Magnetic field induced phases in $\mathrm{Cu}\mathrm{Cr}{\mathrm{O}}_2$: Monte Carlo and analytical investigations

Motivated by the strong magnetoelectric coupling in the multiferroic geometrically frustrated triangular antiferromagnet $\mathrm{Cu}\mathrm{Cr}{\mathrm{O}}_2$ and the high sensitivity of its magnetic structure to external fields, we investigate induced magnetic phases at very low temperatures under high magnetic fields ($B$) up to 325 T applied along the [001] direction. Analytical calculations and Monte Carlo (MC) simulations based on a realistic three-dimensional classical Heisenberg model are used to reveal these magnetic phases. Interestingly, our model mimics a real distorted crystal which considers exchange interactions up to third-nearest neighbors in the $ab$ plane and an interplane interaction, in addition to hard and easy axes anisotropies along the [110] and the [001] directions, respectively. For $B\ge 70\mathrm{T}$, both our MC and analytical results are in an excellent agreement and evidence three commensurate phases, namely, the commensurate Y (CY), the up-up-down (UUD), and the V phases as the magnetic field increases. The field dependence of the characteristic angles of the CY and V phases is determined. Moreover, the saturation field is estimated at 325 T, indicating that the previously predicted values obtained by extrapolation of experimental data are too small. Below 70 T, our MC results indicate that the CY phase is no more stable and several incommensurate Y phases appear. Overall, the observed magnetic phases at nearly 0 K are in a good agreement with a recently published experimental phase diagram. It should be noted that our MC data reject the incommensurate umbrella phase at very low temperatures, which was reported in previous studies.

Figure 1

Magnetic moment configurations in the commensurate Y (CY), the up-up-down (UUD), and the V phases seen in $\mathrm{Cu}\mathrm{Cr}{\mathrm{O}}_2$.

Figure 2

Schematic representation of the intraplane exchange interactions within the $ab$ plane ($J_1$ is along [100] and [010], and $J_1^{\prime }$ is along [110] with $|J_1/J_1^{\prime }|<1$) and the interplane exchange interaction $J_4$.

Figure 3

Size dependence of the energy per spin of the ${120}^{\circ }$ magnetic configuration [$\mathit{q}$ = (1/3, 1/3)] in a finite triangular lattice [the energy per spin of the ${120}^{\circ }$ configuration in the infinite triangular lattice, $E_{\mathrm{GS}}^{\infty }/\left(k_B{S}^2\right)=-41.682$ K, is shown by the red horizontal line].

Figure 4

Energy per spin variation of the magnetic configuration of propagation vector $\mathit{q}$ = ($k, k$, 0) versus $k$ in a finite triangular lattice of size $L=22$ (the vertical dashed line indicates the location of the minimum corresponding to $k=0.31818$).

Figure 5

Energy per spin of the various commensurate phases (CY, UUD, V, and FM) versus the applied magnetic field. Note that the energy of the UUD and FM phases decreases linearly as the magnetic field increases.

Figure 6

Magnetic field dependence of $\alpha$ (CY phase), and $\phi$ and $\theta$ (V phase). Continuous lines represent the theoretical values and filled symbols indicate the MC values.

Figure 7

Comparison of the simulated GS energy per spin $E_{\mathrm{GS}}^{\mathrm{sim}}(L=49)$ with the energy per spin of the CY phase versus the applied magnetic field for $B\le 70\mathrm{T}$.

Figure 8

Applied field dependence of the simulated magnetization per ${\mathrm{Cr}}^{3+}$ ion (in ${\mu }_B$) at $\sim 0$ K.