Order and disorder, crossovers, and phase transitions in dipolar artificial spin ice on the Cairo lattice

We study the thermodynamic properties of the magnetic dipolar spin ice on a 2D pentagonal Cairo lattice by using the numerical Metropolis and the complete enumeration methods. We use the model of point Ising-like dipoles considering long-range interactions with up to 100 nearest neighbors and with periodic boundary conditions. There are two explicit peaks both in the temperature behavior of the heat capacity and in the magnetic susceptibility. The low-temperature peak is caused only by long-range interactions and is not present in the model where each dipole interacts only with four nearest neighbors. The height of the peak depends logarithmically on the quantity of dipoles, which indicates a phase transition. The nature of the low-temperature phase transition is related to the transformation from order to disorder in orthogonal sublattices while maintaining the spin ice state and the spin ice rule in the sublattice of crosses. The high-temperature heat capacity peak is associated with the melting of spin ice, i.e., with the crossover from spin ice to paramagnetic chaos. Its height is constant and does not depend on the quantity of dipoles. It is shown that the choice of the radius of the dipole-dipole interaction has a significant effect on the statistical properties of the model. The model may even show the appearance of the long-range order and the phase transition in the case of long-range interaction or its absence in the case of short-range interaction.

Figure 1

2D Dipolar spin ice on a Cairo lattice. (a) Ferromagnetic islands are shown as gray ovals. Letters $a, b, c$ denote lattice parameters. (b) One of the possible GS configurations of Cairo dipolar spin ice with periodic boundary conditions both in short- and long-range interaction models. The point dipoles are in the middle of the arrows; the direction represents the magnetic moment. The key structural elements of the lattice are horizontal $\alpha$, vertical $\beta$ spins, and crosses $\gamma$ consisting of $\delta$ spins. The unit cell of the lattice $\epsilon$ is shown as a square. The blue (silver) arrows are the moments directed along the field induced by its four nearest neighbors. The red (dim gray) arrows are the moments for which the magnetic field induced by the nearest neighbors is compensated.

Figure 2

Temperature dependence of the heat capacity $C\left(T\right)$ of the dipolar Cairo spin ice with $N=5120$ dipoles and $c=376$ nm in long-range (upper) and short-range (lower) models with PBCs. The dashed lines indicate the peaks of the heat capacity at $T_1^p\approx 4.62$ K, $T_2^p\approx 372.79$ K, and the critical temperature $T_{\lambda }\approx 130.00$ K.

Figure 3

Temperature dependence of the heat capacity $C\left(T\right)$ (upper) and entropy $S\left(T\right)$ (lower) of the dipolar Cairo spin ice with $N=20$ dipoles at $c=376$ nm in long-range interaction and short-range interaction models. The dashed vertical lines indicate temperatures of heat capacity peaks.

Figure 4

The value of the heat capacity peak as a function of quantity of dipoles $N$ for the Cairo lattice $c=376$ nm in both long-range interaction and short-range interaction models.

Figure 5

Ground-state configurations for the dipolar spin ice on the Cairo lattice $L=2$ with periodical boundary conditions in long-range (row 1) and short-range (rows 1–3) models. We illustrate only half of the configurations, implying that the other half is similar but with all spins inverted.

Figure 6

Temperature behavior of the thermodynamic averaged magnetization of the long-range dipolar spin ice on the Cairo lattice with $N=5120$ spins and $c=376$ nm. The green solid line $\langle \left|\vec{m}\right|\rangle$ is the total magnetization. The blue dashed line $\langle \left|m_x\right|\rangle$ and the red dash-dotted line $\langle \left|m_y\right|\rangle$ are the projections of magnetization on the $X$ and $Y$ axes for the same system, respectively.

Figure 7

Temperature behavior of total $\chi$, longitudinal ${\chi }_x$, and transverse ${\chi }_y$ magnetic susceptibility of the long-range dipolar spin ice on the Cairo lattice with $N=5120$ spins and $c=376$ nm.

Figure 8

Temperature behavior of the correlation function for the horizontal $G^{\alpha }$ and vertical $G^{\beta }$ spins of the long-range dipolar spin ice on the Cairo lattice with $N=5120$ spins and $c=376$ nm.

Figure 9

The correlation function for the nearest neighboring spins of the crosses sublattice $\gamma$ of the long-range dipolar spin ice on the Cairo lattice with $N=5120$ spins and $c=376$ nm.

Figure 10

A part of the dipolar spin ice on the Cairo lattice with $c=376$ nm. (a) One of the GSs of the long-range model. (b) One of configurations for $T_{\lambda }$. The vertical and horizontal spins have the following color scheme: blue (silver) $E=-4u$, green (light gray) $E=-2u$, and red (dim gray) $E=0$. The blue circle indicates a pentagon with $E=-5v$, and a red cross is $E=-1v$. The dotted outlines around arrows show the spins that differ from the GS configuration (a).

Figure 11

Temperature behavior of the correlation functions for dipole Cairo spin ice at $c=376$ nm and $N=5120$ in the long-range model. $G_p$ is the correlations of spins in all pentagons of the lattice; $G_p^{\alpha }$ is the correlations of spins in pentagons that contain a horizontal spin from the $\alpha$ sublattice; $G_p^{\beta }$ is the correlations of spins in pentagons that contain a vertical spin from the $\beta$ sublattice.