Timescales in the thermal dynamics of magnetic dipolar clusters

The collective behavior of thermally active structures offers clues on the emergent degrees of freedom and the physical mechanisms that determine the low-energy state of a variety of systems. Here, the thermally active dynamics of magnetic dipoles at square plaquettes is modeled in terms of Brownian oscillators in contact with a heat bath. Solution of the Langevin equation for a set of interacting $x\text{−}y$ dipoles allows the identification of the timescales and correlation length that reveal how interactions, temperature, damping, and inertia may determine the frequency modes of edge and bulk magnetic mesospins in artificial dipolar systems.

Figure 1

Dipolar square plaquette of lattice constant $\frac{\sqrt{2}}{2}(L+2\mathrm{\Delta })$ in the vortex configuration. The dipoles with magnetic moment ${\mathit{m}}^i$, length $L$, and moment of inertia $I$, are represented by black arrows. They rotate with angle ${\alpha }^i$ in the $x\text{−}y$ plane. The system is under finite temperature $T$, and the viscous rotation of the magnets is illustrated by the parameter $\eta$. Dotted (cyan) lines joining dipoles illustrate the magnetic dipolar interaction between them. This interaction gives rise to the magnetic field ${\mathit{B}}^1=(B_{\left|\right|}^1,B_{\perp }^1)$ at the position of the dipole ${\mathit{m}}^1$ yielding the magnetic torque ${\mathcal{T}}_z^1$ responsible of the rotation of ${\mathit{m}}^1$ respect to the $\widehat{z}$ axis.

Figure 2

$\mathcal{C}\left(s\right)$ of the square plaquette relaxing into the vortex configuration shown in Fig. 1. $s=\frac{t}{{\tau }_1}$ is the dimensionless time. (a) Numerical evaluation (ns) of $\mathcal{C}\left(s\right)$ with $\mathrm{\Delta }=0.85$ (in units of $\ell$). Different curves correspond to different values of $T$ [in units of $\mathcal{K}\left(L\right)$]. The different stages of the evolution of $\mathcal{C}\left(s\right)$ are associated to the system time and energy scales as explained in the text. $A$ denotes the amplitude of the early time oscillations. (b) Evaluation of Eq. (9) comparing the early evolution of $\mathcal{C}\left(s\right)$ for three values of $\epsilon$. Panel (c) is like panel (a), but now $T={10}^{-1}$ and different curves correspond to $\mathcal{C}\left(s\right)$ computed at different $\mathrm{\Delta }$. (d) Comparison of the numerical solution of $\mathcal{C}\left(s\right)$ (in red and cyan) with Eq. (9) (in blue and black) during the early stage of thermal relaxation for two values of $\mathrm{\Delta }$ and $T={10}^{-1}$.

Figure 3

(a) Antiferromagnetic vortex state ($\mathrm{AV}$) and (b) ferromagnetic vortex state ($\mathrm{FV}$) for a small cluster made out of four square plaquettes after relaxation. In panels (a) and (b) the red dipoles highlight edge and bulk magnets examined in the text. (c) Dipolar energy density (in units of $\mathcal{K}\left(L\right)$) of the square cluster versus its chirality for several values of $\mathrm{\Delta }.$ (d) Numerical results comparing $\mathcal{C}\left(s\right)$ of edge (in red) and bulk (in blue) dipoles at the sites of the cluster in the $\mathrm{FV}$ state with $\mathrm{\Delta }=1.1$ and at $T=0.06$. The inset shows evaluation of $\mathcal{C}\left(s\right)$ from Eq. (9) of edge (using ${\mathcal{K}}^{\left(E\right)}$) and bulk dipoles (using ${\mathcal{K}}^{\left(B\right)}$) with $\mathrm{\Delta }=1.1$ and $T=0.06$.

Figure 4

Magnetization parallel to the applied field direction. Blue curve shows the result for the cluster in the $\mathrm{AV}$ [Fig. 3], with $\mathrm{\Delta }=1.1$ and $T=6\times {10}^{-2}$, while the red curve shows the results of the lattice in the $\mathrm{FV}$ [Fig. 3], with $\mathrm{\Delta }=1$ and $T=2.2\times {10}^{-2}$. Magnetization $m_x$ is in $m_0$ units and the magnetic field $B_x$ is in units of $\frac{\mathcal{K}\left(\mathrm{\Delta }\right)}{m_0}$.

Figure 5

(a) Magnetization dynamics along the direction parallel to the applied field, of dipoles in the bulk of the clusters in the $\mathrm{AV}$ (blue) and $\mathrm{FV}$ (red) magnetic orders. Panel (b) is like panel (a) but for dipoles located at the edge. In all cases $m_x$ is in $m_0$ units and the magnetic field $B_x$ is in $\frac{\mathcal{K}\left(\mathrm{\Delta }\right)}{m_0}$ units.

Figure 6

Magnetic field ramp used in numerical simulations in units of $\mathcal{K}\left(\mathrm{\Delta }\right)/m_0$.

Figure 7

Normalized magnetization of the cluster versus $\chi$ at $T=0$.

Figure 8

(a) Difference between the dipolar energy of the magnetic configurations $\mathrm{FV}$ and $\mathrm{AV}$ and (b) Difference between the maximum energy density of the system (the energy barrier reached when $M=1$ and $\chi =0.26$) and the energy of the magnetic configuration $\mathrm{FV}$ in terms of $\mathrm{\Delta }$ at $T=0$. All energies are normalized by $\mathcal{K}\left(L\right)$.

Figure 9

(a) The amplitude $A$ of the earliest oscillation of $\mathcal{C}\left(s\right)$ of a single square plaquette of dipoles is shown as a function of temperature $T$. (b) $A$ in terms of $\mathrm{\Delta }$. Panels (a) and (b) are obtained from the numerical solution of Eq. (3). $T$ is normalized by $\mathcal{K}\left(L\right)$.