Soft and isotropic phonons in PrFeAsO${}_{1-y}$

Phonons in single crystals of PrFeAsO${}_{1-y}$ are investigated using high-resolution inelastic x-ray scattering and ab initio pseudopotential calculations. Extensive measurements of several samples ($y\sim 0$, 0.1 and 0.3) at temperatures spanning the magnetic ordering temperature ($T_N\sim 145$ K for $y\sim$ 0) and the superconducting transition temperature ($T_c=36$ K for $y\sim 0.1$ and $T_c=45$ K for $y\sim$ 0.3) show that there are some changes in phonon spectra with temperature and/or doping. We compare our measurements with several ab initio pseudopotential models (nonmagnetic tetragonal, oxygen-deficient O${}_{7/8}$ supercell, magnetic orthorhombic, and magnetic tetragonal) and find that the experimentally observed changes are much smaller than the differences between the experimental data and the calculations. Agreement is improved if magnetism is included in the calculations via the local spin density approximation, as the Fe atomic motions parallel to the ferromagnetic ordering direction are softened. However, the antiferromagnetically polarized modes remain hard, and in disagreement with the experimental data. In fact, given the increasing evidence for anisotropy in the iron pnictide materials, the phonon response is surprisingly isotropic. We consider several modifications of the ab initio calculations to improve the agreement with the experimental data. Improved agreement is found by setting the matrix to zero (clipping the bond) between nearest-neighbor antiferromagnetically aligned Fe atoms in the magnetic calculation, or by softening only the in-plane nearest-neighbor Fe-As force constant in the nonmagnetic calculation. We discuss these results in the context of other measurements, especially of phonons, for several FeAs systems. Fluctuating magnetism may be a partial explanation for the failure of the calculations, but seems incomplete in the face of the similarity of the measured phonon response in all the systems investigated here including those known to have static magnetism.

Figure 1

Crystal structure of $Re$FeAsO ($Re$: rare earth). The arrows on Fe atoms show the magnetically ordered pattern observed for $T<T_N$ (Refs. 1, 2, 3, 4). Dashed (red) arrows indicate the direction of ferromagnetic (F) [antiferromagnetic (AF)] alignment of iron spins in the plane.

Figure 2

Electric resistivity ($\rho$) and the derivative ($d\rho /dT$) of a PrFeAsO sample near the structural and magnetic transition temperatures. Solid (dashed) lines correspond to the measurements with decreasing (increasing) temperature. Inset shows a larger temperature range for $\rho$. The deviation in $\rho$ from slow increase with decreasing temperature or an abrupt increase in $d\rho /dT$ corresponds to the structural transition temperature $T_s$ from tetragonal to orthorhombic symmetry, while the inflection point in $\rho$ or the maximum in $d\rho /dT$ corresponds to the magnetic transition temperature $T_N$. A small upturn below 12 K indicated by an arrow in the inset corresponds to the ordering of Pr magnetic moments.

Figure 3

Typical IXS spectra of doped PrFeAsO${}_{1-y}$ (doped-2; open circle) and parent PrFeAsO (closed circle; shifted vertically for clarify) at room temperature (a) near $\Gamma$ point ($Q=$ (3.03 0 0.06)) and (b) at Brillouin zone boundary ($Q=$ (3.50 0 0.00)). The lines show the fit to the data using Eqn. (1). The individual peaks are also shown for the doped-2. The arrows below each plot show the peak positions based on the fits. Generally the elastic intensity is larger near to the $\Gamma$ point.

Figure 4

Dispersion relations along some high-symmetry directions of doped PrFeAsO${}_{1-y}$ (open circle) and parent PrFeAsO (closed circle) at room temperature. The $Q=(3$ $q$ 0) data of (b) are from the sample with $y\sim$0.1 ($T_c=36$ K; doped-1), and the other ones are from the sample with $y\sim$0.3 ($T_c=45$ K; doped-2). The area of the symbols is proportional to the peak intensity [$B_i$ in Eq. (1)] and the error bars are the intrinsic peak widths estimated by subtracting the experimental resolution from the fit result.

Figure 5

Dispersion relations along some symmetry directions for the parent PrFeAsO. The figure style is the same as that of Fig. 4. The data at room temperature are identical to those in Fig. 4.

Figure 6

No magnetic splitting is observed in the phonon spectra for PrFeAsO at 10 K. (a) 2$\theta$ scans near (3 3 0)/($3\overline{3}$ 0) Bragg point. The (3 3 0)/($3\overline{3}$ 0) correspond to the scattering from antiferromagnetically/ferromagnetically ordered (orthorhombic $a$/$b$) direction, respectively. (b) Phonon spectra at $Q=(3.223.21-0.11)$. The lines are the spin polarized ab initio “magnetic” orthorhombic calculation, (see Sec. 3b) convoluted with the instrumental resolution. (c) Dispersion relations.

Figure 7

Comparison of the measured dispersion for PrFeAsO${}_{1-y}$ ($T_c=45$ K; doped-2) at room temperature at $Q=(3+q$ 0 0) and various first principles calculations. The same data are repeated in each panel for comparison with the different calculations: (a) “original” calculation using a nonmagnetic tetragonal structure, (b) the “O${}_{7/8}$” calculation with an oxygen deficiency, (c) the “magnetic” first principles calculation, and (d) the tetragonal magnetic calculation. See also the text and Table  for details. The area of the experimental data points (line height of the calculations) show the measured (calculated) mode intensity for IXS spectra. For clarity, a cutoff has been used in plotting acoustic mode intensity which becomes large near Bragg points. The low energy acoustic mode appearing at ($4-q$ 0 0) is probably a transverse mode. See text at the end of Sec. 3b.

Figure 8

The bottom right is a schematic of the Fe-As layer: As atoms are located above or below the Fe atomic plane as specified. The arrows on Fe atoms specify the magnetic spin directions for “magnetic” and “mag. tetra” calculations, where the ferromagnetic (antiferromagnetic) propagation vector corresponds to be $<$110$>$ ($<1\overline{1}0>$) as shown in the red dashed arrow F (AF). In (a)–(i) arrows specify atomic displacement patterns in an Fe-As layer at several points in Figs. 7 and 9. Atomic movements along out-of-plane direction are specified by “up” or “down” near the atoms.

Figure 9

Dispersion along $Q=(3+q$ $3+q$ 0) for the parent sample PrFeAsO. In this direction of (c) “magnetic” and (d) “mag. tetra” calculations there are two domains originated from crystal structure and/or magnetic order, and the phonon dispersions from each domains are plotted. See the caption of Fig. 7 for details. The weak, low energy, acoustic mode that can be seen in the data is a transverse acoustic mode (dashed line) that appears in the measured spectra, probably due to the finite momentum acceptance of the analyzers. See the text at the end of Sec. 3b for discussion.

Figure 10

Schematic of model modifications. (a) The “soft Fe-As” model softens both the in-plane and the out-of-plane components, while the “in-plane soft” reduces only the in-plane correlation. (b) Strength of the force constants between Fe atoms on an Fe-As layer in the “clipped” model. A strong correlation exists along the ferromagnetic direction as shown by the thick lines. (c) Effective magnetic exchange coupling constants between Fe atoms in an Fe-As layer suggested by spin wave measurement in SrFe${}_2$As${}_2$ (Ref. 65) and CaFe${}_2$As${}_2$ (Ref. 66). Strong interaction (large $J$) is along the direction of antiferromagnetic order.

Figure 11

(a) Comparison of the experimental data of PrFeAsO${}_{1-y}$ ($T_c=45$ K; doped-2) at $Q=(3+q$ 0 0) at room temperature with the “soft Fe-As” model, (b) the comparison with “in-plane soft” model, and (c) the comparison with “clipped” model. The area of the experimental data as well as the line width of the calculations show the peak intensity on IXS spectra. See text for details.

Figure 12

As Fig. 11 but the data are from parent PrFeAsO and along $Q=(3+q$ $3+q$ 0). In (c) “clipped” model the $<$110$>$ and $<1\overline{1}0>$ directions are inequivalent and are shown in different colors (gray scales).

Figure 13

IXS spectra at $Q=(3.223.22-0.11)$ in the parent PrFeAsO sample at 10 K around the energy transfer of 30 meV; magnification of Fig. 6b. The symbols are also the same as those in Fig. 6. While the red open circles show the measurement at a mostly single domain region, blue open squares show the measurement at a multidomain region. Two modified models as well as the “original” first principles calculation are also plotted with lines.