Effect of dipolar interactions on the magnetization of a cubic array of nanomagnets

We investigated the effect of intermolecular dipolar interactions on an ensemble of 100 three-dimensional systems of $5\times 5\times 4$ nanomagnets, each with spin $S=5$, arranged in a cubic lattice. We employed the Landau-Lifshitz-Gilbert equation to solve for the magnetization curves for several values of the damping constant, the induction sweep rate, the lattice constant, the temperature, and the magnetic anisotropy. We find that the smaller the damping constant, the stronger the maximum induction required to produce hysteresis. The shape of the hysteresis loops also depends on the damping constant. We find further that the system magnetizes and demagnetizes at decreasing magnetic field strengths with decreasing sweep rates, resulting in smaller hysteresis loops. Variations of the lattice constant within realistic values (1.5–2.5 nm) show that the dipolar interaction plays an important role in the magnetic hysteresis by controlling the relaxation process. The temperature dependencies of the damping constant and of the magnetization are presented and discussed with regard to recent experimental data on nanomagnets. Magnetic anisotropy enhances the size of the hysteresis loops for external fields parallel to the anisotropy axis, but decreases it for perpendicular external fields. Finally, we reproduce and test a previously reported magnetization curve for a two-dimensional system [M. Kayali and W. Saslow, Phys. Rev. B 70, 174404 (2004)]. We show that its hysteretic behavior is only weakly dependent on the shape anisotropy field and the sweep rate, but depends sensitively upon the dipolar interactions. Although in three-dimensional systems, dipole-dipole interactions generally diminish the hysteresis, in two-dimensional systems, they strongly enhance it. For both square two-dimensional and rectangular three-dimensional lattices with $\mathbf{B}\Vert (\widehat{\mathbf{x}}+\widehat{\mathbf{y}})$, dipole-dipole interactions can cause large jumps in the magnetization.

Figure 1

Magnetization curves for $N_c=100,a=1.5\mathrm{nm},\Delta B∕\Delta t=0.005$. From left to right for $\mathcal{M}>0,\alpha ∕\gamma =3\times {10}^{-10}$ (dashed), $3\times {10}^{-11}$ (thin dark gray), $3\times {10}^{-12}$ (light gray), $3\times {10}^{-13}$ (thick black). The inset highlights the hysteretic region of the first three of these curves.

Figure 2

Shown is the upper hysteretic region of the normalized magnetization curves at $T=0.7\mathrm{K}$ (gray) and $T=0.1\mathrm{K}$ (black, circles). The $T$-independent damping constants $\alpha ∕\gamma$ are $3\times {10}^{-12}$ (a), $3\times {10}^{-11}$ (b), and $3\times {10}^{-10}$ (c). The other parameters are the same as in Fig. 1.

Figure 3

The magnetization curves for $N_c=100$ at $T=5\mathrm{K},a=1.5\mathrm{nm}$, and $\Delta B∕\Delta t=3000\mathrm{T}∕\mathrm{s}$ with $B_{\max }=22.5\mathrm{mT}$ and $\alpha \left(T\right)∕\gamma =T_0∕T$ for $\mathit{\hslash }g{\mu }_B{B}^{\mathrm{eff}}∕\left(k_{B}T\right)⪡1$ and $T_0=0.3\mathrm{K}$ are shown (Ref. 50). Left-hand inset, details of the upper portion of the curve. Right-hand inset, details of the central hysteretic portion of the curve shown, along with the central portion of the corresponding curve at $T=0.25\mathrm{K}$. The thin curves beginning near to the origin represent the magnetization onsets. These curves are offset for clarity, with the scales on the right-hand (left-hand) sides corresponding to the lower (upper) curves, respectively.

Figure 4

The central loop and starting magnetization curves for two separate configurations, each with $N_c=1$ (open and filled circles) at $T=10\mathrm{K},a=1.5\mathrm{nm}$, and $\Delta B∕\Delta t=3000\mathrm{T}∕\mathrm{s}$ with $\alpha \left(T\right)∕\gamma =T_0∕T$ for $\mathit{\hslash }g{\mu }_B{B}^{\mathrm{eff}}∕k_{B}T⪡1$ and $T_0=0.3\mathrm{K}$ are shown (Ref. 50). The thin gray and thick black curves represent the identical behaviors of the central hysteretic loop portion of the magnetization for the same two configurations obtained after saturation. The arrows indicate the direction of the magnetization hysteresis. Here $B_{\max }=22.5\mathrm{mT}$. See text.

Figure 5

The central loops (solid curves) of the magnetization curve for $N_c=50,N=5\times 5\times 4=100$, with $\mathbf{B}$ along the [110] direction $[\mathbf{B}\Vert (\widehat{\mathbf{x}}+\widehat{\mathbf{y}})∕\sqrt{2}]$ at $T=10\mathrm{K}$, with $a=1.5\mathrm{nm}$, $B_{\max }=22.5\mathrm{mT}$, and $\Delta B∕\Delta t=3000\mathrm{T}∕\mathrm{s}$ with $\alpha \left(T\right)∕\gamma =T_0∕T$, and $T_0=0.3\mathrm{K}$. The dashed curve is the starting magnetization curve. The arrows indicate the direction of the hysteresis. Inset, the full magnetization curve. See text.

Figure 6

Hysteretic region of $\mathcal{M}\left(B\right)$ at 0.7 K, $\alpha ∕\gamma =3\times {10}^{-12}$, and $a=1.5\mathrm{nm}$, for the sweep rates $\Delta B∕\Delta t=0.04\mathrm{T}∕\mathrm{s}$ (thin black), $0.02\mathrm{T}∕\mathrm{s}$ (dark gray), and $0.005\mathrm{T}∕\mathrm{s}$ (thick light gray). The inset shows the entire curves.

Figure 7

Magnetization curves for lattice constants $a=1.5\mathrm{nm}$ (gray) and $a=2.5\mathrm{nm}$ (black). $\Delta B∕\Delta t=0.04\mathrm{T}∕\mathrm{s},\alpha =3\times {10}^{-12}\gamma ,T=0.7\mathrm{K}$.

Figure 8

Parallel 3D magnetization curves including different anisotropy field ${\mathbf{H}}_A=H_A\widehat{\mathbf{x}}$ strengths, with the external induction $\mathbf{B}\Vert {\mathbf{H}}_A$. ${\mu }_0{H}_A=0$ (thin black), 0.2 T (dark gray), and 1.0 T (thick dashed), respectively. For each curve, $N_c=100,$ $N=5\times 5\times 4=100,$ $\alpha ∕\gamma =3\times {10}^{-12},$ $a=1.5\mathrm{nm},$ $\Delta B∕\Delta t=0.04\mathrm{T}∕\mathrm{s},$ $T=0.7\mathrm{K}$, and $B_{\max }=2.0\mathrm{T}$.

Figure 9

Upper region of the 3D perpendicular magnetization curves with the external induction $\mathbf{B}=B\widehat{\mathbf{x}}\perp {\mathbf{H}}_A=H_A\widehat{z}$, for different values of $H_A$. Curves (a)–(e) correspond to ${\mu }_0{H}_A=0,1\times {10}^{-3},1.2\times {10}^{-2}$, 0.5, 1.0 T, respectively. The other parameters are the same as in Fig. 7. The arrows indicate the directions of the field sweeps.

Figure 10

(Left) Hysteresis loop $\mathcal{M}\left(B\right)$ in units of $M_s$, for a weakly coupled array of $5\times 5$ ferromagnetic nanodots in a square lattice on the $xy$ plane, from Fig. 2(i) of KS. The external induction is applied along the array diagonal (45° from the $x$ axis) (Ref. 45). (Right) Our results calculated for $N_c=1$ with $5\times 5$ nanomagnets on a square lattice, $\alpha ∕\gamma =0.6,T=0\mathrm{K},\Delta B∕\Delta t=3000\mathrm{T}∕\mathrm{s},{\mu }_0{H}_A=7.5\times {10}^{-4}\mathrm{T},\mathbf{B}=B(\widehat{\mathbf{x}}+\widehat{\mathbf{y}})∕\sqrt{2}$.

Figure 11

Hysteresis loops for different strengths of $H_A$ for $5\times 5$ nanomagnets on a square lattice with $N_c=1$. $S=5,T=0\mathrm{K},a=1.5\mathrm{nm},\Delta B∕\Delta t=3000\mathrm{T}∕\mathrm{s},\alpha ∕\gamma =0.6,\mathbf{B}=B(\widehat{\mathbf{x}}+\widehat{\mathbf{y}})∕\sqrt{2}$. The thin gray and thick black curves with ${\mu }_0{H}_A=0,0.75\mathrm{mT}$, respectively, are nearly indistinguishable. The small gray circles and dashed curves correspond to ${\mu }_0{H}_A=4.0$,5.5 mT, respectively. The inset shows the entire curves, which are symmetric with respect to the origin.

Figure 12

Hysteresis loops for different induction sweep rates with $5\times 5$ nanomagnets on a square lattice with $N_c=1$, $S=5,T=0\mathrm{K},a=1.5\mathrm{nm},{\mu }_0{H}_A=0.75\mathrm{mT},\alpha ∕\gamma =0.6$. The dashed gray, thick black, and light solid gray curves correspond to $\Delta B∕\Delta t=300,1500,6000\mathrm{T}∕\mathrm{s}$, respectively. The inset shows the entire curves. $\mathbf{B}=B(\widehat{\mathbf{x}}+\widehat{\mathbf{y}})∕\sqrt{2}$.

Figure 13

Hysteresis loops for lattice parameters $a=2.5\mathrm{nm}$ (solid black), $a=2.0\mathrm{nm}$ (dashed black), $a=1.5\mathrm{nm}$ (solid gray), and $a=1.25\mathrm{nm}$ (dotted-dashed black), for $5\times 5$ nanomagnets on a square lattice with $N_c=1$, $S=5,T=0\mathrm{K},\Delta B∕\Delta t=3000\mathrm{T}∕\mathrm{s},{\mu }_0{H}_A=7.5\times {10}^{-4}\mathrm{T},\alpha ∕\gamma =0.6$. The inset shows the entire curves. $\mathbf{B}=B(\widehat{\mathbf{x}}+\widehat{\mathbf{y}})∕\sqrt{2}$.