$\mathrm{PBE}+U$ calculations of the Jahn-Teller effect in $\mathrm{Pr}{\mathrm{O}}_2$

The Jahn-Teller distortion in praseodymium dioxide $\left(\mathrm{Pr}{\mathrm{O}}_2\right)$ has been studied using density functional theory with the $\mathrm{PBE}+U$ functional for the exchange-correlation energy. Various possible distortions were suggested based on recent neutron diffraction experiments [C. H. Gardiner et al., Phys. Rev. B 70, 024415 (2004)]. We could identify the most likely distorted structure with a magnitude of the distortion in very good agreement with the experiment. This structure is the one with the lowest total energy and the corresponding calculated electronic and magnetic properties agree well with the experiment. We investigated the dependency of various properties on the Coulomb parameter $U_{\mathrm{eff}}=U-J$ and suggest $U_{\mathrm{eff}}=6\mathrm{eV}$ as the most appropriate value, for which about 1.65 $4f$ electrons (one localized plus 0.65 delocalized) per Pr atom are found. We also show that magnetism (antiferromagnetism or ferromagnetism) has only a minor influence on the energy and magnitude of the distortion, which is consistent with the fact that the Néel temperature $\left(13.5\mathrm{K}\right)$ is much smaller than the temperature of $120\mathrm{K}$ below which the distortion is observed.

Figure 1

(Color online) The four considered distorted structures: (a) JT1, (b) JT2, (c) JT3, and (d) JT4. The bigger (blue) spheres are the Pr atoms and the smaller (green) are the O atoms. Three types of Pr–O bonds occur, namely, a short (thick), an intermediate (thin), and a long (no bond shown) distance.

Figure 2

(Color online) The different types of distorted oxygen cages found in structures (a)–(d) of Fig. 1. The bigger (blue) spheres are the Pr atoms and the smaller (green) are the O atoms. Three types of Pr–O bonds occur, namely, a short (thick), an intermediate (thin), and a long (no bond shown) distance. The displacements of the O atoms are parallel to the sheet plane and are exaggerated for clarity.

Figure 3

(Color online) Isosurface of the spin magnetization density $({\rho }_{\uparrow }-{\rho }_{\downarrow })$ of $\mathrm{Pr}{\mathrm{O}}_2$ in the AF1 magnetic phase. The red and blue colors correspond to positive and negative values $(\pm 0.07\text{electrons}∕{\mathrm{\AA }}^3)$, respectively. The small red balls are O atoms.

Figure 4

Variation with respect to the Coulomb parameter $U_{\mathrm{eff}}$ is shown for (a) the fundamental band gap, (b) the Pr magnetic moments, (c) the distortion, and (d) the number of $4f$ electrons per Pr atom. These $\mathrm{PBE}+U+\mathrm{SO}$ results were obtained for the JT2 distorted structure and AF1 magnetic phase. The second and third data points are for $U_{\mathrm{eff}}=0.2$ and $0.5\mathrm{eV}$, respectively. The lines between the calculated points are only guides for the eyes.

Figure 5

(Color online) $\mathrm{PBE}+U+\mathrm{SO}$ spin-up DOS of $\mathrm{Pr}{\mathrm{O}}_2$ in the AF1 magnetic phase and JT2 structure for (a) $U_{\mathrm{eff}}=0.2\mathrm{eV}$, (b) $U_{\mathrm{eff}}=2\mathrm{eV}$, and (c) $U_{\mathrm{eff}}=6\mathrm{eV}$. The Fermi energy is set at zero. No broadening was applied.

Figure 6

(Color online) $\mathrm{PBE}+U+\mathrm{SO}$ spin-up DOS of $\mathrm{Pr}{\mathrm{O}}_2$ in the AF1 magnetic phase and JT2 structure for $U_{\mathrm{eff}}=6\mathrm{eV}$. The Fermi energy is set at zero. No broadening was applied.

Figure 7

(Color online) $\mathrm{PBE}+U+\mathrm{SO}$ $(U_{\mathrm{eff}}=6\mathrm{eV})$ electron density (at an isovalue of $1.1\text{electrons}∕{\mathrm{\AA }}^3$) corresponding to the electronic states (a) at the position of the $\mathrm{Pr}4f$ peak at $-4\mathrm{eV}$ (see Fig. 6) and (b) from the region extending from $-3\text{to}0\mathrm{eV}$ (see Fig. 6). The electron density is plotted only inside the cube enclosing a Pr atom (the central atom) and its eight nearest-neighboring O atoms.

Figure 8

Real part of the orientationally averaged optical conductivity $\sigma$ of $\mathrm{Pr}{\mathrm{O}}_2$ (AF1 magnetic phase and JT2 distorted structure) obtained with $\mathrm{PBE}+U+\mathrm{SO}$ $(U_{\mathrm{eff}}=6\mathrm{eV})$. A Lorentzian broadening $(\Gamma =0.03\mathrm{eV})$ was applied.

Figure 9

(Color online) $\mathrm{PBE}+U+\mathrm{SO}$ $(U_{\mathrm{eff}}=6\mathrm{eV})$ total energy $E$ vs distortion $d$ from the perfect cubic fluorite structure. JT1, JT2, JT3, and JT4 refer to the distorted structures in Fig. 1. AF1 and AF2 refer to the two antiferromagnetic phases considered in this work (see text), and F to the ferromagnetic phase. The minimum of the lowest-energy curve was set at $E=0$. The range of experimental values is represented by the shaded region. The lines between the calculated points are only guides for the eyes.