Hysteresis in Two Dimensional Arrays of Magnetic Nanoparticles

We perform computer simulations to probe the magnetic hysteresis in a two-dimensional ($L^{}_x\times L^{}_y$) assembly of magnetic nanoparticles as a function of dipolar interaction strength $h^{}_d$, temperature $T$, aspect ratio $A^{}_r=L^{}_y/L^{}_x$, and the applied alternating magnetic field's direction. In the absence of magnetic interaction ($h^{}_d\approx0$) and thermal fluctuations ($T=0$ K), the hysteresis follows the Stoner and Wohlfarth model, as expected. For weak dipolar interaction and substantial temperature, the hysteresis has the dominance of superparamagnetic behaviour, irrespective of the applied magnetic field's direction and $A^{}_r$. Interestingly, the hysteresis curve has all the characteristics of antiferromagnetic interaction dominance for $A^{}_r\leq6$ and considerable dipolar interaction strength ($h^{}_d>0.2)$, which is independent of applied magnetic direction. When the magnetic field is applied along the system's shorter axis ($x$-direction), a non-hysteresis straight line is observed with large $h^{}_d$. In the case of the magnetic field applied along the long axis of the sample ($y$-direction), ferromagnetic interaction dominates the hysteresis for large $h^{}_d$ and $A^{}_r>6$. Irrespective of $h^{}_d$ and applied magnetic field's direction, the coercive field $\mu^{}_oH^{}_c$ and remanence $M^{}_r$ are minimal for $A^{}_r\leq6$ and significant temperature. They are found to increase with $h^{}_d$ when $A^{}_r$ is enormous. Remarkably, the variation of hysteresis loop area $E^{}_H$ as a function of these parameters is the same as that of the coercive field variation. We believe that the concepts presented in this work are relevant in various technological applications such as spintronics and magnetic hyperthermia, in which such self-assembled nanoparticle arrays are ubiquitous.