Thermodynamically stable skyrmion lattice in a tetragonal frustrated antiferromagnet with dipolar interaction

Oleg I. Utesov

Phys. Rev. B 103, 064414 – Published 8 February 2021

Abstract

Motivated by recent experimental results for ${GdRu}_{2} {Si}_{2}$ [Khanh et al., Nat. Nanotechnol. 15, 444 (2020)], in which a nanometric square skyrmion lattice was observed, we propose a simple analytical mean-field description of the high-temperature part of the phase diagram of centrosymmetric tetragonal frustrated antiferromagnets with dipolar interaction in the external magnetic field. Dipolar forces provide momentum-dependent biaxial anisotropy in reciprocal space. It is shown that in a tetragonal lattice, in the large part of the Brillouin zone, for mutually perpendicular modulation vectors in the $a b$ plane this anisotropy has mutually perpendicular easy axes and collinear middle axes, which leads to double- $Q$ modulated spin structure stabilization. In the large part of its stability region, the latter turns out to be a square skyrmion lattice with a topological charge of $\pm 1$ per magnetic unit cell, which is determined by the frustrated exchange coupling and thus nanometer sized. Easy and middle axes can be swapped in the presence of additional single-ion easy-axis anisotropy. This results in the different phase diagram. It is argued that the latter case is relevant to ${GdRu}_{2} {Si}_{2}$ .

Received 18 December 2020
Revised 22 January 2021
Accepted 22 January 2021

DOI:https://doi.org/10.1103/PhysRevB.103.064414

Physics Subject Headings (PhySH)

Dipolar interaction Frustrated magnetism Phase diagrams Phase transitions Skyrmions

Mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Oleg I. Utesov ^*

B.P. Konstantinov Petersburg Nuclear Physics Institute, National Research Center “Kurchatov Institute,” Gatchina 188300, Russia; Department of Physics, Saint Petersburg State University, St. Petersburg 198504, Russia; and St. Petersburg School of Physics, Mathematics, and Computer Science, HSE University, 190008 St. Petersburg, Russia

^*utiosov@gmail.com

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Vol. 103, Iss. 6 — 1 February 2021

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Figure 1
(a) Tetragonal structure of ${GdRu}_{2} {Si}_{2}$ relevant to the present study; only magnetic ${Gd}^{3 +}$ ions are shown. (b) Using lattice parameters $a = 4.165$ Å and $c = 9.61$ Å of Ref. [28], one can numerically calculate the Fourier transform of the dipolar tensor [see Eq. (7)] for $| q | \neq 0$ . Not taking into account other possible anisotropic interactions, we found that for in-plane modulation vectors $q = (q_{x}, q_{y}, 0)$ the $c$ axis is the easy, middle, and hard one in the blue, gray, and white regions of the Brillouin zone, respectively. (c) For vectors $\pm k_{x} = (k, 0, 0)$ and $\pm k_{y} = (0, k, 0)$ , in a wide range of $k$ , easy axes are mutually perpendicular, and middle ones are collinear, oriented along $c$ (additional single-ion anisotropy can swap the easy and middle axes). This property of the dipolar interaction, which sometimes is referred to as the compass anisotropy [19, 29, 30], leads to the square skyrmion lattice stabilization in a certain part of the phase diagram.
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Figure 2
Possible magnetic structures at zero external field in the $a b$ plane; a region of $2 \times 2$ cells (with size $2 π / k$ ) is shown. (a) Sinusoidal single-modulated spin-density wave ( $1 S$ ). (b) Vortical double spin-density wave ( $2 S$ ). (c) Elliptical spiral ( $1 Q$ ). (d) Double- $Q$ elliptical phase, which consists of alternating merons and antimerons ( $2 Q$ ). In the first two structures spins lie in plane. External field uniformly magnetizes them and transforms them into simple and double-fan structures, respectively. For $1 Q$ and $2 Q$ spin orderings, $z$ components of spins are shown by rainbow colors (from red, spin-up state, to magenta, spin-down state).
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Figure 3
High-temperature part of the phase diagram for a centrosymmetric tetragonal frustrated antiferromagnet with two possible mutually perpendicular modulation vectors and dipolar interaction (see Fig. 1). The parameters in (14) were used. Depending on magnetic field and temperature, the $2 Q$ phase can be either topologically trivial or not (see text). The conical XY phase emerges beyond the theory applicability region and is shown only for illustration purposes.
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Figure 4
Sketch of the $2 Q$ spin ordering in applied magnetic field; part of the $a b$ plane with a size of $2 \times 2$ magnetic unit cells is shown. (a) At small magnetic fields in comparison with the zero-field case [see Fig. 2], additional small core-up merons emerge, providing topological charge $Q = - 1$ per unit cell. (b) At larger fields, the boundary between core-up merons and antimerons vanishes, and the magnetic ordering represents a square skyrmion lattice.
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Figure 5
Analytically obtained phase diagram for parameter set (69). In comparison with Fig. 3, the easy axes for both modulation vectors are along $c$ due to the single-ion anisotropy, which leads to crucial differences. In this case the square SkL (red region) emerges only at finite external magnetic field and not very close to the ordering temperature $T_{c}$ . The conical XY phase appears at $T ≲ 30$ K, where the mean-field approach is inapplicable. This phase diagram captures important features of the experimentally observed one for ${GdRu}_{2} {Si}_{2}$ [17].
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