Magnetoentropic mapping and computational modeling of cycloids and skyrmions in the lacunar spinels ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_8$

We report the feasibility of using magnetoentropic mapping for the rapid identification of magnetic cycloid and skyrmion phases in uniaxial systems, based on the ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ model skyrmion hosts with easy-axis and easy-plane anisotropies, respectively. We show that these measurements can be interpreted with the help of a simple numerical model for the spin Hamiltonian to yield unambiguous assignments for both single-phase regions and phase boundaries. In the two lacunar spinel chemistries, we obtain excellent agreement between the measured magnetoentropic features and a minimal spin Hamiltonian built on Heisenberg exchange, single-ion anisotropy, and anisotropic Dzyaloshinskii-Moriya interactions. In particular, we identify characteristic high-entropy behavior in the cycloid phase that serves as a precursor to the formation of skyrmions at elevated temperatures and is a readily measurable signature of this phase transition. Our results demonstrate that rapid magnetoentropic mapping guided by numerical modeling is an effective means of understanding the complex magnetic phase diagrams innate to skyrmion hosts. One notable exception is the observation of an anomalous, low-temperature high-entropy state in the easy-plane system ${\mathrm{GaV}}_4{\mathrm{Se}}_8$, which is not captured in the numerical model. Possible origins of this state are discussed.

Figure 1

At $T_S=42\mathrm{K}, {\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ distort rhombohedrally to $R3m$, shown above. This ferroelectic transition allows cycloidal modulations with propagation vectors in the plane perpendicular to $\langle 111\rangle$.

Figure 2

Constant-field magnetoentropic measurements of ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_8$. The magnetization $M$ versus temperature $T$ is measured at even intervals of external field for ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ as shown in (a) and (c), respectively. Data points ($+$) are extremely dense, so a statistical smoothing routine is applied (solid lines) before taking the derivative to obtain the derivatives ${(dM/dT)}_H={(dS/dH)}_T$ as shown for ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ in (b) and (d), respectively.

Figure 3

Skyrmion and cycloid phase boundaries in the ${(dM/dT)}_H$ phase diagrams of ${\mathrm{GaV}}_4{\mathrm{S}}_8$. When plotted as a heat map against applied magnetic field and temperature, the skyrmion (Sk) and cycloidal (Cyc) phases are clearly visible in $dM/dT$ in several crystallographic directions $\langle 111\rangle$ (a), $\langle 110\rangle$ (b), and $\langle 100\rangle$ (c). Starred (*) phases correspond to phases in equivalent $\langle 111\rangle$ domains that are not completely aligned with the field.

Figure 4

Large skyrmion and cycloid phase boundaries are apparent in the ${(dM/dT)}_H$ phase diagrams of ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ in different orientations $\langle 111\rangle$ (a), $\langle 110\rangle$ (b), and $\langle 100\rangle$ (c). Starred (*) phases again correspond to phases in equivalent $\langle 111\rangle$ domains that are not completely aligned with the field.

Figure 5

Magnetoentropic phase diagrams measured along the $\langle 111\rangle$ direction compared with previously reported phase boundaries determined by other techniques for (a) ${\mathrm{GaV}}_4{\mathrm{S}}_8$ and (b) ${\mathrm{GaV}}_4{\mathrm{Se}}_8$

Figure 6

Magnetic phase stability and entropic susceptibility $d{S}^{*}/d{H}^{*}=d{M}^{*}/d{T}^{*}$ in lacunar spinels in the low-$T$ limit, across possible values of uniaxial anisotropy $K_u^{*}$

Figure 7

Computed magnetic phase diagram of an easy-plane (a)–(c) and an easy-axis (c)–(f) lacunar spinel ferromagnet, representative of ${\mathrm{GaV}}_4{\mathrm{Se}}_8$ and ${\mathrm{GaV}}_4{\mathrm{S}}_8$, respectively. The magnetic field $H$ is oriented along the high-symmetry $c$ axis of the low-$T R3m$ structure ($\langle 111\rangle$ axis in Fig. 1). (a), (b), (d), (e) Phase stability as a function of normalized temperature $T^{*}$ and applied field or resulting magnetization, respectively. The color map denotes the entropic susceptibility $d{S}^{*}/d{H}^{*}=d{M}^{*}/d{T}^{*}$ obtained from constant-field heating Monte Carlo runs. Phase boundaries are drawn on the basis of magnetization, magnetic structure factor, the topological index ($t$), and heat capacity at each point. (c), (f) Correlation between $dM/dT$ and the topological index as a function of field. In the easy-plane case (c), the data are given only at $T^{*}=1.5$ for clarity, with phase boundaries marked in gray.