Thermodynamic theory of square skyrmion lattice in tetragonal frustrated antiferromagnets

TL;DR

Problem: elucidate the high-temperature phase diagram of tetragonal frustrated antiferromagnets and the stabilization mechanism of the square skyrmion lattice. Approach: a mean-field Landau expansion of a Hamiltonian that includes frustrated exchange, anisotropic exchange, magnetodipolar interaction, single-ion anisotropy, and biquadratic exchange, analyzed for competing spin structures and higher order harmonics. Key findings: the square SkL (2Q) is stabilized by the interplay of anisotropy and biquadratic exchange with essential contributions from higher harmonics (k_pm); field renormalizes effective parameters and enables phase boundaries consistent with experiment; in GdRu2Si2, K is estimated to be on the order of tens of percent of the Heisenberg scale, and a possible topologically nontrivial 2Q' phase can arise. Significance: provides a tractable analytical framework to refine microscopic parameters against experiments and to anticipate similar SkL physics in related tetragonal frustrated magnets.

Abstract

High-temperature part of the phase diagram of tetragonal frustrated antiferromagnets is discussed within the framework of the mean-field approach. Based on recent experimental findings, we generalize previous theoretical studies by considering a model that includes frustrated Heisenberg exchange, biquadratic exchange, magnetodipolar interaction, anisotropic exchange, and single-ion anisotropy. It is analytically demonstrated that a subtle interplay among these interactions results in a variety of phase diagrams in the temperature-magnetic field plane. We argue that one of the proposed diagrams reproduces all crucial features of the phase diagram experimentally observed for~\gdru~compound. Besides magnetodipolar interaction and additional easy-axis contribution, it requires moderate biquadratic exchange. We show that despite the remarkable square skyrmion lattice being stable even if only magnetodipolar interaction/compass anisotropy or biquadratic exchange is included, their ``symbiosis'' allows for greatly enhancing its stability region. It is also demonstrated that higher-order harmonics play an important role in the stabilization of the square skyrmion lattice. The developed analytical approach can be useful for the refinement of the microscopic model parameters when comparing its predictions with experimental findings.

Figures (7)

• Figure 1: (a) Tetragonal crystalline lattice of GdRu$_2$Si$_2$ where only magnetic Gd$^{3+}$ ions are shown. (b) According to the experimental observations khanh2020khanh2022, GdRu$_2$Si$_2$ has ordered magnetic phases with modulation vectors $\mathbf{k}_{x,y}$. Here we also show vectors $\mathbf{k}_\pm = \mathbf{k}_x \pm \mathbf{k}_y$. The respective harmonics play an important role in the square skyrmion lattice stabilization. (c) Momentum-dependent biaxial anisotropy (consisting of magnetodipolar, single-ion, and compass anisotropic exchange contributions) favors vertical screw spirals for $\mathbf{k}_{x,y}$ modulations.
• Figure 2: Sketch of main spin structures in the $xy$ plane, which can appear in typical phase diagrams of tetragonal frustrated antiferromagnets. Color encodes $\hat{z}$ projections of spins (red -- "spin up", violet -- "spin down", green -- spin in the $xy$ plane). (a) Superposition of two spin-density waves with modulation vectors $\mathbf{k}_x$ and $\mathbf{k}_y$ and perpendicular in-plane polarizations (2SXY). In real space, it corresponds to a vortex lattice. (b) Superpositions of two SDWs with in-plane and out-of-plane polarizations (S2YZ). (c) Superposition of screw helicoid with $\mathbf{k}_y$ and SDW with $\mathbf{k}_x$ (1Q+1S); meron -- anti-meron lattice. (d) Topologically-nontrivial square skyrmion lattice made of two screw helicoids with $\mathbf{k}_x$ and $\mathbf{k}_y$, harmonics with $\mathbf{k}_\pm$ polarized along $\hat{z}$, and uniform magnetization (2Q). Its topological charge per unit cell is $n_\textrm{sk}= 1$ (e) Topologically-trivial version of 2Q structure at stronger external field (all $s^z_i>0$), $n_\textrm{sk}=0$. (f) Alternative topologically-nontrivial structure 2Q$^\prime$ with $n_\textrm{sk}=2$.
• Figure 3: Phase diagrams for simplified models, i.e., for which we include only one of the crucial ingredients. (a) Without the biquadratic exchange, but including anisotropy due to magnetodipolar interaction, the skyrmionic 2Q phase is stable only in a tiny region. At a lower $T$, it loses competition to the transverse conical spiral. (b) For nonzero biquadratic exchange with $K=0.05$ K but without anisotropy, TC is stable in a large region of moderate fields. (c) The same as (b), but $K=0.1$ K. Larger $K$ allows substituting TC with vortical phase 2SXY and makes the square skyrmion lattice stable in the intermediate field region.
• Figure 4: Phase diagrams for the model with both momentum-dependent anisotropy and biquadratic exchange. (a) Even relatively weak biquadratic exchange suppresses single-modulated transverse conical structure TC [cf. Fig. \ref{['fig3']}(a)]. (b) Decrease of $\Lambda_\pm$ expands SkL stability domain. (c) Further decrease of $\Lambda_\pm$ and increase of $K$ essentially changes the phase diagram, stabilizing topologically nontrivial 2Q$^\prime$ spin structure with $n_\textrm{sk}=2$.
• Figure 5: Phase diagrams for the easy axes along the high-symmetry direction $\hat{z}$. (a) Without the biquadratic exchange, helicoid 1Q is stable at weak magnetic fields. (b) Introducing small biquadratic exchange with $K=0.03$ K substantially changes the phase diagram. (c) For $K=0$, enhanced anisotropic exchange and competition among $\mathbf{k}_{x,y}$ and $\mathbf{k}_\pm$ (decreased $\Lambda_\pm$), the well-pronounced skyrmion phase can also be observed.