Plane-wave based electronic structure calculations for correlated materials using dynamical mean-field theory and projected local orbitals

The description of realistic strongly correlated systems has recently advanced through the combination of density functional theory in the local density approximation (LDA) and dynamical mean field theory (DMFT). This $\text{LDA}+\text{DMFT}$ method is able to treat both strongly correlated insulators and metals. Several interfaces between LDA and DMFT have been used, such as ($N\text{th}$ order) linear muffin-tin orbitals or maximally localized Wannier functions. Such schemes are, however, either complex in use or additional simplifications are often performed (i.e., the atomic sphere approximation). We present an alternative implementation of $\text{LDA}+\text{DMFT}$, which keeps the precision of the Wannier implementation, but which is lighter. It relies on the projection of localized orbitals onto a restricted set of Kohn–Sham states to define the correlated subspace. The method is implemented within the projector augmented wave and within the mixed-basis pseudopotential frameworks. This opens the way to electronic structure calculations within $\text{LDA}+\text{DMFT}$ for more complex structures with the precision of an all-electron method. We present an application to two correlated systems, namely, ${\text{SrVO}}_3$ and $\beta$-NiS (a charge-transfer material), including ligand states in the basis set. The results are compared to calculations done with maximally localized Wannier functions, and the physical features appearing in the orbitally resolved spectral functions are discussed.

Figure 1

LDA band structure for ${\text{SrVO}}_3$.

Figure 2

(Color online) LDA total and projected density of states for ${\text{SrVO}}_3$.

Figure 3

(Color online) LDA band structure for ${\text{SrVO}}_3$ computed in PAW, with “fatbands” to show the amplitude of the projection of each band on a given atomic orbital ($\text{O}p$, $\text{V}{t}_{2g}$, and $\text{V}{e}_g$).

Figure 4

(Color online) Local impurity spectral function of ${\text{SrVO}}_3$ for $U=4\text{eV}$ within $\text{LDA}+\text{DMFT}$ using the MLWF basis and the PLO scheme $\left(N_b=3\right)$. Inset: Green’s function in imaginary time. Note that when using such a small $\left(t_{2g}\right)$ energy window, the impurity spectral function and Bloch-resolved spectral function coincide.

Figure 5

(Color online) Local impurity spectral function of ${\text{SrVO}}_3$ for $U=6\text{eV}$ within $\text{LDA}+\text{DMFT}$ using the PLO scheme $\left(N_b=12\right)$. Inset: Green’s function in imaginary time for both the PLO and MLWF schemes.

Figure 6

(Color online) Total Bloch spectral function for ${\text{SrVO}}_3$, and spectral functions of $\text{O}p$ and $V{t}_{2g}$ states for $U=4\text{eV}$ (a) and $U=6\text{eV}$ (b) within $\text{LDA}+\text{DMFT}$ using the PLO scheme $\left(N_b=12\right)$. These spectral functions are not the local orbital projected spectral functions (see text): The $t_{2g}$ local impurity spectral functions are plotted in Fig. 5.

Figure 7

(Color online) Local impurity spectral function of ${\text{SrVO}}_3$ for $U=6\text{eV}$ within $\text{LDA}+\text{DMFT}$ using the PLO scheme $\left(N_b=21\right)$. Inset: Green’s function in imaginary time for both the PLO and MLWF schemes.

Figure 8

(Color online) Total Bloch spectral function for ${\text{SrVO}}_3$, and spectral functions of $\text{O}p$, $\text{V}{t}_{2g}$, and $\text{V}{e}_g$ states for $U=6\text{eV}$ within $\text{LDA}+\text{DMFT}$ using the PLO scheme $\left(N_b=21\right)$. These spectral functions are not the local orbital projected spectral functions (see text): The $t_{2g}$ and $e_g$ local impurity spectral functions are plotted in Fig. 7.

Figure 9

(Color online) Spectral function of $t_{2g}$ Bloch states for ${\text{SrVO}}_3$ within the PLO scheme of $\text{LDA}+\text{DMFT}$. For $N_b=12$ and $N_b=21$, $U=6\text{eV}$. For $N_b=3$, $U=4\text{eV}$.

Figure 10

(Color online) Spectral functions of $\text{O}p$ and $\text{V}{t}_{2g}$ states for $U=6\text{eV}$ within $\text{LDA}+\text{DMFT}$ using the PLO scheme $\left(N_b=12\right)$. Two values of the double-counting shift are used: 6.6 (AMF) and 3.6 eV.

Figure 11

(Color online) Projected $\beta -\text{NiS}$ structure with a local $\text{Ni}\left(e_g\right)$ orbital obtained from a MLWF construction using the block of 16 $\text{Ni}\left(3d\right)/\text{S}\left(3p\right)$ bands. The $c$ axis is identical to the vertical axis.

Figure 12

(Color online) LDA data for $\beta \text{-NiS}$. (a) Band structure. (b) DOS. For the local $\text{Ni}\left(3d\right)/\text{S}\left(3p\right)\text{-DOS}$, the cutoff radius was one-half the nearest-neighbor distance.

Figure 13

(Color online) LDA fatband decomposition for $\beta \text{-NiS}$. (red/gray) $\text{Ni}\left(E_g\right)$ and (blue/dark) $\text{Ni}\left(A_{1g}\right)$. The local $\text{Ni}\left(3d\right)/\text{S}\left(3p\right)$ projection cutoff radius was one-half the nearest-neighbor distance.

Figure 14

(Color online) The $\text{LDA}+\text{DMFT}$ local spectral function for $\beta \text{-NiS}$ derived from the local Green’s function in Bloch basis at $\beta =10{\text{eV}}^{-1}$. The contribution from the upper ten bands of the $\text{Ni}\left(3d\right)/\text{S}\left(3p\right)$ block was encoded red, and the one from the lower six bands was encoded blue. This guidance to the eyes should roughly separate dominant $\text{Ni}\left(3d\right)$ from dominant $\text{S}\left(3p\right)$ character.

Figure 15

(Color online) The corresponding $\text{LDA}+\text{DMFT}$ local impurity spectral function for $\beta \text{-NiS}$ at $\beta =10{\text{eV}}^{-1}$.