Magnetic correlations in the triangular antiferromagnet ${\mathrm{FeGa}}_2{\mathrm{S}}_4$

The crystal structure and magnetic correlations in triangular antiferromagnet $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ are studied by x-ray diffraction, magnetic susceptibility, neutron diffraction, and neutron inelastic scattering. We report significant mixing at the cation sites and disentangle magnetic properties dominated by major and minor magnetic sites. The magnetic short-range correlations at $0.77{\AA }^{-1}$ correspond to the major sites and being static at base temperature they evolve into dynamic correlations around 30–50 K. The minor sites contribute to the magnetic peak at $0.6{\AA }^{-1}$, which vanishes at 5.5 K. Our analytical studies of triangular lattice models with bilinear and biquadratic terms provide the ratios between exchanges for the proposed ordering vectors. The modeling of the inelastic neutron spectrum within linear spin-wave theory results in the set of exchange couplings $J_1=1.7,J_2=0.9,J_3=0.8\mathrm{meV}$ for the bilinear Heisenberg Hamiltonian. However, not all features of the excitation spectrum are explained with this model.

Figure 1

(a) Projection of the $A, B, C$ layers on the $ab$ plane. (b) View on the $M\mathrm{Ga}{}_2\mathrm{S}{}_4$ unit cell with labeled (from left to right) sequence of S, Ga, and $M$ atoms, coordination of the voids between the S layers and order of the $A,B,C$ layers.

Figure 2

Top: The $hk0$ (a) and $h0l$ (b) reciprocal layers reconstruction based on the x-ray single-crystal diffraction of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$. Bottom: (c) scanning electron microscopy image of a single crystal of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ and (d) zoom into its left marked corner.

Figure 3

Temperature evolution of the (a) lattice constants, (b) selected bond distances, and (c), (d) thermal parameters of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ from synchrotron x-ray powder diffraction.

Figure 4

(a) Magnetic susceptibility $\chi$ measured at 0.1 T on a single crystal of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ (mass $m=2.87\mathrm{mg}$) along the three principal directions $\left[100\right],[-120]$, and [001]. ZFC and FC $\chi$'s are plotted by empty and filled circles, respectively. The inset: zoom of the low-temperature region, (b) derivative of ZFC magnetic susceptibility highlighting deflection points. (c) Inverse susceptibility $1/\chi$ for $H\parallel \left[100\right]$ (blue) and $H\parallel \left[001\right]$ (red) and fit to the Curie-Weiss law in yellow. (d) The ratio of ZFC ${\chi }_{ab}/{\chi }_c$.

Figure 5

Neutron-diffraction DMC patterns of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ at (a) 1.3 and 50 K. Temperature dependence of the magnetic peak at (b) $Q_1=0.6{\AA }^{-1}$.

Figure 6

Magnetic diffuse scattering measured at 1.5 K (red) and 60 K (cyan) on $D7$ ($E_i=3.8\mathrm{meV}$) and fit to the Warren function (a) and calculated by the reverse Monte Carlo (RMC) method (b) with resulting radial spin-correlation function $\langle S\left(0\right)·S\left(r\right)\rangle$ shown in the inset.

Figure 7

Top: Excitation spectrum of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ measured on LET with $E_i=24.3\mathrm{meV}$ at the lowest (a) $T=2\mathrm{K}$ and the (b) highest 180 K temperatures. Bottom: Temperature evolution of the low-$Q$ feature with the signal integrated over $Q=[0.3\cdots 1.5]{\AA }^{-1}$ with the (c) $E_i=24.3\mathrm{meV}$ rep-mode ($\delta E_{0\mathrm{meV}}=1.15\mathrm{meV}$) and (d) over $Q=[0.4\cdots 0.8]{\AA }^{-1},E_i=7.7\mathrm{meV}(\delta E_{0\mathrm{meV}}=0.22\mathrm{meV})$. The Al can was subtracted, and data were corrected for the Bose factor.

Figure 8

Top: Part of the excitation spectrum of $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ measured on LET with $E_i=3.8\mathrm{meV}(\delta E_{0\mathrm{meV}}=0.1\mathrm{meV})$ at (a) $T=2\mathrm{K}$, (b) 38 K, and (c) 60 K. Bottom: (d) 2 K elastic signal summed over $E_0=[0\cdots 0.2]\mathrm{meV}$ (green symbols, signal divided by 50) and inelastic signals $E_1=[0.2\cdots 0.5]\mathrm{meV}$ (blue symbols), $E_2=[0.5\cdots 3]\mathrm{meV}$ (red symbols). (e) Temperature evolution of the elastic (blue squares) and inelastic (red and black circles) features. The elastic signal is integrated over $Q=[0.6\cdots 1]{\AA }^{-1}$ and $E_0=[-0.1\cdots 0.1]\mathrm{meV}$, the inelastic signals are corrected for the Bose factor and integrated over $Q=[0.44\cdots 2]{\AA }^{-1},E_1=[0.5\cdots 2]\mathrm{meV}$ and $Q=[0.3\cdots 1.5]{\AA }^{-1},E_2=[11\cdots 13]\mathrm{meV}$. The last curve is extracted from the $E_i=24.3\mathrm{meV}$ rep mode.

Figure 9

The relation between (a) $J_1/J_3$ and $J_2/J_3$ and (b) $J_1/J_3$ and $K_B/J_3$ to obtain the ordering wave vectors for $\mathrm{Ni}\mathrm{Ga}{}_2\mathrm{S}{}_4$ (blue) and $\mathrm{Fe}\mathrm{Ga}{}_2\mathrm{S}{}_4$ (red and green) within the $J_1–J_2–J_3$ and $J_1\text{−}{J}_3\text{−}{K}_B$ models. The critical points corresponding to a local maximum, a local minimum, and a global minimum of the energy at the given values of the ordering wave vectors are distinguished by the dotted, dashed, and solid lines, respectively.

Figure 10

Maximum logarithm likelihood after optimizing the magnitude of the spin-wave contributions.

Figure 11

Predicted $E_i=24.3\mathrm{meV}$ (left) and $E_i=7.7\mathrm{meV}$ (right) INS spectra at 2 K, constructed from an empirical background model measured at 180 K and spin-wave simulations for the set of the exchange values $J_1=1.7,J_2=0.9,J_3=0.8\mathrm{meV}$. The margin plots show comparisons with data measured at 2 K.

Figure 12

The red arrows represent the $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$ unit vectors in all the cases. (a) Basis vectors $\{{\mathbf{a}}_1,{\mathbf{a}}_2\}$ (in green) for the real space of the triangular lattice. (b) The first Brillouin zone for the triangular lattice. The $M$ point (green circles), $K$ point (purple circle), and $\mathrm{\Gamma }$ point (black circle). The blue arrows represent the basis vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$ in the reciprocal space. (c) Some examples of interest from the $\mathrm{\Gamma }$ point (0, 0, 0) in light blue to the $K$ point $(\frac{1}{3},\frac{1}{3},0)$ in light green we have the points (expressed in EN): $(\frac{1}{6},\frac{1}{6},0)$ in purple, (0.1737, 0.1737,0) in red, and (0.222, 0.222,0) in brown.