Magnetic order in ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ investigated using neutron scattering and muon spin relaxation

The rare-earth-based ternary intermetallic compounds $R_{2}T{X}_3$ ($R$ = rare earth, $T$ = transition metal, $X$ = Si, Ge, Ga, In) have attracted considerable interest due to a wide range of interesting low-temperature properties. Here we investigate the magnetic state of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ using neutron diffraction, muon spin relaxation ($\mu \mathrm{SR}$), and inelastic neutron scattering (INS). This compound appears anomalous among the $R_2{\mathrm{PdSi}}_3$ series since it was proposed to order ferromagnetically, whereas others in this series are antiferromagnets. Although some members of the $R_{2}T{X}_3$ series have been reported to form ordered superstructures, our data are well described by ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ adopting the ${\mathrm{AlB}}_2$-type structure with a single Nd site, and we do not find evidence of superlattice peaks in neutron diffraction. Our results confirm the onset of long-range magnetic order below $T_0=17$ K, where the whole sample enters the ordered state. Neutron diffraction measurements establish the presence of a ferromagnetic component in this compound, as well as an antiferromagnetic one which has a propagation vector ${\mathbf{k}}_{\mathbf{2}}=(1/2,1/2,1/4-\delta )$ with a temperature-dependent $\delta \approx 0.02–0.04$, and moments orientated exclusively along the $c$ axis. $\mu \mathrm{SR}$ measurements suggest that these components coexist on a microscopic level, and therefore the magnetic structure of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ is predominantly ferromagnetic, with a sinusoidally modulated antiferromagnetic contribution which reaches a maximum amplitude at 11 K and becomes smaller upon further decreasing the temperature. INS results show the presence of crystalline electric field (CEF) excitations above $T_0$, and from our analysis we propose a CEF level scheme.

Figure 1

High-resolution powder neutron diffraction measurements of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ at three temperatures, measured on the D2B instrument. At 25 K, the solid red line shows the refinement of the crystal structure, while the ticks and blue lines below the data show the peak positions and difference plot, respectively. At 1.5 and 10 K, the data were refined while taking into account the crystal structure and a ferromagnetic component, where the scale factor for the nuclear component was fixed from the 25 K refinement. Here the ticks correspond to the position of both the structural Bragg peaks and the magnetic Bragg peaks corresponding to the ferromagnetic component.

Figure 2

(a) Diffraction patterns of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ measured using the D20 instrument at various temperatures below 25 K. At the magnetic transition there is a clear increase in intensity of some of the nuclear peaks, which corresponds to the onset of an ordered ferromagnetic component. In addition, there is the emergence of peaks away from the nuclear peaks, corresponding to an antiferromagnetic component with a propagation vector of about ${\mathbf{k}}_2\approx (1/2,1/2,1/4)$. (b) Thermodiffractogram across a limited angular range for temperatures between 2.5 and 20 K, where the color plot represents the intensity. The emergence of two antiferromagnetic Bragg reflections below the ordering temperature can clearly be resolved. (c) Magnetic contributions to the (100) and $(1/2,-1/2,1/4)$ Bragg reflections as a function of temperature after subtracting the high-temperature data, corresponding to the ferromagnetic and antiferromagnetic components respectively. Note that the magnetic contributions to both peaks start at the same temperature, and while the intensity of the (100) peak continues to increase with decreasing temperature, the intensity of $(1/2,-1/2,1/4)$ reaches a maximum at around 11 K.

Figure 3

(a) Diffraction patterns of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ measured with the D20 instrument after subtracting high-temperature (25 K) data at (a) 1.7 and (b) 11 K. The solid red lines show refinements to a magnetic structure with both a ferromagnetic component and an antiferromagnetic component with a propagation vector of ${\mathbf{k}}_{\mathbf{2}}=(1/2,1/2,1/4-\delta )$ ($\delta \approx 0.02–0.04$). The ticks show the theoretical positions of the peaks corresponding to the two components, while the blue line shows the difference plot. (c) The crystal structure of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ along with the magnetic structure corresponding to the antiferromagnetic component, with Nd atoms in blue and Pd/Si atoms in red. Note that for some Nd layers, the antiferromagnetic component is either very small or zero.

Figure 4

(a) Zero-field $\mu \mathrm{SR}$ spectra of ${\mathrm{Nd}}_2{\mathrm{PdSi}}_3$ measured at four temperatures, both above and below the magnetic transition. The solid lines correspond to the fitting described in the text, where (b) shows the initial asymmetry of the component corresponding to the sample ($A_1$ in Eq. (1)), while (c) shows the Lorentzian relaxation rate $\lambda$.

Figure 5

Color-coded plot of the inelastic neutron scattering intensity as a function of energy transfer and momentum transfer at (a) 5 and (b) 30 K, with an incident neutron energy of $E_i=15$ meV. The scattering intensity scale is in arbitrary units. (c) Cuts of the neutron scattering intensity as a function of energy transfer obtained from integrating across a low-$\left|\mathbf{Q}\right|$ range ($0–3{\AA }^{-1}$) and a high-$\left|\mathbf{Q}\right|$ range ($6–9{\AA }^{-1}$) for $E_i=45$ meV at 5 K. (d) Cuts of the neutron scattering intensity as a function of energy transfer obtained from integrating across a low-$\left|\mathbf{Q}\right|$ range ($0–3{\AA }^{-1}$) at various temperatures for $E_i=15$ meV.

Figure 6

Magnetic contribution to the inelastic neutron scattering intensity at 30 K, estimated from subtracting the scaled high-$\left|\mathbf{Q}\right|$ data ($3–5{\AA }^{-1}$) from low-$\left|\mathbf{Q}\right|$ data ($0–2{\AA }^{-1}$). The solid lines show a fit to the CEF model described in the text.