Designing Hysteresis with Dipolar Chains

Materials that have hysteretic response to an external field are essential in modern information storage and processing technologies. A myriad of magnetization curves of several natural and artificial materials have previously been measured and each has found a particular mechanism that accounts for it. However, a phenomenological model that captures all the hysteresis loops and at the same time provides a simple way to design the magnetic response of a material while remaining minimal is missing. Here, we propose and experimentally demonstrate an elementary method to engineer hysteresis loops in metamaterials built out of dipolar chains. We show that by tuning the interactions of the system and its geometry we can shape the hysteresis loop which allows for the design of the softness of a magnetic material at will. Additionally, this mechanism allows for the control of the number of loops aimed to realize multiple-valued logic technologies. Our findings pave the way for the rational design of hysteretical responses in a variety of physical systems such as dipolar cold atoms, ferroelectrics, or artificial magnetic lattices, among others.

Figure 1

(a) Experimental realization built out of neodymium magnets hinged on top of a PTFE plate. Each cylindrical magnet has a length $2a=5\times {10}^{-3}\mathrm{m}$, radius $r=0.25\times {10}^{-3}\mathrm{m}$, mass $0.007\times {10}^{-3}\mathrm{kg}$, and saturation magnetization $M_s=0.92\times {10}^6\mathrm{A}/\mathrm{m}$. Their north and south poles are highlighted in red and blue, respectively. The distance between the center of mass of two consecutive rods is $2a+\mathrm{\Delta }$. (b) Diagram showing the angular variable of each dipole. Each dipole is modeled as two magnetic charges $+Q$, $-Q$ at the poles of the rod, where $Q=M_s\pi r^2$. This is known as the dumbbell model [20].

Figure 2

(a),(b) Magnetization loops of a system with $N=10$ (Supplemental Material, movies 1, 2 [21]). Experimental and numerical data in blue dots and red circles, respectively. In (a) and (b) the chain is oriented along the $x$ axis and $y$ axis, respectively. Blue arrows indicate evolution of the hysteresis cycle. (c) Energy landscapes showing local minima for $b_1<b_2<b_3<b_{s2}$ and $b_4>b_{s2}$. (e)–(g) Points correspond to the real space configurations shown in (a). Inset shows the energy landscape around point (e). (d) Energy landscapes showing local minima $b_1<b_2<b_3<b_{sw}$ and $b_4>b_{sw}$. (p,q) correspond to the real space configurations shown in (b). The energy is normalized by the energy of two interacting magnets with $\mathrm{\Delta }=3\mathrm{mm}$.

Figure 3

(a) Experimental hysteresis loops of chains with $N=2$ (magenta squares), 8 (red filled circles), 10 (green triangles), and 12 (blue open circles) rotors, for $\mathrm{\Delta }=6a/5$. The inset corresponds to hysteresis loops from numerical simulations. Black arrows indicate the positions of the first order estimates for $b_{s1}^{N=2}$ and $b_{s2}^{N=2}$. (b) Experimental and numerical (inset), hysteresis loops for chains with $N=10$ and $\mathrm{\Delta }=6a/5$ (red dots), $8a/5$ (green open triangles), $2a$ (blue open circles).

Figure 4

(a) Hysteresis loop obtained for a system of dipoles made of two linear chains (${\mathrm{\Delta }}_1=3\times {10}^{-3}\text{m}$ and ${\mathrm{\Delta }}_2=4.0\times {10}^{-3}\mathrm{m}$) arranged as a $T$. Real space configurations for the initial state and fully polarized state are shown in the top and bottom insets, respectively. The red painted side of each magnet is the magnetic north pole (Supplemental Movie 3, [21]). (b)–(d) Numerical simulations for a $T$ system when ${\mathrm{\Delta }}_1=3.0\times {10}^{-3}\mathrm{m}$, $h=1.5{\mathrm{\Delta }}_1$, and ${\mathrm{\Delta }}_2/{\mathrm{\Delta }}_1=1,4/3,2$, respectively.