Spin-wave excitations in the ferromagnetic metallic and in the charge-, orbital-, and spin-ordered states in Nd${}_{1-x}$Sr${}_x$MnO${}_3$ with $x\approx 0.5$

We report inelastic neutron-scattering experiments on single crystals of Nd${}_{1-x}$Sr${}_x$MnO${}_3$ with $x=0.5$ and $x=0.49$. The spin-wave dispersion in the charge-, orbital-, and spin-ordered state in Nd${}_{0.5}$Sr${}_{0.5}$MnO${}_3$ exhibits a strongly anisotropic stiffness. The sign of the anisotropy is characteristic for the site-centered model for charge and orbital ordering in half doped manganites. Within this model, linear spin-wave theory yields a perfect description of the experimental dispersion. In the ferromagnetic metallic state of Nd${}_{1-x}$Sr${}_x$MnO${}_3$ with $x=0.49$ and $x=0.50$ magnetic excitations exhibit nearly the same magnon dispersion. High intense signals near the zone boundary over a wide energy level overlap with a sharp spin-wave dispersion, which can be described with a Heisenberg model including nearest-neighbor interactions.

Figure 1

Schematic representation of a half doped perovskite manganite with charge, orbital, and antiferromagnetic CE-type ordering (Goodenough model). The pattern shows zigzag chains propagation in [1,1,0] direction (cubic notation). The $e_g$ orbital ordering on Mn${}^{3+}$ ions is shown by $d_{3x^2-r^2}$ and $d_{3y^2-r^2}$ orbitals. Mn${}^{4+}$ ions are drawn by balls while arrows stand for the magnetic alignment of the spins. The orbital unit cell is indicated by dashed black lines and the magnetic unit cell of Mn${}^{3+}$ by black dotted lines.

Figure 2

Temperature dependence of transport measurements and structural investigations of Nd${}_{0.5}$Sr${}_{0.5}$MnO${}_3$. Experimental data of electric resistivity upon cooling show a kink at the transition into the (half) metallic state at $T_C=255$ K (a), and the occurrence of the COSO state at $T_{CO}\approx 160$ K causes a sudden jump in resistivity by three orders of magnitude pointing to insulating behavior. Magnetization data indicating a ferromagnetically ordered state below $T_C$ and an antiferromagnetically ordered state below $T_{CO}$ (d). X-ray powder diffraction from $T=15$ K up to $T=850$ K shows remarkable changes of the orthorhombic structure at various temperatures in accordance with electric and magnetic properties (b),(c). Neutron diffraction at the superstructure reflection ${\mathbf{Q}}_{FM}=(1,0,0)$ indicates ferromagnetic ordering between $T_{CO}$$<$$T$$<$$T_C$. Furthermore, superstructure reflections referring to antiferromagnetic ordering of Mn${}^{3+}$ can be detected below $T_{CO}$ (e). For more details see text.

Figure 3

Dispersion of the magnetic excitations in Nd${}_{0.5}$Sr${}_{0.5}$MnO${}_3$ along the main-symmetry directions (a). The path $\Gamma$-$A$ stands for investigations in [1,0,0] direction (diagonal to the chains), while $\Gamma$-$B$ measures the dispersion perpendicular and $\Gamma$-$C$ parallel to the zigzag chains. Investigations parallel to the $c$ axis are presented by $\Gamma$-$Z$. Solid curves are fits as described in the text and crystal-field excitations of the Nd ions are indicated by gray-shaded bars. The different paths along the main-symmetry directions around an antiferromagnetic zone center for chains propagation in [1,1,0] directions is shown in (b).

Figure 4

Representative energy and constant-energy scans at $T=12$ K in Nd${}_{0.5}$Sr${}_{0.5}$MnO${}_3$ along the main-symmetry directions. Two constant-Q scans that measures the dispersion along $\Gamma$-$B$ are depicted in (a). Constant-energy scans to study the acoustic branch parallel to the chains ($\Gamma$-$C$), 45${}^{\circ }$ diagonal to the chains ($\Gamma$-$A$), and in the $c$ direction ($\Gamma$-$Z$) are shown in (b) and (c). Solid lines correspond to fits with Gaussians.

Figure 5

Energy scans in the first Brillouin zone at the antiferromagnetic center $\mathbf{Q}=(0.75,0.75,0.5)$ (a) and at the zone boundary $B$ along the acoustic branch $\Gamma$-$B$ (b). The gap and the splitting of the excitations at the zone center is clearly observable (indicated by black arrows). The signal around $E=1$ meV corresponds to CEF excitations of the Nd ions. (The elastic line is not shown.) The excitation at the end point $B$ can be observed at 7.5(1) meV while the signal around 12 meV can be ascribed to a CEF as well (b). Dashed lines correspond to individual fits with Gaussians and the solid line to the resulting fit.

Figure 6

Illustration of the effective spin-only model for the AFM CE-type structure. Spins along the zigzag chains are coupled by ferromagnetic nearest- and next-nearest-neighbor exchanges $J_{FM}$ and $J_{FM,2}$, the latter acting only between Mn${}^{4+}$ sites. The chains are coupled antiferromagnetically by exchanges $J_{\mathrm{AFM}}$ and $J_{\mathrm{AFM},c}$ between neighbors in the $ab$ plane and along the $c$ axis, respectively.

Figure 7

Full spin-wave dispersion along the main-symmetry directions in the FMM state in Nd${}_{0.5}$Sr${}_{0.5}$MnO${}_3$ at $T=174$ K. Experimental data were taken along the $\Gamma$-$R$, $\Gamma$-$M$, and $\Gamma$-$X$ directions, respectively. Solid lines refer to a fit with a Heisenberg ferromagnet with only nearest-neighbor interactions and dotted lines to a fit with nearest- and forth-nearest-neighbor interactions. A significant improvement of the fit to the data by taking additionally $J_4$ into account cannot be observed. An enhanced signal around the zone boundary might be given by several magnetic levels (b). For more details see text.

Figure 8

Constant-energy scans around the zone center ${\mathbf{Q}}_{FM}=(1,0,0)$ in Nd${}_{0.5}$Sr${}_{0.5}$MnO${}_3$ along the symmetry direction [1,0,0]. Data at $E=1$ and 3 meV are recorded with the spectrometer 1T with fixed $E_f=8.05$ meV and at $E=8$ meV with fixed $E_f=14.7$ meV, respectively (a). A decrease of the signal with increasing scattering vector demonstrates its magnetic origin (b).

Figure 9

Observed excitation at high energies along the symmetry direction [1,1,1]. The maximum shifts to increasing $q$ with increasing energy indicating the steeper dispersion.

Figure 10

Characterization of the single crystal Nd${}_{0.51}$Sr${}_{0.49}$MnO${}_3$ by macroscopic and microscopic studies. A ferromagnetic metallic state can be observed by electric resistivity measurements and magnetization data (a),(c). Elastic neutron scattering at ${\mathbf{Q}}_{FM}=$ (1,1,0) confirms ferromagnetic ordering below $T_C$ (d). A sudden increase in intensity of the superstructure reflection ${\mathbf{Q}}_{OO}=$ (2.25,0.25,0) below $T_{CO}$ refers to orbital ordering (b),(d).

Figure 11

Dispersion of the magnon excitation in the FMM state in Nd${}_{0.51}$Sr${}_{0.49}$MnO${}_3$ along the symmetry directions [1,0,0] and [0,1,1] at T = 151 K (a). Solid lines refer to fits with only nearest-neighbor interactions $J_1$ and dashed lines point to calculations with nearest and forth-nearest-neighbor interactions $J_1$ and $J_4$ as depicted in (b).

Figure 12

Raw-data scans with fixed energy observed in [1,0,0] direction. Data at 2 meV are taken with $E_f=8.05$ meV and above with $E_f=14.7$ meV. Intensities are normalized to the same monitor but shifted as a guide to the eyes (a). Signals along the symmetry direction [0, 1,1] exhibit a loss in intensity and an increase of width with increasing energy, too (b).