(${\mathrm{Fe}}_{1-x}{\mathrm{Ni}}_x{)}_5{\mathrm{GeTe}}_2$: An antiferromagnetic triangular Ising lattice with itinerant magnetism

Based on first-principles calculations, an antiferromagnetic Ising model on a triangular lattice has been proposed to interpret the order of $\mathrm{Fe}\left(1\right)$-Ge pairs and the formation of $\sqrt{3}\times \sqrt{3}$ superstructures in the ${\mathrm{Fe}}_5{\mathrm{GeTe}}_2$ (F5GT) as well as to predict the existence of similar superstructures in Ni doped F5GT (Ni-F5GT). Our paper suggests that F5GT systems may be considered as a rare structural realization of the well-known antiferromagnetic Ising model on a triangular lattice. Based on the superstructures, a Heisenberg-Landau Hamiltonian, taking into account longitudinal spin fluctuations, is implemented to describe magnetism in both F5GT and Ni-F5GT. We unveil that frustrated magnetic interactions associated with Fe(1), tuned by a tiny Ni doping, is responsible for the experimentally observed enhancement of the $T_c$ to 478 K in Ni-F5GT. Itinerant magnetism, reflected by longitudinal spin fluctuations, are found to only affect the $T_c$'s mildly with a modification of $\sim 5\%$ with respect to that obtained with standard Heisenberg interactions. Our calculations show that at low doping levels, monolayer Ni-F5GT has almost the same magnetic phase diagram as that of the bulk, which indicates a pervasive beyond room-temperature ferromagnetism in this Ni-doped two-dimensional system.

Figure 1

(a) Average crystal structure of F5GT sublayer. The white, red, orange, blue, and dark green spheres represent Fe(1)${}_{up}$, Fe(1)${}_{dn}$, Fe(2)(3), Ge, and Te atoms, respectively. (b) Illustration of the Fe(1) sublattice, half white and red indicate split sites of Fe(1) on a triangular lattice. (c) Specific heat as a function of temperature simulated for $x=0$ and $x=0.2$. The inset depicts the analysis of the discrete Fourier transformation of the structure at 153 K for $x=0$. (d) Schematic of the short-range ordered (SRO) superstructure. The $3\times 3$ superstructure is marked by the blue axis with the corresponding primitive $\sqrt{3}\times \sqrt{3}$ superstructure marked by the red axis.

Figure 2

Illustrations of the TGT plane structure in Ni-F5GT systems. The pink, orange, blue, and dark green spheres represent Ni, Fe, Ge, and Te atoms, respectively. The optimized duu structure at ($\mathrm{a})x=0.2$ and ($\mathrm{b})x=0.667$. The optimized uuu structure at ($\mathrm{c})x=0.2$ and ($\mathrm{d})x=0.667$.

Figure 3

Ni-doping level $x$-dependent magnetic moments and $T_c$ of bulk (solid symbol) and monolayer (open symbol) Ni-F5GT systems marked by blue and red curves, respectively.

Figure 4

${\mathbf{M}}_i$-dependent energy in (a) bulk and (b) monolayer F5GT.

Figure 5

The major exchange interactions of bulk (solid symbol) and monolayer (open symbol) of Ni-F5GT as a function of Ni-doping level $x$.