Magnetic ordering in GdNi${}_2$B${}_2$C revisited by resonant x-ray scattering: Evidence for the double-$q$ model

Recent theoretical efforts aimed at understanding the nature of antiferromagnetic ordering in GdNi${}_2$B${}_2$C predicted double-$q$ ordering. Here we employ resonant elastic x-ray scattering to test this theory against the formerly proposed, single-$q$ ordering scenario. Our study reveals a satellite reflection associated with a mixed-order component propagation wave vector, viz., ($q_a,2q_b,0$) with $q_b=q_a\approx 0.55$ reciprocal lattice units, the presence of which is incompatible with single-$q$ ordering but is expected from the double-$q$ model. A ($3q_a,0,0$) wave vector (i.e., third-order) satellite is also observed, again in line with the double-$q$ model. The temperature dependencies of these along with that of a first-order satellite are compared with calculations based on the double-$q$ model and reasonable qualitative agreement is found. By examining the azimuthal dependence of first-order satellite scattering, we show the magnetic order to be, as predicted, elliptically polarized at base temperature and find the temperature dependence of the “out of $a$-$b$ plane” moment component to be in fairly good agreement with calculation. Our results provide qualitative support for the double-$q$ model and thus in turn corroborate the explanation for the “magnetoelastic paradox” offered by this model.

Figure 1

(a) Single-$q$ and (b) double-$q$ representations of AFM order associated with a commensurate (principal) propagation wave vector, of magnitude $q=0.5$ reciprocal lattice units. The arrows represent ordered magnetic moments positioned in a square array. The lattice parameter is the separation between two adjacent moments.

Figure 2

(a) Schematic of the vertical scattering geometry used for the comparative study of first- and higher-order satellites: The ${\mathbf{b}}^{☆}$ reciprocal lattice axis lies in the scattering plane (yellow shaded region) in all measurements and ${\mathbf{c}}^{☆}$ is along the sample surface normal. (b)–(d) Data taken at $T=3$ K in $\sigma \to {\pi }^{\prime }$: (b) scans (at fixed $\mathbf{Q}$) as a function of the incident x-ray energy, and (c) and (d) scans parallel to the [100] and [010] crystal axes, respectively. The symbols (black circle, magenta square, and green triangle) in (a) indicate the different satellite reflection positions in $\mathbf{Q}$ space and refer to same (central) legend as the plots. The upper abscissa in (c) [(d)] gives the $h$ [$k$] Miller index corresponding to the scan of the ($\overline{1}+3q_a,0,5$) [($q_a,0,4$)]. Solid lines are peak fits to Gaussian [(c)] and pseudo-Voigt [(d)] line shapes. An $\arctan$ function is included to fit the steplike background in the scan of the ($q_a,\overline{1}+2q_b,5$) satellite in (c).

Figure 3

Temperature dependence of integrated intensity for the first- and higher-order satellites. The solid lines are simulations based on calculated Fourier components from the double-$q$ model (see Appendices  and ). For clarity, data and simulation have been normalized at $T=3$ K to unity [($q_a,0,4)$], to 0.8 [($\overline{1}+3q_a,0,5$)], and to 0.6 [($q_a,\overline{1}+2q_b,5$)]. The vertical dashed lines indicate the magnetic transition temperatures from previous studies (see text).

Figure 4

(a) Horizontal scattering geometry used to study the azimuthal ($\psi$) dependence of the ($q_a,0,6$) satellite scattering. (b) Temperature dependence of $R$ [Eq. (1)] measured for the sample orientation ($\psi$ setting) depicted in (a). The lower (gray) solid line in (b) gives the $m_c/m_b$ values obtained from the fit to $R\left(T\right)$ (the upper black solid line), and the stars are the $m_c/m_b$ values from model calculations (both curves refer to the right axis). (c) and (d) $\psi$ dependence of $R$ measured above and below $T_R$, respectively. The different “fits” in (d) are described in the text.

Figure 5

Calculated magnetic structure of GdNi${}_2$B${}_2$C at $T=3$ K. The arrows represent projections onto the $xy$ (or $ab$) plane of the spins of an $xy$ plane of Gd ions.