Cooperative Jahn-Teller phase transition in PrAlO3

The perovskite PrAl${\mathrm{O}}_3$ is known to exhibit an apparently complicated series of structural phase transitions from cubic to rhombohedral orthorhombic to monoclinic and ultimately tetragonal symmetry at temperatures of 1320, 205, and 151 K, respectively. The 1320-K cubic-rhombohedral transition is a simple soft $R_{25}$ phonon transition as observed in LaAl${\mathrm{O}}_3$ and SrTi${\mathrm{O}}_3$. In PrAl${\mathrm{O}}_3$ it results in a crystal structure in which the Al${\mathrm{O}}_6$ octahedra are rotated by about 9° in a staggered sense about the [111] axis. Harley et al. have suggested that the first-order 205-K and second-order 151-K transitions occur as a result of a delicate interplay between the anharmonic lattice interactions which prefer a rhombohedral structure and the $R_{25}$ phonon-${\mathrm{Pr}}^{3+}$ $4f^2$ coupling which prefers an orthorhombic and ultimately a tetragonal structure. In this paper we report an extensive elastic and inelastic neutron scattering study of PrAl${\mathrm{O}}_3$ with special emphasis on the 151-K transition. We also have carried out a detailed crystal-field analysis of the ${\mathrm{Pr}}^{3+}$ energy levels in all phases. Our crystallographic measurements of the lattice constants and angles as a function of temperature are consistent with a structural model in which the over-all Al${\mathrm{O}}_6$ rotation angle is constant in magnitude, but the rotation axis itself moves continuously from [101] to [001] as the temperature is lowered from 151 to ∼70 K. Neutron scattering measurements of the crystal-field levels are reported and by combining these with previous results a full crystal-field level scheme is constructed for the ${\mathrm{Pr}}^{3+}$ ion. The temperature dependence of the crystal-field splittings is then semiquantitatively predicted via a simple model with fixed fourth- and sixth-order terms and a constant electric-field gradient tensor in which the principal axis rotates in unison with the Al${\mathrm{O}}_6$ axis from [101] to [001]. With the above as a basis, it is shown that the 151-K transition corresponds to a simple cooperative Jahn-Teller transition involving a single electronic mode (in this case a quadrupole exciton), a $B_1\left(C_{2v}\right)$ optical phonon, and a $B_1\left(C_{2v}\right)$ acoustic mode, the perovskite ${\Sigma }_3{\mathrm{}\left[101\right]\mathrm{}}_{\mathrm{T}2}$ phonon. Accurate measurements of the acoustic-phonon order parameter are reported; it is found that the strain follows the mean-field power law $e_{\mathrm{zz}}-e_{\mathrm{xx}}\sim {(1-\frac{T}{T_c})}^{0.50\pm 0.02}$ for ${10}^{-1\mathrm{}}<1-\frac{T}{T_c}<{10}^{-3\mathrm{}}$. In addition, the normalized order parameter is shown to coincide at all temperatures with independently measured electronic and condensed-optical-phonon order parameters. The coupled exciton-acoustic-phonon dispersion relations have been measured through $T_c$ and analyzed using the theory of Elliott et al. The principal results are the following: (a) The ${\mathrm{}\left[101\right]\mathrm{}}_{\mathrm{T}2\mathrm{}}$ phonon is the soft mode of the system. (b) The quadrupole exciton is observed directly over a wide range of $k$ in the region where it anticrosses the phonon. (c) An analysis of the coupled dispersion relations at $T_c$ yields separately the $k$-dependent phonon-induced ${\mathrm{Pr}}^{3+}$ quadrupole-quadrupole interactions arising from the coupling to the $B_1$ acoustic and optic modes; the former is found to account for about 1/3 of the total interaction at $k=0\mathrm{}$ and to have a mean range of 30 Å. (d) The theory of Elliott et al. is found to predict quantitatively the temperature dependence of the soft-acoustic-phonon dispersion relation above $T_c$. This work thence established the 151-K transition in PrAl${\mathrm{O}}_3$ as a prototype coupled pseudospin-phonon structural phase transition.