Spin excitations used to probe the nature of exchange coupling in the magnetically ordered ground state of ${\mathrm{Pr}}_{0.5}{\mathrm{Ca}}_{0.5}{\mathrm{MnO}}_3$

We have used time-of-flight inelastic neutron scattering to measure the spin wave spectrum of the canonical half-doped manganite ${\mathrm{Pr}}_{0.5}{\mathrm{Ca}}_{0.5}{\mathrm{MnO}}_3$ in its magnetic and orbitally ordered phase. The data, which cover multiple Brillouin zones and the entire energy range of the excitations, are compared with several different models that are all consistent with CE-type magnetic order, but arise through different exchange coupling schemes. The Goodenough model, i.e., an ordered state comprising strong nearest-neighbor ferromagnetic interactions along zigzag chains with antiferromagnetic interchain coupling, provides the best description of the data, provided that further neighbor interactions along the chains are included. We are able to rule out a coupling scheme involving formation of strongly bound ferromagnetic dimers, i.e., Zener polarons, on the basis of gross features of the observed spin wave spectrum. A model with weaker dimerization reproduces the observed dispersion but can be ruled out on the basis of discrepancies between the calculated and observed structure factors at certain positions in reciprocal space. Adding further neighbor interactions results in almost no dimerization, i.e., recovery of the Goodenough model. These results are consistent with theoretical analysis of the degenerate double exchange model for half-doping, and provide a recipe for how to interpret future measurements away from half-doping, where degenerate double exchange models predict more complex ground states.

Figure 1

(a) Illustration of the Goodenough model (see text). The spins in different one-dimensional chains (zigzags) are indicated by either red (dark) or purple (light) arrows. The orbitals on the ${\mathrm{Mn}}^{3+}$ sites are $d_{3x^2-r^2}$ or $d_{3y^2-r^2}$ in character, where $x$ and $y$ point along the two perpendicular Mn-Mn directions along the zigzags. In the model as originally proposed by Goodenough [10] there is a ferromagnetic interaction between nearest neighbors along a zigzag, $J_{F1}$, and antiferromagnetic coupling between nearest neighbors in adjacent zigzags, $J_A$. To explain our data it is necessary to allow for second-neighbor interactions, $J_{F2}$ and $J_{F3}$, within zigzags (see main text). Nearest-neighbor antiferromagnetic coupling between planes, $J_c$, is not shown. (b) The ZP model, indicating exchange interactions between rigidly coupled $S=7/2$ units in the $ab$-plane. The ZP units are indicated by dashed ovals. Parallel units have AFM coupling, $J_A$, and perpendicular units have ferromagnetic couplings, either $J_{FU}$ or $J_{FD}$. (c) The dimer model, as proposed by Johnstone et al. [39], in which spins are not rigidly coupled as in the ZP case, but rather have a stronger ferromagnetic intradimer coupling, $J_{FS}$, and weaker inter-dimer couplings, FM $J_{FW}$ and AFM $J_{A1}$ and $J_{A2}$. The orbitals are shown to have a mixed character of a superposition of $d_{y^2-z^2}$ $\left(d_{x^2-z^2}\right)$ and $d_{3x^2-r^2}$ $\left(d_{3y^2-r^2}\right)$, following the conventions used in model calculations for extensions to the DDEX model [31, 40] discussed in the text. As originally proposed the model was purely phenomenological without reference to an electronic model, and the orbitals should be considered purely schematic in this panel. (d) An alternative more general dimer model (see text) in which we distinguish between the Mn sites with different proportions of $d_{y^2-z^2}$ $\left(d_{x^2-z^2}\right)$ and $d_{3x^2-r^2}$ $\left(d_{3y^2-r^2}\right)$ character, indicated here by purple and light blue coloring respectively. The orbital character together with Goodenough-Kanamori rules lead to FM $J_{FW2}$ and AFM $J_A$, different from the model shown in (c).

Figure 2

Calculated dispersion relations for the Goodenough model (dashed red line) and ZP model (solid blue line). The parameters for the Goodenough model are those obtained as the global best fit in our analysis (see main text). The parameters for the ZP model are $J_{FU}=1.6\mathrm{meV},J_{FD}=-0.11\mathrm{meV},J_A=0.11\mathrm{meV}$ and the inter-planar coupling is 1.05 meV. These parameters were chosen to show that there exists a set of parameters for the ZP model such that at low energies it is practically indistinguishable from the Goodenough model, but that the dispersion at higher energies serves as an effective discriminant. The $x$-axis labels denote high-symmetry positions in the first Brillouin zone: $G=(0,0,0),X=(\frac{1}{4},0,0),Y=(0,\frac{1}{4},0),Z=(0,0,\frac{1}{2}),M=(\frac{1}{4},\frac{1}{4},0)$, and $L=(\frac{1}{4},\frac{1}{4},\frac{1}{2})$. Note that we have chosen to index reciprocal space according to space group $Pbnm$, described in the main text.

Figure 3

(a) The reciprocal space coverage of the detectors in the (equatorial) $(H,0,0)/(0,0,L)$ plane, with $E_i=70\mathrm{meV}$, zero energy transfer, and for a multiangle dataset with the sample $c$ axis rotated stepwise from parallel to perpendicular to the incident beam. (b) Reciprocal space coverage for a single-shot dataset with the orientation fixed to be the $c$ axis parallel to the incident neutron beam.

Figure 4

The magnetic Bragg peaks ${\mathbf{k}}_{\mathbf{1}}=(0,1/2,L)$ and ${\mathbf{k}}_{\mathbf{2}}=(1/2,1/2,L)$, and structural superlattice peak $\mathbf{q}=(1/2,0,L^{\prime })$, indicated by red circles, blue squares, and green triangles respectively. Here we have taken account of the sixfold structural twinning of the PCMO crystal. Notice that the ${\mathbf{k}}_{\mathbf{2}}$ peaks do not overlap with the structural superlattice peaks, whereas the ${\mathbf{k}}_{\mathbf{1}}$ peaks do.

Figure 5

Neutron scattering intensity maps of the dispersion along the $(H,0)$ direction, which is parallel to the zigzag chains. The maps are taken from the single-shot datasets with (a) $E_i=35\mathrm{meV}$, (b) $E_i=70\mathrm{meV}$, and (c) $E_i=140\mathrm{meV}$. The data have been smoothed by convolution with a Gaussian with FWHM equal to two bin sizes along each dimension for clarity. The map in panel (d) is taken from the multiangle dataset with $E_i=70\mathrm{meV}$, hence the value of $L$ can be explicitly given. The white dashed rectangles in panels (b)–(d) indicate the position of the gap between upper and lower branches of the excitations, which is consistent with the Goodenough model. Note that the data in all four panels have been multiplied by $f\left(E\right)=\frac{E/E_0}{1-e^{-E/E_0}}$, where $E_0=k_{\mathrm{B}}\times 10\mathrm{K}$. $f\left(E\right)$ is therefore constant when $E\ll E_0$ and $f\left(E\right)\propto E$ when $E\gg E_0$. This rescaling is to display the full spectrum more clearly on a single intensity scale. For panel (a) the signal was integrated over $\pm 0.05$ r.l.u. in the $(0,K)$ direction, whereas for panels (b)–(d) the integration was over $\pm 0.1$ r.l.u.

Figure 6

Panels (a)–(f) show intensity maps taken with energies in the ranges $9\le \mathit{E}\le 12,15\le \mathit{E}\le 20,20\le \mathit{E}\le 25,35\le \mathit{E}\le 40,50\le \mathit{E}\le 60$, and $70\le \mathit{E}\le 80$ meV, and incident energies of 35, 70, 70, 70, 140, and 140 meV respectively. The data have been smoothed by convolution with a Gaussian with FWHM equal to two bin sizes along each dimension for clarity. Panels (g)–(l) show corresponding simulations of those data using the Goodenough model and the global best fit parameters (see main text). All of the data shown here are from single-shot datasets with the $c$ axis parallel to the incident beam. We have made use of the symmetry of the crystal, and the symmetry and extent of the detector array by folding the data about the $(1,0,0)$ and $(0,1,0)$ axes to effectively increase the overall count rate by a factor of 4. The calculated $S(\mathbf{Q},\omega )$ were convoluted with Gaussians with FWHM matched to the spectrometer's energy resolution at each energy and $E_i$.

Figure 7

Low energy dispersion, the extent in energy of which allows a reasonable estimate of $J_A$ to be obtained before the global fitting procedure.

Figure 8

Cut across part of the top of the lower band of excitations. The red (solid) line shows the global best fit to the data in this region, whereas the blue (dashed) line shows the structure factor if the signs of both $J_{F2}$ and $J_{F3}$ are reversed. Such a reversal preserves the magnitude of the gap between the lower and upper bands of excitations, and also the dispersion relation, but changes the structure factor as shown.

Figure 9

Decomposition of some of the fits presented in Fig. 6 into contributions from specific sets of twins. (a) Contribution from twins 1-4, which should be compared to Figs. 6 and 6; (b) contribution from twins 5 and 6, i.e., those which are not included in panel (a); (c) contribution from twins 1, 2, 5, and 6, which should be compared to Figs. 6 and 6; (d) contribution from twins 3 and 4, i.e., those not included in panel (c); (e) contribution from twins 3–6, which should be compared to Figs. 6 and 6; (f) contribution from twins 1 and 2, i.e., those not included in panel (e). Our convention for labeling the twins is described in the main text.

Figure 10

(a) Intensity map of the dispersion along $(-1.5,K,1)$, taken from the $E_i=70\mathrm{meV}$ multiangle dataset. The energy-dependent background arising from the tail of the incoherent elastic line has been subtracted. The signal was integrated over $\pm 0.1$ r.l.u. in the perpendicular $(H,0,0)$ and $(0,0,L)$ directions. (b) Simulation of the data in panel (a) using the global best fit parameters of the Goodenough model. (c) Simulation of the data in panel (a) using the global best fit parameters for the weak dimer model. These parameters give an essentially identical dispersion to the Goodenough model, with the only difference being the relative spectral weight at $(-1.5,0,1)$-type positions compared to that at $(-1.5,0.5,1)$-type positions at the top of the lower spin wave band.

Figure 11

Cut through the data shown in Fig. 10, integrating the signal over $29<E<34\mathrm{meV}$. The data are indicated by black circles. The simulations using the Goodenough and dimer models are indicated by dashed (red) and solid (blue) lines respectively.