---
title: 'GdRu2X2: Skyrmion Magnetism in Intermetallics'
url: https://www.emergentmind.com/topics/gdru2x2
type: topic
---

# GdRu2X2: Skyrmion Magnetism in Intermetallics

Searching arXiv for recent papers on GdRu2X2 and related compounds to ground the article in published work.
GdRu\(_2\)X\(_2\) denotes a family of rare-earth intermetallics with \(X=\) Si, Ge, or Sn that crystallize in the ThCr\(_2\)Si\(_2\)-type structure and have emerged as a model platform for centrosymmetric skyrmion physics. Within this family, GdRu\(_2\)Si\(_2\) is established as a magnet with a short-period skyrmion square lattice in the absence of Dzyaloshinskii-Moriya interaction, while subsequent work on GdRu\(_2\)Ge\(_2\) and comparative analyses across Si, Ge, and Sn connect skyrmion formation to RKKY-type exchange frustration, interlayer magnetic modulation, \(p\)–\(d\) hybridization, Fermi-surface nesting, and chemical-bonding-driven electronic instability [2409.06736], [2502.21169], [2507.18904].

## 1. Crystal structure and family definition

GdRu\(_2\)X\(_2\) (\(X=\) Si, Ge, Sn) crystallizes in the well-known ThCr\(_2\)Si\(_2\)-type structure, space group \(I4/mmm\) (No. 139) [2502.21169]. In GdRu\(_2\)Si\(_2\), the Gd atoms form a body-centered tetragonal sublattice with lattice constants
\[
a=b\approx 4.20\;\text{Å},\quad c\approx 9.87\;\text{Å},
\]
and occupy fractional coordinates
\[
\mathbf{Gd}_1:(0,0,0),\qquad \mathbf{Gd}_2:\bigl(\tfrac12,\tfrac12,\tfrac12\bigr)
\]
[2409.06736]. For GdRu\(_2\)Ge\(_2\), the experimentally determined lattice parameters are
\[
a=4.2316(1)\;\text{Å},\qquad c=9.8797(5)\;\text{Å},
\]
with Wyckoff positions
\[
\text{Gd}:2a\,(0,0,0),\quad
\text{Ru}:4d\bigl(\tfrac12,0,\tfrac14\bigr),\quad
X:4e\,(0,0,z),
\]
and \(z_{\rm Ge}=0.13007\) [2502.21169].

Structurally, the family consists of square nets of Gd\(^{3+}\) ions separated by edge-sharing RuX\(_4\) tetrahedra [2502.21169]. In GdRu\(_2\)Si\(_2\), the Ru–Si tetrahedral network separates the Gd planes by roughly half a unit cell along \(c\), yet the Gd–Gd spacing along the body diagonal is only slightly larger than the in-plane nearest-neighbor distance [2409.06736]. This crystallographic detail is central to the current understanding of the family: although the square-planar Gd layers suggest a quasi-two-dimensional magnet, the actual Gd sublattice geometry favors substantial three-dimensional magnetic coupling [2409.06736].

A plausible implication is that the formal layered appearance of GdRu\(_2\)X\(_2\) can obscure the dominant magnetic pathways unless the body-centered geometry is treated explicitly.

## 2. Electronic structure, orbital character, and chemical bonding

Spin-polarized DFT (\(\mathrm{GGA}+U\), \(U_{4f}=6.7\) eV) shows that both GdRu\(_2\)Si\(_2\) and GdRu\(_2\)Ge\(_2\) are metallic, with multiple bands crossing the Fermi level \(E_F\) [2502.21169]. The Gd \(4f\) states are split by Hund’s rule, appearing as a sharp majority-spin peak at \(-8\) eV and a minority peak at \(+4\) eV, effectively localized, whereas near \(E_F\) the density of states is dominated by Ru \(4d\) and \(X\,p\) orbitals [2502.21169]. The total DOS at \(E_F\) is \(n(E_F)\approx 2.5\) states eV\(^{-1}\) f.u.\(^{-1}\), which gives the bare Sommerfeld coefficient
\[
\gamma_0=\tfrac{\pi^2 k_B^2}{3}\,n(E_F)\simeq 5.9\;\text{mJ mol}^{-1}\text{K}^{-2}
\]
[2502.21169].

A key trend across the series is the increasing spatial extent of the \(X\)-site \(p\) orbitals from Si-\(3p\) to Ge-\(4p\) to Sn-\(5p\) [2507.18904]. In GdRu\(_2\)Ge\(_2\), the \(X\,p\) bandwidth is broader than in the Si compound, and the qualitative hybridization parameter
\[
V_{pd}\sim \langle p_X | \mathcal{H} | d_{\rm Ru}\rangle
\]
is inferred to be larger for \(X=\)Ge than for \(X=\)Si [2502.21169]. The later comparative study formalizes this trend using crystal-orbital Hamilton population (COHP) and integrated COHP (ICOHP),
\[
COHP_{ij}(E)=\sum_k c_{i,k}H_{ij}c_{j,k}\delta(E-E_k),\qquad
ICOHP_{ij}=\int_{-\infty}^{E_F} COHP_{ij}(E)\,dE,
\]
where negative ICOHP values indicate net bonding character [2507.18904].

The reported ICOHP values show that Gd–Ru bonding strengthens substantially from Si to Ge to Sn, while Ru–X and Gd–X bonding remain sizable throughout the series [2507.18904]. The main values are summarized below.

| Bond | ICOHP trend across \(X\) |
|---|---|
| Gd–Ru | \(-0.10\) eV in Si, \(-0.50\) eV in Ge, \(-0.58\) eV in Sn |
| Ru–X | \(-0.56\) eV in Si, \(-0.53\) eV in Ge, \(-0.57\) eV in Sn |
| Gd–X | \(-0.56\) eV in Si, \(-0.55\) eV in Ge, \(-0.59\) eV in Sn |

These data support a family-level picture in which the [Ru\(_2\)X\(_2\)] conduction layer increasingly mediates the Gd moments as \(X\) becomes chemically heavier [2507.18904]. This suggests that chemical bonding is not merely a structural background variable but a direct control parameter for the magnetic instability landscape.

## 3. Exchange hierarchy and three-dimensional magnetic order

The low-energy magnetism of the Gd sublattice is governed by competing RKKY-type exchange interactions [2409.06736], [2502.21169]. In GdRu\(_2\)Si\(_2\), first-principles calculations of isotropic exchange constants up to the fifth shell yield the following hierarchy in meV:
\[
J_1\approx +0.18,\quad
J_2\approx +0.82,\quad
J_3\approx +0.12,\quad
J_4\approx -0.05,\quad
J_5\approx +0.10,
\]
with \(J_2\) corresponding to \(\mathbf{R}_2=(\pm \tfrac12,\pm \tfrac12,\pm \tfrac12)\), i.e. the [111] body-diagonal directions [2409.06736]. The dominant coupling is therefore along the body diagonal, and \(J_2\approx 0.8\) meV is almost five times larger than the in-plane nearest-neighbor \(J_1\) [2409.06736].

This exchange hierarchy leads to a magnetic-ordering description that cannot be reduced to a purely two-dimensional model. The principal helical modulation vectors in reciprocal-lattice units are
\[
\vec{Q}_{[100]}=(\delta,0,0),\qquad
\vec{Q}_{[010]}=(0,\delta,0),\qquad
\vec{Q}_{[111]}=(\eta,\eta,\eta),
\]
with
\[
\delta\approx 0.22,\qquad \eta\approx 0.17
\]
in GdRu\(_2\)Si\(_2\) [2409.06736]. The associated real-space pitches are \(\lambda_{[100]}\approx 19\,a\) and \(\lambda_{[111]}\approx 14\,a\) [2409.06736]. Sarkar et al. therefore describe the

Source: https://www.emergentmind.com/topics/gdru2x2