Self-interaction correction in multiple scattering theory

We propose a simplified version of self-interaction corrected local spin-density (SIC-LSD) approximation, based on multiple scattering theory, which implements self-interaction correction locally, within the KKR method. The multiple scattering aspect of this new SIC-LSD method allows for the description of crystal potentials which vary from site to site in a random fashion and the calculation of physical quantities averaged over ensembles of such potentials using the coherent potential approximation. This facilitates applications of the SIC to alloys and pseudoalloys which could describe disordered local moment systems, as well as intermediate valences. As a demonstration of the method, we study the well-known $\alpha \text{−}\gamma$ phase transition in Ce, where we also explain how SIC operates in terms of multiple scattering theory.

Figure 1

(Color online) Phase shifts of the SI-corrected and uncorrected $f$ states in Ce from the SIC-LSD calculation. The energies are relative to the Fermi level.

Figure 2

(Color online) Density of states and integrated density of states of the SI-corrected $f$ channel.

Figure 3

(Color online) The calculated total energies for Ce from LDA and SIC-LSD, with different $f$ states localized, as functions of volume, given in atomic units ${\left(\mathrm{a.u.}\right)}^3$.

Figure 4

(Color online) Spin-resolved density of states of Ce in the $\alpha$ (a) and $\gamma$ phase (b) with ferromagnetic arrangement of local moments.

Figure 5

(Color online) Spin- and symmetry-resolved DOS (in states/eV) for the states originating from $s\left(A_{1g}\right)$, $p\left(T_{1u}\right)$, and $d\left(T_{2g}\text{and}{E}_g\right)$ channels. As in Fig. 4, the SIC-LSD calculation refers to the ferromagnetic arrangement of the local moments.

Figure 6

(Color online) Spin- and symmetry-resolved DOS (in states/eV) for the states originating from $f$ channels. Note the different scale of the plot with respect to Fig. 5. As in Fig. 4, the SIC-LSD calculation refers to the ferromagnetic arrangement of the local moments.

Figure 7

Total energies $(T=0\mathrm{K})$ of the Ce $\alpha \text{−}\gamma$ pseudoalloy as a function of the concentration of localized states. The curves correspond to the lattice constants, indicated in the figure. The labels of 8.85 and 9.05, corresponding to the remaining curves, have been omitted for readability.

Figure 8

Spectral functions along $\Gamma \text{−}X$, relative to the Fermi level, for the $\gamma$-phase concentration 0 in panel (a), 0.5 in (b), and 1 in (c), respectively. The spectral functions were calculated at the lattice constant $a=8.65\mathrm{a.u.}$

Figure 9

(Color online) Spectral functions at the $\Gamma$ point for the $\gamma$-phase concentrations 0 (a), 0.5 (b), and 1 (c). The character of the respective peaks is marked above the upper horizontal line of each panel.

Figure 10

(Color) Calculated free energies for the temperatures $T=0$, 800, and $1600\mathrm{K}$. The plots represent fits to the calculations, which have been performed for concentrations from 0 (corresponding to the pure $\alpha$ phase) to 1 (corresponding to the pure $\gamma$ phase), in steps of 0.1, and for lattice constants from $8.25\mathrm{a.u.}$ to $9.65\mathrm{a.u.}$, in steps of $0.2\mathrm{a.u.}$ A constant of $17717\mathrm{Ry}$ has been added to all energies.

Figure 11

(Color) Gibbs free energies for $T=0$, 800, and $1600\mathrm{K}$. In order to enhance the readability of the plots the energies have been calibrated by a linear term, proportional to pressure.

Figure 12

(Color online) The free energies as function of the volume for the temperatures 0 (highest curve), 200, 400, 600, 800, 1000, 1200, 1400, and $1600\mathrm{K}$ (lowest curve). A constant of $17717\mathrm{Ry}$ has been added to all energies.

Figure 13

(Color) Calculated isotherms for the temperatures $T=0$ (lowest curve), 200, 400, 600, 800, 1000, 1200, 1400, and $1600\mathrm{K}$ (highest curve). The color indicates the fraction of localized electrons, blue is all localized ($\gamma$ phase), and red is all delocalized ($\alpha$ phase).

Figure 14

(Color) Phase diagram obtained for the pseudoalloy, composed of $\alpha$- and $\gamma \text{−}\mathrm{Ce}$. The crosses indicate the calculated and experimental critical points. The color indicates the fraction of localized electrons, blue is all localized ($\gamma$ phase), and red is all delocalized ($\alpha$ phase).

Figure 15

Discontinuities of the total energy (thick solid line), the total entropy $TS$ (thick dashed line), and the $pV$ term (thick dashed-dotted line) over the phase separation line as function of the temperature. The entropy term is further decomposed into the electronic (thin solid line), the mixing (thin dashed line), and the magnetic (thin dashed-dotted line) contribution.

Figure 16

Temperature dependence of the electronic free energy, as defined in the text.

Figure 17

Critical points of the phase transition, obtained when the concentration of trivalent Ce atoms in the $\alpha$ phase is artificially fixed at a finite $c_0$. The points are marked by their corresponding value of $c_0$, where $c_0=0$ represents purely tetravalent Ce.