Double-$Q$ ground state with topological charge stripes in the centrosymmetric skyrmion candidate ${\mathrm{GdRu}}_2{\mathrm{Si}}_2$

${\mathrm{GdRu}}_2{\mathrm{Si}}_2$ is a centrosymmetric magnet in which a skyrmion lattice has recently been discovered. Here, we investigate the magnetic structure of the zero-field ground state using neutron diffraction on single crystal and polycrystalline ${{}^{160}\mathrm{GdRu}}_2{\mathrm{Si}}_2$. In addition to observing the principal propagation vectors ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$, we discover higher-order magnetic satellites, notably ${\mathbf{q}}_1+2{\mathbf{q}}_2$. The appearance of these satellites is explained within the framework of a double-$Q$ constant-moment solution. Using powder diffraction, we implement a quantitative refinement of this model. This structure, which contains vortexlike motifs, is shown to have a one-dimensional topological charge density.

Figure 1

(a) Crystal structure of ${\mathrm{GdRu}}_2{\mathrm{Si}}_2$, space group: $I4/mmm$. (b) H-T phase diagram of ${\mathrm{GdRu}}_2{\mathrm{Si}}_2$ with $H\parallel c$. Phase boundaries have been established using a combination of longitudinal resistivity (${\rho }_{\mathrm{xx}}$), DC magnetometry ($M_{\mathrm{dc}}$), and AC susceptibility (${\chi }_{\mathrm{ac}}$). This phase diagram is consistent with previous determinations [16]. Phases 1 and 3 have been nominally designated as meron lattices and Phase 2 the skyrmion lattice, with no current interpretation of Phase 4 [12].

Figure 2

Single-crystal neutron diffraction data in the $(h,k,0)$ plane showing magnetic satellites surrounding the $(1,1,0)$ nuclear peak at $1.5\mathrm{K}$. Intensity has been scaled so that the nuclear peak and sets of incommensurate satellites of the form ${\mathbf{q}}_1, {\mathbf{q}}_2$ and ${\mathbf{q}}_1+2{\mathbf{q}}_2$ (circled in black) can all be simultaneously observed. The arcs are Al powder diffraction lines from the sample holder.

Figure 3

Single-crystal neutron diffraction data showing the splitting between inequivalent ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ at $1.5\mathrm{K}$. The peaks have been fit with a pseudo-Voigt function. These measurements are of the $(0+q_1,0,0)$ and $(0+q_2,0,0)$ satellites.

Figure 4

Rietveld refinement of ${\mathrm{GdRu}}_2{\mathrm{Si}}_2$ at $1.5\mathrm{K}$. Red points are the data, the black curve is the model, and the blue line is the difference curve. Green and purple bars indicate the nuclear and magnetic Bragg peaks, respectively. This pattern is from the high-resolution bank of the WISH diffractometer, where ${\mathbf{q}}_1$ and ${\mathbf{q}}_2$ can be resolved, as highlighted in the inset. $R_{\mathrm{wp}}=15.3\%$ and $R_{\text{exp}}=8.73\%$ and $\text{GOF}=1.75$.

Figure 5

A model of the magnetic structure, shown over $8\times 8$ unit cells. The out-of-plane component of the magnetic moment $m_c$ is shown with the blue-red color scale. The topological charge density is represented with the black-white color scale, showing the topological charge stripes.