Mapping spin-wave dispersions in stripe-ordered ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Ni}{\mathrm{O}}_4$ ($x=0.275$, 0.333)

Using the MAPS spectrometer at the ISIS spallation source, we have measured the magnetic excitations of single-crystal samples of stripe-ordered ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Ni}{\mathrm{O}}_4$ with $x=0.333$ and 0.275. The full two-dimensional spin-wave dispersions were obtained using incident energies of 60 and 160 meV. To analyze the excitations, we have evaluated a spin-only Hamiltonian describing diagonal, site-centered stripes in the linear spin-wave approximation. Besides the superexchange energy $J$ within antiferromagnetic domains, we have considered effective exchange couplings $J_1$ and $J_2$ across a charge stripe coupling second-neighbor Ni sites along Ni-O bond directions and along the plaquette diagonal, respectively. From least-squares fits of the model to the measurements on the $x=1∕3$ sample at $T=10\mathrm{K}$, we find that the dispersions are well described by a model using just $J$ and $J_1$, but not $J$ and $J_2$. Consistent with an analysis of previous measurements, we find that $J$ is about 90% of the superexchange energy of undoped ${\mathrm{La}}_2\mathrm{Ni}{\mathrm{O}}_4$ and $J_1∕J\approx 0.5$. The excitations observed for $x=0.275$ are surprisingly similar to those for $x=1∕3$, despite the differing magnetic-ordering wave vectors; the main difference is a broadening of the excitations for $x=0.275$. For both samples, we find that one spin-wave branch has a gap of $\sim 20\mathrm{meV}$, confirming a previous observation for $x=1∕3$. We discuss the possible origin of this gap.

Figure 1

(Color online) (a) Schematic diagram of spin and charge stripe order within a $\mathrm{Ni}{\mathrm{O}}_2$ plane for the case of $x=1∕3$. Arrows indicate ordered $S=1$ spins on ${\mathrm{Ni}}^{2+}$ sites, open circles are nominally ${\mathrm{Ni}}^{3+}$ sites within charge stripes. $J$, $J_1,J_2$ label the effective exchange interactions discussed in the text. (b) Similar to (a), but showing a twin domain with stripes rotated by ${90}^{\circ }$. (c), (d) Reciprocal lattices corresponding to the order shown in (a), (b), respectively. Circles (squares) indicate superlattice Bragg peak positions. A subset of these (filled symbols) coincide with fundamental Bragg peaks from the average crystal structure. (e) Diagram indicating experimental configuration, with superimposed reciprocal lattices of orthogonal stripe domains. Axes are rotated by 45° relative to (c) and (d).

Figure 2

(Color) (upper panel) Representative constant-energy slice for the $x=1∕3$ sample, measured with $E_i=160$ meV and integration of data over the energy range $50<E<55\mathrm{meV}$. (lower panel) Simulation, using the spin-wave model and parameter values of fit 1 in Table , with instrumental resolution taken into account. The coordinate system corresponds to Fig. 1e. The circles correspond to a slice through cones of spin waves centered on the Bragg vectors of the magnetic reciprocal lattice.

Figure 3

(Color online) Diagram indicating paths of cuts through the data used in the analysis, superimposed on the simulation of Fig. 2 after folding the “data” into a single quadrant.

Figure 4

(Color online) Data slices in $E$ vs $\mathbf{Q}$ planes, with $\mathbf{Q}$ along Cut A of Fig. 3 in (a) and (c), and along Cut B in (b) and (d), illustrating the superposition of spin-wave and phonon dispersions. The intensity is multiplied by the energy to compensate for the energy dependence of the spin-wave and phonon cross sections. The incident energy was 60 meV for (a) and (b); 160 meV for (c) and (d). (e) and (f) are simulations of the magnetic scattering corresponding to (c) and (d), respectively, calculated using the spin-wave model discussed in the text and the parameter values of fit 1 in Table .

Figure 5

(Color) Comparison of constant energy slices for the $x=1∕3$, (a)-(f), and $x=0.275$, (g)-(l), samples. Each panel is labeled by the energy range over which the data have been integrated, with the excitation energy increasing from bottom to top. (a), (b), (g), and (h) correspond to $E_i=60\mathrm{meV}$, and the rest to $E_i=160\mathrm{meV}$. The $\mathbf{Q}$ range is restricted to the area of an antiferromagnetic Brillouin zone.

Figure 6

Constant-energy cuts for the $x=1∕3$ sample along the direction labeled Cut A in Fig. 3. Data have been averaged over the indicated energy ranges; all were measured with $E_i=60\mathrm{meV}$. Solid lines indicate fitted spin-wave model (including convolution with instrumental resolution function) using the parameter values of fit 1 in Table . Dashed lines indicate fitted background.

Figure 7

Data and fits along Cut B (see caption of Fig. 6 for description). Incident energies of 60 meV (160 meV) were used for energy transfers below (above) 50 meV.

Figure 8

Data and fits along Cut C (see caption of Fig. 6 for description). The incident energy was 160 meV for all cuts.

Figure 9

Data and fits along Cut D (see caption of Fig. 6 for description). The incident energy was 160 meV for all cuts.

Figure 10

(Color) Comparison of measured and simulated constant energy slices for the $x=1∕3$ sample. (a)–(c) Data, (d)–(f) simulations using the parameter values of fit 1, (g)–(i) simulations using fit 3.

Figure 11

(Color online) Constant-$\mathbf{Q}$ cuts at ${\mathbf{Q}}_{\mathrm{AF}}$; data averaged over $0.495<Q_h<0.505$ and $-0.05<Q_k<0.05$). The intensity is multiplied by the energy. Filled squares: $x=1∕3$ sample at 10 K; filled circles: $x=0.275$ sample at 10 K; open circles: $x=0.275$ sample at 300 K. The low-temperature features at 5, 20, and 45 meV are due to phonons, while the peak at about 80 meV is due to spin waves.

Figure 12

(Color online) Similar to Fig. 11, but at a fundamental Bragg point, $\mathbf{Q}=(2,0,0)$; data averaged over $1.95<Q_h<2.05$ and $-0.05<Q_k<0.05$).

Figure 13

(Color online) Intensity vs energy at a magnetic zone center. For $x=1∕3$ (squares) the data have been integrated over the window $0.25<Q_h<0.41,-0.15<Q_k<0.15$; for $x=0.275$ (circles) the window is $0.27<Q_h<0.43,-0.15<Q_k<0.15$. The step at $\sim 20\mathrm{meV}$ is evidence of a gapped spin-wave mode in both samples.

Figure 14

(Color online) Constant energy cuts (Cut A) comparing data for $x=0.275$ (open circles) with $x=1∕3$ (filled squares); the lines are guides to the eye. Data are averaged over $-0.05<Q_k<0.05$. The incident energy was 60 (160) meV for excitation energies less than (greater than) 50 meV.