absence of force
at the two boundaries, while the last two conditions on $\bm{S}$ guarantee the
total angular momentum of the system cannot flow out from the boundaries.

Since the iron disk is ferromagnetic, the following initial conditions are
taken—at the beginning, all spins on the disk are uniformly aligned, and both
atomic displacement and velocity are zero (i.e., the disk is stationary). It
should be emphasized that our approach extends beyond the initial
circumstances, and as will be seen later, we intentionally opt for other
initial conditions to effectively illustrate the energy transfer process.

####  IV.1.1 Undamped and damped cases

The selected magnetic material for this study, Fe, possesses an exchange
parameter of $I=4.29\times 10^{-21}\mathrm{~{}J}$, a lattice constant of
$a=2.87\times 10^{-10}\mathrm{~{}m}$, a mass density of $\rho=7.9\times
10^{3}\mathrm{~{}kg/m^{3}}$, Young’s modulus of $E=1.85\times
10^{11}\mathrm{~{}Pa}$, and Poisson’s coefficient of $\nu=0.32$, as reported
by [22]. The disk radius is set to $R=10^{-7}~{}\mathrm{m}$. Unless otherwise
noted, these parameters are used consistently in the following calculations.

Figure 2 shows the results of angular momentum transfer with and without
damping. The variable $\dot{J}_{d}$, as shown in Fig. 2, represents the rate
of change of angular momentum loss attributed to the displacement damping
($\eta$). According to Eq. (21), the damping term introduces a resistive force
acting on the atom, described by $F_{\mathrm{d}}^{\theta}=-\eta~{}{\partial
u_{\theta}}/{\partial t}$. This leads to the emergence of a torque in the
$z$-direction, which can be equivalently expressed as the time derivative of
the angular momentum,

$\dot{J}_{\mathrm{d}}=-\eta\int d^{3}r~{}r\frac{\partial u_{\theta}}{\partial
t}.$ (28)

At $t=0$, the rates of change of spin angular momentum and mechanical angular
momentum are both zero as a result of the initial conditions of the system.
However, at the next moment, the magnetic field $B=1\mathrm{~{}T}$ is applied
to the disk, leading to non-zero rates of change of angular momentum [see Eqs.
(21), (22), (24), and (25)]. Especially in the presence of damping, both the
driving force ($\overline{f}_{\theta}$) and the first-order time derivative of
the displacement field ($\dot{u}_{\theta}$) are no longer zero after the
introduction of the magnetic field, causing an abrupt increase in the second-
order time derivative of the displacement field ($\ddot{u}_{\theta}$), i.e., a
sharp jump in the rate of change of the mechanical angular momentum in Fig. 2.
This phenomenon indicates that the external magnetic field acts as the driving
source for the EdH effect. If the driving source is the rotation of the magnet
(no external magnetic field in this case), the evolution of the displacement
field will result in spin-flips, similar to the Barnett effect. Notably,
between $t=0.25~{}\mathrm{ns}$ and $t=0.35~{}\mathrm{ns}$, there is a strange
peak, which can be suppressed by a stronger magnetic field, suggesting its
intrinsic cause is the competition between the external magnetic field and the
intrinsic elastic dynamics. From an overall perspective, $\dot{J}_{L}$ and
$\dot{J}_{S}$ exhibit a high-frequency periodic behavior in Fig. 2. The
correlation between the oscillation frequency and the strength of the external
magnetic field as well as material properties will be discussed in subsequent
sections.

The timescale of angular momentum transfer is another pivotal factor closely
associated with spin-mechanical applications. Ref. [2] reveals that when a
magnetized material is impinged by an ultrafast laser, the electronic spins
transfer their angular momentum to lattice atoms within a few hundred
femtoseconds. This transfer leads to the generation of circularly polarized
phonons, which then propagate the angular momentum across the material at the
speed of sound, ultimately contributing to the macroscopic EdH rotation. Based
on this research, we make a tentative estimation of the time required for the
angular momentum to transfer from local rotation to macroscopic rotation,
which approximates to $t\approx R/v=0.02~{}\mathrm{ns}$, with the speed of
sound $v\propto\sqrt{E/\rho}$. This timescale is reflected in our theoretical
results. In Fig. 2, at $t=0.02~{}\mathrm{ns}$, the rate of change of the
angular momentum is large enough to indicate the occurrence of macroscopic
rotation. Considering the coefficient factor of $v$ is undetermined, we
estimate the transfer timescale of spin angular momentum to the whole lattice
to be on the order of 0.01 ns. Nevertheless, accurately evaluating the
femtosecond timescale for the transfer of angular momentum from spin to local
rotation is currently unfeasible due to the challenge of precisely identifying
the exact moment when local rotation emerges.

Throughout the coupling process, the rate of change of the spin angular
momentum ($\dot{J}_{S}$) and the rate of change of the lattice angular
momentum ($\dot{J}_{L}$, or the sum of $\dot{J}_{L}$ and
$\dot{J}_{\mathrm{d}}$ in the damped case) evolve inversely, with their sum
approximately (but not strictly) equal to zero. This situation is not improved
by enhancing computational accuracy. One possible explanation for this
phenomenon is the exclusion of phonon spin angular momentum from Eq. (14). The
total mechanical angular momentum should incorporate the contribution of the
phonon spin angular momentum [17, 18], which is defined as
$\bm{J}_{\mathrm{p-s}}=\int d^{3}r~{}\rho(\bm{u}_{\mathrm{local}}\times\
\dot{\bm{u}}_{\mathrm{local}})$, where $\bm{u}_{\mathrm{local}}$ represents
the atomic displacement around the equilibrium position, corresponding to
local rotations in Fig. 1. The displacement $\bm{u}$ described in Eq. (12)
includes both local and overall rotations, without separate kinetic equations
to describe $\bm{u}_{\mathrm{local}}$. Additionally, it is challenging to
distinguish the contributions of the local displacement and overall
displacement within $\bm{u}$. Therefore the calculation of
$\bm{J}_{\mathrm{p-s}}$ is currently unavailable. Eq. (14) differs from the
total angular momentum by $\bm{J}_{\mathrm{p-s}}$, leading to
$\dot{J}_{\mathrm{total}}$ not being strictly zero, as shown in Fig. 2.
However, considering that the local displacement is much smaller than $r$, the
angular momentum $J_{\mathrm{p-s,z}}$ (the $z$-component of
$\bm{J}_{\mathrm{p-s}}$) is much smaller compared to $J_{L,z}=\int
d^{3}r~{}\rho(\bm{r}\times\ \dot{\bm{u}})_{z}$. This renders the contribution
of it negligible in our analysis of the interconversion between spin angular
momentum and lattice angular momentum within the EdH effect.

As our approach is not constrained by initial conditions, it allows exploring
the spin-lattice coupling processes where the system start from a
nonequilibrium or nonferromagnetic configuration. First-principle calculations
can assist in determining the initial spin configurations of diverse
materials. Besides, the dynamic behavior of magnetic systems under external
perturbations, including magnetic field and stress, can also be investigated.
By incorporating the effects of these perturbations into the coupling
equations, one can analyze the response and sensitivity of the system to
different stimuli and predict their influences on the dynamical evolution from
lattice to spin or from spin to lattice.

\begin{overpic}[scale={0.15}]{magnetic_frequency.png}
\put(0.0,73.0){\large{(a)}} \end{overpic}

\begin{overpic}[scale={0.15}]{M_z.png} \put(0.0,75.0){\large{(b)}}
\end{overpic}

\begin{overpic}[scale={0.15}]{relaxation_time.png} \put(0.0,75.0){\large{(c)}}
\end{overpic}

Figure 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
$\eta=0.5$ and $\zeta=0.5$. The inset shows the case without damping, but with
magnetic field $B=1~{}\mathrm{T}$. (c) Spin-lattice relaxation time versus
$B$. The solid line (shown in red) represents an inverse proportional fitting
with $a=0.19$.

####  IV.1.2 Effect of magnetic field

In the following we will study the effect of a constant magnetic field on the
evolution of the system. An alternating magnetic field can also be employed
and without changing the main conclusions. We define the oscillation frequency
of angular momentum as the number of oscillations per second, and its
relationship with the magnetic field is depicted in Fig. 3. When there is no
damping, the frequency is proportional to the magnitude of the magnetic field,
$f\propto B$, and the proportionality constant is the slope of the linear
fitting, $k_{1}$. Eq. (21) reveals two distinct types of displacement field
evolution: intrinsic and forced. The former is governed by the purely elastic
dynamics equation, and its oscillation frequency is determined by the material
elasticity coefficients, while the latter is driven by the change of spin,
with an oscillation frequency determined by the spin torque,
$\dot{\bm{S}}=\bm{S}\times[Ia^{2}\nabla^{2}\bm{S}/\hbar+\gamma\bm{B}+(\nabla\times\dot{\bm{u}})/2]$.
The linear relationship between the oscillation frequency and the external
magnetic field suggests that the displacement field mainly follows forced
oscillations. Under different magnetic fields, we compare the magnitude of
spin torques generated by the Heisenberg term $Ia^{2}\nabla^{2}\bm{S}/\hbar$,
the external magnetic field $\gamma\bm{B}$, and the displacement field
$(\nabla\times\dot{\bm{u}})/2$. We find that the contribution from the
displacement field is significantly smaller than that from the magnetic field.
Moreover, although the Heisenberg interaction torque varyies with the spin
itself, it is also lower than the magnetic field torque across most of the
disk. Consequently, the external magnetic field dominates the evolution
frequency of the spin and displacement fields, while the other two serve as
modulating factors. This reminds us that in purely precessional dynamics of
spin, the precession frequency is proportional to the effective field that
induces spin precession [38],
$\dot{\bm{S}}=\bm{S}\times\gamma\bm{H}_{\mathrm{eff}}$,
$w=\lvert\gamma\bm{H}_{\mathrm{eff}}\rvert=2\pi f^{\prime}$, where
$\bm{H}_{\mathrm{eff}}$ is the effective field, $w$ is the angular frequency,
and $f^{\prime}$ is the frequency. We discover that $k_{1}$ and
$\lvert\gamma/(2\pi)\rvert/10^{9}$ are not equal, with $k_{1}$ being
approximately one-sixth of the latter. However, if the spin torques solely
contain the magnetic field term, the spins will precess around $\bm{B}$, and
the spin angular momentum in the $z$-direction remains unchanged and is not
transferred to the lattice system. In other words, despite the inferior effect
on the oscillation frequency, the Heisenberg term and the displacement field
term play a crucial role in facilitating the transfer of angular momentum
between the spins and lattice.

In the scenario of damping, our investigation aims at the dependence of the
spin-lattice relaxation time on the magnetic field. For this purpose, we
denote the $z$-component of magnetization intensity as $M_{z}$ and the
saturation magnetization intensity as $M_{0}$. Figure 3 illustrates the
dynamic behavior of $M_{z}$ under varying magnetic fields. When both the
magnetic field and damping are present, all the spins end up pointing in the
direction of the magnetic field, with the higher magnetic field leading to
faster saturation of the magnetization. In contrast, without damping (as shown
in the inset), the spins exhibit periodic oscillations and fail to align with
the z-axis direction, despite the action of the magnetic field. Damping is
essential in achieving a stable magnetization. The spin-lattice relaxation
time is defined as the duration required for $M_{z}$ to reach $M_{0}$, which
is considered achieved when the deviation is less than or equal to 0.02, i.e.,
$\Delta M^{z}/\Delta M\leqslant 0.02$, where $\Delta M^{z}=M_{0}-M_{z}$,
$\Delta M=M_{0}-M_{z}^{t=0}$. As shown in Fig. 3, the relaxation time
decreases at a slower and slower rate as the magnetic field increases.
Moreover, through data fitting, we observe an near-inverse correlation between
the relaxation time and the magnetic field. This feature also appears in the
Landau-Lifshitz-Gilbert (LLG) equation for the typical dissipation time
$\tau_{LLG}$ [42], $\tau_{LLG}=1/(w_{L}\zeta)$, where $w_{L}$ is the
precession frequency and $\zeta$ is the damping factor.

Figure 4: The oscillation frequency of angular momentum versus $E/(1+\nu)$ and
$I$. Here, $E_{0}=1.85\times 10^{11}\mathrm{~{}Pa}$, $\nu_{0}=0.32$,
$I_{0}=4.29\times 10^{-20}\mathrm{~{}J}$, and $B=0~{}\mathrm{T}$.

\begin{overpic}[scale={0.15}]{energy_nodamping.png}
\put(-2.0,68.0){\large{(a)}} \end{overpic}

\begin{overpic}[scale={0.15}]{energy_damping.png} \put(-2.0,68.0){\large{(b)}}
\end{overpic}

Figure 5: Evolution of each part of energy. $E_{\mathrm{ex}}$ represents the
Heisenberg energy, $E_{u}$ includes the kinetic energy of the displacement
field and elastic energy, $E_{z}$ is the Zeeman energy, $E_{\mathrm{d}}$
illustrates the effect of spin damping on the Zeeman term, and
$E_{\mathrm{total}}$ is their sum. $\Delta$ denotes their change. (a) Undamped
case with $B=2~{}\mathrm{T}$. (b) Damped case with $B=50~{}\mathrm{T}$,
$\zeta=0.5$, and $\eta=0.5$. $B$ is set to $50~{}\mathrm{T}$ to amplify the
variation of $E_{z}$.

####  IV.1.3 Effect of material parameters

The variations in material parameters, such as the exchange coefficient ($I$),
Young’s modulus ($E$), and Poisson’s coefficient ($\nu$), can also affect the
systerm. But our calculations demonstrate that when the system is exposed to a
magnetic field, these variations can significantly impact the magnitude of the
angular momentum while having minimal effect on the frequency. Therefore, we
exclude the magnetic field ($B=0$) and utilize the rotation of the disk as the
driving force to explore the influence of these material parameters on the
evolution frequency. The chosen initial angular velocity is the same as that
used in the energy transfer case discussed below. The oscillation frequency of
angular momentum in Fig. 4 ranges from $4.5\times 10^{9}$ Hz to $5\times
10^{9}$ Hz, with negligible dependence on $I$. This behavior is expected since
$I$ does not directly affect the coupling coefficient $K$ for isotropic
objects. However, for the elasticity coefficients $E$ and $\nu$, the frequency
exhibits a positive correlation with the ratio of $E$ to $\nu$, not in a
linear manner like the magnetic field. The Young’s modulus of a material is a
manifestation of its interatomic bonding energy and lattice structure. A
larger $E$ to $\nu$ ratio indicates a harder material, which is more likely to
move as a whole and less prone to local elastic deformation. This information
is useful for selecting suitable elastic materials as potential candidates for
magneto-mechanical devices.

###  IV.2 Transfer of energy

The EdH process also involves an energy exchange between the spin subsystem
and lattice subsystem. Noether’s theorem determines the total energy as

$\displaystyle H$ $\displaystyle=\int
d^{3}r~{}\left[\frac{\partial\mathcal{L}}{\partial\dot{\Phi_{a}}}\Phi_{a}-\mathcal{L}\right]$
$\displaystyle=\int
d^{3}r~{}\bigg{[}\frac{1}{2}\rho\dot{\bm{u}}^{2}-\frac{I^{\prime}}{2}\bm{S}^{\prime}\cdot(\bm{S}^{\prime}+a^{2}\nabla^{2}\bm{S}^{\prime})+g\mu_{B}\bm{B}\cdot\bm{S}^{\prime}$
$\displaystyle\quad+\frac{1}{2}C_{\alpha\beta\gamma\rho}u_{\gamma\rho}u_{\alpha\beta}\bigg{]},$
(29)

with $\Phi_{a}=u_{a}$,$S^{\prime}_{a}$. If the initial conditions mentioned in
Section IV.1 persist, the lattice energy will start at a level lower than the
spin energy due to the absence of lattice kinetic energy and the non-zero spin
energy of the ferromagnetic ground state. To balance the consideration of spin
and lattice, we adopt an initial state where the angular velocity disk is set
at one $m~{}\mathrm{rad/s}$, with $m$ being the dimensionless factor of time
in Eq. (17).

Figure 5 describes the energy conversion with and without damping. In the
undamped scenario, we are concerned with the changes in energy of each
component, denoted as $\Delta E$, which are all much smaller than the
components themselves. As depicted, the Zeeman term $\Delta E_{\mathrm{Z}}$
evolves opposite to both the exchange term $\Delta E_{\mathrm{ex}}$ and the
lattice term $\Delta E_{\mathrm{u}}$, representing the transfer of energy from
the former to the latter two. Similar to angular momentum, $\Delta
E_{\mathrm{total}}$ is not constant at zero. Eq. (IV.2) represents the
conserved total energy derived from Noether’s theorem, where $\bm{u}$ is the
displacement vector of the atom. However, as we only consider $u_{\theta}$ in
relation to the EdH rotation, the kinetic energy $\rho{\dot{u}}^{2}_{\theta}$
term deviates from $\rho{\dot{\bm{u}}}^{2}$, making the calculated
$E_{\mathrm{total}}$ not strictly constant in Fig. 5. It is worth emphasizing
that our focus is more on comprehending the mechanisms and patterns of angular
momentum and energy transfers between the spin and lattice systems than the
strict conservation of these quantities. In Figs. 2 and 5, the evolution of
angular momentum and energy within the spin subsystem exhibits an opposite
trend compared to that of the lattice subsystem. This observation strongly
indicates that the coupled Eqs. (12) and (13) effectively capture the transfer
of angular momentum and energy.

Given that the Heisenberg energy $E_{\mathrm{ex}}$ defined in Eq. (II) is
directly proportional to $\bm{S}\cdot\nabla^{2}\bm{S}$, we have
$(\partial{E}_{\mathrm{ex}}/{\partial
t})\propto\dot{\bm{S}}\cdot\nabla^{2}\bm{S}$, where the term
$\nabla^{2}\dot{\bm{S}}$ is neglected as it is small compared to
$\dot{\bm{S}}$. With this in mind, the energy dissipation arising from the
magnetic field response to spin damping, $E_{\mathrm{d}}$, satisfies
$(\partial E_{d}/\partial t)\propto-\dot{\bm{S}}_{\mathrm{damping}}\cdot
g\mu_{B}\bm{B}$. Therefore, the energy loss can be estimated as

$E_{d}=-\int dt~{}g\mu_{B}\bm{B}\cdot[\frac{\zeta}{\hbar
S}(\bm{S}\times\bm{\Gamma})\times\bm{S}].$ (30)

The energy losses resulting from damping effects are discussed in Appendix B,
and their total is represented as $E_{\mathrm{damping}}$ in Fig. 5. Under
large damping conditions, the kinetic energy $E_{\mathrm{u}}$ rapidly
diminishes, eventually reaching zero. It is clear that $E_{\mathrm{u}}$ is the
primary contributor to energy loss, combined with the trend of the loss term
$E_{\mathrm{damping}}$. The Zeeman energy is nearly balanced out by the
damping term $E_{\mathrm{d}}$, as evidenced by their constant sum shown in the
inset. The change in exchange energy $E_{\mathrm{ex}}$ is relatively
insignificant compared to the other energy terms, thus its behavior is shown
separately in the inset. Initially, $E_{\mathrm{ex}}$ rises sharply, then
falls rapidly and stabilize. It should be noted that the damping factors used
in the theoretical calculations are higher than the actual values,
necessitating the use of realistic damping factors to explore the specific
transfer of each energy term.

The effect of the magnetic field on the oscillating frequency of energy is
illustrated in Fig. 6. When the magnetic field is weak, the oscillation
frequency decreases as the magnetic field strength increases. However, once
the magnetic field reaches approximately $1~{}\mathrm{T}$, the frequency
starts to linearly increase with the magnetic field. In energy calculations,
we consider a non-zero initial velocity for the disk, which drives the
intrinsic evolution of the system. At relatively low magnetic fields, the
initial rotation and the magnetic field jointly determine the frequency of the
system, with the former contributing more. But, as the magnetic field
continues to grow to about $1~{}\mathrm{T}$, the frequency approaches 0,
revealing a competitive relationship between these two driving sources in
determining the frequency. With further increase in the magnetic field, the
field will surpasse the influence of the initial conditions and become the
dominant factor in determining the frequency. Different magnetic materials and
initial conditions result in distinct critical magnetic fields. In addition,
the slope of this linear correlation differs from that of the zero initial
velocity case depicted in Fig. 3. In combination with the discussion in
Section IV.1.2, it can be suggested that the linear slope between the
frequency and magnetic field is determined by the Heisenberg interaction, the
spin-lattice coupling term, and the intrinsic evolution of the system.

Figure 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 $1~{}\mathrm{T}$.

###  IV.3 Barnett effect

As the inverse effect of EdH, the Barnett effect is another manifestation of
SRC. In this context, the driving source of the system is attributed to the
rotation of the disk, not the magnetic field. Consequently, the dynamic
equations take the following form,

$\displaystyle\rho\frac{\partial^{2}\bm{u}}{\partial
t^{2}}-\nabla\cdot\bm{\sigma}+\frac{1}{2}\nabla\times\hbar\dot{\bm{S}^{\prime}}=0,$
(31)
$\displaystyle\dot{\bm{S}}=\bm{S}\times[Ia^{2}/(\hbar\nabla^{2}\bm{S})+\bm{\Omega}],$
(32)

with angular velocity $\bm{\Omega}=(\nabla\times\dot{\bm{u}})/2$. Eq. (32)
indicates that the magnetization generated by rotating a ferromagnetic object
at $\bm{\Omega}$ in the absence of a magnetic field is equivalent to that
produced when the object is not rotating but subjected to a magnetic field
$\bm{B}_{\mathrm{eff}}=\bm{\Omega}/\gamma$. This demonstrates the Barnett
effect, and the effective field $\bm{B}_{\mathrm{eff}}$ is referred to as the
Barnett field. The measurement of the Barnett effect is commonly performed
using the magnetic resonance technique, which can detect the frequency shift
signal $\bm{\Omega}/\gamma$ originating from the Barnett field [43, 44]. The
study of the Barnett effect has developed rapidly, including the observation
of the electronic Barnett effect in paramagnetic states [45] and the first
reported observation of the nuclear Barnett effect [46]. Considering that the
paramagnetic and ferromagnetic states are distinguished by the Curie
temperature, we can extend the existing spin-rotation theory by incorporating
temperature modulation based on the Landau phase transition theory. This
approach enables us to explore the spin-lattice dynamics behavior of the
paramagnetic state when high-temperature conditions are taken. However, the
transfer mechanism of angular momentum within nuclear systems is significantly
complicated by the presence of strong interactions and electromagnetic
interactions, although the principle of angular momentum conservation remains
valid at any level. A possible approach to exploring the Barnett effect in
nuclear systems is through the equation of motion for nuclear spin, which
incorporates nuclear spin-orbit interactions. Of course, the specific issue
falls beyond the scope of this paper, and requires further in-depth research
in conjunction with knowledge of atomic nuclear physics [47, 48, 47, 49].
Besides, Ref. [50] has experimentally demonstrated the coupling between the
mechanical rotation of the liquid metal mercury and the electron spin. The
form of the spin-rotation coupling term is consistent with that of the Barnett
effect in solids, which can be regarded as the Barnett effect in liquids.
During this process, the angular momentum is transferred from the fluid to the
spin, resulting in the generation of a spin current [51].

##  V CONCLUSIONS

In summary, we have successfully employed the SRC mechanism to establish a
spin-lattice dynamical system capable of demonstrating the transfers of
angular momentum and energy associated with the EdH effect and the Barnett
effect. Through the utilization of classical field theory, our proposed
approach exhibits remarkable versatility, allowing for its application to
various elastic materials and accommodation of different initial conditions,
including both damped and undamped scenarios.

Our calculations reveal that the transfer of angular momentum from spins to
the entire lattice occurs on a timescale of approximately 0.01 ns for a disk
with a radius of 100 nm. Furthermore, we observe that the evolution frequency
of the system exhibits a linear dependence on the strength of the magnetic
field, with the slope determined by the Heisenberg interaction, the spin-
lattice coupling, and the initial state. In the presence of damping, the spin-
relaxation time shows an inverse relationship with the magnetic field,
resembling the typical dissipation time described in the LLG equation.
Additionally, when the rotation of the disk and the external magnetic field
are simultaneously used as the driving source of the system, the two effects
compete with each other, and the critical magnetic field is determined by the
intrinsic evolution of the system. We also find that increasing the ratio of
Young’s modulus to Poisson’s coefficient can quantitatively raise the
frequency, while the exchange interaction has no impact on it. Recently,
density functional theory has been applied to investigate the influence of
displacement field or strain on the spin exchange interaction [52, 53, 54,
55], which can be taken into account in our approach to make the calculations
more realistic. In addition, the incorporation of phonon spin angular momentum
into our system would be valuable to further complement the microscopic
mechanisms behind the ultrafast demagnetization phenomenon.

The present work offers an intuitive and fundamental study on the spin-
rotation effect, which has the potential to inspire further insights and
ideas. For instance, the spin-rotation Hamiltonian $H_{\mathrm{s-r}}$ in Eq.
(9) currently considers only low-order coupling, while higher-order effects,
including but not limited to those from the Zeeman term, exchange term [27],
need to be further investigated. In this regard, we suggest that the vibration
of the disk in the $z$-direction should be duly considered to conserve the
three angular momentums in the $r$, $\theta$, and $z$ directions. The
additional spin-lattice mechanisms are also allowed by our approach, as long
as they can be characterized by the spin field, displacement field, stress, or
strain. On the other hand, the existence of the EdH effect in topological
magnon insulator [56] and the mechanical spin currents [57, 58, 59, 60]
generated by the spin-rotation effect have been predicted theoretically, and
both are currently active research areas.

## Acknowledgements

We thank Kun Cao, Muwei Wu, and Shanbo Chow for the helpful discussions. X.
N., J. L., and D. X. Y. are supported by NKRDPC-2022YFA1402802,
NKRDPC-2018YFA0306001, NSFC-92165204, NSFC-11974432, and Leading Talent
Program of Guangdong Special Projects (201626003). T.D. acknowledges
hospitality of KITP. Apart of this research was completed at KITP and was
supported in part by the National Science Foundation under Grant No.
NSFPHY-1748958.

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##  Appendix A Elastic dynamics equations in polar coordinates

In polar coordinates, we have the transformation matrix

$\displaystyle\mathcal{R}(\theta)=\left(\begin{array}[]{cccccc}\cos{\theta}&\sin{\theta}\\\
-\sin{\theta}&\cos{\theta}\end{array}\right),$ (S3)

and

$\displaystyle\left(\begin{array}[]{cccccc}\nabla_{r}\\\
r^{-1}\nabla_{\theta}\end{array}\right)=\mathcal{R}(\theta)\left(\begin{array}[]{cccccc}\nabla_{x}\\\
\nabla_{y}\end{array}\right),~{}\left(\begin{array}[]{cccccc}\bm{e}_{r}\\\
\bm{e}_{\theta}\end{array}\right)=\mathcal{R}(\theta)\left(\begin{array}[]{cccccc}\bm{e}_{x}\\\
\bm{e}_{y}\end{array}\right),~{}\left(\begin{array}[]{cccccc}u_{r}\\\
u_{\theta}\end{array}\right)=\mathcal{R}(\theta)\left(\begin{array}[]{cccccc}u_{x}\\\
u_{y}\end{array}\right).$ (S16)

The stain tensor in this coordinate system is just the rotated result using
$\mathcal{R}(\theta)$,

$\displaystyle\left(\begin{array}[]{cccccc}\tilde{u}_{rr}&\tilde{u}_{\theta
r}\\\ \tilde{u}_{r\theta}&\tilde{u}_{\theta\theta}\end{array}\right)$
$\displaystyle=\mathcal{R}(\theta)\left(\begin{array}[]{cccccc}u_{xx}&u_{xy}\\\
u_{yx}&u_{yy}\end{array}\right)\mathcal{R}^{T}(\theta)$ (S21)
$\displaystyle=\mathcal{R}(\theta)\left[\left(\begin{array}[]{cccccc}\nabla_{x}\\\
\nabla_{y}\end{array}\right)\left(u_{x},u_{y}\right)\right]\mathcal{R}^{T}(\theta)$
(S24)
$\displaystyle=\left(\begin{array}[]{cccccc}\nabla_{r}u_{r}&\nabla_{r}u_{\theta}\\\
(\nabla_{\theta}u_{r}-u_{\theta})/r&(\nabla_{\theta}u_{\theta}+u_{r})/r\end{array}\right).$
(S27)

In general, $\tilde{u}_{\theta r}\neq\tilde{u}_{r\theta}$, because the
rotation generated by the coordinate transformation is a field rather than a
uniform rotation everywhere, thus the following symmetric strain tensor $u$ is
defined as

$u_{ab}=(\tilde{u}_{ab}+\tilde{u}_{ba})/2,$ (S28)

where $a$, $b$ represent the direction indexes. Explicitly,

$\displaystyle u_{rr}$ $\displaystyle=\frac{\partial u_{r}}{\partial r},$
(S29) $\displaystyle u_{\theta\theta}$
$\displaystyle=\frac{u_{r}}{r}+\frac{1}{r}\frac{\partial
u_{\theta}}{\partial\theta},$ (S30) $\displaystyle u_{r\theta}=u_{\theta r}$
$\displaystyle=\frac{1}{2}\left(\frac{1}{r}\frac{\partial
u_{r}}{\partial\theta}+\frac{\partial u_{\theta}}{\partial
r}-\frac{u_{\theta}}{r}\right).$ (S31)

Under the assumption of plain strain conditions and considering only
$\theta$-invariant solutions, the above expressions hold $u_{az}=u_{za}=0$,
$\nabla_{\theta}u_{a}=0$, where $a\in\\{r,\theta,z\\}$. Since $C$ is
rotationally invariant, we still have $\sigma=Cu$. i.e.

$\displaystyle\sigma_{rr}$
$\displaystyle=\frac{E}{(1-\nu^{2})}\left[\frac{\partial u_{r}}{\partial
r}+\nu\frac{u_{r}}{r}\right],$ (S32) $\displaystyle\sigma_{\theta\theta}$
$\displaystyle=\frac{E}{(1-\nu^{2})}\left[\nu\frac{\partial u_{r}}{\partial
r}+\frac{u_{r}}{r}\right],$ (S33) $\displaystyle\sigma_{r\theta}$
$\displaystyle=\frac{E}{2(1+\nu)}\left(\frac{\partial u_{\theta}}{\partial
r}-\frac{u_{\theta}}{r}\right).$ (S34)

Therefore, the equations of motion in polar coordinates are

$\displaystyle\rho\frac{\partial^{2}u_{a}}{\partial t^{2}}-f^{(R)}_{a}=$
$\displaystyle\mathcal{R}_{a\alpha}\nabla_{\beta}\sigma^{(M)}_{\alpha\beta}$
$\displaystyle=$
$\displaystyle\nabla_{\beta}\sigma^{(M)}_{a\beta}-(\nabla_{\beta}\mathcal{R}_{a\alpha})\sigma^{(M)}_{\alpha\beta}$
$\displaystyle=$ $\displaystyle(\mathcal{R}^{T})_{\beta
b}\nabla_{b}(\sigma^{(M)}_{ac}\mathcal{R}_{c\beta})-(\mathcal{R}^{T})_{\beta
b}(\nabla_{b}\mathcal{R}_{a\alpha})\sigma^{(M)}_{\alpha
c}\mathcal{R}_{c\beta}$ $\displaystyle=$
$\displaystyle\nabla_{b}\sigma^{(M)}_{ab}+(\mathcal{R}^{T})_{\alpha
b}(\nabla_{b}\mathcal{R}_{c\alpha})\sigma_{ac}^{(M)}-(\mathcal{R}^{T})_{\alpha
c}(\nabla_{b}\mathcal{R}_{a\alpha})\sigma^{(M)}_{cb},$ (S35)

here $a$ and $b$, $\alpha$ and $\beta$ refer to the direction indexes in the
polar and Cartesian coordinate systems, respectively. Note that
$\nabla_{a}\mathcal{R}_{b\alpha}$ is non-zero only when $a=\theta$, then

$\displaystyle(\mathcal{R}^{T})_{\alpha
b}(\nabla_{b}\mathcal{R}_{c\alpha})\sigma_{ac}^{(M)}=$ $\displaystyle
r^{-1}\sigma^{(M)}_{ac}(\nabla_{\theta}\mathcal{R}_{c\alpha})(\mathcal{R}^{T})_{\alpha\theta}$
$\displaystyle=$ $\displaystyle
r^{-1}\sigma^{(M)}_{ac}\begin{pmatrix}-\sin\theta&\cos\theta\\\
-\cos\theta&-\sin\theta\\\
\end{pmatrix}_{c\alpha}\begin{pmatrix}-\sin\theta\\\
\cos\theta\end{pmatrix}_{\alpha\theta}$ $\displaystyle=$ $\displaystyle
r^{-1}\begin{pmatrix}\sigma_{ar}^{(M)}\\\ 0\end{pmatrix},$ (S36)
$\displaystyle-(\mathcal{R}^{T})_{\alpha
c}(\nabla_{b}\mathcal{R}_{a\alpha})\sigma^{(M)}_{cb}=$
$\displaystyle-r^{-1}\sigma^{(M)}_{c\theta}\mathcal{R}_{c\alpha}(\nabla_{\theta}(\mathcal{R}^{T})_{\alpha
a})$ $\displaystyle=$
$\displaystyle-r^{-1}\sigma^{(M)}_{c\theta}\begin{pmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\\\
\end{pmatrix}_{c\alpha}\begin{pmatrix}-\sin\theta&-\cos\theta\\\
\cos\theta&-\sin\theta\\\ \end{pmatrix}_{\alpha a}$ $\displaystyle=$
$\displaystyle r^{-1}\begin{pmatrix}0&-\sigma_{\theta\theta}\\\
\sigma_{r\theta}&0\\\ \end{pmatrix}.$ (S37)

Finally, we get

$\displaystyle\frac{\partial\sigma_{rr}}{\partial
r}+\frac{1}{r}(\sigma_{rr}-\sigma_{\theta\theta})+f_{r}=\rho\frac{\partial^{2}u_{r}}{\partial
t^{2}}$ (S38) $\displaystyle\frac{\partial\sigma_{r\theta}}{\partial
r}+2\frac{\sigma_{r\theta}}{r}+f_{\theta}=\rho\frac{\partial^{2}u_{\theta}}{\partial
t^{2}}.$ (S39)

By substituting Eq. (S34) into Eq. (S39), we can get the equation of motion
for $u_{\theta}$,

$\displaystyle\frac{\partial^{2}u_{\theta}}{\partial
r^{2}}+\frac{1}{r}\frac{\partial u_{\theta}}{\partial
r}-\frac{1}{r^{2}}u_{\theta}=\frac{2(1+\nu)}{E}\left(\rho\frac{\partial^{2}u_{\theta}}{\partial
t^{2}}-f_{\theta}\right),$ (S40)

where $f_{\theta}=\frac{\hbar}{2}\frac{\partial\dot{S}^{\prime}_{z}}{\partial
r}$ is determined by spin evolution.

##  Appendix B Energy loss from damping

The energy losses resulting from displacement damping and spin damping need to
be compensated by calculation. In Section IV.1.1, the drag force induced by
$\eta$ is $F_{\mathrm{d}}^{\theta}=-\eta~{}{\partial u_{\theta}}/{\partial
t}$. By utilizing the principle that energy power is equal to the product of
force and velocity, we can estimate the energy loss as follows,

$\displaystyle{E}_{1}=-\eta\int dt\int d^{3}r(\frac{\partial
u_{\theta}}{\partial t}\times\frac{\partial u_{\theta}}{\partial t}).$ (S41)

The energy loss directly brought about by spin damping $\zeta$ is solved like
the solution of $E_{d}$ in Section IV.2,

$E_{2}=-Ia^{2}\int dt~{}~{}[\frac{\zeta}{\hbar
S}(\bm{S}\times\bm{\Gamma})\times\bm{S}]\cdot\nabla^{2}\bm{S}.$ (S42)

Besides, as the driving force is determined by the spin evolution, the energy
of the displacement field is affected by the spin damping. This part of energy
loss can be estimated in a similar manner as $E_{1}$,

$E_{3}=\int dt\int d^{3}r\frac{\zeta}{2S^{\prime}}(\frac{\partial}{\partial
r}[(\bm{S}\times\Gamma)\times\bm{S}]_{z})\frac{\partial u_{\theta}}{\partial
t}.$ (S43)