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# Low-Frequency Signature of Magnetization Nutation in Nanomagnets

M. P. Adams, R. Bastardis, A. Michels, H. Kachkachi

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## To cite this version:

M. P. Adams, R. Bastardis, A. Michels, H. Kachkachi. Low-Frequency Signature of Magnetization Nutation in Nanomagnets. 2024. hal-04750967

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Distributed under a Creative Commons Attribution 4.0 International License
Low-frequency signature of magnetization mutation in nanomagnets

M. P. Adams¹,*, R. Bastardis²,† A. Michels¹,‡ and H. Kachkachi²§

¹Department of Physics and Materials Science, University of Luxembourg,
162A avenue de la Faiencerie, L-1511 Luxembourg, Grand Duchy of Luxembourg and

²Université de Perpignan via Domitia, Lab. PROMES CNRS UPR8521,
Rambla de la Thermodynamique, Tecnosud, 66100 Perpignan, France

In this work, we show that surface anisotropy in nanomagnets induces a nutational motion of their magnetization at various frequencies, the lowest of which can be described by the macrospin model whose dynamics is governed by an effective energy potential. We derive analytical expressions for the precession and nutation frequencies and amplitudes as functions of the size of the nanomagnet and its atomistic parameters, such as the exchange coupling and the onsite anisotropy. Our analytical model predicts a reduction of the precession frequency with increased surface anisotropy. We also simulate the dynamics of the corresponding atomistic many-spin system and compare the results with the effective model. We thereby show that the first nutation mode induced by the finite size and surface anisotropy occurs at a frequency that is four times larger than the precession frequency, thus lending itself to a relatively easy detection by standard experiments of magnetic resonance.

# I. INTRODUCTION

Nutation is the well known motion of a gyroscope described by classical mechanics; it develops whenever an external force tilts the rotation axis away from the direction of the gravity field, since then the rotation axis no longer coincides with the direction of the angular momentum.

In the case of a magnetic material, magnetization mutation occurs whenever a time-dependent magnetic field (rf or microwave field) is present in the system. Accordingly, it has been demonstrated that there appears the fundamental effect of transient nutations in NMR [1], EPR [2–4] and optical resonance [5]. Spin mutation was first predicted in Josephson junctions [6–10] and was later developed using various approaches based on relativistic quantum mechanics, first principles [11, 12], and electronic structure theory [13–17]. A more recent macroscopic approach [18–21] deals with the magnetization mutation by accounting for magnetic inertia through the addition of a second-order time derivative of the magnetization in the Landau-Lifshitz-Gilbert equation [22, 23]. On the experimental side, evidence of the effects of nutation has been reported for thin films [24–27] in the THz regime. Indeed, magnetization mutation meets with a strong and expanding interest within the magnetism community owing to the real possibility of addressing and controlling the ultra-fast magnetization dynamics, in view of potential practical applications in high-speed data processing and storage technologies based on magnetic materials. Considering the strong potential of achieving high storage densities, this issue acquires still more enthusiasm for nanoscaled magnetic systems.

The dynamic behavior of nanomagnets requires a good understanding of the intricate interplay between differ-
ent factors influencing the spin dynamics, such as finite-size, boundary, and surface effects. In particular, surface
anisotropy emerges as a crucial parameter that can significantly influence the dynamics of nanomagnets as it affects
the potential energy and thereby the relaxation rates [28, 29]. Recently, it was shown [30, 31] that spin misalignment,
induced by surface effects, triggers nutational motion of the magnetization of a nanomagnet, with frequencies ranging
from tens of GHz to THz, depending on various physical parameters, the predominant among them being the spin-
spin exchange coupling and the on-site surface anisotropy. For example, in Refs. [31, 32] it was shown that surface
anisotropy in nanocubes triggers absorption modes of higher frequency than the ordinary uniform precessional motion
with frequencies f_p ~10 GHz (conventional ferromagnetic resonance). More precisely, the authors of Ref. [31] observed
a resonance at a frequency f_c that is 4 times f_p and another frequency f_n in the THz regime. While f_n is related to
atomic spin fluctuations and their exchange coupling, the frequency f_c emerges from the cubic anisotropy induced by
spin inhomogeneities at the surface of the nanomagnet [33–35]. As such, f_c corresponds to the nutational mode with
the lowest frequency and should be more easily detected in resonance experiments on, e.g. arrays of nanomagnets
with acute surface effects, such as platelets, pillars, spheres, or cubes.

In the present work, we focus on the frequency *f*<sub>c</sub> and derive its analytical expression using the effective macrospin model whose potential energy is a polynomial in the components of the net magnetic moment, with coefficients that are

<footnote>* Electronic address: michael.adams@uni.lu</footnote>

<footnote>† Electronic address: roland.bastardis@univ-perp.fr</footnote>

<footnote>‡ Electronic address: andreas.michels@uni.lu</footnote>

<footnote>§ Electronic address: hamid.kachkachi@univ-perp.fr</footnote>
functions of the spin-spin exchange coupling and on-site anisotropy constants [33–36]. The results obtained using the effective model are compared with those of numerical simulations for the corresponding many-spin nanomagnet. The effective model, thus validated, allows us to derive analytical expressions for the precession and nutation frequencies and the nutation amplitude and to study them in terms of the nanomagnet size and surface anisotropy.

Plan of the paper: We first define the many-spin and macrospin approaches and establish the connection between them. Then, we discuss the numerical results from both models for the time trajectories of the net magnetic moment. In particular, we discuss the results regarding the magnetization nutation and its dependence on the particle size and surface anisotropy. This study allows us to validate the effective model which is then used to derive the analytical expressions for the precession and nutation frequencies and amplitudes. From these expressions and further simulations, we draw the main conclusions of the present work.

## II. MODEL FOR THE MAGNETIZATION DYNAMICS

### A. Many-spin approach (MSP)

In the many-spin problem (MSP) or many-spin approach, or still atomistic approach, the nanomagnet is regarded as a crystallite of a given shape and size, containing $N$ atomic spins $S_i = \mu_a s_i$ located at the nodes of a given crystalline structure; $\mu_a$ denotes the atomic magnetic moment and $s_i$ is a unit vector in the direction of $S_i$. The magnetic state of such a system is studied with the help of the atomistic Hamiltonian [37–46]:

$$ \begin{aligned} \mathcal{H} &= \mathcal{H}_{\text{exch}} + \mathcal{H}_{\text{Z}} + \mathcal{H}_{\text{A}} \\ &= -\frac{1}{2} J \sum_{\langle i,j \rangle} s_i \cdot s_j - \mu_a \mathbf{B}_0 \cdot \sum_{i=1}^{N} s_i + \sum_{i=1}^{N} \mathcal{H}_{\text{A},i}, \end{aligned} \quad (1) $$

where $\mathcal{H}_{\text{exch}}$ is the nearest-neighbor (n.n.) exchange energy with $J > 0$, $\mathcal{H}_{\text{Z}}$ denotes the Zeeman energy with $\mathbf{B}_0 = \mu_0 \mathbf{H}_0$ being the homogeneous externally applied magnetic field, and $\mathcal{H}_{\text{A}}$ the magnetic anisotropy energy. For the core spins, we assume the anisotropy to be of uniaxial symmetry with constant $K_c$, while for surface spins we adopt the model proposed by Néel [47] with constant $K_s$. $\mathcal{H}_{\text{A},i}$ is then given by

$$ \mathcal{H}_{\text{A},i} = \begin{cases} -K_c (\mathbf{s}_i \cdot \mathbf{e}_z)^2, & i \in \text{core} \\ +\frac{1}{2} K_s \sum_{j \in \text{n.n.}} (\mathbf{s}_i \cdot \mathbf{u}_{ij})^2, & i \in \text{surface}, \end{cases} \quad (2) $$

where the unit vectors $\mathbf{u}_{ij} = (\mathbf{r}_i - \mathbf{r}_j)/\|\mathbf{r}_i - \mathbf{r}_j\|$ connect the nearest-neighbor spins “i” and “j”. All physical parameters are measured in units of energy per atom.

The (undamped) dynamics of the many-spin system is governed by the following Larmor equation:

$$ \frac{ds_i}{d\tau} = s_i \times \mathbf{b}_{\text{eff},i} \quad (3) $$

with the (normalized) local effective field $\mathbf{b}_{\text{eff},i}$ acting on $s_i$ being defined by $\mathbf{b}_{\text{eff},i} = -\delta\mathcal{H}/\delta s_i$; $\tau$ is the reduced time defined by:

$$ \tau \equiv \frac{t}{\tau_s}, \quad (4) $$

where $\tau_s = \mu_a/(\gamma J)$ is a characteristic time of the system’s dynamics. By way of example, for cobalt $J = 8$ meV, leading to $\tau_s = 70$ fs. Henceforth, we will only use the dimensionless time $\tau$. Accordingly, the frequency $\omega = 2\pi f = 2\pi/T$ is measured in units of $\tau_s^{-1}$ and, as such, in the sequel $\nu \equiv \tau_s\omega$. Note that in these units, $b_{\text{eff}}$ is equal to the effective field (in Tesla) multiplied by $\mu_a/J$ and is thus dimensionless.

To study the dynamics of this MSP, for arbitrary values of all physical parameters, one resorts to numerical methods for solving Eq. (3) using, for instance, the iterative routine based on the 4th-order Runge-Kutta scheme in combination with the projection step $s_i^{\nu+1} = s_i^{\text{RK4}}/\|s_i^{\text{RK4}}\|$ to preserve the constraint $\|s_i\| = 1$. The net magnetic moment $\mathbf{m}$ (unit vector) is then computed using

$$ \mathbf{m} = \frac{\sum_i s_i}{\| \sum_i s_i \|}. \quad (5) $$
For the initial state of the system, we choose a coherent spin configuration along the orientation $(\theta_0, \phi_0)$, i.e. $s_i = (\sin \theta_0 \cos \phi_0, \sin \theta_0 \sin \phi_0, \cos \theta_0)$, for all $i = 1, \dots, N$.

Nutational motion was investigated in Ref. [31] using this MSP model and the results were compared with those of the macrospin model studied with the help of the augmented Landau-Lifshitz-Gilbert equation. In the present work, our aim is to derive analytical expressions for the precession and nutation frequencies in terms of the atomistic parameters $J, K_c, K_s$ and the size of the nanomagnet. For this purpose, we resort to the effective one-spin problem derived in Refs. [33–36] and compare the corresponding results with those obtained for the MSP in Ref. [31] in regard with the nutational motion.

## B. Effective macrospin approach (EOSP)

The effective macrospin approach, or effective one-spin problem (EOSP), is obtained from the many-spin problem under certain conditions regarding the surface anisotropy, which should not be too strong with respect to the exchange coupling, and the particle size, which should not be too small (see Refs. [33–36] for details). The EOSP consists of the net magnetic moment Eq. (5) evolving in an effective energy potential given by

$$ \mathcal{H}_{\text{eff}} \approx -k_2 m_z^2 + k_4 \sum_{\alpha=x,y,z} m_{\alpha}^4. \qquad (6) $$

The values and signs of the (effective) coefficients $k_2$ and $k_4$ are functions of the atomistic parameters $J, K_c, K_s$, in addition to the size and shape of the nanomagnet and its underlying crystal lattice [35]. In the following, we will use the dimensionless constants defined by $k_c = K_c/J$ and $k_s = K_s/J$, so that $k_2$ and $k_4$ are dimensionless.

For a nanomagnet cut out of a simple cubic lattice, we have [33, 48]

$$ k_2 = k_c \frac{N_c}{N}, \quad k_4 \approx \begin{cases} \kappa \frac{k_s^2}{z}, & \text{sphere,} \\ (1 - 0.7/N^{1/3})^4 \frac{k_s^2}{z}, & \text{cube,} \end{cases} \qquad (7) $$

where $z = 6$ is the coordination number, $\kappa$ represents a (dimensionless) surface integral, and $N_c$ is the number of atoms in the core of the nanomagnet (with full coordination), while $N$ is the total number of atoms (including both the core and surface). Note that $\kappa$ was derived in Ref. 33 in the absence of core anisotropy. As shown in Ref. [49], in the presence of the latter, the spin misalignment caused by surface anisotropy does not propagate to the center of the nanomagnet; it is “fended off” by the uniaxial anisotropy in the core which tends to align all spins together. The result of this competition is that $k_4$ (or $\kappa$) is multiplied by the factor $N_c/N$. Likewise, for both the cube and sphere, $k_2$ may be approximated [34] by the first expression in Eq. (7).

From the effective Hamiltonian in Eq. (6), we derive the following effective field that drives the dynamics of the magnetic moment $\mathbf{m}$

$$ \mathbf{b}_{\text{eff}} = 2k_2 m_z \mathbf{e}_z - 4k_4 (m_x^3 \mathbf{e}_x + m_y^3 \mathbf{e}_y + m_z^3 \mathbf{e}_z) \qquad (8) $$

through the Larmor equation of motion for $\mathbf{m}$ as follows:

$$ \frac{d\mathbf{m}}{d\tau} = \mathbf{m} \times \mathbf{b}_{\text{eff}}. \qquad (9) $$

We would like to emphasize that the EOSP described here is not some variant of a single-domain macrospin model. Indeed, it is a macrospin model, but with coefficients that are functions of the atomistic physical parameters. In other terms, it has the advantage to be a "simple" macroscopic model which inherits (to some extent) the nanomagnet atomistic features, especially spin inhomogeneities.

# III. MAGNETIZATION NUTATION

## A. MSP versus EOSP

In this section, we compare the dynamics of the net magnetic moment of different many-spin nanomagnets with that of the corresponding effective models for spheres [50], cubes [32, 51–56] and truncated octahedrons [35, 57, 58].
<figure><img src="image_4.png" /><figcaption>Figure 1. Components of the net magnetic moment for both MSP (continued lines) and EOSP (dashed lines and symbols) for the spherical nanomagnet with a sc lattice and size $N = 1088$. The constants $k_c$ and $k_s$ are indicated in the legend and the effective coefficients are given in Table I. On the right, $T_p$, $T_c$, $T_n$ denote the periods corresponding to the respective frequencies $f_p$, $f_c$, $f_n$ mentioned in the text.</figcaption></figure>

<table><caption>Table I. Effective parameters used for the simulations on the nanospheres [from Eq. (7)].</caption><thead><tr><th>$N = 1088$ ($N_c = 706$)</th><th>$k_2$</th><th>$k_4$</th><th>$\omega_c$ (GHz)</th></tr></thead><tbody><tr><td>$k_c = 0.001, k_s = 0.01$</td><td>$6.5 \times 10^{-4}$</td><td>$5.8 \times 10^{-6}$</td><td>51.5</td></tr><tr><td>$k_c = 0.01, k_s = 0.1$</td><td>$6.5 \times 10^{-3}$</td><td>$5.81 \times 10^{-4}$</td><td>515</td></tr></tbody></table>

Before discussing the results in more detail, let us summarize the procedure followed for the comparison between the MSP and EOSP. For a given nanomagnet with a given size, shape, underlying lattice, and energy parameters ($J$, $K_c$, $K_s$), we solve the undamped Larmor equation (3) for all atomic spins $s_i$ starting from the initial state ($\theta_0$, $\phi_0$) taken here to be $(\pi/4, \pi/4)$. Then, we plot the components $m_\alpha$ (with $\alpha = x, y, z$) of the net magnetic moment (of the MSP) as functions of time<fnref n="1" />. Next, for a cube or a sphere with a simple cubic lattice, we use the expressions for

<figure><img src="image_5.png" /><figcaption>Figure 2. Components of the net magnetic moment for both MSP (continued lines) and EOSP (dashed lines and symbols) for a cube-shaped nanomagnet with a sc lattice and sizes $N = 15^3$ (left) and $N = 9^3$ (right). The constants $k_c$ and $k_s$ are indicated in the legend and the effective coefficients are given in Table II.</figcaption></figure>

<footnote n="1">¹ Note that in all plots we denote by $s_\alpha$, $\alpha = x, y, z$, the Cartesian components of the net magnetic moment of the MSP and by $m_\alpha$, $\alpha = x, y, z$, those of the magnetic moment of the EOSP.</footnote>
<table><caption>Table II. Effective parameters used for the simulations on the nanocubes [from Eq. (7)].</caption><thead><tr><th><i>N</i></th><th><i>N</i><sub>c</sub></th><th><i>k</i><sub>2</sub></th><th><i>k</i><sub>4</sub></th><th><i>ω</i><sub>c</sub>(GHz)</th></tr></thead><tbody><tr><td>729 = 9<sup>3</sup></td><td>343</td><td>4.7 × 10<sup>-4</sup></td><td>12 × 10<sup>-6</sup></td><td>355</td></tr><tr><td>3375 = 15<sup>3</sup></td><td>2197</td><td>6.5 × 10<sup>-4</sup></td><td>13.8 × 10<sup>-6</sup></td><td>52</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Figure 3. Components of the net magnetic moment for both MSP (continued lines) and EOSP (dashed lines and symbols) for the truncated octahedron with fcc lattice and size $N = 2075$. The constants $k_c$ and $k_s$ are indicated in the legend and the effective coefficients are given in Table III.</figcaption></figure>

$k_2$ and $k_4$ in Eqs. (7) to solve the Larmor Eq. (9) for the EOSP and obtain the components $m_\alpha$ as functions of time. For other shapes and/or lattice structures, such as the truncated octahedron or cubes with e.g. body-centered-cubic lattice [see Fig. 3], $k_4$ is obtained by fitting the MSP data. Note that in the case of an underlying bcc or fcc lattice, $k_4$ becomes negative [35].

The results of such a comparison confirm the following facts: the net magnetic moment of the many-spin nanomagnet exhibits three oscillation modes with different frequencies [see Fig. 1 right]: the precession frequency $f_p$ for the $x, y$ components and the nutation frequencies $f_c$ and $f_n$ for the $z$ component. The data in Figs. 1, 2, 3 are plotted as continued lines, in black and red for $s_{x,y}$ and in blue for $s_z$. The corresponding EOSP magnetic moment exhibits only two oscillation frequencies: the precession frequency $f_p$ for $m_{x,y}$ (in red and black symbols) and the (smaller) nutation frequency $f_c$ for $m_z$ (in magenta). This result obtains in all cases of size, shape and other physical parameters, as confirmed in Figs. 1, 2 and 3. We see that as far as $f_p$ and $f_c$ are concerned, and this corresponds to the validity domain of the EOSP (i.e., relatively weak surface anisotropy), the dynamics of EOSP perfectly matches that of the MSP. However, the latter exhibits (in blue) an extra wiggling motion in the component $s_z$ (with frequency $f_n$) on top of a signal with frequency $f_c$. As mentioned in the introduction and discussed in Ref. [31], the appearance of the frequency $f_c$ is due to the quartic term in the Hamiltonian (6), which is a consequence of spin disorder induced by surface anisotropy in the EOSP regime; the coefficient of this term ($k_4$) is a function of $K_s$, as can be seen from Eq. (7). On the other hand, $f_n$ is due to the fluctuations of the individual atomic spins with frequencies on the order of the exchange coupling $J$.

Further analysis of Figs. 1, 2, 3 reveals further useful information. First, the two frequencies $f_p$ and $f_c$ change with size, shape, and energy parameters. As the size decreases, the ratio $N_c/N$ decreases, and the surface contribution to the overall energy increases [49]. Second, as the surface anisotropy constant $K_s$ increases, the nutation frequency $f_n$ is more clearly identified with an increasing amplitude. This effect is more clearly seen in Figs. 1 and 3 (plots on the right). However, the value of the frequency itself changes with the exchange coupling. Likewise, when going from the cube to the more rounded geometry of the sphere, through the truncated octahedron, we see that the nutational motion is progressively enhanced and its amplitude increases.
<table><caption>Table III. Effective parameters used for the simulations on the truncated octahedrons obtained by a fitting procedure. Note that $k_4$ is negative for the fcc lattice [35].</caption><thead><tr><th>N = 2075</th><th>k<sub>2</sub></th><th>k<sub>4</sub></th><th>ω<sub>c</sub>(GHz)</th></tr></thead><tbody><tr><td>k<sub>c</sub> = 0.001, k<sub>s</sub> = 0.01</td><td>6.7 × 10<sup>-4</sup></td><td>-4.5 × 10<sup>-6</sup></td><td>54.6</td></tr><tr><td>k<sub>c</sub> = 0.01, k<sub>s</sub> = 0.1</td><td>6.7 × 10<sup>-3</sup></td><td>-2.9 × 10<sup>-4</sup></td><td>546</td></tr></tbody></table>

## B. Nutation frequency and amplitude: analytical approach

From the previous section we conclude that the effective model (EOSP) perfectly describes the dynamics of the many-spin nanomagnet as far as the validity conditions are met (i.e. small surface anisotropy). On the other hand, we have seen that, in addition to the nutation frequency $f_c$, the MSP exhibits another nutation mode with a much higher frequency $f_n$; the amplitude of this mode increases with the enhancement of surface effects, either through an increase of the anisotropy constant or that of the number of surface spins.

Now, with the macroscopic EOSP model at hand, we can derive analytical expressions for the frequency and amplitude of the precession and nutation modes and study their behavior in terms of surface anisotropy and size. Instead of the Cartesian components $m_x, m_y, m_z$ of $\mathbf{m}$, it is more convenient to parametrize the vector $\mathbf{m}$ using the spherical coordinates $\mathbf{q} = (\theta, \phi)$, with $\theta$ being the polar angle and $\phi$ the azimuth angle, i.e., $\mathbf{m}(\mathbf{q}) = (\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$. Then, in this coordinate system, the Larmor Eq. (9) becomes

$$ \frac{d\mathbf{q}}{d\tau} = \mathbf{b}'_{\text{eff}}(\mathbf{q}), \qquad (10) $$

where we have defined $\mathbf{b}'_{\text{eff}} = (h_\theta, h_\phi)$ as the new effective driving field. More explicitly, the components of the spherical driving field $\mathbf{b}'_{\text{eff}}$ are given by

$$ \mathbf{b}'_{\text{eff},\theta}(\mathbf{q}) = k_4 \sin(4\phi) \sin^3\theta, \qquad (11) $$

$$ \mathbf{b}'_{\text{eff},\phi}(\mathbf{q}) = 2k_2 \cos\theta - k_4 \cos\theta[7\cos^2\theta - \sin^2\theta \cos(4\phi) - 3]. $$

In Eq. (11), the appearance of trigonometric functions with the argument $4\phi$ already hints to the fact that we should observe a mode with a frequency four times that of the “fundamental” mode, namely the precession mode.

To analytically solve the Larmor equation (10), we make the following Ansatz for the angular motion:

$$ \mathbf{q}_p(\tau) = \begin{pmatrix} \theta_p(\tau) \\ \phi_p(\tau) \end{pmatrix} = \begin{pmatrix} \theta_0 \\ \phi_0 + \nu_p \tau \end{pmatrix}, \qquad (12) $$

where $\omega_p$ is the (circular) precession frequency. Note that this Ansatz is only valid for $k_4 = 0$. For finite but small $k_4$, we assume that the dynamics is approximately governed by the equation

$$ \frac{d\mathbf{q}}{d\tau} \approx \mathbf{b}'_{\text{eff}}[\mathbf{q}_p(\tau)]. \qquad (13) $$

More explicitly, this yields:

$$ \frac{d\theta}{d\tau} \approx -k_4 \sin^3 \theta_0 \sin(4\nu_p \tau), \qquad (14) $$

$$ \frac{d\phi}{d\tau} \approx -\cos\theta_0[2k_2 + 3k_4 - 7k_4\cos^2\theta_0] - k_4\cos\theta_0\sin^2\theta_0\cos(4\nu_p\tau). $$

From these approximate solutions, we infer the following equations:

$$ \nu_p = \cos \theta_0 [2k_2 + (3 - 7 \cos^2 \theta_0) k_4], \qquad (15) $$

$$ \nu_c = 4\nu_p, $$

$$ a_{\phi} = \frac{1}{4} \frac{k_4 \sin^2 \theta_0}{2k_2 + (3 - 7 \cos^2 \theta_0) k_4}, \quad a_{\theta} = a_{\phi} \tan \theta_0, $$

where $\nu_p$ and $\nu_c$ are the (dimensionless) precession and nutation frequencies, and $a_\theta, a_\phi$ the amplitudes. Note that in situations where $k_4$ becomes negative, the latter should be replaced by $|k_4|$ in the expression of the amplitude $a_\phi$.
<figure><img src="image_7.png" /><figcaption>Figure 4. EOSP parameter $k_4$, nutation frequency $\omega_c$, and amplitude $a_\theta$ as a function of the linear size $N$ of a bcc iron cube of size $N \times N \times N$ ($k_c = 0.001$ and $k_s = 0.01$). The parameters were obtained from a fit of the many-spin nanomagnet results to Eqs. (15) and (16). The full curves are guides to the eye.</figcaption></figure>

Integrating Eqs. (14) yields the time trajectories of the net magnetic moment:

$$ q(\tau) \approx \begin{pmatrix} \theta_0 \\ \phi_0 + \nu_p \tau \end{pmatrix} - \begin{pmatrix} a_\theta [1 - \cos(\nu_c \tau)] \\ a_\phi \sin(\nu_c \tau) \end{pmatrix}. \quad (16) $$

The set of Eqs. (15) and (16) constitutes the original result of this work. Before discussing the conclusions we can draw from them, we summarize how they can be used in practice. Given a (many-spin) nanomagnet of a certain size and shape and a chosen set of energy parameters $J, K_c, K_s$, so that the spin configuration is not too much disordered, we can characterize the (modified) precession and the first nutation modes by their frequencies and amplitudes given by the corresponding effective macrospin model. If the nanomagnet is a nanocube or a nanosphere with the underlying simple-cubic lattice, the effective coefficients $k_2$ and $k_4$ are given by Eqs. (7). In the general case, one may fit the MSP curves $m_\alpha(\tau)$ to Eq. (16) and infer the quantities in Eqs. (15). In fact, once the constant $k_4$ is obtained, all the other quantities ($\nu_p, \nu_c, a_\theta, a_\phi$) can easily be derived. By way of illustration, we have applied the latter procedure to the much studied iron nanocubes [51, 52, 59–61] with an underlying bcc lattice. The results are shown in Fig. 4, where $k_4, \omega_c$ and $a_\theta$ are plotted against the (linear) cube size $N$; the fit for $k_4$ yields $k_4 \approx -12.5k_s^2/8(1+N)^{5/3}$.

From Eq. (15) and the numerical results in Fig. 4, we make the following observations:

* The frequencies $\nu_p$ and $\nu_c$ increase when the size increases but decrease with surface anisotropy, i.e. when $K_s$ ($k_4 \propto k_s^2$) increases, for a small initial polar angle $\theta_0$ (precession angle). For a relatively large angle $\theta_0$ ($\gtrsim 49^\circ$), the frequency becomes an increasing function of $K_s$ (plot not shown).

* The nutation frequency $\nu_c$ is four times the precession frequency $\nu_p$, independently of the nanomagnet's size.

Regarding the amplitude, from Eqs. (15) we see that it increases with the surface anisotropy constant $K_s$, as already concluded from the results of the MSP simulations in Figs. 1, 2, 3. On the other hand, its behavior as a function of the size depends on the shape. In Fig. 4, we see that it is a decreasing function of $N$.

For a cube with simple-cubic lattice, $N_c/N = (1-2/N)^3$ and thereby $a_\phi, a_\theta \sim 1/N$, which means that when the size increases, the nutation mode dies out. This has a very important implication. In general, any bulk or thin magnetic film presents an anisotropy energy that is an expansion in the components of its net magnetic moment, similar to the Hamiltonian in Eq. (6). However, the dynamics of a bulk magnetic material with such a Hamiltonian does not necessarily exhibit magnetic nutation. Here, we see that nanoscaled magnetic systems, such as nanocubes, do develop such a nutational motion at a frequency that should be observable in resonance experiments on arrays of such objects. For a sphere, using $k_2$ and $k_4$ from Eq. (7), Eq. (15) implies that the amplitudes $a_\phi, a_\theta$ do not depend on the size, as has also been confirmed by the MSP numerical simulations. For an arbitrary shape and/or lattice structure, it is not easy to derive simple formulae for the coefficients $k_2, k_4$. Nevertheless, as was discussed above, they may be obtained by fitting the dynamics of the net magnetic moment of the many-spin nanomagnet to Eqs. (16). This is what has been done for the bcc cubic nanomagnets, studied by experimental groups [32, 51–56], and the results are shown in Fig. 4.

IV. CONCLUSION

We have shown that the magnetization of a nanoscaled magnetic material may exhibit nutational motion with frequencies in the GHz to THz range, the lowest of which is four times the precession frequency. We have provided
analytical expressions for precession and nutation frequencies and the amplitude of the nutational oscillations as functions of the size of the nanomagnet and atomistic energy parameters, such as the spin-spin exchange coupling and the on-site surface and core magneto-crystalline anisotropies.

This has been possible owing to the correspondence we have firmly established between the many-spin approach and the effective macrospin approach, a correspondence that has been validated through numerical simulations of the dynamics of various nanomagnets (cubes, spheres, and truncated octahedrons) with different energy parameters and lattice structures. The results of these simulations have confirmed that the effective model recovers very well the dynamics of the many-spin nanomagnet in the low-frequency regime, when surface anisotropy is not too strong. In this regime, we observe the precession frequency $f_p$ and the lowest nutation frequency $f_c$ that is four times $f_p$ and whose existence is related to the effective cubic anisotropy induced by spin inhomogeneities. The many-spin nanomagnet also exhibits a much higher frequency $f_n$ that can be related to the spin-spin exchange coupling. Even the precession frequency $f_p$ turns out to be affected by surface anisotropy, among other parameters. Within the effective macrospin model, we have demonstrated that the amplitude of the nutational motion is enhanced by surface anisotropy, but its behavior with the size depends on the shape of the nanomagnet.

All in all, in this work, we want to emphasize the fact that surface-induced magnetization mutation in nanoscaled magnets, such as nanocubes and nanospheres, already occurs at the lowest frequency that is four times the precession frequency, which itself is altered by spin misalignments. This low-frequency nutational oscillation lends itself to a detection and characterization using standard experimental techniques of magnetic resonance, such as the network analyzer with varying field and frequency, at low temperature. An array of well separated platelets or nanocubes should provide the adequate conditions for such measurements. Inelastic neutron spectroscopy [62] might also be able to resolve the predicted nutational spin dynamics in nanomagnets.

ACKNOWLEDGMENTS

M.P. Adams and A. Michels thank the National Research Fund of Luxembourg for financial support (AFR Grant No. 15639149). H. Kachkachi thanks A. Michels and the Department of Physics (Univ. Luxembourg) for the hospitality extended to him during his stay.

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