Numerical simulations of the magnetodielectric response in Ising pyrochlores

In this paper, we examine the magnetoelectric response of Ising pyrochlores, focusing on both the ordered antiferromagnetic state and the frustrated ferromagnetic case known as “spin ice.” We employ a model which accounts for magnetoelastic effects by considering the interplay between oxygen distortions and superexchange magnetic interactions within pyrochlores. This, together with numerical simulations, provides a tool to make quantitative comparisons with experiments. Our main target is then to see how to extract relevant information from this simple model, and to explore its limitations. We obtain a direct estimation of quantities such as the electric dipole moment, the central oxygen displacement, and the effective magnetoelastic energy for the canonical spin-ice material ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. We also inquire about the possibility of using the electric dipole carried by magnetic monopoles to obtain a direct measure of their density. In each studied scenario the correlations between monopoles, induced by their number or by the magnetic background, render these findings less straightforward than initially anticipated. Furthermore, the coupling between electrical and magnetic degrees of freedom provides additional tools to investigate magnetic order in these systems. As an example of this we discuss the phase diagram of the antiferromagnetic pyrochlore under applied magnetic field along the [111] direction. We find an instance where the phase stability at nonzero temperatures is not dictated by the energy associated with different ground states but (akin to the phenomenon of “order by disorder”) is instead determined by their accessibility to thermal fluctuations.

Figure 1

Structure, magnetic monopoles, and ${\mathrm{O}}^{2-}$ distortions. (a) Pyrochlore structure, with Ising spins in the shared vertices of up tetrahedra and down tetrahedra. The three-in–one-out configuration in the up tetrahedron (embedded in a cube) has associated a positive single monopole in its center (small green sphere); we also show single and double negative monopoles (small and big red spheres, respectively) and one neutral site (two-in–two-out). (b) For single monopoles, the displacement $\delta \mathbf{r}$ of the central oxygen ion (cyan sphere) along the cube diagonals decreases the exchange constant's value along the three magnetic bonds it approaches (red lines and surfaces), and strengthens the other three (green lines and surfaces). No displacement occurs for neutral sites or double monopoles. Within the model, the rare-earth ions ($R$) are assumed to be fixed.

Figure 2

Zero field magnetoelastic behavior for the AIAO antiferromagnet from Monte Carlo simulations ($L=4$). (a) Density of magnetic charges as a function of temperature for single monopoles (${\rho }_{\mathrm{s}}$), double monopoles (${\rho }_{\mathrm{d}}$), and neutral sites (${\rho }_{\mathrm{n}}$) for $\mu B=0$. We also include the order parameter for the zinc-blende crystal of double monopoles, ${\rho }_d^{\mathrm{stag}}$, and an estimate for ${\rho }_{\mathrm{s}}$ taken from the electric response in (b), ${\chi }^{\mathrm{mon}}/D_q*T$. This estimation is excellent above the ordering temperature of the antiferromagnet, $T_C/J_0\approx 4.2$; below $T_C$, the electric response is accounted for by the density of neutral sites [see (b)]. (b) Electrical susceptibility due to monopoles, ${\chi }_{\mathrm{MC}}^{\mathrm{mon}}$ (open symbols), calculated from Monte Carlo simulations compared with the single tetrahedron approximation ${\chi }^{1\mathrm{tet}}$ based on the single monopole density (red line), and with a similar estimate based on the density of neutral sites (orange symbols). See text for details.

Figure 3

$\mathit{B}\parallel$ [111] magnetoelastic behavior for the AIAO antiferromagnet. (a) Magnetization, (b) density of monopoles, (c) specific heat, and (d) electric susceptibility due to monopoles as a function of magnetic field for three different temperatures below $T_C$. The curves were measured for decreasing field [black arrows in (a)], and show a clear asymmetry around $B=0$. The sharp features at $B<0$ correspond to phase transitions (see dashed orange lines in Fig. 7). (b) Three schematic views of the spin configurations that are stable at low temperature in a two-dimensional projection. Spins can be divided into apical [colored brown on the scheme on the right of (b)], parallel to the applied field, and basal (colored blue on the same configuration).

Figure 4

Magnetoelastic behavior for the AIAO antiferromagnet with parallel electric and magnetic fields along different directions. (a) Magnetization, (b) density of monopoles, (c) electric susceptibility due to monopoles, and (d) staggered charge density for double monopoles as a function of magnetic field for three different magnetic field directions at $T/|J_0|=1,67$. The curves were measured for decreasing field [black arrows in (a)]. Interestingly, ${\chi }_{\widehat{h}}^{\mathrm{mon}}$ [panel (c)] is bigger for field orientations where the density of neutral sites [panel (b)] is bigger, and the staggered charge density [panel (d)] is smaller.

Figure 5

Magnetodielectric behavior for ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ spin ice from Monte Carlo (MC) simulations (zero magnetic field), together with experimental measurements adapted from Saito et al. [27]. (a) Single monopole density from simulations, compared with estimations using ${\chi }^{\mathrm{mon}}$ from MC simulations and experiments in the lower panels. Electric susceptibility due to monopoles, ${\chi }^{\mathrm{mon}}$ for (b) [100] and (c) [111] electric field directions. Monte Carlo simulations (open circles) are compared with experimental data (solid diamonds), after background subtraction. The shape of the experimental curves coincides for directions, as do the simulated ones, giving us confidence on the subtraction procedure. The value of the displacement $\delta r^{\mathrm{eq}}$ in the simulations for each direction, specified in the figures, was chosen so that the curves coincide in the high-$T$ regime.

Figure 6

Simulated magnetodielectric behavior due to monopoles for ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ spin ice for $\mathit{B}\parallel \left[111\right]$. (a) Magnetization vs field at different temperatures above the critical point, with a sharp crossover at low $T$ near 0.9 T. (b) The change into a crystal of single monopoles is shown at low temperatures by a sharp increase in the density of single monopoles, ${\rho }_s$. At low fields ${\rho }_s$ initially decreases with increasing field; this has a marked effect in the magnetodielectric response. (c) Electric susceptibility due to monopoles as a function of field. It is nearly zero at low $T$, with a pronounced maximum near $B=0$ at higher $T$. There is no trace in ${\chi }^{\mathrm{mon}}$ of the sharp crossover near 0.9 T. (d) Electric susceptibility due to monopoles on up tetrahedron only (solid circles), and the negative of the covariance of the polarization due to monopoles in up and down tetrahedra (lines). Both quantities are almost identical (and ${\chi }_{111}^{\mathrm{down}}={\chi }_{111}^{\mathrm{up}}$) across the peak, explaining the absence of a peak in ${\chi }^{\mathrm{mon}}$ at the metamagnetic transition.

Figure 7

Phase diagram for a simple AIAO antiferromagnet in a [111] field. The AIAO and AOAI phases at positive and negative fields are separated at nonzero temperatures by a coexistence line at $\mu B=0$ (blue horizontal line) terminating in a critical end point indicated by a blue circle. The sign (and not the magnitude) of the average magnetic charge in each diamond sublattice is the feature that can be used to label each phase. Since a magnetic field selects a phase through their excitations, there is coexistence between these two phases at $T=0$ even at nonzero magnetic field. The features we observed for negative fields in Fig. 3 are related to the limit of phase metastability, indicated here by a dashed orange line. Inset: Phase diagram for an Ising ferromagnet. The thick blue line for $B=0$ indicates the region of coexistence (a first-order transition line) between the two polarized phases. Their distinction disappears above the Curie temperature, indicated by a blue circle. This phase diagram is similar—but not identical—to that of the AIAO antiferromagnet.