Structural evolution and skyrmionic phase diagram of the lacunar spinel ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$

In the $A{B}_4{Q}_8$ lacunar spinels, the electronic structure is described on the basis of inter- and intracluster interactions of tetrahedral $B_4$ clusters, and tuning these can lead to myriad fascinating electronic and magnetic ground states. In this work, we employ magnetic measurements, synchrotron x-ray and neutron scattering, and first-principles electronic structure calculations to examine the coupling between structural and magnetic phase evolution in ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$, including the emergence of a skyrmionic regime in the magnetic phase diagram. We show that the competition between two distinct Jahn-Teller distortions of the room temperature cubic $F\overline{4}3m$ structure leads to the coexistence of the ground-state $R3m$ phase and a metastable $Imm2$ phase. The magnetic properties of these two phases are computationally shown to be very different, with the $Imm2$ phase exhibiting uniaxial ferromagnetism and the $R3m$ phase hosting a complex magnetic phase diagram including equilibrium Néel-type skyrmions stable from nearly $T$ = 28 K down to $T$ = 2 K, the lowest measured temperature. The large change in magnetic behavior induced by a small structural distortion reveals that ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$ is an exciting candidate material for tuning unconventional magnetic properties via mechanical means.

Figure 1

(a) The cubic $F\overline{4}3m$ crystal structure of ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$, characterized by tetrahedral clusters of Mo atoms. (b) All subgroups of $F\overline{4}3m$ which keep the primitive unit cell volume the same. $R3m$ and $Imm2$ both follow the $T_2$ irreducible representation but through different order parameter directions. (c) The evolution of peaks upon cooling in synchrotron XRD data. At 50 K the (111) cubic peak splits into three, while the (200) cubic peak stays undistorted. Because the middle peak of the (111) split shifts relative to the peak in the cubic phase while the (200) peak does not shift, it is not possible to explain the low-temperature structure by a combination of the cubic phase and a single low-temperature phase. (d) The distortion of the ${\mathrm{Mo}}_4$ tetrahedron as it goes into the $R3m$ and $Imm2$ space groups. Both are polar, but the $Imm2$ cluster has lower symmetry, with three unique bond lengths, compared to two in the $R3m$ cluster.

Figure 2

(a) At 10 K, there is about 70% of the rhombohedral phase and 30% of the orthorhombic phase. (b) The complex peak shapes can be fit with two $R3m$ phases with different particle sizes and an $Imm2$ phase with $hkl$-dependent strain broadening.

Figure 3

(a) DFT energies along the transition pathways from the cubic phase to the rhombohedral and orthorhombic phases. The $R3m$ structure has lower energy, making it the ground-state structure, but it has a higher formation energy barrier, highlighted in the left panel. The energy difference is 20 meV/${\mathrm{Mo}}_4$ cluster. (b) The phase evolution of ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$. Upon cooling, the percentage of the $R3m$ phase slightly increases, indicating it is likely the ground-state crystal structure.

Figure 4

(a) At $T=2$ K, the saturation magnetization is $1{\mu }_B/{\mathrm{Mo}}_4$ cluster, in agreement with DFT and molecular orbital theory. Multiple magnetic phase transitions are evident in field-dependent magnetization measurements. (b) Phase transitions are shown more clearly in the derivative of magnetization with respect to field. These transitions persist down to $T$ = 2 K, the lowest temperature measured.

Figure 5

(a) Using multiple bulk magnetization measurement techniques, we suggest a field vs temperature phase diagram for ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$. Based on theoretical calculations and the phase diagrams for the related V compounds, we posit that the low-field transition is a transition from cycloidal into canted cycloid or ferromagnetic ordering in grains (denoted $ab)$ where the field is mostly perpendicular to the polar axis. Upon increasing field, grains (denoted $c)$ where the field is mostly aligned with the polar axis transition into a skyrmion phase and then become field polarized. Near the Curie temperature there is a fluctuation-disordered (fd) phase. (b) The transition into the proposed skyrmion phase in $c$-oriented grains is associated with an increase in magnetic entropy, shown as a red feature near the Curie temperature that broadens upon cooling. (c) SANS data show the presence of approximately 16-nm magnetic periodicity at multiple applied fields at $T$ = 15 K. The red points are in the cycloidal phase, and the blue and green points are in the skyrmion phase. The gray lines are Gaussian fits to guide the eye.

Figure 6

(a) Computed magnetocrystalline anisotropy in the $Imm2$ and $R3m$ phases of ${\mathrm{GaMo}}_4{\mathrm{Se}}_8$. Gray circles denote Mo atoms, and black arrows illustrate the direction of the Jahn-Teller distortion required to form each phase. The plotted color and distance from the tetrahedron center denote the energy of a ferromagnetic spin configuration oriented along the corresponding crystallographic direction. (b) Energy of spin wave configurations in the $\left(ab\right)$ plane of the $R3m$ phase as a function of their $q$ vector, revealing the family of cycloid ground states with a predicted wavelength of $\approx 18$ nm. (c) Computed magnetic phase diagram of the $R3m$ phase for fields oriented along the $c$ axis or the $\left(ab\right)$ plane. Color denotes the magnetic structure factor $S\left(q\right)$, while phase boundaries are drawn to best capture discontinuities in structure factor, susceptibility, heat capacity, and magnetization. Phase labels correspond to cycloid (cyc), canted cycloid (c. cyc), skyrmion (SkX), ferromagnet (fm), fluctuation-disordered (fd), and paramagnet (pm) regions. Solid and dashed lines denote first- and second-order phase transitions, respectively, while dotted lines denote a transition that is continuous within the resolution of our data.